Transcript Chapter 5 Chow test and dummy variable group test

Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 5)
Slideshow: Chow test and dummy variable group test
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 5). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
In the dummy variable sequences and in the Chow test sequence we investigated whether
the cost functions for occupational and regular schools are different.
1
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
In each case we performed tests that showed that the functions are significantly different.
Could the two approaches have led to different conclusions?
2
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
The answer is no. The Chow test is equivalent to an F test testing the explanatory power of
the dummy variables as a group.
3
CHOW TEST AND DUMMY VARIABLE GROUP TEST
. reg COST N
Source |
SS
df
MS
---------+-----------------------------Model | 5.7974e+11
1 5.7974e+11
Residual | 8.9160e+11
72 1.2383e+10
---------+-----------------------------Total | 1.4713e+12
73 2.0155e+10
Number of obs
F( 1,
72)
Prob > F
R-squared
Adj R-squared
Root MSE
=
74
=
46.82
= 0.0000
= 0.3940
= 0.3856
= 1.1e+05
-----------------------------------------------------------------------------COST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------N |
339.0432
49.55144
6.842
0.000
240.2642
437.8222
_cons |
23953.3
27167.96
0.882
0.381
-30205.04
78111.65
------------------------------------------------------------------------------
With both approaches the starting point is a simple regression of annual recurrent
expenditure on the number of students enrolled, using the entire sample. We make a note
of RSS.
4
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
L3
The regression line is shown graphically.
5
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
We now make a distinction between occupational schools and regular schools.
6
CHOW TEST AND DUMMY VARIABLE GROUP TEST
. reg COST N OCC NOCC
Source |
SS
df
MS
---------+-----------------------------Model | 1.0009e+12
3 3.3363e+11
Residual | 4.7045e+11
70 6.7207e+09
---------+-----------------------------Total | 1.4713e+12
73 2.0155e+10
Number of obs
F( 3,
70)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
74
49.64
0.0000
0.6803
0.6666
81980
-----------------------------------------------------------------------------COST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------N |
152.2982
60.01932
2.537
0.013
32.59349
272.003
OCC | -3501.177
41085.46
-0.085
0.932
-85443.55
78441.19
NOCC |
284.4786
75.63211
3.761
0.000
133.6351
435.3221
_cons |
51475.25
31314.84
1.644
0.105
-10980.24
113930.7
------------------------------------------------------------------------------
With the dummy variable approach, we take one type of school as the reference category.
We will choose regular schools for this category, but it makes no difference to the test.
7
CHOW TEST AND DUMMY VARIABLE GROUP TEST
. reg COST N OCC NOCC
Source |
SS
df
MS
---------+-----------------------------Model | 1.0009e+12
3 3.3363e+11
Residual | 4.7045e+11
70 6.7207e+09
---------+-----------------------------Total | 1.4713e+12
73 2.0155e+10
Number of obs
F( 3,
70)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
74
49.64
0.0000
0.6803
0.6666
81980
-----------------------------------------------------------------------------COST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------N |
152.2982
60.01932
2.537
0.013
32.59349
272.003
OCC | -3501.177
41085.46
-0.085
0.932
-85443.55
78441.19
NOCC |
284.4786
75.63211
3.761
0.000
133.6351
435.3221
_cons |
51475.25
31314.84
1.644
0.105
-10980.24
113930.7
------------------------------------------------------------------------------
We add an intercept dummy and a slope dummy to allow the overhead and marginal costs
of the occupational schools to be different. Again we make a note of RSS, which is smaller
than before.
8
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
Here are the regression lines for the two subsamples.
9
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Whole sample
^ = 24,000 + 339N
COST
RSS = 8.91x1011
Whole sample
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
To see if the cost functions are significantly different, we investigate whether there is a
significant reduction in RSS when the dummy variables are added.
10
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Whole sample
^ = 24,000 + 339N
COST
RSS = 8.91x1011
Whole sample
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
4.71 1011 / 70
We perform the F test described in the sequence on slope dummy variables. The numerator
of the test statistic is the reduction in RSS on adding the dummy variables, divided by the
cost in terms of degrees of freedom.
