Temporal disaggregation with dynamic models

Adapting quarterly national a ounts te hnique tosto k variablesAurélien Poissonnier(aurelien.poissonnierinsee.fr)Insee-CrestO tober 2012A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 1 / 27

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Why is it interesting for a ma roe onomist ?Many quarterly time series for ows (output, wage, onsumption...)Few sto k time series (wealth, apital...)Then how toestimate a simple Cobb-Douglas fun tion ?estimate a wealth ee t in a onsumption equation...Why I needed this in the rst pla e ?To study the greenhouse gaz emissions of households :Monthly data for fuel onsumption, monthly pri es, monthly weather ondition, monthly ar registrations...But annual ar eetA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 2 / 27

Instead of using 12 times less points

1.8e+07

2.0e+07

2.2e+07

2.4e+07

2.6e+07

2.8e+07

3.0e+07

1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009

monthly car fleetcar fleet on January first

Figure : Fren h households' ar eetA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 3 / 27

How?(mi ro) Literature alled temporal disaggregationex : annual data + quarterly indi ator ⇒ quarterly dataThe best pra ti e in QNA ompilation is based on Chow-Lin (1971)and its developments : Litterman (1981), Fernandez (1983),Bournay-Laroque (1979)A tra table optimization of likelihood with missing information but noKalman lter, developed for ow variables.Can be generalized to sto k variables !A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 4 / 27

Plan1 What is Chow and Lin's method ?

A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 5 / 27

Plan1 What is Chow and Lin's method ?2 Adapting this framework to dynami modelsA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 5 / 27

Plan1 What is Chow and Lin's method ?2 Adapting this framework to dynami models3 An example : Fren h household's ar eetA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 5 / 27

2 questions in oneFrom an annual a ount Ca and quarterly indi ators Ia,t, one seeksCa,t, assuming the following relation :

Ca,t = Ia,tβ + εa,t (1)with εa,t white noise or AR or random walk...To estimate the quarterly prole of the a ount, one needs both :1 β2 the quarterly prole of εa,tA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 7 / 27

Missing informationWith only the annual a ount, a dire t regression an not be performedDenote M the matrix summing the four quarters of ea h year. One anestimate β from equation :MCa,t = Ca = MIa,tβ +Mεa,t (2)i.e. on annual data.One then also a ess εa = Mεa,t the annual residual. Withoutadditional information, the quarterly prole of the residual an only befound when assuming a sto hasti stru ture for it ; for instan e AR(1)(εa,t = µεa,t−1 + ηt).


SolutionDenote Ω the varian e- ovarian e matrix assumed for εa,t. One hasΩa = MΩM ′ the varian e- ovarian e matrix of the annual residual εa.The solution to this problem reads

Ca,t = Ia,tβ + Lεa (3)with β =

(

I ′aΩ−1

a Ia)

−1 (

I ′aΩ−1

a Ca

)

the GLS estimator on annual data(4)and L = ΩM ′Ω−1

a a smoothing matrix (5)A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 9 / 27

I onsider the following equation :Ca,t = Ca,t−1ρ+ Ia,tβ + εa,t (6)Under this general form an be des ribed apital dynami withdepre iation rate τ

Ka,t = Ka,t−1(1− τ) + Ia,t + εa,tor a sto k for whi h net inow indi ator are knownSa,t = Sa,t−1 + Fa,tβ + εa,t


"How mu h dieren e does it make ?" you may ask...It's now almost impossible to isolate the annual dataFor a sto k measured at the end of the year :Ca,4 = ρ4Ca−1,4 +

4∑

i=1

ρ4−i (Ia,iβ + εa,i) (7)4

∑

i=1

ρ4−iεa,i = Ca,4 − ρ4Ca−1,4 −4

∑

i=1

ρ4−iIa,iβ (8)I an not isolate the annual residual anymoreA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 12 / 27

Maximum Likelihood EstimatorI denote Θa =∑

4

i=1ρ4−iεa,i = M2(ρ)εa,i, a linear ombination of arandom variable whose draws are unknownThe linear ombination an be written :Θa = M1(ρ)Ca +M2(ρ)Ia,tβThe problem to be solve is then :

MaxE,ρ,β,Ω

1√2π

4N |Ω|exp

(

−1

2E′Ω−1E

) (9)s. . M2E = Θa = M1(ρ)Ca +M2(ρ)Ia,tβ (10)with E the (4N × 1) ve tor of εa,t.A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 13 / 27

The formal solution for the quarterly residualWhatever the values for the parameters (ρ, β,Ω), the solution for Ealways reads :

E = ΩM ′

2

(

M2ΩM′

2

)

−1Θa (11)whi h is similar to Chow-Lin's solution when M2 is repla ed by M .


