Strong Consistency of the Conditional Least Squares Estimator for a Nonstationary Process. Example of the Garch Model

2000 Mathematics Subject Classification: Primary: 62M10, 62J02, 62F12, 62M05, 62P05, 62P10; secondary: 60G46, 60F15.

We consider the Conditional Least Squares Estimator (CLSE) of a unknown parameter θ0 ∈ Rp of the conditional expectation of a real stochastic process {Yn} having finite first two conditional moments E(Yn|Fn-1)< ∞, E(Yn2 | F n-1)< ∞ at each time n, where E(Yn|Fn-1) is Lipschitz and may be nonlinear in θ0 and {Fn} is an increasing sequence of σ-algebra. We generalize to this class of processes the necessary and sufficient condition got for the strong consistency of the CLSE of θ0 in the particular linear deterministic (or linear stochastic if p = 1) model E(Yn|Fn-1) = θT0Wn. We illustrate this theoretical result with examples, mainly a nonstationary GARCH (1,1) model.