Mixed Portmanteau Test for Diagnostic Checking of Time Series Models

Abstract

Model criticism is an important stage of model building and thus goodness of fit tests provides a set of tools for diagnostic checking of the fitted model. Several tests are suggested in literature for diagnostic checking. These tests use autocorrelation or partial autocorrelation in the residuals to criticize the adequacy of fitted model. The main idea underlying these portmanteau tests is to identify if there is any dependence structure which is yet unexplained by the fitted model. In this paper, we suggest mixed portmanteau tests based on autocorrelation and partial autocorrelation functions of the residuals. We derived the asymptotic distribution of the mixture test and studied its size and power using Monte Carlo simulations.

1. Introduction

In practice, there are many possible linear and nonlinear models for a problem under study, for example, autoregressive, moving average, mixed ARMA models, threshold autoregressive. Box and Jenkins [1] have described time series model building as a three-stage iterative procedure that consists of identification, estimation, and validation.

Time series models should be able to describe the dependence among the observations; see, for example, Li [2]. It is a well-discussed issue that in time series model criticism the residuals obtained from fitting a potential model to the observed time series play a vital role and can be used to detect departures from the underlying assumptions, [1, 2].

In particular, if the model is a good fit to the observed series then the residuals should behave somewhat like a white noise process. So, taking into account of the effect of estimation, the residuals obtained from a good fit should be approximately uncorrelated. While looking at the significance of residual autocorrelations, one approach is to test the significance of each individual residual autocorrelation which seems to be quite cumbersome. Another approach is to have some portmanteau test capable of testing the significance of the first, say, 𝑚, residual autocorrelations [1, 2], an approach we now describe.

Since Box and Pierce [3] paper, the portmanteau test has become the vital part of time series diagnostic checking. Several modifications and versions of Box and Pierce [3] have been suggested in the literature; see, for example, Ljung and Box [4], McLeod and Li [5], Monte [6], Katayama [7], and Katayama [8]. These tests are capable of testing the significance of the autocorrelations (partial autocorrelations) up to a finite number of lags.

The residuals are very commonly used as a diagnostic tool to test the goodness of fit of models. In a time series context, if the fitted model is good then it should be able to explain the dependence pattern among successive observations. In other words, all the dependence in terms of autocorrelations and partial autocorrelations of the data generating process (DGP) should be explained by the fitted model so there should be no significant autocorrelation and partial autocorrelation in successive terms of the residuals.

In practice, the most popular way for diagnostic checking of a time series model is the portmanteau test, which tests whether any of the first 𝑚 autocorrelations (𝑟₁̂,…,𝑟_𝑚̂) of a time series are significantly different from zero. This type of test was first suggested by Box and Pierce [3], in which they studied the distribution of residual autocorrelations in ARIMA processes.

Based on the autocorrelations of the residuals obtained by fitting an ARMA(𝑝,𝑞) model, Box and Pierce [3] suggested the following portmanteau test: 𝑄_𝑚=𝑛^𝑚_𝑘=1𝑟²_𝑘̂, (1) where 𝑟_𝑘̂ is the residual autocorrelation at lag 𝑘. They suggested that 𝑄_𝑚∼𝜒²_{𝑚−𝑝−𝑞}, for moderate values of 𝑚, and the fitted model is adequate, under the following conditions: (1)𝜓_𝑗≤𝑂(𝑛^−1/2) for 𝑗≥𝑚−𝑝,(2)𝑚/𝑛=𝑂(𝑛^−1/2),where 𝜓_𝑗 are the weights in the MA(∞) representation. This approximation requires substitution of residuals, 𝜀_𝑡̂, for the error term, 𝜀_𝑡, in the model but this sort of substitution can be a serious underestimation of significance level in diagnostic checking [9]. Many other researchers have also questioned the distribution of 𝑄_𝑚 (see, e.g., [10] and references therein). The choice of 𝑚 is an important issue.

