Large sample inference based on multiple observations from nonlinear autoregressive processes (Q1315406)

Let \(\{Y_ t(i) \mid t = 1,\dots,n\}\) be a time series for individual \(i\), \(i = 1,\dots,m\). The nonlinear autoregressive model \(Y_ t(i) = H(Y_{t - 1}(i); \theta_ i) + \varepsilon_ t(i)\) is considered, where the \(\theta_ i\)'s are \(p \times 1\) vectors of unknown parameters, \(H(y;\theta)\) is a known function and \(\varepsilon_ t(i)\) is a sequence of iid random variables for each \(i\). In section 2 of the paper the case ``\(n \to \infty\), \(m\) fixed'' is dealt with. The Wald statistic based on an asymptotically optimal one-step maximum likelihood estimator is proposed for testing the hypothesis \(H_ 0 : \theta_ 1 = \dots \theta_ m\). The null and non-null limiting distributions are deduced from the LAN property. Section 3 studies a special threshold autoregressive model of order 1 for the case ``\(n\) fixed, \(m \to \infty\)''. This section is mainly concerned with the derivation of the least squares estimator of the parameter vector and its limiting distribution. Section 4 discusses briefly some related topics.