Portmanteau test statistics for seasonal serial correlation in time series models
- Esam Mahdi^{1}Email authorView ORCID ID profile
Received: 5 June 2016
Accepted: 26 August 2016
Published: 5 September 2016
Abstract
The seasonal autoregressive moving average SARMA models have been widely adopted for modeling many time series encountered in economic, hydrology, meteorological, and environmental studies which exhibited strong seasonal behavior with a period s. If the model is adequate, the autocorrelations in the errors at the seasonal and the nonseasonal lags will be zero. Despite the popularity uses of the portmanteau tests for the SARMA models, the diagnostic checking at the seasonal lags \(1s,2s,3s,\ldots ,ms\), where m is the largest lag considered for autocorrelation and s is the seasonal period, has not yet received as much attention as it deserves. In this paper, we devise seasonal portmanteau test statistics to test whether the seasonal autocorrelations at multiple lags s of time series are different from zero. Simulation studies are performed to assess the performance of the asymptotic distribution results of the proposed statistics in finite samples. Results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of this test.
Keywords
Mathematics Subject Classification
Background
In the next section, a brief review of commonly univariate portmanteau tests employed for diagnostic checking in ARMA models is given. In "Portmanteau test statistics for SARMA models" section, we modify the usual portmanteau test statistics suggested by Box and Pierce (1970), Ljung and Box (1978), Peña and Rodríguez (2002, 2006), Fisher and Gallagher (2012), Gallagher and Fisher (2015) to the SARMA class. The approximation distributions of the proposed tests are derived in "Asymptotic distributions" section. In "Simulation studies" section provides simulation experiments demonstrating the behaviour of the asymptotic distributions of the proposed test statistics. We close this article with "An empirical application" section by introducing an illustrative application of seasonal data demonstrating the usefulness of the devised tests. We conclude in "Conclusion" section with a discussion.
Portmanteau test statistics for ARMA models
Portmanteau test statistics for SARMA models
Replacing \(\hat{r}_{\ell}, \ell =1, 2, \ldots , m\) by \(\hat{r}_{\ell s}\), where \(\hat{r}_{1 s},\hat{r}_{2 s},\ldots ,\hat{r}_{ms}\) are the residual autocorrelations at the multiple period lags \(1s, 2s, \ldots ,ms\), will easily extend the classical portmanteau test statistics to test for seasonality at lags multiple of period s. This modification is justifiable under the conditions indicated by McLeod (1978) that we mentioned in the introduction of this article. We devise a list of new portmanteau tests for diagnostic checking of seasonal time series.
Asymptotic distributions
The limiting distribution of the resulting seasonal tests are obtained by a straightforward extension of those obtained in Box and Pierce (1970), Ljung and Box (1978), Fisher and Gallagher (2012), Gallagher and Fisher (2015) and Mahdi and McLeod (2012) and are summarized in the following theorems.
Theorem 1
Assume that the SARMA \((p,q)\times (p_{s},q_{s})_{s}\) model specified as in (1) has i.i.d. innovations \(\{a_{t}\}\) with mean zero and finite constant variance. For constants m and s, as \(n\rightarrow \infty\), where \(ms\le (n-1)\), \(p, q \ll s\), and the roots of the equation \(\phi ({B})\theta ({B})=0\) are not close to the unit circle. When the model has adequately been identified, the test statistics for lack of SARMA fit models, \(Q_{m}(s)\) and \(\hat{Q}_(s)\), would for large n approximately distributed as \(\chi ^{2}_{m-\nu}\), where \(\nu =p_{s}+q_{s}\).
