Fitting a pth Order Parametric Generalized Linear Autoregressive Multiplicative Error Model

A Minimum Distance Lack-of-Fit Test in a Markovian Multiplicative Error Model

“…Let (1), and sup z≥o |G n (z) − G(z)| = o p (1) , the continuity of B implies that analogous to Lemma 2.2, that conditionally, given the original sample, in Skorokhod space and uniform metric. Hence This fact is useful in showing that Δ * n ∶= n 1∕2 (̂ * n −̂ n ) = O p * (1) , in probability, which in turn is needed for deriving the asymptotic distribution of T * n .…”

“…Since V * n is a vector of the sums of martingale difference arrays satisfying (4.6), by extending the arguments in the proof of Theorem 5.4.1 of [15] to a triangular array setup, one verifies that (4.4), (B.3.) and (4.7) imply ‖n 1∕2 (̂ * n −̂ n )‖ = O p * (1), and that ‖n 1∕2 (̂ * n −̂ n ) −t * n ‖ = o p * (1), in probability. By the positive definiteness of G * , we clearly have t * n = (G * ) −1 V * n .…”

“…The assumptions (C.1) and (C.2) are used to show that the defining dispersion M n ( ) is AULQ (asymptotically uniformly locally quadratic) in n 1∕2 ( − ) for ∈ N n (b) , for every 0 < b < ∞ , while assumption (C.3) is used to show that ‖n 1∕2 (̂ n − )‖ = O p (1).…”

“…2.1, while adopting the 'Warp-Speed' Monte Carlo method of [10] to reduce the computational burden. More precisely, as formally proved by [10], since the test statistic satisfies (4.1) below, it is sufficient to generate only one bootstrap sample in each Monte Carlo replication, say r = 1, … , R , and then evaluate the test against the empirical distribution of the R bootstrap statistics T * (1) n , … , T * (R) n . To this end, at any given level , 0 < < 1 , the empirical rejection rate of the null hypothesis, say n,R , is computed as where T (r) n denotes the test statistic at the rth Monte Carlo replication, r = 1, … , R. Note that, in the standard Monte Carlo bootstrap, at each Monte Carlo replication, one generates K bootstrap samples and compute the -level bootstrap critical value satisfying (2.6), say z * (K,r) , r = 1, … , R , and then compute the empirical rejection rate of the null hypothesis as Clearly one can compute n,R much more efficiently than computing (K) n,R , because in the computation of n,R the bootstrap statistic is only computed R times.…”

P. C. Mahalanobis in the Context of Current Econometrics Research*

“…The proposed measure is based on pair-wise cross-section correlations and is therefore related to the Mahalanobis D 2 statistic; the use of averaging, bootstrap and spatial thinking in this paper also bears the legacy of Mahalanobis, which we discuss later. Balakrishna et al (2019) develop omnibus tests of a parametric linear autoregressive time series model with multiplicative errors. As discussed above, a common use of Mahalanobis distance and related divergence measures is in testing a parametric null hypothesis against an omnibus alternative, and this provides a nice contrast with the current approach developed in this paper.…”