Econ 301
Econometrics
Bilkent University
Department of Economics
Taskin
Mid Term Exam II
December 13, 2014
Name ___________________________
For each hypothesis testing in the exam complete the following steps: Indicate the test statistic,
its critical value, your decision on the null hypothesis and its economic interpretation for
each hypothesis.
In your numerical calculations: Show all your computations, complete the calculations to
receive full points.
Please, answer individual sections of each question in the order they are asked.
Please do not write on the margins, you may use the back of each sheet.
1. (24 points) SCALING OF VARIABLES
Consider the following equation:
Yi = b1 + b2 Xi + ui
which is estimated to be:
Yˆi = 4.40 + 0.869Xi
se. (1.23) (0.117)
R 2  0.756
a) If the values of the Xi are multiplied by 2 such as X i *  X i x 2 , find the numerical
values of intercept and the slope coefficient of the following regression using the
coefficient estimates above (show all your steps)
Yi  1  2 X *2i ui
b) How will the residual, ûi , Vaˆr (ˆ2 ) , t-stat and R2 be affected by this scaling. Illustrate
step by step.
c) If the values of the X i and Yi are both multiplied by 2 such as X i *  X i x 2 , and
Yi *  Yi x 2 ; what will be the numerical values of intercept and the slope coefficient of
the following regression (show all your steps).
Y *i  1   2 X *2i ui
Name ___________________________
Question 1 answers (con’t)
2
Name ___________________________
2. (30 points) (FUNCTIONAL FORM, HYPOTHESIS TESTING AND RESTRICTIONS)
The production function for the air transport is given by the following equation
log(Yi ) = b1 + b2 log(Li )+ b3 log(Ki )+ b4t + ui
where Yi is the output; Li is the labor input; K i is the capital input; and t is the time
trend that takes the values 1, 2 ,3, …, 32. The estimated equation is:
Dependent Variable: LOG(Y)
Method: Least Squares
Date: 12/09/10 Time: 23:21
Sample: 1 32
Included observations: 32
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(L)
LOG(K)
TIME
-0.206271
1.555910
-0.062296
0.020077
0.090808
0.144506
0.056165
0.007897
-2.271503
10.76710
-1.109171
2.542197
0.0310
0.0000
0.2768
0.0168
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.993578
0.992890
0.076893
0.165553
38.82121
0.945304
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
a) Before the estimation, explain the expected signs of the
= ¶log(Yi ) / ¶t ).
b2 , b3 and b 4
1.969834
0.911935
-2.176325
-1.993108
1444.083
0.000000
(which is )
b) Test the hypothesis that  2 are individually significant. (Your formal test should include
null and the alternative hypothesis, test statistics, conclusion and interpretation.)
c) What are the mathematical and economic interpretation of
b2 and b3 coefficients.
d) What is the economic interpretation of the b 4 coefficient? (What is the graphical
interpretation of b 4 for the production function in a Y and L space?) Test the hypothesis
that H 0 : b4 = 0 ; against the alternative that H a : b4 > 0. What is your conclusion?
log(Yi / Ki ) = b1 + b2 log(Li / Ki )+ ui ,
what are the restrictions on b2 , b3 and b 4 . (You need to do simplifications in the first
e) If you estimate
equation.)
f) If the sum of squared residual of the above restricted equation (in e) is calculated to be
0.387831, formally test this restriction.
3
Name ___________________________
Question 2 answers (con’t)
4
Name ___________________________
Question 3 answers (con’t)
5
Name ___________________________
3. (30 points) DUMMY VARIABLES
The following is the statistical model used to estimate the consumption expenditure function
with quarterly data, for the period 1974 Q1 to 1984 Q4, in United Kingdom (ie. n=44)
Ct  1  1 D1t   2 D2t   3 D3t   2Yt  1 ( D1t *Yt )   2 ( D 2t *Yt )   3 ( D3t *Yt )  ut
where
C t is the real total consumption expenditures for the quarter,
Yt is the real personal disposable income for the quarter,
D1t is a dummy variable that takes the value of 1 for the first quarter and 0 otherwise.
D2t is a dummy variable that takes the value of 1 for the second quarter and 0 otherwise.
D3t is a dummy variable that takes the value of 1 for the third quarter and 0 otherwise.
The results of the estimation is:
============================================================
LS // Dependent Variable is CONSUMPTION
Sample: 1974:1 1984:4
Included observations: 44
============================================================
Variable
Coefficien
Std. Error
t-Statistic
Prob.
============================================================
C
-3022.704
4792.427
-0.630725
0.5322
D1
1277.218
6303.654
0.202615
0.8406
D2
3370.955
6916.018
0.487413
0.6289
D3
1386.167
6266.817
0.221192
0.8262
INCOME
1.035047
0.090584
11.42641
0.0000
D1*INCOME
-0.009703
0.122342
-0.079313
0.9372
D2*INCOME
-0.076366
0.131093
-0.582536
0.5638
D3*INCOME
-0.020470
0.120583
-0.169759
0.8661
============================================================
R-squared
0.941246
Mean dependent var
50234.07
Adjusted R-squared
0.929822
S.D. dependent var
4689.827
S.E. of regression
1242.388
Akaike info criter
14.41255
Sum squared resid
55567012
Schwarz criterion
14.73695
Log likelihood
-371.5093
F-statistic
82.38969
Durbin-Watson stat
1.267609
Prob(F-statistic)
0.000000
============================================================
Coefficient Covariance Matrix: Diagonal elements are VAR( b i ) and off-diagonal ones are Cov( bi , b j )
====================================================================================








