ECON 837 - Econometrics
Assignment #1
due in class Jan. 28th
Provide detailed calculations and justications to get full credit.
Partial credit may be given.
1 Theoretical exercises
Exercise 1:
Consider the following a linear regression model with two groups of regressors,
y = Xα + Zβ + ϵ
where y is a vector of dependent variables of length n, and X and Z are matrices
of explanatory variables of dimension (n, k1 ) and (n, k2 ) respectively. In addition, we
suppose that X and Z are xed and the error term ϵ has a mean equal to zero and a
variance-covariance matrix equal to σ 2 In .
We want to compare the properties of the following three estimators of β :
1. β̂ obtained by regressing y on X and Z ;
2. β̂ ⋆ obtained by regressing y on Z only;
3. β̂ ⋆⋆ obtained in two steps: rst, regress y on X only to obtain α̂; second, regress
the associated residuals, (y − X α̂), on Z .
(a) Find the expression for the expected value and the variance of each estimator. To
get full credit, you need to provide their matricial expressions (no summations!).
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(b) For the case k1 = k2 = 1, show that the expected value of β̂ ⋆ does not necessarily
have the same sign as β but β̂ ⋆⋆ has the same sign as β .
(c) For the case k1 = k2 = 1, show that var(β̂ ⋆⋆ ) ≤ var(β̂ ⋆ ) ≤ var(β̂).
(d) What is the intuition for the results found in (c) and (d)?
You might nd useful the part on partitioned matrices in the appendix A of the
textbook.
Note:
Exercise 2:
Consider the following model, yi ∼ N (α + βxi , σ 2 ) where the yi 's are independent.
(a) Derive the OLS estimator of α under the restriction that β = 1.
(b) Find the expected value and variance of this estimator when the restriction is true
and when it is false.
(c) How do the expressions found in (b) compare to those obtained for the OLS estimator when there are no restriction?
Exercise 3:
Under which conditions will the OLS estimator of β in the model yi = βxi + ϵi will be
identical to the OLS estimator of β ⋆ in the model yi = α + β ⋆ xi + ui ?
2 Computational exercises
Exercise 4:
In this exercise, you perform a Monte Carlo study. It means that we generate a large
number of random samples with similar (statistical) properties, compute the associated
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estimators of the quantities of interest, and study their statistical properties (see the
appendix below for additional information about Monte Carlo study).
We consider the linear model of Exercise 1,
yi = x′i α + zi′ β + ϵi
where xi is a vector of k1 regressors, zi is a vector of k2 regressors and ui are independent
and normally distributed with mean 0 and variance σ 2 .
The le dataset1.txt contains observations on xi (rst 2 columns) and zi (last 3
columns).
(a) In this question, we consider the case k1 = k2 = 1 with X the rst column and Z
the last column of the dataset. Take also α = β = 1 and σ 2 = 1.
(i) Calculate the Monte-carlo mean and variance of each estimator.
(ii) How do your ndings compare to your answers to questions (b) and (c) in exercise
1?
(b) In this question, we consider the case where k1 = k2 = 1 (as in exercise 1) with X
the second column and Z the last column of the dataset. Take also α = β = 1 and
σ 2 = 1.
(i) Calculate the Monte-carlo mean and variance of each estimator.
(ii) How do your ndings compare to your answers to questions (b) and (c) in exercise
1?
(iii) How do your ndings compare to your answer to question (a) above? Discuss.
(c) In this question, we consider the case where k1 = 2 and k2 = 3. Specically, take X
as the rst two columns and Z as the last three columns of the dataset. Take also
α as the vector of ones of size 2, and β the vector of ones of size 3. (i) Calculate the
Monte-carlo mean and variance of each estimator of vector β .
(ii) Do you see any connection with your answers questions (b) and (c) in exercise 1?
with your answers to questions (a) and (b) above? Discuss.
(d) Summarize your ndings.
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Appendix
• Generating the random samples:
(i) Draw 1 vector of n observations for U from the normal distribution with mean 0
and variance σ 2 .
(ii) Compute the associated vector of dependent variables Y.
(iii) "Forget" what you just did (ie forget U ): you now have 1 sample of n observations
for (yi , xi , zi ).
(iv) Compute the three estimators β̂ , β̂ ⋆ , and β̂ ⋆⋆ .
(v) Repeat (i) to (iv) 10,000 times! (vi) You now have 10,000 estimates for the 3
estimators β̂ , β̂ ⋆ , and β̂ ⋆⋆ . You can now compute the Monte-Carlo mean and variance
for each over these 10,000 replications (you could even look at their distributions...).
• Some useful MATLAB commands:
load data.txt -ascii import the le data; a matrix called data is created in your workspace
rand(m,n)
generate a (m,n)-matrix of numbers distributed uniformly over (0,1)
randn(m,n)
generate a (m,n)-matrix of numbers distributed normally N (0,1)
mean(x)
calculate the mean of the columns of a matrix x
var(x)
calculate the variance of the columns of a matrix x
sum(x)
calculate the sum of the columns of a matrix x
abs(x)
calculate the absolute value of a matrix x (entry-by-entry)
hist(x,n)
plot the histogram of a vector x using n bins
plot(x,y)
plot the couples (xi , yi ) [x and y need to be of the same size].
hold on/off
option to plot another curve on the same/dierent gure
disp(x)
print the variable x in the command window
disp('my result is:')
print the text "my result is:" in the command window
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