Use the data in NYSE to answer these questions.
$\begin{array}{l}{\text { (i) Estimate the model in equation }(12.47) \text { and obtain the squared OLS residuals. Find the average, }} \\ {\text { minimum, and maximum values of } \hat{u}_{t}^{2} \text { over the sample. }}\end{array}$
(ii) Use the squared OLS residuals to estimate the following model of heteroskedasticity:
$$\operatorname{Var}\left(u_{t} | \operatorname{return}_{t-1}, \text { return}_{t-2}, \ldots\right)=\operatorname{Var}\left(u_{t} | r e t u r n_{t-1}\right)=\delta_{0}+\delta_{1} r e t u r n_{t-1}+\delta_{2} r e t u r n_{t-1}^{2}$$
Report the estimated coefficients, the reported standard errors, the $R$ -squared, and the adjusted
$R$ -squared.
(iii) Sketch the conditional variance as a function of the lagged return $_{-1} .$ For what value of return $_{-1}$
is the variance the smallest, and what is the variance?
$\begin{array}{l}{\text { (iv) For predicting the dynamic variance, does the model in part (ii) produce any negative variance }} \\ {\text { estimates? }}\end{array}$
$\begin{array}{l}{\text { (iv) For predicting the dynamic variance, does the model in part (ii) produce any negative variance }} \\ {\text { estimates? }}\end{array}$
$\begin{array}{l}{\text { (v) Does the model in part (ii) seem to fit better or worse than the ARCH(1) model in }} \\ {\text { Example } 12.9 ? \text { Explain. }}\end{array}$
$\begin{array}{l}{\text { (vi) To the ARCH(1) regression in equation }(12.51), \text { add the second lag, } \hat{u}_{t-2}^{2} . \text { Does this lag seem }} \\ {\text { important? Does the ARCH(2) model fit better than the model in part (ii)? }}\end{array}$