Consider the following two models (refer to Pesaran and Deaton 1978):
One particular way of choosing between the two hypotheses is the J-test, suggested by Davidson and Mackinnon (1981). Estimate the comprehensive model
This example illustrates a way to test the specification of a consumption function. The extension of this example to other non-nested models and multiple hypotheses is straightforward.c, income by y, and wealth by w, you postulate
An alternative is to estimate an equation containing lagged consumption as an explanatory variable. This can be thought of as a variant of Duesenberry's (1949) relative income hypothesis. The model can be postulated as
As seen above, the H1 and H2 models are non-nested.
The comprehensive models
The data set consists of seasonally adjusted quarterly time series of real 1958 prices, consumers' expenditure on non-durable goods and personal disposable income. These were collected from the Survey of Current Business and are used by Pesaran and Deaton (1978).
data spec; input yr qr c y w; c_lag=lag(c); date = yyq( yr, qr ); format date yyqc.; datalines; 1954 2 253 270 0 1954 3 257 274 17 1954 4 262 279 34 1955 1 268 282 51 ... ; run;
Before the J-test can be performed, the fitted values from the two models must be calculated. The AUTOREG procedure can fit several different regressions with the inclusion of multiple model statements. In this example, two model statements are used, corresponding to the two hypotheses. For each MODEL statement an output data set is created with the OUT= option. These data sets contain the predicted values CHAT1 and CHAT2, as specified by the P= option. The DATA step merges the original data set with the two predicted data sets for use in the J-test.
proc autoreg data=spec ; model c = y w; /* model for H1 */ output out=model1 p=chat1; model c = y c_lag; /* model for H2 */ output out=model2 p=chat2; run; data spec2; set spec ; set model1; set model2;
Because of the limits of the J-test, it is often wise to test twice for the significance of the mixing parameter by reversing the roles of the hypotheses. The AUTOREG procedure estimates the two comprehensive models. The coefficients of CHAT1 and CHAT2 correspond to a test of the significance of the mixing parameter, . To reject a model requires that the mixing parameter be insignificantly different from 0.
proc autoreg data=spec2; model c = y c_lag chat1; model c = y w chat2; run;
The first comprehensive model yields the following parameter estimates:
The second comprehensive model yields the following parameter estimates:
Cox, D.R. (1961), ``Tests of Separate Families of Hypotheses," Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, Berkeley: University of California Press.
Davidson and Mackinnon (1981), ``Several Tests for Model Specification in the Presence of Alternative Hypotheses," Econometrica, 49, 781-793.
Duesenberry, J.S. (1949), Income, Saving, and the Theory of Consumer Behavior, Cambridge, MA: Harvard University Press.
Greene, W.H. (1993), Econometric Analysis, Second Edition, New York: Macmillan Publishing Company.
McAleer M. (1995), ``The Significance of Testing Empirical Non-nested Models," Journal of Econometrics, 67, 150-171.
Pesaran, M.H. (1974), ``On the General Problem of Model Selection," Review of Economic Studies, 41, 153-171.
Pesaran, M.H. and Deaton, A.S. (1978), ``Testing Non-nested Nonlinear Regression Models," Econometrica, 46, 677-694.
SAS Institute Inc. (1993), SAS/ETS Users's Guide, Version 6, Second Edition, Cary, NC: SAS Institute Inc.
Stone, J.R.E. (1973), ``Personal Spending and Saving in Post-War Britain," in Economic Structure and Development: Essays in Honour of Jan Tinbergen, eds. H.C. Bos, H. Linnemann, and P. De Wolff, Amsterdam: North Holland, 79-98.