# Portmanteau Test Statistics Univariate or multivariate portmanteau test statistics of `BoxPierce`, `MahdiMcLeod`, `Hosking`, `LiMcLeod`, `LjungBox`, and possibly any other test statistic using Monte-Carlo techniques or asymptotic distributions. ```r portest(obj,lags=seq(5,30,5),test=c("MahdiMcLeod","BoxPierce","LjungBox", "Hosking","LiMcLeod","other"),fn=NULL,sqrd.res=FALSE,MonteCarlo=TRUE, innov.dist=c("Gaussian","t","bootstrap"), ncores=1,nrep=1000, model=list(sim.model=NULL,fit.model=NULL),pkg.name=NULL,set.seed=123,...) ``` ## Arguments - `obj`: if `obj` is an object of class `"ar"`, `"arima0"`, `"Arima"`, `("ARIMA forecast ARIMA Arima")`, `"lm"`, `("glm" "lm")`, `"varest"`, or `"list"` then a portmanteau goodness-of-fit test is done on the fitted model. Otherwise, for `obj` with class `"ts"`, `"numeric"`, `"matrix"`, or `("mts" "ts")`, a test of randomness is done. - `lags`: vector of lag values is used for portmanteau test. - `test`: portmanteau test statistic type. - `sqrd.res`: as described in `BoxPierce`, `MahdiMcLeod`, `Hosking`, `LiMcLeod`, and `LjungBox`. - `fn`: a function calculates the test statistic that is associated with `test = "other"`. For example, `fn` can be a function returns the generalized Durbin-Watson test statistic values calculated at different lags. This function has at least two inputs: `obj` and `lags`, where `obj` and `lags` are described as above. - `MonteCarlo`: if `TRUE` then apply the Monte-Carlo version of portmanteau statistic. Otherwise, apply the asymptotic distribution. - `innov.dist`: distribution to generate univariate or multivariate innovation process. This could be `Gaussian`, `t`, or `bootstrap` using resampled errors rather than distributed errors. Default is `Gaussian`. - `ncores`: number of cores needed to use in parallel calculations. Default is a single CPU. - `nrep`: number of replications needed for Monte-Carlo test. - `model`: additional argument defined as a list with two specified functions, `sim.model` and `fit.model`. This argument is needed when the class of `obj` is `"list"` (see details and following example). - `pkg.name`: the name of the required library to be loaded if the Monte-Carlo significance test is used with an object `obj` with class `"list"`. - `set.seed`: `set.seed` is initialized. Default seed is 123, but users can use any seed they wish. - `...`: arguments to be passed to methods, such as `dft` degrees of freedom needed to generate innovations with univariate/multivariate series with t-distribution innovations, or `trunc.lag` used in `varima.sim` function, or `fitdf` and `season` as described in `BoxPierce`, `LjungBox`, `Hosking`, `LiMcLeod` and `MahdiMcLeod`. ## Details The portmanteau test statistics, `MahdiMcLeod`, `BoxPierce`, `LjungBox`, `Hosking`, and `LiMcLeod` are implemented based on the Monte-Carlo techniques and the approximation asymptotic distributions as described in Mahdi and McLeod (2012). Any other possible test statistic is also implemented in this function by selecting the argument `test = "other"` and providing the test statistic as a function passing the argument `fn`. The null hypothesis assuming that the fitted model is an adequate model and the residuals behave like white noise series. This function can be used for testing the adequacy in the nonseasonal fitted time series models. this function can be used to check for randomness as well as to check for `ARCH`-`GARCH` effects. Any other fitted model, for example, threashold autoregression model, may also be tested for adequacy. In this case, two functions, `sim.model()` and `fit.model()`, must be provided via the argument `func`. The object `obj` is the output of the fitted model coded in the function `fit.model` and it is a `"list"` with at least `res`, the residuals from the fitted model in `fit.model()`, and `fitdf`, the fitdf of this fitted model. The output from the function `sim.model()` is a simulated univariate or multivariate series from the fitted model obtained from the function `fit.model()`. The argument `pkg.name` represents the name of the `R` package where the fitted model build in (see the last given example). The parallel computing using the `portes` package proposed by Gonzalo Vera, Ritsert