11
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Whole sample
^ = 24,000 + 339N
COST
RSS = 8.91x1011
Whole sample
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
4.71 1011 / 70
The denominator is the RSS remaining after adding the dummy variables, divided by the
number of degrees of freedom remaining.
12
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Whole sample
^ = 24,000 + 339N
COST
RSS = 8.91x1011
Whole sample
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
4.71 1011 / 70
F (2,70)crit,0.1%  7.6
The critical value of F at the 0.1% level with 2 and 70 degrees of freedom is 7.6. Hence we
conclude that the dummy variables do have significant explanatory power and the cost
functions are different.
13
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
L3
With the Chow test approach we also start by running a regression using the whole sample,
and make a note of the RSS.
14
CHOW TEST AND DUMMY VARIABLE GROUP TEST
. reg COST N if OCC==0
Source |
SS
df
MS
---------+-----------------------------Model | 4.3273e+10
1 4.3273e+10
Residual | 1.2150e+11
38 3.1973e+09
---------+-----------------------------Total | 1.6477e+11
39 4.2249e+09
Number of obs
F( 1,
38)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
40
13.53
0.0007
0.2626
0.2432
56545
-----------------------------------------------------------------------------COST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------N |
152.2982
41.39782
3.679
0.001
68.49275
236.1037
_cons |
51475.25
21599.14
2.383
0.022
7750.064
95200.43
------------------------------------------------------------------------------
We then split the sample into occupational and regular schools, and run separate
regressions, again making a note of RSS. This is the regression output when COST is
regressed on N for the subsample of 40 regular schools.
15
CHOW TEST AND DUMMY VARIABLE GROUP TEST
. reg COST N if OCC==1
Source |
SS
df
MS
---------+-----------------------------Model | 6.0538e+11
1 6.0538e+11
Residual | 3.4895e+11
32 1.0905e+10
---------+-----------------------------Total | 9.5433e+11
33 2.8919e+10
Number of obs
F( 1,
32)
Prob > F
R-squared
Adj R-squared
Root MSE
=
34
=
55.52
= 0.0000
= 0.6344
= 0.6229
= 1.0e+05
-----------------------------------------------------------------------------COST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------N |
436.7769
58.62085
7.451
0.000
317.3701
556.1836
_cons |
47974.07
33879.03
1.416
0.166
-21035.26
116983.4
------------------------------------------------------------------------------
And this is the regression output when COST is regressed on N using the subsample of 34
occupational schools.
16
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
The graph shows the regression lines.
17
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
The regression equations are as shown.
18
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
The cost functions are identical to those implicit in the dummy variable regression with both
intercept and slope dummies. This is because the dummy variable regression has a dummy
variable for each component of the original model (here, the constant and N).
19
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
Implicit cost function for regular schools
^ = 51,000 + 152N
COST
The intercept and the coefficient of N in the dummy variable regression are then chosen so
as to minimize the residual sum of squares for the reference category, the regular schools.
Hence they must be the same as for the regression with regular schools only.
20
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 – 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
Implicit cost function for regular schools
^ = 51,000 + 152N
COST
Implicit cost function for occupational schools
^ = 47,000 + 436N
COST
The intercept and slope dummies then allow the intercept and slope coefficient to be
modified so as to give the best possible fit for the occupational schools. Hence the implicit
cost function must be the same as for the regression with occupational schools only.
21
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
The cost function for regular schools implicit in the dummy variable regression must
coincide with the regression line for the regular schools only.
22
CHOW TEST AND DUMMY VARIABLE GROUP TEST
700000
600000
COST
500000
400000
300000
200000
100000
0
0
200
400
600
800
1000
1200
1400
N
Occupational schools
Regular schools
Similarly, the cost function for occupational schools implicit in the dummy variable
regression must coincide with the regression line for the occupational schools only.
