Score-driven non-linear multivariate dynamic location models€¦ · 09/11/2017 · SZABOLCS...

Score-driven non-linear multivariate dynamic location modelsSZABOLCS BLAZSEK (FRANCISCO MARROQUIN UNIVERSITY)

ALVARO ESCRIBANO (CARLOS III UNIVERSITY OF MADRID)

ADRIAN LICHT (FRANCISCO MARROQUIN UNIVERSITY)

GESG SEMINAR, UNIVERSIDAD FRANCISCO MARROQUIN, 9 NOVEMBER 2017 1

Contribution


Contribution

In this paper, we extend the dynamic conditional score (DCS) model of the multivariate �-distribution that was introduced in the work of Harvey (2013, Chapter 7.2.2).

Motivated by Harvey (2013, Chapter 3.2.1), we name the new model quasi-vector autoregressive (QVAR) model.

QVAR with lag-order �, denoted as QVAR(�), is a score-driven non-linear multivariate dynamic location model, in which the conditional score vector of the log-likelihood (LL) updates the dependent variables.


Contribution

QVAR(�) is an extension of the DCS model for the multivariate �-distribution that is QVAR(1) under our notation.

For QVAR, we present the details of the econometric formulation, the computation of the impulse response function, and the maximum likelihood (ML) estimation and related conditions of consistency and asymptotic normality.


Contribution

We compare the statistical performance of QVAR and that of two benchmark multivariate dynamic location models:

VAR and VARMA (vector autoregressive moving average).

We estimate QVAR by using the ML method.

We estimate VAR and VARMA by using the quasi-ML (QML) method.


Contribution

The likelihood-based model performance metrics suggest that the statistical performance of QVAR is superior to that of VAR and VARMA.

The residual and conditional score diagnostic test results suggest that each residual and conditional score variable of QVAR(2) forms a multivariate i.i.d. time series.

The conditions of consistency and asymptotic normality of ML are satisfied for QVAR.


QVAR model


Quasi-vector autoregressive modelReduced-form representation of QVAR( )

�� = � + �� + �

�� = Φ�� + ⋯ + Φ�� + Ψ��

where ��, �, ��, � and �� are (� × 1) and

Φ�, … , Φ� and Ψ� are (� × �)

For the first � observations, we initialize �� by using its unconditional mean �� = � �� = 0�×�.

For QVAR(�), the conditional expectation of �� is � �� , … , �� = � + ��.



�~��(0, Σ, ν) is the multivariate i.i.d. reduced-form error term that updates ��. Σ is positive definite and ν > 2.

The log conditional density of �� is

ln " �� , … , �� = ln Γ$%�

&− ln Γ

$

&−

�

&

−�

&ln Σ −

$%�

&ln 1 +

)*+,-.)*

$



The partial derivative of the log of the conditional density with respect to �� is

/ 01 2 �� , … , ��

/3*=

$%�

$Σ�� × 1 +

)*+,-.)*

$

��

� =

=$%�

$Σ�� × ��

where the last equality defines the scaled score function ��, which is the representation of �� by using the reduced-form error term.



Harvey (2013, Chapter 7) shows that �� is multivariate i.i.d. with mean zero and covariance matrix

Var �� =$%�

$%�%&Σ��


Quasi-vector autoregressive modelStructural-form representation of QVAR(�)

For the reduced-form error term we have � � = 0 and Var � = Σν/(ν − 2)

We factorize Var � as

Var � =$

$�&

�/&Ω�� Ω�� 9 $

$�&

�/&


Quasi-vector autoregressive modelStructural-form representation of QVAR(�)

We introduce the multivariate i.i.d. structural-form error term :�:

� =$

$�&

�/&Ω��:�

where � :� = 0�×�, Var :� = ;�, and :�~�� 0, ;� ×$�&

$, ν .

Therefore, the representation of the score function �� by using the structural-form error term is

�� = (ν − 2)ν �/&Ω�� ×<*

$�&%<*+<*


Quasi-vector autoregressive modelSummary

Reduced-form representation:

�� = � + Φ�� + ⋯ + Φ�� + Ψ�� + �

where � � = 0�×�and Var � = [ν/(ν − 2)]Ω�� Ω�� 9.

Structural-form representation:

?@�� = ?@� + ?@Φ�� + ⋯ + ?@Φ�� + ?@Ψ�� + :�

where ?@ = [ν/(ν − 2)]��/&Ω��:�, and

� :� = 0�×� and Var :� = ;�.


