Structural Econometrics: Consumption Dynamics
Jean-Marc Robin
Plan 1. The life-cycle model — Hall (JPE, 1978) 2. Estimation of Hall’s lifecycle model — Hall and Mishkin (Econometrica, 1982) 3. Liquidity constraints — Zeldes (JPE, 1986)
1 The life-cycle model
Hall (JPE, 1978).
Discrete time. • At the beginning of each period t, a household decides about consumption for period t given • disposable income.
Notations: •
at savings Rt disposable income
ct consumption H life duration
yt income flow ρ discount rate
rt interest rate Ut VNM utility function
Budget constraint: consumption + savings = disposable income •
ct + at = Rt
= yt +(1+rt)at 1. −
2 State variables: everything that is known by household at decision time t. Includes current • disposable income Rt and present and past income flows and interest rates: yt =(yt,yt 1, ...) − and rt =(rt,rt 1, ...). In the sequel, we omit the reference to exogenous realisations yt and rt − (shocks).
Consumption strategy: afunctionct (Rt; yt, rt) of state variables. •
3 Value function and Bellman equation
Value function = Expected sum of future utility flows for optimal consumption strategy: • H t − i Vt(Rt)= max Et ρ Ut+i(ct+i) , ct( ),ct+1( ),... · · Ã i=0 ! X where Et( )=E ( yt, rt) isexpectationwithrespecttofutureshocksgivenpresentandpast · ·| shocks.
Terminal condition =nobequest: •
RH+1 =(1+rH+1)aH =0.
Bellman equation: given that tomorrow I will behave optimally, what does that mean to • behave optimally today?
Vt(Rt)=maxUt(c)+ρ tVt+1 (Rt+1) c E s.t. Rt+1 = yt+1 +(1+rt+1)(Rt c). −
4 Euler equation
Optimisation problem: •
Vt(Rt)=maxUt(c)+ρEtVt+1 (yt+1 +(1+rt+1)(Rt c)) . c −
First Order Condition: optimal consumption ct (Rt) solves •
Ut0(ct)=ρEt(1 + rt+1)Vt0+1 (yt+1 +(1+rt+1)(Rt ct)) − = ρEt(1 + rt+1)Vt0+1 (Rt+1) (1)
Apply the envelope theorem: •
Vt0(Rt)=ρEt(1 + rt+1)Vt0+1 [yt+1 +(1+rt+1)(Rt ct)] − = ρEt(1 + rt+1)Vt0+1 (Rt+1)
= Ut0(ct). (2)
Euler equation: use (2) to replace V 0 (Rt+1) by U 0 (ct+1) in (1). Then • t+1 t+1
Ut0(ct)=ρEt [(1 + rt+1)Ut0+1 (ct+1)] .
5 Sort of random walk equation for consumption: best predictor of tomorrow’s consumption is today’s consumption.
Hypothesis usually not satisfied by macro data: regress ct ct 1 on ct 1 and yt;significant effect − − − of yt.
6 Estimation of Hall’s model using panel data
Hall et Mishkin (Ecta, 1982).
Construction of an econometric model generally requires additional specification assumptions:
2 Quadratic utility function: Ut(ct)= (c∗ ct) ,ct c∗. • − t − ≤ t Constant interest rate: rt = r. • 1 Discount rate: ρ =(1+r)− . • D P T Income process: yt = y + y + y ,where • t t t D — yt is a deterministic trend, — P = P + = P + + + + permanent income yt yt 1 εt yt0 εt εt 1 ... εt0+1 is (t0 beginning of lifecycle; − − innovation εt is a white noise), T — yt = Φ(L)ηt = φ0ηt + φ1ηt 1 + ... + φqηt q is transitory income (ηt is a white noise): − − For η to be an innovation (i.e. 1 yT only depends on the past of yT ), the roots of ∗ t Φ(L) t t the characteristic polynomial are strictly outside the unit circle. Assume φ =1for normalisation, as Var (η ) and Φ are not separately identifiable. ∗ 0 t 7 Euler equations
Euler equations become: • Ut0(ct)=Et [Ut0+1 (ct+1)] or
Etct+1 ct∗+1 = ct ct∗ − − as
U 0(ct)=2(c∗ ct) . t t − And the Law of iterated expectations implies •
Etct+i ct∗+i = Et (Et+1 (... (Et+i 1 (ct+i ct∗+i)))) − − − = ct c∗. − t
8 Analytical solution
Apply Euler equations to the sequence of budget constraints.