Proof
Box and Pierce (1970) showed that the vector of the residual autocorrelations at nonseasonal lags \(\sqrt{n}\varvec{\hat{r}_{m}}\) from a correctly identified and fitted ARMA (p, q) model can be asymptotically distributed as a multivariate normal distribution with mean vector zero and covariance matrix \(\varvec{({\mathbb {I}}_{m}-Q)}\), where \({\mathbb {I}}_{m}\) is an identity matrix and \(\varvec{Q}\) is a matrix with rank \(p+q\). Consider the SARMA model where \(p\ll\) and \(q \ll s\) and the roots of the equation \(\phi ({B})\theta ({B})=0\) are not close to the unit circle. McLeod (1978) indicated that the vector of the residual autocorrelations at seasonal lags \(1s,2s,\ldots ,ms\), has approximately the same distribution of the vector of the residual autocorrelations at nonseasonal lags \(1,2,\ldots ,m\). Thus, the vector \(\sqrt{n}\varvec{\hat{r}_{ms}}\) from a correctly identified and fitted SARMA \((p,q)\times (p_{s},q_{s})_{s}\) model would for large n be distributed as a multivariate normal with mean vector zero and covariance matrix \(\varvec{({\mathbb {I}}_{m}-Q_{s})}\), where \(\varvec{Q_{s}}\) is a matrix with rank \(p_{s}+q_{s}\). It follows that both \(Q_{m}(s)\) and \(\hat{Q}_(s)\) have the same asymptotic distribution as \(\chi ^{2}_{m-\nu}\), where \(\nu =p_{s}+q_{s}\). \(\square\)
Theorem 2
Under the assumptions of Theorem 1, \(\tilde{Q}_{m}(s)\) converges in distribution to \(\sum _{i=1}^{m}\lambda _{i}\chi _{i}^{2}\), where \(\{\chi _{i}^{2}\}\) denotes a sequence of independent chi-squared random variables, each with one degree of freedom, and \(\lambda _{1},\ldots ,\lambda _{m}\) are the eigenvalues of \(\varvec{({\mathbb {I}}_{m}-Q_{s})M}\) with \({\mathbb {I}}_{m}\) an identity matrix, \(\varvec{Q_{s}}\) is a projection matrix defined as \(\varvec{Q_{s}}=\varvec{X}\Sigma ^{-1}\varvec{X^{\prime}}\), where \(\Sigma ^{-1}\) is the information matrix for the parameters \(\Phi _{1},\ldots ,\Phi _{ps}\) and \(\Theta _{1},\ldots ,\Theta _{qs}\), X is an \(m\times (p_{s}+q_{s})\) matrix defined similar to McLeod (1978, Eq. (16)) with elements \(\Phi ^{\prime}\), and \(\Theta ^{\prime}\) defined by \(1/\Phi (B)=\sum _{i=1}^{\infty}\Phi _{i}^{\prime}B^{i},\) and \(1/\Theta (B)=\sum _{i=1}^{\infty}\Theta _{i}^{\prime}B^{i}\), and \(\varvec{M}\) is an \(m\times m\) diagonal matrix with diagonal weights \(\{1, (m-1)/m, \ldots ,2/m,1/m\}\).
Proof
Theorem 3
Under the assumptions of Theorem 1, \(\mathfrak {D}_{m}(s)\) converges in distribution to \(\sum _{i=1}^{m}\lambda _{i}\chi _{i}^{2}\), where \(\{\chi _{i}^{2}\}\) denotes a sequence of independent chi-squared random variables, each with one degree of freedom, and \(\lambda _{1},\ldots ,\lambda _{m}\) are the eigenvalues of \(\varvec{({\mathbb {I}}_{m}-Q_{s})M}\), where \(Q_{s}\) is given in Theorem 2 and \(\varvec{M}\) is a diagonal matrix of size m with diagonal elements \(\{m, m-1, \ldots ,1\}\).
Proof
It is worth noting that the \(\mathfrak {D}_{m}(s)\) statistic may be seen as a weighted Ljung and Box (1978) considering of the residual autocorrelations at the seasonal lags \(1s, 2s, \ldots ,ms\). It essentially has the same characteristics as \(\tilde{Q}_{m}(s)\) with standardizing weights \(3m(2m+1)^{-1},3(m-1)(2m+1)^{-1},\ldots ,3(2m+1)^{-1}\) using the seasonal residuals at lags \(1s, 2s, \ldots ,ms\).
Simulation studies
The objective of our simulations is to explore the performance of the proposed portmanteau seasonal tests, \(Q_{m}(s),\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\), and \(\mathfrak {D}_{m}(s)\), in finite samples and when the sample size grow. We study the empirical type I and type II error rates demonstrating the accuracy of the approximation distributions of the proposed seasonal tests in producing the correct sizes and conducting a power comparison studies. For each simulation experiment, we determine the critical values from the corresponding asymptotic distributions of the proposed seasonal test statistics. One can use the Monte-Carlo test procedures, as described by Lin and McLeod (2006) and Mahdi and McLeod (2012), to compute these critical values instead of using the approximation distributions. The simulations were run on a modern quad-core personal computer using the R package portes (Mahdi and McLeod 2015) and WeightedPortTest (Fisher and Gallagher 2012) that are available from the CRAN website (R Development Core Team 2015).
