Solutions to the Review Questions at the End of Chapter 6
1. (a) This is simple to accomplish in theory, but difficult in practice as a
result of the algebra. The original equations are (renumbering them (1),
(2) and (3) for simplicity)
y1t   0  1 y 2 t   2 y 3t   3 X 1t   4 X 2 t  u1t
(1)
y 2 t  0  1 y 3t  2 X 1t  3 X 3t  u2 t
(2)
y 3t   0   1 y1t   2 X 2 t   3 X 3t  u3t
( 3)
The easiest place to start (I think) is to take equation (1), and substitute in for
y3t, to get
y1t   0  1 y 2 t   2 ( 0   1 y1t   2 X 2 t   3 X 3t  u3t )   3 X 1t   4 X 2 t  u1t
Working out the products that arise when removing the brackets,
y1t  0  1 y 2 t  2  0  2  1 y1t   2  2 X 2 t  2  3 X 3t   2 u3t  3 X 1t   4 X 2 t  u1t
Gathering terms in y1t on the LHS:
y1t   2  1 y1t  0  1 y 2 t  2  0  2  2 X 2 t   2  3 X 3t  2 u3t   3 X 1t   4 X 2 t  u1t
y1t (1   2  1 )   0  1 y 2 t   2  0   2  2 X 2 t  2  3 X 3t   2 u3t  3 X 1t  4 X 2 t  u1t
(4)
Now substitute into (2) for y3t from (3).
y 2 t  0  1 ( 0   1 y1t   2 X 2 t   3 X 3t  u3t )  2 X 1t  3 X 3t  u2 t
Removing the brackets
y 2 t  0  1 0  1 1 y1t  1 2 X 2 t  1 3 X 3t  1u3t  2 X 1t  3 X 3t  u2 t
(5)
Substituting into (4) for y2t from (5),
y1t (1   2  1 )   0  1 ( 0  1 0  1 1 y1t    2 X 2 t  1 3 X 3t  1u3t  2 X 1t 
3 X 3t  u2 t )   2  0   2  2 X 2 t   2  3 X 3t   2 u3t   3 X 1t   4 X 2 t  u1t
Taking the y1t terms to the LHS:
y1t (1   2  1  1 1 1 )  0  1 0  1 1 0  1   2 X 2 t  1 1 3 X 3t  1 1u3t  1 2 X 1t 
1 3 X 3t  1u2 t  2 0  2 2 X 2 t  2 3 X 3t  2 u3t  3 X 1t  4 X 2 t  u1t
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Gathering like-terms in the other variables together:
y1t (1  2  1  1 1 1 )  0  1 0  1 1 0  2  0  X 1t (1 2   3 )  X 2 t (1 1 2  2  2   4 ) 
X 3t (1 1 3  1 3  2  3 )  u3t (1 1  2 )  1 u2 t  u1t
(6)
Multiplying all through equation (3) by (1   2  1  1 1 1 ) :
y 3t (1   2  1  11 1 )   0 (1   2  1  11 1 )   1 y1t (1   2  1  11 1 ) 
 2 X 2 t (1   2  1  11 1 )   3 X 3t (1   2  1  11 1 )  u3t (1   2  1  11 1 )
Replacing y1t (1  2 1  11 1 )
(7)
in (7) with the RHS of (6),
 0  1 0  1 1 0   2  0  X 1t (1 2   3 )



y 3t (1   2  1  11 1 )   0 (1   2  1  11 1 )   1  X 2 t (1 1 2   2  2   4 )  X 3t (1 1 3  1 3 
  2  3 )  u3t (1 1   2 )  1u2 t  u1t

  2 X 2 t (1   2  1  11 1 )   3 X 3t (1   2  1  11 1 )  u3t (1   2  1  11 1 )
(8)
Expanding the brackets in equation (8) and cancelling the relevant terms
y3t (1  2  1  11 1 )   0   10   11 0  X 1t (1 2 1   1 3 )  X 2 t ( 2   14 ) 
X 3t ( 11 3   3 )  u3t   11u2 t   1u1t
(9)
Multiplying all through equation (2) by (1   2  1  1 1 1 ) :
y 2 t (1   1  1 1 1 2 )  0 (1   1  1 1 1 2 )  1 y 3t (1   1  1 1 1 2 ) 
2 X 1t (1   1  1 1 1 2 )  3 X 3t (1   1  1 1 12 )  u2 t (1   1  1 1 1 2 )
(10)
Replacing y3t (1  2 1  11 1 )
in (10) with the RHS of (9),
 0   1 0   11 0  X 1t (1 2  1   1 3 )  


y 2 t (1   1  1 1 1 2 )  0 (1   1  1 1 12 )  1  X 2 t ( 2   1 4 )  X 3t ( 3   11 3 )  u3t  
 11u2 t   1u1t

