The graphical output next presents four charts

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The graphical output next presents four charts showing the recursive t ratio coefficients for each coefficient. Look out here for substantial shifts in the magnitude and significance of t statistics over the recursion. These ones are clearly unstable. The next chart shows the “1-step recursive residuals”, that is the difference between actual and fitted values for each set of coefficient estimates calculated recursively along the sample. These are shown with a 95% confidence interval, given by +/- 2 SER (remembering that it is the recursive estimates of the SER being used here). Points outside the 95% confidence interval are either outliers or are associated with parameter changes (although this does not help very much as we really wish to know which of these is the case!). Hendry argues that in this example: “Further, the 1-step residuals show major outliers around 1974” and that “the 1-step residuals show an increase in regression equation variance after 1974” (which is itself a form of parameter instability). He also comments on the fact that the 1-step Chow test amply reflects this change. The last set of graphs are all variants of the Chow (1960) type 2 F test statistics, and have been scaled in such a way that critical values at each point in the sample are equal to unity. Details of the formulae used to calculate the test statistics are given on page 233 of the PcGive manual. Of these Chow test charts, the graph labelled “1 up CHOW” is the 1 step Chow test. These are one step (one period ahead) forecast F tests. This appears to show an “outlier” around 1974.
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24 The last two charts are also for particular forms of recursive CHOW test statistics, the terminology for which is potentially confusing. PcGive labels these as the Ndwn (that is N down), and Nup (that is N up) Chow tests. We could also think of them in terms of forwards vs. backwards versions. Consider the N down (or forwards) graph first. Hendry calls this the Break point F tests , or N down- Step Chow tests. Here the chart shows a sequence of Chow forecast tests running down from N = T-M+1 to N=1 (that is, the forecast horizon is decreasing and hence the use of the term down ). The sequence begins by using observations 1 to M-1 to predict the remaining observations from M to T (i.e. N = T-M+1 forecasts). Then one observation is added to the estimation period, so that observations 1 to M are used to predict the remaining observations from M+1 to T. This continues until observations 1 to T-1 are used to forecast the T th period. So at each point in time in the sample, the chart shows the value of the Chow forecast F test for that date against the final period. The values shown have been scaled by the appropriate critical value (we have chosen 1% here). This implies that the horizontal line at unity becomes the critical value to use for making inference about stability.
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