Intertemporal budget constraint: • Rt = yt +(1+r)at 1 − Rt+1 yt+1 ⎧ = + Rt ct ⎫ Rt+1 = yt+1 +(1+r)(Rt ct) − ⎪ 1+r 1+r − ⎪ ⎪ Rt+2 yt+2 Rt+1 ct+1 ⎪ ⎧ Rt+2 = yt+2 +(1+r)(Rt+1 ct+1) ⎫ ⎪ = + − ⎪ − ⎪ 2 2 ⎪ ⎪ . ⎪ ⎪ (1 + r) (1 + r) 1+r ⎪ ⎪ . ⎪ = ⎪ . ⎪ ⎪ ⎪ ⇒ ⎪ . ⎪ ⎪ R = y +(1+r)(R c ) ⎪ ⎪ ⎪ ⎨ H H H H ⎬ ⎨ RH yH RH 1 cH 1 ⎬ − = + − − − RH+1 =0=(1+r)(RH cH) (1 + r)H t (1 + r)H t 1+r ⎪ − ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ RH+1 RH cH ⎪ ⎪ ⎪ ⎪ =0= ⎪ ⎪ ⎪ ⎪ H t+1 − H t ⎪ ⎩ ⎭ ⎪ (1 + r) − (1 + r) − ⎪ ⎪ ⎪ ⎪ ⎪ Summing up, ⎩⎪ ⎭⎪ H t H t − yt+i − ct+i 0= +(1+r)at 1 . i − i i=0 (1 + r) − i=0 (1 + r) X X
9 Take expectation: • H t H t − Etyt+i − Etct+i 0= +(1+r)at 1 i − i i=0 (1 + r) − i=0 (1 + r) XH t XH t − Etyt+i − ct∗+i + ct ct∗ = +(1+r)at 1 − i − i i=0 (1 + r) − i=0 (1 + r) X X using Euler equations.
Hence the consumers demands • 1 ct = ct∗ + γt− [(1 + r)at 1 + Yt Ct∗] − − with H t H t H t − 1 − y − c γ = ,Y= Et t+i ,C= t∗+i . t (1 + r)i t i t∗ i i=0 i=0 (1 + r) i=0 (1 + r) X X X Consumption smoothing: expected wealth (1 + r)at 1 + Yt Ct∗ is spread evenly over the • − − lifecycle.
10 Expected wealth
The income process implies
Etyt = yt D P T = yt + yt + yt D P = yt + yt 1 + εt + ηt + φ1ηt 1 + ... + φqηt q, − − − D P T Etyt+1 = yt+1 + Etyt+1 + Etyt+1 D P = yt+1 + yt 1 + εt + φ1ηt + ... + φqηt+1 q, − − D P T Etyt+i = yt+i + Etyt+i + Etyt+i D P yt+i + yt 1 + εt + φiηt + ... + φqηt+i q (i
Euler equations show that the consumption process is a random walk with a deterministic drift, • that is ct ct∗ ct 1 ct∗ 1 is a martingale difference sequence. − − − − − ¡ ¢ (Note that a process (ut) is called a martingale difference sequence if Et 1ut =0.) −
Expectation of ct given time-(t 1) information is • − 1 Et 1ct = Et 1 ct∗ + γt− [(1 + r)at 1 + Yt Ct∗] − − − − 1 = ct∗ + γ©t− [(1 + r)at 1 + Et 1Yt Ct∗] . ª − − − Hence • min H t,q { − } 1 1 φi ct Et 1ct = γt− (Yt Et 1Yt)=γt− γtεt + ηt − − (1 + r)i − − ⎛ i=0 ⎞ X or ⎝ ⎠
ct ct∗ = ct 1 ct∗ 1 + εt + βtηt − − − − with min H t,q { − } φ β = γ 1 i . t t− (1 + r)i i=0 X 12 Data
Panel Study of Income Dynamics (PSID) 1969-75
Consommation = food.
