# How to get standard errors of fitted noise variance and constant for random process, e.g., ARProcess from TimeSeriesModelFit

I want to know how to get the standard errors (and hopefully the CovarianceMatrix) for the constant and noise variance parameters of (say) an ARMAProcess fitted by TimeSeriesModelFit.

## Example

So, for example, lets suppose we make an AR(1) process, sample it, and then do a fit:

SeedRandom[1];
testProcess = ARProcess[1, {.5}, 2];
samples = RandomFunction[testProcess, {1, 10000}];
timeSeriesModel = TimeSeriesModelFit@samples;


Above, in the second line, 1 is the constant offset of the AR(1) model, 0.5 is the coefficient, and 2 is the noise variance. We can look at the fitted model using Normal@timeSeriesModel, and we get

ARProcess[0.969334, {0.512352}, 1.94524]


so TimeSeriesModelFit has made an estimate of all three parameters, with some precision that it can presumably estimate. My question is how do I get these uncertainties in the model parameters as well as the correlations in uncertainties?

## What I've tried

I tried to use timeSeriesModel@"ParameterTable", but all I got was the following table:

$\begin{array}{l|llll} \text{} & \text{Estimate} & \text{Standard Error} & \text{t-Statistic} & \text{P-Value} \\ \hline \mathit{a}_1 & 0.512352 & 0.00858776 & 59.6607 & \text{4.88712458388898319$\times10^{-664}$} \\ \end{array}$

Similarly if I want the correlations of errors in the model, I can use timeSeriesModel@"CovarianceMatrix", and I get

{{0.737496}}


which again only concerns the coefficient.

I think this is how you can get the covariances of the parameter estimates of $c$, $\rho$, and $v$:

logL = LogLikelihood[ARProcess[c, {ρ}, v], samples]

sol = Flatten[(EstimatedProcess[samples, ARProcess[c, {ρ}, v]]) /. ARProcess -> List]
(* {0.9693337798192243, 0.5123516081122499, 1.9452424330398612} *)

cov = -Inverse[(D[logL, {{c, ρ, v}, 2}]) /. Thread[{c, ρ, v} -> sol]] // MatrixForm

(* {{0.000485985, -0.000146633, 1.63498*10^-7},
{-0.000146633, 0.00007376, -6.23427*10^-8},
{1.63498*10^-7, -6.23427*10^-8, 0.000756962}} *)