Econ 9720: Econometrics II
GSU Department of Economics, Spring 2016

Practice Midterm Questions
(No Solution will be Provided)
1. Suppose the data generating process (the true relationship) is y = Xβ + ε, where E[ε|X] = 0, E[εε |X] = σ 2 I n ; and X includes an intercept term. You do not observe the data set Z = [y X]. Instead you observe


150 15 50
Z Z =  15 25 0 
50 0 100
2
Compute the least squares estimators β, s2 , R2 and RAdj (the adjusted R2 ). Is there anything to be gained by observing the full data set?

2. Suppose you have the simple regression model with no intercept: yi = xi β+ i for i = 1, 2. Suppose further that the true value of β is 1, the values of xi observed in the sample are x1 = 2 and x2 = 3, and the distribution of i is
Pr( i = −2) = Pr( i = 2) = 1/2 with 1 independent of 2 .
(a) Find the least squares estimator of β.
(b) What is it mean and variance? Is it BLUE?
(c) Consider the alternative estimator β ∗ = y /¯, where y is the sample mean
¯ x
¯
of yi and x is the sample mean of xi . What is the mean and variance of
¯
β ∗ ? Is it unbiased?
(d) Which estimator is more efficient, the least squares estimator or β ∗ ?
3.
Suppose x1 , x2 . . . xn is an independent but not identically distributed random sample from a population with E[xi ] = µ and Var[xi ] = σ 2 /i for i =
1, 2, . . . , n. Consider the following class of estimators for the population mean µ: n µ=
ˆ

ci xi

where

c1 , . . . , c n

are constants

i=1

Each sequence {c1 , c2 , . . . , cn } defines an estimator for µ.
(a) Give a necessary and sufficient condition on the ci for µ to be an unbiased
ˆ
estimator of µ.
(b) Find the best unbiased estimator µ∗ in the class of estimators µ.
ˆ
ˆ
1
(c) Compare the relative efficiency of µ∗ and the sample mean, x =
ˆ
¯ n Explain.

1

n i=1 xi .

Some possibly useful results:

i

i = n(n + 1)/2

i

i2 = n(n + 1)(2n + 1)/6.

4. Suppose we have the linear regression model y = Xβ + ε
(a) Show that if E[ε] = 0 then the least squares estimator of β is biased.
(b) Suppose E[ε] = 0 and E[εε ] = σ 2 I n . Let X = [i X2 ] where i is an nvector of ones and X2 is n×(k − 1) and measured in deviations from means.
Show that the least squares estimator of the intercept (the parameter on i) is uncorrelated with the least squares estimator of the slopes (the parameters on X2 ).
5. Suppose you observe a random sample of n observations (y, X) from a population that satisfies y = Xβ + ε with E[ε|X] = 0 and E[εε |X] = σ 2 I n . Let β denotes the least squares estimate of β based on the sample (y, X). Now you observe the vector of independent variables, x∗ for one more observation that is not part of your estimation sample but is sampled from the same population. You do not observe the dependent variable y∗ for the new observation. Let y∗ = x∗ β
ˆ
denote the OLS prediction of y∗ .
(a) Is y∗ an unbiased estimator of y∗ ? Explain/prove your claim.
ˆ
(b) What is the variance of the OLS prediction error, y∗ − y∗ ?
ˆ
(c) What is the best linear (in y) unbiased estimator of y∗ ? Explain/prove your claim.
(d) Now suppose that ε|X ∼ N (0, σ 2 I n ). Derive a test statistic (and its sampling distribution) to test the null hypothesis that y∗ = y0 .
6. Suppose the data generating process is yi = xi β + εi where the errors are spherical and have mean zero. The data fall into one of two groups of equal size.
In the first group (group 1) of n/2 observations, x i = [1 1]. In the second group (group 2), x i = [1 − 1]. Despite your knowledge of least squares, you
¯
devise a new estimator β by noting that y 1 = β1 + β2 + ε1
¯
¯

(1)

y 2 = β1 − β2 + ε2
¯
¯

(2)

¯ where y 1 is the sample mean of yi in group 1, y 2 is the sample mean of yi in
¯
group 2, ε1 is the sample mean of εi in group 1, and ε2 is the sample mean of εi
¯
¯
¯
in group 2. Since E[¯1 ] = E[¯2 ] = 0 you define your estimator β as the solution ε ε to the linear equations (1) and (2) with ε1 and ε2 set to zero.
¯
¯
¯
(a) Give a formula for β
¯ unbiased? If yes, prove it. If not, derive and sign the bias.
(b) Is β
¯
(c) What is the sampling variance of β?
(d) How do its finite sample properties compare to the least squares estimator?
Explain carefully.

