RDP 8701: The Australian Demand Function for Money: Another Look at Stability 4. Stability Tests: Some Results

Glenn Stevens, Susan Thorp and John Anderson

January 1987

Figures 1 through 3 show the results of the CUSUM and CUSUMSQ procedures, for estimated versions of the three models where these tests could be conducted, from both the forward and the backward recursions, over both the original and extended sample periods. Five per cent confidence intervals are plotted. Table 6 gives the results of the Quandt procedure to determine the most likely break point, and the results of the Chow test and Watt's Wald test conducted at those points, as well as the homogeneity statistic. Results for each model are discussed in turn.

(a) Sharp and Volker (1977)

Over the original sample period, equation SV-2 exhibited some tendency to instability. Both CUSUM and CUSUMSQ statistics remain well within the confidence intervals, for the forward recursions. But in the backward recursions, the CUSUMSQ test points to evidence of structural change some time in the mid 1960s.

Extending the sample period to 1986(1), in the forward recursions there is again evidence of instability, in the mid 1960s, from the CUSUMSQ. In addition, the CUSUM indicates the possibility of further structural change in the last year or two of the sample. The backward recursions point to the possibility of parameter changes in the first half of the 1970s, the mid 1960s and even the mid 1950s.

FIGURE 1
Figure 1
FIGURE 2
Figure 2
FIGURE 3
Figure 3
Table 6
Results of Stability Tests
Equation no:
 		 Sample Period Minimum Quandt1 Chow Test2
Statistic		 Critical 1% Value 5% Wald Test3
Statistic		 Critical 1% Value 5% Homogeneity Statistic4 Critical 1% Value 5%
SV-2 52(3)–72(3) 57(4) 7.91 3.07 2.18 39.09 18.48 14.07 42.99 1.77 2.24
SV-3 52(3)–86(1) 71(3) 4.32 1.82 1.53 28.21 18.48 14.07 119.05 1.795 2.265
PO-2 66(3)–79(2) 72(3) 3.23 2.91 2.11 26.77 20.10 15.51 19.75 2.26 3.15
PO-3 69(3)–86(1) 82(3) 2.95 2.50 1.91 26.44 20.10 15.51 10.95 1.91 2.6
FR-2 67(3)–83(2) 75(3)6 1.47 2.47 1.88 13.97 16.81 12.59 n.a.    
FR-3 67(3)–85(1) 75(3)6 1.27 2.38 1.84 12.69 16.81 12.59 n.a.    
PV-2 67(4)–78(2) 74(3) 3.53 3.13 2.23 33.61 20.1 15.51 0.70 2.32 3.29
PV-3 67(4)–85(4) 74(4) 2.43 2.63 1.96 22.49 20.1 15.51 1.49 2.11 2.85
  1. Point at which Quandt's likelihood ratio statistic is minimised.
  2. Under the null hypothesis, the distribution of the Chow statistic is Fk,T−2k, where k is the number of regressors, and T the sample size.
  3. Under the null hypothesis, the asymptotic distribution of the Wald statistic is Inline Equation, where k is the number of regressors.
  4. Non-overlapping sample size is 20 observations, unless otherwise stated.
  5. Non-overlapping sample size is 40 observations.
  6. The Quandt ratio technique could not be applied. 75(3) was arbitrarily chosen as the break point.

The Chow test applied at 1957(4) in the original sample rejects the null hypothesis of stability, as does the Wald test, quite convincingly. Both tests also reject stability in the extended sample.

The homogeneity statistic overwhelmingly rejects the null hypothesis of no instability at the 1 per cent level of significance, for a range of non-overlapping sample sizes, both in the original sample and the extended sample.

In the case of Sharpe and Volker's distributed lag models, we found that the large number of dummy variables tended to induce matrix singularity in the sub-samples, so most of the stability tests could not be conducted.

(b) Porter (1979)

Over the original sample period, the CUSUMSQ statistics point to possible instability in the early to mid 1970's. This is supported by the Chow and Wald tests conducted with the sample break at 1972(3), and the homogeneity test, in all of which the null hypothesis is convincingly rejected.

Over the sample 1969–86, there is again some evidence from the CUSUMSQ plots of instability in the first half of the 1970's. In addition to this, it also seems possible that there was some shift during the first half of the 1980's. The point at which Quandt's likelihood ratio is minimised is 1982(3). The Wald test conducted on this assumption rejects the null hypothesis, and although the Chow test statistic is not significant at the 5 per cent level, it is not really low enough to allow one to entertain the hypothesis of stability with great confidence. The homogeneity test over this extended sample also rejects stability.

(c) Preeland (1984)

The presence in the specification of two bank deposit interest rates suggests looking for some effects the deregulation of those rates in December 1980. Unfortunately, the three interest rates in the specification are highly collinear, making for difficulty in inverting the X'X matrix for a small number of observations. Consequently, the CUSUM, CUSUMSQ and homogeneity tests could not be conducted, and the Quandt likelihood ratio procedure could not be applied.

We are left then with the Chow and Wald tests. 1975(3) was arbitrarily chosen as the break point, for both the original and extended samples. Although the Chow test at this point cannot reject the null hypothesis, the Wald test rejects it at the 5 per cent level of significance.

(d) Pagan and Volker (1981)

The original authors contended, on the basis of the CUSUM plot, that “the evidence is not obviously indicative of Instability”. This appears reasonable on the basis of the plots reported in their paper. However, the differences between the data set they used and the one we used, even over the same period, appear to be such that confidence in that conclusion is diminished somewhat. Yet again, the mid 1970s appear as a problem period. Over the longer sample, the equation appears to perform a little better in the CUSUM and CUSUMSQ tests, staying inside the confidence intervals.

On the other hand, the Chow and Wald tests for a break at the minimum Quandt point in the second half of 1974 are not particularly supportive of stability. The most surprising outlier in this set of test results is the homogeneity statistic, which cannot reject the null.

So, while Pagan and Volker's conclusion against instability, cannot be held unswervingly, there is probably enough variation in the results that it cannot be dismissed out of hand.

(e) Summary

On balance, the conclusion we draw from the above results is that in the case of M3, it is difficult to accept the proposition that equations have not suffered from instability, as detected by the tests employed here. The verdict on M1 is perhaps less clear.

The recurrence of the period from the end of 1971 through to the middle of 1975 as the period in which structural change is most likely to have occurred is notable. But it is also likely that there has been instability at other times, including the 1980's. The next section looks more closely at this period, which is one of considerable recent interest for monetary theory and policy.