QUESTION 1

Observation Hourly Earnings Years of Education

1 30 20

2 20 16

3 22 12

Consider the following sample:

(a) What is the sample size?

(b) Calculate the sample average hourly earnings.

(c) Calculate the sample median hourly earnings.

(d) Calculate the sample variance of hourly earnings. What is its unit of measurement?

(e) Calculate the sample standard deviation of hourly earnings. What is its unit of measurement?

(f) Calculate the sample covariance between hourly earnings and years of education. What is its unit of measurement?

(g) Calculate the sample correlation between hourly earnings and years of education.

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QUESTION 2

Consider the following regression:

X has a sample average of 15 and a sample standard deviation of 2. Y has a sample average of 25 and a sample standard deviation of 5. The

sample correlation between X and Y is 0.30.

(a) Calculate the OLS estimate of .

(b) Calculate the OLS estimate of .

(c) Assume the standard error of is 6 and the standard error of is 0.3.

(i) You want to test vs. . Calculate the relevant t-statistic. Will you reject the null hypothesis at 1%

signi”cance level?

(ii) Calculate the lower limit and the upper limit of the 95% con”dence interval for .

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QUESTION 3

The following regression is estimated for a sample of 150 countries:

where ChildMort is the number of deaths of children under 5 per 1,000 live births and GDPpC is GDP per capita. Let’s denote the true value of

the coe#cient on GDPpC by . We want to test vs. .

(a) Calculate the actual value of the relevant t-statistic.

(b) What is the relevant critical value for the test statistic (use the large-sample normal approximation) at 5% signi”cance level?

(c) Would you reject H0 in favor of H1 at 5% signi”cance level?

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QUESTION 4

You decide to estimate the following three regressions using the same sample of data (assume that sample size is 10,000):

wagei = b0 + b1 femalei + ui (1)

wagei = c0 + c2 malei + vi (2)

wagei = d1 femalei + d2 malei + ei (3)

where wage refers to average hourly earnings, u, v, and e are the regressions’ error terms, and

femalei = 1 if observation i refers to a female, and = 0 if observation i refers to a male

malei = 1 if observation i refers to a male, and = 0 if observation i refers to a female

(a) How much is the expected wage of a female according to each regression?

(b) How much is the expected wage of a male according to each regression?

(c) Interpret the vertical intercept and the slope in regression (1).

(d) Interpret the vertical intercept and the slope in regression (2).

(e) Interpret the coe#cients d1 and d2 in regression (3).

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QUESTION 5

You estimate the linear regression:

and “nd that the standard error of the regression, SER, equals 0.7. The total sum of squares, TSS, equals 200.

(a) Calculate the residual sum of squares (RSS), also referred to as the sum of squared residuals (SSR).

(b) Calculate R2.

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QUESTION 6

Consider the following regression output, where ahe refers to average hourly earnings and yrseduc refers to years of education.

(a) What fraction of the sample variance of ahe is explained by yrseduc?

(b) How much is the standard error of the regression (SER)?

(c) What is the sample size? [Hint: Check the degrees of freedom of the SER and note that we have lost 2 degrees of freedom when

estimating the two coe#cients of the regression.]

(d) What is the OLS estimate of the slope?

(e) What is the standard error of the OLS estimator of the slope?

(f) What is the t-statistic corresponding to the two-sided test with null hypothesis that the slope equals 0.

(g) Will you reject the null hypothesis that the slope equals 0 in favor of the two-sided alternative at 5% signi”cance level?

(h) Find the lower and the upper limit of the 95% con”dence interval for the slope of the regression (use the normal approximation,

which is justi”ed since the sample size is large enough).

(i) Calculate the predicted wage (i.e., average hourly earnings) of a person with 16 years of education.

(j) What would be the predicted increase in the wage of a high-school graduate if he/she obtains a college degree? In answering this

question assume that college takes 4 years.

(k) Give an example of a variable that can directly increase a person’s wage and can be positive correlated with years of education.

(l) In view of (k), do you expect the OLS estimator of the slope to be unbiased? In particular, do you think that the expected value of the

OLS estimator of the slope is greater, smaller, or equal to the true slope?

Sample Solution

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