Maximum Likelihood Estimator 2Repla ing E by its expression :

Maxρ,β,Ω

1√2π

4N |Ω|exp

(

−1

2Θ′

a

(

M2ΩM′

2

)

−1Θa

) (12)with Θa = M1Ca +M2Iaβ a fun tion of only known variables(indi ators, annual a ount) and parameters (ρ, β).This problem an thus be solved with any optimization softwareA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 15 / 27

From annual data and households' ar eet omputed by CCFA, one an estimate the monthly ar eet.New registrations are used as an indi ator of inows3 modelling of outows are tested : proportional to the eet (ρ < 1),proportional to new registrations (ν < 1) or ontant (κ 6= 0).Fleeta,t = ρF leeta,t−1 + νRegista,t − κ− εa,t (13)

with εa,t = cste+ µεa,t−1 + ηa,t (14)or ∆εa,t = µ∆εa,t−1 + ηa,t (15)


Estimation pro edureS rappage programs : ad ho orre tion in De ember 1992, June 1995,De ember 1996 and November-De ember 2009 and 2010 (1.5 million ars beneted from the s rappage program between 2009 and 2011)The likelihood to be maximized, depends on the exit rate (ρ), theauto orrelation of the exit sho k (µ), the standard error of itsinnovation (σ) and the ve tor β = [ν, κ]. I estimated the dynami modelfor ars with the following onstraints on the parameters :0 < ρ < 1

0 < µ < 1

0 < σ < ∞0 < ν < 1

−200000 < κ ≤ 0A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 18 / 27

Estimation resultsexit sho k AR(1) I(1) onstraints ρ = 1 ρ = 1 ρ = 1 ρ = 1 ρ = 1ν = 1 κ = 0 µ = 0

ρ 0.999 - - - 1 - -µ 0.999 0.998 0.455 0.998 0.987 0.987 -σ 917.611 1131.63 700503.72 1131.74 185.41 185.41 1145.49ν 0.277 0.246 - 0.247 0.159 0.159 0.137κ 0.000 -2.607 0.000 - - - -log-likelihood -5984.3* -5950.8* -10356.3* -5950.8 -4603.1 -4603.1 -5954.3* : no optimum foundTable : Results for the optimization of a sto k model on fren h ar eetExits from fren h ar eet an best be modelled as being proportionalto the new registrations. 84% of the new registrations are bought inrepla ement of a ar being s rapped.The model's residual is found to be an ARIMA(1,1,0) with highauto orrelation (but not I(2)).A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 19 / 27

Figure : Fren h households' ar eet

1.8e+07

2.0e+07

2.2e+07

2.4e+07

2.6e+07

2.8e+07

3.0e+07

1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009

monthly car fleetcar fleet on January first


Figure : Fren h households' ar eet (monthly hanges)

0

20000

40000

60000

1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010

estimated residualnew registrations’ net contribution

monthly changes in households’ car fleetA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 21 / 27

Hessian at the optimumThe hessian matrix at the optimum when minimizing the opposite ofthe log-likelihood of the I(1) model for ars with ρ = 1 reads :

H =

34799.5197094 0.8281052 −3337.2970.8281052 0.05506399 −0.00001477929

−3337.2966141 0.00001477929 0.0001530383

(16)Its eigenvalues are 35355, 14748 and 0.055.A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 22 / 27

0.95 0.96 0.97 0.98 0.99 1.00

−46

25−

4620

−46

15−

4610

−46

05

mu

log−

likel

ihoo

d

Figure : Log-Likelyhood on ar data for the I(1) model with ρ = 1 at theoptimum as a fun tion of µA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 23 / 27

0.0 0.2 0.4 0.6 0.8 1.0

−10

000

−90

00−

8000

−70

00−

6000

−50

00

nu

log−

likel

ihoo

d

Figure : Log-Likelyhood on ar data for the I(1) model with ρ = 1 at theoptimum as a fun tion of νA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 24 / 27

100 200 300 400 500

−70

00−

6500

−60

00−

5500

−50

00

sigma

log−

likel

ihoo

d

Figure : Log-Likelyhood on ar data for the I(1) model with ρ = 1 at theoptimum as a fun tion of σA. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 25 / 27

Why ρ = 1 ?

rho0.0 0.2 0.4 0.6 0.8 1.0

nu 0.00.2

0.40.6

0.81.0

−6e+08

−4e+08

−2e+08

Figure : Log-Likelyhood on ar data for the I(1) model with µ = 1 at theoptimum as a fun tion of ρ, ν between 0 and 1A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 26 / 27

Con lusionWhenever needed, you an now estimate more solid data for sto kvariables to estimate your ma roe onomi models using a simpleoptimization fun tion !A. Poissonnier (Insee-Crest) Temporal disaggregation O t. 2012 27 / 27

Temporal disaggregation with dynamic models

Economy & Finance

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