Sir please explain the part
Time varying volatility ARCH Model
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Whenever we estimate variances, we tend to assume that their value will not change over time. Basically, we find the unconditional value of the variance generally i.e., we do not impose any condition on our calculations when estimating variance. Say, currently we are at a particular stage of the business cycle. Our estimation of volatility is generally the one that does not regard any stage in the business cycle. Note that business cycle is not the only factor that influences the current variance.
So, basically we take an averaged value of variance that holds in the long term. However, when estimating variance in the short term we need to look into the current conditions that affect volatility i.e., we need to find the conditional volatility to make decisions.
It is a known fact that financial asset returns exhibit volatility clustering. This means that volatility of asset returns tends to cluster around previous values and so there are marked periods of high and low volatility. This means that current volatility estimate is conditional on the previous volatility figure. In order to measure such conditional volatility, time varying volatility ARCH models are useful.
The general formula of this model states that Variance of period t is a function of the variance of period (t-1) and an unexpected return component i.e., a shock which is conditional on the information in period (t-1). This means time t’s variance is dependent on time (t-1)’s variance and a shock component. The level of dependence on these two components is exhibited by the figures alpha and beta of the equation. Alpha is the coefficient of (t-1)’s variance and beta is the coefficient on the shock component. If alpha is greater than beta it can obviously be concluded that current variance will be closer to the previous period’s variance. If beta is greater, then there would obviously be a significant difference between current and previous variance.
In simple words, think of this model as being analogous to a regression model where we are finding out how strongly current variance (the dependent variable) is determined by previous variance (an independent variable) and a shock component (another independent variable). Using this model we can understand what should be the current period’s variance estimate conditional to the current scenario. If shock component i.e., the unexpected return component is not very high, logically thinking current variance should not markedly differ from the previous variance. Current variance is different from previous variance only because of the current period’s return figure. This figure is not a part of previous variance’s calculation but it is a part of current variance’s calculation. This means that if current period’s return is not unexpectedly different, current variance will be close to previous variance.
This is essentially what time varying volatility ARCH model tries to achieve. Hope you have got an idea of how to interpret the model. Also, please check the formula and the ancillary write-up on it in the core.