The Breush-Pagan test of heteroscedasticity assumes that it is explained by some set of variables z that may be included in or excluded from the original regression. It is an LM test, with test statistic ESS/2, where ESS is the explained sum of squares from the auxiliary regression of normalized squared residuals16 on the variables in z. It is asymptotically distributed as chi-squared with p degrees of freedom, where p is the number variables in z - that is, the number of variables thought to influence the heteroscedasticity.

The Goldfeld-Quandt test assumes that the /th residual ef has variance o2xj for some known variable x. The data are rearranged so that the first observation is the one with the smallest value for x, the second observation has the second smallest value of x, and so on. Following this, the sample is divided into three parts: under the alternative hypothesis of heteroscedasticity the residuals in the first part will have smaller variance than those in the last part. This is depicted in Figure A.8. The test statistic is the ratio, RSS,/RSS2, of the sum of squared residuals from the first part of the data, to the sum of squared residuals from the last part of the data. It is F-distributed (under the null hypothesis) with - k,n2 - degrees of freedom, where is the number of observations in the first part and n2 is the number of observations in the last part. Excluding observations from the test in the middle part of the reordered data decreases the probability of failing to reject a false null hypothesis (§A.2.1). But if too many observations are omitted, the power of the test to reject false null hypotheses is diminished. A rule of thumb is to exclude the middle third of the reordered data.

A.3.3 Generalized Least Squares

If there is autocorrelation and/or heteroscedasticity in the errors then ( ) a2I. In this case OLS will still give consistent estimators (§A.1.3), but this is only useful when sample sizes are large. What should be done if autocorrelation or heteroscedasticity is present in model residuals when the data set is relatively small? OLS estimators will still be unbiased if the explanatory variables are not correlated with the errors. But they will be inefficient, so parameter estimates will not be robust to small changes in the data.

When disturbances are not spherical, efficient estimators are provided by the GLS estimator

bGLS = (X1X)-1X0-1y, (A.3.1)

where denotes the covariance matrix of the errors. The assumption V(z) = a2I has been replaced by ( ) = a2fi to reflect the fact that errors are either autocorrelated or heteroscedastic, or both. The GLS estimator covariance matrix is

,- :: . :n :hi> case is dividing by the residual variance estimate, Te2/T.

(boLs) = cAXirX)-1,

and estimates of this covariance matrix may be used in the hypothesis tests of linear restrictions outlined in §A.2.2.

A few lines of algebra verify that applying formula (A.3.1) directly on the data X and is actually equivalent to applying OLS to the transformed data X* = TX and * = , where T is the Cholesky matrix of H~l (so T = C~, where CC - . For example, if there is heteroscedasticity and of = o2Wf for some known variable W, the transformation matrix is T - diag(Wr. . ., W~l). So the mth row of data in X and is divided by Wm to get the transformed data X* and y*, and then OLS is applied to these data. This form of GLS is called weighted least squares.

The precise form of fi depends on the nature of the errors process. For example, when there is heteroscedasticity but no autocorrelation, fi will be a diagonal matrix, diag(a2, . . ., of). The hypothesis of homoscedasticity, viz.

2 2 2

°1 = °2 = • • • =On,

will have been rejected using one or other of the tests described in §A.3.2 above. If there is autocorrelation but no heteroscedasticity, fi will take a form that is determined by the type of autocorrelation. For example, if the autocorrelation is AR(1), so

e, = pe, ! + v„

where p is the first-order autocorrelation coefficient in the residuals, then

Although the form of fi can be specified, depending on the type of autocorrelation or heteroscedasticity that is assumed, the actual values in fi are of course unknown. A consistent estimate of fi will normally be used in (A.3.1), to give feasible GLS estimators.17 Although feasible GLS estimators may be more efficient than OLS when autocorrelation or heteroscedasticity is present, feasible GLS does not necessarily give the most efficient estimators in small samples.

Another case where GLS is often applied is to systems of seemingly unrelated regression equations (SURE). In general simultaneous models the explanatory variables of some equations become the dependent variable in another

17Cochrane and Orcutt (1949) describe one such consistent estimator when there is first-order autocorrelation in the residuals.

equation. In SURE models equations are related only through their disturbance terms. The arbitrage pricing model of §8.1.2 falls into this category, its general formulation being a system of equations

- X,p - j: (7=1, n), (A.3.2)

where y. denotes the returns to asset j, X, is a matrix of risk factor returns relevant for asset j, p. is the vector of risk factor sensitivities and e;- is the disturbance term specific to asset j.

For domestic portfolios it is common to assume that risk factors are the same for all assets, so X; = X for all j. In this case OLS is an appropriate estimation procedure. But for large international portfolios risk factors will be country-specific. In this case the more general formulation (A.3.2) should be employed.

Consider now a situation where y- denotes the returns to the portfolio in country j, Xj is a matrix of risk factor returns relevant for country j, p, is the vector of risk factor sensitivities and e, is the error process that is specific to country j. The whole system may be written:

= + ,

where is the stacked vector of data on the n different country stock returns, X is the block-diagonal matrix of country risk factor returns, p is the stacked vector of factor sensitivities according to country and to risk factor within that country, and £ is the vector of disturbances.

Applying GLS to this system will provide unbiased and efficient estimators. The assumption of country-specific risks being contemporaneously correlated becomes

( $) = a,yI,

where I is the T x T identity matrix, or in terms of a Kronecker product

£(££) = V = S cg> I,

where £ = (a,,). Then the inverse matrix V-1 = S"1 cg> I should be used to calculate GLS estimators as

bGLS = (Xv-xrxv-y.

Of course in practice the cross-market covariances a,-,- will not be known and