A COMPANION TO Theoretical Econometrics

Hypothesis Testing

Much has been written on the problem of testing for serial correlation, particu­larly in the disturbances of the linear regression model. Given the nonexperimental nature of almost all economic data and also the strong potential for disturbances to be autocorrelated in economic time series applications, it is extremely import­ant to test for autocorrelation in this context.

3.1 The Durbin-Watson test

The von Neumann (1941, 1942) ratio is an important test of the independence of successive Gaussian time series observations, with unknown but constant mean. Its test statistic is of the form

nX (yt - Vt-iY

where у is the sample mean. Hart (1942a, 1942b) tabulated the distribution and critical values of n under the null hypothesis of independent Gaussian yts.

The most well known test for autocorrelation in regression disturbances is the Durbin-Watson (DW) test. Durbin and Watson (1950, 1951, 1971) considered the problem of testing H0 : p = 0 in the linear regression model (3.12) with stationary AR(1) disturbances (3.13) and Gaussian errors, i. e. et ~ IN(0, a2). The Durbin - Watson test statistic is of the form

n n

d = X (et - et-1)2 X e2

t=2 t=1

where e is the OLS residual vector from (3.12). Unfortunately the null distribu­tion of d1 is a function of the X matrix through the projection matrix M = In - X(X'X)-1X'. This meant that a single set of critical values could not be tabulated as was the case for the von Neumann ratio.

Durbin and Watson (1951) overcame this problem by tabulating bounds for the critical values. When the calculated value of the test statistic is between the two bounds, the test is inconclusive. A number of approximations to the DW critical values have been suggested, see King (1987). There are also extensive alternative tables of bounds, see Savin and White (1977), Farebrother (1980) and King (1981). These days computer methods such as simulation or Imhof's (1961) algorithm allow p-values for the test to be calculated as a matter of routine.

The Durbin-Watson test has a number of remarkable small sample power properties. The test is approximately uniformly most powerful invariant (UMPI) when the column space of X is spanned by k of the eigenvectors of the tridiagonal matrix,

and approximately locally best invariant (LBI) against AR(1) disturbances. The exact LBI test can be obtained by adding the first and last OLS residual squared to the numerator of the DW statistic. This is known as the modified DW test. The modified DW test is also LBI against MA(1) errors and is uniformly LBI against ARMA(1, 1) disturbances and sums of independent ARMA(1, 1) error compon­ents. Consequently, the DW test is approximately LBI or approximately uni­formly LBI against these disturbance processes (see King and Evans, 1988). In summary, the literature suggests that the DW test generally has good power against any form of serial correlation provided there is a strong first-order com­ponent present.

3.2 Some other tests

The DW test has low power when testing against the simple AR(4) disturbance process given by (3.14) with Gaussian errors. Wallis (1972) and Vinod (1973) separately developed the fourth-order analogue to the DW test which has the test statistic

n n

d4 = X (ЄІ - Є^-4)2 X Є2.

t=5 t=1

This has been a popular test for use with quarterly time series data and, as Vinod (1973) demonstrated, can be easily generalized to test against any simple AR( p) disturbance process.

Another weakness of the DW test against AR(1) disturbances is the potential for its power to decline to zero as p1 tends to one. This was first noted by Tillman (1975) and explained convincingly by Bartels (1992). It means that the DW test can be at its weakest (in terms of power) just when it is needed most. The class of point optimal tests (King, 1985) provide a solution to this problem. They allow the researcher to fix a value of p1 at which power is optimized. For some X matrices (similar to those for the DW test), this test is approximately UMPI, so optimizing power at a particular p1 value does not mean the test has low power for other p1 values.

The Lagrange multiplier (LM) test is a popular test for AR(p), MA(q) or ARMA(p, q) regression disturbances. A strength of the DW test is that it is one­sided in the sense that it can be applied to test for either positive or negative autocorrelation by use of the appropriate tail of the distribution of d1 under H0. The LM test is a general two-sided test and therefore cannot be expected to be as powerful against specific forms of autocorrelation. For further discussion, see Godfrey (1988) and the references therein.

Of course any of the classical tests, such as the likelihood ratio, Wald, and LM tests, can be applied to these testing problems. They can also be used to test one form of disturbance process against a more general form. Again there is mount­ing evidence to suggest that in terms of accuracy (more accurate sizes and better centered power curves), it is better to construct these tests using the marginal likelihood rather than the full likelihood function (see Rahman and King, 1998; Laskar and King, 1998).

With respect to non-nested testing of disturbance processes, there is a growing literature on testing MA(1) disturbances against AR(1) disturbances and vice versa. See for example King and McAleer (1987), Godfrey and Tremayne (1988) and Silvapulle and King (1991).