This option computes the Ordinary Least Squares (OLS) estimates of together with the corresponding standard errors, t-ratios, and probability values. (See Sections 6.1.2 and
6.1.3). It also computes a number of summary statistics and diagnostic test statistics (with probability values) aimed at checking for possible deviations from the classical normal as- sumptions (A1 to A5). The summary statistics include R2, R2, Akaike information criterion (AIC), Schwartz Bayesian criterion (SBC), residual sum of squares, standard error of re- gression, and the maximized value of the log-likelihood function. The formulae used for the computation of these and other statistics are given in Sections 21.6and 21.7.
The diagnostic statistics included in the OLS regression results are for testing the fol- lowing hypotheses:
- Residual serial correlation.
- Functional form misspeci…cation.
- Normality of residuals.
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- Heteroscedasticity.
- Predictive failure.
- Structural stability.
For each of these hypotheses the program computes two test statistics: a Lagrange mul- tiplier (LM ), or score statistic, and an F statistic. The LM statistic is asymptotically distributed as a chi-square ( 2) variate. For a comprehensive review of the use of LM tests in econometrics, see Godfrey (1988). The F -statistic, also known in the literature as ‘LM F ’or ‘modi…ed LM ’statistic, is taken approximately to have the F distribution: seeHarvey (1981), p. 277. The LM and the F statistics have the same distribution asymptotically. But, on the basis of Monte Carlo results, Kiviet (1986) has shown that in small samples the F version is generally preferable to the LM version. In what follows we provide a brief account of these diagnostic tests. For further details of the econometric methods involved and the relevant references to the literature, see Section21.6.2.
6.6.1 Tests of residual serial correlation
The program provides the following tests of the non-autocorrelated error assumption, A3:
- Durbin-Watson test (Durbin and Watson (1950),Durbin and Watson (1951)).
- Durbin’s h-test (Durbin (1970))5.
- Lagrange multiplier (LM) tests6.
The program always supplies the DW statistic, but reports the h-statistic only when the regression equation is explicitly speci…ed to include a single, one-period lag of the dependent variable. The LM statistic is included in the diagnostic tests table, and is applicable to models with and without lagged dependent variables (Godfrey 1978b,1978c). It is applicable in testing the hypothesis that the disturbances, ut, are serially uncorrelated against the
alternative hypothesis that they are autocorrelated of order p (either as autoregressive or moving average processes). In the diagnostic tests table the following values are chosen for p:
p = 1 for undated, annual and daily data p = 2 for half-yearly data
p = 4 for quarterly data p = 12 for monthly data
Other values for p can be speci…ed using option 1 in the Hypothesis Testing Menu (see Section6.23).
5
See alsoGodfrey (1978a).
6For example, seeGodfrey (1978b),Godfrey (1978c),Breusch and Pagan (1980), andBreusch and Godfrey
(1981).
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6.6.2 Ramsey’s RESET test for functional form misspeci…cation
The RESET test (Ramsey 1969) reported in the diagnostic tests table refers to the simple case where only the square of …tted values (^yt2) are included in the extended regression of et = yt x0tb (or ^yt) on xt and ^yt2. A pth order RESET test can be carried out by using
Option 6 in the Hypothesis Testing Menu (Section 6.23), with ^y2t; ^yt3; : : : ; ^ypt speci…ed as additional variables. Notice that to carry out such a test, you …rst need to save the …tted values of the regression of y on X by means of Option 7 in the Display/Save Residuals and Fitted Values Menu (see Section6.21).
6.6.3 The normality test
This is the test proposed by Jarque and Bera (1980) for testing the normality assumption, A5, and is valid irrespective of whether or not the regression equation includes an intercept term.
6.6.4 Heteroscedasticity test
This is a simple test of the (unconditional) homoscedasticity assumption, A4, and provides an LM test of = 0 in the model
E u2t = 2t = 2+ x0t 2; t = 1; 2; :::; n
See Koenker (1981), where it is also shown that such a test is robust with respect to the non-normality of the disturbances.
6.6.5 Predictive failure test
This is the second test discussed in Chow (1960), and is applicable even if the number of available observations for the test is less than the number of unknown parameters. As shown in Pesaran, Smith, and Yeo (1985)the predictive failure test can also be used as a general speci…cation error test.
6.6.6 Chow’s test of the stability of regression coe¢ cients
This is the …rst test discussed inChow (1960), and tests the equality of regression coe¢ cients over two sample periods conditional on the equality of error variances. In the statistics literature this test is known as the analysis of covariance test: see Sche¤e (1959). The program computes this test if the number of observations available after the estimation period is greater than k, the number of regressors included in the model.
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6.6.7 Measures of leverage
In the classical linear regression model (6.2), the leverage (or the in‡uence) of points in the regression design is measured by the diagonal elements of the matrix7
A= X(X0X) 1X0= (aij) (6.16)
The relevance of the leverage measures, aii; i = 1; 2; :::; n in regression analysis has been
discussed in detail by Belsley, Kuh, and Welsch (1980) Chapter 2, andCook and Weisberg (1982) Chapter 2.
The program provides plots of aii; i = 1; 2; :::; n, and allows you to save them for subse-
quent analysis. In the plot of the leverage measures, the average value of aii, which is equal
to k=n, is also displayed.8
The leverage measures also provide important information on the extent of small sample bias that may be present in the heteroscedasticity-consistent estimators of the covariance matrix of ^OLS (Section 21.22). As shown by Chesher and Jewitt (1987), substantial down- ward bias can result in the heteroscedasticity-consistent estimators of the variance of the least squares estimators, if regression design contains points of high leverage.