Abstract: This paper deals with the statistical inference of an au toregressive conditional heteroscedasticity (ARCH) model under restriction. We g ave a criteria function to compute a least squares estimation for the nonnegativ e parameters (α₀,α₁,α₂,…,α_q) of the ARCH model, and showed the strong consistency of the estimation. By discuss ing the exact expression and the asymptotic normality of the estimation under ordered restriction (α₁≥α₂≥…≥α_q), we obtained the form of the test statist ical quantity, then solved the testing problem with the parameter space under or dered restriction.

Key words: least squares estimator, strong consistency, asymptoti c normality property, ordered restriction