Seminar Lecture Introduction to Structural Estimation: A Stochastic Life Cycle Model James Zou School of Economics (HUST) September 29, 2017 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 1 / 34
1 Introduction 2 Model 3 Solution Method 4 Estimation 5 First-Step Estimation Results 6 Second-Step Estimation Results 7 Data 8 Model Extension 9 Reference James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 2 / 34
Introduction Structural Estimation Structural estimation is a methodological approach in empirical economics explicitly based on economic theory. A requirement of structural estimation is that economic modeling, estimation, and empirical analysis be internally consistent. Structural estimation can be defined as theory-based estimation: the objective of the exercise is to estimate an explicitly specified economic model that is broadly consistent with observed data. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 3 / 34
Introduction Why Structural Model? Theoretical model can not make quantitate predictions, even often can not make qualitive predictions. Numerical simulations rely on parameter assumptions. Its predictions are not reliable. Reduced form methods lack theoretical foundation, and can not make counterfactual simulation and welfare analysis. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 4 / 34
Introduction Why Structural Estimation? With a structural model, we have two branches of method: Calibration Estimation: include MSM, SML, SMD, Indirect Inference, and Bayesian estimation. Estimation is more reliable than calibration. MSM (or SMM) is a general approach to estimate dynamic stochastic model. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 5 / 34
Introduction Stochastic Life Cycle Model Stochastic life cycle model is a branch of general models that research agents life cycle behavioral, include consumption, wealth accumulation, health insurance, portfolio choice, retirement, tax policy etc. It perfectly connects economic theory, econometrics, and micro data. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 6 / 34
Introduction For this seminar We will introduce a basic stochastic life cycle model and use the method of simulated moment (MSM) to estimate the model. We want to use this method to research the consumption effect and labor supply effect of the China Public Institutions Reform, analyze its welfare effect and do counterfactual simulation. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 7 / 34
Model Preferences We assume that preferences take the form u(c t ) + E t { T j=t+1 β j t u(c j ) + β T +1 t b(a T +1 )} The within-period utility function is of the form u(c t ) = C1 ρ t 1 ρ workers who die value bequests of assets, A t, according to the function b(a t ) = θ (A t + k P t ) 1 ρ 1 ρ James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 8 / 34
Model Constraints The asset accumulation equation is where A t 0 A t+1 = R(A t C t ) + Y t+1 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 9 / 34
Model Income Dynamic The income consist of permanent component P t, and transitory component U t Y t = P t U t P t = G t P t 1 N t where U t and N t are both independently and identically log-normally distributed, lnu t N(0, σ 2 u), lnn t N(0, σ 2 n), James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 10 / 34
Model Recursive Formulation In recursive form, the individuals problem can be written as or V t (X t ) = max{u(c t ) + βe t V t+1 (X t+1 )} V t (A t, P t ) = max{u(c t ) + βe t V t+1 (A t+1, P t+1 )} normalize the necessary variables by dividing them by permanent income, we can get: V t (a t ) = max{u(c t ) + βe t (G t+1 N T +1 ) 1 ρ V t+1 (a t+1 )} James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 11 / 34
Model Optimal Consumption Behavior use the permanent component of income to normalize The following Euler equation holds c T +1, is linear in a T +1 : a t+1 = R(a t c t ) G t+1 N t+1 + U t+1 u (c t (a t )) = βre[u (c t+1 (a t+1 )G t+1 N t+1 )] c T +1 = γ 0 + γ 1 a T +1 The solution to the consumer problem consists of a set of consumption rules {c t (a t )} 1 t T James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 12 / 34
Solution Method Gauss-Hermite Quadrature Gauss-Hermite Quadrature u (c t (a t )) = βre[u R (c t+1 ((a t c t ) G t+1 N + U)G t+1n)] i,j f t (n i, u j )w ij where f t (n, u) = 1 R π u (c t+1 ((a t c t ) e 2σnn + e 2σuu )G t+1 e 2σnn ) G t+1 w i,j are the weights, and n i, u j are nodes James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 13 / 34
Solution Method The Method of Endogenous Gridpoints x t = a t c t, discretize x t into {x k }(k = 1, 2...K) get c k, and a k = c k + x k use linear interpolation method to get {c t (a t )} James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 14 / 34
Estimation Moment Conditions we employ a two-stage estimation procedure, we use additional data and moments to estimate χ in a first stage. by making the simulated moments as close as possible to theoretical mements g t (θ; ˆχ) = 1 N t lnĉi,t(θ; ˆχ) ln N C t t i=1 Our second-stage estimation procedure is then a method of simulated moments estimator (MSM) that minimizes over θ: ˆθ = argmin g(θ; ˆχ) W g(θ; ˆχ) James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 15 / 34
