Econ 301.04
Econometrics
Bilkent University
Department of Economics
Taskin
Fall 2014
Lab Exercise 4_answers
The data set WAGE1 contain information on 526 workers. Educ ( years of education), exper (years of labor
market experience), tenure ( years with current employer) and female (data which takes the value of one if
the worker is female and describes a qualitative property-gender), are reported for these workers. The
standard wage equation explains the wage relationship such as:
1.
Expected signs in the model: The following relationship explains the change in log(wage) as a
function education, experience, number of years worked in one job, ie tenure and gender.
Log (wage)i  0  1educi   2 exp eri  3tenurei  4 femalei  ui
The expected signs for all variables, other than gender, is positive. If education increases the
percentage change in wage is expected to increase, if experience increases the wage increase will
be larger, and with longer tenure the wage change will also be larger. However wage increases less
for female and hence its coefficient b 4 is expected to be negative.
2. Interpretations of the coefficients:
(log( wage))
 exp er
. 1
  wage
 2
wage   exp er

Hence  2 is the percentage change in wage for one unit (one more year) increase in experience.
The same interpretation can be used for education and tenure variables as well.
The estimated model is:
Dependent Variable: LOG(WAGE)
Method: Least Squares
Date: 10/22/14 Time: 14:22
Sample: 1 526
Included observations: 526
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
EDUC
EXPER
TENURE
FEMALE
0.501348
0.087462
0.004629
0.017367
-0.301146
0.101902
0.006939
0.001627
0.002976
0.037246
4.919885
12.60463
2.845087
5.835235
-8.085414
0.0000
0.0000
0.0046
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
0.392270
0.387604
0.415959
90.14445
-282.4592
84.07213
0.000000
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
1.623268
0.531538
1.093001
1.133545
1.108876
1.775217
TEST OF SIGNIFICANCE OF EACH VARIABLE:
1
3.
Test the null hypothesis that H 0 :  j  0, against the alternative hypothesis H a :  j  0 for
each coefficient. What is your conclusion regarding the presence of each variable in the model?
In testing this hyothesis
(i)
write the null and the alternative hypothesis;
(ii)
compute the test statistic;
(iii)
determine the critical values for the test statistic;
(iv)
decide for the hypothesis,
(v)
interpret your decision economically.
For b1 :
H 0 : 1  0, and H a : b1 ¹ 0
ˆ1 0.0875

 12.61
test statistic for this hypothesis is t  stat 
ˆ ˆ
0.102
(i)
the null and alternative hypothesis are
(ii)
1
(iii)
critical values for the test
statistic: Pr( t / 2
 t  stat  tcritical
/ 2 )  1  0.05
Pr( 1.96  t  stat  1.96)  0.95
critical
(iv)
(v)
Decision: Reject the null hypothesis since 12.61 is out of the range defined by
the t-distribution.
The explanatory variable education is a statistically significant variable in
explaining the percentage change in wages..
For b 2 : In a similar test t-stat for b 2 is 2.85, we can conclude that exper is a is a statistically
significant variable in explaining the percentage change in wages.
For
b3 : In a similar test t-stat for b3 is 5.84, we can conclude that tenure is a is a statistically
significant variable in explaining the percentage change in wages.
For b 4 : In a similar test t-stat for b 4 is -8.09 we can conclude that female is a is a statistically
significant variable in explaining the percentage change in wages.
4.
What is your expectation, a priori, about the sign of
hypothesis against the alternative
H a : 2  0 .
2
coefficient? Now test the above null
For this one-sided hypothesis, the critical value for the statistic is:
Pr(t  stat  tcritical )  1  0.05 or Pr(t  stat  1.64)  0.95
ˆ 2
 2.85 confirms to the above critical t boundaries.
Hence we need to see if the t  stat 
ˆ ˆ
2
Since, the t-stat is not within the interval we can reject the null hypothesis in favour of the
alternative that the coefficient is positive.
5. Calculate a 95% confidence interval for  2 is as follows:


