Assignment 1
Econometrics 1
Jinhyun Lee
Deadline: 09:00 October 14 Friday (before class)
Instructions
– Write your name and student number on the front page.
– Make your answers into one pdf …le.
– Submit your pdf …le to LearnUs by Deadline: 09:00 October 14
Friday (before class).
– Do not "copy and paste" others’answers. If detected, your grade will be
automatically set to zero. Your answers should re‡ect your own understanding of the course material.
– Use the same notation used in the lectures (e.g. for the coe¢ cients of the
regression function, errors, residuals, R2 etc). You do not need to de…ne
each in your answers.
1. (De…nitions and concepts) De…ne or describe the followings. Stick to the notation from the lectures. If you use mathematical expressions, add your explanation in words. You could use graphs to elaborate your answers.
(a) Homoskedasticity
(b) Unbiasedness of the OLS estimators of the regression coe¢ cients. Specify
what assumption(s) are required to guarantee the unbiasedness of OLS
estimators.
(c) E¢ ciency of the OLS estimators of the regression coe¢ cients. Specify what
assumption(s) are required to guarantee the e¢ ciency of OLS estimators.
2. (True/False) Identify whether the following statements are true or false and
present reasons why you believe so.
(a) The normality assumption of the regression error can be informally tested
by examining the normality of the dependent variable
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(b) Homoskedasticity is required to show the unbiasedness of the OLS estimators.
(c) Unbiasedness of an estimator of a population parameter guarantees that
a realized value of the estimator is likely to be very close to the true
population parameter.
(d) Suppose the variance of an estimator is n52 : Then if the estimator is unbiased, the estimator is consistent. That is it is likely that the realized values
of the estimator is very close to the true population parameter. [Hint: use
graphs].
bi ; not errors, Ui ) and one of the
(e) The sample covariance of OLS residuals (U
regressors (X1 ; X2 ; :::; Xk ) is zero.
(f) In a simple regression, the OLS estimator, c1 ; contains the information of
the covariance between the dependent variable, Y; and the regressor, X.
[Hint: It would be best if you could …nd the sample version ("analog") of
Cov(Y; X) and relate this with c1 . Then discuss your conclusion based on
this.]
(g) Given the population regression, Yi = 0 + 1 Xi + Ui ; the sample means of
fXi g and fYi ); X and Y , are on the population regression funcion, That
is
Y = 0 + 1 X:
3. (In continuation of Quiz 1: Refer to your answers to Quiz 1) The
following table contains education and log wage. Suppose we are interested in
the e¤ect of education on log wage.
(a) Based on your answers to Quiz 1, interpret the estimated intercept (b0 )
and slope (b1 ); bearing mind the log-level relation of the regression.
(b) Estimate the variance of the error.
(c) Calculate the standard error of the regression (SER).
(d) Calculate the standard error of the slope parameter (b1 ).
(e) Report the 95% con…dence interval for the slope parameter (b1 ).
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(f) Conduct the hypothesis test on whether the slope is positive at the signi…cance level of 1%.
i.
ii.
iii.
iv.
Write down the null and alternative hypotheses.
Write down the form (formula) of the test-statistic.
Calculate the realized value of the test-statistic.
Find the relevant critical value(s).
v. Draw the distribution of b1 under the null (that is, the null distribution
of b1 ): What is the distribution of b1 ? What is the mean and the
variance of the distribution of b ?
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vi. Draw the distribution of the test-statistic, T =
b
b
1
1
s:e(b1 )
. What is the
1
1
distribution of T = s:e(
b ) ? Indicate the critical value(s) and rejection
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region. Also, indicate the signi…cance level in the graph. What does
the signi…cance level indicate?
vii. Conclude the hypothesis test.
4. Using 209 (n = 209) observations on CEO salary and …rm salaries, the following
regession is estimated:
\
log(salary)
= 4:82 +
0:257
(0:004)
log(sales)
R2 = 0:211
where salary denotes CEO’s annual salary in thousand dollars and sales annual
…rm sales in millions of dollars. Standard errors appear in parentheses below
the estimated coe¢ cients.
(a) Interpret the coe¢ cient on log(sales), bearing in mind the log-log relationship. Is the sign of the estimate what you expect it to be?
(b) Test whether sales is statistically signi…cant (in other words, whether the
coe¢ cient of sales is signi…cantly di¤erent from zero).
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