# The Modified Multivariate Portmanteau Test, Li-McLeod (1981) The modified multivariate portmanteau test suggested by Li and McLeod (1981). ```r LiMcLeod(obj,lags=seq(5,30,5),fitdf=0,sqrd.res=FALSE) ``` ## Arguments - `obj`: a univariate or multivariate series with class `"numeric"`, `"matrix"`, `"ts"`, or `("mts" "ts")`. It can be also an object of fitted time-series model with class `"ar"`, `"arima0"`, `"Arima"`, `("ARIMA forecast ARIMA Arima")`, `"lm"`, `("glm" "lm")`, or `"varest"`. `obj` may also an object with class `"list"` (see details and following examples). - `lags`: vector of lag auto-cross correlation coefficients used for `Hosking` test. - `fitdf`: Default is zero for testing the randomness of a given sequence with class `"numeric"`, `"matrix"`, `"ts"`, or `("mts" "ts")`. In general `fitdf` equals to the number of estimated parameters in the fitted model. If `obj` is an object with class `"ar"`, `"arima0"`, `"Arima"`, `"varest"`, `("ARIMA forecast ARIMA Arima")`, or `"list"` then no need to enter the value of `fitdf` as it will be automatically determined. For `obj` with other classes, the `fitdf` is needed for degrees of freedom of asymptotic chi-square distribution. - `sqrd.res`: if `TRUE` then apply the test on the squared values. This checks for Autoregressive Conditional Heteroscedastic, `ARCH`, effects. When `sqrd.res = FALSE`, then apply the test on the usual residuals. ## Details 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 `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 fitdf of the fitted model (`list(res = ..., fitdf = ...)`). See the following example with the function `FitModel()`. ## Returns The multivariate test statistic suggested by Li and McLeod (1981) and its corresponding p-values for different lags based on the asymptotic chi-square distribution with `k^2(lags-fitdf)` degrees of freedom. ## Author(s) Esam Mahdi and A.I. McLeod. ## References 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. ## See Also `Box.test`, `BoxPierce`, `LjungBox`, `MahdiMcLeod`, `Hosking`, `portest`, `GetResiduals`. ## Examples ```r x <- rnorm(100) LiMcLeod(x) ## univariate test x <- cbind(rnorm(100),rnorm(100)) LiMcLeod(x) ## multivariate test ## ## ## Monthly log stock returns of Intel corporation data: Test for ARCH Effects monthintel <- as.ts(monthintel) LjungBox(monthintel) ## Usual test LjungBox(monthintel,sqrd.res=TRUE) ## Test for ARCH effects ## ## ## 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 (fitdf will be automatically applied) LiMcLeod(fit,lags,fitdf = 2) ## Correct (no need to specify fitdf) LiMcLeod(fit,lags) ## Correct ## Apply the test statistic on the residuals res <- ts((fit$resid)[-(1:2), ]) LiMcLeod(res,lags,fitdf = 2) ## Correct LiMcLeod(res,lags) ## Wrong (fitdf is needed!) ## ## ## Write a function to fit a model: Apply portmanteau test on fitted obj with class "list" FitModel <- function(data){ fit <- ar.ols(data, intercept = TRUE, order.max = 2) fitdf <- 2 res <- res <- ts((fit$resid)[-(1:2), ]) list(res=res,fitdf=fitdf) } 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) LiMcLeod(Fit) ```

LiMcLeod function