Variance of the iid AR(1) shock

[Marcus Grayer]

Hello everyone,

I am currently working on a time series of realized GDP values for a given country (annual frequency data ranging from 1960 till 2023). I need to construct a country specific inflation variance, which should be the estimated variance of the iid AR(1) shock divided by (1-AR(1) persistence^2). To estimate the AR(1) persistence I am currently using the command

arima GDP_values, arima(1,0,0)

My doubt is: how should I get such estimated variance of the iid shock? I would use the approach where I predict the residuals and then I estimate the variance just after the arima command through

    predict residuals, residuals
    summarize residuals, detail

what I get is that the variance should be 5.422983 but I am not sure about autocorrelation of errors in this framework.

An alternative approach is to run

        mat list e(V)

just after the arima command to produce the variance-covariance estimator, but I do not get which should be the correct variance as this is the result of such a command:

symmetric e(V)[3,3]
                      GDP_values:         ARMA:        sigma:
                                              L.              
                            _cons            ar         _cons
GDP_values:_cons     15.625739
          ARMA:L.ar    -.06853216     .00321081
        sigma:_cons    -.36765128    -.00205707     .02638782

Thank you in advance for your time.