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I am using statsmodel package for fitting ARIMA(p,d,q) model to a time series. My question is how exactly does this package estimate confidence intervals of the parameters of this model? statsmodels documentation says that

"The confidence interval is based on the standard normal distribution if self.use_t is False. If self.use_t is True, then uses a Student’s t with self.df_resid_inference (or self.df_resid if df_resid_inference is not defined) degrees of freedom."

Then the question is how is the variance of different parameters estimated to apply the standard Normal or t-distribution method?

Edit: I used the Hessian matrix method to compute the covariance matrix. But the confidence interval obtained using my approach are much wider than those produced by statsmodels. Which means that statsmodels is not using the Hessian matrix approach. Also, I noticed that as I increase the length of my time-series, the confidence intervals obtained by these approaches become similar.

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The documentation for the cov_type argument to fit method describes the options for computing the covariance matrix associated with the parameter estimates.

The default method is the outer product of gradients estimator (cov_type='opg').

If you want to use the numerically approximated Hessian, you can choose cov_type='approx', but note that by default this will use complex step differentiation rather than finite differences, so it may still be different from what you compute by hand.

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