 2 X 1t (1   1  1 1 1 2 )  3 X 3t (1   1  1 1 12 )  u2 t (1   1  1 1 1 2 )
(11)
Expanding the brackets in (11) and cancelling the relevant terms
y2 t (1   1  1
( 
1 12 )  0  02 1  
1 0 
1 10  X 1t
1 1 3  2  22 1 )  X 2 t ( 
1 2  
1 14 ) 
X 3t ( 
1 3  3  32 1 )  1u3t  u2 t (1  2 1 )  
1 1u1t
(12)
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Although it might not look like it (!), equations (6), (12), and (9) respectively
will give the reduced form equations corresponding to (1), (2), and (3), by
doing the necessary division to make y1t, y2t, or y3t the subject of the formula.
From (6),
 0  1 0  1 1 0   2  0
(1 2   3 )
(     2  2   4 )

X 1t  1 1 2
X 2t 
(1   2  1  1 1 1 )
(1   2  1  1 1 1 )
(1   2  1  1 1 1 )
(1 1 3  1 3   2  3 )
u (    2 )  1 u2 t  u1t
X 3t  3 t 1 1
(1   2  1  1 1 1 )
(1   2  1  1 1 1 )
(13)
From (12),
y1t 
y2 t 
0  02 1  1 01 10 ( 1 1 3  2  22 1 )
( 1 2  1 14 )

X1t 
X 
(1   1  11 12 )
(1   1  11 12 )
(1   1  11 12 ) 2 t
( 1 3  3  32 1 )
 u  u (1  2 1 )  1 1u1t
X 3 t  1 3t 2 t
(1   1  11 12 )
(1   1  11 12 )
(14)
From (9),
y 3t 
 0   10   11 0
(1 2  1   1 3 )
( 2   1 4 )