Jansen, and Remo Suppi (2008) will run if one decide to choose the argument `MonteCarlo=TRUE` provided that `ncores` equals to a positive integer more than 1. ## Returns The portmanteau test statistic with the associated p-values for different lag values. When the argument `MonteCarlo` is set to be `FALSE` then the degrees of freedom will be an additional output. ## 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. Fox, J and Weisberg, S and Adler, D and Bates, D and Baud-Bovy, G and Ellison, S and Firth, D and Friendly, M and Gorjanc, G and Graves, S and Heiberger, R and Laboissiere, R and Monette, G and Murdoch, D and Nilsson, H and Ogle, D and Ripley, B and Venables, W and Zeileis, A and R-Core (2019). car: Companion to Applied Regression. R package version 3.0-3, [https://CRAN.R-project.org/package=car](https://CRAN.R-project.org/package=car). Fraley C, Leisch F, Maechler M, Reisen V, Lemonte A (2012). fracdiff: Fractionally differenced ARIMA aka ARFIMA(p,d,q) models. R package version 1.4-2, [https://CRAN.R-project.org/package=fracdiff](https://CRAN.R-project.org/package=fracdiff). Hosking, J. R. M. (1980). "The Multivariate Portmanteau Statistic". Journal of American Statistical Association, 75, 602-608. John Haslett and Adrian E. Raftery (1989). "Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource (with Discussion)". Applied Statistics, 38, 1-50. 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. Lin, J.-W. and McLeod, A.I. (2008). "Portmanteau Tests for ARMA Models with Infinite Variance". Journal of Time Series Analysis, 29, 600-617. Ljung, G.M. and Box, G.E.P (1978). "On a Measure of Lack of Fit in Time Series Models". Biometrika, 65, 297-303. Mahdi, E. and McLeod, A.I. (2012). "Improved Multivariate Portmanteau Test". Journal of Time Series Analysis, 33(2), 211-222. McLeod A.I, Li W.K (1983). "Distribution of the Residual Autocorrelation in Multivariate ARMA Time Series Models". Journal of Time Series Analysis, 4, 269-273. 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. Pfaff B, Stigler M (2018). vars: VAR Modelling. R package version 1.5-3, [https://CRAN.R-project.org/package=vars](https://CRAN.R-project.org/package=vars). Rob J Hyndman with contributions from George Athanasopoulos Slava Razbash DSZZYKCB, Wang E (2019). forecast: Forecasting Functions for Time Series and Linear Models. R package version 8.7, [https://CRAN.R-project.org/package=forecast](https://CRAN.R-project.org/package=forecast). Tierney, L., Rossini, A. J., Li, N., and Sevcikova, H. (2018). snow: Simple Network of Workstations. `R` package version 0.4-3. [https://CRAN.R-project.org/package=snow](https://CRAN.R-project.org/package=snow). Trapletti A, Hornik K, LeBaron B (2019). tseries: Time Series Analysis and Computational Finance. R package version 0.10-47, [https://CRAN.R-project.org/package=tseries](https://CRAN.R-project.org/package=tseries). Gonzalo Vera and Ritsert C. Jansen and Remo L. Suppi (2008). R/parallel - speeding up bioinformatics analysis with R. BMC Bioinformatics, 9:390. Wuertz D, core team members R (2019). fGarch: Rmetrics - Autoregressive Conditional Heteroskedastic Modelling. R package version 3042.83.1, [https://CRAN.R-project.org/package=fGarch](https://CRAN.R-project.org/package=fGarch). ## See Also `acf`, `ar`, `ar.ols`, `ar.burg`, `ar.yw`, `ar.mle`, `arima0`, `arima`, `lm`, `glm`, `Box.test`, `BoxPierce`, `LjungBox`, `MahdiMcLeod`, `LiMcLeod`, `portest`, `ToeplitzBlock`, `GetResiduals`, `Arima`, `auto.arima`, `VAR`, `fracdiff`, `garchFit`, `garch`, `varima.sim`. ## Examples ```r ## Not run: ################################################################################# #### #### #### Portmanteau Tests #### #### #### ################################################################################# ## Monte-Carlo (MC) and asymptotic tests for randomness series ## ################################################################################# data("DEXCAUS") returns <- log(DEXCAUS[-1]/DEXCAUS[-length(DEXCAUS)]) portest(returns) ## MC using one CPU takes about 24.16 seconds portest(returns, ncores=4) ## MC using 4 CPUs takes about 9.51 seconds portest(returns, MonteCarlo=FALSE) ## asymptotic MahdiMcLeod