Problem 4.1. For the model \(Y=\beta _0+\beta _1X_1+\beta _2X_2+e\), state precisely what \(\beta _1\) measures, and explain why its value may change — even in sign — when \(X_2\) is removed from the model.

Problem 4.2. Show that \(SST=SSR+SSE\) for a model containing an intercept, and explain which term of the derivation fails if the intercept is omitted.

Problem 4.3. The columns of \(X\) are orthogonal. Show that \(X^tX\) is diagonal, and deduce that each \(\widehat {\beta }_j\) is then the same whether or not the other regressors are in the model. Explain why an experimenter can arrange this and a regression analyst usually cannot.

Problem 4.4. Define the variance inflation factor and show that it equals \(1/\left (1-R_j^2\right )\), where \(R_j^2\) is from the regression of \(X_j\) on the remaining regressors. What does \(VIF=10\) say about the standard error of \(\widehat {\beta }_j\)?

Problem 4.5. The condition number of \(X^tX\) for a fitted model is \(1{,}450\). State what this indicates by Table 5, name two remedies from Section 4.5, and say what each one costs.

Problem 4.6. A residual plot against the fitted values shows a funnel widening to the right. Which assumption is in question, which is not, and what would you do about it?

Problem 4.7. Explain why residuals are plotted against fitted values rather than against the observed \(Y_i\). Where to start: \(\widehat {e}\) and \(\widehat {Y}\) are uncorrelated by construction; \(\widehat {e}\) and \(Y\) are not.

Problem 4.8. A regression of ice cream sales on drowning deaths over 52 weeks gives \(R^2=0.79\) and a coefficient significant at the \(0.1\%\) level. Using Section 4.9, explain what is wrong with concluding anything causal, and name the feature of the data that produces this.