Highly correlated covariates added

Author: HWagn

vignettes/Chapter06.Rmd

@@ -153,7 +153,7 @@ knitr::kable(round(cbind(qinvgamma(0.025,a=cN,b=reg.improp$CN), sigma2.hat,
              col.names = c("2.5% quantile", "posterior mean", "97.5% quantile"))
 ```
 
-# Exercise 6.3.
+### Exercise 6.3.
 
 We now are interested in predicting of  the  box office  sales on the opening 
 weekend. We compute the  predicted box office sales  for a film with  an average number of \emph{Screens} for a range of values for \emph{Budget}.
@@ -397,6 +397,7 @@ the effects of the Weeks and Screens. Together with the high value of the poster
 almost unshrunk.
 
 ## 6.3 Regression Analysis under the Semi-Conjugate Prior
+
 We now include all available covariates in the regression analysis. As there is only one film with MPAA rating ``G'', we
 merge the two ratings ``G'' and ``PG'' into one category which we define as our
 baseline. 
@@ -496,6 +497,19 @@ sigma2.sc<-post.draws$sigma2s
 res_sigma2.sc <- res.mcmc(sigma2.sc)
 names(res_sigma2.sc) <- colnames(res_beta.sc)
 knitr::kable(t(round(res_sigma2.sc, 3)))
+```
+
+ However the different signs of the
+ effects of \emph{Vol-4-6} and  \emph{Vol-1-3} deserve some further comment. The 
+ two covariates are highly correlated.  Due to this high correlation the usual interpretation of the effect by changing the value of one covariate at the time does not make sense. Hence, we predict the change in box office sales for a film with twitter volume scores  are 1 unit higher both in weeks 4-6 and weeks 1-3. 
+
+```{r}
+cor(X[,"Vol-4-6"], X[,"Vol-1-3"])
+
+par(mfrow=c(1,1))
+plot(X[,"Vol-4-6"], X[,"Vol-1-3"])
+
+print(res_beta.sc["Vol-4-6","Mean"]+res_beta.sc["Vol-1-3","Mean"])
 ```
 
 # 6.4 Regression Analysis Based on the Horseshoe Prior