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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.

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  • $\begingroup$ 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, $B_n = \sqrt{n}$. $\endgroup$
    – Michael
    May 28, 2020 at 0:41
  • $\begingroup$ 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? $\endgroup$ May 28, 2020 at 1:31
  • $\begingroup$ @DavidWaterworth, yes, that is basically it. Here are a few answers that you may find useful: stats.stackexchange.com/questions/222221/… stats.stackexchange.com/questions/312341/… stats.stackexchange.com/questions/60942/… stats.stackexchange.com/questions/153444/… $\endgroup$ May 28, 2020 at 12:22
  • $\begingroup$ 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. $\endgroup$ May 29, 2020 at 3:48
  • $\begingroup$ I think in case of prediction intervals, one usually exploits the MA structure of the forecast errors. $\endgroup$ May 29, 2020 at 6:11

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