Springer Texts in Business and Economics

Applications of Recursive Residuals

Recursive residuals have been used in several important applications:

(1) Harvey (1976) used these recursive residuals to give an alternative proof of the fact that Chow’s post-sample predictive test has an F-distribution. Recall, from Chapter 7, that when the second sample nt had fewer than k observations, Chow’s test becomes

where e'e = RSS from the total sample (n1 + nt = T observations), and e'1e1 = RSS from the first n1 observations. Recursive residuals can be computed for t = k + 1, ...,n1, and continued on for the extra nt observations. From (8.36) we have

Therefore,

£ 1 wt/nt

TZ k+1 wt/(n 1 — k)

But the wt’s are ~IIN(0, a2) under the null, therefore the F-statistic in (8.38) is a ratio of two independent chi-squared variables, each divided by the appropriate degrees of freedom. Hence, F ~ F(nt, n1 — k) under the null, see Chapter 2.

(2) Harvey and Phillips (1974) used recursive residuals to test the null hypothesis of homoskedasticity. If the alternative hypothesis is that a2 varies with Xj, the proposed test is as follows:

1) Order the data according to Xj and choose a base of at least k observations from among the central observations.

3) Under the null hypothesis, it follows that

F = w2W2/wlwi ~ Fm, m (8.41)

Harvey and Phillips suggest setting m at approximately (n/3) provided n > 3k. This test has the advantage over the Goldfeld-Quandt test in that if one wanted to test whether varies with some other variable Xs, one could simply regroup the existing recursive residuals according to low and high values of Xs and compute (8.41) afresh, whereas the Goldfeld-Quandt test would require the computation of two new regressions.

(3) Phillips and Harvey (1974) suggest using the recursive residuals to test the null hy­pothesis of no serial correlation using a modified von Neuman ratio:

This is the ratio of the mean-square successive difference to the variance. It is arithmetically closely related to the DW statistic, but given that w ~ N(0,a2IT-k) one has an exact test available and no inconclusive regions. Phillips and Harvey (1974) provide tabulations of the significance points. If the sample size is large, a satisfactory approximation is obtained from a normal distribution with mean 2 and variance 4/(T — k).

(4) Harvey and Collier (1977) suggest a test for functional misspecification based on recursive residuals. This is based on the fact that w N(0,a2IT-k). Therefore,

w/(sw/^T — k) ~ tT-k-l (8.43)

where w = ^2T=k+l wt/(T — k) and s2w = Y)'T=k+l(wt — w)2/(T — k — 1). Suppose that the true functional form relating y to a single explanatory variable X is concave (convex) and the data are ordered by X. A simple linear regression is estimated by regressing y on X. The recursive residuals would be expected to be mainly negative (positive) and the computed t-statistic will be large in absolute value. When there are multiple X’s, one could carry out this test based on any single explanatory variable. Since several specification errors might have a self-cancelling effect on the recursive residuals, this test is not likely to be very effective in multivariate situations. Wu (1993) suggested performing this test using the following augmented regression:

y = X@ + zy + v (8.44)

where z = CiT-k is one additional regressor with C defined in (8.34) and iT-k denoting a vector of ones of dimension T — k. In fact, the F-statistic for testing Ho; y = 0 turns out to be the square of the Harvey and Collier (1977) t-statistic given in (8.43), see problem 15.

Alternatively, a Sign test may be used to test the null hypothesis of no functional misspecifi - cation. Under the null hypothesis, the expected number of positive recursive residuals is equal to (T — k)/2. A critical region may therefore be constructed from the binomial distribution. However, Harvey and Collier (1977) suggest that the Sign test tends to lack power compared with the t-test described in (8.43). Nevertheless, it is very simple and it may be more robust to non-normality.

(5) Brown, Durbin and Evans (1975) used recursive residuals to test for structural change over time. The null hypothesis is

where et is the vector of coefficients in period t and a2 is the disturbance variance for that period. The authors suggest a pair of tests. The first is the CUSUM test which computes

Wr — Yn=k+1 Wt/sw for r — k + 1,...,T (8.46)

where s2w is an estimate of the variance of the wt’s, given below (8.43). Wr is a cumulative sum and should be plotted against r. Under the null, E(Wr) — 0. But, if there is a structural break, Wr will tend to diverge from the horizontal line. The authors suggest checking whether Wr cross a pair of straight lines (see Figure 8.1) which pass through the points {k, ±a/T — k}and {T, ±3a/T — k} where a depends upon the chosen significance level a. For example, a — 0.850, 0.948, and 1.143 for a — 10%, 5%, and 1% levels, respectively.

If the coefficients are not constant, there may be a tendency for a disproportionate number of recursive residuals to have the same sign and to push Wr across the boundary. The second test is the cumulative sum of squares (CUSUMSQ) which is based on plotting

Wr* — E r=k+i w2/ E T=k+i w2 for t — k + 1,...,T (8%7)

against r. Under the null, E(Wr*) — (r — k)/(T — k) which varies from 0 for r — k to 1 for r — T. The significance of the departure of Wr) from its expected value is assessed by whether W. Г crosses a pair of lines parallel to E(Wr*) at a distance co above and below this line. Brown, Durbin and Evans (1975) provide values of c0 for various sample sizes T and levels of significance a.

The CUSUM and CUSUMSQ should be regarded as data analytic techniques; i. e., the value of the plots lie in the information to be gained simply by inspecting them. The plots contain more information than can be summarized in a single test statistic. The significance lines constructed are, to paraphrase the authors, best regarded as ‘yardsticks’ against which to assess the observed plots rather than as formal tests of significance. See Brown et al. (1975) for various examples. Note that the CUSUM and CUSUMSQ are quite general tests for structural change in that they do not require a prior determination of where the structural break takes place. If this is known, the Chow-test will be more powerful. But, if this break is not known, the CUSUM and CUSUMSQ are more appropriate.

Example 2: Table 8.5 reproduces the consumption-income data, over the period 1959-2007, taken from the Economic Report of the President. In addition, the recursive residuals are computed as in (8.30) and exhibited in column 5, starting with 1961 and ending in 2007. The CUSUM given by Wr in (8.46) is plotted against r in Figure 8.2. The CUSUM crosses the upper 5% line in 1998, showing structural instability in the latter years. This was done using EViews 6.

The post-sample predictive test for 1998, can be obtained from (8.38) by computing the RSS from 1950-1997 and comparing it with the RSS from 1950-2007. The observed F-statistic is 5.748 which is distributed as F(10,37). Using EViews, one clicks on stability diagnostics and then selects Chow forecast test. You will be prompted to enter the break point period which

Specification: CONSUM C Y

Test predictions for observations from 1998 to 2007

in this case is 1998. EViews gives the back up regression which is not shown here, and also performs a likelihood ratio test, see Table 8.4.

The reader can verify that the same F-statistic can be obtained from (8.40) using the recursive residuals in Table 8.5. In fact,

F = (£wt2/10)/(£ 1997)6! wf/37) = 5.748