calc_lrvar_bartlett: Long-Run Variance Estimation with Bartlett Kernel

In Westerlund: Panel Cointegration Tests Based on Westerlund (2007)

Long-Run Variance Estimation with Bartlett Kernel

Description

Computes a Bartlett-kernel (Newey–West style) long-run variance estimate for a univariate series using Stata-like conventions: missing values are removed (na.omit), autocovariances are scaled by 1/n (not 1/(n-j)), and optional centering is controlled by nodemean.

Usage

calc_lrvar_bartlett(x, maxlag, nodemean = FALSE)

Arguments

x	A numeric vector. Missing values are removed prior to computation.
maxlag	Non-negative integer. Maximum lag order m used in the Bartlett kernel. If maxlag=0, the function returns the sample variance estimate \gamma_0.
nodemean	Logical. If FALSE (default), the series is centered by subtracting its mean before computing autocovariances. If TRUE, the function assumes x is already centered and does not de-mean it.

Details

Let x_t be the input series after removing missing values. If nodemean=FALSE, the function replaces x_t with x_t - \bar{x}.

Define the lag-j autocovariance using the Stata-style scaling:

\gamma_j = \frac{1}{n}\sum_{t=j+1}^{n} x_t x_{t-j}, \qquad j=0,1,\dots,m,

where n is the length of the cleaned series and m=\code{maxlag}. Note that the scaling uses 1/n for all j (rather than 1/(n-j)).

The Bartlett kernel weight at lag j is:

w_j = 1 - \frac{j}{m+1}.

The long-run variance estimate is then:

\widehat{\Omega} = \gamma_0 + 2\sum_{j=1}^{m} w_j \gamma_j.

If the cleaned series has zero length, the function returns NA.

Value

A single numeric value:

• The Bartlett-kernel long-run variance estimate \widehat{\Omega} (scalar),

• or NA if x contains no non-missing values.

Technical Notes

This section illustrates how calc_lrvar_bartlett() computes a Bartlett-kernel long-run variance (LRV) estimate and how its options map to common time-series preprocessing choices.

Stata-like conventions

This helper function is designed to match Stata-style calculations:

• Missing values are dropped: na.omit(x) is applied first.

• Scaling by 1/n: autocovariances at all lags divide by n, not by n-j.

• Optional de-meaning: controlled by nodemean.

Centering

By default (nodemean=FALSE), the series is centered: x_t \leftarrow x_t - \bar{x}. Set nodemean=TRUE when you have already centered the series elsewhere.

Choosing maxlag

maxlag sets the truncation point m. Larger m captures more serial correlation but increases estimation noise.

See Also

westerlund_test

Examples

## Example 1: Basic usage
x <- rnorm(200)
calc_lrvar_bartlett(x, maxlag = 4)

## Example 2: maxlag = 0 returns gamma_0
calc_lrvar_bartlett(x, maxlag = 0)

## Example 3: Handle missing values (they are removed)
x_na <- x
x_na[c(5, 10, 50)] <- NA
calc_lrvar_bartlett(x_na, maxlag = 4)

## Example 4: Compare centering choices
## Default: de-mean internally
lr1 <- calc_lrvar_bartlett(x, maxlag = 4, nodemean = FALSE)

## Pre-center and skip de-meaning
x_centered <- x - mean(x)
lr2 <- calc_lrvar_bartlett(x_centered, maxlag = 4, nodemean = TRUE)

Westerlund documentation built on Feb. 7, 2026, 5:07 p.m.