Ljung and Box [4] suggested the use of the modified statistic 𝑄^∗_𝑚=𝑛(𝑛+2)^𝑚_𝑘=1𝑟²_𝑘̂.𝑛−𝑘 (2) They have shown that the modified portmanteau statistic 𝑄^∗_𝑚 has a finite sample distribution which is much closer to 𝜒²_{𝑚−𝑝−𝑞}. Their results also show that 𝑄^∗_𝑚 is insensitive to the normality assumption of 𝜀_𝑡. As pointed out by many researchers, for example, Davies et al. [11], Ansley and Newbold [12], the true significance levels of 𝑄_𝑚 tend to be much lower than predicted by the asymptotic theory and though the mean of 𝑄^∗_𝑚 is much closer to the asymptotic distribution, this corrected version of the portmanteau test has an inflated variance. But Ljung and Box [4] pointed out that approximate expression of variance given by Davies et al. [11] overestimates the variance of 𝑄^∗_𝑚.

McLeod [10, Theorem 1] has shown that 𝐫̂=(𝑟₁̂,…,𝑟_𝑚̂) is approximately normal with mean 𝟎 and Var(𝐫̂)=(𝐈−𝐂)/𝑛, where 𝐂=𝐗𝕀⁻¹𝐗^𝑇, 𝐈 is the identity matrix, and 𝕀 is the Fisher information matrix. The superscript 𝑇 stands for transposition of matrix. We noticed that approximation of 𝐂 by 𝐃=𝐗(𝐗^𝑇𝐗)⁻¹𝐗^𝑇, especially when 𝑚 is small, is a source of bias in approximating the asymptotic distribution of portmanteau tests.

In practice, the optimal choice of 𝑚 is difficult as the use of the 𝜒²_{𝑚−𝑝−𝑞} approximation and diagnostic checking require large values of 𝑚 which results in less power and unstable size of test, as noticed by Ljung [13] and Katayama [7]. Katayama [8] suggested a multiple portmanteau test to overcome this problem. His suggested test is based on several portmanteau tests for a range of small to medium values of 𝑚. He showed using some numerical examples that his suggestion leads to a superior test. He also discussed the linkage between his suggested multiple test and the test due to Peña and Rodríguez [14]. He suggested a method based on some iterative procedure to approximate joint distribution of the multiple test as the computation of the distribution is very hard due to correlated elements.

The portmanteau tests 𝑄_𝑚 and 𝑄^∗_𝑚 are based on the autocorrelations. Monti [6] suggested a portmanteau test 𝑄^∗_𝑚(𝜔)=𝑛(𝑛+2)^𝑚_𝑘=1𝜔²_𝑘,𝑛−𝑘 (3) where 𝜔_𝑘 is the residual partial autocorrelation at lag 𝑘. She showed that 𝑄^∗_𝑚(𝜔), analogously to 𝑄^∗_𝑚, has an asymptotic null distribution 𝜒²_{𝑚−𝑝−𝑞} and that 𝑄^∗_𝑚(𝜔) is more powerful than 𝑄^∗_𝑚 especially when the order of the moving average component is understated. Chand and Kamal [15] had studied the size of Monti’s test.

Portmanteau tests, for example, Box-Pierce test, which are based on residuals autocorrelations, have good power against the understatement of autoregressive component while the tests using residuals partial autocorrelations have shown good power against the understatement of moving average term. A new test using both autocorrelation and partial autocorrelations is expected to perform better in both of these two scenarios and will be capable of picking up understatement in moving average and autoregressive components.

2. Materials and Methods

In this section, we give the algorithms for the Monte Carlo method used to compute the empirical size and power of the diagnostic tests defined in Section 2. For each Monte Carlo run, a sample time series {𝑦_𝑡}^𝑛_𝑡=1 is simulated under the model ℳ. For empirical size, ℳ is the null model while for the computation of power it is the alternative model. In both of the situations, we estimate the null model for the simulated sample time series and 𝑇 is calculated from the residuals, 𝜀_𝑡̂.

