What is the long run variance?

How is long run variance in the realm of time series analysis defined?

I understand it is utilized in the case there is a correlation structure in the data. So our stochastic process would not be a family of X1,X2 i.i.d. random variables but rather only identically distributed?

Could I have a standard reference as an introduction to the concept and the difficulties involved in its estimation?

Answer

It is a measure of the standard error of the sample mean when there is serial dependence.

If Yt is covariance stationary with E(Yt)=μ and Cov(Yt,Ytj)=γj (in an iid setting, this quantity would be zero!) such that j=0|γj|<. Then
lim
where the first equality is definitional, the second a bit more tricky to establish and the third a consequence of stationarity, which implies that \gamma_j=\gamma_{-j}.

So the problem is indeed lack of independence. To see this more clearly, write the variance of the sample mean as
\begin{align*}
E(\bar{Y}_T- \mu)^2&=E\left[(1/T)\sum_{t=1}^T(Y_t- \mu)\right]^2\\
&=1/T^2E[\{(Y_1- \mu)+(Y_2- \mu)+\ldots+(Y_T- \mu)\}\\
&\quad\{(Y_1- \mu)+(Y_2- \mu)+\ldots+(Y_T- \mu)\}]\\
&=1/T^2\{[\gamma_0+\gamma_1+\ldots+\gamma_{T-1}]+[\gamma_1+\gamma_0+\gamma_1+\ldots+\gamma_{T-2}]\\
&\quad+\ldots+[\gamma_{T-1}+\gamma_{T-2}+\ldots+\gamma_1+\gamma_0]\}
\end{align*}

A problem with estimating the long-run variance is that we of course do not observe all autocovariances with finite data. Kernel (in econometrics, "Newey-West" or HAC estimators) are used to this end,


\hat{J_T}\equiv\hat{\gamma}_0+2\sum_{j=1}^{T-1}k\left(\frac{j}{\ell_T}\right)\hat{\gamma}_j

k is a kernel or weighting function, the \hat\gamma_j are sample autocovariances. k, among other things must be symmetric and have k(0)=1. \ell_T is a bandwidth parameter.

A popular kernel is the Bartlett kernel
k\left(\frac{j}{\ell_T}\right) = \begin{cases}
\bigl(1 - \frac{j}{\ell_T}\bigr)
\qquad &\mbox{for} \qquad 0 \leqslant j \leqslant \ell_T-1 \\
0 &\mbox{for} \qquad j > \ell_T-1
\end{cases}

Good textbook references are Hamilton, Time Series Analysis or Fuller. A seminal (but technical) journal article is Newey and West, Econometrica 1987.

Attribution
Source : Link , Question Author : Monolite , Answer Author : Christoph Hanck

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