LRVar - Long-run Variance (Bartlett Kernel)

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


LRVar(X, k)

is the input data sample (a one-dimensional array of cells (e.g., rows or columns)).
is the input Bartlett kernel window size. If omitted, the default value is the cubic root of the sample data size.


  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:
    $$\sigma^2=\frac{1}{T}\sum_{t=k}^{T-k}\sum_{i=-k}^k w_i(x_t-\bar{x})(x_{t-i}-\bar{x})$$
    • $x_{t} \in X$ 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.

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