OLS and ML method for Estimating ARCH Models

In financial time series modeling, the model consist of two equation. One of them is mean equation and other is variance equation which are defined following equation.

RY_t=c+u_t   (1)

  

Equation (1) and (2) shows mean and variance equation, respectively. For estimating these equations we can use OLS and ML methods. But there is some limitation about the estimated coefficients. So we must be careful chosing the estimate method. There are two limitation for estimating these equations.

1           1. Both equation must be estimated siumultaneaously

             2. All coeffiecients of the variance equation must be positive

A drawback of the least squares approach to estimate an ARCH model is that there is no guarantee that all the estimated ARCH coefficients will be positive.  Another reason the least squares method is not appropriate for estimating the ARCH model is that we need to estimate both the mean function and the variance function simultaneously.

One advantage of the ML method is that we can estimate the mean and variance functions simultaneously, instead of separated as under OLS. Another advantage of ML method is that it guaranteed that all estimated ARCH coefficients will be positive.[1]

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[1] Source: Damodor Gujarati Econometrics by Example p244-245