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I'm looking for assistance in understanding/implementating the following paper Covariance Matrix Estimation in Time Series Where I need help is Eq 33

Assume $EX_i = 0$. Using the idea of lag window spectral density estimate, we estimate the covariance matrix $\Sigma_n = var(Sn)$ of the sum $S_n = \sum_{i=1} X_i$ by

$\Sigma_n = \sum_{1<i,j<n} K\left( \frac{i-j}{B_n} \right) X_i X_j^T$

K is a window function, K(0)=1, K(u)=0 for |u|>1. What I don't understand is what $B_n$ is the lag sequence satisfying $B_n \to \infty$ and $B_n/n \to 0$ means? What I assume this is doing is effectively applying a weighted sum of the variance and covariances but it's not clear to me how to implement in practice. I'm looking for an explanation which would allow implementation for a scalar $X_i$ or references.

kjetil b halvorsen
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David Waterworth
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  • You should fix typos in question, and $\Sigma_n$ is not the variance of $S_n$---there should be a normalizing factor $\frac{1}{\sqrt{n}}$. Computationally, $B_n$ is the width of the rolling window used to compute sample autocovariance. $B_n \rightarrow 0$ and $B_n/n \to 0$ are conditions ensuring consistency. In practice, where sample size is necessarily finite, one might choose, say, $B_n = \sqrt{n}$. – Michael May 28 '20 at 00:41
  • I don't think it's a typo - I copied it from the linked paper. Sn is the sum so we're talking about the variance of the sum not the variance of the sample? So if $X_i$ was iid then the variance of the sum of $X_i$ would be $n \sigma^2$ which is the sum of the elements of the nxn covariance matrix $\sigma^2 I$ right? – David Waterworth May 28 '20 at 01:31
  • @DavidWaterworth, yes, that is basically it. Here are a few answers that you may find useful: https://stats.stackexchange.com/questions/222221/using-hac-standard-errors-although-there-might-be-no-autocorrelation/222240#222240 https://stats.stackexchange.com/questions/312341/comparison-between-newey-west-1987-and-hansen-hodrick-1980/366238#366238 https://stats.stackexchange.com/questions/60942/newey-west-t-stats-and-critical-values/139379#139379 https://stats.stackexchange.com/questions/153444/what-is-the-long-run-variance/153543#153543 – Christoph Hanck May 28 '20 at 12:22
  • Thanks @ChristophHanck. Do you have any references where HAC estimators are used to estimating the sum of future residuals in order to estimate a prediction interval (in particular of the sum of time series observations? Most of the literature seems to revolve around computing the standard error of a parameter estimator. – David Waterworth May 29 '20 at 03:48
  • I think in case of prediction intervals, one usually exploits the MA structure of the forecast errors. – Christoph Hanck May 29 '20 at 06:11

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