Three Essays on Time Series and Macroeconomics

Alejandro Pérez Laborda

Three Essays on Time Series and Macroeconomics

Alejandro Pérez Laborda

Advisors: Prof. Mª Dolores Guilló Fuentes, Prof. Fidel Pérez Sebastián

A Dissertation Submitted to Departamento de Fundamentos del Análisis Económico

Universidad de Alicante

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

June 2012

A mi familia.

Acknowledgements

Writing this thesis has been, perhaps, the most daunting part of my education.

Fortunately, I have received help, advice and encouragement from a large number of

people.

First I would like to thank my advisers, Mª Dolores Guilló and Fidel Pérez-Sebastian for

my insightful supervision. To Juan Carlos Conesa, Antonio Manresa and Tim Kehoe who

were the first to believe that this was possible and also to my colleagues in the CREB: Ina

Stoyanova and Luis Díaz-Serrano who have never stopped to give me support. To

Máximo Camacho and Gabriel Pérez-Quirós, who first introduced me in the analysis of

time series econometrics and also advised me during my MA dissertation. I am also

indebted to Mª Angeles Carnero, Luis A. Gil-Alana and Elena Martinez-Sanchís, who

carefully red all the chapters of this thesis. I would also like to thank Lola Collado and

Juan Mora as well as all the administrative staff of the department: Lourdes Garrido,

Cristina Ramirez, Mariló Rufete and Josefa Zaragoza, for their patience solving many of

the problems I had during these years.

This thesis would not have been possible without the support of all my colleges and

friends of the QED. Many thanks for your friendship, for keeping a stoic attitude to my

oaths and curses each time my Matlab programs did not want to run and also for letting

me encroach on your apartments from time to time.

Finally, but most important, I owe this thesis to my family. To my parents, Sol y Paco;

obviously nothing would have ever started without them. To my wife, Yuliya, who has

always helped and supported me during this long process and has obtained many days my

bad temperament in exchange. And to my dog, Sprocket, who has carefully collected, has

hidden and has destroyed an enormous amount of discarded pieces of work. I am sure this

has prevented me to get completely disappointed.

Contents

Introduction

Introduction (English)…………………………………………………………………….7

Introducción ampliada (Español)………………….……………………………………..10

CHAPTER 1 A fractionally integrated approach to monetary policy and inflation

dynamics

1.1 Introduction……………………………………………..…………………….…....29

1.2 Econometric framework………………...………………………………………….33

1.2.1 Estimation of the time varying spectrum………………………………...33

1.2.2 Model description and frequency domain estimation……………………34

1.3 Empirical evidence…………….………...…………………………………………37

1.3.1 Inflation persistence...................................................................................39

1.3.2 The pattern of exogenous shocks...………………………………………41

1.3.3 Non-systematic monetary policy………………………………………...41

1.3.4 Systematic monetary policy…….………………………………………..43

1.4 Conclusions……..………………………………………………………………….45

1.5 Appendix: tables and figures……………………………………………..…...........48

1.6 Bibliography…………………...………………………………………………...…54

CHAPTER 2 On the invertibility of seasonally adjusted series

2.1 Introduction………………………………………………………………..….……61

2.2 The problem……………………………………………………..…………….……64

2.3 Seasonal fractional integration……………..………………………………………65

2.4 Simulation study…………….………………………………………………….….69

2.4.1 Simulation setup........................................................................................69

2.4.2 Description of the results...........................................................................72

2.5 Empirical examples……………………………..…………………………………..76

2.6 Conclusions………………………….…………………………………….….……78

2.7 Appendix: Tables and Figures………………..………………………….……...….80

2.8 Bibliography………………………..………………………………………………84

CHAPTER 3 Explaining wage premium changes across sectors: evidence from the

Translog model

3.1 Introduction………………..………………………………………………….……89

3.2 The transcendental logarithmic cost function……………………….……….....….91

3.2.1 Model description…………………………………….……………….…91

3.2.2 The decomposition of the growth rate of the skilled premium…………..93

3.3 Empirical strategy.................................................................................................…95

3.3.1 Econometric framework……………………..………………………..…95

3.3.2 Data description.........................................................................................98

3.4 Results………………………………………..………….........……………………99

3.5 Conclusions..............................................................................................................103

3.6 Appendix: Tables and Figures………………………..……………….………...…104

3.7 Bibliography…………………………………………………………………..…...114

7

Introduction

Introduction

This thesis consists of three separate chapters. In the first chapter, I investigate the

possible causes of the loose control of U.S. inflation during the 70 and early 80’s, and the

role of the Fed in the observed changes afterwards. There is no consensus in the literature

on whether the higher stability period following Volker’s and prolonging through first

half of Greenspan’s chairmanship, was a consequence of a better monetary policy from

the part of the Fed (Boivin and Gianonni (2006), Clarida et al. (2000), Cogley and

Sargent (2002), or Lubick and Shorftheide (2007)) or rather, just the consequence of a

decay in the volatility of the exogenous disturbances (Bernanke and Mihov (1998),

Canova and Gambetti (2009), Primiceri (2005) or Sims and Zha (2006)). The objective

here is to relax the strong assumptions on the persistence of inflation typically made in

previous literature, which has mainly relied on the analysis of vector autorregresions

(VAR). Despite their different conclusions, all previous research has restricted the VAR

parameter space to deliver stationary results. The assumption of stationarity contrasts to a

large stream of different literature which agrees that inflation is much more persistent

than that, and is better characterized by an ( )I 1 process (see e.g. Benati and Surico

(2008), Cogley, Primiceri and Sargent (2010), Pivetta and Reis (2007) or Stock and

Watson (2007)). To address these issues, this paper considers a wider statistical

framework that encompasses both ( )I 0 and ( )I 1 assumptions as well as other

fractionally integrated possibilities. My target is to incorporate FI into a multivariate and

time varying model of the US economy. To do so, first a time-varying multivariate

spectrum is estimated for post WWII US data. Then, a structural fractionally integrated

VAR (VARFIMA) is fitted to each of the resulting time dependent spectra. In this way,

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both the coefficients of the VAR and the innovation variances are allowed to evolve

freely. The model is employed to analyze inflation persistence and to evaluate the stance

of US monetary policy. My findings indicate a strong decline in the innovation variances

during the great disinflation, consistent with the view that the good performance of the

economy during the 80’s and 90’s is in part a tale of good luck. However, I also find

evidence of a decline in inflation persistence together with a stronger monetary response

to inflation during the same period. This last result suggests that the Fed may still play a

role in accounting for the observed differences in the US inflation history. Finally, I

conclude that previous evidence against drifting coefficients could be an artifact of

parameter restriction towards the stationary region.

The second chapter has been written in co-authorship with Luis Alberiko Gil-

Alaña and Yuliya Lovcha. In the chapter, we examine the implications of the ARIMA

model based approach (AMB) to seasonally adjust time series in the context of seasonal

fractional integration. Gomez and Maravall (2001) call attention to the fact that the

spectrum of the seasonally adjusted series presents dips at seasonal frequency. According

to the authors, these dips are the counterpart of MA unit roots and therefore, the process is

not invertible and in general does not accept autoregressive AR (or VAR) approximations

as is usually done in practice. In the chapter we analyze in detail the dips at the seasonal

frequencies and the apparent non-invertibility produced by AMB approach within the

fractional integration framework, which admits a wider representation of the invertibility

condition than the one applied by Gomez and Maravall (2001). In this particular study we

center in the TRAMO-SEATS adjustment program, which makes a part of the data

adjustment programs that have been intensively employed by Eurostat since 1994, and

nowadays their use has been extended to various European countries. By mean of a

simulation study, we find that if the true data generating process follows the default of the

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program Airline model (ARIMA with unit roots at seasonal frequencies), the adjusted

series produced by TSW are indeed non-invertible. However, if the series is fractionally

integrated at the seasonal frequencies, which is less restrictive and very plausible in some

cases according to the empirical evidence (see e.g. Arteche and Robinson (2000), Gil-

Alana and Robinson (2001), Arteche (2003), or Hassler et al. (2009)), the adjusted series

may be invertible, depending on the stationarity of the original series. Thus, if the original

series is seasonally stationary with coefficients of fractional integration at seasonal

frequencies smaller than 0.5, the adjusted series estimated by TSW is invertible. On the

contrary, if the original series is seasonally non-stationary, the adjusted series is not

invertible. We also illustrate this issue with an empirical example.

The last chapter of the thesis has been written in co-authorship with my advisor,

Fidel Perez-Sebastian. The objective in this last chapter is to examine the evidence for

capital-skilled labor complementarity (CSC) (Griliches (1969), Krusell et al. (2000)) and

for the skilled biased technological change (SBTC) (Bound and Johnson (1992))

hypotheses at a sectoral level. To do so, we employ time series data from the EU-KLEMS

dataset and we estimate a translog (Christensen et al. (1971, 1973), Ruiz-Arranz (2002))

model with 4 inputs (skilled and unskilled labor, and ICT and Non-ICT capital) for seven

different sectors of the US, UK and Japan economies. Our main finding is that, although

we find evidence in favor of the CSC hypothesis, this one appears to be both a sector and

country specific phenomenon. Conversely, the SBTC explains a substantial fraction of

wage premium changes in (almost) all sectors and countries.

10

Introducción ampliada en Español

Bajo el titulo “Tres ensayos en series temporales y macroeconomía” se recogen

tres artículos expuestos en tres capítulos diferenciados. Los dos primeros capítulos, que

configuran la mayor parte de la tesis doctoral, configuran un tronco común que aplica el

análisis espectral al estudio de las series macroeconómicas. En el primer capítulo, titulado

“Una aproximación fraccionalmente integrada al análisis da la política monetaria de

EEUU y a la dinámica de la inflación”, relajo los supuestos I(0) e I(1) típicamente

establecidos en el análisis VAR de la macroeconomía monetaria. Para ello considero un

marco más amplio que engloba los dos supuestos anteriores así como otras alternativas

fraccionalmente integradas. Primero estimo un espectro multivariante y variante en el

tiempo de la economía de EEUU y luego ajusto un modelo VAR fraccionalmente

integrado (VARFIMA) a cada uno de los resultantes espectros. De esa manera, todos los

parámetros del modelo, incluyendo la varianza de las innovaciones y los órdenes de

integración de todas las series, varían libremente de acuerdo con los cambios en el

espectro. El modelo es empleado para analizar la persistencia de la inflación y la postura

de la política monetaria de la Reserva Federal. Mis resultados muestran la existencia de

una fuerte disminución en la varianza de las perturbaciones exógenas, manifestada en una

reducción en las varianzas de las innovaciones del modelo, desde mediados de los años

80. Este resultado concuerda con la visión de que el periodo de mayor estabilidad en la

inflación durante los años 80 y 90 es, en parte, simplemente una cuestión de suerte. No

obstante también encuentro una respuesta de la reserva federal a la inflación mucho más

agresiva así como una disminución de la persistencia de la inflación durante el mismo

periodo. Por tanto, la política monetaria de la Reserva Federal puede también haber

jugado un papel importante en la reciente historia de la inflación en los EEUU. El

segundo capítulo, titulado “Sobre la invertibilidad de las series ajustadas

11

estacionalmente”, también aplica el análisis espectral al estudio de las series

macroeconómicas. En particular, el capitulo analiza en detalle los ceros en las frecuencias

estacionales así como la aparente no invertibilidad de las series ajustadas mediante

métodos basados en modelos ARIMA (enfoque AMB) en un contexto de integración

fraccionalmente integrada. El marco de la integración fraccional es menos restrictivo y

más en acuerdo con la evidencia empírica de las series macroeconómicas. El capitulo

muestra que los ceros en las frecuencias estacionales en la series ajustadas no

necesariamente corresponden a raíces MA unitarias cuando la serie original es

(estacionalmente) fraccionalmente integrada. Por tanto, aunque la serie ajustada presente

ceros en las frecuencias estacionales puede, todavía, ser invertible y admitir

representación autorregresiva dependiendo de la estacionariedad o no de la serie original.

El tercer capítulo titulado “Explicando los cambios en la brecha salarial a nivel sectorial:

un enfoque Translog” constituye un bloque diferenciado y pretende examinar la validez y

de las hipótesis de progreso tecnológico sesgado en favor del trabajo cualificado (SBTC)

y la complementariedad entre capital y trabajo cualificado (CSC), a nivel sectorial. Para

ello se emplea un modelo translog, que permite descomponer y cuantificar el papel de

ambas hipótesis en la variación de la brecha salarial nivel sectorial. Los resultados indican

que la validez de la hipótesis de CSC depende tanto del sector como del país en concreto.

Contrariamente el SBTC explica una parte sustancial de las variaciones del la brecha

salarial en todos los países y sectores.

A continuación expongo de manera más detallada el contenido de cada uno de los

capítulos.

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Capítulo 1: Una aproximación fraccionalmente integrada al análisis da la

política monetaria de EEUU y a la dinámica de la inflación.

Durante los últimos 40 años, la economía de los EEUU ha estado caracterizada

por diversos episodios marcadamente diferentes tanto en el nivel de la inflación como en

su volatilidad. La inflación media en los EEUU disminuyó de un 4.5% en el periodo

1960-1984 a un 3% en 1985-2009. Asimismo, la desviación estándar de la inflación

disminuyó un tercio entre esos dos periodos. Estos hechos han generado una cantidad

considerable de literatura macroeconómica. Una buena aparte de esa literatura trata de

analizar el papel de la política monetaria de la Reserva Federal en las diferencias

observadas. Si bien el episodio de elevada inflación y alta volatilidad, es el resultado

directo de una pésima política monetaria por parte de la Reserva Federal durante ese

periodo o tan sólo es consecuencia de la “mala suerte”, es todavía motivo de controversia

en la literatura. Mientras unos autores afirman que la política monetaria se ha llevado a

cabo más eficientemente desde el mandato de Volker (véase Boivin y Gianonni (2006),

Clarida et al. (2000), Cogle y Sargent (2002), o Lubick y Shorftheide (2007)), otros no

encuentran diferencias sustanciales en la política monetaria desde mediados de los 60.

Las referencias principales de esta otra visión incluyen Bernanke y Mihov (1998),

Canova y Gambetti (2009), Primiceri (2005) o Sims y Zha (2006). De acuerdo con este

último grupo de autores, la causa que explica las diferencias en el comportamiento de la

inflación es simplemente una reducción progresiva en la volatilidad de las perturbaciones

exógenas desde el mandato de Volker. Este hecho forma parte de un proceso más amplio

de estabilización de toda la economía de los EEUU desde principios de los 80, conocido

como la “Gran Moderación” (Kim y Nelson (1999), McConnell y Perez-Quirós (2000)).

Este fenómeno se manifiesta en una reducción de la varianza de las innovaciones en

modelos autorregresivos tipo VAR de la economía de EEUU. En particular, Sims y Zha

13

(2006) no encuentran evidencia de cambios en los parámetros autorregresivos de un

modelo VAR de la economía de EEUU, una vez la variabilidad en las perturbaciones

estructurales es tenida en cuenta.

Como he expuesto anteriormente, este primer capítulo investiga las posibles

causas de la inestabilidad en la economía de los EEUU durante los años 70 y primeros 80

así como el papel de la Reserva Federal en la posterior estabilización. Más

concretamente, el objetivo del presente estudio es relajar los fuertes supuestos sobre la

persistencia de la inflación mantenidos en la literatura previa. La mayoría de ella se ha

basado en el análisis de vectores autorregresivos (VAR). Puesto que la naturaleza de la

cuestión es dinámica, los modelos VAR son estimados en diferentes periodos, o bien los

parámetros del modelo pueden variar en el tiempo para poder detectar los cambios. De

hecho, los modelos VAR con parámetros variantes en el tiempo (time-varying VAR)

están recibiendo actualmente mucha atención en la literatura. Sin importancia de la

conclusión obtenida, toda la literatura previa restringe el espacio paramétrico para obtener

VAR estacionarios. El supuesto de estacionariedad es útil en el análisis posterior del

modelo VAR, pues permite obtener respuestas impulso-reacción no explosivas. No

obstante, este supuesto de estacionariedad de la inflación contrasta con otras corrientes de

investigación que consideran que la inflación está mucho mejor representada por un

proceso de raíz unitaria (I(1)) (véase Benati y Surico (2008), Cogley et al. (2010), Pivetta

y Reis (2007) o Stock y Watson (2007)), al menos en algunas partes de la muestra. Más

allá, al restringir el modelo VAR para obtener modelos estacionarios no se impone

solamente estacionariedad en la inflación (y en las otras variables del modelo) sino

también memoria corta (I(0)). La distinción entre los procesos de raíz unitaria I(1) y los

de memoria corta I(0) no es menor. Aunque ambas formulaciones pueden proporcionar

predicciones similares a corto con los parámetros apropiados, las predicciones a medio y

14

largo plazo (frecuentemente el objeto de interés en macroeconomía) son radicalmente

distintas. La función de autocorrelación de los procesos de memoria corta I(0) decae

exponencialmente, con el efecto de las perturbaciones desapareciendo en el corto plazo.

Los procesos de raíz unitaria I(1) están caracterizados por funciones de autocorrelación

planas, revelando el hecho de que los efectos de las perturbaciones exógenas son

permanentes. Asimismo, si los parámetros del modelo VAR son forzados hacia la región

estacionaria pero la inflación no sigue un proceso de memoria corta I(0), uno puede

preguntarse hasta qué punto los cambios en los parámetros del VAR pueden estar

subestimados frente a cambios en la varianza en la literatura previa.

Para analizar estas y otras cuestiones, este trabajo considera un marco estadístico

más amplio que engloba los supuestos I(0) y I(1) así como otras alternativas

fraccionalmente integradas. Es decir, mi objetivo en el presente estudio es incorporar la

integración fraccionada a un modelo multivariante y cambiante en el tiempo de la

economía de los EEUU. En los modelos univariates (tipo ARFIMA), la estimación en el

dominio de frecuencias es la manera estándar de solventar los problemas asociados con la

complicada función de verosimilitud cuando el componente autorregresivo AR está

presente. En un marco multivariante, la inclusión de interdependencias entre las

diferentes series imposibilita totalmente la estimación en el dominio temporal, siendo la

estimación en dominio de frecuencias la única alternativa viable. Por el otro lado, si la

estimación es llevada a término en el dominio de frecuencias, la dimensión temporal se

pierde y los parámetros del modelo no pueden ser modelados como variantes en el

tiempo. Para subsanar este problema circular, primero estimo un espectro multivariante y

cambiante en el tiempo de la economía de EEUU y luego ajusto vectores autorregresivos

fraccionalmente integrados (VARFIMA) en el dominio espectral a cada uno de los

resultantes espectros estimados. De esta manera, todos los parámetros del modelo

15

VARFIMA, incluyendo la varianza de las perturbaciones, evolucionan en el tiempo de

acuerdo con los cambios en el espectro, sin necesidad de establecer leyes de movimiento

para todos los parámetros del modelo. Esto a su vez constituye una ventaja respecto a los

modelos time-varying VAR, que necesitan de previa especificación dinámica de todos los

parámetros. Sin embargo, la característica más importante del marco estadístico empleado

es que permite relajar los supuestos sobre el orden de integración de las diversas series

que se incluyen el modelo. No me restrinjo a orden enteros de integración (I(0) o I(1)),

sino que el orden de integración (posiblemente fraccional) de cada una de las series se

estima junto a los restantes parámetros del modelo. La integración fraccional I(d) permite

modelizar situaciones en que los supuestos I(0) y I(1) son demasiado restrictivos. Si el

orden de integración “d” está situado entre 0 y 1, la serie presenta memoria larga y el

efecto de las perturbaciones exógenas aunque no es permanente como en el caso I(1),

decae a una tasa menor que exponencial (caso I(0)). Es relevante el caso particular donde

la serie no es estacionaria pero si revierte a su media: el coeficiente de integración

fraccional de las serie “d” es menor que 1, pero mayor o igual a 0.5. En este caso

particular, aunque el efecto de las perturbaciones exógenas sigue siendo eventual, decae

tan lentamente que la serie no es estacionaria. El caso de no estacionariedad pero

reversión a la media está en acuerdo con la evidencia empírica de muchas series

macroeconómicas (véase Henry y Zaffaroni (2002)). De hecho, los modelos de

integración fraccional ya se han empleado satisfactoriamente para modelizar la inflación,

siempre mediante enfoques univariantes (y sin variación en los coeficientes). Las

principales referencias incluyen: Baillie et al. (1996), Baum et al. (1999), Bos et al.

(2002), Hassler y Wolters (1995), Franses y Ooms (1997), Gil-Alana (2005) o Gadea and

Mayoral (2006). La gran mayoría de los estudios anteriormente citados evidencian

precisamente el carácter no estacionario (pero revertiendo a su media) de la inflación.

16

El marco estadístico discutido anteriormente es empleado para analizar la

persistencia de la inflación de EEUU, así como investigar la presencia de cambios en la

manera que la política monetaria se ha llevado a término en las últimas décadas. La

persistencia de la inflación es un factor clave en el diseño de la política económica puesto

que determina el mecanismo de trasmisión de la política monetaria (cuanto más

persistente es la inflación, mayores serán los esfuerzos necesarios para conducir la

inflación a su nivel objetivo) así como puede influir en las creencias de la Reserva

Federal sobre la existencia de una tasa natural de desempleo. El marco estadístico

empleado es particularmente apropiado para analizar cambios en la persistencia de la

inflación, pues viene medido por la evolución de su parámetro de integración fraccional a

frecuencia cero.

