Estimating an Euler Equation Using GMM

In one of my grad. econometrics courses we cover Generalized Method of Moments (GMM) estimation. I thought that some readers might be interested in the material that I use for one of the associated lab. classes.

The lab. exercise involves estimating the Euler equation associated with the "Consumption-Based Asset-Pricing Model" (e.g., Campbell, 1993, 1996.) This is a great example for illustrating GMM estimation, because the Euler equation is a natural "moment equation".

The basic statement of the problem is given below, taken from the handout that accompanies the lab. class exercises:

The Canadian data that are used are real per capita non-durable consumption expenditure (CND), and R = [1 + TBILL/CPIND], where TBILL is the 91-day T-bill rate, and CPIND is the non-durables CPI. The sample period is 1962Q1 to 2005Q3. The data are available in an Excel file on the Data page for this blog.

Here are the plots of the two series that are used in the model:

To estimate the model using EViews, we select "Quick", "Estimate equation", and select "GMM" as the method of estimation. Then we enter the equation specification, as shown below. Note that the expression that we enter is the expression whose expectation is zero. That is, we enter the moment condition:

In the equation specification, the coefficient C(1) corresponds to the parameter β; and the coefficient C(2) corresponds to the parameter γ.

You'll also see that a list of instruments has been supplied. These instruments are all part of the information set, at time "t". To be honest, the results that I'm going top show you are actually somewhat sensitive to the instruments that are used. 

Next, we choose some options for the estimation of the model:

The results that we get are:

The p-values for the reported t-statistics are based on a 2-sided alternative hypothesis. We expect both beta and gamma to be positive, so the appropriate p-value for the t-test associated with C(2) (i.e., γ) is actually 0.0301. So, we reject the hypotheses that β = 0 and that γ = 0, at the 5% significance level.

The "J-statistic" is for testing the validity of the over-identifying restrictions that arise because we have 9 instruments, but we are estimating only 2 parameters. The associated p-value of 60% supports the validity of these over-identifying restrictions.

Finally, let's interpret the estimates obtained for the two parameters, β and γ.

• β is the subjective time discount factor. Our estimate of 0.82 is pretty typical of other results to be found in the literature.
• γ is the reciprocal of the inter-temporal elasticity of substitution of future consumption for current consumption. Estimates of γ reported in the literature range between roughly 3 and 85 in value! (e.g., see Hall, 1988) Our estimate of γ implies an elasticity of substitution of 0.21. Hall's estimate in the case of T-bills, based on U.S. data, is 0.346.

References

Campbell, J. Y., 1993. Intertemporal asset pricing without consumption data. American Economic Review, 83, 487-512.

Campbell, J. Y., 1996. Understanding risk and return. Journal of Political Economy, 104, 298-345.

Hall, R. E., 1988. Intertemporal substitution in consumption. Journal of Political Economy, 96, 339-357. 

© 2013, David E. Giles