Asymptotic properties for the first-order bilinear time series model

For the first-order bilinear time series model where {et} is a sequence of independent normal random variables with mean 0 and variance σ2, the asymptotic distribution of the sample autocorrelation function is obtained and shown to follow a normal distribution. The variance of the asymptotic distribution is of a complicated form and hence a bootstrap estimate of the variance is proposed for large sample inference. This result can be used to distinguish between different bilinear models. Finally, we obtain moment estimators of the parameters and derive their asymptotic distribution.