X 1t 
X 
(1   2  1  11 1 ) (1   2  1  11 1 )
(1   2  1  11 1 ) 2 t
( 11 3   3 )
u   11u2 t   1u1t
X 3t  3t
(1   2  1  11 1 )
(1   2  1  11 1 )
(15)
Notice that all of the reduced form equations (13)-(15) in this case depend on
all of the exogenous variables, which is not always the case, and that the
equations contain only exogenous variables on the RHS, which must be the
case for these to be reduced forms.
(b) The term “identification” refers to whether or not it is in fact possible to
obtain the structural form coefficients (the , , and ’s in equations (1)-(3))
from the reduced form coefficients (the ’s) by substitution. An equation can
be over-identified, just-identified, or under-identified, and the equations in a
system can have differing orders of identification. If an equation is underidentified (or not identified), then we cannot obtain the structural form
coefficients from the reduced forms using any technique. If it is just identified,
we can obtain unique structural form estimates by back-substitution, while if
it is over-identified, we cannot obtain unique structural form estimates by
substituting from the reduced forms.
There are two rules for determining the degree of identification of an
equation: the rank condition, and the order condition. The rank condition is a
necessary and sufficient condition for identification, so if the rule is satisfied,
it guarantees that the equation is indeed identified. The rule centres around a
restriction on the rank of a sub-matrix containing the reduced form
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coefficients, and is rather complex and not particularly illuminating, and was
therefore not covered in this course.
The order condition, can be expressed in a number of ways, one of which is the
following. Let G denote the number of structural equations (equal to the
number of endogenous variables). An equation is just identified if G-1
variables are absent. If more than G-1 are absent, then the equation is overidentified, while if fewer are absent, then it is not identified.
Applying this rule to equations (1)-(3), G=3, so for an equation to be
identified, we require 2 to be absent. The variables in the system are y1, y2, y3,
X1, X2, X3. Is this the case?
Equation (1): X3t only is missing, so the equation is not identified.
Equation (2): y1t and X2t are missing, so the equation is just identified.
Equation (3): y2t and X1t are missing, so the equation is just identified.
However, the order condition is only a necessary (and not a sufficient)
condition for identification, so there will exist cases where a given equation
satisfies the order condition, but we still cannot obtain the structural form
coefficients. Fortunately, for small systems this is rarely the case. Also, in
practice, most systems are designed to contain equations that are overidentified.
(c). It was stated in Chapter 4 that omitting a relevant variable from a
regression equation would lead to an “omitted variable bias” (in fact an
inconsistency as well), while including an irrelevant variable would lead to
unbiased but inefficient coefficient estimates. There is a direct analogy with
the simultaneous variable case. Treating a variable as exogenous when it really
should be endogenous because there is some feedback, will result in biased
and inconsistent parameter estimates. On the other hand, treating a variable
as endogenous when it really should be exogenous (that is, having an equation
for the variable and then substituting the fitted value from the reduced form if
2SLS is used, rather than just using the actual value of the variable) would
result in unbiased but inefficient coefficient estimates.
If we take the view that consistency and unbiasedness are more important that
efficiency (which is the view that I think most econometricians would take),
this implies that treating an endogenous variable as exogenous represents the
more severe mis-specification. So if in doubt, include an equation for it!
(Although, of course, we can test for exogeneity using a Hausman-type test).
(d). A tempting response to the question might be to describe indirect least
squares (ILS), that is estimating the reduced form equations by OLS and then
substituting back to get the structural forms; however, this response would be
WRONG, since the question tells us that the system is over-identified.
A correct answer would be to describe either two stage least squares (2SLS) or
instrumental variables (IV). Either would be acceptable, although IV requires
the user to determine an appropriate set of instruments and hence 2SLS is
simpler in practice. 2SLS involves estimating the reduced form equations, and
obtaining the fitted values in the first stage. In the second stage, the structural
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form equations are estimated, but replacing the endogenous variables on the
RHS with their stage one fitted values. Application of this technique will yield
unique and unbiased structural form coefficients.
2. (a) A glance at equations (6.97) and (6.98) reveals that the dependent
variable in (6.97) appears as an explanatory variable in (6.98) and that the
dependent variable in (6.98) appears as an explanatory variable in (6.97). The
result is that it would be possible to show that the explanatory variable y2t in
(6.97) will be correlated with the error term in that equation, u1t, and that the
explanatory variable y1t in (6.98) will be correlated with the error term in that
equation, u2t. Thus, there is causality from y1t to y2t and from y2t to y1t, so that
this is a simultaneous equations system. If OLS were applied separately to
each of equations (6.97) and (6.98), the result would be biased and
inconsistent parameter estimates. That is, even with an infinitely large
number of observations, OLS could not be relied upon to deliver the
appropriate parameter estimates.
(b) If the variable y1t had not appeared on the RHS of equation (6.98), this
would no longer be a simultaneous system, but would instead be an example
of a triangular system (see question 3). Thus it would be valid to apply OLS
separately to each of the equations (6.97) and (6.98).
(c) The order condition for determining whether an equation from a
simultaneous system is identified was described in question 1, part (b). There
are 2 equations in the system of (6.97) and (6.98), so that only 1 variable
would have to be missing from an equation to make it just identified. If no
variables are absent, the equation would not be identified, while if more than
one were missing, the equation would be over-identified. Considering
equation (6.97), no variables are missing so that this equation is not identified,
while equation (6.98) excludes only variable X2t, so that it is just identified.
(d) Since equation (6.97) is not identified, no method could be used to obtain
estimates of the parameters of this equation, while either ILS or 2SLS could be
used to obtain estimates of the parameters of (6.98), since it is just identified.
ILS operates by obtaining and estimating the reduced form equations and
then obtaining the structural parameters of (6.98) by algebraic backsubstitution. 2SLS involves again obtaining and estimating the reduced form
