Inference for nonstandard MA and noncausal VAR models

Assuming that {(Un, Vn)} is a sequence of cadlag processes converging in distribution to (U, V) in the Skorohod topology, conditions are given under which {&int &int fn( beta, u, v)dUndVn} converges weakly to &int &int f(beta, x, y) dUdV in the space C( R ), where fn(beta, u, v) is a sequence of "smooth" functions converging to f( beta, u, v). Integrals of this form arise as the objective function for inference about a parameter beta in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of beta which optimizes a particular objective function.One important application of this kind of convergence result is to derive the asymptotic theory for the estimators for the unit root and near unit root cases of a moving average model. Previous studies of the MA(1) unit root problem rely on the special autocovariance structure of the MA(1) process, in which case, the eigenvalues and eigenvectors of the covariance matrix of the data vector have known analytical forms. In this thesis, we take a different approach to first consider the joint likelihood by including an augmented initial value as a parameter and then recover the exact likelihood by integrating out the initial value. This approach bypasses the difficulty of computing an explicit decomposition of the covariance matrix and can be used to study unit root behavior in moving averages beyond first order. The asymptotics of the generalized likelihood ratio (GLR) statistic for testing unit roots are also studied. The GLR test has operating characteristics that are competitive with the locally best invariant unbiased (LBIU) test of Tanaka for sonic local alternatives and dominates for all other alternatives.We also consider a general vector autoregressive (VAR) process generated from non-Gaussian noise. In particular, the VAR is allowed to have possibly noncausal components. Such noncausal VAR models appear to have applications in empirical economic research. Assuming a non-Gaussian distribution for the noise, we show how to compute an approximation to the likelihood function. Under suitable conditions, it is shown that the maximum likelihood estimator (MLE) of the vector of AR parameters is asymptotically normal. The estimation procedure is illustrated with a simulation study for a VAR(1) process and with two real data examples.