Processes possessing this property are called compound
autoregressive processes (CaR) (see Darolles, Gourieroux, and Jasiak, 2006); they are the discrete time counterparts of the (continuous time) affine processes, which is widely used in finance and insurance (see, e.g., Duffie, Filipovic, and Schachermayer, 2003; Schrager, 2006).
By using (4) we generated 100 multivariate
autoregressive processes with known causality structures.
By taking into account time-lagged soil water content, time-lagged soil temperature,
autoregressive processes and seasonality, the model provides more-detailed information on the nature of the relationship between [N.sub.2]O and the environmental drivers and the effects of temporal resolution on [N.sub.2]O emissions, obtained from fitting the model with weekly or monthly data.
Among the topics are gradient-based algorithms with applications to signal-recovery problems, graphical models of
autoregressive processes, convex analysis for non-negative blind source separation with applications in imaging, robust broadband adaptive beamforming using convex optimization, and cooperative distributed multi- agent optimization.
To examine the nature of various nonlinear estimates, we generated a large number of series from second-order
autoregressive processes. The best nonlinear model was compared with the best linear model using in-sample and out-of-sample criteria.
Likelihood ratio statistics for
autoregressive processes. Econometrica 1981;49:1057-72.
9B and 9C are not a pair of first-order
autoregressive processes due to the dependence of [[Phi].sub.2](t) and [[Phi].sub.3](t) on other variables.
A second approach is to build dynamics into the unobserved factors themselves by modeling them as
autoregressive processes.
Johansen's tests for cointegration (1988, 1991, Johansen and Juselius (1990)) are a logical multivariate extension of Dickey-Fuller unit root tests for
autoregressive processes. The Johansen tests use the canonical correlation of residuals from a reparameterized model to estimate the space of cointegration vectors and test the dimensions of the space.
Estimation of the above relationship requires specification of the maximum number of lags, [Rho] and [Gamma] (or [Gamma]' and [Rho]' in reverse causality), for the
autoregressive processes. As noted by Hsiao (1981), artificial selection of the length of the lags may bias the estimates and induce inefficiency in the inferential procedure.