“How much do you spend on food in an average week ?” • Question asked in March. •
Income is last calendar year’s income.
13 Econometric model
(Simpler version of Hall and Mishkin’s)
H = ,soγ = γ =1+1 and β = β constant. • ∞ t r t T Transitory income is MA(1): yt = ηt + φηt 1. • − To take into account that food is not total consumption and the different timing of income and • consumption measures, the econometric model assumes:
∆ct = ∆ct∗ + α[δ(εt+1 + βηt+1)+(1 δ)(εt + βηt)], D − ( ∆yt = ∆yt + εt + ηt (1 φ)ηt 1 φηt 2. − − − − − where
— α is the share of total weath spent on food and — δ models the advance of information on future income.
D ∆c∗ and ∆y are linear functions of age, age squared, demographics and log price changes. • t t
14 Remarks:
Hall and Mishkin add another MA(1) component to ∆ct to account for measurement errors. • Transitory income is a MA(2) process. • Note that there is no individual specificeffect although the data are panel data. •
15 Estimation
D Use OLS to estimate ∆ct∗ and ∆yt functions of exogenous variables.
Then maximise likelihood of residuals to estimate α, δ, β, φ and variances of εt and ηt. Results:
α =0.11 (mean propensity to consume on raw data: 0.19.) • β =0.29 (slightly above the theoretical value that can be computed using standard interest • rate variables).
δ =0.25 (if consumption observed in March; should depend for 3/4 of past income and for 1/4 • of current year income; bingo!)
16 Excess sensitivity to transitory income
Model predicts covariances well except for Cov(∆ct, ∆yt 1) which is theoretically nil as • −
∆ct = εt + βηt and ∆yt 1 = εt 1 + ηt 1 +(φ 1) ηt 2 φηt 3 − − − − − − −
Yet, regressing ∆ct on ∆yt 1 yields: • −
∆ct = 4.95 0.010∆yt 1. −(6.16) − (.002) − Negative correlation can be explained if household population is heterogenous: •
— afractionμ of households track income, i.e. ∆ct = α∆yt. — afraction1 μ behavesasinthetheory. − 2 2 Then for constrained households, Cov(∆ct, ∆yt 1)= α (φ 1) ση < 0. − − − Hall and Mishkin estimate μ = .20. • Note that this ad hoc change in the model does not solve the problem completely. Predicted • Cov(∆ct, ∆yt 1) is 0.032 instead of 0.077. − − −
17 Liquidity constraints
Zeldes (JPE, 1989)
Structural model of households constrained by credit rationing.
Liquidity constraint: Households cannot borrow more than a certain amount: •
at a pour tout t, ≥ t
where at < 0. Implies the following constraint on consumption: •
at = Rt ct a − ≥ t or
ct ct = Rt a ≤ − t where Rt = yt +(1+rt)at 1. −
18 Optimal consumption at t given optimal behaviour at t +1
Bellman equation: • Rt+1 = yt+1 +(1+rt+1)(Rt c) Vt(Rt)=maxUt(c)+ρ tVt+1 (Rt+1) s.t. − c E ( c ct ≤ Kuhn and Tucker conditions: •
Ut0(ct) ρEt [(1 + rt+1)Vt0+1 (yt+1 +(1+rt+1)(Rt ct))] μt =0, − − − μt (ct ct)=0. − e where μ 0 is Kuhn and Tucker’s constant. t ≥ e Two cases: • e
— Corner solution: μt > 0 and ct = ct,
— Interior solution: μt =0and ct = ct ct,wherect solves e ≤ Ut0(ct) ρEt [(1 + rt+1)Vt0+1 (yt +(1+rt)at 1 ct)] = 0. e − e e − −
