## Wednesday, May 1, 2013

### 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.

1. My impression is that we stopped estimating Euler equations using GMM decades ago. The sample sizes in macro are simply too small to have any confidence in the asymptotics.

1. Really? Here I have about 200 observations. I'll do some bootstrapping to see if the asymptotics have kicked in. Keep in mind that this is just a lab. exercise to get practice using GMM - not a piece of serious research!!

2. Dear Professor Dave

In this case, it is possible that both series may be I(1). How may I proceed? Should I evaluate if residuals are I(0)?

Best Regards
João Paulo - Brazil

1. If they are I(1) and cointegrated, there's no problem - you just continue to work with the levels of the data. If they are I(1) but NOT cointegrated, you may have a problem. Specifically, if you difference the consumption data and there are any negative changes, you'll strike a problem when you go to raise that value to the power of minus gamma.

3. Hopefully you will get more readers downunder.

Keep them coming

1. 2. Actually about 1,000 hits from Oz last month.

4. Thanks for this, very helpful for a CCAPM project that I am running. I have used similar data but the mean equation is generating an error, kindly assist.

5. Greetings Professor Dave,

I am an undergraduate economics student and in understanding how to model consumption-based asset pricing models, I stumbled on your blog and found this content helpful. I downloaded the data file and have tried running it on Eviews 5 but no result could be obtained because I keep getting a dialog box that says 'no coefficient specified'...Could you kindly help out? Would really appreciate.

1. Was GMM estimation an option in EViews 5? Probably not. You need to use an up to date version of the package.

6. Dear Prof. Dave,
I have a panel data and in Eviews equation estimation dialogue there is another pane named panel options. it has a number of options to select. For example in cross sectional effects there are 5 options differenced, orthogonality conditioned, fixed, random and none. similarly Period effects, GMM weights, GLS Wieghts and coeffeicient of covariance has a number of options. Please if you can give description about them. when to use each option? Thank you in advance. Yasir Riaz

1. Use the HELP tab at the top of the EViews workfile - it gives you access to the full manual with the descriptions you need.

7. Dear Prof. Dave,
I run a similar model but for a different country, I got a negative sign for gamma, how should I explain this result? Is it possible acording to the theory?
Thank you very much
Mart

8. Dear Dave, why would you calculate real rate as tbill/cpi? Shouldn't it be tbill-inflation?

Thanks
Bob

1. Bob - that would make more sense!