- A Test statistics A.1 Test statistic for TVV-model specification In order to specify gt we not only test constancy but even specify the number of transitions before estimating the GARCH component of the model. Amado and TeraÌsvirta (2013) showed that maximum likelihood estimators of the corresponding time-varying variance (TVV) model, assuming that there is no conditional heteroskedasticity, are consistent and asymptotically normal. This forms the base for constructing Lagrange multiplier type tests for testing r against r + 1 transitions. For notational simplicity consider testing one transition against two. Omitting the subscript i for simplicity, the TVV model is (9) with ht = 1, and gt = δ0 + δ1G1(t/T, γ1, c1) + δ2G2(t/T, γ2, c2), γi > 0, i = 1, 2.
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- B Simulations of test statistics B.1 Tests of GARCH equations The test for slow moving baseline volatility has a statistic whose distribution is sensitive to the high frequency, GARCH, volatility. For this reason, one cannot use the asymptotic distribution, rather the distribution must be generated via simulation. Further, Silvennoinen and TeraÌsvirta (2016) showed that the size of the test is distorted if the GARCH parameterisation deviates from the true one. For this reason a few alternative approaches to estimate the GARCH parameters, and especially the persistence, are investigated. It should be noted that estimating GARCH without taking the nonstationarity into account will yield overestimated persistence, thereby impacting the test statistic distribution and thus rendering the test outcomes unreliable. These estimates are given in Table 3.
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- Overall, it is clear that using simply the GARCH estimates from the entire sample to calibrate the test statistic distribution for the specification of the deterministic component of the volatility is not recommended. For comparison, Table 3 reports also the GARCH estimates from a TV-GARCH model where the TV specification has been completed. The estimated persistence is higher than the ones obtained from the calm period or rolling window variance targeting method, however, as discussed in Silvennoinen and TeraÌsvirta (2016), underestimation of persistence has less severe impact on the performance of the TV specification test than overestimation does.
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- Silvennoinen, A. and TeraÌsvirta, T.: 2021, Consistency and asymptotic normality of maximum likelihood estimators of the multiplicative time-varying smooth transition correlation GARCH model, Econometrics and Statistics (in press) .
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- The simulation uses 2000 observations on a bivariate TVGARCH model parameterised as ht = 0.10 + 0.05ε2 tâ1/gtâ1 + 0.85htâ1, gt = 1 + 3(1 + exp{âe3(t/T â 0.5)})â1. These are coupled with a CCC model with Ï = 0.5, and then with an STCC model parameterised as Ï(1) = 0.3, Ï(2) = 0.7, Gt = (1 + exp{âe2.5(t/T â 0.5)})â1. The noise terms are iid standard normal. Two estimation procedures were used, a two-step and a multi-step one. 1st step The individual TVGARCH models are estimated, assuming the series are uncorrelated. 2nd step Estimate the correlation model conditional on the volatility model estimates from the previous step. Then, estimate the TVGARCH models conditional on the correlation estimates.
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Tse, Y. K. and Tsui, K. C.: 2002, A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations, Journal of Business and Economic Statistics 20, 351â362. Online Appendix This Appedix contains additional material to the paper. Section A provides details of the TVV-model specification, the MTV-GARCH model evaluation, the test of constant correlations, and finally the test for an additional transition in the correlations. The simulations studies in Section B explore aspects of the specification and evaluation of the GARCH equations, and the size and sensitivity of the test of constant correlations. Proof of Lemma 1 is presented in Section C. Section D presents the details of maximisation by parts. Estimated deterministic components of the Four Banksâ transition equations are presented in Section E. Finally, Sections F and G provide tabulated results and figures related to the simulation studies.