— Corner solution optimal iff ct > ct = Rt a . e − t e 19 e Euler equation
Value function: • Vt(Rt)=Ut(ct)+ρEtVt+1 (yt+1 +(1+rt+1)(Rt ct)) − Ut(Rt at)+ρEtVt+1 (yt+1 +(1+rt+1)at) if ct > ct = Rt at, = − − ¯ Ut(ct)+ρEtVt+1 (yt+1 +(1+rt+1)(Rt ct)) if ct Rt at. ¯ − ≤ − ¯ e Envelope theorem¯ : • ¯ e e e Ut0(Rt at)=Ut0(ct) if ct > ct = Rt at, Vt0(Rt)= − − ¯ ρEt [(1 + rt+1)Vt0+1 (yt+1 +(1+rt+1)(Rt ct))] = Ut0(ct) if ct Rt at. ¯ − ≤ − ¯ e Hence: Vt0(R¯ t)=Ut0 (ct). ¯ e e e Use Kuhn and Tucker conditions again to derive Euler equation: • Ut0(ct)=ρEt [(1 + rt+1)Vt0+1 (Rt+1)] + μt
m e Ut0(ct)=ρEt [(1 + rt+1)Ut0+1 (ct+1)] + μt or ρ(1 + rt+1)Ut0+1 (ct+1) μt 1 e Et =1 (μt 0). U (c ) − U (c ) ≡ 1+μ ≥ ∙ t0 t ¸ t0 t t 20 e Data and econometric specification
PSID •
Consumption = Food
Assumption: utility function additively separable between food and other other consumptions. •
Implies that previous theory applies exactly the same to food consumption.
Econometric specification: household i,periodt, • c1 α U (c )= it− exp θ it it 1 α it − where θit is a linear index of exogenous variables:
2 θit = b0Ageit + b1Ageit + b2 ln(nit)+ui + ut + uit,
where nit is household size.
21 Econometric model of consumption dynamics
For non constrained households, Euler equations take the form: • α cit−+1 exp [θit+1 θit] ρ(1 + rit+1)(1 + μit) − α =1+εit+1 cit− where εit+1 is a martingale difference sequence (Etεit+1 =0). Take logs: • 1 ∆ ln cit+1 = [∆θit+1 +lnρ +ln(1+rit+1)+ln(1+μ ) ln(1 + εit+1)] . α it − Second order approximation: • 1 2 1 ln(1 + εit+1)=εit+1 εit+1 Et ln(1 + εit+1) Var tεit+1. − 2 ⇒ ' −2 Final specification: • ln(1 + μ ) ∆ ln c = κ + 1 ln(1 + r )+2b1 Age + b2 ∆ ln n + it + η it+1 t α it+1 α it α it+1 α it+1 where 1 1 κ = ln ρ + b + b + ∆u + Var ε t α 0 1 t+1 2 t it+1 1 ∙ ¸ ηit+1 = [∆uit+1 ln(1 + εit+1)+Et ln(1 + εit+1)] MA(1) process. α − ∼
22 Estimation
rit+1 possibly endogenous. •
— Assumption: ln yit,ait 1/yit,whereait 1 is measured by household assets are valid in- − − struments.
— Estimation:2SLSon∆ ln cit (∆ ln cit) with time dummies. − i Assume that most of the individual fixed effect results from ln(1+μit) and • α ln(1 + μ ) — estimate it as mean residuals α µ ¶i
1 2b1 b2 ln(1 + μit) (∆ ln cit) κt + ln(1 + rit+1)+ Ageit + ∆ ln nit+1 + i − α α α α µ ¶i
23 Test of liquidity constraints
Partition sample into poor and rich household, according to last period’s assets ai,t 1. −
Test 1: Estimate model on both subsamples incorporating ln yit in the equation but without • ln(1 + μit).
Result:thecoefficient of ln yit comes out significantly only for the poor. Test 2 : • 1. Estimate the model on the sample of rich households. ln(1 + μ ) 2. Compute it from poor subsample using parameters estimated at stage 1. α µ ¶i ln(1 + μ ) Résultat :meanof it positive (.017) but not significant. α µ ¶i Test 3 : Regress estimated residuals (ln(1 + μ )+η )onln yit. • it it Result : negative coefficient, not significant.
Note that different ways of partitioning the sampe gave similar results.
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