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Seemingly Unrelated Regressions
Anton Parlow
Lab session Econ 710UWM Econ Department
03/04/2011
Anton Parlow Lab session Econ 710 UWM Econ Department ()
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Our plan
Overview Sur-Regression.
Stata commands.Example 1.
Example 2.
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Seemingly Unrelated regression
If equations in a system of equations are related by their error-terms,estimating them together using GLS is more efficient compared to OLS.(or the variance is ”more” minimized)
We have: Y i = X i β i + i with i = 1 . . . L equations
If the error-terms across the equations are linked, so-called
”contemporaneous correlation” s.t. E i
j
= σ2ij I j
then Zellner (1962) shows that a GLS-estimator is more efficient.
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Seemingly Unrelated regression continued
Let the above model in matrix notation be: Y = z δ + e.g. Y is a nTxL
matrix (and so on)
that δ GLS = z
V −1z −1
z
V −1Y
and the variance-covariance matrix E =
= V with V = Σ⊗ I T
For the SUR-model Σ has to be non-diagonal or the off-diagonal termsshould not be zero. If they are zero there is no correlation between the
error-terms.
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Seemingly Unrelated regression continued
This can be tested after the Sur-Regression using a LaGrange-Multipliertest suggested by Breusch Pagan:
λLM = T
ni =2
i −1 j =1
r 2ij
The H 0 = σij has to be rejected, if we want to use SUR.
Otherwise every equation can be regressed by itself using OLS (there is nogain in using a GLS-estimator)
This is also true (most of the time) if the explanatory variables are thesame across the equations and if the regressors of one equation are asubset of another (Greene, p.343)
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Stata commands
Imagine you have two models:
y1=x1+x2 and y2=x1+x3 using OLS in Stata will be
reg y1 x1 x2
reg y2 x1 x3You suspect they are related by their error-terms → use sureg
sureg (y1 x1 x2) (y2 x1 x3)
in general
sureg (model1) (model2) (model3), options
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Example 1
from the Stata-website. Load hsb2.dta into Stata
We want to explain test-scores for science and writing for female students:
reg science math female
reg write read femaleand suspect the error-terms are related e.g. because of the school they arein
sureg (science math female) (write read female), corr
Do you see any difference to the OLS-results???
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Example 1 continued
Sur-output → you see the output for the 2 models
Note: Stata adds the BP-test to the output! In this case the error-termsare correlated.
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Example 1 continued
Let us test the effect of type of program prog on scores. There are threeprograms, a dummy would be nice:tabulate prog, gen(prog)Consider following three OLS-regressions (having the same regressors!):read female prog1 prog3
write female prog1 prog3math female prog1 prog3Still we can assume the errors are related:sureg (read female prog1 prog3) (write female prog1 prog3) (mathfemale prog1 prog3), corr
sames as:sureg (read write math = female prog1 prog3), corrbecause the regressors are the same
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Example 2
Consider a system of cost-share equations derived from a translog costfunction having four inputs (KLEM-model):
S K = αK + γ KK lnP K + γ KLlnP L + γ KE lnP E + γ KM lnP M
S L = αL + γ KLlnP K + γ LLlnP L + γ LE lnP E + γ LM lnP M
S E = αE + γ KE lnP K + γ LE lnP L + γ EE lnP E + γ EM lnP M
S M = αM + γ KM lnP K + γ LM lnP L + γ EM lnP E + γ MM lnP M
We estimate only three share equations because the shares add up to 1.And let us ignore homogeneity and symmetry constraints on that system
for simplicityLoad the bw.dta file
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Example 2 continued
Use sureg (lk ll le = lpk lpl lpe), corr
and compare it to OLS-estimates for each equation.
What do you see?
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