Question

I. In a regression analysis of banks, four types of banks were involved, namely, commercial banks, mutual banks, community banks and savings banks. In this study, Y is the previous year's profit (in millions of dollars), X, is the size of the bank (in millions of dollars) and type of bank = X, X, X, which are indicator variables as coded below. Type of Bank X X X , Commercial 1 0 Mutual Community Savings 0 0 0 The first-order response function to be used in the study is: EXY) =R+BX+B,X, +B,X,+B.X. 1. Develop the response function for each type of bank (10pts) 2. Specify the null and alternative hypotheses for the following questions: a) With size of bank fixed, do commercial banks have higher expected profits than savings banks? (5pts) b) With size of bank fixed, do commercial banks and mutual banks have the same expected profits? (5pts) 3. Interpret the value B, and what it represents. (Spts) 4. Why is the size of bank included in the regression model, when the interest is in estimating the profits for different types of banks? (5pts) II. You are given the following linear regression model, where X, represents a qualitative variable with two outcomes {0,1} and X1 is a quantitative variable: Y = Bo + ,Xi1 + R2X2 +8 1. The response function for Y using X2 = 0 is: (5pts) 2. The response function for Y using Xiz = 1 is: (5pts) 3. Explain theoretically what does a) A indicates. (10pts) b), indicates (10pts) c), indicates (10pts) III. Use the data set from HW6. The following model was fit: Y+BX+B.X2+BX+.XX.+B,X, Xa+B.XX+6, a) Provide the R output for the summary of the upper mentioned regression model: (5pts) b) Provide the Routput for the anova table for the upper mentioned regression model: (5pts) Test whether the interaction tems can be dropped from the model, by answering the following questions, c) State the null and alte mative hypotheses. Identify any symbols you use(5pts) d) What is the value of the test statistic for this test? (5pts) e) Using significance level a = 0.05, the critical F(0.95, d.f.=(df1,df 2)), where (df1,df2) are the appropriate degrees of freedom for this test. Find the F-critical value and justify should the null hypothesis be rejected? Explain. (5pts) 6) Draw a conclusion regarding the test and the model. (5pts)

Answer 1

1.

For Commercial Bank, X2 = 1, X3 = 0, X4 = 0, the response function is,

E(Y) = Bo + 32 + B1X1

For Mutual Bank, X2 = 0, X3 = 1, X4 = 0, the response function is,

E(Y) = Bo + B3 + B1X1

For Community bank, X2 = 0, X3 = 0, X4 = 1, the response function is,

E(Y) = Bo +84 + B1X1

For Savings bank, X2 = 0, X3 = 0, X4 = 0, the response function is,

E(Y) = Bo + B1X1

2.

(a)

Difference between profits of Commercial Bank and Saving Banks is $\beta_2$ .

The appropriate hypotheses are,

Null hypothesis H0: $\beta_2$ $\le$ 0

Alternative hypothesis Ha: $\beta_2$ > 0

(b)

Difference between profits of Commercial Bank and Mutual Banks is .

32 - 3

The appropriate hypotheses are,

Null hypothesis H0:
32 - 3
=  0 or
B2 = B3

Alternative hypothesis Ha:
32 - 3
$\ne$ 0 or  
Eg 7 g
​​​​​​​

3.

For Community bank, X2 = 0, X3 = 0, X4 = 1

The regression equation becomes,

E(Y) = Bo +84 + B1X1

For Savings bank, X2 = 0, X3 = 0, X4 = 0

The regression equation becomes,

E(Y) = Bo + B1X1

Thus, the value $\beta_4$ can be interpreted as the difference between profits of community and savings bank of the same size.

4.

For the same type of bank, the profit of the bank may vary based on the size of the bank. In other words, for the same type of bank, the profit of the bank is significantly correlated with the size of the bank. If we do not include the size of the bank in the model, the model will predict same profit for a given type of bank which is obviously not viable.