Para analizar la política monetaria y la persistencia de la inflación en el marco

anteriormente citado, considero un pequeño modelo trimestral de la economía de EEUU

compuesto por las series de inflación y desempleo (como variables control) y el tipo de

interés (como variable de política). La inflación es medida como la primera diferencia del

logaritmo del índice de precios encadenados del Producto Nacional Bruto (PNB) de la

economía norteamericana, como en Stock y Watson (2007). El desempleo es medido por

el valor a mitad de trimestre de la serie mensual de desempleo civil. Por último el tipo de

interés nominal es medido por la tasa secundaria de los bonos del tesoro, seleccionando el

primer mes de cada trimestre como en Cogley y Sargent (2002,2005). Las dimensión

temporal de las series empleadas en el presente estudio va desde el primer trimestre de

1948 hasta el último trimestre del 2009 y todas ellas han sido descargadas de la base de

datos de la reserva Federal (FRED). La estimación del espectro variante en el tiempo es

realizada sin la parametrización del espectro. Para ello, divido el rango de frecuencias [0,

π] en M=100 puntos equidistantes. Los valores iniciales del espectro son obtenidos

17

mediante partición de las series temporales en bloques de 40 observaciones y computando

el periodograma en cada bloque. Todos los puntos temporales en un mismo bloque tienen

el mismo valor inicial del espectro. Luego aplico el modelo bayesiano SS–ANOVA (Gu y

Wahba (1993) y Gu (2002)) donde el valor inicial del espectro en cada periodo es

suavizado simultáneamente en la frecuencia y en el tiempo, y así obtengo un estimador

del la densidad espectral multivariante para cada momento del tiempo. El modelo SS-

ANOVA ha sido recientemente empleado por Qin y Guo (2006) o Koopman y Wong

(2011). Una vez obtenida una estimación del espectro para cada momento del tiempo,

ajusto un modelo VARFIMA a cada densidad espectral mediante el estimador de máxima

verosimilitud aproximada (también llamado “Whittle”) en el dominio de frecuencias. Para

poder comparar mis resultados con la literatura previa, considero un modelo VAR con un

número fijo de retardos. En particular obtengo que un modelo VAR con un solo retardo

en la parte autorregresiva más el componente de integración fraccional proporciona el

mejor ajuste en todos los periodos. Para identificar los shocks monetarios, ordeno el tipo

de interés nominal como última serie en el (FI) VAR. Es decir, asumo que los shocks

monetarios toman al menos un trimestre en filtrarse a las otras dos variables. Este

supuesto de identificación a corto plazo es estándar en la literatura (véase Sims y Zha

(2006) o Primiceri (2005)).

Mis resultados incluyen lo siguiente. La varianza de las perturbaciones se ha

reducido drásticamente desde los 80, lo que sugiere que, la mayor estabilidad de la

economía desde principios de los años 80 es, en parte, una cuestión de buena suerte. No

obstante, en contraste con los resultados obtenidos por Primiceri (2005), encuentro

evidencia de una respuesta mucho más activa a la inflación por parte de la Reserva

Federal. Este cambio en la magnitud de la respuesta a la inflación empieza en la

presidencia de Volker y se prolonga hasta los dos primeros tercios de la presidencia de

18

Greenspan. En línea con este último resultado, y también encuentro una disminución en la

persistencia de la inflación durante el mismo periodo, lo que en la mayoría de modelos

monetarios se interpreta como consecuencia directa de una mayor atención a la inflación

por parte del Banco Central. No obstante, y a pesar de la disminución en su persistencia,

la inflación en EEUU se caracteriza por un comportamiento no estacionario pero

revirtiendo a la media durante todo el periodo. Este resultado cuestiona la idoneidad del

marco I(0) empleado en la literatura previa y sugiere que la falta de evidencia en favor de

cambios en los parámetros de los modelos VAR puede ser debida a la restricción del

modelo hacia la región estacionaria. En líneas generales, mis conclusiones están en línea

con los resultados de Cogley y Sargent (2005) o de Fernández-Villaverde et al. (2010).

Capítulo 2: Sobre la invertibilidad de las series ajustadas estacionalmente.

El segundo capítulo de la tesis está escrito en coautoría con los doctores Luis

Alberiko Gil-Alaña y Yuliya Lovcha, ambos pertenecientes a la Universidad de Navarra.

En este segundo capítulo examinamos las implicaciones del ajuste estacional mediante

métodos basados en modelos ARIMA (AMB) en la invertibilidad de las series resultantes

en un contexto de estacionalidad fraccionalmente integrada. Dada la naturaleza estacional

de las series macroeconómicas, el ajuste estacional es una práctica habitual para millones

de series económicas. Muchas de ellas, como las empleadas en el capitulo anterior, no

pueden ni siquiera ser obtenidas en la versión no ajustada estacionalmente. El ajuste

estacional permite eliminar las fluctuaciones en las frecuencias estacionales sin

(presumiblemente) realizar cambios sustanciales en otras frecuencias (especialmente en la

parte baja del espectro). De esta manera, el ajuste estacional hace los datos más tratables

para una posterior modelización y análisis. No obstante, en realidad, las propiedades de la

serie ajustada dependen crucialmente del método de ajuste empleado así como de las

19

propiedades de la serie original, y pueden resultar tan poco atractivas para el posterior

análisis como la serie estacional original.

En este trabajo analizamos una de las más importantes características de los datos

ajustados: ceros en el periodograma en las frecuencias estacionales y la resultante no

invertibilidad de la serie ajustada. Los ceros espectrales son producidos por todos los

métodos de ajuste empleados en la práctica, sin importancia si es un ajuste naif

empleando variables categóricas estacionales, o bien técnicas de extracción de señal

mucho más sofisticadas mediante programas especializados. Nerlove (1964) aplica el

programa Census X-11 y el método modificado ‘Hannan’ y concluye que ambos métodos

modifican mucho más que el componente estacional. Grether y Nerlove (1970) muestran

que el fenómeno observado en Nerlove (1964), creación de ceros cerca de las frecuencias

estacionales, también es producido mediante métodos óptimos de ajuste.

Consecuentemente, las rutinas de Census X-11 ARIMA y X-12 producen el mismo

resultado por construcción. Ooms y Hassler (1999) muestran que la regresión con

variables categóricas estacionales generan ceros en las frecuencias estacionales del

periodograma que pueden llevar a problemas de singularidad en regresiones log-

periodogram. Gomez y Maravall (2001) apuntan que la aplicación de métodos basados

en modelos ARIMA (AMB), que constituyen la base de Tramo-Seats, también generan

ceros en las frecuencias estacionales cuando el modelo que el programa identifica para la

serie original contiene raíces unitarias estacionales. De acuerdo con Gomez y Maravall

(2001), los ceros espectrales son la contrapartida a raíces MA unitarias y, por tanto, la

serie ajustada no es invertible y no acepta aproximación AR (o VAR) a su representación

de Wold. Aunque mayoritariamente ignorado, ésta es posiblemente la mayor implicación

que el ajuste estacional tiene sobre las series macroeconómicas, pues la aproximaciones

20

AR y VAR son típicamente empleadas en el trabajo econométrico aplicado como ha

quedado manifiesto en el capitulo anterior.

En este trabajo, analizamos en detalle los ceros en frecuencias estacionales así

como la aparente no invertibilidad de las series resultantes producidas por el ajuste AMB

en un marco de integración fraccionada, que admite una representación de la

invertibilidad más amplia que la usada por Gomez y Maravall (2001). Nuestros

resultados muestran que los ceros en las frecuencias estacionales no necesariamente

corresponden a raíces MA unitarias. Un proceso fraccionalmente integrado es

(estacionalmente) invertible cuando los coeficientes de integración fraccional en las

frecuencias estacionales son mayores que -0.5. Dado que los ceros en frecuencias

estacionales corresponden a valores negativos de integración fraccional, un proceso puede

tener ceros en esas frecuencias pero aún así ser invertible. En el presente estudio

concentramos nuestra atención al ajuste estacional mediante métodos AMB y para ello

empleamos TRAMO-Seats (TSW) como programa representativo. TSW forma parte de

de los programas de ajuste estacional empleados por Eurostat desde 1994, y cuyo uso se

ha extendido actualmente a muchos países Europeos (Gomez y Maravall (2001), ESS

Guidelines on Seasonal Adjustment (2009)).

Para examinar la invertibilidad de las series ajustadas por TSW, simulamos un

conjunto de procesos. No nos restringimos a procesos con órdenes enteros de integración

en cero y las frecuencias estacionales puesto que la evidencia empírica ha puesto de

manifiesto que la integración fraccional en las frecuencias estacionales es un fenómeno

habitual en las series económicas (Porter-Hudak (1990); Gil-Alana y Robinson (2001);

etc.). No obstante, también simulamos datos para procesos del tipo Airline, que es el tipo

de proceso por defecto identificado por TSW. Para cada modelo, simulamos 500 series,

las ajustamos mediante TSW y luego estimamos los parámetros de integración fraccional

21

en la serie ajustada mediante el estimador semi-paramétrico Gaussiano propuesto por

Hurvich y Chen (2000). Para realizar la estimación, aplicamos un “taper” de valor

complejo. El uso de la técnica de “tappering” permite subsanar los efectos de la (sobre)

diferenciación y el sesgo en los estimadores basados en el periodograma con datos

diferenciados.

Nuestros resultados muestran que si el proceso original (DGP) sigue realmente un

modelo de tipo Airline, la serie resultante del ajuste por TSW es no invertible. No

obstante, si la serie original es (estacionalmente) fraccionalmente integrada, lo que es

menos restrictivo y en acorde con la evidencia empírica, la serie ajustada puede ser

invertible o no dependiendo de la estacionariedad de la serie original. En particular, si la

serie original es generada mediante un modelo estacionario ARFISMA y luego ajustada

por TSW, la serie ajustada presenta seros en las frecuencias estacionales, pero esos ceros

corresponden a coeficientes de integración fraccional negativa en las frecuencias

estacionales mayores que -0.5, y no a raíces MA unitarias. Por el contario, si la serie

original es generada mediante un modelo ARFISMA no estacionario, la serie resultante

presenta coeficientes de integración negativa menores que -0.5, y por tanto, no es

invertble. En general, nuestros resultados están en línea con Gomez y Maravall (2001)

pero para un contexto más amplio de invertibilidad que incluye integración fraccional.

Este hecho es ilustrado con un ejemplo empírico para diversas series de datos trimestrales

de la economía Española. En particular empleamos el Índice de Producción Industrial

(IPI), la series de empleo (EMP) y de pasajeros aéreos (AIR) así como tres indicadores

de actividad económica para España: el consumo de cemento (CC) y los registros de

vehículos (CR) y de viviendas iniciadas (HS). El índice de producción industrial IPI y los

tres indicadores de actividad económica son considerados por Leamer (2009) como

conductores del ciclo económico y han sido recientemente empleados por Bujosa et al.

22

(2010) para la construcción de un indicador avanzado de actividad económica para la

economía española.

Capítulo 3: Explicando los cambios en la brecha salarial a nivel sectorial: un

enfoque Translog

El último capítulo de la tesis ha sido realizado en coautoría con mi director de

tesis, Fidel Pérez-Sebastián. El objetivo de este último capítulo es examinar la evidencia

en favor de la hipótesis de complementariedad entre el capital y el trabajo cualificado y la

hipótesis de progreso tecnológico sesgado, a nivel sectorial. Durante los últimos 40 años,

la oferta de trabajadores cualificados se ha incrementado en la mayoría de economías

desarrolladas. No obstante, este hecho no se ha visto acompañado por una disminución

drástica de la brecha salarial entre trabajadores cualificados y no cualificados.

Contrariamente, para algunas economías, como EEUU, el salario relativo de los

trabajadores cualificados ha incluso aumentado sensiblemente. La estructura salarial en

Europa ha permanecido más estable aunque ningún país europeo presenta una

disminución sustancial en la brecha salarial. La mayoría de los estudios encuentran que

un proceso latente de progreso tecnológico sesgado hacia el trabajo cualificado es la

solución a esta aparente contradicción. El progreso tecnológico sesgado hacia el capital se

define como un cambio en tecnología de producción que favorece el trabajo cualificado

sobre el no cualificado incrementando su productividad relativa y, consecuentemente, su

demanda relativa. Por ejemplo, Bound y Johnson (1992), concluyen que la mayoría de la

variación en la brecha salarial puede ser explicada simplemente por una tendencia

residual. Como explicación alternativa, la hipótesis de complementariedad entre capital y

trabajo cualificado (Griliches (1969), Krusell et al. (2000)) también ha recibido mucha

atención en la literatura. De acuerdo con esta hipótesis, el capital y el trabajo cualificado

son más complementarios como inputs productivos que lo son el capital y el trabajo no

23

cualificado. Como consecuencia, un incremento en el stock de capital incrementa la

productividad marginal del trabajo cualificado por más que la del no cualificado. En

acuerdo con Krussell et al (2000), la (relativamente mayor) complementariedad entre

capital y trabajo cualificado es un factor importante para entender la evolución de la

brecha salarial, pues el stock de capital (especialmente en infraestructuras de la

información y comunicación (ICT)) ha aumentado notoriamente en estos últimas décadas.

La validez de las hipótesis de complementariedad entre capital y trabajo

cualificado y de progreso tecnológico sesgado en favor del trabajo cualificado es de suma

importancia, pues tiene implicaciones centrales en el las teorías de crecimiento

económico, de comercio internacional y de desigualdad (véase Stokey (1996)). Los

estudios empíricos realizados encuentran soporte bien para una o la otra de las dos

hipótesis empleando para ello sólo datos del sector manufacturero o datos completamente

agregados (véase Fallon and Layard (1975), Berman et al. (1998), Flug y Herkowitz

(2000) , Krusell et al (2000), Ruiz-Arranz (2002) o Duffy et al. (2004) entre otros). De

entre ellos, solamente Ruiz-Arranz (2002) intenta separa y cuantificar la contribución de

las dos hipótesis a la vez, empleando datos completamente agregados para la economía

de EEUU.

El propósito de este capítulo es examinar las dos hipótesis a nivel sectorial. En

este sentido, nuestro trabajo está relacionado con Fallon y Layard (1975). Fallon y Layard

emplean datos de corte transversal de 9 países desarrollados y 13 economías en desarrollo

en el año base 1963, para estimar la forma reducida de las ecuaciones derivadas de una

tecnología del tipo CES-anidada. Los autores encuentran una ligera evidencia en favor de

la hipótesis CSC (aunque no estadísticamente significativa) a nivel agregado pero fuerte

evidencia a nivel sectorial. Sus datos, no obstante, no contienen datos ni de precios ni de

stock de capital a nivel sectorial. Para solventar ese problema asumen mercados

24

perfectamente competitivos así como igualdad en el parámetro de eficiencia entre

sectores de manera que las condiciones de primer orden puedan ser usadas para estimar

las ecuaciones derivadas de la forma reducida. Contrariamente, la base de datos empleada

en este trabajo si contiene datos de precios y stock de capital a nivel sectorial permitiendo

diferencias de eficiencia entre sectores productivos. Asimismo, la dimensión temporal de

los datos permite analizar el sesgo en el progreso tecnológico, que no es analizado por

Fallon y Layard.

La metodología empleada en este trabajo se deriva de los estudios de Christensen

et al. (1971, 1973) y de Ruiz-Arranz (2002), y nos permite distinguir y cuantificar los

efectos que ambas hipótesis tienen en la evolución de la brecha salarial a nivel sectorial.

Para ello estimamos un modelo del tipo translog con 4 inputs (trabajo cualificado y no

cualificado, capital de tipo ICT y no ICT) empleando datos de siete sectores (primario

(PRIM), minero (MIN), manufacturas (MAN), construcción (CON), y tres diferentes

agregaciones del sector servicios (servicio de suministros (SUP), servicios financieros y

empresariales (FIN), y resto de servicios (OTHER)) para tres países desarrollados

diferentes (EEUU, UK, y Japón).

El modelo translog proporciona un sistema de ecuaciones tipo SURE de las

participaciones de los inputs en los costes totales de producción en función del

(logaritmos) de los precios. Asimismo, el sistema está sujeto a un conjunto de

restricciones en los parámetros como consecuencia de los supuestos habituales sobre la

las propiedades de la función de costes: homogeneidad de primer grado en los precios de

los factores, simetría y concavidad. El modelo translog no garantiza la concavidad de la

función de costes. En nuestro caso, la concavidad resulta ser un problema serio pues la

estimación del modelo no restringido no satisface la concavidad en una parte importante

de la muestra. Evidentemente el fallo en el supuesto de concavidad puede dar lugar a

25

errores de inferencia. Para asegurar la curvatura correcta de la función de costes sin

afectar a la flexibilidad del modelo translog, este estudio emplea el método desarrollado

por Ryan y Wales (2000) que impone la concavidad en un punto referencia. La elección

apropiada de ese punto de referencia deriva en la satisfacción del supuesto de concavidad

en todos los puntos. Asimismo, con datos agregados, los precios determinados por oferta

y demanda no pueden ser tratados como variables exógenas y, por tanto, la estimación

del sistema de participaciones en el coste derivadas del modelo translog tiene que ser

instrumentado. En este trabajo empleamos el método de estimación de los momentos

generalizado en su versión no lineal (non-linear GMM) donde retardos de las variables

endógenas y de la variable dependiente son empleadas como instrumentos. Esta

metodología, inicialmente importada de los modelos de crecimiento por Caselli et al.

(1996), se ha convertido en el procedimiento habitual en la estimación de las funciones de

producción. La no linealidad es consecuencia de las restricciones en los parámetros que

aseguran la concavidad de la función de costes.

La estimación del modelo translog para cada uno de los siete sectores de los tres

países analizados, proporciona cierta evidencia empírica en favor de la hipótesis CSC.

Los resultados indican una relativamente mayor complementariedad entre trabajo

cualificado y el capital en equipos de información y comunicación. Asimismo, el trabajo

no cualificado parece ser más complementario con el otro tipo de capital. Este hecho es

importante pues el stock de capital ICT ha aumentado relativamente más rápido en estos

últimos años en casi todos los sectores. No obstante, esta evidencia es débil y varía en

función del sector y país analizado. Contrariamente, el progreso tecnológico sesgado en

favor del trabajo cualificado explica una importante proporción de las variaciones en la

brecha salarial en todos (o casi todos) los sectores y países analizados.

26

27

CHAPTER 1:

A Fractionally Integrated Approach to Monetary Policy and Inflation Dynamics

28

29

1.1 Introduction

During the last forty years, the U.S. economy has been characterized by markedly

different episodes concerning both the level and the volatility of inflation. As can be seen

in Figure 1, the U.S. annual average rate of inflation declined from a 4.5% in the period

1960-1984 to 3% in 1985-2009. Also its standard deviation decreased by one third

between these two periods. This has led to research trying to assess the role of the Fed in

accounting for these observed differences. Whether the high inflation episode had been a

consequence of bad policy or rather, the result of bad luck is still controversial in the

literature. While some authors assert that monetary policy has been conducted more

efficiently starting from Volker’s chairmanship (see e.g. Boivin and Gianonni (2006),

Clarida et al. (2000), Cogley and Sargent (2002), or Lubick and Shorftheide (2007))

other’s found small or null evidence of drastic changes in the monetary policy from about

fifteen years prior to Volker (main references include Bernanke and Mihov (1998),

Canova and Gambetti (2009), Primiceri (2005) or Sims and Zha (2006). According to the

last group of authors, the main factor driving the differences between the two periods was

a reduction of the volatility of the exogenous shocks. For instance, Sims and Zha (2006)

do not find any evidence of coefficient drifting once time variability in the structural

disturbances is taken into account.

This paper investigates the possible causes of the poor economic performance of

the 70 and early 80’s, and the role of the Fed in the observed changes. The objective here

is to relax the strong assumptions on the persistence of inflation typically made in

previous literature. Most of it has relied on the analysis of vector autorregresion (VAR).

Given that the nature of the question is fully dynamic, the models have to be estimated at

different sample periods or the parameters must be able to change. In fact, time-varying

30

VAR’s have recently received a lot of attention in the literature1. Despite their different

conclusions, previous research has restricted the VAR parameter space to deliver

stationary results. The assumption of stationarity contrasts to a large stream of different

literature which agrees that inflation is better characterized by an ( )I 1 process (see e.g.

Benati and Surico (2008), Cogley, Primiceri and Sargent (2010), Pivetta and Reis (2007)

or Stock and Watson (2007))2. Furthermore, by restricting the VAR parameter space, one

is imposing not only stationarity, but also short memory ( ( )I 0 ). The distinction between

( )I 0 and ( )I 1 cases is not minor. Although both formulations can deliver similar short

term predictions if appropriate parameters are chosen, the medium and long run

implications (frequently the object of interest in macroeconomics) are drastically

different. While the autocorrelation function of ( )I 0 process show exponential decay

with the effect of the shocks dying in the short run, ( )I 1 process are characterized by a

flat autocorrelation function revealing that shocks are permanent. Besides, if the VAR

parameters are forced towards the stationary region and inflation is not I(0), one may

question up to which extent the coefficient drifting found in previous literature may be

underestimated in front of changes in the variance.

To address these issues, this paper considers a wider statistical framework that

encompasses both ( )I 0 and ( )I 1 assumptions as well as other fractionally integrated

possibilities. Thus my target in this paper is to incorporate FI into a multivariate and time

varying model of the US economy.

