Asymptotic properties of QMLE for periodic asymmetric strong and semi-strong GARCH models.

In this paper, we propose a natural extension of time-invariant coefficients threshold GARCH (TGARCH) processes to periodically time-varying coefficients (PTGARCH) one. So some theoretical probabilistic properties of such models are discussed, in particular, we establish firstly necessary and sufficient conditions which ensure the strict stationarity and ergodicity (in periodic sense) solution of PTGARCH. Secondary, we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) for estimating the unknown parameters of the model. More precisely, the strong consistency and the asymptotic normality of QMLE are studied in cases when the innovation process is an i.i.d (Strong case) and/or is not (Semi-strong case). The finite-sample properties of QMLE are illustrated by a Monte Carlo study. Our proposed model is applied to model the exchange rates of the Algerian Dinar against the U.S-dollar and the single European currency (Euro)

QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations

summary:This paper develops an asymptotic inference theory for bilinear $\left( BL\right)$ time series models with periodic coefficients $\left( PBL\text{ for short}\right)$. For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator $\left( QMLE\right)$ under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of $PBL$ model exists and is finite, under which the strong consistency and asymptotic normality of $QMLE$ for $PBL$ are proved. Moreover, we consider also the periodic $ARMA$ $\left( PARMA\right)$ models with $PBL$ innovations and we prove the consistency and the asymptotic normality of its $QMLE$

<article-title xmlns:mml="http://www.w3.org/1998/Math/MathML">The <inline-formula id="I1"><mml:math id="m1"><mml:msub><mml:mi mathvariant="double-struck">L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>−</mml:mo></mml:math></inline-formula> Structure of Su

The models of stochastic subordination, or random time indexing, has been recently applied to model financial returns Xtt≥0 exhibiting some characteristic periods of constant values for instance exchange rate. In reality, sharp and large variations for X(t) do occur. These sharp and large variations are linked to information arrivals and/or represent sudden events and hence we have a model with jumps. For this purpose, by substituting the usual deterministic time t as a subordinator Ttt≥0 in a stochastic process Xtt≥0 we obtain a new process XTtt≥0 whose stochastic time is dominated by the subordinator Ttt≥0. Therefore we propose in this paper an alternative approach based on a combination of the continuous-time bilinear (COBL) process subordinated by a Poisson process (that it is a Levy process) which permits us to introduce further randomness for the phenomena which exhibit either a speeded up or slowed down behavior. So, the main probabilistic properties of such models are studied and the explicit expression of the higher-order moments properties are given. Moreover, moments method (MM) is proposed as an estimation issue of the unknown parameters. Simulation studies confirm the theoretical findings and show that the MM method proposal can effectively reduce both the bias and the mean square error of parameter estimates

A note on the stability and causality of general time-dependent bilinear models

In this paper, sufficient conditions are given for the existence of a causal stable solution for general bilinear time series with time-dependent coefficients.Time-dependent bilinear models Markovian representation Volterra series Stability Causality