2

7. Suppose we have the linear regression model y = Xβ + ε
(a) Show that if E[ε|X] = 0 then the least squares estimator of β is biased.
(b) Suppose we write the model as y = X1 β1 + X2 β2 + ε and E[ε|X] = X1 γ for some γ = 0. Is the least squares estimator of β 2 biased? Prove your claim. 8.
Consider the regression model y = X1 β1 + X2 β2 + ε. For mysterious reasons, you are mainly interested in β 2 . Let M1 = I − X1 (X1 X1 )−1 X1 and
P1 = I − M1 . For even more mysterious reasons, you estimate the following regressions: (a) y = X1 β1 + X2 β2 + ε.
(b) P1 y = X2 β2 + ε
(c) P1 y = P1 X2 β2 + ε
(d) M1 y = X2 β2 + ε
(e) y = M1 X2 β2 + ε
(f) M1 y = X1 β1 + M1 X2 β2 + ε
(g) M1 y = M1 X1 β1 + M1 X2 β2 + ε which gives you a variety of estimates of β 2 . How many different estimates are there? How are they related?
9. Consider the translog production function ln Qi = β1 + β2 ln Li + β3 ln Ki + β4

(ln Ki )2
(ln Li )2
+ β5
+ β6 ln Li ln Ki + εi
2
2

(a) Show that the condition for constant returns to scale is
∂ ln Qi
∂ ln Qi
+
=1
∂ ln Li
∂ ln Ki
(b) What restrictions on the coefiicients correspond to constant returns to scale?
(c) How would you estimate the restricted model?
(d) How would you test the hypothesis of constants returns to scale?
10. Suppose the data generating process is given by y = X1 β1 + X2 β2 + ε where X1 is n × k1 , X2 is n × k2 , and the other quantities are vectors. Suppose you estimate this model (call it the “long” model) via OLS, and you also estimate the “short” model, which excludes X2 .
(a) Derive the sum of squared residuals in both models, and sign their difference.
(b) Derive the expected sum of squared residuals in both models, and sign their difference. 3

(c) Suppose β2 = 0. Does this change your answers to parts a and b? Explain.
11. As part of your dissertation research, your senior supervisor suggests you estimate the linear regression model y = Xβ + Gθ + ε where X is n × k, G is n × p, and β and θ are conformable parameter vectors.
The model has no intercept. G is a matrix that indicates whether an observation belongs to one of p mutually exclusive and collectively exhaustive groups. So each column of G is a vector of ones and zeros. The value in column g is one if the observation belongs to group g and zero otherwise.
(a) While writing code to estimate the regression, another graduate student warns you: “don’t forget about the dummy variable trap! If there are p groups, you can only include p − 1 columns of G in the regression!” Is she right? Explain.
(b) (A freebie) It turns out that p is a very large number. Despite your best efforts, you can’t convince your computer to calculate the least squares estimates because it requires inverting a (k + p) × (k + p) matrix. Write down the system of equations your computer is attempting to solve, and identify the problem matrix.
(c) Another helpful graduate student (who has already taken ECON 9720) says
“ no problem! You can calculate the least squares estimate of β without inverting that matrix!? She’s right. Prove it, and show there is a very easy way to compute the least squares estimate of β.
(d) Feeling proud of your accomplishment, you show your estimates β to your advisor. He says “that’s great. But what would be really interesting is an estimate of θ”. He assures you there is a very easy way to compute the least squares estimate of θ (i.e., you could do it by hand if you had to).
What is it?
(e) Your advisor is impressed with your effort to this point, but he has one more question for you: “what proportion of the variation in y is explained by group membership?” Answer the question. (Decompose R2 into a proportion of variation explained by X and a proportion explained by group membership). 12. Consider the linear regression model yi = βxi + εi , where xi is a scalar, E[εi |xi ] = 0, E[ε2 |xi ] = σ 2 . However, the data on xi i have outliers and the researcher would like to avoid that those observations have impact on the estimation of β, so they perform the OLS only on data whose magnitude are less than some chosen constant c. The estimator is given by
˜
β=

n i=1 xi yi 1[|xi | < c]
,
n
2
i=1 xi 1[|xi | < c]

4

where 1[·] is an indicator function whose value is 1 if the statement in bracket is true and 0 otherwise.
˜
(a) Is β a consistent estimator of β? prove your claim.
˜
(b) What is the asymptotic distribution of β?
˜
(c) Compare β with β, the usual OLS estimator obtained from the whole sample.
(d) Suppose that the researcher wants to exclude outliers on the dependent variable yi instead. The estimator is
˜
β=

n i=1 xi yi 1[|yi | < c]
.
n
2
i=1 xi 1[|yi | < c]

Show that this estimator is inconsistent in general.
13.
(a) Lecture notes
(b) Homework 1
(c) Homework 2
(d) Homework 3
(e) Homework 4

Partitionned inverse formula:
A11
A21

A12
A22

−1

= with A−1 (I − A12 F2 A21 A−1 ) −A−1 A12 F2
11
11
11
−F2 A21 A−1
F2
11
F2 = (A22 − A21 A−1 A12 )−1
11

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