equations, and then estimating the structural equations but replacing the
endogenous variables on the RHS of (6.97) and (6.98) with their reduced form
fitted values.
Comparing between ILS and 2SLS, the former method only requires one set of
estimations rather than two, but this is about its only advantage, and
conducting a second stage OLS estimation is usually a computationally trivial
exercise. The primary disadvantage of ILS is that it is only applicable to just
identified equations, whereas many sets of equations that we may wish to
estimate are over-identified. Second, obtaining the structural form coefficients
via algebraic substitution can be a very tedious exercise in the context of large
systems (as the solution to question 1, part (a) shows!).
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(e) The Hausman procedure works by first obtaining and estimating the
reduced form equations, and then estimating the structural form equations
separately using OLS, but also adding the fitted values from the reduced form
estimations as additional explanatory variables in the equations where those
variables appear as endogenous RHS variables. Thus, if the reduced form
fitted values corresponding to equations (6.97) and (6.98) are given by y1t and
y2t respectively, the Hausmann test equations would be
y1t   0   1 y 2t   2 X 1t   3 X 2t   4 y 2t 'u1t
y 2t   0   1 y1t   2 X 1t   3 y1t ' u1t
.
Separate tests of the significance of the y1t and y2t terms would then be
performed. If it were concluded that they were both significant, this would
imply that additional explanatory power can be obtained by treating the
variables as endogenous.
3. An example of a triangular system was given in Section 6.7. Consider a
scenario where there are only two “endogenous” variables. The key distinction
between this and a fully simultaneous system is that in the case of a triangular
system, causality runs only in one direction, whereas for a simultaneous
equation, it would run in both directions. Thus, to give an example, for the
system to be triangular, y1 could appear in the equation for y2 and not vice
versa. For the simultaneous system, y1 would appear in the equation for y2,
and y2 would appear in the equation for y1.
4. (a) p=2 and k=3 implies that there are two variables in the system, and that
both equations have three lags of the two variables. The VAR can be written in
long-hand form as:
y1t   10   111 y1t 1   211 y 2t 1   112 y1t  2   212 y 2t  2   113 y1t 3   213 y 2t 3  u1t
y 2t   20   121 y1t 1   221 y 2t 1   122 y1t  2   222 y 2t  2   123 y1t 3   223 y 2t 3  u 2t
 10 
 y1t 
 u1t 
where 0   , y t   , ut    , and the  coefficients on the lags of yt
20 
 y2t 
u 2 t 
are defined as follows: ijk refers to the kth lag of the ith variable in the jth
equation. This seems like a natural notation to use, although of course any
sensible alternative would also be correct.
(b) This is basically a “what are the advantages of VARs compared with
structural models?” type question, to which a simple and effective response
would be to list and explain the points made in the book.
The most important point is that structural models require the researcher to
specify some variables as being exogenous (if all variables were endogenous,
then none of the equations would be identified, and therefore estimation of
the structural equations would be impossible). This can be viewed as a
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restriction (a restriction that the exogenous variables do not have any
simultaneous equations feedback), often called an “identifying restriction”.
Determining what are the identifying restrictions is supposed to be based on
economic or financial theory, but Sims, who first proposed the VAR
methodology, argued that such restrictions were “incredible”. He thought that
they were too loosely based on theory, and were often specified by researchers
on the basis of giving the restrictions that the models required to make the
equations identified. Under a VAR, all the variables have equations, and so in
a sense, every variable is endogenous, which takes the ability to cheat (either
deliberately or inadvertently) or to mis-specify the model in this way, out of
the hands of the researcher.
Another possible reason why VARs are popular in the academic literature is
that standard form VARs can be estimated using OLS since all of the lags on
the RHS are counted as pre-determined variables.
Further, a glance at the academic literature which has sought to compare the
forecasting accuracies of structural models with VARs, reveals that VARs seem
to be rather better at forecasting (perhaps because the identifying restrictions
are not valid). Thus, from a purely pragmatic point of view, researchers may
prefer VARs if the purpose of the modelling exercise is to produce precise
point forecasts.
(c) VARs have, of course, also been subject to criticisms. The most important
of these criticisms is that VARs are atheoretical. In other words, they use very
little information form economic or financial theory to guide the model
specification process. The result is that the models often have little or no
theoretical interpretation, so that they are of limited use for testing and
evaluating theories.
Second, VARs can often contain a lot of parameters. The resulting loss in
degrees of freedom if the VAR is unrestricted and contains a lot of lags, could
lead to a loss of efficiency and the inclusion of lots of irrelevant or marginally
relevant terms. Third, it is not clear how the VAR lag lengths should be
chosen. Different methods are available (see part (d) of this question), but
they could lead to widely differing answers.
Finally, the very tools that have been proposed to help to obtain useful
information from VARs, i.e. impulse responses and variance decompositions,
are themselves difficult to interpret! – See Runkle (1987).
(d) The two methods that we have examined are model restrictions and
information criteria. Details on how these work are contained in Sections
6.12.4 and 6.12.5. But briefly, the model restrictions approach involves
starting with the larger of the two models and testing whether it can be
restricted down to the smaller one using the likelihood ratio test based on the
determinants of the variance-covariance matrices of residuals in each case.
The alternative approach would be to examine the value of various
information criteria and to select the model that minimises the criteria. Since
there are only two models to compare, either technique could be used. The
restriction approach assumes normality for the VAR error terms, while use of
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“Introductory Econometrics for Finance” © Chris Brooks 2008
the information criteria does not. On the other hand, the information criteria
can lead to quite different answers depending on which criterion is used and
the severity of its penalty term. A completely different approach would be to
put the VARs in the situation that they were intended for (e.g. forecasting,
making trading profits, determining a hedge ratio etc.), and see which one
does best in practice!
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“Introductory Econometrics for Finance” © Chris Brooks 2008