Estimation Asymptotic Variance Covariance Matrix θ is is distributed asymptotically as normal distribution: var(ˆθ) = 1 + τ N (G θ W G θ) 1 G θ W V W G θ(g θ W G θ) 1 And the statistic is distributed asymptotically as Chi-squared with T #θ degrees of freedom: χ = N 1 + τ g(ˆθ; ˆχ) W g(ˆθ; ˆχ) James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 16 / 34
First-Step Estimation Results Equally-Weighted Minimum Distance Estimator Run a regression lny it = f(x it ) + y it Decompose residual y it Take first order difference y it = p it + u it, u it N(0, σ 2 u) p it = g it + p it 1 + n it, n t N(0, σ 2 n) y it = g it + n it + u it u it 1 Use the variance and covariance of y it to generate moments for estimation var( y it ) = σ 2 n + 2σ 2 u... cov( y it, y it 1 ) = σ 2 u James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 17 / 34
First-Step Estimation Results First-Step Parameter Table 1 Parameter Value R 1.0344 σu 2 0.044 σn 2 0.0212 w 2.794 σ w 1.784 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 18 / 34
First-Step Estimation Results Income and Cunsumption Profile James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 19 / 34
Second-Step Estimation Results Estimation Result Table 2 MSM Estimation Equally Weighting β 0.9596 (0.0511) ρ 0.5403 (0.2140) γ 0 0.0006 (0.0002) γ 1 0.0758 (0.0273) χ 2 (36) 128.8 Standard errors in parentheses James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 20 / 34
Second-Step Estimation Results James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 21 / 34
Second-Step Estimation Results James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 22 / 34
Second-Step Estimation Results James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 23 / 34
Second-Step Estimation Results James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 24 / 34
Second-Step Estimation Results James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 25 / 34
Data Data CHARLS 2011-2013-2015 data hould comsunption C: exclude medical expenditure, education and training household income Y: household wage income keep 45 age 75 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 26 / 34
Data Income and Consumption Profiles 5 6 7 8 9 10 40 50 60 70 80 age Income Consumption James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 27 / 34
Data Equally-Weighted Minimum Distance Estimator y it = ( 2 y i,2013, 2 y i,2015, 4 y i,2015 ) Moment condition: E(m k m j ), k, j {1, 2, 3} Then we can get six moment condition James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 28 / 34
Data Missing Data d i = (d i,2011, d i,2013, d i,2015 ) and d it = 1{ y it is not missing}. we can drive N N m = vech{( y i y i) ( d i d i)} i=1 i=1 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 29 / 34
Model Extension Health status and Medical expense The within-period utility function is of the form Health status Survival rate s t Medical expense M t is government transfers u(c t, H t ) = δ(h t ) C1 ρ t 1 ρ π j,k = P (H t+1 = k H t = j), j, k {1, 0} lnm t = m(h t, t) + σ(h t, t)ψ t T r t = max{0, C + M t (A t + Y t )} James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 30 / 34
Model Extension Labor Supply The within-period utility function is of the form u(c t, L t ) = (Cγ t Lγ t )1 ρ 1 ρ The asset accumulation equation is A t+1 = RA t + W t (L L t ) C t wage dynamic lnw t = αlnn t + ω t James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 31 / 34
Model Extension Portfolios Choice cash on hand evolves as X t+1 = (1 + rt+1)s e t + (1 + r)b t + Y t+1 the excess return of the risky asset is assumed to be i.i.d.: rt+1 e r = µ + ε t+1 James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 32 / 34
Reference Reference Gourinchas, PierreOlivier, and Jonathan A. Parker. Consumption over the life cycle. Econometrica 70.1 (2002): 47-89. Cagetti, Marco. Wealth accumulation over the life cycle and precautionary savings. Journal of Business & Economic Statistics 21.3 (2003): 339-353. French, Eric. The effects of health, wealth, and wages on labour supply and retirement behaviour. The Review of Economic Studies 72.2 (2005): 395-427. Laibson, David, Andrea Repetto, and Jeremy Tobacman. Estimating discount functions with consumption choices over the lifecycle. No. w13314. National Bureau of Economic Research, 2007. De Nardi, Mariacristina, Eric French, and John B. Jones. Why do the elderly save? The role of medical expenses. Journal of Political Economy 118.1 (2010): 39-75. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 33 / 34
Reference Reference French, Eric, and John Bailey Jones. The Effects of Health Insurance and SelfInsurance on Retirement Behavior. Econometrica 79.3 (2011): 693-732. Alan, Sule. Do disaster expectations explain household portfolios?. Quantitative Economics 3.1 (2012): 1-28. De Nardi, Mariacristina, Eric French, and John Bailey Jones. Medicaid insurance in old age. The American Economic Review 106.11 (2016): 3480-3520. Blundell, Richard, Luigi Pistaferri, and Itay Saporta-Eksten. Consumption inequality and family labor supply. The American Economic Review 106.2 (2016): 387-435. Blundell, Richard, et al. Female labor supply, human capital, and welfare reform. Econometrica 84.5 (2016): 1705-1753. James Zou (School of Economics (HUST)) Seminar Lecture September 29, 2017 34 / 34