# Long Run Variance Estimates the long run variance respectively covariance matrix of the supplied time series. ```r lrv(x, method = c("kernel", "subsampling", "bootstrap", "none"), control = list()) ``` ## Arguments - `x`: vector or matrix with each column representing a time series (numeric). - `method`: method of estimation. Options are `kernel`, `subsampling`, `bootstrap` and `none`. - `control`: a list of control parameters. See 'Details'. ## Details The long run variance equals the limit of $n$ times the variance of the arithmetic mean of a short range dependent time series, where $n$ is the length of the time series. It is used to standardize tests concering the mean on dependent data. If `method = "none"`, no long run variance estimation is performed and the value 1 is returned (i.e. it does not alterate the test statistic). The `control` argument is a list that can supply any of the following components: - **`kFun`**: Kernel function (character string). More in 'Notes'. - **`b_n`**: Bandwidth (numeric > 0 and smaller than sample size). - **`gamma0`**: Only use estimated variance if estimated long run variance is < 0? Boolean. - **`l`**: Block length (numeric > 0 and smaller than sample size). - **`overlapping`**: Overlapping subsampling estimation? Boolean. - **`distr`**: Tranform observations by their empirical distribution function? Boolean. Default is `FALSE`. - **`B`**: Bootstrap repetitions (integer). - **`seed`**: RNG seed (numeric). - **`version`**: What property does the CUSUM test test for? Character string, details below. - **`loc`**: Estimated location corresponding to `version`. Numeric value, details below. - **`scale`**: Estimated scale corresponding to `version`. Numeric value, details below. Kernel-based estimation The kernel-based long run variance estimation is available for various testing scenarios (set by `control$version`) and both for one- and multi-dimensional data. It uses the bandwidth $b_n = $ `control$b_n` and kernel function $k(x) = $ `control$kFun`. For tests on certain properties also a corresponding location `control$loc` ($m_n$) and scale `control$scale` ($v_n$) estimation needs to be supplied. Supported testing scenarios are: * `"mean"` * 1-dim. data: $$ \hat{\sigma}^2 = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^2 + \frac{2}{n} \sum_{h = 1}^{b_n} \sum_{i = 1}^{n - h} (x_i - \bar{x}) (x_{i + h} - \bar{x}) k(h / b_n). $$ If `control$distr = TRUE`, then the long run variance is estimated on the empirical distribution of $x$. The resulting value is then multiplied with $\sqrt{\pi} / 2$. Default values: `b_n` = $0.9 n^{1/3}$, `kFun = "bartlett"`. * multivariate time series: The $k,l$-element of $\Sigma$ is estimated by $$ \hat{\Sigma}^{(k,l)} = \frac{1}{n} \sum_{i,j = 1}^{n}(x_i^{(k)} - \bar{x}^{(k)}) (x_j^{(l)} - \bar{x}^{(l)}) k((i-j) / b_n), $$ $k, l = 1, ..., m$. Default values: `b_n` = $\log_{1.8 + m / 40}(n / 50)$, `kFun = "bartlett"`. * `"empVar"` for tests on changes in the empirical variance. $$ \hat{\sigma}^2 = \sum_{h = -(n-1)}^{n-1} W \left( \frac{|h|}{b_n} \right) \frac{1}{n} \sum_{i = 1}^{n - |h|} ((x_i - m_n)^2 - v_n)((x_{i+|h|} - m_n)^2 - v_n). $$ Default values: $m_n =$ `mean(x)`, $v_n = $ `var(x)`. * `"MD"` for tests on a change in the median deviation. $$ \hat{\sigma}^2 = \sum_{h = -(n-1)}^{n-1} W \left( \frac{|h|}{b_n} \right) \frac{1}{n} \sum_{i = 1}^{n - |h|} (|x_i - m_n| - v_n)(|x_{i+|h|} - m_n| - v_n). $$ Default values: $m_n =$ `median(x)`, $v_n = \frac{1}{n-1} \sum_{i = 1}^n |x_i - m_n|$. * `"GMD"` for tests on changes in Gini's mean difference. $$ \hat{\sigma}^2 = 4 \sum_{h = -(n-1)}^{n-1} W \left( \frac{|h|}{b_n} \right) \frac{1}{n} \sum_{i = 1}^{n - |h|} \hat{\phi}_n(x_i)\hat{\phi}_n(x_{i+|h|}) $$ with $\hat{\phi}_n(x) = n^{-1} \sum_{i = 1}^n |x - x_i| - v_n$. Default value: $v_n =$ $\frac{2}{n(n-1)} \sum_{1 \leq i \< j \leq n} |x_i - x_j|.