armaprestotal

(1)

The Dynamics of Car Sales: A Discrete

Choice Approach

J´erˆome Adda

University College London

and Russell Cooper

(2)

Motivation

• How can we understand the aggregate dynamics of durables?

– role of income and price shocks.

– role of evolution of stock.

Aggregate Level:

• How can we resolve the Mankiw (1982) puzzle?

– PIH implies an ARMA(1,1) speciﬁcation for durables expenditures.

– MA coeﬃcent is parameterized by the rate of depreci-ation.

– From US data, estimated rate of depreciation is close to 100%!

Micro Level:

• From micro studies, infrequent adjustment and lumpiness. Lam (1991), Attanasio (1997).

(3)

Key Points

• Document the time series of durable goods.

We show that the representative agent model (PIH)

– Fails to explain time series properties of durable goods.

– Structural interpretation of parameters is diﬃcult (de-preciation rate).

• Specify and estimate a dynamic discrete choice model of car ownership.

=⇒ No need to rely on PIH or ad hoc (S,s) bands.

• D.D. Choice model predicts aggregate behavior well.

• Match aggregate sales with estimated micro model.

– sensible depreciation rate at the micro level (3 to 6%).

(4)

Evidence on Time Series Dynamic

DATA:

• Two countries: US, France.

• Aggregate durables, new car registrations, car expenditure.

• Relative prices and household income.

TOOLS:

• ARMA(1,1) for sales. Mankiw (1982).

• ARMA(1,q) for sales. Caballero (1990).

(5)

ARMA(1,1)

Mankiw (1982): PIH, quadratic utility and durable goods im-plies:

et+1 = δα0 + α1et +εt+1 −(1−δ)εt. δ: depreciation rate.

Table 1: ARMA(1,1) Estimates on US and French Data

Speciﬁcation No trend Linear trend

α1 δ α1 δ

Quarterly Data

US durable expenditures 1.00(.001) 1.006 (.08) 0.97 (0.00) 0.92(0.08)

US car registration 0.85(.08) 0.53 (.13) 0.84 (0.08) 0.53(0.13)

France durable expenditures 0.996 (0.008) 0.8(0.1) 0.92 (0.05) 0.78 (0.11)

France car expenditures 0.98(0.01) 0.71 (0.1) 0.88 (0.06) 0.74 (0.12)

France car registrations 0.95 (0.03) 0.97 (0.11) 0.88 (0.6) 1.00 (0.12) Annual Data

US durable expenditures 1.00(.03) 1.5 (.15) 0.76 (0.12) 1.42 (0.17)

US car registration 0.36(.29) 1.34 (.30) 0.33 (0.30) 1.35(0.31)

France durable expenditures 0.98 (0.04) 1.20 (0.2) 0.56 (0.24) 1.2 (0.36)

France car expenditures 0.97(0.06) 1.3 (0.2) 0.49 (0.28) 1.20 (0.32)

France car registrations 0.85 (0.13) 1.00 (0.26) 0.41 (0.4) 1.20 (0.41) Notes: For the US, source FRED database, 1959:1-1997:3. French

(6)

ARMA(2,1)

Bernanke (1985): PIH with durable goods and cost of adjust-ments implies:

et = α0 + α1et−1 +α2et−2 +εt −(1−δ)εt−1

Table 2: ARMA(2,1) Estimates on US and French Data

Speciﬁcation No trend Linear trend

α1 α2 δ α1 α2 δ

US durable expenditures 1.95 (0.07) -0.95 (0.07) 0.13 (0.1 ) 0.62(0.7) 0.33(0.7) 1.24 (0.7) US car registration 0.89 (0.31) -0.04 (0.21) 0.5 (0.3) 0.89(0.31) -0.04(0.21) 0.5 (0.3) France durable expenditures 0.7(0.4) 0.3(0.4) 1.03 (0.4 ) 0.7(0.4) 0.2 (0.4 ) 1.0 (0.45) France car expenditures 0.8 (0.3) 0.2(0.3) 0.9(0.3) 0.65 (0.2) 0.2 (0.3) 0.96 (0.4) France car registrations 0.85 (0.24) 0.09 (0.22) 1.43(0.23) 0.72 (0.36) 0.09 (0.23) 1.35 (0.35) Notes: Estimation done on quarterly data. For the US, source FRED database, 1959:1-1997:3.

(7)

ARMA(1,q)

Caballero (1990): PIH with durables and slow adjustment. et = α0 +α1et−1 +εt+ ρ1εt−1 +. . .+ρqεt−q

According to Caballero: _δ = 1 +

q

l

ρl

On US data, Caballero ﬁnds _δ = 5% annually.

