Identification of Mixed Causal-Noncausal Models in Finite Samples

Mixed causal–noncausal autoregressions with exogenous regressors

“…• Step 1: Determining total autoregressive order. In a first step, we identify the so-called pseudo-causal model (see, e.g., Hecq et al, 2016) by estimating ARDL models as in Equation (5) by ordinary least squares (OLS) and identify by information criteria such as Akaike (AIC), Bayesian (BIC), and Hannan-Quinn (HQ) the autoregressive order p that makes the residuals free of serial correlation. 7 Additionally, we find the lowest lag orders of the exogenous regressors to consider in the estimation of all models.…”

“…A conventional estimator of Σ is based on the Hessian of the log-likelihood, but nonlinear optimization of this function often involves complicated numerical methods, which are relatively unstable. Similar to Hecq et al (2016), we provide an alternative way to approximate standard errors based on the asymptotic distribution of the Student t MLE estimated parameters in the finite variance framework. If > 2, the MLE is √ n consistent and asymptotically normal.…”

“…The usefulness of mixed causal-noncausal autoregressive (MAR) models-that is, time series specifications with both lag and lead components-can be explained by at least three findings. First, Gouriéroux and Zakoïan (2016), Fries and Zakoian (2019), and Hecq, Lieb, and Telg (2016) demonstrate how noncausal autoregressive terms can generate dynamics such as speculative bubbles and asymmetric cycles that instead could only be generated using complex nonlinear models. Second, Lanne, Luoto, and Saikkonen (2012) and Lanne, Nyberg, and Saarinen, (2012) show that noncausal models might improve forecast performances.…”

“…The null of normality should be tested on the residuals of the pseudo-causal model to justify the non-Gaussian ML estimator and to look for signs of noncausality. The model selection procedure for MAR models can be found inHecq et al (2016).…”

Selecting between causal and noncausal models with quantile autoregressions

“…3 Once a distribution or a group of distributions is chosen, the parameters in π(L)ϕ(L −1 ) can be estimated. Assuming for instance a non-standardized t-distribution for the innovation process, the parameters of mixed causal-noncausal autoregressive models of the form (1) can be consistently estimated by the approximate maximum likelihood (AML) method (Hecq, Lieb, and Telg 2016). Let (ε 1 ,…,ε T ) be a sequence of i.i.d.…”