Algorithm 1 gives the bootstrap procedure used in our numerical study. From this algorithm, we obtain the bootstrap approximation of the distribution of the test. We will use this algorithm to compute empirical size in the following algorithm of our simulation study consisting of 𝑁 Monte Carlo runs.

Step  1. Generate bootstrap sample 𝑦∗𝑡 using resamples of residuals, 𝜀𝑡̂, obtained by fitting the model under null hypothesis to the simulated time series. We denoted these resampled residuals as 𝜀∗𝑡. Step  2. Fit the null model to the bootstrap sample 𝑦∗𝑡 and obtain residuals as 𝜀∗𝑡̂=𝑦∗𝑡−𝑦∗𝑡, where 𝑦∗𝑡 is the fitted series. Step  3. Using the residuals, 𝜀∗𝑡̂, obtained in Step  2 above, calculate test-statistic 𝑇. We denote it as 𝑇∗. Step  4. Repeat Step  1–3 for each of the 𝐵 bootstrap samples.

Algorithm 1 

Bootstrap sampling procedure.

3. Results

Suppose we have an observed time series {𝑦_𝑡∶𝑡=1,2,…,𝑛} generated by an ARMA(𝑝,𝑞) model given by 𝜙(𝐵)𝑦_𝑡=𝜃(𝐵)𝜀_𝑡, (4) where 𝐵 is the backward shift operator, 𝜙(𝐵)=1−𝜙₁𝐵−𝜙₂𝐵²−⋯−𝜙_𝑝𝐵^𝑝 is the autoregressive polynomial of order 𝑝, and 𝜃(𝐵)=1+𝜃₁𝐵+𝜃₂𝐵²+⋯+𝜃_𝑞𝐵^𝑞 is the moving average polynomial of order 𝑞. Let an appropriately identified model be fitted to the observed time series and let residuals 𝜀₁̂,𝜀₂̂,…,𝜀_𝑛̂ be obtained. These residuals play an important role in the diagnostic checking of the fitted model. The residuals autocorrelation at lag 𝑘 can be defined as 𝑟_𝑘̂=∑^𝑛_𝑡=𝑘+1𝜀_𝑡̂𝜀̂_𝑡−𝑘∑^𝑛_𝑡=1𝜀²_𝑡̂𝑘=1,2,…, (5) where {𝜀̂_𝑡−𝑘} is the series at lag 𝑘 of {𝜀_𝑡̂}. Let 𝜷=(𝜙₁,…,𝜙_𝑝,𝜃₁,…,𝜃_𝑞) be the vector of model parameters and let 𝜷=(𝜙₁,…,𝜙_𝑝,𝜃₁,…,𝜃_𝑞) be the corresponding estimator of 𝜷.

Let us define autocorrelation of error term, 𝜀_𝑡, that is, white noise at lag 𝑘, as 𝑟_𝑘=∑^𝑛_𝑡=𝑘+1𝜀_𝑡𝜀_𝑡−𝑘∑^𝑛_𝑡=1𝜀²_𝑡𝑘=1,2,…. (6)

The root mean square error of 𝜷 can be defined as 𝐸𝜷−𝜷². (7)

As the above term is of order 1/√𝑛, thus we can use Taylor series of first order to approximate 𝑟_𝑘̂ around 𝑟_𝑘. We can write 𝑟_𝑘̂=𝑟_𝑘+^𝑝+𝑞_𝑗=1𝛽_𝑗−𝛽_𝑗²𝜆_𝑗𝑘+𝑂_𝑝1𝑛, (8) where 𝜆_𝑗𝑘=−𝜕𝑟_𝑘𝜕𝛽̇_{𝑗̇𝜷𝜷=},𝜆_𝑗𝑘𝜕𝑟̂=−^𝑘𝜕𝛽̇_{𝑗̇𝜷𝜷=}. (9)

