# LRVar - Long-run Variance (Bartlett Kernel)

Returns the long-run variance using a Bartlett kernel with window size k.

## Syntax

LRVar(X, k)

X is the input data sample (a one dimensional array of cells (e.g. rows or columns)).

k is the input Bartlett kernel window size. If omitted, the default value is the cubic root of the sample data size.

## Remarks

1. The input time series data may include missing values (e.g. #N/A, #VALUE!, #NUM!, empty cell), but they will not be included in the calculations.
2. The long-run variance is computed as follows:
$$\sigma^2=\frac{1}{T}\sum_{t=k}^{T-k}\sum_{i=-k}^k w_i(x_t-\bar{x})(x_{t-i}-\bar{x})$$
Where:
• $x_{t} \in X$ is a value from the input time series data
• $\bar{x}$ is the mean of the input time series data
• The weight ($w_i$) in Bartlett kernel is defined as follows:

$$w_i= 1- \frac{\left | i \right |}{k+1}$$
• $k$ is the input window size for the Bartlett kernel

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A B
Date Data
1/1/2008 #N/A
1/2/2008 -1.28
1/3/2008 0.24
1/4/2008 1.28
1/5/2008 1.20
1/6/2008 1.73
1/7/2008 -2.18
1/8/2008 -0.23
1/9/2008 1.10
1/10/2008 -1.09
1/11/2008 -0.69
1/12/2008 -1.69
1/13/2008 -1.85
1/14/2008 -0.98
1/15/2008 -0.77
1/16/2008 -0.30
1/17/2008 -1.28
1/18/2008 0.24
1/19/2008 1.28
1/20/2008 1.20
1/21/2008 1.73
1/22/2008 -2.18
1/23/2008 -0.23
1/24/2008 1.10
1/25/2008 -1.09
1/26/2008 -0.69
1/27/2008 -1.69
1/28/2008 -1.85
1/29/2008 -0.98

Formula Description (Result)
=LRVar($B$2:$B$30,3) Long-run variance (2.084)