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where Y represents total output in cars
washed per hour and X is number
of men. The study was based on 22
observations over a onemonth period
during which time the number of workers
varied from a minimum of 3
to a maximum of 10. Given the equation, the
t
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tors) of a production process and the output
(product) that results. Hence
it is also sometimes called the "inputoutput"
relation. Like a demand
function, a production function can also be
expressed in the form of a
schedule or a graph as shown subsequen
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estimational purposes may be expressed
conceptually by writing it in the
form Y = f(X). However, since the product Y
will be the result of com
bining the input factor X (e.g., labor) with
other factors (such as capital,
land, management, etc.), the funct
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Study Guide
Unit 1
Name:
The study guide is to help you focus your studying for the Unit 1 exam. Submit for 10 Extra Credit points! The
Extra credit is available for 1 week after the Unit Test. It will be available again during the last week before the
se
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1.01 Scientific Method Lab Simulation Report
Bio202A
Turn on Comments (Review>Show Comments Simple Markup) to see the rubric for each graded item.
When using the underlined key terms in your answers, EXPLAIN them. For example: The hypothesis, which is
an
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15.4 METHODS OF ESTIMATION
It is possible to estimate the reducedform parameters, and , consistently by ordinary least squares. But
except for forecasting y given x, these are generally not the parameters of interest; ,B, and are. The
ordinary least squa
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A = advertising/S, D = durable goods industry(0/1), C = concentration, Gr = industry growth rate, Cd=
consumer demand/S, K = capital stock/S, MES= efcient scale/S, Gd= geographic dispersion. Since the
only restrictions are exclusions, we may check identic
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is relatively large.
15.5.6 TWOSTAGE LEAST SQUARES IN MODELS THAT ARE NONLINEAR IN VARIABLES The analysis of
simultaneous equations becomes considerably more complicated when
theequationsarenonlinear.Amemiyapresentsageneraltreatmentofnonlinearmodels.18 A
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Ourchoiceofnoninformativepriorforln ledtoaconvenientpriorfor2 inourderivationoftheposterior for .
The idea that the prior can be specied arbitrarily in whatever form is mathematically convenient is very
troubling; it is supposed to represent the accumulat
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CHAPTER 16 Estimation Frameworks in Econometrics
Therearedifferentwaysthatonemightcharacterizethelackofpriorinformation.The implication of a at prior
is that within the range of valid values for the parameter, all intervalsofequallength
hence,inprinciple,
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isequationbyequationordinaryleastsquaresandisinconsistent.Butevenifordinary least squares were
consistent, we know from our results for the seemingly unrelated regressions model in the previous
chapter that it would be inefcient compared with an estimat
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Three techniques are generally used for joint estimation of the entire system of equations: threestage
least squares, GMM, and full information maximum likelihood.
15.6.1 THREESTAGE LEAST SQUARES Consider the IV estimator formed from
W= Z=diag[X(XX)1XZ
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400 CHAPTER 15 SimultaneousEquations Models
columnofZj onX.Thisequationalsoresultsinausefulsimplicationoftheestimated asymptotic covariance
matrix, Est.Asy.Var[ j,2SLS]= jj[ Zj Zj]1. It is important to note that jj is estimated by
jj =
(yj Zj j)(yj Zj j
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The limited information maximum likelihood (LIML) estimator is based on a single equation under the
assumption of normally distributed disturbances; LIML is efcient among singleequation estimators. A
full (lengthy) derivation of the loglikelihood is pro
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where j is a degrees of freedom parameter and 2 is the Bayesian estimate of 2. The prior degrees of
freedom m is a parameter of the prior distribution for 2 that would havebeendeterminedattheoutset.
(Seethefollowingexample.)Onceagain,itisclear
8Note that
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that will be an imposter for equation j. Many fjs are already precluded. 1. f1 must be the rst column of
an identity matrix. The rst equation is identied and normalized on y1. 2. In all remaining columns of F,
all elements below the diagonal must be zero,
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sentially involves only making an assessment of which of two hypotheses has a higher probability of
being correct.
Greene50240 book June 20, 2002 18:2
438 CHAPTER 16 Estimation Frameworks in Econometrics
The Bayesian approach to hypothesis testing bears
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The rank condition ensures that there is exactly one solution for the structural parameters given the
reducedform parameters. Our alternative approach to the
identicationproblemwastousethepriorrestrictionson[,B]toeliminateallfalsestructures.Anequivalentc
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equationsareestimatedoneatatimeorjointly.Wedividethemintotwoclasses,limited information or full
information, on this basis.
15.5.3 TWOSTAGE LEAST SQUARES Themethodoftwostageleastsquaresisthemostcommonmethodusedforestimating simultaneousequations models.
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will be consistent and have asymptotic covariance matrix
Asy.Var[ j,IV] = jj T
plim
1T
WjZj1
1T
WjWj
1T
ZjWj1
=
jj T 1 wzww1 zw. (1516) A consistent estimator of jj is
jj =
(yj Zj j,IV)(yj Zj j,IV) T
, (1517)
whichisthefamiliarsumofsquaresoftheestimate
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E[xtjt]= E[xt(yjt zjtj)]=0. If xt is taken to be the full set of exogenous variables in the model, then we
obtain the criterion for the GMM estimator, q =
e(zt,j)X T W1 jj
Xe(zt,j) T = m(j)W1 jj m(j), where
m(j) =
1T
T t=1
xt(yjt zjtj) and W1 jj =the GMM
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joint density the likelihood for and 2 given the data, so L(,2y,X) =[22]n/2e[(1/(22)(yX)
(yX)]. (163) For purposes of the results below, some reformulation is useful. Let d = n K (the degrees
of freedom parameter), and substitute yX =yXbX( b) =eX( b) in
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which is Msets of equations, each one of the form
plim
1T
M j=1
ij Zjj =0.
Each is the sum of vectors all of which converge to zero, as we saw in the development of the 2SLS
estimator. The second requirement, that
plim
1T
Z(1I)Z=0, andthatthematrixbenons
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in . A useful case is that of zero covariances across the disturbances.14 Once again, it is most convenient
to consider this case in terms of a false structure. If the structure is [,B,], then a false structure would
have parameters [ , B, ]=[F,BF,FF]. If
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for identication purposes, essentially the same as an exclusion.13 By an appropriate manipulation
thatis,bysolvingouttherestrictionwecanturntherestrictioninto one more exclusion. The order
condition that emerges is nj M1, wherenj isthetotalnumberofrestric
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[In terms of nonzero parameters, sij is ij of (1532).] In maximizing lnL, it is necessary to impose all the
additional restrictions on the structure. The trace may be written in the form tr(1S) = M i=1M j=1 ij(yi
Yii Xii)(yj Yj j Xj j) T . (1534) Maximi
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is to obtain an estimate of . In estimation of the multivariate regression model, for efcient estimation
(that remains to be shown), any consistent estimator of will do. The designers of the 3SLS method,
Zellner and Theil (1962), suggest the natural choic
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CHAPTER 15 SimultaneousEquations Models 397
havethevirtueofcomputationalsimplicity,althoughwithmodernsoftware,thisvirtue is extremely modest.
For better or worse, OLS is a very commonly used estimator in this context. We will return to this issue
later i
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CHAPTER 15 SimultaneousEquations Models
variables that the conditions are met trivially. In practice, it is simple to check both conditions for a small
model. For a large model, frequently only the order condition is veried. We distinguish three cases: 1