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<q), − − D P = ¯ yt+i + yt 1 + εt + φqηt (i = q), ¯ − ¯ D P ¯ yt+i + yt 1 + εt (i>q). ¯ − ¯ Hence process Yt can be represented¯ as ¯ H t q min H t,q j D { − − } − yt+i P φi+j Yt = i + γtyt 1 + γtεt + i ηt j. (1 + r) − (1 + r) − i=0 j=0 ⎛ i=0 ⎞ X X X ⎝ ⎠ 11 Consumption innovation 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.