Non-linear vector MA representations and impulse response functions



The first-order representation of the reduced-form QVAR(A) is

B� = C + D� + E�

D� = ΦD�� + ΨF��

where

B� =

��

��

⋮��%� (��×�)

C =

��⋮� (��×�)

D� =

��

��

⋮��%� (��×�)



E� =

�

��

⋮��%� (��×�)

Φ =

Φ� Φ& … Φ�� Φ�

;� 0�×� … … 0�×�

0�×� ;� 0�×� … …… … … … …

0�×� … 0�×� ;� 0�×� (��×��)



Ψ =

Ψ� 0�×� … 0�×�

0�×� 0�×� … 0�×�

… … … …0�×� … … 0�×� (��×��)

F�� =

��

0⋮0 (��×�)



The corresponding structural-form non-linear vector MA(∞) representation of �� is:

�� = � + ∑ JΦKJ′Ψ� ν − 2 ν �/&Ω�� <*-.-M

$�&%<*-.-M+ <*-.-M

NKO@ +

+$

$�&

�/&Ω��:�

where J = (;� , 0�×� , … , 0�×�) (� × ��).



The impulse response function IRFK = S��%K/S:� for T = 0,1, … , ∞is given by:

IRF@ =$

$�&

�/&Ω��

IRFK� = JΦKJ′Ψ� ν − 2 ν �/&Ω��U��K for T = 1, … , ∞,

where

U� =/

V*

W-XYV*+V*

/<*(this derivative is evaluated in closed form)



As IRFK� for T = 1, … , ∞ depends on �, we evaluate its

unconditional mean and use:

IRFK = JΦKJ′Ψ� ν − 2 ν.

XΩ��(U��K)

If all elements of U��K form covariance stationary time series,

then �(U��K) can be estimated by using the sample average.

We test the covariance stationarity of U��K by using the

augmented Dickey-Fuller (1979) (ADF) unit root test.


Maximum likelihood (ML) estimation


Maximum likelihood (ML) estimation

The parameters of QVAR are �, Φ�, … , Φ�, Ψ�, Ω�� and ν.

We estimate these parameters by using the ML method:

Θ[\] = arg maxa

LL ��, … , �c =

= arg maxa

∑ ln "(��|��, … , ��)c�O�

We use the inverse information matrix to estimate the standard errors of parameters (Harvey 2013). We use Harvey (2013,

Chapters 2.3, 2.4 and 3.3) to find the conditions under which the ML estimates of QVAR(�) are consistent and asymptotically Gaussian.


Conditions of consistency and asymptotic normality of ML


Condition 1

If all eigenvalues of Φ are within the unit circle, then �� is covariance stationary.

Let C� denote the maximum modulus of all eigenvalues.

C� < 1 supports Condition 1.


Condition 2

We use Condition 2 from the work of Harvey (2013, p. 35, Condition 2).

Condition 2 is that the score function �� (� × 1) and its first derivative S��/S�� (� × �) have finite second moments and covariance that are time-invariant and do not depend on ��.

We test Condition 2 by using the ADF test (C& metric).


Condition 3

In order to obtain Conditions 3 and 4 for QVAR, we use the arguments of the proof of Theorem 5 from the work of Harvey (2013, p. 49).

Recall the first-order representation of QVAR(�):

D� = ΦD�� + ΨF��

We consider the representative element ΨfK from the

matrix Ψ.


Condition 3The derivative of the previous equation with respect to ΨfK is:

/g*

/hiM= Φ

/g*-.

/hiM+ Ψ

/j*-.

/hiM+ kfKF��

Where the element (l, T) of kfK (�� × ��) is one and the rest of the

elements are zero. From here we express:

/g*

/hiM= Φ + Ψ

/j*-.

/g*-.+

/g*-.

/hiM+ kfKF�� = m�

/g*-.

/hiM+ kfKF��

Condition 3 is that all eigenvalues of E m� are inside the unit circle (Cometric). We estimate this expectation by using sample average (ADF test).


Condition 4

From

/g*

/hiM= m�

/g*-.

/hiM+ kfKF��

we express:

vec/g*

/hiM

/g*+

/hst= m� ⊗ m� vec

/g*-.

/hiM

/g*-.+

/hst+

vec m�/g*-.

/hiMkfK

9 F�� + vec F��9 kvw

/g*-.+

/hstm�

9 +

vec kfKF��F��9 kvw

9


Condition 4

Condition 4 is that all eigenvalues of � m� ⊗ m� are inside the unit circle (Cx metric: maximum modulus of all eigenvalues).

We estimate this expectation by using sample average.

We use the ADF test for each element of m� ⊗ m�.


Benchmark models


Benchmark models

We compare QVAR with the benchmark VAR(�) and VARMA(�, 1) models.

We estimate both benchmark models by using the QML estimator (Gourieroux, Monfort and Trognon 1984a-b).

We use the Gaussian distribution as auxiliary distribution for QML.

Covariance stationarity and invertibility of VARMA ensures consistency and asymptotic normality of QML.


Data


DataWe use macroeconomic data from the book of Kilian and Lütkepohl (2017). This dataset includes the following variables:

(i) monthly West Texas Intermediate (WTI) price of crude oil for period December 1972 to June 2013;

(ii) quarterly US GDP deflator for period 1959Q1 to 2013Q2;

(iii) quarterly US real GDP level for period 1959Q1 to 2013Q2.