Comparison of type I error rates
The empirical 1, 5 and 10 % significance levels for different fitted SAR \((1)_{s}\) models, with different SAR coefficients \(\Phi _{1}= 0.1,0.3, 0.5, 0.7,\) and 0.9, for the seasonal portmanteau test statistics \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\), and \(\mathfrak {D}_{m}(s)\), where \(s=4,12\), \(n=200\) and lags \(m=5,15\)
\(\Phi _{1}\) | \(\hat{Q}_{5}(s)\) | \(\tilde{Q}_{5}(s)\) | \(\mathfrak {D}_{5}(s)\) | \(\hat{Q}_{15}(s)\) | \(\tilde{Q}_{15}(s)\) | \(\mathfrak {D}_{15}(s)\) | |
---|---|---|---|---|---|---|---|
\((s=4)\,\alpha =0.01\) | 0.1 | 0.010 | 0.013 | 0.007 | 0.013 | 0.012 | 0.008 |
0.3 | 0.007 | 0.017 | 0.007 | 0.009 | 0.010 | 0.007 | |
0.5 | 0.009 | 0.010 | 0.008 | 0.014 | 0.007 | 0.009 | |
0.7 | 0.010 | 0.017 | 0.009 | 0.014 | 0.015 | 0.006 | |
0.9 | 0.016 | 0.009 | 0.009 | 0.019 | 0.005 | 0.007 | |
\(\alpha =0.05\) | 0.1 | 0.050 | 0.041 | 0.034 | 0.051 | 0.038 | 0.033 |
0.3 | 0.043 | 0.051 | 0.039 | 0.035 | 0.040 | 0.037 | |
0.5 | 0.049 | 0.049 | 0.042 | 0.055 | 0.039 | 0.041 | |
0.7 | 0.049 | 0.070 | 0.041 | 0.068 | 0.050 | 0.041 | |
0.9 | 0.060 | 0.055 | 0.044 | 0.064 | 0.043 | 0.042 | |
\(\alpha =0.10\) | 0.1 | 0.091 | 0.081 | 0.077 | 0.109 | 0.070 | 0.070 |
0.3 | 0.092 | 0.105 | 0.089 | 0.084 | 0.084 | 0.088 | |
0.5 | 0.095 | 0.087 | 0.082 | 0.101 | 0.081 | 0.081 | |
0.7 | 0.093 | 0.134 | 0.090 | 0.106 | 0.102 | 0.086 | |
0.9 | 0.122 | 0.107 | 0.085 | 0.113 | 0.090 | 0.088 | |
\((s=12)\,\alpha =0.01\) | 0.1 | 0.018 | 0.018 | 0.006 | 0.014 | 0.011 | 0.005 |
0.3 | 0.015 | 0.010 | 0.007 | 0.018 | 0.008 | 0.007 | |
0.5 | 0.015 | 0.018 | 0.010 | 0.016 | 0.012 | 0.008 | |
0.7 | 0.019 | 0.014 | 0.012 | 0.015 | 0.010 | 0.011 | |
0.9 | 0.023 | 0.015 | 0.008 | 0.022 | 0.013 | 0.009 | |
\(\alpha =0.05\) | 0.1 | 0.070 | 0.061 | 0.031 | 0.046 | 0.041 | 0.030 |
0.3 | 0.068 | 0.063 | 0.044 | 0.067 | 0.041 | 0.040 | |
0.5 | 0.072 | 0.070 | 0.040 | 0.050 | 0.047 | 0.041 | |
0.7 | 0.075 | 0.076 | 0.039 | 0.069 | 0.054 | 0.037 | |
0.9 | 0.073 | 0.069 | 0.043 | 0.072 | 0.060 | 0.038 | |
\(\alpha =0.10\) | 0.1 | 0.119 | 0.126 | 0.083 | 0.088 | 0.091 | 0.080 |
0.3 | 0.129 | 0.142 | 0.090 | 0.118 | 0.089 | 0.121 | |
0.5 | 0.133 | 0.140 | 0.102 | 0.091 | 0.104 | 0.099 | |
0.7 | 0.150 | 0.144 | 0.074 | 0.114 | 0.104 | 0.088 | |
0.9 | 0.141 | 0.140 | 0.088 | 0.130 | 0.120 | 0.111 |
Power comparisons
Here, we conduct a power comparison simulation study between the proposed seasonal \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s),\mathfrak {D}_{m}(s)\) statistics where the critical values are calculated from the corresponding asymptotic distributions. Table 2 below provides the empirical power of these statistics when a series of length \(n=200\) is generated from a 20 Gaussian SARMA \((2,2)\times (2,2)_{s}\) processes are inadequately fitted by SAR \((1)_{s}\) or SMA \((1)_{s}\), \(s=4\) and 12, and tested at lag \(m=10\). In each case, the test statistic with the largest power has been put in italic to assist the reader. The results in Table 2 indicate that the proposed tests are competitors to each others with no absolute known optimal test that is determined.
