0.05 0.1 0.15 0.2

0.05

-0.05

(b) -0.1

Figure A.7 (a) Structural break in data; (b) residuals from regression without dummy.

A.3.1 Autocorrelation

Common causes of autocorrelation in residuals are:

5* Structural breaks. Structural breaks are very common in financial markets, since a market crash or move to a different regime may signify a structural change in the data generation process. If one attempts to fit a linear model to data with structural breaks, residuals may be autocorrelated (Figure A.7). The solution is to use a dummy variable in the regression, whose precise form depends on the type of structural break. In the example of Figure A.7, the residual autocorrelation is removed if one adds another explanatory variable to the model: a dummy variable taking the value 0 before the structural break and the value 1 at and after the structural break. The single model becomes equivalent to two models with different constant terms and/or different slope parameters, depending on how this dummy is incorporated (§A.4.4).

5* Omitted variables. If the dependent variable has characteristics that are not explained by any of the included explanatory variables, those characteristics will show up in the residuals.15

5* Over- or under-differencing the data. The usual method for transforming financial price, rate or yield data into stationary series is first differencing (§11.1.4). If this is not done, the data may have a very high degree of positive autocorrelation, and if there is no similar explanatory variable to balance the model, the autocorrelation will show up in residuals. On the other hand, if first differencing is performed when it is not necessary, the data will have high negative autocorrelation. In either case, it is by no means certain that the model could explain such structural autocorrelation. It is a problem with the basic data that should already have been identified when plotting the data prior to building the model.

The Durbln-Watson test for autocorrelation is standard output in many statistical packages. The test statistic is

d= Y.(et-et ,JlY.e].

Small values of d indicate positive autocorrelation in the OLS residuals, and large values indicate negative autocorrelation. In large samples d 2(1 - r), where r is the first-order autocorrelation coefficient estimate of the residuals (§11.3). So the approximate range for d is [0, 4], and d has an expected value of approximately 2 under the null hypothesis of zero autocorrelation. No exact distribution is available, but upper and lower limits dv and dL for the significance levels of d are found in Durbin-Watson tables. The decision rule is: if d < dL then reject the null in favour of the alternative of positive autocorrelation; if d > du then do not reject the null; and if dL < d < dv the test is inconclusive.

Although positive autocorrelation is the natural hypothesis to test (being the prime indication of omitted explanatory variables), negative autocorrelation is also a concern for the application of OLS. Durbin-Watson tests of negative autocorrelation proceed in a similar way, replacing d by 4 - d in the decision rule above.

The basic Durbin-Watson test has limited applicability; in particular, the explanatory variables cannot be stochastic. There is an extension to the case where lagged dependent variables are included in the regression (Durbins h-test). A more general framework for testing autocorrelation, in residuals from a regression model or in more general time series, is explained in §11.3.2.

If OLS residuals are found to be autocorrelated, the first step is to respecify the linear model to try to remove the autocorrelation. Assuming the data on the

The >:r-c:jral break dumrm just described is an example of an omitted variable causing autocorrelation. If one : :-e .-trijr.jior. \unables had the same type of structural break as the dependent variable, the residuals would r.o: Sc j-tocorrelated But if there is nothing else in the model to explain that sort of variation, it has to be

0.35

-0.2

Figure A.8 Reordered data for the Goldfeld-Quandt test.

dependent variable are not over- or under-differenced, this would normally involve including more explanatory variables on the right-hand side. If none of the steps that are taken can remove the autocorrelation in OLS residuals then an alternative estimation method should be considered (§A.3.3).

A.3.2 Unconditional Heteroscedasticity

If the dependent variable has properties that are not present in any of the included explanatory variables then potential explanatory variables may have been omitted from the model. When relevant explanatory variables are omitted from the model the only term that can model those effects is the error term. Thus residuals will pick up the properties of the dependent variable that have not been captured because relevant variables have been omitted. If the omitted variables are heteroscedastic so also will be the residuals. Because of this, many tests for heteroscedasticity are based on an auxiliary regression of the squared OLS residuals on other (included or excluded) variables that are thought to be linked to the heteroscedasticity. The following are examples of such tests.

Whites test of the null hypothesis of homoscedasticity is an LM test that is performed by regressing the squared residuals on all the squares and cross products of the explanatory variables (White, 1980, 1984). The LM test statistic is TR2, where is the number of data points used in this regression and R2 is the squared correlation (ESS/TSS). It is asymptotically distributed as chi-squared with p degrees of freedom, where p is the number of explanatory variables in the auxiliary regression, including the constant. Although very general, Whites test may have rather low power. For example, a rejection of the null hypothesis of homoscedasticity may be due to an incorrect linear functional form. The test also helps little in identifying the variables that cause heteroscedasticity.