Basic Econometrics, Gujarati and Porter
CHAPTER 19:
THE IDENTIFICATION PROBLEM
19.1
Using the definitions of M, m, K, and k, and letting R equal
the number of variables (endogenous as well as predetermined)
excluded from a given equation, then, by Definition 19.1,
R = ( M − m) + ( K − k ) ≥ ( M − 1)
Subtracting ( M − m) from each side, we obtain
( K − k ) ≥ m − 1, which is Definition 19.2.
19.2
The structural coefficients are:
β 0 = π 3 − β1π 0
α 0 = π 3 − α1π 0
β1 =
π4
π1
β2 = π 5 −
19.3
α1 =
π 2π 4
π1
π5
π2
α2 = π 4 −
π 1π 5
π2
(a) The reduced form equations are:
Yt = π 0 + π 1 I t + wt
(1)
Ct = π 2 + π 3 I t + wt
(2)
For this system M = 2 (C,Y) and K =1 (I). The order condition
applied to (2) shows that it is exactly identified. The income
identity is identified by definition.
(b) The reduced form equations are:
Wt = π 0 + π 1UN t + π 2 M t + wt
P = π + π UN + π R + π M
(1)
(2)
6 t
7
t
For this system, M = 2 ( W , P ) and K = 3 ( UN , R , M ) .
By the order condition, Eq. (1) overidentified, but Eq. (2) is
just identified.
t
4
5
t
(c) This problem is designed to show the tedious nature of
developing reduced form equations. The solution is left to
the reader.
19.4
See Exercise 19.3. The rank condition test provides the same result.
19.5
The reason that the supply equation is overidentified is that the
demand equation contains two predetermined variables, I and R .
If it contained just one, the supply equation would be just identified.
Thus, if α 2 = 0 or α 3 = 0 , the supply equation would be just
identified.
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Basic Econometrics, Gujarati and Porter
19.6
(a) For this system, M = 2 (Y1, Y2) and K = 2 (X1, X2). By the order
condition, Y1 and Y2 are both exactly identified.
(b) In this case Y1 is identified, but not Y2.
19.7
(a) Following the system (19.2.12) and (19.2.22), it can be shown
that:
πˆ
πˆ
βˆ10 = [πˆ 20 − 22 πˆ10 ] = −3; βˆ12 = 22 = 1.25
πˆ12
πˆ12
πˆ
πˆ
βˆ20 = [πˆ 20 − 21 πˆ10 ] = −6; βˆ21 = 21 = 2
πˆ11
πˆ11
πˆ πˆ
πˆ πˆ
γˆ11 = [πˆ 21 − 11 21 ] = 2.25; γˆ12 = [πˆ 22 − 12 21 ] = −6
πˆ11
πˆ11
(b) To test this hypothesis, we need the standard error of γˆ11 . But as
you can see from (a), γˆ11 is a nonlinear function of the πˆ coefficients
and it is not easy to estimate its standard error.
19.8
(a) In this example, Y1 is not identified but Y2 is. This system is
similar to the system (19.2.12) and (19.2.13). Thus,
πˆ
βˆ21 = 3 = 1.5; βˆ20 = (πˆ 2 − βˆ21πˆ 0 ) = −4
πˆ1
The other structural coefficients cannot be identified.
(b) In this case both Y1 and Y2 are identified.
19.9
In this system M = 4 and K = 5. Here all the equations are
overidentified.
19.10 Here M = 4 and K = 4. By the order condition, Y1 and Y2 are not
identified, but Y3 and Y4 are just identified.
19.11 Here M = 5 and K = 4. By the order condition, Y1 , Y2 , and Y5 are
just identified, Y3 is not identified and Y4 is overidentified.
To show how the rank condition works, consider the first equation.
It excludes variables Y3 , Y5 , X 2 and X 3 . For this equation to be
identified, there must be at least one 4 x 4 non-zero determinant
from coefficients of the variables excluded from this equation but
included in the remaining equations. One such determinant is:
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Basic Econometrics, Gujarati and Porter
β 23 0 γ 22 γ 23
1
β 35 0 λ33
≠0
0 0 0 γ 43
0 1 γ 52 γ 53
Thus by the rank condition also the first equation is identified.
Follow a similar procedure for the other equations.
19.12 For this model, M = 4 and K = 2. By the order condition,
the equation for C is identified, and those for I and T are
overidentified.
With r treated as exogenous, M = 4 and K = 3. By the order
condition now the equations for C, T, and I are all overidentified.
