- Use the data in the ps6.xls workbook on the class web page to do these problems. Use the worksheet page

- Use the data in the ps6.xls workbook on the class web page to do these problems.

Use the worksheet page labeled CAPM for this question. It contains observations on monthly returns for a set of stocks: AT&T, GE, GAP, $\mathrm{HD}$ (Home Depot), IBM and GM; a proxy for the overall market return: the return for a value-weighted portfolio of NYSE, Nasdaq, and Amex stocks; and a proxy for the risk-free rate given by the 30-day T-Bill rate. All these returns have been adjusted for inflation so they are what's known as real returns. These returns cover the time period from January 1982 to December $2001 .$

(12 points) First, choose your favorite stock among those in the worksheet and your favorite 5 year period from this time span. Use the data from only the 60 months in your favorite time span for parts (a), (b), and (c) of this question. Next, construct what are called excess returns for your stock by subtracting the risk-free rate from that for your stock. In other words, if your stock is GE construct a column containing
$r_{t}^{\text {excess } G E}=r_{t}^{G E}-r_{t}^{\text {risk free }}$
Do the same for the market return forming $r_{t}^{\text {excess Market }}=r_{t}^{\text {Market }}-$ $r_{t}^{\text {risk }}$ free . Now estimate a measure of how your stock comoves with the market return called beta in the Capital Asset Pricing Model by estimating $\beta$ in the following regression:
$r_{t}^{\text {excess YourStock }}=\alpha+\beta r_{t}^{\text {excess } \text { Market }}+u_{t}$
Report parameter estimates and $95 \%$ confidence intervals for $\alpha$ and $\beta$ along with the R-squared from this regression. Also construct the residuals, $\hat{u}_{t}$, from this regression for use below.
(12 points) What do your estimates imply about your stock's average excess return when the market excess return goes up or down $1 \%$ ?
(12 points) Plot your residuals versus their lagged values; plot $\hat{u}_{t}$ versus $\hat{u}_{t-1}$ for the 59 pairs in your data (you have only 59 since there is no lagged observation for the first data point in your sample). Can you see any evidence of serial correlation in these residuals? What does this imply about the reliability of the $95 \%$ confidence intervals for $\beta$ that you have estimated?
(12 points) Now choose another 60 month time span between January 1982 and December 2001 that does not overlap with your favorite time span used in parts (a), (b), and (c). Construct excess returns and re-estimate the regression $\left({ }^{*}\right)$ of excess returns for your favorite stock upon excess market returns. Report a point estimate and $95 \%$ confidence interval for $\beta$ for this new time span. How do they compare to those from part (a) when you used your favorite time span?
(12 points) Assume that the data from your two time spans are independent and investigate whether the two betas you have estimated in parts (a) and (d) are equal by constructing a $95 \%$ confidence interval for their difference. (Hint: the stats facts referred to seventh week regarding linear combinations of normal random variables and variances of linear combinations may be useful) Does beta appear to have changed for your favorite stock?

2. (60 points) The worksheet page labeled $X$ contains 450 observations on a time series $X_{t}$. Estimate an AR1, AR3, and AR5 model for this time series using the first 300 data points. Report point estimates, $95 \%$ confidence intervals, and R-squared for each of these three specifications. Test each model with an out-of-sample prediction exercise by using each estimated model to predict each of the last 100 data points given the prior 1,3 , or 5 realizations of $X$ (for the AR1, AR3, and AR5, respectively). Report sums of squared prediction errors for each of the three models. Which model performs the best in this prediction test? How do differences in R-squared across models compare to differences in their forecasting performance?

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Deliverable: Word Document