Reference no: EM132168764

Problem 1: Sampling Frequency and Portfolios

Get the 11 stock monthly and daily returns. Make sure to download the daily file again because I modified vwret which had a timing issue. Make the 11 stock portfolio. Now regress them all on the market return and fill Table 1

In Row Average, put the average of the statistics for the 11 stocks. Use the row Average to build intuition to answer the questions below

a) Compare the 12 monthly and daily returns Betas.
Are the βs from both frequencies markedly different?
Are their standard errors markedly different?
Which frequency seems best as an estimationg strategy?
For other stocks than these very large US stocks, what could be a problem with estimating βs from daily returns regressions?

b) Compare the 12 monthly and daily returns intercepts. You can help your intuition by looking at the t-statistics outputs for the monthly vs daily returns regressions.
Are the estimates very different (after appropriate transformations)? Feel free to transform the α and s_α columns as you think makes more sense for the reader.
Are the standard errors very different (after appropriate transformation)? Explain why the result differs from the β estimation.

c) Monthly regression: portfolio vs stock regression precision
What is the average β^ standard error for the 11 stock regression? What is the standard error for the EW portfolio regression β^?
Use equation [2] in Problem 4 to infer an average cross-correlation of the 11 β^.
If the correlation ρb was exactly zero, how smaller do you expect the standard deviation of theβ ^of an 11 stock portfolio to be than that of a typical stock.
Get the 11 residual vectors (of the 11 stock regressions) in one data matrix. Compute their 11x11 correlation matrix, and compute the average cross-correlation. Make sure to remove the ones. What is it? Compare to your answer above.

Problem 2: Typical Output Analysis

Consider the AAPL regression on daily returns. For each diagnostic plot, indicate what it is supposed to detect, and if you find anything suspicious for the regression. You don't need to show a normal probability plot but check for yourself that it is very non normal.

a) Standardized residuals vs the market return in Figure 1.

b) Absolute value of the standardized residuals vs the market return in Figure 2.

d) Autocorrelation function of the standardized residuals in Figure 3. Do not use the standard acf command. Instead, use the library: forecast, and use the Acfcommand (with uppercase A).

f) Fill in the table and conclude whether heteroskedasticity is an issue. Load packages lmtest (to run coeftest and coefci) and sandwich (to run the adjusted covariance matrices). The (2,2) element of vcov is the variance of β ^, the slope coefficient.
Relevant commands are: vcov(model) vcovHC(model) vcovHAC(model) coeftest(model) coeftest(mm,vcov=vcovHC) etc... coefci(model,...) for a confidence interval

Table 2: Sandwich estimates of the slope standard error for the APPL daily regression

For Problem 6, you can loop around lm for the 12 regressions. You can also do them all with:
mymod<- lm( stkrets~rm) # creates the 12 regressions
modsum<-summary(mymod) # creates the summary object with all the good stuff in it.
coefficients(mymod) # also works to print the output with estimates and std.errs.

But mymod is a multivariate linear model object, a list where each regression is an item in the list, then each regression is itself a list. Same for modsum. It makes it hard to retrieve in a vector for example, all the standard errors. To see the problem, do names(mymod), names(modsum).
Some commands don't work. You can't use things like: confint(mymod)
What we want is not two steps down in the hierarchy of the list ( of list)
You can extract it with the lapply (ell apply!) and sapplycommands. They are similar toapply but work to extract components of lists. In coefficients(mymod), you see in what order the output is. Now try this for example, you will see what it does:

lapply(modsum, coefficients) is exactly the same as coefficients(modsum). But is is now a list with 12 items. This takes what we need from it:
sapply(lapply(modsum,coefficients),'[',c(2,4))
coefficients output for all ‘[‘ says to c(2,4) takes elements 2 and 4
regressions go down one level

Attachment:- Problem.rar