Table 3: ARMA(1,q) Representations for Annual Data

Series Sample Period Implied _δ

A. MA(5) on Changes. Exponentially Detrended Series.

US, Durable Expenditures 1959-97 0.1 (0.25)

US, New Car Registrations 1968-95 -0.8 (0.76)

France, Durable Expenditures 1970-97 1.07 (0.27)

France, Car Expenditures 1970-97 1.07 (0.25)

France, New Car Registrations 1968-97 0.44 (0.18)

B. ARMA(1,5) on Levels. Exponentially Detrended Series.

US, New Car Registrations 1968-95 0.54 (1.08)

France, Car Expenditures 1970-97 0.99 (1.01)

C. MA(5) on Changes. No Detrending.

US, New Car Registrations 1968-95 -0.78 (0.73)

France, Car Expenditures 1970-97 -1.82 (1.32)

(8)

ARMA(1,q) II

(9)

Summary and Modeling Strategy

DATA:

• From VAR:

– income and prices matter.

– ﬂuctuations in sales in response to income shocks.

• From ARMA(1,q): PIH is not a good framework, even in the long run. Not robust.

• ARMA(1,1) seems to be a robust ﬁnding.

MODEL:

• Model does not rely on PIH.

• Directly specify the individual optimization problem.

• Lumpiness: Discrete choice at the individual level.

• Heterogeneity in age of cars.

(10)

Dynamic Discrete Choice Model

• agent with a car of age _i in state _Z = (_{p, Y, ε})_{, z} = (_y).

• either KEEP or REPLACE.

• probability _δ of breakdown.

Value:

Vi(z, Z) = max[Vik(z, Z), Vir(z, Z)]

where

V_ik(_{z, Z}) = _u(_s_i_{, y}+ _{Y, ε}) +_β(1−_δ)_EV_i+1(z, Z) +

βδ{EV1(z, Z)−u(s1, y+ Y, ε) +u(s1, y +Y −p +π, ε)}

and

V_ir(_{z, Z}) = _u(_s1, y +Y −p+ π, ε) +β(1−δ)EV2(z, Z) +

βδ{EV1(z, Z)−u(s1, y+ Y, ε) +u(s1, y +Y −p +π, ε)}.

Utility function:

u(_s_i_{, c}) =

i−γ + _ε(c/λ)

1−ζ

1−_ζ

=⇒ Hazard function at the individual level:

(11)

Exogenous Process

Yt = µY +ρY YYt−1 +ρY ppt−1 + uY t

pt = µp+ ρpYYt−1 +ρpppt−1 + upt

εt = µε+ρεYYt−1 +ρεppt−1 +uεt

The covariance matrix of the innovations _u = {_u_{Y t}_{, u}_pt_{, u}εt} is :

Ω =



 ω_ωY_pY ω_ωY p_p 00

0 0 _ω_ε

 

• estimate the process for (_p_t_{, Y}_t) in ﬁrst stage

(12)

Aggregate Sales

• Aggregate hazard function:

H(_{i, Z};_θ) = _h(_{i, z, Z};_θ)_φ(_z)_dz

• Aggregate Sales:

St =

i

H(_{i, Z}_t;_θ)_f_t(_i)

• Cross sectional distribution:

  

ft+1(1) = St

ft+1(i) = [1−H(i, Zt;θ)]ft(i−1)

(13)

Estimation: Methodology Stochastic Dynamic Programming Problem ? Predicted Hazard Function ? Aggregation over Households ?

Predicted Sales and C.D.F.

?

Parameters _θ

Simulated Moments from Aggregate Data

6 ? Match ? 1 PPP_PPP q No Yes

Actual Moments from

(14)

Estimation Method

Parameters to estimate: _θ = {_{γ, δ, λ, ζ, σ}_y_{, ρ}_εY_{, ρ}_εp_{, ω}ε}

Objective function to minimize:

LN(_θ) =_αL1_N(_θ) + L2_N(_θ)

• Matching predicted and observed sales (conditional on prices and income):

L1

N(θ) =

1 T T t=1

(_S_t −_S¯_t(_θ))2 − 1 N(_N −1)

N

n=1

(_S_tn(_θ)−_S¯_t(_θ))2

where

– _S_t: aggregate sales of new cars.

– _N number of draws for the unobserved taste shock.

• Matching Moments:

L2

N(θ) =

i={5,10,15,AR,MA}

αi( ¯Fi−F¯i(θ))2

where

– _F¯i_{, i} = 5_,10_,15: average fraction of cars of age _i across all periods.

(15)

Estimation and Results

Table 4: Estimated Parameters for Discrete Choice Model

U.S.

Non Linear Case Linear Case

Parameters Estimates S.E. Estimates S.E.

