Politis and White (2004) Spectral Density "PWSD" Automatic Block-Length Selection

Run the Automatic Block-Length selection method proposed by Politis and White (2004) and corrected in Patton, Politis, and White (2009). The method is based on spectral density estimation via flat-top lag windows of Politis and Romano (1995). This code was adapted from b.star to add functionality and include correlogram support including an S3 method, see Hayfield and Racine (2008).

pwsd(
  data,
  K_N = NULL,
  M_max = NULL,
  m_hat = NULL,
  b_max = NULL,
  c = NULL,
  round = FALSE,
  correlogram = TRUE
)

Arguments

data: an \(n x k\) data.frame, matrix, or vector (if \(k = 1\)) where the optimal block-length will be computed for each of the \(k\) columns.
K_N: an integer value, the maximum lags for the auto-correlation, \(rho_k\), which to apply the implied hypothesis test. Defaults to max(5, log(N)). See Politis and White (2004) footnote c.
M_max: an integer value, the upper-bound for the optimal number of lags, \(M\), to compute the auto-covariance for. See Theorem 3.3 (ii) of Politis and White (2004).
m_hat: an integer value, if set to NULL (the default), then m_hat is estimated as the smallest integer after which the correlogram appears negligible for K_N lags. In problematic cases, setting m_hat to an integer value can be used to override the estimation procedure.
b_max: a numeric value, the upper-bound for the optimal block-length. Defaults to ceiling(min(3 * sqrt(n), n / 3)) per Politis and White (2004).
c: a numeric value, the constant which acts as the significance level for the implied hypothesis test. Defaults to qnorm(0.975) for a two-tailed 95% confidence level. Politis and White (2004) suggest c = 2.
round: a logical value, if set to FALSE then the final block-length output will not be rounded, the default. If set to TRUE the final estimates for the optimal block-length will be rounded to whole numbers.
correlogram: a logical value, if set to TRUE a plot of the correlogram (i.e. a plot of \(R(k)\) vs. \(k\)) will be output to the console. If set to FALSE, no interim plots will be output to the console, but may be plotted later using the corresponding S3 method, plot.pwsd.

Value

an object of class 'pwsd'

References

Andrew Patton, Dimitris N. Politis & Halbert White (2009) Correction to "Automatic Block-Length Selection for the Dependent Bootstrap" by D. Politis and H. White, Econometric Review, 28:4, 372-375, DOI: doi: 10.1080/07474930802459016

Dimitris N. Politis & Halbert White (2004) Automatic Block-Length Selection for the Dependent Bootstrap, Econometric Reviews, 23:1, 53-70, DOI: doi: 10.1081/ETC-120028836

Politis, D.N. and Romano, J.P. (1995), Bias-Corrected Nonparametric Spectral Estimation. Journal of Time Series Analysis, 16: 67-103, DOI: doi: 10.1111/j.1467-9892.1995.tb00223.x

Tristen Hayfield and Jeffrey S. Racine (2008). Nonparametric Econometrics: The np Package. Journal of Statistical Software 27(5). DOI: doi: 10.18637/jss.v027.i05

Examples

# Generate AR(1) time series
sim <- stats::arima.sim(list(order = c(1, 0, 0), ar = 0.5),
                        n = 500, innov = rnorm(500))

# Calculate optimal block length for series
pwsd(sim, round = TRUE)

#> $BlockLength
#>      b_Stationary b_Circular
#> data            7          8
#> 
#> $Acf
#> $Acf$data
#> 
#> Autocorrelations of series 'data[, i]', by lag
#> 
#>      0      1      2      3      4      5      6      7      8      9     10 
#>  1.000  0.435  0.218  0.080  0.007  0.049  0.064  0.044 -0.006  0.039  0.060 
#>     11     12     13     14     15     16     17     18     19     20     21 
#>  0.074  0.087  0.032  0.031 -0.012  0.003 -0.011  0.049  0.016  0.010  0.021 
#>     22     23     24     25     26     27     28 
#>  0.010 -0.015 -0.007 -0.040 -0.072  0.029  0.034 
#> 
#> 
#> $parameters
#>        n k        c K_N M_max b_max m_hat M rho_k_critical
#> [1,] 500 1 1.959964   5    28    68     2 4      0.1439999
#> 
#> $Call
#> pwsd(data = sim, round = TRUE)
#> 
#> attr(,"class")
#> [1] "pwsd"


# Use S3 Method
b <- pwsd(sim, round = TRUE, correlogram = FALSE)
plot(b)