Assignment 8

Exercise 1 (5 points)

State the number of degrees of freedom that are associated with each of the following extra sums of squares:

Note on notation. \(\text{SSReg}(A | B)\) is the extra sum of squares that appeared as a result of including variables \(A\) into the regression model that already had variables \(B\) in it. Thus it is used to compare the full model with both \(A\) and \(B\) in it against the reduced model with only \(B\).

Exercise 2 (5 points)

Explain in what sense the regression sum of squares \(\text{SSReg}(X_1)\) is an extra sum of squares.

Exercise 3 (5 points)

For a multiple regression model with five \(X\) variables, what is the relevant extra sum of squares for testing

  1. whether or not \(\beta_5 = 0\)
  2. whether or not \(\beta_2 = \beta_4 = 0\)

Exercise 4 (5 points)

Continue working with the Brand preference data, which is available here.

  1. Obtain the ANOVA table that decomposes the regression sum of squares into extra sum of squares associated with \(X_1\) and with \(X_2\), given \(X_1\).
  2. Test whether \(X_2\) can be dropped from the model while \(X_1\) is retained.
  3. Fit first-order simple linear regression for relating brand liking ( \(Y\) ) to moisture content ( \(X_1\) ).
  4. Compare the estimated regression coefficient for \(X_1\) with the corresponding coefficient obtained in (a).
  5. Does \(\text{SS}_{reg}(X_1)\) equals \(\text{SS}_{reg}(X_1 | X_2)\) here? Is the difference substantial?
  6. Regress \(Y\) on \(X_2\) and obtain the residuals. Regress \(X_1\) on \(X_2\) and obtain the residuals. Regress residuals from the “\(Y\) on \(X_2\)” on residuals from the model “\(X_1\) on \(X_2\)”; compare the estimated slope, error sum of squares with #1. What about \(R^2\).

Exercise 5 (5 points)

Consider a regression model \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon\), where \(X_1\) is a numerical variable and \(X_2\) is a dummy variable. Sketch the response curves (the graphs of \(E(Y)\) as a function of \(X_1\) for different values of \(X_2\)), if \(\beta_0 = 25\), \(\beta_1 = 0.2\), and \(\beta_2 = -12\).

Exercise 6 (5 points)

Continue the previous exercise. Sketch the response curves for the model with interaction, \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \varepsilon\), given that \(\beta_3 = -0.2\).

Exercise 7 (5 points)

In a regression study, three types of banks were involved, namely, (1) commercial, (2) mutual savings, and (3) saving and loan. Consider the following dummy variables fr the types of bank:

Type of Bank \(X_1\) \(X_2\)
Commercial 1 0
Mutual saving 0 1
Saving and loan 0 0
  1. Develop the first-order linear regression model (no interactions) for relating last year’s profit or loss ( \(Y\) ) to size of bank ( \(X_1\) ) and type of bank ( \(X_2, X_3\) ).
  2. State the response function for the three types of banks.
  3. Interpret each of the following quantities:
    1. \(\beta_2\)
    2. \(\beta_3\)
    3. \(\beta_2 - \beta_3\)