Least squares estimator for regression models with some deterministic time varying parameters (Q1915122)

Models with time varying parameters are usually taken to be stochastic and used in applied works of adaptive control, for example. Is the relevance of a regression model with deterministically changing parameters doubtful? Suppose we use a simple regression model \(y_n= bu_{n-1}+ \varepsilon_n\) instead of a true model with a trend evolution \(y_n= (b+ 0.001n) u_{n-1}+ \varepsilon_n\) in an applied problem of control. For small values of \(n\), the error in the model is not serious; but what happens if the evolution is assumed to continue indefinitely? It seems important to detect and identify quickly the trend coefficient. At least for this reason this type of model seems to be of practical importance. To the best of the authors' knowledge, there are few works about almost sure convergence of the least squares estimator of deterministic time varying parameters in stochastic regression models. Here we study the case where the evolution of the parameter is given by \[ \theta_n= \beta_0 \varphi_{n,0}+ \beta_1 \varphi_{n,1}+ \cdots+ \beta_r \varphi_{n,r}, \] where \((\varphi_{n,i} )_{0\leq i\leq r}\) is a given deterministic functional basis time varying and \(\beta_i\) are the parameters to be estimated. We consider two kinds of evolution, an explosive (polynomial) evolution, \(\theta_n= \beta_0+ \beta_1 n+\cdots+ \beta_r n^r\), and a stable (sinusoidal) evolution, \(\theta_n= A\sin (n\omega+ \varphi)\). We prove strong consistency, and establish the rate of convergence of the least squares estimates.