====================================================================================

22967355 -22967355 -22967355
-22967355
-432.78
432.7877 432.7877
432.7877

-22967355
39736049 22967355
22967355
432.7877 -768.1134 -432.7877
432.7877

-22967355
22967355 47831308
22967355
432.7877 -432.7877 -903.9751
-432.7877

-22967355
22967355 22967355
39272999
432.7877 -432.7877 -432.7877
-752.7958

-432.7877
432.7877
432.7877
432.7877
0.008205 -0.008205 -0.008205
-0.008205

432.7877 -768.1134 -432.7877
-432.7877
-0.008205 0.014968 0.008205
0.008205

432.7877 -432.7877 -903.9751
-432.7877
-0.008205 0.008205 0.017185
0.008205

432.7877 -432.7877 -432.7877
-752.7958
-0.008205 0.008205 0.008205
0.014540
============================================================================== =====
6
Name ___________________________
a) Which one is the base category and write the equation that represents this category?
b) Formulate and test the hypothesis that the autonomous consumption is the same in the
first and the fourth quarter.
c) Formulate and test the hypothesis that the autonomous consumption is the same in the
second and the third quarter.
d) Formulate and test the hypothesis that marginal propensity to consume in the first
quarter is greater than the fourth quarter.
e) Formulate the hypothesis and describe (but DO NOT TEST) how you will test that
marginal propensity to consume is the same in all quarters.
f) Formulate the hypothesis and describe (but DO NOT TEST) how you will test that
quarters do not affect consumption behaviour.
[HINT: INTERCEPT SHOWS AUTONOMOUS CONSUMPTION AND SLOPE SHOWS MARGINAL
CONSUMPTION]
[HINT: YOUR FORMAL TESTS SHOULD INCLUDE HYPOTHESIS, TEST STATISTICS,
CRITICAL STATICS AND INTERPRETATION]
7
Name ___________________________
8
Name ___________________________
4. (16 points) TRUE FALSE ON MULTICOLLINEARITY
Indicate and explain whether the following statements are True or False? Your explanations
should involve a clear explanation and/or formal proof of your statement. The completeness of
your answer will determine the points you will receive.
1) In a regression model
Yi = b1 + b2 X2i + b3 X3i + ui ; as the correlation coefficient between X2i
and X3i increases, the Var(b̂2 ) and Var(b̂3 ) declines, because more the variation in
explained by the explanatory variables.
Yi can be
2) You will not obtain a high R2 value in a multiple regression if all the partial slope coefficients
are individually statistically in significant on the basis of t-tests.
3) Even with perfect multicollinearity the OLS estimates are B.L.U.E.
9
Name ___________________________
FORMULA SHEET
ˆ 2 =
nå X iYi - å X i åYi
nå X - (å X i )
2
i
å
ˆ1 = Y - b2 X =
2
=
å( X - X )(Y -Y )
å( X - X )
i
i
2
i
where i = 1….n
X i2 åYi - å X i å X iYi
nå X i2 - (å X i )2
where i = 1….n


 X 2i

Var ( ˆ1 )  ˆ 2 
2 
n

(
X

X
)
i




1

Var ( ˆ2 )  ˆ 2 
2 
 ( X i  X ) 


X

Cov( ˆ1 ˆ 2 )  ˆ 2 
2 
 ( X i  X ) 
sˆ 2 =
t stat 
R2 
å ûi2
n-K
ˆ j   j
seˆ( ˆ j )
SSE
SSR
 1
SST
SST
R 2  1
SSE is sum of squared explained, SSR is sum of squared residuals
SSR /( n  K )
TSS /( n  1)
Var ( x  y )  Var ( x)  Var ( y )  2Cov( xy)
F  stat 
SSRR  SSRU /( J )
SSRU /( n  K )
  ei 2  2k
AIC  ln 

 n  n
10
Name ___________________________
11
Name ___________________________
12