1 See among others the work of Boivin and Gianonni (2006), Canova and Gambetti (2009), Cogley and Sargent (2002, 2005a) or Primiceri (2005). 2 Pivetta and Reis (2007) calculate that imposing such a restriction leads to more than 40% of parameter rejections. From the other side, evidence on unit root tests is mixed and strongly period specific. However several authors were not able to reject the unit root null for long period of time (see e.g. Murray et al. (2008) and references there in).

31

In the univariate framework, the frequency domain approach has become a standard way

of circumventing the problems associated with the complicated likelihood function

arising from univariate ARFIMA when the autoregressive component is present. In the

multivariate framework, the inclusion of additional interdependencies among the

different time series makes the time domain estimation virtually unfeasible. From the

other side, if the estimation is carried out in the frequency domain, the time dimension is

lost and parameters cannot be modeled as time varying.

To overcome this circular problem, I first estimate a time varying spectrum for post

WWII US data and then I fit a fractionally integrated (FI) vector autoregressive model

(VARFIMA) in the frequency domain to each of the resultant time dependent spectrums.

Thus, the entire model’s parameters change with the time varying spectrum, obtaining

smooth parameter transitions without the need of parametric specification of the laws of

motion. This is also an advantage with respect to the standard time varying VAR

literature, where a dynamic specification must be assumed at the outset.

Fractional integration account for situations where the ( )I 0 and ( )I 1 assumptions

are too restrictive, allowing the effect of the shocks not to be permanent but decaying at a

rate lower than exponential, which is the actual behavior of many macroeconomic time

series3 (see e.g. Henry and Zaffaroni (2002) for significant references). In fact,

fractionally integrated models have been successfully employed for modeling inflation,

typically by univariate approaches4. Most of these studies report evidence of a non-

stationary but mean reverting behavior of inflation.

The statistical framework above is employed to assess the degree of stability of

inflation persistence and to investigate changes in the way the monetary policy has been

3 As shown in Gadea and Mayoral (2006), fractional integration may appear in inflation as the result of price aggregation over heterogeneous agents. 4 Main references include Baillie et al. (1996), Baum et al. (1999), Bos et al. (2002), Hassler and Wolters (1995), Franses and Ooms (1997), Gil-Alana (2005) or Gadea and Mayoral (2006).

32

conducted by the Fed. Inflation persistence is a key factor in the design of monetary

policy since it determines the monetary transmission mechanism and may affect Fed

believes about the natural rate hypothesis. My framework is particularly well suited to

evaluate inflation persistence, since it can be measured by the fractional integration

parameter at zero frequency.

My findings include the following. The variance of the shocks reduced drastically

from the 80’s, suggesting that a sizable part of the great performance in the economy

during last 80’s and 90’s could be just a tale of good luck. However, in contrast with the

results of Primiceri (2005), I find overwhelming evidence of a more active monetary

policy towards inflation starting from Volker’s and extending through first half of

Greenspan’s chairmanships. In line with the previous result, I find that inflation

persistence also have been falling from the early 80’s, which most monetary models

interpret as the result of a more vigorous attention to inflation on the part of the Fed.

Nevertheless, inflation has remained very persistent during the whole period,

characterized by a non-stationary but mean reverting behavior. This result questions the

adequacy of the ( )I 0 framework and suggests that previous evidence against drifting

coefficients sometimes may be an artefact of parameter space restriction towards the

stationary region. Overall, my conclusions are more similar to Cogley and Sargent (2005)

or Fernandez-Villaverde et al. (2010).

The outline of the article is as follows: Section 2 describes the econometric

framework. Empirical analysis may be found in section 3. Section 4 concludes.

33

1.2 Econometric Framework

1.2.1 Estimation of the time varying cross-spectrum.

As a first step of the analysis, I estimate a time-varying cross-spectrum of the

variables. In the statistical literature, various methods have been proposed to estimate

time-varying spectra, both parametric and non-parametric.

Examples of parametric methods for the estimation of time varying spectrum are

given by Jansen et al. (1981), Kitagawa and Gersch (1996) and Davis et al. (2006). First,

the time series is divided into blocks, and after the parameters of autoregressive models

are estimated at each block. By substituting the estimated parameters in the spectral

density of the corresponding autoregressive models, the spectrum of each block is

obtained. Another solution, in principle, is to fit an autoregressive model with time-

varying parameters to the data (see e.g. Cogley and Sargent (2002, 2005)). In any case,

autoregressive models focus on fitting short run dynamics, and they are not expected to

produce good estimates of the spectrum at low frequencies as required for fractionally

integration estimation (see e.g. Christiano et al. (2006)).

From the other side, non-parametric estimation of the time-varying spectra is

produced without parameterization of the spectrum. Thus, Cohen (1989) and Adak (1998)

divide the time series into blocks and compute the sample spectrum for each block of

observations. All time points in each block have the same estimate of the spectrum in this

approach. The time-varying spectrum only evolves over the blocks. By using the

smoothing spline ANOVA (SS-ANOVA) method (Gu and Wahba (1993) and Gu (2002)),

the time series is smoothed over these initial block spectra. Smoothing is produced

simultaneously over time and frequency. This procedure has been recently adopted by

Koopman and Wong (2011) and is also the approach I follow in the present study.

34

Let ,j k

x be the initial log-sample spectrum for frequency k

λ at time j

t (plus a

constant). The log-sample spectrum is related to an unknown two dimensional function

representing the true log-spectrum ( , )j k

G t λ as:

( ) ( ), , ,, , 0, , 1... , 1...

j k j k j k j kx G t N j T k Mζλ ζ ζ σ= + = =∼ ,

, where ,j k

ζ is an error term. Omitting interaction terms between k

λ and j

t , the unknown

function ( , )j k

G t λ is modeled as a tensor product between two cubic spline functions

1 2( , ) ( ) ( )

j k j kG t G t Gλ λ= + , being

1( )

jG t and

2( )

kG λ the smoothed main effects over time

and frequency. Defining a Bayesian stochastic model for ( , )j k

G t λ and the recursive Bayesian

model, one can cast the SS-ANOVA into a State Space and obtain the estimates of

( , )j k

G t λ by employing the Kalman filter and smoother recursions (see Qin and Guo

(2006) or Koopman and Wong (2011), for details).

Once the time varying cross spectral density is obtained, a VARFIMA model is

fitted to each of the time dependent resultant cross-spectra estimates. The next sections

review the VARFIMA model and its estimation procedure.

1.2.2 Model description and frequency domain estimation:

The VARFIMA model

Univariate ARFIMA models can be generalized to multivariate settings leading to

the VARFIMA model. More specifically, the autoregressive VARFIMA model can be

written as:

( ) t tD L y e= (1.1)

( )t t te F L e ε= + (1.2)

where ty is a 1N × vector of variables for 1,...,t T= ; L is the backward shift operator;

and ( )D L is the diagonal N N× matrix with the diagonal elements given by ( )1 ndL− ,

35

1,...,n N= . The memory parameter nd is the parameter of fractional integration at zero

frequency of the seriesnty . As larger nd is, more persistent variable is, and stronger

policy actions are required to bring the variable to the targeted value. Stationarity requires

( )0.5,0.5nd ∈ − which can always be achieved by taking a proper number of integer

differences. Short memory occurs when nd =0, and the autocorrelations falling at

exponential rate. If ( )0,0.5nd ∈ the process has long memory, and the autocorrelations

show hyperbolic decay with the effect of the shocks taking more time to disappear than in

the short memory case. The case [ )0.5,1nd ∈ is of a lot of interest in macroeconomics.

The process is not stationary but presents mean reversion: the effect of the shocks

eventually disappears. Shocks have permanent effect whenever 1nd ≥ . ( )F L is the

stationary autoregressive polynomial with p lags capturing short run dynamics. The

1N × vector of errors tε is assumed to be( )0,N Ω . Examples of empirical application of

VARFIMA models may be found in Gil-Alana and Moreno (2008), Lovcha (2009) or

Halbleib and Voev (2011).

The reduced and structural form of the VARFIMA model

The VARFIMA model defined as in (1.1) and (1.2) is a reduced form model.

Substitution of (1.1) into (1.3) leads after arrangement to the reduced-form MA (∞)

representation ofty :

( ) ( )( ) 11t N ty D L I F L ε−−= − (1.3)

The structural model includes the contemporaneous relationships between

variables, and it is given by:

( ) t tAD L y u= (1.4)

( )t t tu Q L u ξ= + (1.5)

where A is N N× matrix of structural relationships. The vector of structural error terms tξ is assumed to be( )0, NN I . Substitution of (1.4) into (1.5) and pre-multiplication

of both sides by 1A− leads to: ( ) ( ) ( )1 1

t t tD L y A Q L AD L y A ξ− −= +

( )( ) ( )1 1N t tI A Q L A D L y A ξ− −− =

36

Implying:

( ) ( )( ) 11 1 1t N ty D L I A Q L A A ξ

−− − −= − (1.6)

, which is the structural MA(∞) representation of ty .

To identify structural errors I apply Sims (1989) short-run assumptions: I

assume that the matrix of contemporaneous relationships is lower-triangular5. It follows

from (1.3) and (1.6) that 1t tA ξ ε− = . Once the variance-covariance matrix Ω of the

reduced–form model errors is estimated, entries of the matrix of contemporaneous

responses can be found.

Impulse responses

Substitution of 1t tA ξ ε− = into (1.3) leads to:

( ) ( )( ) ( )11 1t N t ty D L I F L A Lξ ξ−− −= − = Λ

Since the matrix ( )D L is diagonal, its inverse is also diagonal, with elements given

by ( ) ( )1 nd

nD L L−= − . The operator ( )nD L can also be defined by its infinite Taylor

expansion:

( ) ( )( ) ( ), ,

0

1 , 1

nd nkn k n k

k n

k dL d L d

k d

∞−

=

Γ −− = − =

Γ + Γ −∑

where ( ).Γ denotes the gamma function, satisfying ( ) ( )1z z zΓ + = Γ . The computation of

the elements of the matrix ( )( ) 1

NI F L−

− is straightforward6.

The matrix ( )LΛ is N N× and its elements are infinite MA polynomials which

coefficients are impulse responses of variables to the structural shocks. Thus, the

coefficients of the polynomial ( )lm LΛ are impulse responses of the variable l to a shock

in m .

5It is, the order of variables in the model matters; and a variable ,n ty is not contemporaneously influenced

by any variable ,n l ty + , 1,...,l n N= + , but may be influenced by variables ,l ty , 1,...,l n= . 6 The Taylor expansion can be applied or the coefficients can be computed as in Hamilton (1994, p.260)

37

The approximate maximum likelihood estimation of VARFIMA model

To estimate the process given by (1.1) and (1.2) I use approximate frequency

domain maximum likelihood, also known as Whittle estimation (Boes et al. (1989)). The

discussion of the multivariate version of the estimation procedure can be found in Hosoya

(1996).

An approximate log-likelihood function of θ based on ty , 1,...,t T= , is given

up to constant multiplication, by

( ) ( ) ( ) ( )( )1

1

ln , ln det , , ,T

j j T jj

L y f tr f I yθ ω θ ω θ ω−

=

= − + ∑

, for equispaced frequencies2

, 1,2,..., / 2 1j

jj T

T

πω = = − . The N N× matrix ( ),T jI yω

is the estimated spectrum and the (theoretical) spectrum of the VARFIMA process (2), (3)

at frequency jω is given by:

( ) ( ) ( ) ( )( ) ( )( ) ( )1 11 11, 2 j j j ji i i i

j N Nf D e I F e I F e D eω ω ω ωω θ π− −− −− − −= − Ω −

, where i is the imaginary unit, ( )jiD e

ω− is the complex conjugate of( )jiD e

ω ,

( ) 2

1 2 ...j j j ji i i ip

pF e Fe F e F eω ω ω ω= + + + and ( )ji

F eω− is its complex conjugate.

Following standard practice, frequency 0ω = is excluded from estimation in

order to avoid singularity problems in the sample periodogram in the fractionally

integrated framework. Confidence intervals for the model parameters are computed by

frequency domain bootstrap (see e.g. Franke and Härdle (1992)).

1.3 Empirical Evidence

In this section I apply the framework discussed above for the estimation of a small

quarterly model of the U.S. economy. Inflation is measured as the log-difference of the

GDP chain-type price index, as in Stock and Watson (2007). For the VARFIMA I also

38

condition on unemployment and nominal interest rate7. Unemployment is measured by

civilian unemployment rate. The original monthly series is converted to quarterly by

sampling the middle month of each quarter. Nominal interest rate is measured by the

secondary market rate three-month Treasury bill expressed as yield to maturity. Monthly

data is also converted to quarterly by selecting the first month of each quarter, in order to

align the interest rate data with inflation (see e.g. Cogley and Sargent (2002,2005)). All

data span from 1948:1 to 2009:4 and were downloaded from the Federal Reserve

Economic Database (FRED)8. For the estimation of the time-varying (multivariate)

spectrum, the frequency grid [0, π] is fragmented into M = 100 equidistant points. The

initial sample spectra are computed by partitioning the time series into blocks into blocks

of 40 observations and by computing the periodogram for each block. All points in a

block have the same initial estimate of the spectrum. Consequently, I apply the SS–

ANOVA model to fit the initial spectra and to obtain the time-varying spectrum9 as

explained in the section above.

For the sake of comparability with the existent time varying VAR literature, I

work with a fixed lag-length VAR. I found that one lag in the autoregressive part plus the

7 In this way our system is comparable to Cogley and Sargent (2002, 2005), Cogley et al. (2010), Primiceri (2005) or Benati and Surico (2006). 8 http://research.stlouisfed.org/fred2/ 9 By a matter of robustness, the estimation of the time varying spectrum has also been repeated by estimating a cross spectrum with a moving regression window of 15 years, as in Benati (2009), without obtaining significant changes. The procedure that I employed for the moving window is the following. The estimation step is one year (4 observations). It is, at each estimation step, I include 4 new observations and exclude 4 observations from the beginning of the sample. At each step I estimate the multivariate spectra

from the sample autocovariance matrices j

Γ as ( ) ( )1

1ˆ2

Ti j

X j Mj T

f w j e λλπ

−

=− +

= Γ∑ , where

( ) 1,...M j

w j=± is a sequence of kernel (Parzen) weights. As a result, the estimated cross-spectrum also

varies smoothly over time and frequency.

39

fractional integration component( )D L , provides the overall best fit10 with the significant

autoregressive component at each of the time dependent frequency domain estimations.

I identify monetary policy shock by ordering interest rates down in the VAR. It is,

I assume that policy shocks require at least one quarter percolating through non-policy

variables. This short-run identifying assumption is standard in the VAR literature (see e.g.

Leeper, Sims and Zha (1996), Christiano, Eichenbaum and Evans (1999) or Primiceri

(2005)). I further assume that unemployment does not affect inflation contemporaneously

as in Primiceri (2005) so the variables are arranged in the order ( ), , 't t t ty u iπ= . Different

order inside the non-policy block does not alter results.

With the intention to shed some light on the contribution of the Fed to the

different inflation episodes, I organize the results around four general themes: (i) what

was the degree of inflation persistence and did it remain constant?; (ii) were the

exogenous perturbations higher during the great inflation?; (iii) were differences in the

transmission mechanism of policy shocks?; (iv) was the systematic part of monetary

policy different during the great disinflation

1.3.1 Inflation persistence

The persistence of inflation is a central concern of macroeconomics (see e.g.

Fuhrer (2009) for a good review of the topic). Persistence is a key factor in the design of

monetary policy since it determines the monetary policy transmission mechanism and

also influences the behavior of private agents. Inflation persistence may also affect Fed

believes about the existence of an exploitable trade-off between inflation and

unemployment since tests of the natural rate hypothesis are very sensitive to it11.

10 Note that the FI component delivers an infinite VAR representation. 11 Solow-Tobin type tests have been criticized to tend to reject the natural rate when inflation persistence is low (see e.g. Sargent 1971)

40

Recently, several authors have looked for evidence that changes in the behavior of central

banks have resulted in a reduction of persistence. For instance, Cogley and Sargent (2002,

2005) using a multivariate time-varying VAR find that persistence increased in the early

1970’s, remained high for around a decade and declined afterwards. On the other hand,

Sims (2002) and Stock (2002) find that persistence remained constant and high over the

past 40 years. This view is also supported by Pivetta and Reis (2007). From the other

side, Benati and Surico (2008) and Cogley et al (2010) find a reduction in persistence in

inflation gap12, although they agree that inflation has remained highly persistent. Already

in the FI framework, Gadea and Mayoral (2006) test for changes in persistence using a

Lagrange Multiplier test on the stability of the memory parameter of inflation. The

authors find weak support to the hypothesis that persistence has changed.

The evolution of the memory parameter for inflation is provided in Figure 2.

Several results stand out. First, as in Cogley and Sargent (2002, 2005), persistence

increased during the 60’s and 70’s, reaching its top during early 80’s, and have decrease

during second half of the 80’s and 90’s, coinciding with the disinflation period. A new re-

point of persistence seems to have appeared from the beginning of the millennium, but

the statistical evidence about it is weak. Second, inflation has remained highly persistent

during the whole period, with the long memory parameter always over 0.5. This means

that inflation is characterized by a non-stationary but mean-reverting behavior; the effect

of the shocks, although eventually disappear, decay at a slow hyperbolic rate. This last

result is consistent with the findings of Baillie et al. (1996) or Gadea and Mayoral (2006)

in the FI framework, and provides evidence that inflation is not good characterized by an

I(0) process.

12 They define the gap as the difference between inflation and the Federal Reserve’s long-run target for inflation.

41

1.3.2 The pattern of exogenous shocks

Kim and Nelson (1999) and McConnell and Perez-Quiros (2000) found evidence

that U.S. economy from the early 80’s experienced a growing period of stability, which

they characterized in terms of a decline of the VAR innovation variances, phenomenon

that is known as the “Great Moderation”. It has been argued that neglecting the

heteroskedasticity of the innovations may lead to an exaggerated parameter drift (see e.g.

Sims (2002) or Stock (2002)). In the method proposed in this paper, the entire models

parameters, including the parameters of the variance covariance matrix of the innovations

are allowed to vary. Figure 4 presents the time profile for the standard deviation of the

VAR innovations together with their confidence intervals. The pattern resembles other

estimates in the literature (see Bernanke and Mihov (1998), Cogley and Sargent (2005) or

Primiceri). The standard deviations of the innovations of inflation and unemployment

(Fig. 4.a and 4.b) have been falling strongly from the middle 80’s, coinciding with the

disinflation period. This result suggests that changes in the variances of exogenous shocks

may be responsible of a considerable amount of the observed inflation stability during the

80’s and 90’s. Nevertheless, the standard deviation of the inflation innovation presents a

less pronounced peak during the last 70’s beginning of 80’s compared to other works. As

shown in the previous section, inflation almost acquired a unit root during the same

period. Therefore, the strong peak found in previous literature may be in part explained

by an exaggerated variance response due to the I(0) restriction.

1.3.3 Non-systematic monetary policy

As in Primiceri (2005), the term non-systematic policy captures the responses of

interest rate to policy mistakes and to variables other than inflation or unemployment. The

identified monetary shock is therefore the logical measure of the non-systematic policy.

42

Figure 4.c depicts the evolution of the standard deviation of the innovations to the interest

rate. As can be seen in the figure, the standard deviation of the exogenous shocks has

been also decreasing since the end of Volker’s monetary aggregate targeting. Fixed policy

rules, as the one stated in this paper13, are better approximations for monetary policy

actions from the 80’s on, which is consistent with the abandonment of discretionary

macroeconomic policy by the Fed and the adoption of a rules-based macroeconomic

policy.

The effects of monetary policy shocks on inflation and unemployment are

summarized in Table 1. The table reports the IRF of the non-policy variables to the

monetary shock for selected years. The selected years are 1975, 1984, 1996 and 2008.

The dates coincide with the middle of the Burns-Miller’s, Volker’s, Greenspan’s, and

Bernanke’s chairmanships period. Note also that 1984 coincides with the adoption of

inflation-targeting regime by Volker and the end of the great inflation. As can be seen in

the table, the responses of inflation and unemployment to a positive interest rate shock

have the expected sign; negative for inflation and positive for unemployment for all

selected years. Also, there are not significant differences in the size of the responses. In

general all are small and statistically not different from zero. Nevertheless, there is some

evidence of an increase of the long run responsiveness of unemployment to the policy

shock in recent years. However, once one takes into account the decrease in the size of

the shocks, as depicted in Figure 4.c, this increase become in line with the size of non-

systematic policy actions.

13 See equation (8) in next section for an interpretation of the third equation of the VARFIMA as an augmented Taylor Rule.

43

1.3.4 Systematic monetary policy.

Along this section, I try to evaluate the degree of activism of the monetary policy,

it is, how much the interest rate respond to inflation and unemployment movements. I

find that the multivariate coherence provides a good measure of the systematic part of

monetary policy. If tX is a process with 2N > components, it may be the case that the

most of the power in a given series can be removed by subtracting a linear function of

several components. This would indicate a relationship among the components which

might arise, for example, as a result of a common “driving mechanism”.

