Constrained spline regression in the presence of AR(p) errors

Extracting the trend from the pattern of observations is always difficult, especially when the trend is obscured by correlated errors. Often, prior knowledge of the trend does not include a parametric family, and instead the valid assumptions are vague, such as 'smooth' or 'monotone increasing'. Incorrectly specifying the trend as some simple parametric form can lead to overestimation of the correlation. The proposed method uses spline regression with shape constraints, such as monotonicity or convexity, for estimation and inference in the presence of stationary AR(p) errors. Standard criteria for selection of penalty parameter, such as Akaike information criterion (AIC), cross-validation and generalised cross-validation, have been shown to behave badly when the errors are correlated and in the absence of shape constraints. In this article, correlation structure and penalty parameter are selected simultaneously using a correlation-adjusted AIC. The asymptotic properties of unpenalised spline regression in the presence of correlation are investigated. It is proved that even if the estimation of the correlation is inconsistent, the corresponding projection estimation of the regression function can still be consistent and have the optimal asymptotic rate, under appropriate conditions. The constrained spline fit attains the convergence rate of unconstrained spline fit in the presence of AR(p) errors. Simulation results show that the constrained estimator typically behaves better than the unconstrained version if the true trend satisfies the constraints.