Simultaneous Equation Models Exercises
Exercise 13.2
From Section 13.4 of Ramanathan, the ILS estimate of 
is given by 
=
SCI
/ ( SCI + SII ). Applying OLS to
the
reduced form equation for
Yt , we get 
=
SYI / SII (compare this with
equation
(3.12) in Ramanathan). From equation (13.14) we see that =
1 / (1 - 
).
Hence the ILS estimator is
Because Y = C + I , we have
Hence, =
(
SYI - SII ) / SYI
) = SCI / ( SCI + SII
), which is the same as .
Therefore ILS can be applied to the reduced form of either Ct
or Yt .
Exercise 13.4
a. Because there are three endogenous variables, at least two variables
(either endogenous or exogenous) must be absent in each equation. In
the second equation A and X are absent. In the third equation
M, Y, and U are excluded. The third equation
is therefore overidentified (Y
and U are available as
instrumental variables for A).
The second equation is exactly identified (there are no endogenous
variables that need instruments).
b. The equation for M is
already in the reduced form. For A
we have
.
Solving for A we get the
reduced form for A as
.
The reduced from for X is
.
c. First regress A against a
constant, Y, P, and U and save 
.
Next regress X against a
constant, P, and
to obtain estimates of 1, 2, and 3.
d. The second equation has no other endogenous variables and so is
already in the reduced form. OLS will bive estimates that are unbiased
and consistent.
e. Because A is correlated
with v (see b above), OLS
estimates of the third equation will be biased and inconsistent.
Exercise 13.5
a. I would expect 
2>0, 4>0,
5<0, 2<0, 3<0, 
2>0,
and , 3>0. The sign of 3 is ambiguous: if POPDEN
is high there are more houses to burgle and hence POPCRIME might go up;
on the other hand, if an area is dense, more people might be alert and
report crimes. Simlilarly, the effect of age on the incidence of
violent crime ( 4) is
ambiguous.
b. We need two variables (endogenous or exogenous) to be absent from
each equation. The first equation has DEATH, MEDAGE, and VIOLNTCRIME
missing; the first two are exogenous variables that can act as
instruments for POLICE, which is endogenous. The second equation has
MEDHOME, POPDEN, UNEMP, and PROPCRIME missing; the first three of these
are exogenous variables that can act as instruments for POLICE. The
third equation has MEDHOME, POPDEN, UNEMP, DEATH, and MEDAGE missing;
all of these are exogenous variables that can be used as instruments
for PROPCRIME and VIOLNTCRIME. The order condition is therefore
satisfied by all equations.
c. First regress each of POLICE, PROPCRIME, and VIOLNTCRIME against a
constant, MEDHOME, POPDEN, UNEMP, DEATH, and MEDAGE, and save the
predicted values. In the second stage use the predicted values in place
of the actual values and estimate the three equations. In obtaining
residuals and standard errors, however, actual values for these
variables will be used.
Exercise 13.6
Substituting for y1
from the first equation into the second and solving for y2, we get
= 
1 x1 + 2
x2 + 3
x3 + 
1
Substituting this into the first equation, we have
y1 = (1 1 + 2) x1 + 1
2 x2 + 1
3 x3 + 2
= 
1 x1 + 2
x2 + 3
x3 + 2
Note that 2 = 1
2 and 3
= 1 3. Hence, 1 can be estimated two
ways as 2 / 2
and as 3 / 3. Thus the model is overidentified
and the appropriate estimation procedure is 2SLS. In the first stage,
regress y1 and y2 against x1, x2, and x3 and compute 
1
and 2 . Next estimate the
two structural equations using i instead of yi
(i - 1,2).
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Updated April 7, 2006