5: California Schools Example 0n the text website, http://wwwpearsonhighered.com/stock_watson], you will find a data set CASchools_EEili_lnSample that contains a subset of n=5CIIJ schools from the data set used in this chapter. Included are data on test scores and 2|] of the primitive predictor variables; see CASchoolsjElLDescription for a description of the variables. In this exercise, you will construct prediction models like those described in the text and use these models to predict test scores for 500 out-of-sample schools (Please read EE14L SoftwareNotes on the text website before solving the exercise.) a. From the 20 primitive predictors, construct squares of all the predictors, along with all of the interactions (that is, the cross products % for all jand k). Collect the 20 primitive predictors, their squares, and all interactions into a set of kpredictors. Verify that you have 20+20+(ZDX19)/2=23|] predictors One of the primitive predictors is the binary variable charter} Drop the predictor (abarferjz from the list of 230 predictors, leaving 29 predictors for the analysis. Why should (mane-.32 be dropped from as a predictor? b. Compute the sample mean and standard deviation of each of the predictors, and use these to compute the standardized regressors. Compute the sample mean of TestScore and subtract the sample mean from TesLScoreto compute its demeaned value. c. Using OLS, regress the demeaned value of TesLS'core on the standardized regressors. i. Did you include an intercept in the regression? Why or why not? ii. Compute the standard error of the regression. d. Using ridge regression with Humming\" ress the demeaned value of TesLS'coneon the standardized regressors. Compare the and ridge estimates of the standardized regression coefficients. e. Using Lasso with 16 0L 1. L=L Elaimress the demeaned value of TestScomon the standardized regressors. How many of the estimated Lasso coefcients are different from I]? Which predictors have a nonzero coefcient. f. Compute the scree plot for the 229 predictors. How much of the variance in the standardized regressors is captured by the rst principal component? By the first two principal components? Bythe rst 15 principal components? g. Compute 15 principal components from the 229 predictors. Regress the demeaned value of kscme on the 15 principal components. h. On the text website, you will nd a data set CASchools_EE14_ OutOfSample that contains data from another n=50lJ schools i. Predict the average test score for each of these 500 schools using the OLS, ridge, Lasso, and principal components prediction models that you estimated in (c), (d), (e), and (9). Compute the root mean square prediction error for each of the methods. ii. Construct 4 scatter plots like those in Figure 14.8. What do you team from the plots? i. Estimate lbnxmgfi number of principal components using 10-fold cross validation from the in-sample data set. j. Use the estimated values of lhnxdmgf'flf number of principal components from G) to construct predictions of test scores for the out-of-sample schools. Are these predictions more accurate than the predictions you computed in (h)? Is the difference in line with what you expected from the cross-validation calculations in (i)