Take 𝜀_𝑡=𝑦_𝑡−𝜙₁𝑦_𝑡−1−⋯−𝜙_𝑝𝑦_𝑡−𝑝−𝜃₁𝜀_𝑡−1−⋯−𝜃_𝑞𝜀_𝑡−𝑞. (10) For an AR(𝑝) process 𝜀_𝑡=𝑦_𝑡−𝜙₁𝑦_𝑡−1−⋯−𝜙_𝑝𝑦_𝑡−𝑝, (11) at 𝜷=𝜷̇, that is, for an AR(𝑝), 𝝓=𝝓̇, we can write 𝜀_𝑡̇=𝑦_𝑡−𝜙̇₁𝑦_𝑡−1−⋯−𝜙̇_𝑝𝑦_𝑡−𝑝,𝜕𝜀_𝑡̇𝜕𝜙̇_𝑗=−𝑦_𝑡−𝑗. (12)

From (10) and summing over observed sample, we get 𝜀²_𝑡=𝑦_𝑡−𝜙₁𝑦_𝑡−1⋯−𝜙_𝑝𝑦_𝑡−𝑝². (13) Partially differentiating with respect to 𝜙_𝑗𝜕∑𝜀²_𝑡𝜕𝜙_𝑗=2𝑦_𝑡−𝜙₁𝑦_𝑡−1⋯−𝜙_𝑝𝑦_𝑡−𝑝−𝑦_𝑡−𝑗. (14) At 𝝓=𝝓̇𝜕∑𝜀²_𝑡𝜕𝜙_𝑗=0. (15) Now 𝜆_𝑗𝑘𝜕𝑟̂=−^𝑘𝜕𝜙̇_𝑗𝜕𝑐=−^𝑘/𝑐₀𝜕𝜙_𝑗𝜕=−𝜕𝜙_𝑗∑𝜀_𝑡𝜀_𝑡−𝑘∑𝜀²_𝑡1=−∑𝜀²_𝑡𝜕∑𝜀_𝑡𝜀_𝑡−𝑘𝜕𝜙_𝑗. (16) As ∑𝜀_𝑡𝜀_𝑡−𝑘=∑𝜙(𝐵)𝑦_𝑡𝜙(𝐵)𝑦_𝑡−𝑘=∑_𝑡∑_𝑖∑_{𝑗𝑖=𝑘+𝑗}𝜙_𝑖𝜙_𝑗𝑦_𝑡−𝑖𝑦_{𝑡−𝑘−𝑗}+∑_𝑖∑_{𝑗𝑖≠𝑘+𝑗}𝜙_𝑖𝜙_𝑗𝑦_𝑡−𝑖𝑦_{𝑡−𝑘−𝑗}, (17) thus 𝜆_𝑗𝑘∑=−^𝑡𝑦²_𝑡∑_𝑡𝜀̂²_𝑡𝑖𝑟_{𝑘−𝑖+𝑗}𝑦+𝑟_{𝑘+𝑖−𝑗}𝑦∑=−^𝑖𝑝ℎ𝑖_𝑖𝑟_{𝑘−𝑖+𝑗}𝑦+𝑟_{𝑘+𝑖−𝑗}𝑦∑_𝑖∑_𝑗𝜙_𝑖𝜙_𝑗𝑟_𝑖−𝑗𝑦, (18) where 𝑟_𝑘𝑦=∑𝑦_𝑡𝑦_𝑡−𝑘∑𝑦²_𝑡. (19)

Using Bartlett’s formula, we can write 𝑟_𝑘𝑦=𝜌_𝑘+𝑂_𝑝𝑛^−1/2. (20)

Using 𝜙_𝑗=𝜙_𝑗+𝑂_𝑝(𝑛^−1/2) and the above results 𝜆_𝑗𝑘=𝜆_𝑗𝑘+𝑂_𝑝𝑛^−1/2. (21) Consider the AR(𝑝) process 𝜙(𝐵)𝑦_𝑡=𝜀_𝑡,1−𝜙₁𝐵−𝜙₂𝐵²−⋯−𝜙_𝑝𝐵^𝑝𝑦_𝑡=𝜀_𝑡,𝑦_𝑡−𝜙₁𝑦_𝑡−1−𝜙₂𝑦_𝑡−2−⋯−𝜙_𝑝𝑦_𝑡−𝑝=𝜀_𝑡. (22)

