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# Re: st: rho, sigma_u, sigma_e in xtreg

 From Misha Spisok To statalist@hsphsun2.harvard.edu Subject Re: st: rho, sigma_u, sigma_e in xtreg Date Fri, 30 Apr 2010 13:55:15 -0700

```Here's my take, with some help from the [XT] manual.

The [XT] manual gives the following for the random effects model:

between: corr(x_bar_i*beta_hat, y_bar_i)^2
within: corr{ (x_it - x_bar_i)*beta_hat, y_it - y_bar_i)}^2
overall: corr(x_it*beta_hat, y_it)^2

u refers to the individual component
e refers to the idiosyncratic component

e.g., y_it = alpha + x_it*beta + nu_i + epsilon_it

with u corresponding to nu and e to epsilon.

rho, explained in words in the output, is

rho = (sigma_u)^2/[(sigma_u)^2 + (sigma_e)^2]

sigma_u and sigma_e are estimates of the standard deviation of nu and epsilon.

For example,

display .02692048^2/(.02692048^2 + .0829595^2)
or, more generally,
di e(sigma_u)^2/(e(sigma_u)^2 + e(sigma_e)^2)

ought to give you rho.

On Fri, Apr 30, 2010 at 1:34 PM, Katherine Packman <packman@ualberta.ca> wrote:
> I have somewhat of a basic question.
>
> I am running xtreg assuming random effects for 150 observations
> (cross-section is 10 and timeseries is 15). I was wondering if anyone could
> explain what rho, sigma_u and sigma_e mean. This is an example of my output:
>
>
> . xtreg pmor dumcov dumuni dum_cu lnprd prduni
>
> Random-effects GLS regression                   Number of obs      =
> 150
> Group variable: session                         Number of groups   =
>  10
>
> R-sq:  within  = 0.2093                         Obs per group: min =
>  15
>       between = 0.6968                                        avg =
>  15.0
>       overall = 0.3196                                        max =
>  15
>
> Random effects u_i ~ Gaussian                   Wald chi2(5)       =
> 50.32
> corr(u_i, X)       = 0 (assumed)                Prob > chi2        =
>  0.0000
>
> ------------------------------------------------------------------------------
>        pmor |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
> Interval]
> -------------+----------------------------------------------------------------
>      dumcov |   .1254805   .0344025     3.65   0.000     .0580529
>  .1929082
>      dumuni |  -.0042582   .0392887    -0.11   0.914    -.0812628
>  .0727463
>      dum_cu |  -.1437736   .0444134    -3.24   0.001    -.2308223
> -.0567249
>       lnprd |   -.077336   .0130727    -5.92   0.000     -.102958
>  -.051714
>      prduni |   .0101917    .002951     3.45   0.001     .0044079
>  .0159756
>       _cons |    .886512   .0343942    25.78   0.000     .8191005
>  .9539234
> -------------+----------------------------------------------------------------
>     sigma_u |  .02692048
>     sigma_e |   .0829595
>         rho |  .09526924   (fraction of variance due to u_i)
> ------------------------------------------------------------------------------
>
>
> Any help would be appreciated.
>
> Cheers
>
> --
> Katherine
> University of Alberta
>
>
>
>
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>

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```

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