If an asset has a real rate of return R_{t}, an individual may be able to increase his utility by deferring consumption from the current period and investing in the asset in order to consume in a later period. The relative attractiveness between current and future consumption affects the asset's price as reflected in its return. An important implication is that changes in consumption should mirror changes in asset prices.
Following Hansen and Singleton (1982), the agent maximizes the above utility function subject to the sequence of budget constraints
The first-order conditions yield the following Euler equations
By assuming a specific utility function you can relate the Euler equation that characterizes the agent's optimal solution to the returns that you see in the market.
Assuming a power utility function [Note: this is a restrictive assumption. Refer to Pratt (1964) for a discussion on risk-aversion and utility functions.] with risk aversion coefficient , the above equation can be written in the estimable form
Briefly, the GMM estimator is computed by minimizing the quadratic form
To perform the estimation requires data on some measure of consumption, returns for the assets of interest, and instruments from the public's information set. Of these, measures of consumption are the most difficult to obtain. Ferson and Harvey (1992) provide an informative discussion on the intricacies of consumption data.
The data for this example are the same as those used by Ferson and Harvey (1992) in estimating equation (3) from their paper. They use quarterly observations from the second quarter of 1947 (1947.6) through the fourth quarter of 1987 (1987.12). Seasonally adjusted nondurables deflated by seasonally adjusted personal consumption deflators are used as the measure of consumption; the four assets used are the real returns from government and corporate bonds and the smallest and largest equity decile real returns. Real returns are nominal returns deflated by the price index corresponding to the consumption growth measure. In addition, four lags of the real Treasury bill return and four lags of real consumption growth, both deflated by the overall Con sumer Price Index (CPI), are used as the instruments, Z_{t}.
In the MODEL procedure, the endogenous variable is the percentage change in consumption, given by the variable CONRAT. The exogenous variables are the real returns for government bonds (GB), corporate bonds (CB), and the smallest and largest equity deciles (D1 and D10, respectively). One difference between the MODEL procedure and many other estimation tools is that the parameters to be estimated need to be specified explicitly. For this example, BETA denotes the time discount factor and ALPHA denotes the concavity parameter. Because computing parameter estimates for nonlinear equations requires an iterative process, you can supply the starting values. This example uses starting values of 1.0 for both parameters; the default starting values are 0.0001 for all parameters.
proc model data=harvey; endogenous conrat; exogenous gb cb d1 d10 ; parms beta 1.0 alpha 1.0;
You can also transform variables for use in the estimation. The following statements create the four lagged values of real consumption growth (CINST) and the real Treasury bill return (RINST) to be used as instruments in the model.
lc1 = lag(cinst); lc2 = lag2(cinst); lc3 = lag3(cinst); lc4 = lag4(cinst); ltb1 = lag(rinst); ltb2 = lag2(rinst); ltb3 = lag3(rinst); ltb4 = lag4(rinst);
The requisite moment conditions for the four assets are given by
eq.h1 = beta * (1+conrat)**(-alpha) * (1+gb) - 1 ; eq.h2 = beta * (1+conrat)**(-alpha) * (1+cb) - 1 ; eq.h3 = beta * (1+conrat)**(-alpha) * (1+d1) - 1 ; eq.h4 = beta * (1+conrat)**(-alpha) * (1+d10) - 1 ;
You fit these equations by specifying the iterated GMM option using a Parzen kernel. Iterated GMM re-estimates the variance matrix at each iteration with the parameters determined by the GMM estimation from the previous iteration. See Ferson and Foerster (1994) for a discussion of iterated GMM. Iteration terminates when the variance matrix for the equation errors changes by less than the value of the CONVERGE= option. The default convergence value is 0.001. The KERNEL= option enables you to specify the smoothing function for the variance matrix. The INSTRUMENTS statement specifies the instrumental variables to be used in the ITGMM estimation method. This example uses the lagged variables created above.
fit h1-h4 / itgmm kernel=(parzen,1,0); instruments lc1-lc4 ltb1-ltb4 ; run;
Let r be the number of unique instruments times the number of equations. The value r represents the number of orthogonality conditions imposed by the GMM method. Under the assumptions of the GMM method, r - p linearly independent combinations of the orthogonality conditions should be close to 0. The GMM estimates are computed by setting these combinations to 0. When r exceeds the number of parameters to be estimated, the OBJECTIVE*N, reported at the end of the estimation, is an asymptotically valid statistic to test the null hypothesis that the over-identifying restrictions of the model are valid. The OBJECTIVE*N is distributed as a chi-square with r - p degrees of freedom. The p-value of the test can be calculated with the following DATA step.
data _null_; /* OBJECTIVE*N, degrees of freedom */ p = 1-probchi( 38.4454, 34 ); put p=; run;
In this example, OBJECTIVE*N = 38.445 is distributed as a chi-square with 34 degrees of freedom. This has an associated p-value of 0.275. Thus, you would not reject the null hypothesis that the overidentifying restrictions fit the model.
Ferson, W. and Foerster, S. (1994), ``Finite Sample Properties of the Generalized Method of Moments in Tests of Conditional Asset Pricing Models," The Journal of Financial Economics, 36, 29-55.
Ferson, W. and Harvey, C. (1992), ``Seasonality and Consumption-Based Asset Pricing," The Journal of Finance, 47, 511-552.
Hansen, L. (1982), ``Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, 50, 1029-1084.
Hansen, L. and Singleton, K. (1982), ``Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, 50, 1269-1286.
Lucas, R. (1978), ``Asset Prices in an Exchange Economy," Econometrica, 48, 1149-1168.
Pratt, J. (1964), ``Risk Aversion in the Small and in the Large," Econometrica, 32, 122-137.
SAS Institute Inc. (1993), SAS/ETS User's Guide, Version 6, Second Edition, Cary, NC: SAS Institute Inc.