Estimation of life cycle consumption
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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).

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Estimation of Life-Cycle Consumption

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Estimation of life cycle consumption

Estimation of Life-Cycle Consumption

Zhe Li (PhD Student)

Stony Brook University


Introduction

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

Model

  • Live t= 0----N, and work for periods T<N. T and N are exogenous.

  • The households maximize

  • Utility is of CRRA form, and multiplicatively separable in Z.


Model1

Model

  • When working

  • Income

  • Transitory shock: takes 0 with probability

  • and otherwise.

  • Permanent shock:


Model2

Model

  • After retirement, no uncertainty.

  • Illiquid wealth in the first year of retirement

  • Retirement value function

  • Consumption Rule (Merton (1971))


Solution

Solution

  • Normalization

  • At retirement

  • When working


Solution1

Solution

  • In the last period of working

  • In periods


Numerical method

Numerical method

  • Intertemporal budget constraint

  • Two-dimensional Gauss-Hermite quadrature


Simulation

Simulation


Estimation

Estimation

  • Objective

  • Two step MSM:

  • The first subset:

  • The second subset:

  • Expectation of log consumption,

  • Approximation (Monte-Carlo)


Estimation1

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

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 method1

Estimation method

  • This study,

    • x is the set of parameters

    • Dimension is T (time periods)

    • Objective function is

    • Gradient

    • Hessian matrix


Estimation results

Estimation Results


Estimation of life cycle consumption

Figure 9. Life Cycle Consumption (Thousands in 1987 dollars)

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Raw data

Trust-region Newton

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L-M

Quasi-Newton

Global convergence

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Age


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