The empirical power for a nominal 5 % level test comparing the approximation distributions of the seasonal portmanteau test statistics \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\), and \(\mathfrak {D}_{m}(s)\), at lag \(m=10\) based on \(10^{4}\) simulations. In each simulation, the SAR \((1)_{s}\) and SMA \((1)_{s}\) are fitted to data of series length \(n=200\) generated from SARMA \((2,2)\times (2,2)_{s}\) models where asterisk (*) refers to NULL and \(s=4,12\)
Model | \(\phi _{1}\) | \(\phi _{2}\) | \(\theta _{1}\) | \(\theta _{2}\) | \(\Phi _{1}\) | \(\Phi _{2}\) | \(\Theta _{1}\) | \(\Theta _{2}\) | \(\hat{Q}_{m}(4)\) | \(\tilde{Q}_{m}(4)\) | \(\mathfrak {D}_{m}(4)\) | \(\hat{Q}_{m}(12)\) | \(\tilde{Q}_{m}(12)\) | \(\mathfrak {D}_{m}(12)\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Fitted by SAR (1) | ||||||||||||||
1 | * | * | * | * | * | * | −0.5 | * | 0.111 | 0.133 | 0.129 | 0.100 | 0.113 | 0.119 |
2 | * | * | * | * | * | * | −0.6 | 0.3 | 0.333 | 0.301 | 0.389 | 0.301 | 0.301 | 0.356 |
3 | * | * | * | * | 0.7 | * | −0.4 | * | 0.087 | 0.091 | 0.063 | 0.084 | 0.090 | 0.062 |
4 | * | * | * | * | 0.1 | 0.3 | * | * | 0.102 | 0.119 | 0.120 | 0.092 | 0.095 | 0.089 |
5 | 0.3 | * | * | * | −0.35 | * | * | * | 0.786 | 0.663 | 0.652 | 0.763 | 0.641 | 0.661 |
6 | 0.4 | * | * | * | * | * | −0.8 | * | 0.961 | 0.998 | 0.971 | 0.951 | 0.996 | 0.965 |
7 | * | * | * | * | 0.4 | −0.6 | 0.3 | * | 0.371 | 0.401 | 0.442 | 0.363 | 0.396 | 0.406 |
8 | 0.7 | 0.2 | * | * | −0.5 | * | * | * | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
9 | 0.7 | * | 0.7 | * | −0.8 | * | * | * | 0.124 | 0.096 | 0.075 | 0.119 | 0.094 | 0.066 |
10 | 0.1 | 0.3 | * | * | * | * | −0.8 | * | 0.867 | 0.918 | 0.889 | 0.844 | 0.907 | 0.887 |
Fitted by SMA (1) | ||||||||||||||
11 | * | * | * | * | 0.5 | * | * | * | 0.126 | 0.148 | 0.150 | 0.111 | 0.144 | 0.143 |
12 | * | * | * | * | * | * | −0.6 | 0.3 | 0.175 | 0.172 | 0.264 | 0.161 | 0.165 | 0.244 |
13 | * | * | * | * | 0.7 | * | −0.4 | * | 0.382 | 0.701 | 0.665 | 0.378 | 0.700 | 0.662 |
14 | * | * | * | * | 0.1 | 0.3 | * | * | 0.113 | 0.100 | 0.211 | 0.109 | 0.098 | 0.203 |
15 | 0.3 | * | * | * | −0.35 | * | * | * | 0.760 | 0.689 | 0.662 | 0.755 | 0.679 | 0.654 |
16 | 0.4 | * | * | * | * | * | −0.8 | * | 0.958 | 0.961 | 0.913 | 0.953 | 0.958 | 0.908 |
17 | * | * | * | * | 0.4 | * | 0.3 | * | 0.204 | 0.367 | 0.265 | 0.201 | 0.295 | 0.257 |
18 | 0.7 | 0.2 | * | * | −0.5 | * | * | * | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
19 | 0.7 | * | 0.7 | * | −0.8 | * | * | * | 0.632 | 0.761 | 0.935 | 0.628 | 0.757 | 0.931 |
20 | 0.1 | 0.3 | * | * | * | * | −0.8 | * | 0.779 | 0.705 | 0.700 | 0.770 | 0.698 | 0.698 |
An empirical application
In this section, we make use of the monthly Federal Reserve Board Production Index data. Data is available from the R package astsa with the name prodn from January 1948 to December 1978 with 372 observations (Shumway and Stoffer 2011) and displayed in Fig. 3. All p-values from seasonal and nonseasonal tests suggest rejecting the null hypothesis, at the significance of 5 % level, that the seasonal and nonseasonal autocorrelations of the prodn series are equal to zero. Following Shumway and Stoffer (2011), we take the seasonal difference of the differenced production data \(\nabla _{12}(Z_{t}-Z_{t-1})\) and apply the BIC criteria to select the preferred model SARMA \((2,0)\times (0,3)_{12}\). Here, we are not interested in selecting the best fitted model but the main objective of this application is to demonstrate that the proposed seasonal tests are useful for investigating whether the autocorrelations of the residual SARMA model at the seasonal period are different from zero.
