I was wondering whether there is any test that allows to judge the statistical significance of the change in the variance ratios calculated with Lo/MacKinlay's (1988) test for different subperiods, for each span of differencing (q) on an individual basis and/or jointly.
ISSUE: Significant change
LOW-MCKINLAY VARIANCE RATIO: The ratio test is given by;
VR(q) = sigma2(q) / sigma2(1)
The question asks about significant change of VR(q)i, i.e. VR(q)1, VR(q)2, ...VR(q)T., not about VR(q) calculation itself but the changes of VR(q) in subsequent readings. There are at least 3 readings of observed values in this problem:
(i) ΔVR(q)i which is the change (VR(q)2 - Vr(q)1) or (VR(q)i+1 - VR(q)i) or the difference would just be denoted as d1, d2, ..., dT. Is the dispersion of di significant?
(ii) The actual value of VR(q) from each reading, i.e. VR(q)1, VR(q)2, ..., VR(q)T. The time interval for reading may not be equal. use each VR(q)i values to verify which instance is statistically significant.
(iii) Examine the rate of change: Ri = (VRi - VRi-1) / VRi-1.
In all three instance, one can use the standard score to verify statistical significant for each observation:
Z = (Xi - X^) / S
... where Xi = each observed value specified in (i), (ii) and (iii); X^ = mean; and S = standard deviation. Z is the critical value which may be compared to the Z(0.95) = 1.65; if Z(obs) > 1.65 then significant. This is one possible approach.
The variance-covariance of errors have to be sequentially estimated and tabulated for comparison and selection as I have done in some papers. This requires an algorithmic approach as I have proved. But of course that doesn't mean the data will converge to the selected bound or critical value as in Lo & Mackinlay because of financial market reasons. Please refer to some of my papers.