LAD Estimator For The Linear Regression Model With ARCH Errors

Nonlinear time series widely appeared in the economical fields. Autoregressive conditionally heteroscedastic(ARCH) model has been a representation for the nonlinear time series. It can well distribute the heavy-tailed phenomenon and volatility cluster phenomenon. So there are both economic meanings and statistics meanings to study this work.Linear regression model with heavy-tailed errors model which was proposed by Engel in 1982 is a kind of model existed in the economic field. With the assump-tion of innovations following normal distribution, Engel(1982) used the the maximum likelihood estimation method to estimate the parameters of the model and attached the asymptotic normality for the LAD estimation. In real studies, the innovations can not be specified. Weiss(1986) proposed the quasi-maximum likelihood estimation(QMLE) method. Under the conditions that the eighth moment of the error is finite, he pro-posed the asymptotic normality for the QMLE. However, empirical evidence indicates that LADE method have more accurate estimation than least square estimation for the heavy-tailed errors. In economic fields, data may have heavy tails (without fourth mo-ment). In such case, this paper choose the least absolute deviation (LAD) estimation method to study the linear regression model with heavy-tailed ARCH errors.First, we give the LAD estimation for the linear regression model with heavy-tailed ARCH errors and prove the asymptotic normality property for the LAD estima-tion. By the asymptotic normality properties, we attach the confidence regions for the LAD estimation. Second, we use the same method to discuss the ARCH(∞) model and attach the asymptotic normality properties of the LAD estimation. The confidence regions are constructed.Numerical studies is one of most important part in this paper. We use the Matlab software to do the simulations. In order to get the minimum for the object function, we use the grid search method to get the LAD estimator. At the same time we calculate the estimator and the true parameter. We can get the confidence regions by the theorem. Through the simulations we can get the coverage probabilities of confidence regions to test the accuracy of the conclusions. Also, we plot the area of the LADE coverage regions.For further illustrating the real meanings of the model, we show a real application to this research. We use the monthly average price for the crude oil and petroleum from US Energy Information Administration to show the relationship between crude oil and petroleum. We also test the accuracy of the model. We use LAD method and QML method to fit the linear regression model with ARCH errors. Though the analysis of the errors, we get the conclusion that LAD estimation method have more accuracy than the QML method.