Time-varying influence of household debt on inequality in United Kingdom

Data source: UK Office for National Statistics.

The data is downloadable from: https://​discover.​ukdataservice.​ac.​uk/​series/​?​sn=​200016 and https://​discover.​ukdataservice.​ac.​uk/​series/​?​sn=​2000028.

We would like to thank Professor Haroon Mumtaz for kindly sharing the inequality data.

Complete details of the unit root tests are available upon request from the authors.

Interestingly, the constant parameter-based Granger causality test could not detect causality running from GI1 to GD even at the 10% level of significance, though ExpW, MeanW, Nyblom, and SupLR tests all overwhelmingly rejected the null of no time-varying predictability at the highest level of significance. Complete details of these results are available upon request from the authors.

The recursive-rolling procedure involves intensive recursive calculations of the relevant test statistic (Wald test for Granger causality), for all subsamples in a backward expanding sample sequence in which the final observation of all samples is the (current) observation of interest. Inference regarding the presence of Granger causality for this observation rely on the supremum taken over the values of all the test statistics in the entire recursion. As the observation of interest moves forward through the sample, the subsamples in which the recursive calculations are performed accordingly move forward and the whole sequence of calculations rolls ahead.

While the test of Rossi and Wang (2019) is not necessarily completely comparable with the recursive, rolling, and recursive-rolling approaches of Shi et al. (2018, 2020), the recursive version is probably the closest to a full-fledged time-varying approach, as it is based on an expanding window, whereby one observation at a time is added, and new estimations are conducted. Note, in the case of the rollin-window, the window size remains the same, with the window being continuously moved forward by leaving behind one observation. In other words, barring the initial window-size which is used for the first-set of estimation, in some sense with estimation performed at each point of time under the recursive approach, similarities can be drawn with the time-varying model, which considers each data point as a different state, and captures the underlying heterogeneity in the data. This could be a reason behind the similar inferences drawn under the recursive version of Shi et al.’s (2018, 2020) tests with that of the results derived from Rossi and Wang (2019), but this of course requires more deeper analysis in terms of the statistical properties of the tests, which indeed is beyond the scope of this paper.