Multiple coherence of interest rate ti on inflation πt and unemployment ut, ranges

between 0 and 1, and measures the portion of the power (density) at frequency λ

attributable to the linear regression of i t on mt =πt , ut, and is given by :

( ) ( ) ( ) ( ) ( )1 *2, ,

ˆ ˆ ˆ ˆˆ /j m i m m i m iR f f f fλ λ λ λ λ−⋅ = ,

Figure 5 depicts the (time varying) multiple coherence between the interest rate

and the non-policy block. Results for selected years and frequencies, together with their

confidence intervals may be found in the Table 2. As can be seen in the table, interest rate

was explained by other two variables worse during the great inflation, especially at low

and business cycle frequencies. More generally, the multiple coherence coefficient fall

during Burns-Miller chairmanship, and rise strongly from the early 80’s on. This pattern

can be better seen in Figure 6, which depicts the time profile of the multiple coherence

coefficients at business cycle frequencies.

Now I proceed to check if they have been changes in the size of the responses of

interest rates to changes in the non-policy block during the two periods. To be concrete,

consider the reduced form VARFIMA model given by (1.1) and (1.2). Substitution of the

first equation into the second, and taking into account that 1t tA ξ ε− = , leads to:

44

( ) ( ) ( ) 1t t tD L y F L D L y A ζ−= + .

Multiplication of the both sides of the equation by A imply

( ) ( ) ( )t t tAD L y AF L D L y ζ= + .

Define ( ) ( )AF L G L= . The equation for the interest rate (the third equation in the model)

is given by:

( ) ( ) ( )1 2 2

31 1, 32 2, 3,1 1 1d d d

t t ta L y a L y L y− + − + − =

( ) ( ) ( )1 2 3

31 1, 32 2, 33 3, 3,1 1 1d d d

t t t tG L L y G L L y G L L y ζ= − + − + − +

Rearranging this expression leads to:

( ) ( )1 3 2 331 31 32 323, 1, 2, 3,

33 33

1 11 1

d d d d

t t t t

G L a G L ay L y L y

G L G Lυ− −− −= − + − +

− −

After collecting the polynomials, this expression can be re-written as:

( ) ( )3, 1 1, 2 2, 3,t t t ty L y L y υ= Φ + Φ + (1.7)

where ( ) ( ) 31

3, 33 3,1 1d

t tG L Lυ ζ− −= − − . The equation (1.7) can be considered as an

augmented Taylor Rule. The coefficients of the polynomial ( )1 LΦ are the responses of

the interest rate to the impulse in inflation. The sum ( )1 1Φ is finite if and only if

1 3 1d d− < . In this framework, the Taylor principle14 requires the infinite sum of these

coefficients, ( )1 1Φ is greater than one.

The cumulative response of the interest rate to permanent increase in the non-policy

variables is summarized in the Table 3. The table presents the contemporaneous and

cumulative responses of interest rate after 4, 20 and 40 quarters to a one percent

permanent increase in inflation (up) and to a one percent permanent increase in

14 The Taylor principle specifies that for each one-percent increase in inflation; the central bank should raise the nominal interest rate by more than one percentage point It has been argued that the Taylor Principle is a necessary and sufficient condition under which rational expectations equilibriums exhibits desirable properties.

45

unemployment (down), together with confidence intervals for the same selected years.

Some results can be highlighted. First, the response of interest rate to the non-policy

variables is gradual. This gradual Fed response is also well documented in the literature

(see e.g. Primiceri (2005)). Second, although there are not significant differences in the

interest response to unemployment in the selected years, the interest rate response to

inflation is drastically much stronger for the year 1996. Third, the Fed responses to

inflation have declined strongly in recent years. Figure 7 and 8 illustrate the differences in

the Fed response to inflation further. Figure 7 plots the evolution of the responses of the

interest rate to a one percent over the last forty years and Figure 8 depicts the complete

cumulative response up to 40 quarters, which proxies the long-run response, for the

selected years. As can be seen in Figures, the systematic monetary policy becomes more

reactive to inflation starting from middle 80’s during Volker’s chairmanship, and

coinciding with the beginning of the great disinflation leading to stability period. Note

also that the response to inflation during 90’s, corresponding to the years of higher

stability, is considerably stronger. In particular it reaches the Taylor principle before 20

quarters. Overall, I found strong evidence that the Fed monetary policy during second

half of the 80’s and 90’s was significantly much more active towards inflation, even after

controlling for heteroskedasticity in the disturbances. However, I do not find differences

in the responses of interest rate to inflation under the Burns-Miller and Bernanke

Chairmanships.

1.4 Conclusion

This paper applies frequency domain methods to study inflation persistence and

changes in Fed’s monetary policy without relaying in the standard I(0) assumption over

inflation (and the other variables in the VAR), consistent with the increasing evidence

that inflation is much more persistent.

46

As in the previous literature, I find a strong reduction of the variance of the shocks

which is in line with the view that a sizable amount of the great performance in the

economy during the 80’s and 90’s is just a tale of good luck. However, even controlling

for the (strong) reduction in the variance of the innovations, I find strong evidence of

changes in the way the monetary policy has been conducted during the period considered,

with an increasingly higher policy response to inflation starting from Volker’s and

extending up to middle Greenspan’s chairmanships, as in Cogley and Sargent (2002,

2005) or Fernandez-Villaverde et al. (2010). This period of stronger monetary response to

inflation coincides with a reduction in inflation persistence, which is often interpreted as

the consequence of a more active behaviour towards inflation from the part of the Central

Bank. These results suggest that it still may be a role for the Fed in accounting for the

different inflation episodes.

An interesting extension of this work would be to “plant Greenspan in the 70’s”.

This can be done by generating artificial inflation data employing parameterization of the

90’s but using estimated 70’s disturbances, as in Canova and Gambeti (2009), Primiceri

(2005) or Sims and Zha (2006). However, if monetary policy was indeed different in the

70’s and 90’s, this type of counterfactual exercise is strongly subject to Lucas-critique,

since private agents would have changed their expectations accordingly. The Lucas-

critique is especially relevant in my case, since I do have found evidence of parameter

drifting and also of a decline in inflation persistence.

From the methodological point of view, an interesting extension would be to allow

the autoregressive length at each period to change. I find evidence that fixed lag length

models, as the one I use in the paper and the others in the literature, are not the best

possible approximation to each period dynamics. Contrary to the standard time varying

47

VAR literature, the framework discussed here allows for pre-testing at each step.

Interesting different short-run dynamics may appear of such exercise.

48

1.5 Appendix: tables and figures

Table 1 IRFs of variables to an interest rate shock after 1, 4, and 20 quarters, selected years 1975 1984 1996 2008 IRFs of inflation to an interest rate shock 1 quarter -0.0447

[-0.0841;0.0052] [-0.0768;-0.0021]

-0.0044 [-0.0417;0.0242] [-0.0363;0.0189]

-0.0076 [-0.0320;0.0137] [-0.0276;0.0099]

-0.0159 [-0.0437;0.0136] [-0.0390;0.0089]

4 quarters -0.0141 [-0.0376;0.0153] [-0.0333;0.0110]

-0.0138 [-0.0365;0.0045] [-0.0331;0.0012]

-0.0013 [-0.0126;0.0080] [-0.0109;0.0063]

0.0049 [-0.0146;0.0252] [-0.0114;0.0220]

20 quarters -0.0086 [-0.0285;0.0164] [-0.0248;0.0127]

-0.0118 [-0.0324;0.0044] [-0.0294;0.0014]

-0.0011 [-0.0096;0.0058] [-0.0084;0.0046]

0.0020 [-0.0097;0.0141] [-0.0077;0.0122]

IRFs of unemployment to an interest rate shock 1 quarter 0.0348

[-0.0189;0.0896] [-0.0101;0.0807]

0.0708 [0.0205;0.1128] [0.0280;0.1053]

0.0270 [-0.0059;0.0544] [-0.0010;0.0495]

0.0891 [0.0580;0.1127] [0.0625;0.1082]

4 quarters 0.0638 [-0.0223;0.1531] [-0.0080;0.1388]

0.1097 [0.0244;0.1883] [0.0378;0.1749]

0.0384 [-0.0117;0.0813] [-0.0041;0.0737]

0.1633 [0.0973;0.2206] [0.1073;0.2106]

20 quarters 0.0452 [-0.0180;0.1116] [-0.0074;0.1011]

0.0959 [0.0163;0.1700] [0.0288;0.1575]

0.0678 [-0.0189;0.1234] [-0.0073;0.1118]

0.2247 [0.1226;0.3132] [0.1382;0.2976]

Notes: a) The selected years coincide with the middle period chair at Burns-Miller, Volker, Greenspan and Bernanke chairmanships. b) Numbers inside brackets are the 95 and 90% confidence intervals respectively. Table 2 Multiple coherence, 3 quarters, 1 year, 5 years, 7 years frequency, selected years. 1975 1984 1996 2008

MULTIPLE COHERENCE

7 years frequency 0.4457 [0.2452;0.6864] [0.2812;0.6504]

0.3498 [0.1713;0.5861] [0.2051;0.5523]

0.4701 [0.3251;0.6198] [0.3491;0.5958]

0.6807 [0.5211;0.8284] [0.5462;0.8033]

5 years frequency 0.4072 [0.2291;0.6172] [0.2607;0.5855]

0.3336 [0.1734;0.5403] [0.2034;0.5104]

0.4645 [0.3226;0.6109] [0.3462;0.5874]

0.6616 [0.5053;0.8036] [0.5296;0.7792]

1 year frequency 0.0937 [0.0532;0.1501] [0.0611;0.1422]

0.1362 [0.0814;0.1889] [0.0902;0.1801]

0.2194 [0.1541;0.3069] [0.1665;0.2944]

0.1324 [0.0531;0.2143] [0.0662;0.2011]

3 quarters frequency

0.0404 [0.0169;0.0897] [0.0229;0.0838]

0.0835 [0.0378;0.1367] [0.0459;0.1286]

0.1050 [0.0478;0.2022] [0.0604;0.1896]

0.0309 [0.0053;0.0771] [0.0111;0.0712]

Notes: Numbers inside brackets are the 95 and 90% bootstrapped confidence intervals respectively.

49

Table 3 Responses of interest rate to a 1% permanent increase in inflation and unemployment after 1, 4, and 20 and 40 quarters for specific years 1975 1984 1996 2008 Responses to 1% permanent increase in inflation 0 quarter 0.1650

[0.0232;0.3165] [0.0471;0.2925]

0.1682 [-0.0298;0.3177] [-0.0014;0.2894]

0.0734 [-0.0342;0.1766] [-0.0170;0.1594]

0.0066 [-0.0692;0.0820] [-0.0568;0.0697]

4 quarters 0.2359 [0.0210;0.4346] [0.0547;0.4008]

0.2758 [-0.0203;0.4779] [0.0204;0.4373]

0.3809 [0.1329;0.6222] [0.1728;0.5823]

0.1170 [-0.0762;0.3192] [-0.0439;0.2869]

20 quarters

0.2829 [-0.0046;0.5593] [0.0414;0.5133]

0.3240 [-0.0529;0.5920] [-0.0003;0.5394]

0.8394 [0.2313;1.4733] [0.3327;1.3719]

0.2111 [-0.1431;0.5902] [-0.0833;0.5303]

40 quarters

0.3060 [-0.0289;0.6372] [0.0255;0.5829]

0.3470 [-0.0756;0.6575] [-0.0158;0.5977]

1.1745 [0.2453;2.1775] [0.4030;2.0197]

0.2670 [-0.1985;0.7715] [-0.1193;0.6923]

Responses to a 1% permanent increase in unemployment 1 quarter -0.1150

[-0.2308;0.0024] [-0.2118;-0.0166]

-0.1819 [-0.3055;-0.0329] [-0.2833;-0.0551]

-0.3328 [-0.4223;-0.2594] [-0.4090;-0.2727]

-0.2677 [-0.3421;-0.1859] [-0.3293;-0.1986]

4 quarters -0.4252 [-0.5491;-0.3018] [-0.5289;-0.3220]

-0.4322 [-0.5735;-0.3016] [-0.5513;-0.3238]

-0.4899 [-0.6258;-0.3597] [-0.6041;-0.3815]

-0.4721 [-0.5702;-0.3601] [-0.5531;-0.3772]

20 quarters

-0.5719 [-0.7778;-0.3772] [-0.7451;-0.4099]

-0.4962 [-0.7251;-0.2923] [-0.6898;-0.3277]

-0.5202 [-0.7460;-0.3031] [-0.7098;-0.3393]

-0.3992 [-0.5086;-0.2716] [-0.4892;-0.2909]

40 quarters

-0.6471 [-0.9204;-0.3957] [-0.8776;-0.4385]

-0.5257 [-0.8093;-0.2782] [-0.7659;-0.3215]

-0.5336 [-0.8102;-0.2707] [-0.7661;-0.3148]

-0.3731 [-0.4895;-0.2382] [-0.4690;-0.2587]

Notes: Numbers inside brackets are the 95 and 90% bootstrapped confidence intervals respectively.

50

Fig.1 U.S Quarterly GDP Inflation

1960:I 1970:I 1978:II 1979:III 1987:III 2006:I-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Notes: a) Inflation is measured a the log-difference of a quarterly GDP chain price index. B)Fed chairmanships’: McChesney (1951:II-1970:I), Burns (1970:I-1978:II), Miller (1978:II-1979:III), Volker (1979:III-1987:III), Greenspan (1987:III-2006:I), Bernanke (2006:I-pres.).

Fig.2 Time profile of the estimated long memory parameter

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Notes: Time profile of the fractional integration parameter at zero frequency-BLUE: d. Stationarity requires 0≤d<0.5, strictly greater than 0 for long memory. Non-stationary but mean reversion behavior appears when 0.5≤d<1, implying high persistence but with the effect of the shocks dying in the long run although not fast enough to deliver finite variance. Permanent effect of the shocks appears whenever d≥1. 16% and 84% bootstrapped percentiles-RED

51

Fig.4 Time profiles of the standard deviations of the residuals

Fig.4.a Std. inflation shock

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig.4.b Std. unemployment shock

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig 4.c Std of interest rate shock

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Notes: Standard deviations of VARFIMA innovations-BLUE, 95% bootstrapped

percentiles-RED

52

Fig.5 Multiple coherence at different frequencies for selected years, VARFIMA

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 π/2 π

Notes: a) years: 1975-BLUE, 1984-RED, 1996-GREEN, 2008-PURPLE; b) Multiple coherence estimated from the VARFIMA spectrum.

Fig.6 Time profile, Multiple coherence at 5 and 7 year frequencies, VARFIMA

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Notes: a) Frequencies correspondent to: 7 years-BLUE, 5 years-

RED; b) Multiple coherence, estimated from the VARFIMA spectrum.

53

Fig. 7 Time profile of the responses of interest rate to a 1% permanent increase in inflation, Contemporaneous response, 1 year, 5 year and 10 years, VARFIMA

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010-0.5

0

0.5

1

1.5

2

Notes: Contemporaneous response-BLUE, 1 year response-RED, 5 years response-GREEN, 10 years response-PURPLE.

Fig. 8 Responses of interest rate to a 1% permanent increase in inflation for selected years, VARFIMA

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

Notes: Years: 1975-BLUE, 1984-RED, 1996-GREEN, 2008-PURPLE

54

1.6 Bibliography

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Cogley, T., and T.J. Sargent (2005). “Drift and volatilities: monetary policies and outcomes in the post WWII U.S,” Review of Economic Dynamics, vol. 8(2): 262-302. Cogley, T., Primiceri G.E., and T.J. Sargent (2010). “Inflation gap persistence in the U.S,” American Economic Journal: Macroeconomics, 2(1): 43-69. Cohen L. (1989). “Time–frequency distributions: a review.” Proceedings of the IEEE 77(7): 941–981. Davis R., Lee T., and G. Rodriguez-Yam (2006). “Structural break estimation for non-stationary time series models.” Journal of the American Statistical Association 101(473): 223–239. Fernandez-Villaverde, J., and J.F. Rubio-Ramirez (2008). “How structural are structural parameters?,” NBER Macroeconomics Annual 2007, Volume 22: 83-137. Fernandez-Villaverde, J., Guerron-Quintana, P., and J.F. Rubio-Ramirez (2010). “Fortune or virtue: time-variant volatilities versus parameter drifting in U.S. data,” NBER Working Papers 15928. Franke J., and W. Härdle. (1992). “On bootstrapping kernel spectral estimates,” Annals of Statistics 20: 121–145. Fuhrer, J.C., (2009). “Inflation persistence,” Federal Reserve Bank of Boston WP #09-14. Gadea, M.D. and L. Mayoral (2006). “The persistence of inflation in OECD countries: a fractionally integrated approach,” International Journal of Central Banking vol.2. March. Gil-Alana, L.A. (2005). “Testing and forecasting the degree of integration in the US inflation rate.” Journal of Forecasting vol. 24(3): 173-187. Gil-Alana, L.A., and A. Moreno (2008). “Technology Shocks And Hours Worked: A Fractional Integration Perspective,” Macroeconomic Dynamics, 13(05): 580-604. Gu, C., and G.Wahba (1993). “Semi-parametric analysis of variance with tensor product thin plate splines.” Journal of the Royal Statistical Society, Series B 55: 353–368. Gu, C, (2002). Smoothing Spline ANOVA Models. Springer: New York. Hamilton, J.D, (1994). Time Series Analysis, Princeton University Press. Halbleib, R., and V. Voev. (2011). “Forecasting multivariate volatility using the VARFIMA model on realized covariance Cholesky factors.” Journal of Economics and Statistics (Jahrbücher für Nationalökonomie und Statistik), 231(1): 134-152. Hassler, W., and J. Wolters (1995). “Long memory in inflation rates: international evidence,” Journal of Business and Economic Statistics13:37-45.

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Henry, M., and P. Zaffaroni (2002). “The long range dependence paradigm for macroeconomics and finance,” Discussion Papers 0102-19, Columbia University, Department of Economics. Hosoya, Y. (1996). “The quasi-likelihood approach to statistical inference on multiple time-series with long-range dependence,” Journal of Econometrics, 73: 217-236. Jansen B., Hasman A., and R. Lenten (1981). “Piecewise EEG analysis: an objective evaluation,” International Journal of Bio-Medical Computing 12: 17–27. Judd, J.P., and G.D. Rudebusch (1998). “Taylor's rule and the Fed, 1970 1997,” Economic Review, Federal Reserve Bank of San Francisco, 3-16. Kitagawa G, and W., Gersch (1996). Smoothness Priors Analysis of Time Series. Springer: New York. Kim, C.J., and C., Nelson, (1999). “Has the US economy become more stable? A Bayesian approach based on a Markov Switching model of the business cycle,” Review of Economics and Statistics 81: 608-616. Koopman, S.J., and S.Y. Wong (2011). “Kalman filtering and smoothing for model-based signal extraction that depend on time-varying spectra,” Journal of Forecasting, vol.30: 147-167. Koopmans, L.H. (1995). The spectral analysis of time series, Probability and mathematical statistics, A series of Monographs and Textbooks, Volume 22, Ed. Birnbaum, Z.W. and Lukacs. Leeper E., C. A. Sims and T. Zha (1996). “What does monetary policy do?,” Brookings Papers on Economic Activity 2:1-78. Lovcha, Y. (2009). “Hours worked - Productivity puzzle: identification in fractional integration setting,” Unpublished Typescript. Lubik, T.A., and F. Schorfheide, (2007). “Do central banks respond to exchange rate movements? A structural investigation,” Journal of Monetary Economics, vol. 54(4): 1069-1087. McConnell, M.M., and G. Perez-Quiros (2000). “Output fluctuations in the United States: what has changed since the early 1980’s?,” American Economic Review 90: 1464 – 1476. Murray, C., Nikolsko-Rzhevskyy, A., and D. Papell, (2008). “Inflation persistence and the Taylor principle,” MPRA Paper 11353. Pivetta F., and R. Reis (2007). “The persistence of inflation in the United States.” Journal of Economic Dynamics and control, vol.31:1326-1358. Primiceri, G. (2005). “Time varying structural vector autorregresions and monetary policy,” Review of Economic Studies, vol. 72(3): 821–852.

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Qin, L., and W.Guo (2006). State space representation for smoothing spline ANOVA models. Journal of Computational and Graphical Statistics 15(4): 830–8473. Rose, A. (1988). “Is the Real Interest Rate Stable?.” Journal of Finance. Volume 43:1095-1112. Sargent, T.J. (1971). “A note on the acceleracionist controversy.” Journal of Money, Credit and Banking 8: 721-725. Sims, C. A. (2002). “Comment on “Evolving Post-World War II U.S. Inflation Dynamics,” NBER Macroeconomics Annual16: 373–79. Sims, C.A. and T. Zha, (2006). “Were there regime switches in U.S. monetary policy?,” American Economic Review, vol. 96(1): 54-81. Stock, J. H. (2002). “Evolving post-world war II U.S. inflation dynamics: Comment,” NBER Macroeconomics Annual 16: 379–387. Stock, J.H. and M.W Watson (2007). “Has inflation become harder to forecast?,” Journal of Money, Credit and Banking 39:3-34. Whittle, P. (1953). “The analysis of multiple stationary time series,” Journal of the Royal Statistical Society Series. B. 15 (1): 125–139.

58

59

CHAPTER 2:

On the Invertibility of Seasonally Adjusted Series

60

61

2.1 Introduction

Given the seasonal nature of many macroeconomic time series, seasonal adjustment

is a widespread practice and millions of series are routinely adjusted, some of them are

not even publicly available in the non-adjusted versions. Seasonal adjustment is believed

to remove undesirable fluctuations at seasonal frequencies without producing significant

changes at other frequencies (especially at the low part of the spectrum) making the data

easily tractable thereby simplifying posterior modelling and analysis. However, the

properties of the adjusted series crucially depend on the method used for the adjustment

and the initial properties of the series, and they may result just as unattractive for analysts

as seasonality itself.