19.13 From Eq. (19.1.2), the reduced form of the income equation is:
Yt = π 0 + π 1 I t + ut
The OLS results are:
Yˆt = 10.0000 + 5.0000 I t
t = ( 8.458) (12.503)
R 2 = 0.897
From Eq. (19.1.4), the reduced form for consumption is:
Ct = π 2 + π 3 I t + wt
The OLS results are:
Cˆt = 10.000 + 4.000 I t
t = (8.458) (10.002)
r 2 = 0.848
For this model, M = 2 and K = 1. By the order condition, the
consumption function is just identified. The estimates of the
structural coefficients are:
πˆ
πˆ
βˆ0 = 0 = 2; βˆ1 = 3 = 0.8
πˆ1
πˆ 2
19.14 See Exercise 19.1. From Eq. (19.3.1), with Definition 19.2,
K − k ≥ m −1
Add k to each side. This yields
K ≥ m + k −1
19.15 (a) The reduced form equations are:
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Basic Econometrics, Gujarati and Porter
Y 
Pt = π 1 + π 2  t  + π 3 Ft + π 4Wt + π 5Ct −1 + π 6Tt −1 + π 7 Pt −1 + vt
 Nt 
Y 
Qt = π 8 + π 9  t  + π 10 Ft + π 11Wt −1 + π 12Ct −1 + π 13Tt −1 + π 14 Pt −1 + wt
 Nt 
(b) Here M =2, K = 7. By the order condition, both equations are
overidentified.
Empirical Exercises
19.16 (a) & (b) Here M = 2 and K = 2. By the order condition, the demand
function is not identified and the supply function is overidentified.
(c) The reduced form equations are:
Yt = π 0 + π 1 Rt + π 2 Pt + vt
M t = π 3 + π 4 Rt + π 5 Pt + wt
(d) To test for simultaneity in the supply function,
(1) Estimate the reduced form for Yt and obtain the residuals, vˆt
(2) Regress Mt on Yt and vˆt
(3) The null hypothesis is that there is no simultaneity, i.e., the
coefficient of vˆt in step (2) is not statistically significant.
The Stata results of this exercise are as follows:
Source |
SS
df
MS
-------------+-----------------------------Model | 120078361 2 60039180.3
Residual | 2071202.91 34 60917.7326
-------------+-----------------------------Total | 122149564 36 3393043.43
Number of obs = 37
F( 2, 34) = 985.58
Prob > F = 0.0000
R-squared = 0.9830
Adj R-squared = 0.9820
Root MSE
= 246.82
d-statistic = 0.2538
-----------------------------------------------------------------------------m2 | Coef.
Std. Err.
t P>|t| [95% Conf. Interval]
-------------+---------------------------------------------------------------Y | 0.8055752 0.0184182 43.74 0.000 .7681449 .8430054
vˆt | -0.02237 0.1043517 -0.21 0.832 -.2344382 .1896982
_cons | -2517.945 134.307
-18.75 0.000 -2790.889
-2245
-----------------------------------------------------------------------------Since the coefficient of the residual term is not statistically
significant, do not reject the null hypothesis that there is no
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Basic Econometrics, Gujarati and Porter
simultaneity. Therefore, we can assume there is no simultaneity
present.
(e) Here we use the exogeneity test discussed in the chapter. We
estimate the following regression:
M t = β1 + β 2Yt + β 3Yˆt + ut
(1)
where Yˆ is obtained from the regression of the reduced form for Y
t
given in (c).
If the estimated β3 is statistically different from zero, reject the
hypothesis that Yt is exogenous.
Source |
SS
df
MS
-------------+-----------------------------Model | 120078361 2 60039180.3
Residual | 2071202.9 34 60917.7323
-------------+-----------------------------Total | 122149564 36 3393043.43
Number of obs = 37
F( 2, 34) = 985.58
Prob > F = 0.0000
R-squared = 0.9830
Adj R-squared = 0.9820
Root MSE
= 246.82
-----------------------------------------------------------------------------m2 | Coef.
Std. Err.
t
P>|t| [95% Conf. Interval]
-------------+---------------------------------------------------------------Y | 0.7832051 0.1027135 7.63 0.000 0.5744663 0.991944
Y_hat | 0.0223701 0.1043517 0.21 0.832 -0.1896982 0.2344383
_cons | -2517.945
134.307
-18.75 0.000 -2790.889
-2245
-----------------------------------------------------------------------------Dependent Variable: M2
Variable
C
Yt
Yˆt
Coefficient
-2295.7898
0.3292
0.4791
R2 =0.9928
d = 0.5946
Std. Error
78.9873
0.06825
0.0695
t-Statistic
-29.0652
4.8238 4
6.8929
As these results show, the coefficient of Yˆt is not statistically
significant, leading to the conclusion that Y is exogenous.
19.17 (a) To test this, we can apply the techniques from Section 8.7 for
testing the equality of two regression coefficients using the restricted
least squares F test. The implications of this are left to the reader to
explore.
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Basic Econometrics, Gujarati and Porter
(b) To see if νˆt is correlated with u2t , we can treat νˆt as a regressor
in the Qt model (19.4.7) and, after the equation substitution,
perform another restricted least squares F test.
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