Bartlett formula for confidence interval estimation

[Sara Safari]

Hi!

My collaborators and I are working on a project that involves using tsa module implemented in statsmodels. We regularly use acf functionality, and estimate the confidence intervals using the built-in method of the package that is based on Bartlett’s formulation.

 

We understand that the way it’s done by statsmodels is that at each step it assumes a MA(k-1) process for the data to estimate the error at lag k. 

However, the closed form of Bartlett according to Brokwell and Davis is based on a general ARMA(p,q) model. Is there a reason that the Auto-Regressive part is dropped in the implementation and the true process is assumed to be MA(k-1)?

[josef...@gmail.com]

AFAIR (from a long time ago)

The confints are computed under the null that the acf coefficient is zero (similar to score confidence intervals).

A AR process with non-zero coefficients does not have zero acf.

To get zero acf coefficients we either need independence ARMA(0, 0) or an MA process with order smaller than the acf lag.

AFAIR, I have never seen "Wald confidence intervals", i.e. confidence intervals for the estimated non-zero acf of an ARMA(p, q) process.

Josef

Josef

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[josef...@gmail.com]

On Tue, Aug 6, 2024 at 1:45 PM <josef...@gmail.com> wrote:

AFAIR (from a long time ago)

The confints are computed under the null that the acf coefficient is zero (similar to score confidence intervals).

A AR process with non-zero coefficients does not have zero acf.

To get zero acf coefficients we either need independence ARMA(0, 0) or an MA process with order smaller than the acf lag.

AFAIR, I have never seen "Wald confidence intervals", i.e. confidence intervals for the estimated non-zero acf of an ARMA(p, q) process.

coming back to this.

Wald confidence intervals could be computed for a given model like ARMA(p, q).

ACF is a nonlinear function of the estimated ARMA coefficients, so we could compute standard errors and 

confidence intervals using the Delta method based on the asymptotic distribution of the estimated coefficients.

However, this assumes that the specified ARMA model is the true model. 

Confidence intervals for acf and pacf are simple indicators of which correlations might be non-zero.