In this paper we examine one of the important features of the adjusted data: dips in

the periodogram at seasonal frequencies and the resulting non-invertibility of the adjusted

series. The spectral dips (or zeros) are produced by all seasonal adjustment methods used

in practice; no matter it is a naive adjustment by seasonal dummies or sophisticated signal

extraction produced by specialized programs. Nerlove (1964) applies Census X-11 and

the modified ‘Hannan’ method and concludes that both methods remove more than just

the seasonal component. Grether and Nerlove (1970) show that the phenomenon observed

in Nerlove (1964), namely dips created near the seasonal frequencies after adjustment, is

obtained as a result of ‘optimal’ adjustment procedure as well. The consequent seasonal

adjustment routines of Census, X-11-ARIMA and X-12, produce the same result by

construction. Ooms and Hassler (1999) point out that the regression on seasonal dummies

generates zeros in the periodogram at seasonal frequencies that can lead to the

singularities in the log-periodogram regression. Gomez and Maravall (2001) call attention

to the fact that the application of the ARIMA-model based (AMB) signal extraction,

which constitutes the core of the TRAMO-Seats, generate dips at seasonal frequencies

62

whenever the model identified for the data contains seasonal unit roots. According to

Gomez and Maravall (2001), the spectral zeros are the frequency counterpart of the unit

MA roots and therefore the adjusted series is not invertible and does not accept

autoregressive AR (or VAR) approximation to its Wold representation. Although often

ignored, perhaps this is the most important practical implication of AMB adjustment,

since AR (and VAR) approximations to seasonally adjusted data are typically done in

applied econometric work.

In this paper we analyze in detail the dips at the seasonal frequencies and the

apparent non-invertibility produced by AMB approach within the fractional integration

(FI) framework, which admits a wider representation of the invertibility condition than

the one applied by Gomez and Maravall (2001), and we show that the dips at seasonal

frequencies produced within the AMB approach do not necessarily correspond to MA

unit roots.

A fractionally integrated process is (seasonally) invertible whenever the FI

coefficients at seasonal frequencies are higher than -0.5. In addition, notice that the

negative seasonal FI parameters correspond to the spectral zeros at seasonal frequencies.

Thus, the process can have spectral zeros at seasonal frequencies, but still remain

invertible.

We concentrate our attention on the seasonal adjustment produced within the

AMB approach and use for this end the TRAMO-Seats (TSW) as representative seasonal

adjustment program. TSW makes a part of the data adjustment programs that have been

intensively employed by Eurostat since 1994, and nowadays their use has been extended

to various European countries (Gomez and Maravall (2001), ESS Guidelines on Seasonal

Adjustment (2009)).

63

To check for the invertibility of the series adjusted by TSW, we produce

simulations for a set of processes. We do not restrict the analysis to processes with integer

orders of integration at zero and the seasonal frequencies since it has been shown by

many authors that FI at seasonal frequencies is also a widespread phenomenon in

economics (Porter-Hudak, 1990; Gil-Alana and Robinson, 2001; etc.). However, we also

make simulations for a set of Airline models, which are the default models in TSW. For

each model, we simulated 500 series, we adjust them by TSW and then, we estimate the

fractional differencing parameters at the seasonal frequencies in the adjusted series with

the Gaussian semiparametric estimator, proposed by Hurvich and Chen (2000).

We find that if the true original data generating process (DGP) follows the default

Airline model, the adjusted series produced by TSW are indeed non-invertible. However,

if the original series is fractionally integrated at the seasonal frequencies that is less

restrictive and very plausible in some cases according to the empirical evidence, the

adjusted series may be invertible depending on the stationarity of the original series.

Thus, if the series is generated from a seasonally stationary ARFISMA15 model and after

adjusted by TSW, the resulting adjusted series does contain dips at the seasonal

frequencies, but these dips do not correspond to MA unit roots but rather to negative

seasonal FI with coefficients greater than -0.5. Hence the adjusted series is invertible. On

the contrary, the adjustment with TSW of a series generated from an ARFISMA model

with non-stationary seasonality results in non-invertible negative FI coefficients. Overall,

these results are in line with Gomez and Maravall (2001), but for a wider definition of

invertibility which includes fractionally integration.

The paper is organized as follows. Section 2 describes the problem. Section 3

briefly introduces the ideas behind the concept of seasonal FI. The simulations set-up and

15 Some authors employ the alternative notation SARFIMA (Seasonal AutoRegressive Fractionally Integrated Moving Average).

64

the results are presented in Section 4. Section 5 contains a small empirical application

illustrating the results reported in Section 4. Section 6 concludes the paper.

2.2 The Problem

TRAMO is a pre-adjustment program, while SEATS is a seasonal adjustment

routine based on the "ARIMA-model-based" (AMB) approach. Within the AMB

approach, the seasonal adjustment process starts by identifying an ARIMA model to the

observed data. Once the model for the observed data is identified, ARIMA specifications

for its unobserved components can be derived. All components are orthogonal to each

other and each of them captures variations around the different frequencies in the

spectrum. Thus, the trend-cycle captures the peak around the zero frequency, the seasonal

component captures the peaks around the seasonal frequencies, and the transitory

component picks up stationary transitory variations different from the white noise. In

other words, the AR roots of the identified ARIMA model are distributed between the

corresponding unobserved components. Thus, if the model contains seasonal unit roots,

the seasonal component will be non-stationary. If the aim of the application of the AMB

approach is seasonal adjustment, all the components different from the seasonal are

grouped together in a single component called “signal”, whereas the remaining seasonal

component is defined as “noise”.

The estimate of the signal is obtained by means of the Wiener-Kolmogorov (WK)

filter as the MMSE estimator of the signal given the observed series. An important feature

of the estimated signal is that if the seasonal component is non-stationary (i.e. it contains

unit roots at seasonal frequencies), these unit roots will show up as MA in the model

generating the estimated signal and will produce spectral values of zeros for the

65

associated seasonal frequencies16 According to Gomez and Maravall (2001), “…These

spectral zeros are the frequency counterpart of the unit MA roots. ...". An important

implication of this result according to the authors is that " ... the estimator will be a non-

invertible series ..." and " ... will not accept, in general, an AR (or VAR) approximation to

its Wold representation".

2.3 Seasonal fractional integration

The AMB approach assumes that the data follows an ARIMA-type process. This

assumption restricts the DGP to be stationary I(0) or, alternatively integrated of order one,

I(1), at zero and/or at seasonal frequencies. In this article we extend the seasonal I(1)/I(0)

approach to the fractional case, and examine cases where the original series has non-

integer order of integration at seasonal frequencies. In such a case, the process is said to

be seasonally fractionally integrated or seasonal I(d).

For the purpose of the present work, we first define an I(0) process as a covariance

stationary process with a positive and bounded spectral density at all frequencies in the

spectrum. Then, we say that a process xt is seasonal I(d) if it can be represented as:

(1 )d

t tB x aτ− = (2.1)

where Bτ is the seasonal lag operator (i.e., Bτxt = xt-τ) and τ represents the number of

periods per year (e.g., τ = 4 with quarterly data, τ = 12 in case of monthly data, etc.), d is a

real value and at is an I(0) process that may include seasonal and non-seasonal weakly

autocorrelated (e.g., ARMA) terms. If d > 0 in (2.1), xt is said to be a seasonal long

memory process, so-named because of the strong degree of association between

observations widely (seasonally) separated in time. It may be shown that for this process,

16 A more detailed description of the problem can be found in Maravall and Planas (1999).

66

the spectral density function is unbounded at the zero and the seasonal frequencies, which

is a characteristic of the seasonal long memory processes. In relation with this model, few

applications have been carried out. Porter-Hudak (1990) was the first to use a model of

this form in some quarterly US monetary aggregates, concluding that a fractional ARMA

model could be more appropriate than the usual ARIMA models for these aggregate

data17. Other more recent applications using this model are Gil-Alana (2002; 2005),

Reisen, Rodrigues and Palma (2006), etc. However, the specification in (1) is rather

restrictive in the sense that it imposes the same degree of integration at all frequencies,

noting that (1 - Bτ) can be decomposed into (1 – B)S(B) where S(B) = 1 + B + B2 + … +

Bτ-1 refers exclusively to the seasonal frequencies. Thus, for example, in case of the

polynomial (1 – B4)d, it can be expressed (1 – B)d(1 + B + B2 + B3)d = (1 – B)d (1 + B)d

(1 + B2)d implying the same degree of integration, d, at zero and the seasonal frequencies

π, π/2 (3π/2) (of a 2π cycle). Extending this model, we may consider a more general

specification that permits different degrees of integration at each of the frequencies. In

particular, we will examine in the paper models of form:

20 2 1(1 ) (1 ) (1 )d d d

t tB B B x a− + + = (2.2)

where d0 refers to the order of integration at the long run or zero frequency; d2 is the order

of integration at the semiannual frequency π, and d1 corresponds to the annual frequencies

π/2 and 3π/2. Applications using the flexible model of form as in (2.2) can be found in

Arteche and Robinson (2000), Gil-Alana and Robinson (2001), Arteche (2003), Hassler,

Rodrigues and Rubia (2009), etc.

As an illustration of the models employed in the paper, let’s consider the

following three examples.

17 The notion of fractional processes with seasonality was initially suggested by Abrahams and Dempster (1979) and Jonas (1981), and extended in a Bayesian framework by Carlin et al. (1985) and Carlin and Dempster (1989).

67

Example 1:

Very often the models identified by TSW fall into the class of the so-called Airline Models of Box and Jenkins (1970) that are believed to approximate reasonably well the stochastic properties of many series

1(1 ) (1 ) (1 ) (1 )

t tB B x Q B Q B aτ τ

τ− − = + + (2.3)

with Q1 and/or Qτ always negative18 These two MA parameters, if fairly close to -1,

indicate the presence of a very stable trend (Q1 → -1) and/or a stable or small seasonality

(Qτ → -1). On the contrary, if the parameters are close to 0, the trend and the seasonality

are very strong.

The classical Airline model we consider in this example is given by:

( )( )4 41, 1,1 1 (1 0.7 ) (1 0.6 )t tB B x B B ε− − = − − (2.4)

It has a relatively small, though non-stationary seasonality and not a very strong trend.

We separate the trend and seasonal polynomials of this model as

( )1/4 1/4 1/2 2

1, 1,2

(1 0.7 ) (1 0.6 ) (1 0.6 ) (1 0.6 ),

( )1t t

B B B Bx

S BBε− − + +=

−

with S(B) = (1 + B)(1 + B2). The seasonal impulse responses obtained from the expansion

of (1 + 0.61/4B) (1 + 0.61/2B2) / S(B) are plotted in Figure 1a.

Example 2:

This is a stationary seasonal ARMA model obtained by applying the first order

approximation to (2.4):

4

2, 2,(1 0.3 ) (1 0.4 )

t tB B x ε− − = (2.5)

As before we separate the seasonal and non-seasonal roots:

18 See, for example, the analysis of the quarterly Spanish Industrial Production Index (IPI) in Gomez and Maravall (2001) and the Japanese Balance of Trade series in Maravall (2006). The Japanese Import series in the same paper is an ARFIMA(1,1,0)(0,1,1)s model, with Q1 = -0.38 changed automatically by the Airline model to get an admissible decomposition. In Maravall (2009) 50% of the 500 monthly exports and imports series of 15 European Union countries analyzed accept the default Airline model as appropriate. The majority of the series examined in Maravall (2008) are also approximated with the Airline model.

68

11/4 1/4 1/2 2 1

2, 2,(1 0.3 ) (1 0.4 ) (1 0.4 ) (1 0.4 ) ,t tx B B B B ε−

− = − − + +

and we plot the seasonal contribution to the impulse responses also in Figure 1a.

Example 3:

The third model is a non-stationary but mean reverting (d0 = 0.7 > 0.5) ARFISMA

with d1 = d2,

0.7 0.4 2 0.4

3, 3,(1 ) (1 ) (1 )

t tB B B x ε− + + = (2.6)

The seasonal and non-seasonal roots are already separated in this model. The seasonal

contribution to the impulse responses is again displayed in Figure 1a.

Since the Airline model contains seasonal unit roots, the contribution of the

seasonal component to the responses of the variable to a shock does not decrease,

whereas the seasonal responses of the stationary seasonal ARMA decay relatively quickly

and after 20 lags are not distinguishable from zero. The seasonal contribution to the

responses in the ARFISMA decreases with time but at a much slower rate than in the

ARMA model and it is statistically different from zero even after 50 lags. Note that if the

fractional differencing parameters at the seasonal frequencies in ARFISMA models are

smaller than 0.5, it means that the seasonal component of the model is stationary.

However, the spectrum of the ARFISMA at seasonal frequencies is infinite, as in the

Airline model that constitutes a key difference with the spectrum of the seasonal ARMA

model (see Figure 1b). The total impulse responses of the variables to a shock for the

three models presented in Figure 1c. Responses of the Airline model explode revealing

the non-stationary nature of the series (non-stationary trend), seasonal fluctuations do not

disappear with time (non-stationary seasonality). The responses in ARMA converge to

zero relatively fast since the process is stationary. The responses of ARFISMA do not

69

display explosive behaviour (the process is non-stationary but mean reverting: d0 = 0.7),

and the seasonal fluctuations diminish slowly (stationary fractional seasonality).

The ARFISMA processes allow for a more flexible specification of non-stationarity

at zero and the seasonal frequencies that makes them attractive for seasonality modelling.

An additional advantage of the ARFISMA models is that they do not require making a

priori assumptions about the order of integration at zero and the seasonal frequencies

since they are estimated within the model.

2.4 Simulation study

2.4.1. Simulation setup

To study invertibility of the time series adjusted by TSW, we generate the data for

64 different specifications of ARFISMA model. The parameters for the simulated

ARFISMA processes of the form as in (2.2) are d0 = 0.3, 0.7, 1.0, 1.5, di = 0.1, 0.3,

0.5, 0.7, i = 1, 2 and σ2 = 1. The choice of the values is justified by the empirical

evidence. The number of observations for each series is set T = 500.

To generate the data, the long memory polynomials in (2.2) have to be expanded.

We choose the lag truncation 1000 for each polynomial. Thereafter, we multiply

expanded long memory polynomials (the resulting polynomial has 3000 lags) and

following Bhardwaj and Swanson (2006), we truncate the resulting polynomial when the

coefficients become smaller than 1.0e-004 (the truncation lag is always smaller than

1000). All observations are generated using standard normal errors. For each process and

each replication, we generate 3000 observations and we use just the last T observations to

avoid the initial values problem, especially important when taking into account long-

memory properties of the DGP.

70

To each simulated series we apply TSW. If TSW chooses a seasonally non-

stationary ARIMA model for this series, we collect it for the future analysis. If the model

chosen by TSW contains stationary seasonality we discard the simulated series. We

proceed until we have I=500 simulated series for each specification, identified by TSW as

seasonally non-stationary.

In addition to ARFISMA, we produce simulations for a set of quarterly Airline

models as in (3) with 4τ = and negative values for Q1 and Q4: 0.8, 0.6, 0.3iQ = − − − ,

1,4i = and [ ] [ ] [ ] [ ] 1 4, 1, 0.8 , 1, 0.6 , 1, 0.3Q Q = − − − − − − . In the same way as for

ARFISMA, we collect I=1000 series for each specification identified by TSW as

seasonally non-stationary.

Thereafter, each selected series for each specification is adjusted by TSW and

coefficients of FI at seasonal frequencies are estimated.

It is very important to remark that, even if several series are simulated from the

same ARFISMA specification, the TSW can choose distinct ARIMA models for each

simulated series. Since seasonal filters applied to the data are based on the identified

ARIMA models, different filters may be applied to each of the series simulated from the

same ARFISMA process. In this way, the mean of the estimated parameters of FI at

seasonal frequencies does not have statistical meaning19. Therefore, after estimating the

FI parameters at seasonal frequencies we test if the adjusted series are statistically non-

invertible20: at least one estimated coefficients of seasonal FI is statistically smaller than -

0.5. If it is not the case, we test if the series is statistically invertible: both estimated

coefficients of seasonal FI are statistically greater than -0.5. As a result, for each Airline

19 Even if we restrict TSW to choose always the default Airline model, the estimated coefficients of the model will differ at each replication and the seasonal filters are going to be distinct. Obviously, we could fix an ARIMA model with a given set of parameters to use for the adjustment, but this would produce results not interesting from the practical point of view. 20 We use 5% significance level.

71

and ARFISMA specification, we can compute both the percentage of statistically non-

invertible and the percentage of statistically invertible series (in the adjusted I=500 series

chosen by TSW to be seasonally non-stationary before adjustment).

To estimate the coefficients of FI at the seasonal frequencies we use the Gaussian

semiparametric estimator (Hurvich and Chen (2000)). In order to reduce periodogram

bias due to the strong peaks and thoughts in the spectral density, we apply a complex-

valued taper21 on the simulated data prior to computing the Fourier transform:

( )2 0.5

0.5 1 exp , 1,...,t

i th t T

T

π − = − =

The choice of the Gaussian semiparametric estimator is justified by several

reasons. First, after the application of the TSW, the SA series does not follow any more

the process given by (2). Since we do not know what the correct specification after

adjustment is, we avoid the parameterization of the whole spectrum by choosing a local

estimation method. Second, as suggested by Hurvich and Ray (1995), tapering is

particularly suitable when the estimated coefficients of FI are expected to be negative,

possibly smaller than -0.5. In these circumstances, the estimation results based on the

non-tapered data will have a strong positive bias, making impossible to build conclusions

on the invertibility of the estimated processes. The use of a taper can alleviate the

negative effects of overdifferencing, most importantly, the bias in estimates of

coefficients of FI based on the periodogram of the differenced data. Finally, tapering

reduces the bias that appears due to contamination of the periodogram from the short

memory component of the spectral density (Hurvich and Chen (2000)).

21 A taper is a nonrandom weight sequence with certain desired properties that is multiplied with time series data prior to Fourier transformation or parameter estimation.

72

The detailed description and the asymptotic properties of the method may be found in

Hurvich and Chen (2000)22. The comprehensive discussion of the performance of the

method for the seasonal and cyclical time series with asymmetric long memory properties

is presented in Arteche and Velasco (2005).

Nevertheless, we also perform a small Monte Carlo study to check the

performance of the estimation method for negative FI at seasonal frequencies, as it is

important in this work. After simulating the data and before applying the TSW (i.e. when

we still know the true DGP), we take yearly difference, making sure that the resulting

series are overdifferenced at seasonal frequencies (it has negative coefficients of FI). We

estimate the coefficients of FI at seasonal frequencies and compute the mean for each

specification to assess the estimation bias in the presence of negative fractional

integration.

All the simulations and the estimations are produced in Matlab. All programs are

available from the authors upon request. For the seasonal adjustment we use the last

release of the TSW for Matlab developed by the Bank of Spain. The programs with

instructions can be downloaded from the web-site of the Bank of Spain23.

2.4.2. Description of the results

The results of the simulation study for the different Airline and ARFISMA

specifications are presented in Table 1 and Table 2 respectively. In both tables, the

particular specification from which the data is simulated appears in first column (i.e., the

values for the MA parameters Q1 and Q4 for the Airline model and the parameters of FI at

seasonal frequencies d1 and d2 for ARFISMA).

22 We employ the approximation for the estimator variance given in Hurvich and Chen (2000) pp. 164 since, as shown by the authors, it outperforms the asymptotical variance even for a relatively big (T=500) sample size, as used in the simulation study. 23 http://www.bde.es/servicio/software/interfacese.htm.

73

The results of the Monte Carlo study are presented in the following two columns.

The coefficients in the table are *ˆ ˆ1i id d= + , 1,2i = (in columns two and three

respectively), where *ˆid is the mean of the estimates of the coefficient with yearly

differenced data24. As can be observed in both tables, the bias of the estimation method

employed is very small; the mean of the estimated values ˆid over the replicas, are always

very close to the values used for simulation (one for the Airline specifications and the

different di employed for the ARFISMA). For the Airline specifications, the d1 appear

always a little bit underestimated and d2 slightly overestimated but this pattern is not

reproduced in the ARFISMA. Overall, although slightly biased, the Gaussian

semiparametric estimator with tapered data performs very well for negative seasonal FI

even for coefficients from the non--invertible region and also for the estimation of the

parameter at frequency π/2, were the spectrum is not symmetric. It is also important to

remark that, as shown in Table 1, the method works also very well in the presence of

short memory component. On the whole, the results from the Monte Carlo study confirm

that the Gaussian semiparametric estimator with tapered data is suitable for the purposes

of the present work25.

Next column (column four, both tables) presents the percentage of cases TSW

chooses a seasonally non-stationary model to fit the data for each of the simulated

processes. As can be seen in Table 1, TSW always choose a non-stationary model when

24 Recall that since we expect to have negative seasonal fractional integration after adjustment, to check the performance of the estimation method, the original data was first differenced in the Monte Carlo analysis to ensure that the resulting series are over-differenced at seasonal frequencies with negative coefficients of fractional integration. For the processes with the fractional order of integration at zero 0 1.5d = , we take a

first difference in addition to the yearly difference. The same applies to the Airline model. 25 The variance approximation given in Hurvich and Chen (2000), and employed in this paper, is also very close to the one obtained in the Monte Carlo analysis.

74

the true DGP follows the Airline model. As expected, for the ARFISMA specifications

(Table 2), this percentage increases together with the magnitude of both d1 and d2 .

