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Seemingly Unrelated Regressions

Anton Parlow

Lab session Econ 710UWM Econ Department

03/04/2011

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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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