# Econometrics Exercise

Exercise 7 Econ 2311

1. Consider the equation, Y = β0 + β1X1 + β2X2 + u. A null hypothesis, H0: β2 = 0 states that:

(A) X2 has no effect on the expected value of β2. (B) X2 has no effect on the expected value of Y. (C) β2 has no effect on the expected value of Y. (D) Y has no effect on the expected value of X2.

2. The general t statistic can be written as:

(A) t = Hypothesized value Standard error

(B) t = (estimate-hypothesized value) (C) t = estimate−hypothesized value

variance

(D) t = estimate−hypothesized value standard error

3. Which of the following statements is true?

(A) When the standard error of an estimate increases, the confidence interval for the estimate narrows down. (B) Standard error of an estimate does not affect the confidence interval for the es- timate. (C) The lower bound of the confidence interval for a regression coefficient, say βj, is

given by β̂j- [standard error · β̂j] (D) The upper bound of the confidence interval for a regression coefficient, say βj,

is given by β̂j+ [Critical value · standard error ( β̂j)]

4. Which of the following statements is true of hypothesis testing?

(A) The t test can be used to test multiple linear restrictions. (B) A test of single restriction is also referred to as a joint hypotheses test. (C) A restricted model will always have fewer parameters than its unrestricted model. (D) OLS estimates maximize the sum of squared residuals.

5. Consider an equation to explain salaries of CEOs in terms of return on equity (roe, in percentage form), return on the firm’s stock (ros, in percentage form) and

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firm sales:

log(salary) = β0 + β1roe + β2ros + β3log(sales) + u.

Suppose we use a dataset and obtain the following estimated equation by OLS:

̂log(salary) = 4.32 + 0.0174roe + 0.0124ros + 0.320log(sales) (0.32) (0.0076) (0.0054) (0.1220)

n = 973, R2 = 0.658.

(1) In terms of the model parameters, state the null hypothesis that, after controlling for ros and roe, a 1% increase of sales causes a 0.1% increase of salary. State the alternative that 1% increase of sales causes an increase of salary other than 0.1%. (Be careful! This is a log(y) ∼ log(x) relationship.)

(2) Test the null hypothesis you wrote down in part (2) at the 5% significance level. (If X∼N(0,1), Prob(X>1.65) =0.05, Prob(X>1.96) =0.025, Prob(X>2.58) =0.005.)

(3) Find the 90% confidence interval for β3.

6. Your company has spent considerable resources making its product information available on the Internet. After a year of experience with this approach, it is eval- uating how things are progressing. One of the key question is whether “hits” turn into sales. To explore this issue, you have been given year 2000 data on the firm’s 420 products. For each product, you have data on two variables: SALES (sold number (each purchaser buys only one unit)) and HITs (the number of times the product’s page was accessed). You decide to run a regression of SALES on HITS and a constant. Below you can find a scatter plot of the data.

Here is the regression output you obtain estimating the regression with OLS (the standard errors are in parenthesis):

̂SALESi = 58.9 + 0.074 ·Hits (7.63) (0.0036)

n = 420, R2 = 0.46.

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(1) Calculate the p-value for the slope. Is the slope statistically significant using the standard 1%, 5% and 10% significance values?

(2) You would like to know how many additional sales you would expect to be generated by an increase in hits of 100. Construct a 95% confidence interval for the expected increase in sales.

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