Column five (both tables) presents the percentage of cases that the process are

estimated invertible, it is, the two estimated coefficients of FI at seasonal frequencies

after adjustment are greater than -0.5. Thereafter we compute the percentage of

replication in which the SA series has at least one estimated coefficient of seasonal FI

statistically smaller than -0.5 - that is to say, the series is statistically non-invertible

(column six). If the hypothesis of statistical non-invertibility is rejected, we test statistical

invertibility: both estimated coefficients are statistically greater than -0.5. The percentage

of statistically invertible results is given in column seven (both tables). When the data are

simulated from the Airline model (Table 1), the estimated coefficients of FI at frequencies

π/2 and π are almost always smaller than -0.5, which indicates the (possible) non-

invertibility of the corresponding SA series. Moreover, in a big percentage of the cases

this non-invertibility is statistically significant. This result is not surprising and it is

completely in line with the implications of the TSW for this class of models (Gomez and

Maravall (2001)). For the ARFISMA specifications (Table 2), the result of the application

of TSW depends on the initial properties of the simulated data. Thus, if originally both

coefficients of the seasonal FI are within the stationary region (di < 0.5, i = 1, 2), even

if the TSW identifies a non-stationary seasonal model (that occurs in a relatively small

percentage of cases), the estimated coefficients of seasonal FI of the SA series are greater

than -0.5 in most of the cases. Only a very small percentage of series is (possibly)

statistically non-invertible. The percentage of statistically invertible results decreases as

seasonal FI coefficients of the original series approach to non-stationary region. For

example, for d0 =0.3, if the original series has both coefficients of seasonal FI di = 0.1,

TSW only selects a seasonal non-stationary representation in a 31.8% of the cases. In

75

addition, even if it is the case and a non-stationary ARIMA is chosen, the estimated

coefficients of the SA series lie always (100%) in the invertible region. Moreover, in a

99.5% of the cases both parameters are statistically greater than -0.5 and the series are

statistically invertible. The percentage of statistically invertible results decreases to 80.5%

if d1 and d2 are equal to 0.3. Still, the estimated parameters are greater than -0.5 in a

99.6% of the cases and the percentage statistically non-invertible results is just 0.5%.

On the contrary, if one of the coefficients of seasonal FI in the DGP is greater

than 0.5, TSW fits a seasonally non-stationary model in a relatively higher percentage of

the cases, and for that cases the SA series are often estimated to be non-invertible. Once

more, the percentage of statistically non-invertible results increases with the parameters

of seasonal FI of the original series. Thus, (again for d0 = 0.3) if di = 0.7 for the two

seasonal FI coefficients in the original DGP, TSW always select a non-stationary

representation. Only in a 16% of the cases the SA series are estimated to be invertible

(and only in a 9.5% the invertibility is statistically significant) whereas in an almost 50%

of the replicas the SA series were found statistically non-invertible. It is also interesting to

note that, although the parameter of fractionally integration at zero is not neutral, the

same conclusions are obtained for all simulated d0.

Overall, these results are in line with Gomez and Maravall (2001), but using a

more flexible definition of invertibility: if the process contains strong non-stationary

seasonality (including FI) then the SA series estimated by TSW will be in general non-

invertible. However, if the original series was stationary fractionally integrated at

seasonal frequencies, TSW will choose a non-stationary representation in a smaller

percentage of cases and, even if a non-stationary model is chosen, the resulting SA series

is likely to be invertible, and therefore still admits AR (o VAR) approximation. These

76

results are important because stationary seasonal long memory behaviour is not rare in

practice. We illustrate our results with real data in the following section.

2.5 Empirical examples

To illustrate the simulation results, we consider several quarterly series of Spanish

economy: Industrial Production Index (IPI), airline passengers (AIR), employment (EMP)

and three quarterly cyclical economic indicators for Spain, namely: cement consumption

(CC), car registrations (CR) and housing starts (HS). IPI and these three indicators are

considered to be the cycle drivers for an economy in Leamer (2009) and have been

recently used by Bujosa et al. (2010) to construct a composite leading indicator for the

Spanish economy. All series are strongly seasonal, and cover a span starting from the

beginning of 70th. Monthly data from IPI, CC, CR, HS and AIR can be obtained from the

Bank of Spain26. To convert the IPI to quarterly, we use the simple average of the

observations inside each quarter. Other series are converted to quarterly by adding the

observations inside the quarter. The employment has been obtained quarterly from the

OECD stats database27. We exclude the last years of observations to avoid the influence

of the current crisis28.

Figure 2 (left panels) plots the original series (not seasonally adjusted) and the

series after adjustment by TSW. Non-stationarity in the mean and a strong seasonal

pattern is observed in all the original series.

The right panel in Figure 2 depicts the periodogram of both differenced series:

original and adjusted by TSW. As expected, the periodogram of the differenced original

26 http://www.bde.es/webbde/es/estadis/ estadis.html 27 http://stats.oecd.org/ 28 For IPI, CC, and CR we exclude the last three years of data. For the case of housing starts (HS) an additional year had to be taken since the effects of the crisis manifest earlier for this indicator. Contrary, an additional year could be added for airline passengers (AIR) and employment (EMP).

77

series have strong peaks at the two seasonal frequencies, while those of the differenced

adjusted series present dips at the same frequencies.

TSW identifies the following models for the original series:

Variable ARIMA model chosen by TSW

ln(IPI)

( ) ( )4 41 0.2275 1 0.7146t tB y B ε− ∇∇ = −

changed by Seats to:

( )( )4 41 0.2275 1 0.7146t ty B B ε∇∇ = − −

CC ( )4 41 0.7054t ty B ε∇∇ = −

CR ( )4 41 0.6775t ty B ε∇∇ = −

HS ( ) ( )41 0.4836 1 0.5360t tB y B ε+ ∇ = −

ln(AIR) ( )( )4 41 0.1686 1 0.6158t ty B B ε∇∇ = − −

EMP ( ) ( )4 41 0.7923 1 0.6320t tB y B e+ ∇∇ = −

As can be seen from the above, all the series except IPI , HS and EMP follow a

standard Airline model. For CC and CR the trend is very strong and Q1 is equal to zero.

The model identified for the IPI does not accept the admissible decomposition (NA) and

is modified by SEATS. Given that AR(1) polynomials with Φ1 in the interval (-0.2, -0.4)

are practically indistinguishable from the MA(1) with Q1 = - Φ1, SEATS replaces the NA

model with the corresponding Airline model (Maravall (2009)). TSW chooses a

stationary seasonal model for HS and SARIMA for EMP. According to the simulation

results, when the true DGP followed an Airline model, the estimated coefficients of FI at

seasonal frequencies in the original series were close to one, while for the adjusted series

typically relayed at the non--invertible region. Consequently, we estimate the coefficients

of FI at seasonal frequencies before and after the adjustment by TSW. The results are

presented in Table 3.

It is interesting to note that it seems that the examined series follow neither the

Airline nor the SARIMA model, since the estimated coefficients of FI at the seasonal

frequencies before the adjustment (columns four and five) are all statistically different

78

from one. As can be seen in the table the table CC, CR and EMP seem to be stationary

seasonal before adjustment. Especially, for the case of CC which is statistically seasonally

stationary at usual significance level. For the other two series, non-stationarity can be

rejected at 15%, which seems to be appropriate given the relative short length of the

series. In line with our simulation results, event though TSW has selected a seasonally

non-stationary model for the three series, the estimated coefficients after the adjustment

are substantially bigger than -0.5, suggesting that the adjusted series are indeed invertible.

That also seems to be the case of HS, for which TSW has selected a stationary

representation before adjustment (although we do not find definitive evidence about that

in our estimation). For the IPI and AIR series, the estimated coefficients of FI at

frequency π/2 are larger than 0.5, albeit that the null of invertibility cannot be rejected at

any usual significance level. As expected from the simulation results, although one cannot

draw a definite statistical conclusion with regard to invertibility of the adjusted series, one

of the estimated coefficients after adjustment is smaller than -0.5 and the adjusted series

may be non-invertible.

2.6 Conclusions

In this paper we have analysed the invertibility property of the seasonal series

adjusted by TSW. According to Gomez and Maravall (2001) whenever the process

chosen by TSW to fit the data contains seasonal unit roots, the adjusted series estimated

by the program has MA unit roots and, as a result, it is not invertible and cannot be

approximated by an AR (VAR) process as is usually done in practice.

In this simulation study we have found that if the true DGP follows the default of

the program Airline model (ARIMA with unit roots at seasonal frequencies), the adjusted

series produced by TSW are to be expected non-invertible. However, if the series is

79

fractionally integrated at the seasonal frequencies, which is less restrictive and very

plausible in some cases according to the empirical evidence, the adjusted series may be

invertible, depending on the stationarity of the original series. Thus, if the original series

is seasonally stationary with coefficients of FI at seasonal frequencies smaller than 0.5,

the adjusted series estimated by TSW is likely to be invertible and admit AR (o VAR)

representation even if the model chosen by the program was a seasonal non-stationary.

Invertibility is more probable as further are the seasonal FI parameters of the original

series from the non-stationary region. On the contrary, if the original series is seasonally

non-stationary, the resulting adjusted series is more plausible to be non-invertible. As

shown in the empirical example, these results are important since stationary FI

seasonality is not strange to the data.

80

2.7 Appendix: tables and figures

Table 1: Simulation results, Airline model

True specification

Before adjustment After adjustment

1 4,Q Q 1d 2d %, NS %, I %, SNI %, SI

-0.8,-0.3 -0.8,-0.6 -0.8,-0.8 -0.6,-0.3 -0.6,-0.6 -0.6,-0.8 -0.3,-0.3 -0.3,-0.6 -0.3,-0.8 -1,-0.3b

-1,-0.6b

-1,-0.8b

0.979 0.961 0.958 0.994 0.975 0.992 0.988 0.988 0.992 0.976 0.983 0.995

1.034 1.034 1.041 1.018 1.013 1.021 1.008 1.007 1.016 1.036 1.012 1.020

100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

9.0 9.0 13.0 4.0 3.5 11.5 2.5 8.5 6.5 6.0 2.5 5.0

91.0 88.0 85.0 95.0 94.0 84.0 96.5 90.5 93.5 93.0 97.0 95.0

8.0 8.0 7.5 4.0 2.5 9.5 2.0 8.0 6.5 5.5 2.5 4.0

Notes: (a) Airline model: ( )( ) ( ) ( )4 41 41 1 1 1t tB B y Q B Q B ε− − = + + . If 4 1Q → − , the seasonality

is small or stable; if 1 1Q → − , the trend is small. (b) These models correspond to the case when 1 1Q = −

in the standard Airline model. ∇ and ( )11 Q L+ will be canceled out in this case and the model will have

a special form ( )4 441t ty Q L ε∇ = + . (c) Since the Airline model contains seasonal unit

roots 1 2 1d d= = . (d) NS – percentage of cases the TSW identifies a seasonally non-stationary model for

the simulated data for a given process; I – percentage of cases both estimated coefficients of seasonal fractional integration are greater than -0.5; SNI – percentage of cases at least one coefficient of seasonal fractional integration is statistically smaller than -0.5 (statistically non-invertible); SI - percentage of cases both coefficients of seasonal fractional integration are statistically greater than -0.5 (statistically invertible);

(e) The bandwidth parameter in the estimation is 0.7T . Table 2: Simulation results, ARFISMA model

True specification

Before adjustment After adjustment

1 2,d d 1d 2d %, NS %, I %, SNI %, SI

0 0.3d =

0.1,0.1 0.1,0.3 0.1,0.5 0.1,0.7

0.130 0.109 0.111 0.120

0.117 0.317 0.522 0.714

31.8 52.0 58.6 35.7

100.0 97.5 85.0 8.0

0.0 1.5 4.5 75.0

99.5 84.0 31.5 4.5

0.3,0.1 0.3,0.3 0.3,0.5 0.3,0.7

0.314 0.308 0.300 0.295

0.086 0.312 0.511 0.731

59.8 64.3 78.7 71.4

97.5 99.6 77.4 16.5

0.0 0.5 4.2 54.0

80.5 80.5 36.0 4.5

0.5,0.1 0.5,0.3 0.5,0.5 0.5,0.7

0.497 0.493 0.487 0.480

0.090 0.294 0.508 0.729

96.6 94.8 100.0 99.5

89.0 88.0 74.0 19.5

2.0 2.5 3.5 49.0

62.0 47.5 17.5 6.0

0.7,0.1 0.7,0.3 0.7,0.5 0.7,0.7

0.711 0.716 0.694 0.694

0.083 0.275 0.475 0.707

100.0 100.0 100.0 100.0

62.5 61.5 36.0 16.0

7.0 7.5 19.5 55.0

32.0 22.5 11.5 10.0

81

0 0.7d =

0.1,0.1 0.1,0.3 0.1,0.5 0.1,0.7

0.130 0.124 0.104 0.092

0.112 0.304 0.543 0.723

17.3 22.2 73.0 79.4

100 98.0 61.5 16.5

0.0 0.4 4.0 58.0

99.2 88.4 22.5 7.5

0.3,0.1 0.3,0.3 0.3,0.5 0.3,0.7

0.306 0.292 0.292 0.291

0.094 0.294 0.521 0.737

66.9 76.0 77.2 94.3

94.5 93.5 63.0 18.5

1.5 3.0 14.5 50.5

79.0 68.5 27.5 9.5

0.5,0.1 0.5,0.3 0.5,0.5 0.5,0.7

0.497 0.492 0.488 0.486

0.075 0.297 0.491 0.711

98.0 98.0 97.6 100.0

84.0 40.0 33.5 9.0

6.5 20.0 24.5 75.5

44.5 20.0 5.0 5.0

0.7,0.1 0.7,0.3 0.7,0.5 0.7,0.7

0.701 0.692 0.686 0.688

0.070 0.266 0.459 0.686

100.0 100.0 100.0 100.0

24.5 15.5 10.5 9.5

47.5 57.5 68.0 81.0

11.0 7.0 5.5 6.0

0 1.0d =

0.1,0.1 0.1,0.3 0.1,0.5 0.1,0.7

0.110 0.126 0.117 0.103

0.095 0.302 0.515 0.750

24.9 24.8 30.9 54.6

100.0 99.5 64.5 19.5

0.0 0.0 6.0 67.5

100.0 94.0 41.0 6.5

0.3,0.1 0.3,0.3 0.3,0.5 0.3,0.7

0.304 0.295 0.292 0.297

0.089 0.282 0.494 0.735

53.8 64.5 75.2 88.9

100.0 98.5 79.5 14.0

0.0 0.5 4.5 49.0

97.0 92.5 50.0 3.0

0.5,0.1 0.5,0.3 0.5,0.5 0.5,0.7

0.486 0.477 0.486 0.483

0.072 0.271 0.509 0.713

96.2 97.6 96.2 98.0

90.0 57.0 49.5 20.5

4.0 15.5 20.0 45.0

67.0 28.5 25.0 8.0

0.7,0.1 0.7,0.3 0.7,0.5 0.7,0.7

0.684 0.684 0.687 0.688

0.079 0.265 0.475 0.696

100.0 100.0 100.0 100.0

35.0 26.5 17.5 10.0

34.5 53.0 68.0 77.5

13.0 10.5 10.0 3.0

0 1.5d =

0.1,0.1 0.1,0.3 0.1,0.5 0.1,0.7

0.147 0.118 0.111 0.092

0.112 0.316 0.545 0.750

13.7 39.8 83.0 81.3

98.5 95.5 78.5 22.0

0.0 2.0 4.5 53.0

94.5 81.5 35.5 15.0

0.3,0.1 0.3,0.3 0.3,0.5 0.3,0.7

0.309 0.301 0.309 0.287

0.099 0.304 0.526 0.731

69.0 79.7 90.5 91.7

89.0 86.5 62.0 27.5

0.0 4.0 10,0 36.5

64.0 61.0 22.0 6.5

0.5,0.1 0.5,0.3 0.5,0.5 0.5,0.7

0.497 0.496 0.500 0.481

0.077 0.307 0.502 0.720

92.6 97.6 95.7 98.0

64.5 61.5 52.5 21.5

8.0 6.5 11.0 39.0

30.5 15.0 5.5 4.5

0.7,0.1 0.7,0.3 0.7,0.5 0.7,0.7

0.704 0.713 0.692 0.692

0.081 0.271 0.470 0.700

99.5 100.0 100.0 100.0

40.0 39.5 15.5 11.0

38.5 26.5 39.0 50.5

16.0 9.0 3.0 3.0

Notes: (a) ARFISMA model: ( ) ( ) ( ) 10 2 21 1 1dd d

t tL L L y e− + + = , if 1 2, 0.5d d > the seasonality is

non-stationary; (b) NS – percentage of cases the TSW identifies a seasonally non-stationary model for the simulated data for a given process; I – percentage of cases both estimated coefficients of seasonal fractional integration are greater than -0.5; SNI – percentage of cases at least one coefficient of seasonal fractional integration is statistically smaller than -0.5 (statistically non-invertible); SI - percentage of cases both coefficients of seasonal fractional integration are statistically greater than -0.5 (statistically invertible). (c)

The bandwidth parameter in the estimation is 0.7T .

82

Table 3: Empirical results

T 1 2/m m Before adjustment After adjustment

1d 2d 1ˆad 2

ˆad

ln(IPI) 128 32/16 0.617 (0.144)

0.231 (0.229)

-0.153 (0.144)

-0.646 (0.229)

CC 148 36/18 0.276 (0.133)

0.075 (0.209)

-0.127 (0.133)

-0.284 (0.209)

CR 188 48/24 0.385 (0.112)

0.060 (0.171)

-0.075 (0.112)

-0.330 (0.171)

HS 144 36/18 0.391 (0.134)

0.474 (0.210)

-0.266 (0.134)

-0.132 (0.2100)

ln(AIR) 148 36/18 0.532 (0.133)

0.244 (0.209)

-0.195 (0.133)

-0.537 (0.209)

EMP 140 36/28 0.380 (0.134)

0.054 (0.210)

-0.232 (0.134)

-0.226 (0.210)

Notes: (a) HS- Houses started (1970:1-2006:4); CR – Cars registered (1960:1-2007:4); CC – Cement consumption (1970:1-2007:4); ln(IPI) – natural logarithm of IPI (1975:1-2007:4); ln(AIR) – Airline

passengers (1970:1-2008:4); ICR – industrial cars registration - (1964:1-2007:4); (b) 1 2/m m - bandwidth

parameters for the estimation at frequencies(π/2)/π. For each time series the bandwidth parameters (for both frequencies) were selected based on an examination of the log-log plot of the tapered periodogram of the data in differences. Figure 1. The seasonal contribution to the impulse responses, spectra of the models, and the impulse responses in the Airline, ARFISMA and ARMA models from the Example

-1

-0.6

-0.2

0.2

0.6

1

1 11 21 31 41

Airline ARFISMA

ARMA

0

2

4

6

8

10

pi/2 pi

Airline ARFISMA

ARMA

0

0.4

0.8

1.2

1.6

2

1 11 21 31 41

Airline ARFISMA

ARMA

1a The seasonal contribution to the impulse responses

1b Spectra of the models 1c Impulse responses

83

Figure 2. Non-adjusted and TSW adjusted data of the empirical application and their respective sample periodogram.

3.5

4

4.5

5

1975 1980 1985 1990 1995 2000 2005 2010

lnIPI lnIPI adjusted

0

0.001

0.002

0.003

0.004

0.005

0 pi/ 2 pi

sample periodogram of d(lnIPI)

sample periodogram of d(lnIPI adj)

20

60

100

140

1970 1975 1980 1985 1990 1995 2000 2005 2010

CC CC adjusted

0

1000

2000

3000

0 pi/2 pi

sample periodogram of d(CC)

sample periodogram of d(CC adj)

10

110

210

310

410

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

CR CR adjusted

0

200

400

600

800

1000

0 pi/ 2 pi

sample periodogram of d(CR)

sample periodogram of d(CR adj)

100

600

1100

1600

1970 1975 1980 1985 1990 1995 2000 2005

HS HS adj

0

50

100

150

200

0 pi/ 2 pi

sample periodogram of d(HS)sample periodogram of d(HS adj)

8

9

10

11

12

1970 1975 1980 1985 1990 1995 2000 2005 2010

lnAIR lnAIR adjusted

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

0

sample periodogram of d(CVR)

sample periodogram of d(CVR adj)

84

11000

16000

21000

1973.1 1978.1 1983.1 1988.1 1993.1 1998.1 2003.1 2008.1

Empl Empl. adjusted

0

20000

40000

60000

0 pi/2 pi

sample periodogram of d(Empl)

sample periodogram of d(Empl adj)

Note: Left panel: The original and adjusted by TSW variables; right panel: estimated sample periodograms of the original and adjusted by TSW series.

