Generalized Variance Portmanteau Test
New generalized variance portmanteau test based on the determinant of the Hosking's autocorrelation block
Toeplitz matrix with order m+1 given in the function ToeplitzBlock,
where m represents the order of the block matrix.
Originally, the generalized variance portmanteau test, MahdiMcLeod,
for univariate time series was derived by Pena and Rodriguez (2002)
based on the gamma distribution.
Lin and McLeod (2006) proposed the Monte-Carlo version of this test and
Mahdi and McLeod (2012) extended both methods to the multivariate case.
Simulation results suggest that the Monte-Carlo version of
MahdiMcLeod statistic is more accurate
and powerful than its competitors proposed by Box and Pierce (1970), Ljung and Box (1978),
and Pena and Rodriguez (2002, 2006) in the univariate time series and
Hosking (1980) and Li and McLeod (1981) in the multivariate time series.
MahdiMcLeod(obj,lags=seq(5,30,5),order=0,season=1,squared.residuals=FALSE)
obj |
a univariate or multivariate series with class |
lags |
vector of lag auto-cross correlation coefficients used for |
order |
Default is zero for testing the randomness of a given sequence with
class |
season |
seasonal periodicity for testing seasonality. Default is 1 for testing the non seasonality cases. |
squared.residuals |
if |
However the portmanteau test statistic can be applied directly on the output objects from
the built in R functions ar(), ar.ols(), ar.burg(),
ar.yw(), ar.mle(), arima(), arim0(), Arima(),
auto.arima(), lm(), glm(), and VAR(),
it works with output objects from any fitted model.
In this case, users should write their own function to fit any model they want, where they
may use the built in R functions FitAR(), garch(), garchFit(),
fracdiff(), tar(), etc.
The object obj represents the output of this function.
This output must be a list with at least two outcomes:
the fitted residual and the order of the fitted model (list(res = ..., order = ...)).
See the following example with the function FitModel().
The generalized variance portmanteau test statistic and its associated p-values for different lags based on asymptotic chi-square as given in Mahdi and McLeod (2012).
Esam Mahdi and A.I. McLeod.
Hosking, J. R. M. (1980). "The Multivariate Portmanteau Statistic". Journal of American Statistical Association, 75, 602-608.
Li, W. K. and McLeod, A. I. (1981). "Distribution of The Residual Autocorrelations in Multivariate ARMA Time Series Models". Journal of The Royal Statistical Society, Series B, 43, 231-239.
Lin, J.-W. and McLeod, A.I. (2006). "Improved Generalized Variance Portmanteau Test". Computational Statistics and Data Analysis 51, 1731-1738.
Mahdi, E. and McLeod, A.I. (2012). "Improved Multivariate Portmanteau Test". Journal of Time Series Analysis, 33(2), 211-222.
Pena, D. and Rodriguez, J. (2002). "A Powerful Portmanteau Test of Lack of Test for Time Series". Journal of American Statistical Association, 97, 601-610.
Pena, D. and Rodriguez, J. (2006). "The log of the determinant of the autocorrelation matrix for testing goodness of fit in time series". Journal of Statistical Planning and Inference, 136, 2706-2718.
x <- rnorm(100)
MahdiMcLeod(x) ## univariate test
x <- cbind(rnorm(100),rnorm(100))
MahdiMcLeod(x) ## multivariate test
##
##
## Annual flow of the river Nile at Aswan - 1871 to 1970
fit <- arima(Nile, c(1, 0, 1))
lags <- c(5, 10, 20)
## Apply the univariate test statistic on the fitted model
MahdiMcLeod(fit, lags) ## Correct (no need to specify order)
MahdiMcLeod(fit, lags, order = 2) ## Correct
## Apply the test statistic on the residuals and set order = 2
res <- resid(fit)
MahdiMcLeod(res, lags) ## Wrong (order is needed!)
MahdiMcLeod(res, lags, order = 2) ## Correct
##
##
## Quarterly, west German investment, income, and consumption from 1960 Q1 to 1982 Q4
data(WestGerman)
DiffData <- matrix(numeric(3 * 91), ncol = 3)
for (i in 1:3)
DiffData[, i] <- diff(log(WestGerman[, i]), lag = 1)
fit <- ar.ols(DiffData, intercept = TRUE, order.max = 2)
lags <- c(5,10)
## Apply the test statistic on the fitted model
MahdiMcLeod(fit,lags) ## Correct (no need to specify order)
## Apply the test statistic on the residuals where order = 2
res <- ts((fit$resid)[-(1:2), ])
MahdiMcLeod(res,lags) ## Wrong (order is needed!)
MahdiMcLeod(res,lags,order = 2) ## Correct
##
##
## Monthly log stock returns of Intel corporation data: Test for ARCH Effects
monthintel <- as.ts(monthintel)
MahdiMcLeod(monthintel) ## Usual test
MahdiMcLeod(monthintel,squared.residuals=TRUE) ## Test for ARCH effects
##
##
## Test for seasonality
## Accidental Deaths in the US 1973 - 1978
seasonal.arima <- arima(USAccDeaths, order = c(0,1,1), seasonal = list(order = c(0,1,1)))
MahdiMcLeod(seasonal.arima, lags = 5, season = 12)
## Quarterly U.K. economic time series from 1957 Q3 to 1967 Q4
cd <- EconomicUK[,1]
cd.fit <- arima(cd,order=c(0,1,0),seasonal=list(order=c(0,1,1),period=4))
MahdiMcLeod(cd.fit, lags = c(5,10), season = 4)
##
##
#### Write a function to fit a model: Apply portmanteau test on fitted obj with class "list"
## Example 1
require("FitAR")
FitModel <- function(data){
fit <- FitAR(z=data,p=2)
p <- length(fit$phiHat)
order <- p
res <- fit$res
list(res=res,order=order)
}
Fit <- FitModel(Nile)
MahdiMcLeod(Fit)
detach(package:FitAR)
##
## Example 2
FitModel <- function(data){
fit <- ar.ols(data, intercept = TRUE, order.max = 2)
order <- 2
res <- res <- ts((fit$resid)[-(1:2), ])
list(res=res,order=order)
}
data(WestGerman)
DiffData <- matrix(numeric(3 * 91), ncol = 3)
for (i in 1:3)
DiffData[, i] <- diff(log(WestGerman[, i]), lag = 1)
Fit <- FitModel(DiffData)
MahdiMcLeod(Fit)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.