portest(returns,test="LjungBox", MonteCarlo=FALSE) ## asymptotic LjungBox ################################################################################# ## Monte-Carlo goodness-of-fit arima test using 4 CPUs ## ################################################################################# ## arima() function takes about 11.32 seconds ## Example 1 ans1 <- arima(WWWusage,fitdf=c(3,1,0)) portest(ans1, ncores = 4) # ## arima0() function takes about 11.1 seconds ## Example 2 ans2 <- arima0(WWWusage,fitdf=c(3,1,0)) portest(ans2, ncores = 4) # ## Arima() or auto.arima() functions from forecast package take about 12.1 seconds ## Example 3 ans3 <- Arima(WWWusage,fitdf=c(3,1,0)) portest(ans3, ncores = 4) # ## ar() function takes about 7.39 seconds ## Example 4 ans4 <- ar(Nile,fitdf.max=2) portest(ans4, ncores = 4) # ## ar() function with your own R code takes about 8.75 seconds ## Example 5 fit.model <- function(data){ fit <- ar(data,aic = FALSE, fitdf.max=2) fitdf <- 2 res <- ts(fit$resid[-(1:fitdf)]) phi <- fit$ar theta <- NULL sigma <- fit$var.pred demean <- fit$x.mean list(res=res,phi=phi,theta=theta,sigma=sigma,demean=demean) } sim.model <- function(parSpec){ res <- parSpec$res n <- length(res) innov <- function(n) ts(stats::rnorm(n, mean = demean, sd = sqrt(sigma))) phi <- parSpec$phi theta <- parSpec$theta sigma <- parSpec$sigma demean <- parSpec$demean arima.sim(n = n, list(ar = phi, ma = theta), rand.gen=innov) } ans5 <- fit.model(Nile) portest(ans5,ncores=4,model=list(sim.model=sim.model,fit.model=fit.model),pkg.name="stats") ################################################################################# ## Monte-Carlo test for seasonality ## ################################################################################# ## Accidental Deaths in the US 1973 - 1978 seasonal.arima<-Arima(USAccDeaths,fitdf=c(0,1,1),seasonal=list(fitdf= c(0,1,1))) portest(seasonal.arima,ncores=4,nrep=1000,lags=1:5) ## Quarterly U.K. economic time series from 1957 Q3 to 1967 Q4 cd <- EconomicUK[,1] cd.fit <- Arima(cd,fitdf=c(0,1,0),seasonal=list(fitdf=c(0,1,1),period=4)) portest(cd.fit, lags = c(5,10),ncores=4) ################################################################################# ## Monte-Carlo test for linear models and time series regression ## ################################################################################# ## Linear Model require("car") fit <- lm(fconvict ~ tfr + partic + degrees + mconvict, data=Hartnagel) portest(fit,lags=1:3,ncores=4) ## MC of MahdiMcLeod test ## MC of generalized Durbin-Watson test needs the argument function fn() as follows fn <- function(obj,lags){ test.stat <- numeric(length(lags)) for (i in 1:length(lags)) test.stat[i] <- -sum(diff(obj,lag=lags[i])^2)/sum(obj^2) test.stat } portest(fit,lags=1:3,test="other",fn=fn,ncores=4) detach(package:car) ## Time series regression fit.arima <- Arima(LakeHuron, fitdf = c(2,0,0), xreg = time(LakeHuron)-1920) portest(fit.arima,ncores=4) ################################################################################# ## Monte-Carlo goodness-of-fit VAR test - Multivariate series ## ################################################################################# data("IbmSp500") ibm <- log(IbmSp500[,2]+1)*100 sp500 <- log(IbmSp500[,3]+1)*100 IBMSP500 <- data.frame(cbind(ibm,sp500)) ## ar.ols() function takes about 9.11 seconds ans6 <- ar.ols(IBMSP500, aic=FALSE, intercept=TRUE, fitdf.max=5) portest(ans6,nrep=100,test="MahdiMcLeod",ncores=4,innov.dist="t",dft=5) ## VAR() function takes about 11.55 seconds require("vars") ans7 <- VAR(IBMSP500, p=5) portest(ans7,nrep=100,test="MahdiMcLeod",ncores=4,innov.dist="bootstrap") portest(ans7,test="Hosking",MonteCarlo=FALSE) ## asymptotic Hosking test detach(package:vars) ################################################################################# ## Monte-Carlo test for ARCH/GARCH effects using 4 CPUs ## ################################################################################# ## It takes about 12.65 seconds data("monthintel") returns <- as.ts(monthintel) lags <- c(5, 10, 20, 40) portest(returns, lags = lags, ncores = 4, sqrd.res = FALSE) portest(returns,lags=lags,ncores=4,sqrd.res=TRUE,innov.dist="t",dft=5) ## End(Not run) ```

portest function