2.8 Bibliography

Abrahams, M. and A.P. Dempster (1979). “Research on seasonal analysis. Progress report of the ASA/Census project on seasonal adjustment”, Technical report, Department of Statistics, Harvard University, Cambridge, Massachusetts. Arteche, J. (2003). “Semiparametric robust tests on seasonal or cyclical long memory time series”, Journal of Time Series Analysis 23, 251-285. Arteche, J. and P.M. Robinson (2000). “Semiparametric inference in seasonal and cyclical long memory processes”, Journal of Time Series Analysis 21, 1-25. Arteche, J. and C. Velasco (2005). “Trimming and Tapering Semi-Parametric Estimates in Asymmetric Long Memory Time Series”, Journal of Time Series Analysis 26, 581-611. Bhardwaj, G. and N.R. Swanson (2006). “An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series”, Journal of Econometrics 131, 539-578. Box, G.E.P. and G.M. Jenkins (1970). Time Series Analysis: Forecasting and Control, San Francisco:Holden-Day. Box, G.E.P., Hillmer, S.C. and G.C. Tiao (1978). Analysis and modelling of seasonal time series, in Zellner, A. (ed.), Seasonal Analysis of Economic Time Series, Washington, D.C.: U.S. Dept. of Commerce. Beureau of Census, 309-334. Bujosa, M., García-Ferrer, A. and A. de Juan (2010). “Did we miss the onset of the recent recession? Follow the leaders”, Typescript. Carlin, J.B. and A.P. Dempster (1989). “Sensitivity analysis of seasonal adjustments. Empirical cases studies”, Journal of the American Statistical Association 84, 6-20.

85

Carlin, J.B., A.P. Dempster and A.B. Jonas (1985). “On methods and moments for Bayesian time series analysis”, Journal of Econometrics 30, 67-90. Eurostat (2009). ESS Guidelines on Seasonal Adjustment, European Communities, Luxembourg. Geweke, J. and S. Porter-Hudak (1983) “The estimation and application of long memory time series models”, Journal of Time Series Analysis 4, 221-237. Gil-Alana, L.A. (2002). “Seasonal long memory in the aggregate output”, Economics Letters 74, 333-337. Gil-Alana, L.A. (2005). “Deterministic seasonality versus seasonal fractional integration”, Journal of Statistical Planning and Inference 134, 445-461 Gil-Alana, L.A. and P.M. Robinson, (2001). “Testing of seasonal fractional integration in the UK and Japanese consumption and income”, Journal of Applied Econometrics 16, 95-114. Gomez, V. and A. Maravall (2001). Seasonal adjustment and signal extractions in economic time series. A course in Advanced Time Series analysis. D. Peña, D. G. Tiao and R. S. Tsay. (eds.), NY: J. Wiley and Sons, 171-201, Chapter 8. Grether, D.M. and M. Nerlove (1970). “Some Properties of 'Optimal' Seasonal Adjustment”, Econometrica 38, 682-703. Hassler, U., P. Rodrigues and A. Rubia (2009). “Testing for general fractional integration in the time domain”, Econometric Theory 25, 1793-1828. Hurvich, C.M. and W.W. Chen (2000). “An efficient taper for potentially overdifferenced long-memory time series”, Journal of Time Series Analysis 21, 155-180 Jonas, A. B. (1981). “Long memory self similar time series models”, Unpublished manuscript, Harvard University, Department of Economics, Harvard. Leamer, E. E. (2009). Macroeconomic patterns and stories, Springer-Verlag, Berlin. Maravall, A. (2006). “An application of the TRAMO-SEATS Automatic Procedure; Direct versus Indirect Adjustment”, Computational Statistics and Data Analysis 50, 2167-2190. Maravall, A. (2007). “An application of program TSW to a set of macroeconomic time series”, Bank of Spain, Statistic and Econometric Software. Maravall, A. (2009). “Identification of Reg-ARIMA Models and of Problematic Series in Large Scale Applications Program TSW”, Mimeo. Bank of Spain, Statistic and Econometric Software. Maravall, A. and C. Planas (1999). “Estimation error and the specification of unobserved component model”, Journal of Econometrics 92, 325-353.

86

Nerlove, M. (1965). “A comparison of a modified 'Hannan' and the BLS seasonal adjustment filters”, Journal of the American Statistical Association 60, 442-91. Ooms, M. and U. Hassler (1997). “On the effect of seasonal adjustment on the log-periodogram regression”, Economics Letters 56, 135-141. Pierce, D.A. (1979) “Signal extraction error in non-stationary time series,” Annals of Statistics 7, 1303-1320. Porter-Hudak, S. (1990). “An application of the seasonal fractionally differenced model to the monetary aggregates”, Journal of the American Statistical Association 85, 338-344. Reisen, V.A., A.L. Rodrigues and W. Palma (2006) “Estimation of seasonal fractionally integrated processes,” Computational Statistics and Data Analysis 50, 568-582. Whittle P., 1963, Prediction and Regulation by Linear Least-Square Methods, London: English Universities Press.

87

CHAPTER 3:

Explaining wage premium changes across sectors: Evidence from the translog model

88

89

3.1 Introduction

Over the last 40 years, the supply of skilled workers has increased steadily in most

developed economies. However, this fact has not been accompanied by a drastic decrease

of the wage premium. Contrary, for some countries such as the US, the relative wage of

skilled workers has even increased. The wage structure of European economies has

remained more stable, although none of them shows a strong decline in the wage

premium. Most of the studies found that a latent skilled biased technological change

(SBTC) is the answer to this apparent contradiction. SBTC change is defined as a shift in

the production technology that favors skilled over unskilled labor by increasing its

relative productivity and, therefore, its relative demand. For instance, Bound and Johnson

(1992), conclude that much of the variation in the skill premium is attributed to a residual

trend component.

As an alternative explanation, the capital-skill complementarity (CSC) hypothesis

(Griliches (1969), Krusell et al. (2000) ) has also received substantial attention in the

literature29. According to this hypothesis, capital and skilled labor are more

complementary as inputs than they are capital and unskilled labor. As a consequence, an

increase in the amount of capital raises the marginal productivity of skilled labor by more

than the one of the unskilled. According to Krusell et al. (2000),

the (relatively higher) complementarity between capital and skilled labor may be an

important factor for understanding wage inequality, since the stock of capital equipment

has also grown steady over the last decades.

The validity of the CSC and the SBTC hypotheses has central implications in

economic growth, trade, and inequality (see e.g. Stokey (1996)). Many studies have found

empirical support from either one or the other hypothesis, mainly using cross-sectional or

29 Other explanations as international trade (see e.g. Feenstra and Hanson (2003)) or migration (see e.g. Borjas (1994)) have also been also studied in the literature.

90

time series manufacturing or completely aggregated data (see e.g. Fallon and Layard

(1975), Berman et al. (1998), Flug and Herkowitz (2000) , Krusell et al (2000), Ruiz-

Arranz (2002) or Duffy et al. (2004) among others)30.

The aim of this paper is to examine the SBTC and CSC hypothesis at the sectoral

level. In this sense, our work is closely related to Fallon and Layard (1975). Fallon and

Layard use data for 9 developed and 13 less developed countries pieced together for a

single year, 1963, to estimate reduced form equations derived from two-level CES

production function. They find "mild" (though statistically not significant) evidence in

favor the CSC hypothesis at economy wide level but strong evidence at sectoral level.

Their dataset, however, does not include capital stock and factor price data at sectoral

level. To circumvent this problem they assume perfectly competitive markets and cross-

sector equality of the efficiency parameter, so that the marginal product conditions can be

used to estimate linear reduced forms. On the other side, the dataset employed in this

paper does contain sectoral data on capital prices and stocks allowing for differences in

efficiency across sectors.

The methodology used in this paper follows Christensen et al. (1971, 1973) and

Ruiz-Arranz (2002), and allows us to distinguish and quantify the effects that the CSC

and SBTC have on the evolution of the sectoral wage premium. We estimate a translog

model with 4 inputs (skilled and unskilled labor, and ICT and Non ICT capital)

employing sectoral data for three different countries (US, UK and Japan).

The main result of the paper is that the relatively higher complementarity between

skilled and (ICT) capital inputs is both a sector and country specific phenomenon.

30 Among them only Ruiz-Arranz (2002) tries to separate and quantify the contribution of this two effects. The author develops a framework based on a translog production function with four (and five) inputs and separate trends for the factor biases of technical change, and applies it to time series data from the U.S. over the period 1965-1999. This is also the approach followed in this paper.

91

Conversely, skilled labor using innovation technologies explains a substantial proportion

of the wage premium changes in all or almost all sectors and countries.

The paper is organized as follows. Section 2 briefly describes the translog model.

Section 3 discuses the empirical framework and describes the data. The estimation results

and the decomposition of the growth rate of wage premium for the different

sectors/countries are presented in Section 4. Section 5 concludes the paper.

3.2 The transcendental logarithmic cost function

3.2.1 Model description

The model used in this paper runs on the assumption that technological possibilities faced

at each sector can be summarized by a cost function:

( ) ( , ) min ' : , 0Q

C Q f Q≡ ≥ ≥p p q q q (3.1)

where q is a I x 1 vector of inputs, p is a I x 1 vector of input prices and Q is scalar output.

The translog (Christensen et al. (1971,1973)) is a flexible functional form that

provides a 'second order approximation' (in logs) to an arbitrary cost function.

Under constant returns to scale, the translog cost function can be written as:

20

1ln log log log

2tP t log t tα α β′ ′ ′= + + +p pp pt ttp + p p + pα βα βα βα βΒΒΒΒ (3.2)

where, P is the output price, and the level of technology is represented by t. The factor

shares can be obtained by applying Shephard's lemma:

ln

ln

Plog t

∂= = +∂ p pp pts p +

pα βα βα βα βΒΒΒΒ (3.3)

The (I x I) matrix of share elasticities Βpp gives the response of the value shares of

all inputs to proportional changes in the input prices. The (I x 1) vector of bias of

92

technical change βpt is obtained by differentiation of the input share function with respect

to time. If βit is positive (negative) the value share of input i increases (decreases) with

time and technological progress is input using (saving). Furthermore if βit = 0 for all

inputs, the technological progress is (Hicks) neutral and is given only by the parameters αt

and βtt in equation (3.2).

The theoretical properties of the general cost function (3.1) can be expressed in

terms of the parameters appearing in (3.2). Specifically, linear homogeneity and product

exhaustion are satisfied if:

1 1 1

1, 0 1... , 0I I I

i ij iti j i

= i Iα β β= = =

= ∀ = =∑ ∑ ∑ (3.4)

Symmetry requires:

′=pp ppΒ ΒΒ ΒΒ ΒΒ Β (3.5)

Non-negativity of factor shares is satisfied if:

0i i its log tα β′= + ≥i p +ββββ (3.6)

Finally, concavity will hold if the Hessian matrix of second order derivatives of the

translog function (3.2) is negative semi-definite.

Uzawa (1962) has derived the Allen partial elasticities (AES) between any pair of

inputs, that within the translog framework are:

2

2, i j and ij i j ii i i

ij iii j i

s s s s

s s s

β βσ σ+ + −= ≠ = (3.7)

If σij are positive the factors are substitutes in production; otherwise the factor are

complements. Note that within the translog framework the AES vary in accordance with

the input shares. Thus, the translog production function overcomes the drawbacks of other

93

specifications as the Cobb-Douglas or the (nested) CES specifications, since it does not

impose a priori restrictions over the substitutability among any pair of inputs and also

because it allows the elasticities to evolve over time, which is consistent with the

observed variation of the labor demand.

3.2.2 The decomposition of the growth rate of the skilled premium

The framework discussed above allows us to decompose the growth of the skilled

premium in three different components.

Relative supply effect: It involves the growth of the relative supplies of skilled and

unskilled workers in the sector.

Complementarity effect: It involves the growth of the supplies of the other factors used in

production (capital) and how they interact with the two types of labor. Thus, if high

skilled labor is more complementary with input j in production than it is unskilled, a

growth in the supply of input j (ceteris paribus) will increase the skill premium.

Technological change bias effect: It involves the relative technological biases of all the

types of inputs employed in production.

By Shepherd lemma, PQ P Q PQ

q Q P∂ ∂ ∂= = + =∂ ∂ ∂

sp p p p

and PQ=p sq

. By taking

logarithms and differencing the previous expression with respect to time the vector of

growth rates of input prices can be written as ln

p s q PQtγ∂= = − +

∂p

iγ γ γγ γ γγ γ γγ γ γ

where γx denotes the growth rate of the variable (vector) x and i is a I x 1 vector of ones.

The first term sγγγγ can be expressed as ( )s pp p pt= +Λ ΒΛ ΒΛ ΒΛ Βγ γ βγ γ βγ γ βγ γ β where Λ is a I x I diagonal

matrix containing the inverse of factor shares on the diagonal. Therefore the vector of

growth rates of factor prices can be finally written as:

94

( ) ( )1

p pp IxI q pt PQγ−

= − − iΛΒ ΛΛΒ ΛΛΒ ΛΛΒ Λγ − Ι γ βγ − Ι γ βγ − Ι γ βγ − Ι γ β

Let the vector of growth rates of factor prices be ordered as ( ),, ,

H L H Lp p p pγ γ −=γ γγ γγ γγ γ , where

the parameters ,H Lp pγ γ denote the growth rate of the price of skilled and unskilled labor

respectively, and the vector ,H Lp−γγγγ the growth rates of the price of the remaining inputs

other than labor in production (e.g. capital). Then the growth rate of skilled premium H

L

p

p

, can be decomposed as:

( )1 2 1 2, ,

H L H L j

I I

p p q q j q Ht Lt j jtj H L j H L

γ γ φ γ φ γ φ γ ϕ β ϕ β ϕ β≠ ≠

− = + + + + +

∑ ∑ (3.8)

where the parameter kφ is the element (1,k) minus the element (2,k) of the matrix

( ) 1

pp IxI

−ΛΒΛΒΛΒΛΒ − Ι− Ι− Ι− Ι and the parameter φk is the element (1,k) minus the element (2,k) of the

matrix ( ) 1

pp IxI

−ΛΒ ΛΛΒ ΛΛΒ ΛΛΒ Λ− Ι− Ι− Ι− Ι . The first term in equation (3.8) captures the effect of changes in

the relative supply of both types of workers on the growth rate of skill premium. It is, the

relative supply effect. The complementarity effect is gathered at the second component,

and involves the growth in the supplies of the remaining inputs in production. Finally, the

technological change bias effect is captured by the third term in the equation. It is

important to remark that the parameters kφ and φk governing the size of these effects are

time-varying since they are function of the factor shares

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3.3 Empirical strategy

3.3.1 Econometric framework

To achieve the purposes of the present study, we follow Krusell et al. (2000) and

we describe aggregate output in each sector as a function of four inputs; skilled and

unskilled labor, and ICT and non-ICT capital31. We also include separate trends for the

factor biases of technological change, which are measured as time trend32. We

characterize the structure of technology of each sector by estimating the system of the

four factor shares (skilled and unskilled labor and ICT and Non-ICT capital) as given in

(3.3) together with the restrictions imposed by the theory of production (3.4) to (3.6). In

order to complete the stochastic model, we assume that the deviation of factor shares

from the logarithmic derivatives of the translog cost function in each sector is the result of

random errors in cost minimizing behavior (see e.g. Berndt and Woods (1975)). Thus, we

complete the stochastic model by adding to each of the equations in (3.3) a disturbance

term , , , ,i i H L ICT NICTε = .

Given the homogeneity and symmetry restrictions in (3.4) to (3.5) and the fact that

the cost shares of the four equations add one, the system of the four cost shares given in

(3.3) is over-determined. This implies that the disturbance covariance matrix of the four

equation model is singular and non-diagonal. Following standard practice, we arbitrary

drop one of the equations of the system (we drop the Non-ICT capital equation) and

specify that the disturbance column vector ( ) ( ) ( ) ( )( ), ,H L ICTt t t tε ε ε ′Ε = is

independently distributed with mean zero and non-singular variance-covariance

matrix , 1...t TΩ = . Therefore the system to estimate with (3.4) and (3.5) restrictions

imposed are given by: 31The ICT capital includes Communication, Information and Technology categories. 32 This is also the specification that Ruiz-Arranz (2002) employs as a benchmark to analyze the wage premium in the US GDP.

96

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( )( )

( )

log log log

log log log

log

H t L t ICT tH t H HH HL HICT Ht H t

NICT t NICT t NICT t

H t L t ICT tL t H HL LL LICT Lt L t

NICT t NICT t NICT t

H tICT t H HICT

NICT t

p p ps t

p p p

p p ps t

p p p

ps

p

α β β β β ε

α β β β β ε

α β β

= + + + + +

= + + + + +

= + + ( ) ( )( )

( ) ( )

log logL t ICT tLICT ICTICT ICTt L t

NICT t NICT t

p pt

p pβ β ε+ + +

(3.9)

The system looks like a SURE model subject to parameter restrictions (recall that

each equation shares two parameters with the other two equations33), and all equations

have the same set of regressors. Cross equation parameter restrictions appear to be an

additional advantage of the translog framework over the direct estimation of the

production function since country specific studies are typically done with relative few

data points. The remaining parameters for the Non-ICT capital equation can be easily

obtained from restrictions (3.4) and (3.5).

Imposing local concavity at a reference point t*

Let ppH C= ∇ be the Hessian matrix of the translog cost function. Concavity

of C requires H to be negative semi-definite. The translog function’s curvature property is

data dependent, hence concavity is not guaranteed if the usual estimation procedure is

used. Diewert and Wales (1987) estimate a translog cost function for the US

manufacturing sector for 1947 to 1971 and show that the translog cost function fails to

satisfy concavity at 40% of the sample points. It turns out that in our case concavity of the

estimated cost function is also a serious issue since it is not satisfied at 30% - 50% of the

sample points depending of the sector. Obviously, this may lead to a erroneous inference.

Unfortunately, no parametric restrictions can be imposed to ensure concavity of the

33 So, in addition to the 3 intercepts, only 9 free parameters have to be estimated within the 3 equations.

97

estimated translog cost function without affecting its flexibility34. Alternatively, as

discussed in Ryan and Wales (2000) concavity can be imposed locally without affecting

the flexibility of the translog. Let t* be the reference point. The system in (3.3) can be re-

written as:

*( )log t t= + −p pp pts p +ΒΒΒΒα βα βα βα β

Without loss of generality, we can set all input prices equal to one at the reference

point, which gives ( )*t = ps αααα at this point.

Let G be a matrix with element (ij ) given by

, 1 if and 0 otherwiseij ij i ij i j ijg i jβ α δ α α δ= − + = =

Diewert and Wales (1987) show that the Hessian is negative semi-definite if and only if

the matrix G is also negative semi-definite. The procedure described in Ryan and Wales

(2000) imposes curvature at the reference point t* solving for:

( ')ij ij i ij i jDDβ α δ α α= − + − (3.10)

where G = -DD', D triangular matrix and taking into account that αi = si.. Although this

procedure only ensures concavity at the selected point, it turns that a careful selection of

the point leads to satisfaction of concavity at all or almost all points in the sample.

Estimation procedure

With aggregate data the prices that determine demands and supply cannot be

treated as exogenous and the estimation of the three equation system of factor shares (3.9)

with the concavity restrictions given in (3.10) have to be instrumented. In this paper we

employ Non-Linear GMM were lagged values of endogenous and the dependent variables

34 The standard practise is to impose concavity globally by decomposing the matrix of share elasticities

ppΒΒΒΒ in terms of its Cholesky factorization (Diewert and Wales (1987)). However, as the authors point out,

this procedure imposes a priori unacceptable restrictions and leads to the loss of the flexibility of the functional form.

98

are used as instruments35. This methodology was initially imported from growth literature

by Caselli et al. (1996) and has become the standard procedure for the estimation of

production functions36.

3.3.2 Data description

The data we use in our analysis is obtained from the EU KLEMS database37,

which contains yearly internationally comparable information on productivity, capital

formation and employment creation at the industry level for European member states and

some other OECD countries from 1970 onwards38. Hence, the length of the data is similar

to other country specific studies39.

In this study we consider a 4 input model: two labor inputs and two capital inputs.

In our empirical analysis, we divide labor into two types: unskilled and skilled labor.

Following Krusell et al. (2000), skilled labor is defined as those workers who have at

least college degree. The EU KLEMS categorization of labor by skills is relatively

similar to ours, but instead of two types of skills, they provide data on three types (low,

medium and high). We group the low and medium series into a single series that

corresponds to our definition of unskilled labor, and the remaining data coincides with

our definition of skilled labor.

35 Recall that the procedures to achieve concavity as the one described in (3.10), impose a set of quadratic restrictions to the parameters. 36 More specifically, I employ as instruments for the endogenous variables for the share equation Sit , the log PHt-1, log PLt-1, log PICTt-1, log PNICTt-1 and Sit-1, i=H, L,ICT,NICT. 37 The EU-KLEMS database can be found at http://www.euklems.net/. For a detailed description of the EUKLEMS data, see O’Mahony and Timmer (2009). 38 The availability of all the data depends on the particular country. For Japan, the data runs from 1973 to 2005. UK has data from 1970 to 2005 and, finally, US data may be found from 1977 to 2005. 39 For instance, Betts (1997) estimates a translog model for 17 Canadian manufacturing industries by separately using yearly data from 1961 to 1986. Ruiz-Arranz (2002) also estimates a translog model with data from 1965-1999 for the US, using completely aggregate data. Similar or even shorter lengths can also be found in Krusell et al. (2000) or Katz and Murphy (1992). As an appealing alternative, one could use a panel of countries as in Duffy et al. (2004). However, this strategy would entail the use of equal technology assumption in all countries, which appears to be strong when sectoral data is considered. Furthermore, it may be difficult to find a reference point that ensures concavity for all countries in the sample.

99

We use data on 7 different 1-digit sectors: agriculture, forestry & fishing (PRI),

mining & quarrying (MIN), total manufacturing (MAN), construction (CON), energy, gas

& water supplies (SUP), financial, business & real estate services (FIN), and the

remaining services categories (OTHER).