Multiplying both sides by 𝑦_𝑡−𝑘 and on taking expectation, we get 𝐸(𝑦_𝑡𝑦_𝑡−𝑘)−𝜙₁𝐸(𝑦_𝑡−1𝑦_𝑡−𝑘)−𝜙₂𝐸(𝑦_𝑡−2𝑦_𝑡−𝑘)−⋯−𝜙_𝑝𝐸(𝑦_𝑡−𝑝𝑦_𝑡−𝑘)=𝐸(𝜀_𝑡𝑦_𝑡−𝑘). Without loss of generality, we can assume 𝐸(𝑦_𝑡)=0; moreover it is considered a stationary process; therefore 𝐸(𝑦²_𝑡)=var(𝑦_𝑡)=𝑐₀ and 𝐸(𝑦_𝑡𝑦_𝑡−𝑘)=cov(𝑦_𝑡,𝑦𝑡−𝑘)=𝑐_𝑘. Moreover, 𝐸(𝜀_𝑡𝑦_𝑡−𝑘)=0 for 𝑘≥1. Thus 𝑐_𝑘−𝜙₁𝑐_𝑘−1−𝜙₂𝑐_𝑘−2−⋯−𝜙_𝑝𝑐_𝑘−𝑝=0. (23)

Divide both sides by 𝑐₀ and using 𝑟_𝑘=𝑐_𝑘/𝑐₀, we get 𝑟_𝑘−𝜙₁𝑟_𝑘−1−𝜙₂𝑟_𝑘−2−⋯−𝜙_𝑝𝑟_𝑘−𝑝𝜙(𝐵)𝑟=0,_𝑘=0. (24) Using this result and the structure of AR(𝑝) process, it implies 𝜆_𝑗𝑘=∑_𝑖𝜙_𝑖𝜙_{𝑘−𝑗+𝑖}∑_𝑖𝜙_𝑖𝜌_𝑖. (25)

The AR(𝑝) process can be written in an equivalent moving average representation, that is, MA(∞), given as 𝜙(𝐵)𝑦_𝑡=𝜀_𝑡,𝑦_𝑡=𝜓(𝐵)𝜀_𝑡, (26) where 𝜓(𝐵)=1𝜙(𝐵)=^∞_𝑖=0𝜓_𝑖𝐵^𝑖. (27) Using Box and Pierce [3, page. 1514] and the above results, we can write 𝜆_𝑗𝑘=𝜓_𝑘−𝑗, (28) where 𝑘=1,2,…,𝑚, and thus using (8), we get 𝑟_𝑘̂=𝑟_𝑘+_𝑖𝜙_𝑖−𝜙_𝑖𝜓_𝑘−𝑖+𝑂_𝑝1𝑛. (29)

Using McLeod [10, Theorem 1], the asymptotic distribution of 𝐫̂=(𝑟₁̂,…,𝑟_𝑚̂), where 𝑚≥𝑝+𝑞 is normal with mean 𝟎 and covariance matrix (𝟏−𝐗𝕀⁻¹𝐗^𝑇)/𝑛; that is, 𝐫̂∼𝑁𝟎,𝐈−𝐗𝕀⁻¹𝐗^𝑇𝑛, (30) where for an ARMA(𝑝,𝑞) process 𝜓𝐗=10…0₁𝜓1…0₂𝜓₁𝜓…0⋮⋮⋮_𝑚−1𝜓_𝑚−2…𝜓_𝑚−𝑝𝜋10…0₁𝜋1…0₂𝜋₁𝜋…0⋮⋮⋮_𝑚−1𝜋_𝑚−2…𝜋_𝑚−𝑝. (31)

The elements 𝜓_𝑗 and 𝜋_𝑗 are the coefficients of 𝐵^𝑗 in the infinite expansion of inverse of AR and MA polynomials. For an AR(𝑝) process, the above matrix reduces to 𝜓𝐗=10…0₁𝜓1…0₂𝜓₁𝜓…0⋮⋮⋮_𝑚−1𝜓_𝑚−2…𝜓_𝑚−𝑝. (32)