23
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
Implicit cost function for regular schools
^ = 51,000 + 152N
COST
Implicit cost function for occupational schools
^ = 47,000 + 436N
COST
Since the cost functions implicit in the dummy variable regression coincide with those in
the separate regressions, the residuals will be the same. It follows that RSS for the dummy
variable regression must be equal the sum of RSS for the separate regressions.
24
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
Hence the F statistics for the F tests will be the same. The starting point for both
approaches is the residual sum of squares for the basic regression making no distinction
between types of school.
25
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
In the Chow test approach, RSS is reduced by splitting the sample. In the dummy variable
approach, RSS is reduced by adding the intercept and slope dummies. RSS after making
the change will be the same because the residuals will be the same.
26
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
This also means that the first part of the denominator of the F statistic will be the same.
27
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
The cost of the improvement in the fit is the same, since either way two extra parameters
have to be estimated.
28
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
And either way, the number of degrees of freedom remaining will be 70, since the number of
observations is 74 and 4 parameters have to be estimated.
29
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
F (2,70)crit,0.1%  7.6
Thus all the components of the F statistics are the same, and the outcome of the test must
be the same. In this case, the null hypothesis of identical cost functions for the two types of
school was rejected at the 0.1% level.
30
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
F (2,70)crit,0.1%  7.6
What are the advantages and disadvantages of the two approaches?
31
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
F (2,70)crit,0.1%  7.6
The Chow test is quick. You just run the three regressions and compute the test statistic.
But it does not tell you how the functions differ, if they do.
32
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
F (2,70)crit,0.1%  7.6
The dummy variable approach involves more preparation because you have to define a
dummy variable for the intercept and for each slope coefficient.
33
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Regular schools only
^ = 51,000 + 152N
COST
RSS = 1.22x1011
Occupational schools only
^ = 47,000 + 436N
COST
RSS = 3.49x1011
Whole sample, with dummy variables
^ = 51,000 - 4,000OCC + 152N + 284NOCC
COST
RSS = 4.71x1011
(8.91 1011  [3.49  1011  1.22  1011]) / 2
F (2,70) 
 31.2
(3.49  1011  1.22  1011) / 70
(8.91 1011  4.71 1011) / 2
F (2,70) 
 31.2
11
4.71 10 / 70
F (2,70)crit,0.1%  7.6
However, it is more informative because you can perform t tests on the individual dummy
coefficients and find out where the functions differ, if they do.
34
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
A final note. The Chow test and the dummy variable group test are equivalent only if there
is a full set of dummy variables.
35
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
By this is meant an intercept dummy (here D) and a slope dummy variable for every X (here
DX2, DX3, … DXK).
36
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
If there is a full set of dummy variables, OLS will choose the intercept b1 and the b
coefficients of X2 … XK so as to optimise the fit for the D = 0 observations. The coefficients will be
exactly the same as if the regression has been run with only the subsample of D = 0 observations.
37
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
The coefficient of the intercept dummy D and the slope dummy variables will then be
chosen so as to optimise the fit for the D = 1 observations. (b1+d), (b2+l2), …, (bK+lK) will be
the same as the coefficients in a regression using only the subsample of D = 1 observations.
38
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
Thus with a full set of intercept and slope dummy variables, the improvement in fit on
adding the dummy variables to the basic equation is the same as that obtained by splitting
the sample and running separate subsample regressions.
39
CHOW TEST AND DUMMY VARIABLE GROUP TEST
Y = b1 + b2X2 + b3X3+ … + bKXK + u
Y = b1 + b2X2 + b3X3+ … + bKXK + dD + l2DX2 + l3DX3 + … + lKDXK + u
D=0
Y = b1 + b2X2 + b3X3+ … + bKXK + u
D=1
Y = (b1+d) + (b2+l2)X2 + (b3+l3)X3 + … + (bK+lK)XK + u
It follows that the F statistic for the test of the joint explanatory power of the intercept and
slope dummy variables is equivalent to the F statistic for the Chow test.
40
Copyright Christopher Dougherty 2011.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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11.07.25