The use of these variables is motivated by several works from the body of literature, which study the question of how oil price shocks affect US real GDP and inflation (e.g. Blanchard 2002; Barsky and Kilian 2004; Kilian 2008; Kilian and Lütkepohl 2017).


DataWe define:

(i) variable �� as the quarterly first difference of log real price of crude oil (hereinafter, crude oil);

(ii) variable �&� as the quarterly first difference of log US GDP deflator (hereinafter, inflation);

(iii) variable �o� as the quarterly first difference of log US real GDP level (hereinafter, GDP growth).

We define �� = (�� , �&� , �o�)9, hence, � = 3 for all models in this paper. Furthermore, we use data for period 1987Q1 to 2013Q2 (Kilian and Lütkepohl 2017, Chapter 9.2.1).


INTERNATIONAL CONFERENCE IN HONOR OF LUC BAUWENS, 19-20 OCTOBER 2017 37

Seasonality effects in variables

For each variable we estimate a linear regression:

�� = z�,{�U{�,� + z�,{&U{&,� + z�,{oU{o,� + z�,{xU{x,� + η��

�&� = z&,{�U{�,� + z&,{&U{&,� + z&,{oU{o,� + z&,{xU{x,� + η&�

�o� = zo,{�U{�,� + zo,{&U{&,� + zo,{oU{o,� + zo,{xU{x,� + ηo�

We test the significance of parameter differences.

For crude oil and GDP growth, we find significant differences, which indicates seasonality effects in those variables.


Identification of QVAR, VAR and VARMA


Identification of structural forms

The QVAR, VAR and VARMA models used in this paper are recursively identified structural models (Kilian and Lütkepohl 2017, Chapter 9).

This identification method is supported by the argument that oil price shocks may act as domestic supply shocks for the US economy (Kilian and Lütkepohl 2017, p. 239).


Identification of structural forms

Therefore, we specify

Ω�� =

Ω�� 0 0

Ω&�� Ω&&

�� 0

Ωo�� Ωo&

�� Ωoo��

with Ω�� > 0, Ω&&

�� > 0 and Ωoo�� > 0.

That is Σ is factorized by using the Cholesky decomposition.


Identification of model parameters

For both QVAR and VARMA we use the restriction:

Ψ� = Ψ�,�� × ;�

This implies that Ψ� is diagonal and Ψ�,�� = Ψ�,&& = Ψ�,oo.

Without this restriction, the ML procedure did not converge.


ML estimates and model diagnostics



Firstly, we find that some elements of Φ and Ω�� are significantly different from zero for all.

This suggests significant dynamic and contemporaneous interaction effects, respectively, among crude oil, inflation and GDP growth.



Secondly, for both VAR specifications, the Ljung-Box (1978) (LB) test suggests that :&� and &� each form a non-independent time series.

For all VARMA and QVAR specifications, the LB test suggests that each reduced-form and structural-form error term forms an independent time series.

For QVAR(1), the LB test suggests that the score function �� is a non-independent time series. For QVAR(2), we find that each score function forms an independent time series.



Thirdly, for all models, we find that conditions of consistency and asymptotic normality of ML and QML are supported.


ML estimates and model diagnosticsFourthly, we compare the statistical performance of QVAR, VAR and VARMA, by estimating the following likelihood-based performance metrics:

(i) mean LL; (ii) mean Akaike information criterion (AIC); (iii) mean Bayesian information criterion (BIC); (iv) mean Hannan--Quinn criterion (HQC).

All model performance metrics suggest that the statistical performance of QVAR is superior to that of VAR and VARMA.


Impulse response functions


Impulse response functions

We find that Φ�,�& = Φ�,�o = Φ&,�& = Φ&,�o = 0.

Since by the Cholesky decomposition we impose that the oil

price changes are predetermined with Ω�&�� = Ω�o

�� = 0(Kilian 2008), therefore, we conclude that oil price changes are strictly exogenous (Kilian 2008) for the parameters of interest in the inflation and the GDP growth rate equations.

Furthermore, since Ω&�� = 0, therefore, the inflation rate is

also predetermined (Kilian 2008).


Robustness analysis


Robustness analysis

Perhaps, the most surprising result of the present paper is that for QVAR(2) stochastic annual cyclical effects are identified for the impulse response function.

One may argue that these cyclical effects may be spurious, because they may be due to the non-significant parameters within Φ�, Φ& and Ω��, or they may be due to the order of variables in QVAR. We investigate this concern.


Restricted QVAR(2) and restricted VARMA(2,1)


Variable order: crude oil, GDP growth and inflation for QVAR(2)


Thank you for your [email protected]@[email protected]@UFM.EDU

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Score-driven non-linear multivariate dynamic location models€¦ · 09/11/2017 · SZABOLCS...

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