# [R] Hausman Test

Achim Zeileis Achim.Zeileis at uibk.ac.at
Sun Jan 16 16:37:55 CET 2011

```On Sun, 16 Jan 2011, Arne Henningsen wrote:

> Hi Holger!
>
> On 16 January 2011 15:53, Holger Steinmetz <Holger.steinmetz at web.de> wrote:
>> One follow up question. The Hausman-test always gives me a p-value of 1 - no
>> matter how small the statistic is.
>>
>> I now generated orthogonal regressors (X1-X3) and the test gives me
>>
>>
>>        Hausman specification test for consistency of the 3SLS estimation
>>
>> data:  data
>> Hausman = -0.0138, df = 2, p-value = 1
>>
>> What is confusing to me is the "3SLS". I am just beginning to learn about
>> instrumental variables (I am a psychologist ;) Perhaps that's a problem?
>>
>> As a background, here's the complete simulation:
>>
>> W = rnorm(1000)
>> X2 = rnorm(1000)
>> X3 = rnorm(1000)
>> X1 = .5*W  + rnorm(1000)
>> Y = .4*X1 + .5*X2 + .6*X3 + rnorm(1000)
>> data = as.data.frame(cbind(X1,X2,X3,Y,W))
>>
>> fit2sls <- systemfit(Y~X1,data=data,method="2SLS",inst=~W)
>> fitOLS <- systemfit(Y~X1,data=data,method="OLS")
>>
>> print(hausman.systemfit(fitOLS, fit2sls))
>
> comparing 2SLS with OLS results has not been implemented yet:
>
> ====== part of documentation of hausman.systemfit() =================
> Usage:
>
>        hausman.systemfit( results2sls, results3sls )
>
> Arguments:
>
> results2sls : result of a _2SLS_ (limited information) estimation
>          returned by ?systemfit?.
>
> results3sls : result of a _3SLS_ (full information) estimation
>          returned by ?systemfit?.
>
> Details:
>
>     The null hypotheses of the test is that all exogenous variables
>     are uncorrelated with all disturbance terms.  Under this
>     hypothesis both the 2SLS and the 3SLS estimator are consistent but
>     only the 3SLS estimator is (asymptotically) efficient.  Under the
>     alternative hypothesis the 2SLS estimator is consistent but the
>     3SLS estimator is inconsistent.
>
>     The Hausman test statistic is
>
>               m = ( b_2 - b_3 )' ( V_2 - V_3 ) ( b_2 - b_3 )
>
>     where \$b_2\$ and \$V_2\$ are the estimated coefficients and their
>     variance covariance matrix of a _2SLS_ estimation and \$b_3\$ and
>     \$V_3\$ are the estimated coefficients and their variance covariance
>     matrix of a _3SLS_ estimation.
>
> =========================================
>
> Please don't hesitate to write a new version of hausman.systemfit()
> that can also compare 2SLS with OLS results.

Arne: Unless I'm missing something, hausman.systemfit() essentially does
the right thing and computes the right statistic and p-value (see my other
mail to Holger). Maybe some preliminary check on the input objects could
be used for determining the right order of models.

Best,
Z

> Best regards from Copenhagen,
> Arne
>
> --
> Arne Henningsen
> http://www.arne-henningsen.name
>
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