All of the models that have been discussed in the lessons and text have been I Tear in nature Y . = log of Y that is the model is linear in the param eters . so that there is ere is one parameter multiplied by each ariable in the model . For example , a structural model could be something like dy " - 1 " - - relative change in the exchange rate - By + B 2 2 + Boxes + Baka + 2 al - mean of dy or more compactly ! = XB tu . It was additionally assumed that u - N ( 0 0" ) as" - as One particular non - linear model in widespread usage in finance is known as an ARCH model ( ARCH stands for autoregressive conditionally heteroscedastic ) . To see why this class of Thus , X is the mean- adjusted relative change in the exchange rate . Now we can use X as a models is useful , recall that a typical structural mode . could be expressed by an equation of the measure of volatility . Being a squared quantity quantity , its value will be high in periods when there are form given above with ( 0 0 ) . The assumption of multiple regression that the variance big changes in the prices ces of financial assets and its value will be comparatively small when there of the errors is constant is known as homoscedasticity ( Le it is assumed that var ( 4 ) = 0 " ) . If are modest changes in the pri prices of financial assets Accepting X as a measure of volatility how do we know if it changes over time ? Suppose we consider the following model the variance of the errors is not constant , this would be known as heteroscedasticity . If the errors are heteroscedastic , but assumed homoscedastic , an implication would be that standard error Po + BY + estimates could be wrong . It is unlikely in the context of financial time series that the variance of the errors will be constant over time , and hence it makes sense to consider a model that does not This model postulates that volatility in the current period is related to its value in the previous assume that the variance is constant and which describes how the variance of the errors evolves period plus a white noise error term . If By is positive , it suggests that if volatility was high in the Under the ARCH model , the autocorrelation in volatility is modeled by allowing the previous period it will continue to be high in the current period , indicating volatility clustering conditional variance of the error term of to depend on the immediately pre If Bi is zero , then there is no volatility clustering . The statistical significance of the estimated Ba pious value of the can be judged by the usual I test squared error The estimation of the model provided the following do + age x -0 043 + 0.23036 * The above model is known as an ARCH ! ) , since the conditional variance depends on only 1 = ( 7.71 ) one lagged squared error . Notice that ( 2 ) is only a partial model , since nothing has been said jet R = = 0.0531 d = 10933 about the conditional mean Under ARCH the conditional mean equation ( which describes how the dependent variable , M . varies over time ) could take almost any form that the researcher where It is as defined befor wishes . One example of a full model would be Given ap value for By of 601 what do conclude about the significance of the model ? By + Bike + Bits + Boxat H N ( 0 , Of ) The ARCH model is an important tool for the analyst as it provides a ides a method of measuring time varying volatility In risk management , volatility is often us often used as a measure of risk Let us illustrate this with an exchange rate example . ( See Basic Econometrics Fifth Edition ) by Damodar N . Gujarati and Dawn C Chapter 22 for a more complete discussion ) Let You - U.SUK exchange rate
