Classic Estimator for the Hurst Effect

ClassicEstimators

This module contains functions to estimate the Hurst coefficient, closely related to the long memory parameter, of a time series using the rescaled range (R/S) statistic.

Author

J. Eduardo Vera-Valdés

andrews_bandwidth(x::AbstractArray)

Computes the data-dependent number of lags of Andrews (1991) for the Bartlett long-run variance estimator, based on an AR(1) approximation: floor((3T/2)^{1/3} |2ρ/(1-ρ²)|^{2/3}), where ρ is the first-order autocorrelation. Internal function used by lo_rs_test and vs_test.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-858.

autocorrelation(x::AbstractArray, k::Int=30)

Computes the autocorrelation function of a time series at lags 0, 1, ..., k-1.

Arguments

• x::Array: The time series.
• k::Int: The number of lags (including lag 0). Default is 30.

Output

• acf::Array: The autocorrelation function.

Notes

To plot the autocorrelation function use autocorrelation_plot.

Examples

julia> autocorrelation(randn(100), 10)

autocorrelation_plot(x::AbstractArray, k::Int)

Computes the autocorrelation function of a time series.

Requires Plots.jl to be loaded, e.g. with using LongMemory, Plots.

Arguments

• x::Array: The time series.
• k::Int: The lag of the autocorrelation.

Output

• p1::Plots.Plot: The autocorrelation function.

Examples

julia> autocorrelation_plot(randn(100), 10)

autocovariance(x::AbstractArray, k::Int)

Computes the autocovariance of a time series at lags 0, 1, ..., k-1.

Arguments

• x::Array: The time series.
• k::Int: The number of lags (including lag 0).

Output

• acv::Array: The autocovariances at lags 0, ..., k-1.

Examples

julia> autocovariance(randn(100), 2)

bartlett_lrv(x::AbstractArray, q::Int)

Computes the Bartlett (Newey-West) estimator of the long-run variance of x with q lags. Internal function used by lo_rs_test and vs_test.

brownian_bridge_range_cdf(v::Real)

Distribution function of the range of a Brownian bridge (Kuiper's distribution), 1 + 2 Σ_{k≥1} (1 - 4k²v²) exp(-2k²v²). Internal function used by lo_rs_test.

brownian_bridge_variance_cdf(x::Real)

Distribution function of the variance of a Brownian bridge, 1 + 2 Σ_{k≥1} (-1)^k exp(-2k²π²x). Internal function used by vs_test.

dfa_est(x::AbstractArray; k::Int = 20)

Estimates the long memory parameter of a time series using detrended fluctuation analysis (DFA).

Arguments

• x::Array: The time series.

Optional arguments

• k::Int: The number of box sizes, log-spaced between 10 and T/4. Default is 20.

Output

• d::Real: The estimated long memory parameter.

Notes

The profile (the cumulative sum of the demeaned series) is split into non-overlapping boxes of size n, a linear trend is fitted by OLS in each box, and the fluctuation F(n) is the root mean squared residual. The slope α of the regression of log F(n) on log n estimates the Hurst coefficient H = d + 1/2, so d = α - 1/2. See dfa_fluctuation for the fluctuation function.

References

Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., and Goldberger, A. L. (1994). Mosaic organization of DNA nucleotides. Physical Review E, 49(2), 1685-1689.

Examples

julia> dfa_est(randn(500))

dfa_fluctuation(x::AbstractArray; k::Int = 20)

Computes the fluctuation function of detrended fluctuation analysis (DFA) with linear detrending.

Arguments

• x::Array: The time series.

Optional arguments

• k::Int: The number of box sizes, log-spaced between 10 and T/4. Default is 20.

Output

• nt::Array: The box sizes.
• F::Array: The fluctuation function at the box sizes.

Examples

julia> dfa_fluctuation(randn(500))

dfa_plot(x::AbstractArray; k::Int = 20, slope::Bool = false, slope2::Bool = false)

Produces the detrended fluctuation analysis (DFA) plot of a time series.

Requires Plots.jl to be loaded, e.g. with using LongMemory, Plots.

Arguments

• x::Array: The time series.

Output

• p1::Plots.Plot: The DFA plot.

Optional arguments

• k::Int: The number of box sizes. Default is 20.
• slope::Bool: If true, the slope of the linear regression is displayed.
• slope2::Bool: If true, the theoretical slope for a short memory process is displayed.

Notes

The slope of the regression estimates the Hurst coefficient H = d + 1/2. See dfa_est.

Examples

julia> dfa_plot(randn(500))

lo_rs_test(x::AbstractArray; q::Int = andrews_bandwidth(x))

Computes Lo's (1991) modified rescaled range test of the null hypothesis of short memory.

Arguments

• x::Array: The time series.

Optional arguments

• q::Int: The number of lags in the Bartlett estimator of the long-run variance. By default, it is chosen with the data-dependent rule of Andrews (1991) as suggested by Lo (1991), see andrews_bandwidth.

Output

• V::Real: The test statistic, R / (σ̂(q) √T), where R is the range of the partial sums of the demeaned series and σ̂²(q) the Bartlett estimator of the long-run variance.
• pval::Real: The two-sided p-value from the asymptotic distribution under the null, the range of a Brownian bridge.

