Econ 301.
Econometrics
Bilkent University
Department of Economics
Taskin
Sample questions from the text-book
As a studying exercise solve the following questions in the text-book.
Chapter 2: 2.5, 2.6, 2.8 ((i, ii),
Chapter 3: 3.2, 3.4, 3.5, 3.12 (i, ii), 3.13,
Chapter 4: 4.7.
Sample questions
1.
(25 points) In the statistical model:
y i   1   2 xi   i
i  1...., n,
where the random errors  i ’s are independent
’s are independent variables, and
2.
N (0,  2 ) variables. Yi are dependent variables, and X i
 1 ,  2 are unknown parameters.
E (Yi ) and  i , give graphical and verbal explanations.
a.
What are
b.
If
c.
d.
Find an expression for the sum of squared residuals.
Derive the normal equations for the Ordinary Least Square estimation and the OLS estimators.
Yˆi is the fitted line with the estimated parameters ̂ 1 , ̂ 2 , write the equation for the residual.
(25 points) In the statistical linear model:
yi    xi   i
i  1...., n,
where the random errors  i ’s are independent
N (0,  2 ) variables. Yi are dependent variables, and X i
 ,  are unknown parameters.
Write the formula for ordinary least square estimator for  .
’s are independent variables, and
a.
b.
c.
3.
What do we mean by an unbiased estimator?
Draw a hypothetical probability distribution for i) unbiased estimator, ii) biased estimator, iii) biased
estimator with small variance, iv) unbiased estimator with large variance.
~

0.5Yn  0.5Y1
~
is an alternative estimator, prove that  is an unbiased estimator of  .
0.5 X n  0.5 X 1
d.
If
e.
What is an Mean Square Error? Write the formula for the Mean Square Error.
(25 points) The following data refers to transaction demand for cash balances M (in billion dollars) and
national income Y(billions of dollars) for an economy over 11 years:
M
21.3
24.2
26.4
27.1
28.5
29.2
30.1
33.2
34.7
37.2
39.0
Y
80.6
95.1
103.4
110.3
114.3
117.3
120.8
134.4
139.2
150.3
156.2
( M i  M ) 2  298.9
(Yi  Y ) 2  5386.7 ,
( M i  M )(Yi  Y )  1265.9
M i    Yi   i with the above data.
a. Compute the Least Square Estimate of the parameters,  and  .
The objective is to estimate the money demand equation,
1
b.
c.
What is the total variation in the dependent variable of the model?
If the sum of squared residuals is equal to 1.151518, compute the coefficient of determination.
d.
What is the s , the estimated residual variance.
e.
What is the estimated s ˆ , the estimated coefficient variance?
f.
Construct a 95% confidence interval for
2
2
[Caution: the dependent variable is
.
M i and the independent variable is Yi .
4. (24 points) In the regression model
yt  1  xt 2  et
a.
b.
c.
Equation 1
Write and explain the GAUSS MARKOV Theorem.
What are the necessary assumptions, regarding the error term, for this theorem to be true? Write
and explain in your own words each of these assumptions.
If you learn the following information about the error term et
et  1   2 z t   t ,
z t is a fixed (nonrandom) economic variable,  ’s are the coefficients of this equation, and
 t is a disturbance term with mean zero and constant variance, i.e. E (  t )  0, and
where
Var(t )   2 , what can you say about the properties of the error term et ?
d.
Given the information in (c) does the Gauss Markov Theorem hold for the parameters estimated in
Equation 1.
5. Short answers: MATRIX NOTATIONS, Choose two to answer.
a)
In the model Y  X   where
Show and prove that OLS estimator
 is distributed N (0,  2 ) ,
̂ defined as ( X X ) 1 X Y is an unbiased estimator of  .
E (ee)   2 I
2
1
c) Show that Var ( ˆ )   ( X X ) .
b) Show that
d) What are the assumptions regarding the explanatory variables X matrix. What is the value of E ( X  )
or the covariance between X and
.
2
6. In the statistical model:
Yt  1   2 X t   t
 t is the random error with ~ N (0,  2 ) , Yt is the dependent variable and X t is the explanatory variable
which is fixed (nonstochastic).
a) Explain and graphically plot
E[Yt ] and  t for five hypothetical observations on the graph below.
b) What can you say about the distributions of Yt ,  2 and  t .
c)
If
1 and  2
are the ordinary least square estimators of the parameters of the above model, write the
equation of the fitted line (sample regression line) and draw the fitted line on the below graph.
d) What can you say about the distributions of
e)
2 .
What is the ‘residual’? Write the equation of the residual and show the residual on the graph.
Yt
xt
7.
Given the following regression equations which can be compared using R2 statistics to choose
the best regression and why? Explain.
i. Yt  1   2 X t   t estimated with 50 observations
Yt  1   2 X t   Zi   t estimated with 25 observations.
iii. log(Yt )  1   2 log( X t )   t estimated with 50 observations
iv. Yt  1   2 X t   (1/ Zi )   t estimated with 25 observations.
ii.
3