# Hurst coefficient estimation via spectral regression

### Description

Function to estimate the Hurst parameter H of a time series by linear regression of the log(spectrum) versus log(frequency) with frequency points accumulated into boxes of equal width on a logarithmic scale and spectrum values averaged over each box.

- standard
Given an estimate of the SDF for the input time series, this function estimates the Hurst coefficient of the time series by performing a linear regression of log(SDF) versus log(frequency). The range of frequencies to be included in the regression is specified by the

`dc`

and`freq.max`

input arguments.- smoothed
Given an estimate of the SDF for the input time series, this function estimates the Hurst coefficient of the time series by performing a linear regression of log(SDF) versus log(frequency). The range of frequencies to be included in the regression is specified by the

`dc`

and`freq.max`

input arguments. Frequencies are partitioned into blocks of equal width on a logarithmic scale and the SDF is averaged over each block. The number of blocks is controlled by the`n.block`

argument.- robinson
Estimates the Hurst coefficient by Robinson's SDF integration method. Given an estimate of the SDF for the input time series, this function estimates the Hurst coefficient of a time series by applying Robinson's integral method (typically) to the low- frequency end of the SDF. Use the

`freq.max`

argument to define the low-frequency cutoff.

### Usage

1 2 |

### Arguments

`x` |
a vector containing a uniformly-sampled real-valued time series. |

`...` |
optional SDF estimation arguments passed directly to the |

`dc` |
a logical value. If |

`fit` |
a function representing the linear regression scheme to use in fitting
the resulting statistics (on a log-log scale). Supported functions are: |

`freq.max` |
the largerst normalized frequency to include in the regression scheme.
Default: |

`method` |
a character string indicating the method to be used in estimating the Hurst coefficient (H). Choices are: `"standard"` Regression of SDF estimate. `"smoothed"` Regression of block averages of the SDF estimate taken over dyadic partitions in frequency. `"robinson"` Robinson's SDF integration method.
Default: |

`n.block` |
an integer denoting the number of logarithmic frequency divisions to use
in partitioning the estimated SDF. This input argument is only used if |

`sdf.method` |
a character string denoting the method to use in estimating the SDF.
Choices are |

`weight` |
a function with a single required variable ( |

### Value

an object of class `fractalBlock`

.

### References

P.M. Robinson (1994), Semiparametric analysis of long-memory time series,
*Annals of Statistics*, **22**, 515–539.

I. Lobato and P.M. Robinson (1996), Averaged periodogram estimation of long
memory, *Journal of Econometrics*, **73**, 303–324.

J. Geweke and Susan Porter-Hudak (1983), The Estimation and Application of
Long Memory Time Series Models, *Journal of Time Series Analysis*,
**4**, 221–237.

Murad S. Taqqu, Vadim Teverovsky, and Walter Willinger (1995), Estimators
for Long-Range Dependence: An Empirical Study, *Fractals*,
**3**, 785–798.

### See Also

`hurstBlock`

, `fractalBlock`

, `HDEst`

, `lm`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
## create test series
set.seed(100)
x <- rnorm(1024)
walk <- cumsum(x)
## calculate the Hurst coefficient of a random
## walk series using various techniques. use a
## multitaper SDF
methods <- c("standard","smoothed")
z <- lapply(methods, function(method, walk){
hurstSpec(walk, method=method, sdf.method="multitaper")
},walk=walk )
names(z) <- methods
## plot results
old.plt <- par("plt")
for (i in 1:2){
splitplot(2,1,i)
plot(z[[i]])
}
par(plt=old.plt)
## Robinson's method
hurstSpec(walk, method="robinson", sdf.method="multitaper")
``` |