Pr ˆ2  ˆ ˆ .t / 2,d . f .   2  ˆ2  ˆ ˆ .t / 2,d . f .  1  
2
2
Pr0.00463  (0.001627)(1.96)   2  0.00463  (0.001627).(1.96)  1  0.05
Pr0.00463  (0.00318892)   2  0.00463  (00318892)  0.95
Pr0.00144   2  0.00782  0.95
2
6.
The sum of squared residuals (  uˆ i ) is reported as: 90.14445
2
TEST OF OVERALL SIGNIFICANCE OF THE MODEL:
This is to test whether any of the explanatory variables have contributed in the explanation of the variation
in the dependent variable. If they had no effect, then it will be the same thing as saying that all the
coefficients other than the intercept term is equal to zero.
7.
If you impose the restriction that
1  2  3  4  0 , what will be your new model?
Log ( wage) i   0  ui
Dependent Variable: LOG(WAGE)
Method: Least Squares
Date: 10/22/14 Time: 14:22
Sample: 1 526
Included observations: 526
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
1.623268
0.023176
70.04042
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.000000
0.000000
0.531538
148.3297
-413.4396
1.819214
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
1.623268
0.531538
1.575816
1.583925
1.578991
8.
The estimation of this model is as follows and the sum of squared residuals of the new model is:
The percentage change in the SSR is
%change in SSR  [( SSRrestricted mod el  SSRunresstricted mod el ) / SSRunresstricted mod el ]100 
 [148.3297  90.1445) / 90.1445]100  64.546%
In fact this helps us define the F statistic for the F test to test the joint null hypothesis
H 0 : b1 = b2 = b3 = b4 = 0 against the alternative of
H A : at least one  is none zero
(SSRR  SSRU ) / 4 (148.3297  90.1445) / 4
with the F  stat 

 84.233
(SSRU / n  4)
90.1445 /(526  4)
which should have an F distribution if the hypothesis should not be rejected.
Critical value of the F  stat 4, 522  2.37
Hence we can easily reject the nul hypothesis and conclude that the overall model is significant.
9.
If education and experience have the same effect then the model should be:
3
Log (wage) i   0  1educi  1 exp eri   3 tenurei   4 femalei  u i
Dependent Variable: LOG(WAGE)
Method: Least Squares
Date: 10/22/14 Time: 14:32
Sample: 1 526
Included observations: 526
LOG(WAGE)=C(1)+ C(2)*EDUC+ C(2)*EXPER+ C(3)*TENURE+ C(4)
*FEMALE
C(1)
C(2)
C(3)
C(4)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
Null hypothesis is
Coefficient
Std. Error
t-Statistic
Prob.
1.615159
0.003030
0.016444
-0.345348
0.056546
0.001848
0.003390
0.042245
28.56372
1.639368
4.850849
-8.174977
0.0000
0.1017
0.0000
0.0000
0.209579
0.205036
0.473923
117.2430
-351.5848
46.13577
0.000000
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
1.623268
0.531538
1.352034
1.384469
1.364734
1.779596
H 0 : 1   2 and the alternative hypothesis is H A : 1   2
The relevant estimation is:
(SSRR  SSRU ) / 4 (117.24  90.1445) / 1

 157.90
(SSRU / n  4)
90.1445 /(526  4)
Critical value of the F  stat1,522  3.84
The
F  stat 
Hence we can easily reject the null hypothesis and conclude that the education and experience
have different impacts on the wage changes.
10. If education and experience have the same effect then the model should be:
Dependent Variable: LOG(WAGE)
Method: Least Squares
Date: 10/22/14 Time: 14:33
Sample: 1 526
Included observations: 526
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
EXPER
FEMALE
1.747566
0.003759
-0.392970
0.040644
0.001581
0.042921
42.99673
2.377211
-9.155770
0.0000
0.0178
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
0.148832
0.145577
0.491327
126.2536
-371.0583
45.72486
0.000000
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
1.623268
0.531538
1.422275
1.446602
1.431800
1.810468
4
Log (wage) i   0  1 exp eri   4 femalei  u i
Null hypothesis is
H 0 : 1   3  0 and the alternative hypothesis is H 0 : 1   3  0
The relevant estimation is:
The
F  stat 
( SSRR  SSRU ) / 4 (126.25  90.1445) / 2

 104.54
(SSRU / n  4)
90.1445 /(526  4)
Critical value of the F  stat 2, 522  3.00
Hence we can easily reject the null hypothesis and conclude that the education and tenure
have impacts on the wage changes.
5