Let 𝜔_𝑘 be the partial autocorrelation at lag 𝑘; then using Monti [6, page 777, equation (4)], we have 𝝎=𝐫̂+𝑂_𝑝1𝑛. (33)

According to Monti [6], the 𝐫̂ can be obtained using a singular linear transformation of the noise autocorrelations, that is, 𝐫, given as 𝐫̂=𝐈−𝐗𝐗^𝑇𝐗⁻¹𝐗𝐫+𝑂_𝑝1𝑛. (34) Using (30), we can write 𝝎∼𝑁𝟎,𝐈−𝐗𝕀⁻¹𝐗^𝑇𝑛. (35)

Ljung and Box [4], using (30), suggested a portmanteau test for diagnostic testing of the fitted ARMA(𝑝,𝑞) model given as 𝑄^∗_𝑚(𝑟)̂=𝑛(𝑛+2)^𝑚_𝑘=1𝑟²_𝑘̂𝑛−𝑘∼𝜒²_{𝑚−𝑝−𝑞}. (36)

Following the same lines and using the result in (35), Monti [6] suggested the following portmanteau test: 𝑄^∗_𝑚(𝜔)=𝑛(𝑛+2)^𝑚_𝑘=1𝜔²_𝑘𝑛−𝑘∼𝜒²_{𝑚−𝑝−𝑞}. (37)

4. New Test

We suggest a new test 𝑄^†_𝑚(𝑟,̂𝜔)=𝑄^∗_𝑚(𝑟)̂+𝑄^∗_𝑚(𝜔), that is, the sum of Ljung and Box [4] and Monti [6], given as 𝑄^∗_𝑚(𝑟,̂𝜔)=𝑛(𝑛+2)^𝑚_𝑘=1𝑟²_𝑘̂+𝜔²_𝑘.𝑛−𝑘 (38)

It is obvious that this new statistic 𝑄^∗_𝑚 is the sum of two chi-square variables as discussed above in (36) and (37). Moreover, it can be noticed that both of these random variables will be positively correlated due to their structure of using residual autocorrelation and partial autocorrelation. We have discussed their correlational structure later in this report.

Using the result of Joarder and Omar [16, Theorem 3.1], we can write the probability density function of 𝑍=𝑄^∗_𝑚=𝑄_𝑚(𝑟)̂+𝑄_𝑚(𝜔) as 𝑓_𝑍(𝑧)=1−𝜌²_𝑚^−𝑚/22^𝑚Γ(𝑚)𝑧^𝑚−1exp−𝑧2−2𝜌²_𝑚×₀𝐹₁𝑚+12;𝜌²_𝑚𝑧²4−4𝜌²_𝑚²;0≤𝑧<∞, (39) where 𝜌_𝑚 is the correlation between 𝑄^∗_𝑚(𝑟)̂ and 𝑄^∗_𝑚(𝜔): _𝑝𝐹_𝑞𝑎₁,𝑎₂,…,𝑎_𝑝;𝑏₁,𝑏₂,…,𝑏_𝑞;𝑧=^∞_𝑖=0∏^𝑝_𝑗=1𝑎_𝑗{𝑖}𝑧^𝑖∏^𝑞_𝑗=1𝑏_𝑗{𝑖},𝑎𝑖!_{𝑖}=𝑎(𝑎+1)⋯(𝑎+𝑖−1). (40)

Now the important issue is to estimate 𝜌_𝑚 which is discussed in Section 4.1.

4.1. Estimation of 𝜌_𝑚

We have used Monte Carlo simulation to estimate 𝜌_𝑚. We have considered various choices of sample size and 𝑚 and obtained the following results. Empirical results suggest that, for any 𝑛, we have 0.8≤𝜌_𝑚≤1, (41) where 𝑚≤𝑛/4. Results are given in Table 1.

