′
′
Greene-50240 book June 18, 2002 15:28
286? CHAPTER13? ? ModelsforPanelData?
individuals. This random parameters model was proposed quite early in this literature,
but has only fairly recently enjoyed widespread attention in several fields. It represents
a natural extension in which researchers broaden the amount of heterogeneity across
individuals while retaining some commonalities—the parameter vectors still share a
common mean. Some recent applications have extended this yet another step by allow?
ing the mean value of the parameter distribution to be person-specific, as in
y
it?
=·x
it
(β +·z
i?
+·h
i
)?+·(α? +·u
i
)?+·ε
it
,?
where z
i?
is a set of observable, person specific variables, and  is a matrix of parameters
to be estimated. As we will examine later, this hierarchical?model? is extremely versatile.
5.? Covariance?Structures:? Lastly, we will reconsider the source of the heterogeneity in
the model. In some settings, researchers have concluded that a preferable approach to
modeling heterogeneity in the regression model is to layer it into the variation around
the conditional mean, rather than in the placement of the mean. In a cross-country
comparison of economic performance over time, Alvarez, Garrett, and Lange (1991)
estimated a model of the form
y
it?
=· f (labor organization
it
,?political organization
it
)?+·ε
it?
in which the regression function was fully specified by the linear part, x
′
it
β +·α, but
the variance of ε
it?
differed across countries. Beck et al. (1993) found evidence that the
substantive conclusions of the study were dependent on the stochastic specification and
on the methods used for estimation.
Example13.1? CostFunctionforAirlineProduction?
To? illustrate? the?computations? for? the?various?panel?data?models,?we?will? revisit? the?airline?
cost?data?used?in?Example?7.2.?This?is?a?panel?data?study?of?a?group?of?U.S.?airlines.?We?will?
fitasimplemodelforthetotalcostofproduction:?
ln?cost
it?
=·β
1?
+·β
2?
ln?output
it?
+·β
3?
ln?fuelprice
it?
+·β
4?
loadfactor
it?
+·ε
it
.?
Output? is? measured? in? “revenue? passenger? miles.”? The? load? factor? is? a? rate? of? capacity?
utilization;?it?is?the?average?rate?at?which?seats?on?the?airline’s?planes?are?filled.?More?complete?
models?of?costs?include?other?factor?prices?(materials,?capital)?and,?perhaps,?a?quadratic?term?
in?log?output?to?allow?for?variable?economies?of?scale.?We?have?restricted?the?cost?function?
tothesefewvariablestoprovideastraightforwardillustration.?
Ordinary? least? squares? regression? produces? the? following? results.? Estimated? standard?
errorsaregiveninparentheses.?
ln?cost
it?
=·9.5169(?0.22924)? +·0.88274(?0.013255) ln?output
it?
+·0.45398(?0.020304)?ln?fuelprice
it?
−·1.62751(?0.34540)?loadfactor
it?
+·ε
it?
R
2?
=·0.9882898,? s
2?
=·0.015528,? e e?=· 1.335442193.?
The? results? so? far? are?what? one?might? expect.? There? are? substantial? economies? of? scale;?
e.s.
it?
=·(1/0.88274)? −·1?=·0.1329.? The? fuel? price? and? load? factors? affect? costs? in? the? pre-
dictable?fashions?as?well.?(Fuel?prices?differ?because?of?different?mixes?of?types?of?planes?and?
regionaldifferencesinsupplycharacteristics.)?

Greene-50240 book June 26, 2002 21:55
CHAPTER19? ? ModelswithLaggedVariables? 595?
one-time innovation in v
m
. We could also examine the effect of a one-time innovation
of v
l?
on variable m. The impulse response function would be
φ
ml?
(i?)? =·element (m, l?)? in
i?
.?
Point estimation of φ
ml?
(i?)? using the estimated model parameters is straightforward.
Confidence intervals present a more difficult problem because the estimated functions
ˆ
φ
ˆ
ml?
(i, β)?are so highly nonlinear in the original parameter estimates. The delta method
has thus proved unsatisfactory. Killian (1998) presents results that suggest that boot?
strapping may be the more productive approach to statistical inference regarding im?
pulse response functions.
19.6.7? STRUCTURALVARs?
The VAR approach to modeling dynamic behavior of economic variables has provided
some interesting insights and appears [see Litterman (1986)] to bring some real benefits
for forecasting. The method has received some strident criticism for its atheoretical
approach, however. The “unrestricted” nature of the lag structure in (19-30) could be
synonymous with “unstructured.” With no theoretical input to the model, it is difficult
to claim that its output provides much of a theoretically justified result. For example,
how are we to interpret the impulse response functions derived in the previous section?
What lies behind much of this discussion is the idea that there is, in fact, a structure
underlying the model, and the VAR that we have specified is a mere hodgepodge of all
its components. Of course, that is exactly what reduced forms are. As such, to respond
to this sort of criticism, analysts have begun to cast VARs formally as reduced forms
and thereby attempt to deduce the structure that they had in mind all along.
A VAR model y
t?
=·µ +· y
t?−1
+·v
t?
could, in principle, be viewed as the reduced
form of the dynamic structuralmodel?
y
t?
=·α +·y
t?−1
+·ε
t?
,?
where we have embedded any exogenous variables x
t?
in the vector of constants α. Thus,
 =·
−1
,µ =·
−1
α, v? =·
−1
ε, and
=·
−1
(
−1
)
′
. Perhaps it is the structure,
specified by an underlying theory, that is of interest. For example, we can discuss the
impulse response characteristics of this system. For particular configurations of , such
as a triangular matrix, we can meaningfully interpret innovations, ε. As we explored at
great length in the previous chapter, however, as this model stands, there is not suffi?
cient information contained in the reduced form as just stated to deduce the structural
parameters. A possibly large number of restrictions must be imposed on ,, and 
to enable us to deduce structural forms from reduced-form estimates, which are always
obtainable. The recent work on “structural VARs” centers on the types of restrictions
and forms of the theory that can be brought to bear to allow this analysis to proceed.
See, for example, the survey in Hamilton (1994, Chapter 11). At this point, the literature
on this subject has come full circle because the contemporary development of “unstruc?
tured VARs” becomes very much the analysis of quite conventional dynamic structural
simultaneous equations models. Indeed, current research [e.g., Diebold (1998a)] brings
the literature back into line with the structural modeling tradition by demonstrating
how VARs can be derived formally as the reduced forms of dynamic structural models.
That is, the most recent applications have begun with structures and derived the reduced