# Weighted Portmanteau Test for Fitted ARCH process

### Description

A weighted portmanteau test for testing the null hypothesis of adequately fitted ARCH process. This is essentially a weighted version of the statistic proposed by Li and Mak (1994)

### Usage

1 2 3 | ```
Weighted.LM.test(x, h.t, lag = 1,
type = c("correlation", "partial"),
fitdf = 1, weighted = TRUE)
``` |

### Arguments

`x` |
a numeric vector or univariate time series, or residuals of a fitted time series |

`h.t` |
a numeric vector of the conditional variances |

`lag` |
the statistic will be based on |

`type` |
type of test to be performed, either based on the autocorrelations or partial-autocorrelations. |

`fitdf` |
the number of ARCH parameters fit to the model, default=1 since at least some ARCH model must be fit to find h.t |

`weighted` |
A flag determining if the weighting scheme should be utilized. TRUE by default, if FALSE, it performs the test from Li and Mak (1994) |

### Details

These test can be performed after fitting an ARCH process to a time series. The theoretical work was originally developed in Li and Mak (1994) and has recently been extended in Fisher and Gallagher (2012).

### Value

A list with class "`htest`

" containing the following components:

`statistic` |
the value of the test statistic |

`parameter` |
The approximate shape and scale parameters for the weighted statistic or degrees of freedom of the chi-squared distribution if the weighted flag is set to FALSE. |

`p.value` |
The p-value of the test |

`method` |
a character string indicating which type of test was performed. |

`data.name` |
a character string giving the name of the data |

### Note

Similiar to the `Box.test()`

and `Weighted.Box.test()`

functions

### Author(s)

Thomas J. Fisher

### References

Fisher, T. J. and Gallagher, C. M. (2012), New Weighted Portmanteau Statistics for Time Series Goodness-of-Fit Testing. Journal of the American Statistical Association, accepted.

Li, W. K. and Mak, T. K. (1994), On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15(6), 627-636.

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