$ * `"Qalpha"` for tests on changes in `Qalpha`. $$ \hat{\sigma}^2 = \frac{4}{\hat{u}(v_n)} \sum_{h = -(n-1)}^{n-1} W \left( \frac{|h|}{b_n} \right) \frac{1}{n} \sum_{i = 1}^{n - |h|} \hat{\phi}_n(x_i)\hat{\phi}_n(x_{i+|h|}), $$ where $\hat{\phi}_n(x) = n^{-1} \sum_{i = 1}^n 1_{\{|x - x_i| \leq v_n\}} - m_n$ and $$ \hat{u}(t) = \frac{2}{n(n-1)h_n} \sum_{1 \leq i \< j \leq n} K\left(\frac{|x_i - x_j| - t}{h_n}\right) $$ the kernel density estimation of the densitiy $u$ corresponding to the distribution function $U(t) = P(|X-Y| \leq t)$, $h_n =$ `IQR(x)`$n^{-\frac{1}{3}}$ and $K$ is the quatratic kernel function. Default values: $m_n = \alpha = 0.5$, $v_n =$ `Qalpha(x, m_n)[n-1]`. * `"tau"` for tests in changes in Kendall's tau. Only available for bivariate data: assume that the given data `x` has the format $(x_i, y_i)_{i = 1, ..., n}$. $$ \hat{\sigma}^2 = \sum_{h = -(n-1)}^{n-1} W \left( \frac{|h|}{b_n} \right) \frac{1}{n} \sum_{i = 1}^{n - |h|} \hat{\phi}_n((x_i, y_i))\hat{\phi}_n((x_{i+|h|}, y_{i+|h|}), $$ where $\hat{\phi}_n(x) = 4 F_n(x, y) - 2F_{X,n}(x) 2 - F_{Y,n}(y) + 1 - v_n$ and $F_n$, $F_{X,n}$ and $F_{Y,n}$ are the empirical distribution functions of $((X_i, Y_i))_{i = 1, ..., n}$, $(X_i)_{i = 1, ..., n}$ and $(Y_i)_{i = 1, ..., n}$. Default value: $v_n = \hat{\tau}_n = \frac{2}{n(n-1)} \sum_{1 \leq i \< j \leq n} sign\left((x_j - x_i)(y_j - y_i)\right)$. * `"rho"` for tests on changes in Spearman's rho. Only availabe for $d$-variate data with $d \> 1$: assume that the given data `x` has the format $(x_{i,j} | i = 1, ..., n; j = 1, ..., d)$. $$ \hat{\sigma}^2 = a(d)^2 2^{2d} \left\{ \sum_{h = -(n-1)}^{n-1} K\left( \frac{|h|}{b_n} \right) \left( \sum_{i = 1}^{n-|h|} n^{-1} \prod_{j = 1}^d \hat{\phi}_n(x_i, x_j) \hat{\phi}_n(x_{i+|h|}, x_j) - M^2 \right) \right\} , $$ where $a(d) = (d+1) / (2^d - d - 1)$, $M = n^{-1} \sum_{i = 1}^n \prod_{j = 1}^d \hat{\phi}_n(x_i, x_j)$ and $\hat{\phi}_n(x, y) = 1 - \hat{U}_n(x, y)$, $\hat{U}_n(x, y) = n^{-1}$ (rank of $x_{i,j}$ in $x_{i,1}, ..., x_{i,n})$. When `control$gamma0 = TRUE` (default) then negative estimates of the long run variance are replaced by the autocovariance at lag 0 (= ordinary variance of the data). The function will then throw a warning. Subsampling estimation For `method = "subsampling"` there are an overlapping and a non-overlapping version (parameter `control$overlapping`). Also it can be specified if the observations x were transformed by their empirical distribution function $\tilde{F}_n$ (parameter `control$distr`). Via `control$l` the block length $l$ can be controlled. If `control$overlapping = TRUE` and `control$distr = TRUE`: $$ \hat{\sigma}_n = \frac{\sqrt{\pi}}{\sqrt{2l}(n - l + 1)} \sum_{i = 0}^{n-l} \left| \sum_{j = i+1}^{i+l} (F_n(x_j) - 0.5) \right|. $$ Otherwise, if `control$distr = FALSE`, the estimator is $$ \hat{\sigma}^2 = \frac{1}{l (n - l + 1)} \sum_{i = 0}^{n-l} \left( \sum_{j = i + 1}^{i+l} x_j - \frac{l}{n} \sum_{j = 1}^n