The SARMA \((2,0)\times (0,3)_{12}\) model was fitted to the monthly difference of the differenced federal reserve board production index data
Test | \(m=10\) | \(m=15\) | \(m=20\) | |||
---|---|---|---|---|---|---|
\(s=12\) | \(s=1\) | \(s=12\) | \(s=1\) | \(s=12\) | \(s=1\) | |
\(Q_{m}(s)\) | 0.822 | 0.114 | 0.381 | 0.030 | 0.574 | 0.069 |
\(\hat{Q}_{m}(s)\) | 0.744 | 0.107 | 0.087 | 0.024 | 0.093 | 0.055 |
\(\tilde{Q}_{m}(s)\) | 0.824 | 0.097 | 0.676 | 0.033 | 0.596 | 0.058 |
\(\mathfrak {D}_{m}(s)\) | 0.623 | 0.057 | 0.520 | 0.076 | 0.570 | 0.054 |
We apply the approximation distribution tests for the p-values associated with \(\alpha =5\,\%\) of \(\hat{Q}_{m}(s),\tilde{Q}_{m}(s)\) and \(\mathfrak {D}_{m}(s)\), on the residuals of the SARIMA \((2,1,0)\times (0,1,3)_{12}\) model, where \(m=10,15\), and 20 are the lags at seasonal and nonseasonal periods \(s=12,1\), respectively (Table 3). As seen in Table 3, all seasonal tests indicate that the SARIMA model is good in capturing the seasonal autocorrelations where no period autocorrelations are detected at seasonal lags 10, 15, and 20. On the other hand, as noted by Shumway and Stoffer (2011), we note that the classical nonseasonal tests (except that \(\mathfrak {D}_{m}\)) indicate that the model SARIMA \((2,1,0)\times (0,1,3)_{12}\) is inadequate where it does not capture the nonseasonal autocorrelations at lag \(m=15\).
Conclusion
Despite the popularity of the SARMA models in various economic and financial data, the goodness-of-fit portmanteau tests at multiple period lags \(1s,2s,3s,\ldots ,ms\), where m is the largest lag considered for autocorrelation and s is the seasonal period, has not yet received as much attention as it should deserve. In literature, the classical nonseasonal portmanteau statistics Box and Pierce (1970), Ljung and Box (1978), Peña and Rodríguez (2002, 2006), Mahdi and McLeod (2012), Fisher and Gallagher (2012) and Gallagher and Fisher (2015) for testing the lack of fit of SARMA models would be misleading since they are only implementing at the nonseasonal lags \(1,2,\ldots ,m\) ignoring the possibility of autocorrelations at seasonal lags of multiple period s. In this paper, we devise a new list of portmanteau statistics for seasonal time series using the asymptotic distribution of the residual autocorrelation at seasonal lags of multiple period s. We modify the classical nonseasonal portmanteau tests of the ARMA models mentioned above to the SARMA class with a case of \(p,q\ll s\) and the roots of the equation \(\phi ({B})\theta ({B})=0\) are not close to the unit circle. We provide simulation studies to demonstrate that the asymptotic tests are valid with satisfactorily performance in finite sample. In summary, in order to check the adequacy of time series models, we recommend to use the seasonal and nonseasonal versions of anyone of the portmanteau test statistics Box and Pierce (1970), Ljung and Box (1978), Peña and Rodríguez (2002, 2006), Mahdi and McLeod (2012), Fisher and Gallagher (2012) and Gallagher and Fisher (2015) as complementary to each other.
Declarations
Acknowledgements
The author thanks the Editor, the Associate Editor and the three anonymous referees for helpful comments.
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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