Table 1 

Monte Carlo estimates of 𝜌_𝑚.

4.2. Empirical Size

The size of a test is helpful in assessing how reasonable is our assumption of the null distribution. We can compute the size when the sample is simulated under the null model. This is the probability of rejecting the null model when the null model is true model. The model under null hypothesis is considered an AR(1) process 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡. To see the effect of stationarity and near to unit-root problem, we have considered various choices of 𝜙. The value of 𝜙 near to 1, for example, 𝜙=0.9, is an example of near nonstationary process.

It can be seen from the results shown in Tables 2 and 3 that our new suggested mixture test, that is, 𝑄^∗_𝑚(𝑟,̂𝜔), have close approximation of corresponding nominal sizes. These results show the good approximation of asymptotic distribution of our new suggested test.

Table 2 

Empirical size at 1% nominal size for an AR(1) process: 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡.

Table 3 

Empirical size at 5% nominal size for an AR(1) process: 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡.

4.3. Empirical Power

The power of a test is the probability of rejecting a false null hypothesis. For empirical power, as mentioned earlier, the sample is generated under the alternative model. Algorithms 2 and 3 state the Monte Carlo procedure we use to determine the empirical size and power of test.

Step  1. Obtain 𝑇∗ using Algorithm 1, for each of the 𝐵 bootstrap samples, reject null model if 𝑇∗≥𝑇, otherwise accept it. Step  2. Determine the proportion of 𝐵 bootstrap samples, say 𝑝𝑐, for which the null hypothesis is rejected. Step  3. Repeat Step  1-2 for each of the 𝑁 Monte Carlo runs. Step  4. Empirical size, 𝛼, is determined as the proportion of Monte Carlo runs for which the 𝑝𝑐≤𝛼, where 𝛼 is the level of significance,           𝑝𝛼=#(𝑐≤𝛼)𝑁.

Algorithm 2 

Computation of empirical size.

Step  1. Calculate 100(1−𝛼)th percentile, say 𝑇∗1−𝛼, of the bootstrap distribution of 𝑇∗ obtained using Algorithm 1. Step  2. Reject null model if 𝑇≥𝑇∗1−𝛼 otherwise accept it. Step  3. Repeat Step  1-2 for each of the 𝑁 Monte Carlo runs. Step  4. Empirical power, 1−𝛽, is determined as below,           1−𝛽=#(𝑇≥𝑇1−𝛼)𝑁.

Algorithm 3 

Computation of empirical power.

For power computation, we have simulated under an ARMA(1,1) process 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡+0.8𝜀_𝑡−1 and 𝑦_𝑡=0.8𝑦_𝑡−1+𝜀_𝑡+𝜃𝜀_𝑡−1 as alternative models, while AR(1) process 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡 is the model specified in null hypothesis. Results are provided in Tables 4 and 5. These results show that our suggested mixed portmanteau tests have empirical power comparable with other tests. It can be noticed that our new test has shown power better than other considered tests when moving average part is understated and the moving average parameter is relatively small.

Table 4 

Empirical power by fitting an AR(1) process against an ARMA(1,1) process: 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡+0.8𝜀_𝑡−1 using 1000 Monte Carlo runs.

Table 5 

Empirical power by fitting an AR(1) process against an ARMA(1,1) process: 𝑦_𝑡=0.8𝑦_𝑡−1+𝜀_𝑡+𝜃𝜀_𝑡−1 using 1000 Monte Carlo runs.

5. Conclusion

Portmanteau tests based on residuals autocorrelations and partial autocorrelations have their individual good properties. Our suggested mixture portmanteau tests have a good approximation of their asymptotic distribution. As it is important to study both the residuals autocorrelation and the partial autocorrelations for model diagnostic checking, so this new test is capable of doing it simultaneously instead of running separate diagnostic tests.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and the referees for their helpful comments and suggestions that greatly improved the previous version of this paper. The work of first author was supported by the Research Grant 5227-Est-I-2013 from the University of the Punjab, Lahore, Pakistan.