Notes

Under the null of short memory, V converges to the range of a Brownian bridge, with 2.5% and 97.5% quantiles 0.809 and 1.862. Large values of V indicate long memory and small values antipersistence. In contrast to the classic R/S statistic, the test is robust to short-range dependence through the long-run variance estimator.

References

Lo, A. W. (1991). Long-term memory in stock market prices. Econometrica, 59(5), 1279-1313.

Examples

julia> lo_rs_test(randn(500))

log_variance_est(x::AbstractArray; m::Int=100)

Computes the long memory estimate using the log-variance.

Arguments

• x::Array: The time series.

Output

• d_var::Real: The estimated long memory parameter computed as (beta+1)/2, where beta is the slope of the linear regression of the log-variance plot.

Optional arguments

• m::Int: The number of partitions of the time series. Default is 100. A value of m ≈ length(x)/2 is recommended.

Notes

This function uses the linear regression method on the log-variance to estimate the long memory parameter.

Examples

julia> log_variance_est(randn(100,1); m = 40)

log_variance_plot(x::AbstractArray; m::Int=100, slope::Bool=false, slope2::Bool=false)

Produces the log-variance plot of a time series.

Requires Plots.jl to be loaded, e.g. with using LongMemory, Plots.

Arguments

• x::Array: The time series.

Output

• p1::Plots.Plot: The log-variance plot.

Optional arguments

• m::Int: The number of partitions of the time series. Default is 100. A value of m ≈ length(x)/2 is recommended.
• slope::Bool: If true, the slope of the linear regression is displayed.
• slope2::Bool: If true, the theoretical slope for a short memory process is displayed.

Notes

This function uses the linear regression method on the log of the variance plot to estimate the long memory parameter.

Examples

julia> log_variance_plot(randn(100,1); m = 40)

rescaled_range(x::AbstractArray; k::Int = 20)

Computes the rescaled range (R/S) statistic of a time series on the subsamples x[1:n] for an increasing grid of n.

Arguments

• x::Array: The time series.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 20.

Output

• RS::Array: The rescaled range statistic.

Examples

julia> rescaled_range(randn(100))

rescaled_range_est(x::AbstractArray; k::Int = 20)

Estimates the long memory parameter of a time series using the rescaled range (R/S) statistic.

Arguments

• x::Array: The time series.

Output

• d::Real: The estimated long memory parameter.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 20.

Notes

This function uses the linear regression method on the log of the rescaled range to estimate the long memory parameter. The Hurst coefficient is related to the long memory parameter d by the formula H = d + 1/2.

Examples

julia> rescaled_range_est(randn(100))

rescaled_range_plot(x::AbstractArray; k::Int = 100, slope::Bool = false, slope2::Bool = false)

Produces the rescaled range plot of a time series.

Requires Plots.jl to be loaded, e.g. with using LongMemory, Plots.

Arguments

• x::Array: The time series.

Output

• p1::Plots.Plot: The rescaled range plot.

Optional arguments

• k::Int: The number of partitions of the time series. Default is 100.
• slope::Bool: If true, the slope of the linear regression is displayed.
• slope2::Bool: If true, the theoretical slope for a short memory process is displayed.

Notes

This function uses the linear regression method on the log of the rescaled range to estimate the long memory parameter. The Hurst coefficient is related to the long memory parameter d by the formula H = d + 1/2.

Examples

julia> rescaled_range_plot(randn(300,1))

smean(x::AbstractArray)

Computes the sample mean of a time series.

Arguments

• x::Array: The time series.

Output

• mean::Real: The sample mean.

Examples

julia> smean(randn(100))

sstd(x::AbstractArray; k::Int=0)

Computes the sample standard deviation of a time series.

Arguments

• x::Array: The time series.

Output

• std::Real: The sample standard deviation.

Optional arguments

• k::Int: The bias-correction term.

Notes

This function divides by T instead of T-1 if no bias-correction term is provided.

Examples

julia> sstd(randn(100))

sstdk(x::AbstractArray, m::Int)

Computes the sample variance of the means of the non-overlapping blocks of size m of a time series.

Arguments

• x::Array: The time series.
• m::Int: The block size.

Output

• s2k::Real: The sample variance of the block means.

Notes

Only complete blocks are used, i.e. the last length(x) % m observations are discarded.

Examples

julia> sstdk(randn(100), 20)

vs_test(x::AbstractArray; q::Int = andrews_bandwidth(x))

Computes the V/S test of Giraitis, Kokoszka, Leipus and Teyssière (2003) of the null hypothesis of short memory.

Arguments

• x::Array: The time series.

Optional arguments

• q::Int: The number of lags in the Bartlett estimator of the long-run variance. By default, it is chosen with the data-dependent rule of Andrews (1991), see andrews_bandwidth.

Output

• M::Real: The test statistic, the sample variance of the partial sums of the demeaned series divided by T σ̂²(q).
• pval::Real: The p-value from the asymptotic distribution under the null.

Notes

Under the null of short memory, M converges to the variance of a Brownian bridge, with distribution function 1 + 2 Σ_{k≥1} (-1)^k exp(-2k²π²x). The 5% critical value is 0.1869; large values indicate long memory. The test is a modification of the KPSS test that uses the variance of the partial sums instead of their second moment, which makes it more sensitive to shifts in the variance of the partial sums.

References

Giraitis, L., Kokoszka, P., Leipus, R., and Teyssière, G. (2003). Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics, 112(2), 265-294.

Examples

julia> vs_test(randn(500))