I'm having difficulty understanding conceptually and mathematically what is the variance of a regression coefficient. I have an autoregressive model of the type: $$y_t=\beta_1x_1+\beta_2x_2+\beta y_{t-1}+\beta_0$$ My goal is to calculate the Durbin h-statistic for this model, for which I need the variance of the coefficient on the lagged $y$ variable, $Var(\beta)$. Here's where I'm not sure how to calculate that. Is the variance of the coefficient on the lagged $y$ variable the variance explained by that predictor?
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asked Jan 21, 2017 at 20:56
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$\begingroup$ $\sigma^2 (X'X)^{-1}$ is the formula for the variance covariance matrix. The variances are on the diagonal. $\endgroup$– TonyCommented Jan 21, 2017 at 22:12
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$\begingroup$ @Tony That true for OLS, but this is an autoregressive model and therefore does not assume independent errors, which is a crucial assumption behind the formula you quote. The correct formula to use depends on how $\beta$ is estimated. $\endgroup$– whuber ♦Commented Jan 21, 2017 at 23:01
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$\begingroup$ @whuber Hmm, I have seen many estimate AR models using OLS. If the autocorrelation of errors is included in the estimate, then generally GLS is used and the variance can be obtained from: $\sigma^2 (X'WX)^{-1}$, where W is the inverse of variance/covariance of the errors. Use the following to find W: people.virginia.edu/~sns5r/classes/grad/econ772stf/gls.pdf $\endgroup$– TonyCommented Jan 22, 2017 at 11:59
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I believe this question is based on
It seems that you may be doing your analysis in R, hence my example will uses R code.
library(car)
fit = lm(fconvict ~ tfr + partic + degrees + mconvict, data=Hartnagel)
n = nrow(Hartnagel)
s = summary(fit)
coef(s)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 127.63999736 59.957044218 2.128857 4.081885e-02
tfr -0.04656651 0.008032987 -5.796911 1.754618e-06
partic 0.25341619 0.115131823 2.201096 3.483237e-02
degrees -0.21204914 0.211453792 -1.002816 3.232469e-01
mconvict 0.05910466 0.045145314 1.309209 1.995067e-01
v = n * coef(s)[1, 2] ^ 2 ## std^2 -var
v
[1] 136604.2
d = durbinWatsonTest(model = fit)
durbinH <- (1 - 0.5 * d$dw) * sqrt(n / (1 - n*v))
Warning message:
In sqrt(n/(1 - n * v)) : NaNs produced
As you see here, the durbinH statistic is not defined since $n \times var(\beta_1) \geq $. It seems Durbin H is not always defined so you may need to be cautious with this statistic.
Refer to the wiki page: https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic#Durbin_h-statistic
However, there are other methods for detecting autocorrelation among the residuals. For example acf will produce an autocorrelation plot and statistics.
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$\begingroup$ Thanks, @Jon. Just to confirm, in the case of the model I have,
vwould be the square of the std. error on the y lag one term, right? $\endgroup$ Commented Jan 21, 2017 at 22:50 -
$\begingroup$ It is now the variance. Before I forgot to multiply by n to convert to variance, so in reality it was variance/n. The answer has been updated. $\endgroup$– JonCommented Jan 22, 2017 at 0:12
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$\begingroup$ While multiplying by n to convert to variance is correct, if I use this approach then n*v will almost always be >1. Reading more on the topic it seems that it's appropriate to square the standard error to get variance: academlib.com/1023/economics/durbins_test_statistic. $\endgroup$ Commented Jan 22, 2017 at 14:40
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$\begingroup$ I do not know how valid the reference you linked is, but they do use the standard error; in fact, they use the standard error of for the coefficient belonging to $y_{t-1}$ instead of the intercept term which is different from what I remember Wikipedia had. I suggest looking into at least one more reference (or if you're in school, asking your teacher) to compare and validate procedures. $\endgroup$– JonCommented Jan 24, 2017 at 17:27