# The Univariate-Multivariate Box and Pierce Portmanteau Test The univariate or multivariate Box-Pierce (1970) portmanteau test. ```r BoxPierce(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()`. Note: In `stats` `R`, the function `Box.test` was built to compute the Box and Pierce (1970) and Ljung and Box (1978) test statistics only in the univariate case where we can not use more than one single lag value at a time. The functions `BoxPierce` and `LjungBox` are more accurate than `Box.test` function and can be used in the univariate or multivariate time series at vector of different lag values as well as they can be applied on an output object from a fitted model described in the description of the function `BoxPierce`. ## Returns The Box and Pierce univariate or multivariate test statistic with the associated 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 Box, G.E.P. and Pierce, D.A. (1970). "Distribution of Residual Autocorrelation in Autoregressive-Integrated Moving Average Time Series Models". Journal of American Statistical Association, 65, 1509-1526. ## See Also `Box.test`, `LjungBox`, `MahdiMcLeod`, `Hosking`, `LiMcLeod`, `portest`, `GetResiduals`. ## Examples ```r x <- rnorm(100) BoxPierce(x) ## univariate test x <- cbind(rnorm(100),rnorm(100)) BoxPierce(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 BoxPierce(fit, lags) ## Correct (no need to specify fitdf) BoxPierce(fit, lags, fitdf = 2) ## Correct ## Apply the test statistic on the residuals and set fitdf = 2 res <- resid(fit) BoxPierce(res, lags) ## Wrong (fitdf is needed!) BoxPierce(res, lags, fitdf = 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 BoxPierce(fit,lags) ## Correct (no need to specify fitdf) ## Apply the test statistic on the residuals where fitdf = 2 res <- ts(na.omit(fit$resid)) BoxPierce(res,lags) ## Wrong (fitdf is needed!) BoxPierce(res,lags,fitdf = 2) ## Correct ## ## ## Monthly log stock returns of Intel corporation data: Test for ARCH Effects monthintel <- as.ts(monthintel) BoxPierce(monthintel) ## Usual test BoxPierce(monthintel,sqrd.res=TRUE) ## Test for ARCH effects ## #### Write a function to fit a model: Apply portmanteau test on fitted obj with class "list" ## Example FitModel <- function(data){ fit <- ar.ols(data, intercept = TRUE, order.max = 2) fitdf <- 2 res <- ts(na.omit(fit$resid)) 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) BoxPierce(Fit) ```

BoxPierce function