# Estimation of Life-Cycle Consumption Zhe Li (PhD Student) Stony Brook University.

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Estimation of Life-Cycle Consumption Zhe Li (PhD Student) Stony Brook University

Introduction A CLASSICAL METHOD of moments estimator Instead using analytically form, replace the expected response function by a simulation result ---- the method of simulated moments (MSM). An application of MSM to life-cycle consumption model (Gourinchas and Parker (2002)).

Model Live t= 0----N, and work for periods T { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/13/3876409/slides/slide_4.jpg", "name": "Model Live t= 0----N, and work for periods T

Model When working Income Transitory shock: takes 0 with probability and otherwise. Permanent shock:

Model After retirement, no uncertainty. Illiquid wealth in the first year of retirement Retirement value function Consumption Rule (Merton (1971))

Solution Normalization At retirement When working

Solution In the last period of working In periods

Numerical method Intertemporal budget constraint Two-dimensional Gauss-Hermite quadrature

Simulation ParametersDescriptionValue Rate of time preference0.96 Rate of risk aversion0.514, where h is the ratio of illiquid wealth to the permanent component of income at retirement 0.001 Marginal propensity to consume at retirement0.071 RInterest rate1.0344 pProbability of unemployment0.00302 Variance of permanent shock0.0212 Variance of transitory shock0.0440 Initial log normalized wealth at age 26-2.794 (1.784)

Estimation Objective Two step MSM: The first subset: The second subset: Expectation of log consumption, Approximation (Monte-Carlo)

Estimation Find that minimize Where W is a T*T weighting matrix: –Inverse of the sample counterpart of –Corrected by the variance-covariance matrix for the first-stage estimation

Estimation Method Start at a point x in N-dimensional space, and proceed from there in some vector direction p Any function of N variables f(x) can be minimized along the line p, say finding the scalar a that minimizes f(x+ap) Replace x by x+ap, and start a new iteration until convergence occurs Example: Newton method

Estimation method This study, –x is the set of parameters –Dimension is T (time periods) –Objective function is –Gradient –Hessian matrix

Estimation Results Parameters Trust-Region Newton L-MQuasi-NewtonGlobal convergence 0.95970.95950.95110.9637 (0.0390)(0.0385)(0.0313)(0.0556) 0.49810.52360.94110.2008 (3.9409)(2.6876)(3.6892)(5.0803) Retirement Rule: 0.01160.00010.10690.0511 (7.3221)(6.7996)(6.5953)(1.8346) 0.07650.08220.06500.0895 (0.6604)(0.6086)(0.3639)(0.4741) fmin0.06860.06520.07720.0924 177.8519169.1142200.1481239.5556

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