Model Selection and Post Selection to Improve the Estimation of the ARCH Model

Abstract

The Autoregressive Conditionally Heteroscedastic (ARCH) model is useful for handling volatilities in economical time series phenomena that ARIMA models are unable to handle. The ARCH model has been adopted in many applications that contain time series data such as financial market prices, options, commodity prices and the oil industry. In this paper, we propose an improved post-selection estimation strategy. We investigated and developed some asymptotic properties of the suggested strategies and compared with a benchmark estimator. Furthermore, we conducted a Monte Carlo simulation study to reappraise the relative characteristics of the listed estimators. Our numerical results corroborate with the analytical work of the study. We applied the proposed methods on the S&P500 stock market daily closing prices index to illustrate the usefulness of the developed methodologies.

Liu-type pretest and shrinkage estimation for the conditional autoregressive model

“…We will refer to the estimator in β1 LU in (15) as the full model estimator of β 1 . The Liu estimator of the sub-model in (11) is defined as follows:…”

“…where β * 1 as any of the proposed estimators, and β1 is the MLE of the model in (11). Also, the asymptotic covariance matrix (AC) of β * 1 is defined as:…”

“…Al-Momani, M. and Dawod B.A. [ 11 ] used the idea of pretest, and the shrinkage estimation for the Autoregressive Conditionally Heteroscedastic (ARCH) model. They discovered that the positive shrinkage estimator outperformed the restricted, pretest, and shrinkage estimators regardless of the accuracy of the restriction provided by the linear hypothesis to check whether some of the coefficients of the ARCH model’s parameter vector are not significant is true or not.…”