I estimated a linear bivariate regression model by OLS method.
I did the ARCH effect test. And there is the presence of ARCH effect in residuals.
How can I deal with the presence of ARCH effect while estimating the bivariate linear model?
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Estimate a GARCH model where the conditional mean equation ($\mu_t$ below) corresponds to your bivariate regression. This way you will estimate the regression coefficients efficiently and will have appropriate standard errors that account for the ARCH pattern.
\begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \gamma_0 + \gamma_1 x_{t}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $D$ is some probability distribution with zero mean and unit variance.
(Of course, you do not have to worry about ARCH in cross-sectional data; it only makes sense for time series.)
answered Jul 22 at 15:39
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$\begingroup$ Thank you for your answer. Can you please explain this more in mathematical way? And please suggest any source to understand that? Because my knowledge is very limited. Thank you. $\endgroup$– 1190Commented Jul 22 at 19:32
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$\begingroup$ @1190, here you go. Aside from the fact that the conditional mean of $y$ depends on $x$, this is just the standard GARCH(r,s) model. Regarding a source, I do not remember one that covers this kind of model specifically, sorry. $\endgroup$ Commented Jul 24 at 15:30