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145

H0: a0 = 1, a2 = a3 = 0, a5 = a6 + a7, a8 = 2

in the model Y - a0 + + . . . + cigA + 8. There are 10 parameters in the model so - 10 in (A.2.2), and the unrestricted residual sum of squares RSSrj is obtained by first estimating the model with no restrictions on the parameters. There are 5 equalities in the null, so q - 5, and the restricted model is

Y = 1 + a, Xx + a4X4 + (a6 + a7)X5 + a6X6 + a1X1 + 2X% + agX9 + e

[ Y- 1 - 2 %] = a{X{+ a4X4 + a6(X5 + X6) + av(T5 + X7) + agXg + e.

To estimate the restricted model a new dependent variable [Y- 1 - 2X%] is regressed on five explanatory variables: Xu X4, (X5 + X6), (X5 + X7) and Xg. The residual sum of squares from this model is the RSSR.

For the statistic defined by (A.2.2) to be F-distributed it is necessary to assume that the error process is normally distributed. Otherwise even the asymptotic normality of the OLS estimators is not sufficient to invoke the F-distribution of (A.2.2). When errors are not normal the Wald statistic qF, where Fis given by (A.2.2) and q is the number of restrictions, can be used in large samples (§A.2.5). In fact the F-distribution is a finite-sample approximation to the chi-squared distribution of Wald statistics, just as the -distribution is a finite-sample approximation to the standard normal.

A.2.4 The Analysis of Variance

The analysis of variance (ANOVA) describes the decomposition of the total variance of the dependent variable (as measured by the total sum of squares, TSS) into two components, the explained sum of squares, ESS, and residual sums of squares, RSS. In ANOVA the F-test for the null hypothesis that all parameters except the constant are zero is applied automatically. This is the goodness-of-fit test of the model. The restricted model is just a constant, so RSSR is just TSS and the numerator of (A.2.2) becomes ESS. So the F-statistic for goodness of fit is simply measuring the same thing as the squared correlation coefficient R2 = ESS/TSS.

The ANOVA table and some basic diagnostic statistics for the Eletrobras CAPM that has been estimated in §A.2.2 are shown below (the output is from Excel):

Regression Statistics

R Square

0.800796

Adjusted R Square

0.800542

Standard Error

0.017024

Observations



ANOVA

Degrees of freedom

Sum of squares

Significance F

Regression

0.9122

3145.517

1.9E-276

Residual

0.226916

Total

1.139116

The explained sum of squares is 0.9122, the residual sum of squares is 0.226916 and the total sum of squares is 1.139116. So the F-statistic for goodness of fit is 0.9122/(0.226916/783) = 0.9122/0.00029 = 3145.517 which is very highly significant (1% F, 7S3 = 6.67). Dividing the ESS by the TSS gives the R1 = 0.9122/1.139116 = 0.800796.13 Very few diagnostics are standard Excel output. It does report the standard error of the equation (0.017024 in the example above) and the square of this gives the estimated residual variance s2. More advanced model diagnostic tests that are based on the residuals from the regression follow the procedures outlined in Appendix 3.

A.2.5 Wald, Lagrange Multiplier and Likelihood Ratio Tests

Linear restrictions on the model parameters can be tested in a number of ways other than the simple tests described above. Of course, F- and /-tests have many practical advantages: they are particularly simple to use in the testing down procedure for model specification (§8.4), and their framework is flexible enough to admit quite general linear hypothesis. However, they do have limited applicability and they are not always as powerful as the more general Wald, Lagrange multiplier (LM) or likelihood ratio (LR) tests. The test statistics for Wald, LM and LR hypothesis tests on the parameters of a normal regression model are as follows:14

Test

Statistic

Wald

(T - fc)(RSSR - RSSu)/RSSu

(T- k + <7)(RSSR - RSSu)/RSSR

(1 RSSR - In RSSrj)

13In this simple model with only one explanatory variable (plus a constant) the R2 and the adjusted R2, which is a modification of R2 to take degrees of freedom into account, are roughly the same.

14The unrestricted sum of squares. RSSLl, and the restricted sum of squares, RSSR, were defined in §A.2.3.



Under the null hypothesis these tests are all asymptotically distributed as chi-squared with q degrees of freedom. Wald, LM and LR tests are quite general. They can be applied to parameter testing in statistical distributions, non-linear hypotheses in linear models, or testing restrictions in general covariance matrices (see Greene, 1998; Griffiths et al., 1993). They do not have to assume that errors are normally distributed. Actually the form of LR test given above does assume normal errors, but the general LR test statistic can be defined in terms of any likelihood function.

Appendix 3 Residual Analysis

The simple goodness-of-fit test described in §A.2.4 goes some way towards indicating how well a linear model has been specified. But a full diagnostic analysis of residuals gives much more information about the possible omission of relevant explanatory variables, and the properties of coefficient estimators.

In an ideal world, including sufficient explanatory variables would improve a linear specification to the point that the classical assumption ( ) = a2I can be upheld. In that case unbiased and efficient parameter estimators may be obtained by OLS. This appendix examines the residual diagnostic tests which should indicate whether this assumption is violated. Two questions will be addressed:

5* Is K(e) diagonal? If not the errors will be autocorrelated. 5* Is F(e) homoscedastic? In other words, does the error process have a constant variance?

These two assumptions may be tested using a wide variety of residual diagnostic tests. A brief account of the most common test procedures is given, with a description of the common causes and remedies if autocorrelation or heteroscedasticity is found in the residuals. If residuals are autocorrelated or heteroscedastic an alternative method of estimation such as generalized least squares (GLS) could be used to obtain unbiased and efficient estimators in small samples, and this is described in §A.3.3. However, since OLS is consistent under fairly general conditions, it may be applied without problems to the large samples that many of the financial applications of linear models employ. That is not to say that residual diagnostics are unnecessary for models that use large amounts of data, because the characteristics of the OLS residuals will still reveal a considerable amount about the model specification.



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