γ 0.02 0.0015 0.02 0.002

δ 0.038 0.0025 0.03 0.003

ζ 1.49 0.0037 0

-λ 483.5 10.5 1.6e-5 1.4e-06

σε 0.085 0.015 0.03 0.097

σy 0.52 0.016 0.26 0.013

ρε,p 1.14e-4 4.2e-05 -3.9e-5 1.9e-05

Pseudo-R2: 0.69 0.52

P(Overidentiﬁcation test) 0.84 0.60

FRANCE

Non Linear Case Linear Case

Parameters Estimates S.E. Estimates S.E.

γ 0.017 0.003 0.028 0.014

δ 0.061 0.016 0.06 0.03

ζ 1.46 0.007 0

-λ 108.0 12.5 6e-6 2.8e-06

σε 0.215 0.09 0.06 0.037

σy 0.99 0.003 0.44 0.22

ρε,p 1.2e-4 4.3e-05 -2.9e-5 1.5e-05

Pseudo-R2: 0.61 0.52

P(Overidentiﬁcation test) 0.91 0.67

(16)

More on the Fit

Table 5: Observed and Predicted Moments from CDF

US

Observed Non Linear Linear

Fraction of cars of age 5 8.6 8.5 8.4

(1.1) (1.4) (1.3)

(1.0) (1.1) (1.1)

(0.6) (0.3) (0.3)

FRANCE

Observed Non Linear Linear

(0.8) (0.8) (0.6)

(0.7) (0.5) (0.3)

(17)

ARMA Representation of Simulated Data

Table 6: ARMA Coeﬃcient Observed and Predicted.

AR₁ MA₁

US Observed 0.33 -0.35

(0.30) (0.30)

US Predicted 0.23 -0.30

(0.36) (0.38)

AR₁ MA₁

France Observed 0.51 -0.16

(0.27) (0.31)

France Predicted 0.06 -0.05

(0.63) (0.72)

Note: Monte Carlo results obtained over 100 replications with sample length 21 (France), 28 (US). French estimates on annual car registration, 1972-1994. US estimates on annual car registrations 1968-1995. All ARMA models included a linear trend.

(18)

VAR Representation

• Simulated data from our model

• Unrestricted linear VAR estimation, IRF created

(19)

(20)

More on IRF

Compare:

• IRF computed from simulation of structural model.

• IRF computed from observed data with ARMA(1,1).

• IRF computed from observed data with linear VAR.

Note:

• IRF from ARMA: _ρt−1(_ρ−_α)

– _{ρ >} 0 =⇒ No oscillations possible.

– _ρ−_{α >} 0 =⇒ _{δ >}1−_ρ. With _ρ 0_.6 =⇒ _{δ >} 0_.4

(21)

Decompositions

Recall:

Aggregate Sales:

St =

i

H(_{i, Z}_t;_θ)_f_t(_i)

Oscillations of sales as a result of a shock:

• Endogenous movement in CDF.

• Serial correlations in prices and income: Hazard shifts.

=⇒ How important are these two sources?

• Decompose IRF into shifts in CDF and shifts in hazard.

(22)

IRF Decompositions

Sales: interaction of Hazard function with CDF.

St =

k

(23)

IRF Decomposition

Cross Eﬀect of Prices and Income

Yt = µY +ρY YYt−1 +ρY ppt−1 + uY t

pt = µp+ ρpYYt−1 +ρpppt−1 + upt

(24)

Time Series Decomposition

Sales: interaction of Hazard function with CDF.

St =

k

Hk(zt, θ)ft(k)

Decomposition:

St −St−1 = ∆StHaz + ∆StCDF + ut

=

k

[_H_k(_z_t;_θ)−_H_k(_z_t₋₁;_θ)]_f_t₋₁(_k)

Shifts in Hazard

+

k

Hk(zt;θ)[ft(k)−ft−1(k)]

Shifts in CDF +_u_t

• Total Contribution of ∆_S_tCDF to ∆_S_t ?

• Contribution of ∆_S_tCDF to ∆_S_t at diﬀerent frequencies ?

R2

(25)

Conclusion

Our model is able to :

• reproduce various aspects of the data and time series im-plications:

– reconcile aggregate ARMA(1,1) results with micro ob-servations.

– Impulse response functions.

– ﬁt decomposition shifts Hazard/CDF, at diﬀerent fre-quencies.

Without requiring nonsense values for the micro parame-ters.

• investigate the sources of dynamics:

– role of endogenous evolution of CDF (small)

(26)

Figure 1: Impulse Response Functions To an Orthogonal Shock to Income

(27)

(28)

Figure 3:

(29)

Figure 5:

(30)