ETW2510 Quiz (Week 8 - Week 11)

We have data on hourly wage of 5 individuals, 2 of whom are female and the other 3 are male. We have generated two dummy variables. The first one is female which is equal to 1 if the individual is female and is equal to zero otherwise. The other dummy variable is male is equal to 1 if the individual is male and is equal to zero otherwise. We estimate a regression of wage of these two dummy variables without a constant term. Refer to attached X matrix is on the left. The X'X matrix is

• diagonal matrix

• a square matrix

• an invertible matrix

• 
• all of the above

Answer explanation

We have data on hourly wage of 5 individuals, 2 of whom are female and the other 3 are male. We have generated two dummy variables. The first one is female which is equal to 1 if the individual is female and is equal to zero otherwise. The other dummy variable is male is equal to 1 if the individual is male and is equal to zero otherwise. We estimate a regression of wage of these two dummy variables without a constant term. The estimated coefficient of the female dummy variable is

• the average wage of the 3 male individuals in the sample

• the average wage of all 5 individuals in the sample

• the average wage of the 2 female individuals in the sample

• the mean wage of all females in the population

• the difference between the average wage of the 2 female and the 3 male individuals in the sample

Answer explanation

In order to investigate the relationship between firm performance and CEO compensation, we have collected information on 209 CEOs of firms operating in all sectors of the economy, including the finance and consumer product sectors. Variables in our data set are:

CEO annual salary of the firm's CEO in thousand dollars

SALES annual salary of the firm's CEO in thousand dollars
FINANCE a dummy variable = 1 if the firm is in the finance sector, 0 otherwise
CONSPROD a dummy variable = 1 if the firm is in the consumer product sector, 0 otherwise
OTHER a dummy variable = 1 if the firm is in sectors other than finance and consumer products, 0 otherwise.
Using this dataset, we have estimated the following regression equation:


R² = 0.29 tells us that

• the square correlation of the residual with its first lag is 0.29

• 29% of the variation in log (CEO) in this sample is explained by the variation in log (SALES) and the sector that the firm is operating in

• the sum of square of the sample correlation coefficients between log (CEO) and each of the three explanatory variables is 0.29

• the estimated regression equation predicts the logarithm of the salary 29% of the CEOs perfectly with zero error

• the total sum of squares is 0.29 x 47.22 = 13.6938

Answer explanation

In order to investigate the relationship between firm performance and CEO compensation, we have collected information on 209 CEOs of firms operating in all sectors of the economy, including the finance and consumer product sectors. Variables in our data set are:

CEO annual salary of the firm's CEO in thousand dollars

SALES annual salary of the firm's CEO in thousand dollars
FINANCE a dummy variable = 1 if the firm is in the finance sector, 0 otherwise
CONSPROD a dummy variable = 1 if the firm is in the consumer product sector, 0 otherwise
OTHER a dummy variable = 1 if the firm is in sectors other than finance and consumer products, 0 otherwise.
Using this dataset, we have estimated the following regression equation:

The estimated coefficient of FINANCE tells us that

• the CEO of a financial firm is predicted to earn 25 thousand dollars than the CEO of a firm with the same annual sales revenue in another sector

• the CEO of a financial firm is predicted to earn more than the CEO of a consumer product firm with the same annual sales revenue

• the CEO of a financial firm is predicted to earn 100(e^0.25-1) = 28% more than the CEO of a firm operating in a sector other than finance or consumer products

• the CEO of a financial firm is predicted to earn 100(e^0.25-1) = 28% more than the CEO of a firm with the same annual sales revenue operating in a sector other than finance or consumer products

• for one percentage increase in sales, the salary of the CEO of a financial firm is predicted to increase by 0.25 percent more than the salary of the CEO of a firm in a sector other than finance and consumer products.

Answer explanation

In order to investigate the relationship between firm performance and CEO compensation, we have collected information on 209 CEOs of firms operating in all sectors of the economy, including the finance and consumer product sectors. Variables in our data set are:

CEO annual salary of the firm's CEO in thousand dollars

SALES annual salary of the firm's CEO in thousand dollars
FINANCE a dummy variable = 1 if the firm is in the finance sector, 0 otherwise
CONSPROD a dummy variable = 1 if the firm is in the consumer product sector, 0 otherwise
OTHER a dummy variable = 1 if the firm is in sectors other than finance and consumer products, 0 otherwise.
Using this dataset, we have estimated the following regression equation:

From the estimated regression we can infer that

• CEOs of consumer product firms earn 32% more than CEOs of finance firms on average

• CEOs of consumer product firms earn 7% more than CEOs of finance firms on average

• the CEO of a consumer product firm is predicted to earn 32% more than the CEO of a finance firm with the same annual sales revenue

• the CEO of a consumer product firm is predicted to earn 100(e^0.32-1) = 38% more than the CEO of a finance firm with the same annual sales revenue

• the CEO of a consumer product firm is predicted to earn 7% more than the CEO of a finance firm with the same annual sales revenue

Answer explanation

• we can still use the OLS estimator because it is unbiased, and we can use the usual OLS standard errors to perform t tests

• we can still use the OLS estimator because it is unbiased, but we need to use heteroskedasticity robust standard errors to perform t tests

• we cannot use the OLS estimator because the OLS estimator is biased in this case

• we can still use the OLS estimator because it is the best linear unbiased estimator in this case

• we can still use the OLS estimator because the OLS estimator is the same as the "weighted least squares" estimator in this case

Answer explanation

A researcher obtained the sample autocorrelation functions for the residuals, which are shown in the Figure:

(i) What does the information in Figure 1 suggest with regard to the behaviour of the residuals? Briefly explain.

(ii) Breusch-Godfrey test will be used for testing no serial correlation in errors against the alternative of serial correlation of order 2. Clearly state the steps involved, the null and alternative hypotheses of the test, the statistic(s) of interest and corresponding distribution(s).


(iii) What are the implications of the problem of serial correlation? What standard errors for the OLS estimates would you recommend using in this instance? Briefly explain.

In finance, it is generally believed that asset prices are non-stationary time series. The following graph shows a time series plot of the Standard and Poor's U.S. Composite Stock Price Index (S&P500) from January 1985 to July 2017.

(i) Define a covariance stationary time series
(ii) With reference to the graph, explain why S&P 500 is not a covariance stationary time series.

Answer explanation

(i) A series is covariance stationary if its mean and variance are finite and do not depend on t, and the covariance between y_i and any of its lags only depend on the lag and does not depeng on t.

(ii) It is clear from the graph that the mean depends on t. It is very unlikely that the mean in the first half of the sample is the same as the mean in the second half.

If E(u_i |x_i1, x_i2, . . . , x_ik) = 0 for all i, then

• E(u | X) = 0

• 
• 
• 
• all of the above

Answer explanation

If E(u_i |x_i1, x_i2, . . . , x_ik) = 0 for all i, and the sample is randomly selected, then

• 
• 
• 
• 
• all of the above

Answer explanation