Figure 1 depicts the time profile of the natural logarithm of the wage premium in

all sectors and countries. The pattern of the (log) supplies of the 4 input types is also

plotted in the in Figure 1. Thus, growth rates can be inferred from the slope of the graphs.

As can be seen in the figure, there are country differences in the evolution of the wage

premium. For the US, the wage premium has been growing in most of the sectors. This is

also the case of the UK, although the increase in the wage premium is smaller.

Conversely, the wage premium has decreased in Japan in many sectors, although this

decrease has not been dramatic. Notice also that in most of the sectors, the stock of the

NICT capital has increased sharply which is the argument that supports the CSC

hypothesis. The total percentage changes of the four inputs and the wage premium, over

the whole period considered, are presented in Table 1.

3.4 Results

This section presents the estimation results of the translog model and quantifies

the effect of the three different factors on the growth of skill premium; relative supply,

relative complementarity and unobserved biased technological change.

Table 2 (a, b and c) reports the GMM estimates of the translog system of share

equations with linear symmetry and homogeneity restrictions imposed, as given by (3.9),

subject to the local concavity restriction (3.10) for each sector and country. For each

sector and country, we need to estimate 12 free parameters in total: three

intercepts , ,i i H K ICTα

=, (coinciding with the estimated value shares at the selected

100

reference point), 6 share elasticities , , ,ij i j H K ICT

β=

and three biases of technical change

, ,it i H K ICTβ

=. The remaining 6 parameters , , ,,nict nict j nict tα β β found in the table are

obtained from (3.4).

Special attention deserves the estimated factor biases itβ . Table 3 presents the p-

values of the test of Hicks neutrality; 0 , , , ,: h t l t ict t h tH β β β β= = = together with a test of

skill neutral technical change0 , ,: h t l tH β β= . As can be seen in the table, progress is in

general non-neutral (with the exemption of the PRI sector in the UK) and skill biased.

Nevertheless, the null of skilled neutral technological change is not rejected for the SUP

sector in any country.

The following table (Table 4 a, b and c) reports the estimated Allen Partial

Elasticities (AES) of substitution ,i jξ between any pair of inputs i,j computed as in (3.7).

By definition, the AES is symmetric so only 10 elasticities have to be reported. Recall

that a positive (negative) AES implies substitutability (complementarity) between the

factors. The estimated own elasticities are always negative as implied by concavity. The

last two columns of the table analyze the Griliches hypothesis: the skilled labor is more

complementary with capital (left ICT and right Non-ICT) than it is the unskilled labor.

The values inside the columns are , ,h ict l ictξ ξ− and , ,h nict l nictξ ξ− . Thus, the CSC hypothesis

requires the numbers to be negative. As can be seen in the table, the evidence favoring the

CSC is mild. In general, skilled labor seems to be more complementary in production

with ICT capital than it is the unskilled (the estimated numbers of the previous to the last

column in the table are often negative), but the unskilled labor seems to be relatively

more complementary with the Non ICT capital input. Yet, with the exception of Japan,

this evidence is statistically weak and varies with the sectors and countries.

101

Finally, we compute the decomposition of the growth of skilled premium on the

three components (relative supply effect, complementarity effect and bias of technology

effect) by the methodology explained in the previous section. The decomposition of the

growth of the skilled premium for the period 1980-2005 at each particular sector and

country may be found in the Table 5. The first four columns report the total contribution

(aggregated over the years) of observables. The numbers in the first two columns ate the

total effect of the change in the supply of skilled and unskilled workers respectively

(relative supply effect) while the following two are the contribution of the change in the

supply of ICT and Non ICT capital (complementarity effects). The columns 5 and 6 are

the contribution of the unobservable, due to the relative skill and ICT using technology

innovations (bias of technical change effect). The sum of the columns 1 to 6 gives the

predicted total percentage change in the skilled premium (column 7). I also include the

actual percentage change (in column 8) by a matter of comparison. As can be seen in the

table, the translog model predicts skilled premium changes fairly well, especially if one

takes into account that the table reports the cumulative changes over the specified years.

The total predicted and actual values are in general very similar and the sign is always

correct. The only exception is the construction sector (CON), where the growth of the

wage premium is under-predicted for all countries, especially for the UK. Presumably,

this is just the consequence of the housing bubble for residential markets that has been

taken place in the same period40. Second, the effect of increases (or decreases) in the

supply of skilled and unskilled workers (columns 1 and 2) have also the expected sign in

accordance with the actual increases or decreases of the supplies of both types of labor

(see Table 1). As expected, an increase in the supply of skilled workers (as it is usually

the case) affects negatively the skilled premium (column 1) and an increase (decrease) in

40 For US and UK, the housing bubble peaked around 2006, at the end of the data used in this study. The Japanese housing bubble peaked earlier than the other two countries (1991).

102

the supply of unskilled workers has a positive (negative) effect on the wage premium. In

general, movements in the relative supply of skilled workers are the main factor driving

wage premium movements.

We now turn to quantify the effects of the CSC and SBTC hypotheses. The

magnitude of the CSC effect can be found in columns 3 and 5. The numbers in the

columns are the contribution of the growth of the ICT capital (3rd column) and the Non

ICT capital (4th column) to the growth of the skill premium. I group the unobservable

technology effects in two different components: the fraction of the variation of the skill

premium accounted by relative skill using (column 5) and relative ICT capital using

(column 6) innovations. As can be seen in the table, the complementarity effects

(columns 4th and 5th) usually operate in different directions. The opposite signs reflex

that, in general, the skilled labor appears to be relatively more complementary with ICT

capital whereas the unskilled labor is in general relatively more complementary with

Non-ICT capital, as estimated from the AES. Hence, increases in the supply of ICT

capital (Non-ICT) have a positive (negative) effect on the estimated growth of the skill

premium. Column 5 contains the relative effect of the efficiency parameters of the two

types of labor. As commented before, progress seems to have been skill biased in all the

countries and sectors with the only exception of construction (CON) where have favored

the unskilled. In general, SCBT explains a sizable fraction of the growth of the skill

premium. The numbers in Column 6 involve the efficiency parameters of the two types of

capital and how the capital inputs interact with skill and unskilled labor. On the whole,

controlling for the supply effect of the two types of labor, although we provide evidence

favoring the CSC hypothesis, the skill premium movements cannot be explained without

taking into account the effect of the unobservable technical change in all the countries and

sectors.

103

3.6 Conclusions

In this paper we have examined the CSC and SBTC hypotheses at a sectoral level

for three different OECD countries (Japan, UK and US). Opposite to the dataset

employed by Fallon and Layard (1975), the EU-KLEMS dataset does contain information

on capital stocks and prices, which has allowed us to re-examine the CSC at sectoral level

without relaying in the assumption of equal efficiency across sectors. Furthermore, the

time dimension of the data has permitted us to include a time trend in order to account for

the unobserved SBTC. We have adopted the translog framework (Christensen et al.

(1971, 1973), Ruiz-Arranz (2002)) with 4 inputs (skilled and unskilled labor, and ICT and

Non-ICT capital) in order to be able to decompose the growth of the wage premium in

each country/sector in three main components: the effect of the relative supply of skilled

and unskilled workers, the effect of the supply of ICT and Non-ICT capital, and the

unobserved effect of technology innovations. In the light of our results, most of the

changes in the sectoral wage premium can be explained by labor movements. We have

also found some evidence that the skill labor is more complementary to ICT capital than

it is the unskilled, although this evidence is mild and vary with the country and sector.

However, the SBTC hypothesis appears to be more universal. After clearing for labor

supply movements, the wage premium changes cannot be explained without taking into

account the effect of the unobserved technical innovations in all sectors and countries.

104

3.7 Appendix: tables and figures

Table 1. % Total Change in the wage premium and in the supplies of the 4 input: 1980-2005 construction manufacturing mining finance primaries supplies other

JAP UK USA JAP UK USA JAP UK USA JAP UK USA JAP UK USA JAP UK USA JAP UK USA

QH 30% 150% 65% 39% 63% 43% -10% -43% 3% 119% 170% 109% -23% 39% 63% 19% 60% 34% 58% 138% 72%

QL -26% 23% 74% -48% -50% -19% -124%

-179% -3% 31% 81% 86%

-127% -33% 32% -45% -76% -11% -11% 25% 45%

QICT 248% 297% 622% 285% 279% 316% 103% 56% 285% 389% 380% 443% 27% 285% 505% 232% 302% 298% 216% 324% 244%

QNICT 36% 69% 36% 83% -1% 27% 5% 55% 40% 114% 75% 76% 58% 2% -6% 73% 39% 35% 91% 59% 83%

PH/PL 8% 40% 51% -10% 51% 39% 25% 34% 51% 9% -12% 61% -28% 30% 22% -13% 34% 44% -2% -10% 20%

Notes: QH,QL,QICT and QNICT account for the quantities of skilled labor, unskilled labor, ICT capital and Non-ICT capital. The wage premium (PH/PL) is defined as the rate of the price of the skilled to the unskilled labor. Table 2. Estimates of the Translog price function. 2.a JAPAN ah al ai an bhh bhl bhi bhn bll bli bln bii bin bnn bth btl bti btn

construction 4.437*** 0.313** -0.011 -3.739***

-0.408*** 0.016 0.027 0.365***

-0.053*** 0.011** 0.025* -0.005 -0.033

-0.358*** -0.004* 0.001*** -0.006* 0.010***

manufacturing 0.167 0.370 0.026 0.438 -0.265*** 0.278*** -0.044** 0.031

-0.291*** 0.046* -0.032 -0.007 0.005 -0.004 0.000 0.001 0.000 -0.002

mining 0.164* 0.477 -0.087*** 0.446 -0.031 0.013 0.019** 0.000 -0.005 -0.008 0.000 -0.011 0.000 0.000 0.004*** -0.004 0.000** 0.000

finance 0.337*** 0.054 0.092 0.516* -0.072** 0.052* -0.008 0.029** -0.037 0.004 -0.018 -0.006 0.010 -0.020 0.004*** -0.002 0.001* -0.003*

primaries 1.005*** 0.325 0.013 -0.343 -0.178*** 0.082* -0.001 0.097*** -0.038 0.001 -0.045 0.000 0.001 -0.053 0.000

-0.013*** 0.000 0.013***

supplies 0.104 0.138 0.049 0.709* -0.047 0.043 -0.018** 0.022* -0.039 0.017 -0.020 -0.008 0.009 -0.011 0.000 -0.001 0.000 0.001

other 0.690*** 0.259 0.140** -0.088 -0.121*** 0.078** -0.028** 0.072*** -0.050 0.018 -0.046* -0.007 0.017 -0.042 0.002*** -

0.005*** 0.001*** 0.002**

105

2.b UK

ah al ai an bhh bhl bhi bhn bll bli bln bii bin bnn bth btl bti btn

construction 1.372*** -0.068 0.260*** -0.564*** -0.127** 0.053 -0.014 0.088*** -0.051* 0.008*** -0.010 -0.002 0.008

-0.087***

-0.004*** 0.001

-0.004*** 0.008***

manufacturing -0.007 0.754*** 0.002 0.250** 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.004*** -0.004*** 0.001*** -0.001

mining 0.416*** 0.192** 0.181*** 0.210 -0.264*** 0.261*** -

0.113*** 0.116*** -0.259*** 0.112*** -

0.115*** -0.049*** 0.050*** -0.051 -0.004*

-0.012*** -0.002** 0.018***

finance 0.142 0.575** 0.046 0.237 0.000 0.000 0.000 -0.001 -0.001 -0.002 0.003 -0.009 0.011 -0.013 0.003 -0.008** 0.002 0.004

primaries 1.926 -1.058 0.074 0.058 -1.557 1.261 -0.056 0.352 -1.022 0.045 -0.284 -0.002 0.013 -0.082 0.028 -0.014 0.001 -0.015*

supplies 0.618 1.833 0.242 -1.692 -0.044 -0.100 -0.018 0.162 -0.230 -0.040 0.370 -0.007 0.065 -0.597 0.005** -0.001 0.003*** -0.007

other 0.036 0.736*** 0.012 0.217* 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.005*** -0.004*** 0.001*** -

0.003***

2.c US

ah al ai an bhh bhl bhi bhn bll bli bln bii bin bnn bth btl bti btn

construction 1.233*** -0.114 0.135 -0.253 -0.111* 0.058* -0.024 0.077 -0.049*** 0.013 -0.022 -0.005 0.016 -0.071

-0.007*** 0.002** 0.002 0.004**

manufacturing 0.165* 0.616*** 0.074* 0.145 -0.007 0.003 -0.006 0.009 -0.001 0.002 -0.003 -0.018* 0.022* -0.028 0.004** -0.009*** 0.000 0.004***

mining 0.455 0.086 0.073 0.386 -0.168 0.132 -0.032* 0.068 -0.104 0.025 -0.053 -0.006 0.013 -0.027 0.001 -0.004* 0.001** 0.002

finance 0.195 0.447 0.084 0.274 -0.002 -0.008 -0.001 0.011 -0.039 -0.007 0.054 -0.001 0.010 -0.075 0.006** -0.002 0.001 -0.005

primaries 0.467 -0.066 0.033 0.566 -0.278 0.230 -0.022 0.070 -0.190 0.019 -0.058 -0.002 0.006 -0.018 0.003 0.005 0.000 -0.009***

supplies -0.090 0.421 0.051 0.618 -0.055 0.080 0.013 -0.038 -0.116 -0.019 0.054 -0.003 0.009 -0.026 0.003* -0.005 0.001 0.001

other 0.242 0.576** 0.014 0.168 -0.029 0.040 -0.001 -0.011 -0.055 0.001 0.015 0.000 0.000 -0.004 0.003** -0.004*** 0.002*** 0.000

Notes: The number in the cells are Non-linear GMM estimates of the system of factor shares given in Error! Reference source not found. subject to the local concavity restriction Error! Reference source not found. . *,**,*** account for significance at 10%, 5% and 2% respectively. Data runs from 1973 to 2005 (Japan), 1970-2005 (UK) and 1977-2005 (US).

106

Table 3. Hicks neutrality and Skill neutral technical change tests; p-values.

Japan UK US

Test Hicks Neutral: Ho: Bht=Blt=Bit=Bnt=0

Test skill neutral capital change Ho: Bht-Blt=0

Test Hicks Neutral: Ho: Bht=Blt=Bit=Bnt=0

Test skill neutral capital change Ho: Bht-Blt=0

Test Hicks Neutral: Ho: Bht=Blt=Bit=Bnt=0

Test skill neutral capital change Ho: Bht-Blt=0

construction 0.000 0.038 0.000 0.002 0.000 0.000

manufacturing 0.027 0.779 0.000 0.000 0.000 0.003

mining 0.000 0.000 0.000 0.002 0.001 0.245

finance 0.000 0.000 0.005 0.051 0.013 0.005

primaries 0.000 0.000 0.163 0.550 0.004 0.694

supplies 0.083 0.351 0.002 0.188 0.044 0.111

other 0.000 0.000 0.000 0.000 0.000 0.011

Notes: The numbers in the cells are p-values. Thus, if the number is larger than 0.10, 0.05 and 0.02 the null hypotheses cannot be rejected at the usual 10%, %5 and 2% significance levels. Table 4. Allen Partial Elasticities (AES) 4.a JAPAN ehh ehl ehi ehn ell eli eln eii ein enn ehi-eli ehn-eln

construction -1.95 4.27 1.24 4.28 -1252.68 7.93 16.97 -4.43 -0.01 -15.25 -6.69*** -12.69**

manufacturing -29.21 6.46 -19.58 1.67 -2.74 6.03 0.82 -70.11 1.61 -1.45 -25.61*** 0.84***

mining -29.80 1.51 54.13 0.99 -1.24 -1.28 1.00 -373.05 1.05 -1.22 55.40** -0.01

finance -8.85 2.65 0.02 1.33 -4.86 1.34 0.84 -19.46 1.31 -0.79 -1.33* 0.48***

primaries -39.76 3.22 -1.47 4.09 -1.34 1.19 0.77 -161.33 1.29 -1.95 -2.66 3.32***

supplies -65.16 8.43 -19.65 1.82 -6.65 4.90 0.84 -50.31 1.45 -0.32 -24.56*** 0.98***

other -6.86 1.77 -3.36 2.36 -1.24 2.11 0.65 -36.63 3.03 -3.47 -5.47* 1.72***

107

4.b UK ehh ehl ehi ehn ell eli eln eii ein enn ehi-eli ehn-eln

construction -0.43 15.31 0.82 2.09 -3942.98 21.30 -23.56 -9.45 2.12 -21.05 -20.48 25.65*

manufacturing -11.71 1.00 0.99 1.00 -0.53 1.00 1.00 -39.96 1.02 -3.64 -0.01 0.00

mining -640.66 92.24 -521.48 6.46 -24.96 112.88 -0.06 -983.36 8.89 -0.33 -634.36*** 6.53***

finance -4.23 1.00 1.07 0.99 -1.74 0.85 1.02 -33.01 1.79 -1.86 0.22 -0.03

primaries -1372.26 54.25 -809.24 35.14 -2.85 36.58 -0.56 -1313.79 29.09 -3.95 -845.82 35.70

supplies -32.60 -5.07 -8.89 5.85 -5.95 -2.55 3.24 -32.46 3.95 -2.35 -6.34 2.61

other -5.43 1.00 1.00 1.00 -0.56 1.00 1.00 -29.08 1.00 -5.66 0.00 0.00

4.c US ehh ehl ehi ehn ell eli eln eii ein enn ehi-eli ehn-eln

construction -0.50 42.97 0.60 2.30 -37492.23 136.53 -227.15 -13.36 4.54 -28.06 -135.94 229.45

manufacturing -4.17 1.03 0.13 1.18 -1.11 1.10 0.97 -45.50 3.48 -3.14 -0.97 0.20

mining -20.75 5.18 -13.08 2.04 -3.92 5.44 0.67 -67.17 2.16 -0.88 -18.52* 1.37*

finance -3.35 0.81 0.89 1.09 -5.76 0.31 1.58 -17.02 1.33 -1.21 0.58 -0.49

primaries -42.90 7.93 -125.94 2.41 -3.25 26.04 0.69 -1102.96 5.53 -1.04 -151.98 1.72

supplies -23.81 7.30 5.82 0.27 -8.79 -1.66 1.45 -26.79 1.32 -0.47 7.48 -1.18

other -2.46 1.27 0.95 0.80 -1.38 1.04 1.18 -25.83 0.97 -5.00 -0.09 -0.38

Notes: Allen partial elasticities represent mean values of the AES over the period 1980-2005. The numbers in the last two columns are the differences in the elasticites between the two types of labor and ICT and Non-ICT capital respectively. Hence CSC hypothesis is supported when the sign in the cell is negative. *,*,*** represent significance at 10, 5 and 2 % of a test of the Ho: ehk-elk ≥ 0 if the sign is negative and ehk-elk ≤ 0 if the sign is positive, for k =ICT, N-ICT.

108

Table 5. Decomposition of the growth rate of skill premium 1980-2005; Total aggregated change.

5.a JAPAN

Observable Unobservable

Total estimated effect

Total real effect Relative supply effect Complementarily effect Relative skill technology

Relative ITC technology Sk.lab Unsk.lab ITC NITC

construction -10% 0% 40% 0% -30% 0% -8% 6%

manufacturing -10% -10% 30% -10% 0% 0% -6% -8%

mining 10% -120% -10% 0% 160% -20% 26% 20%

finance -70% 20% 20% -20% 60% 0% 0% 0%

primaries 10% -90% 0% -20% 50% 30% -25% -28%

supplies -10% -20% 50% -20% 0% -10% -5% -13%

other -40% -10% 20% -20% 30% 0% -19% -16%

5.b UK

Observable Unobservable

Total estimated effect

Total real effect Relative supply effect Complementarily effect Relative skill technology

Relative ITC technology Sk.lab Unsk.lab ITC NITC

construction -30% 0% 40% -10% -40% 30% -2% 39%

manufacturing -60% -60% 0% 0% 160% 0% 42% 48%

mining 0% -40% 20% -10% 190% -140% 19% 34%

finance -170% 80% 0% 0% 90% 0% -7% -16%

primaries -10% -10% 10% 0% 40% -20% 14% 15%

supplies -50% -100% 20% -10% 260% -50% 69% 34%

other -140% 30% 0% 0% 110% 0% -7% -9%

109

5.c US

Observable Unobservable

Total estimated effect

Total real effect Relative supply effect Complementarily effect Relative skill technology

Relative ITC technology Sk.lab Unsk.lab ITC NITC

construction -10% 10% 120% -10% -100% 0% 27% 54%

manufacturing -46% -23% 6% -1% 113% 0% 49% 39%

mining -10% 10% 40% -10% 30% -10% 47% 79%

finance -120% 80% -10% 10% 90% 10% 56% 66%

primaries -10% 20% 30% 0% 10% -20% 32% 54%

supplies -20% 0% -40% 10% 80% 10% 41% 41%

other -60% 40% 0% 0% 40% 0% 19% 19%

Notes: Number in the cells is cumulative percentage points over the sample period 1980-2005. The sum of the first 6 columns adds the total estimated percentage change of the skilled premium. Last column report the actual percentage change of the skilled premium.

110

Figure 1. Evolution of the wage premium and the input supplies (in logs)

CON MAN

MIN FIN

111

112

PRI SUP

113

OTHER

Notes: The series are reported in natural logarithms. Thus, growth rates can be inferred from the slopes.

114

3.6 Bibliography

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