x_j \right)^2. $$ If `control$overlapping = FALSE` and `control$distr = TRUE`: $$ \hat{\sigma} = \frac{1}{n/l} \sqrt{\pi/2} \sum_{i = 1}{n/l} \frac{1}{\sqrt{l}} \left| \sum_{j = (i-1)l + 1}^{il} F_n(x_j) - \frac{l}{n} \sum_{j = 1}^n F_n(x_j) \right|. $$ Otherwise, if `control$distr = FALSE`, the estimator is $$ \hat{\sigma}^2 = \frac{1}{n/l} \sum_{i = 1}^{n/l} \frac{1}{l} \left(\sum_{j = (i-1)l + 1}^{il} x_j - \frac{l}{n} \sum_{j = 1}^n x_j\right)^2. $$ Default values: overlapping = TRUE, the block length is chosen adaptively: $$ l_n = \max{\left\{ \left\lceil n^{1/3} \left( \frac{2 \rho}{1 - \rho^2} \right)^{(2/3)} \right\rceil, 1 \right\}} $$ where $\rho$ is the Spearman autocorrelation at lag 1. Bootstrap estimation If `method = "bootstrap"` a dependent wild bootstrap with the parameters $B = $ `control$B`, $l = $ `control$l` and $k(x) = $ `control$kFun` is performed: $$ \hat{\sigma}^2 = \sqrt{n} Var(\bar{x^*_k} - \bar{x}), k = 1, ..., B $$ A single $x_{ik}^*$ is generated by $x_i^* = \bar{x} + (x_i - \bar{x}) a_i$ where $a_i$ are independent from the data `x` and are generated from a multivariate normal distribution with $E(A_i) = 0$, $Var(A_i) = 1$ and $Cov(A_i, A_j) = k\left(\frac{i - j}{l}\right), i = 1, ..., n; j \neq i$. Via `control$seed` a seed can optionally be specified (cf. `set.seed`). Only `"bartlett"`, `"parzen"` and `"QS"` are supported as kernel functions. Uses the function `sqrtm` from package `pracma`. Default values: `B` = 1000, `kFun = "bartlett"`, `l` is the same as for subsampling. ## Note Kernel functions `bartlett`: $$ k(x) = (1 - |x|) * 1_{\{|x| < 1\}} $$ `FT`: $$ k(x) = 1 * 1_{\{|x| \leq 0.5\}} + (2 - 2 * |x|) * 1_{\{0.5 < |x| < 1\}} $$ `parzen`: $$ k(x) = (1 - 6x^2 + 6|x|^3) * 1_{\{0 \leq |x| \leq 0.5\}} + 2(1 - |x|)^3 * 1_{\{0.5 < |x| \leq 1\}} $$ `QS`: $$ k(x) = \frac{25}{12 \pi ^2 x^2} \left(\frac{\sin(6\pi x / 5)}{6\pi x / 5} - \cos(6 \pi x / 5)\right) $$ `TH`: $$ k(x) = (1 + \cos(\pi x)) / 2 * 1_{\{|x| < 1\}} $$ `truncated`: $$ k(x) = 1_{\{|x| < 1\}} $$ `SFT`: $$ k(x) = (1 - 4(|x| - 0.5)^2)^2 * 1_{\{|x| < 1\}} $$ `Epanechnikov`: $$ k(x) = 3 \frac{1 - x^2}{4} * 1_{\{|x| < 1\}} $$ `quatratic`: $$ k(x) = (1 - x^2)^2 * 1_{\{|x| < 1\}} $$ ## Returns long run variance $\sigma^2$ (numeric) resp. $\Sigma$ (numeric matrix) ## Author(s) Sheila ## References Andrews, D.W. "Heteroskedasticity and autocorrelation consistent covariance matrix estimation." Econometrica: Journal of the Econometric Society (1991): 817-858. Dehling, H., et al. "Change-point detection under dependence based on two-sample U-statistics." Asymptotic laws and methods in stochastics. Springer, New York, NY, (2015). 195-220. Dehling, H., Fried, R., and Wendler, M. "A robust method for shift detection in time series." Biometrika 107.3 (2020): 647-660. Parzen, E. "On consistent estimates of the spectrum of a stationary time series." The Annals of Mathematical Statistics (1957): 329-348. Shao, X. "The dependent wild bootstrap." Journal of the American Statistical Association 105.489 (2010): 218-235. ## See Also `CUSUM`, `HodgesLehmann`, `wilcox_stat` ## Examples ```r Z <- c(rnorm(20), rnorm(20, 2)) ## kernel density estimation lrv(Z) ## overlapping subsampling lrv(Z, method = "subsampling", control = list(overlapping = FALSE, distr = TRUE, l_n = 5)) ## dependent wild bootstrap estimation lrv(Z, method = "bootstrap", control = list(l_n = 5, kFun = "parzen")) ```

lrv function