Portmanteau Test Research Articles

Bootstrapping the transformed goodness-of-fit test on heavy-tailed GARCH models

We study the bootstrap inference on the goodness-of-fit test for generalized autoregressive conditional heteroskedastic (GARCH) models. Note that the commonly-used portmanteau tests for model adequacy checking necessarily impose moment conditions on innovations, we hence construct the test on the sample autocorrelations of a bounded transformation of absolute residuals, which are obtained by the least absolute deviation estimation from a fitted GARCH model. Specifically, we employ the empirical distribution function of absolute residuals as the transformation. Thus the corresponding portmanteau tests are applicable for very heavy-tailed innovations with only finite fractional moments. We bootstrap both the estimation equation and sample autocorrelations of transformed residuals to approximate the test statistics. The asymptotic validity of the bootstrap procedure is established. Monte Carlo experiments compare the finite-sample performance of the proposed bootstrap-based test with other existing tests. An empirical analysis of modeling exchange rates illustrates its usefulness.

An ARCH volatility analysis of real GDP, real gross capital formation, and foreign direct investment in Bangladesh

This paper aims to investigate the volatility of the growth rates of Bangladesh's real GDP, real gross capital formation, and net inflows of foreign direct investment. The study used data on these indicators from the World Bank for the period between 1972 and 2020. Autoregressive integrated moving average (ARIMA) and the autoregressive conditional heteroscedastic (ARCH) methods were applied to model the conditional mean and conditional variance components for each growth rate. The validity of the selected volatility models was evaluated using a variety of diagnostic techniques, such as the time series graph of estimated residuals, cumulative periodogram, and the portmanteau test for white noise. The overall performance of the selected models is evaluated using the mean squared error (MSE) and the root mean squared error (RMSE). For all three indicators, the conditional means depend on the growth rates of the previous year or two years, whereas conditional variances depend on the previous year’s rates. The outcomes of the study also indicate the existence of time-varying volatility in Bangladesh's economy. This study may be helpful in understanding the potential risks related to the volatile nature of macroeconomic growth rates.

A portmanteau test for the iid hypothesis

In this paper, we introduce a new portmanteau test for the iid hypothesis, where the elements of the sample are allowed to take values in a general space (e.g., a function space). We study the finite sample properties of our test, evaluating its performance in terms of empirical size and power. In particular, we compare the empirical power of our test with the power of other tests in the literature designed to work in a specific data setting, as the one‐way analysis of variance test used in experimental data analysis and three portmanteau tests used in time series analysis. In every case, we found conditions where our test outperformed in power to the competing test. Finally, to illustrate the usefulness of our test, we implement it on two real‐world applications based on function‐valued data.

Asymptotic behaviour of the portmanteau tests in an integer-valued AR model

The portmanteau test has been popular for diagnostic checking in time series models. Asymptotic properties of portmanteau tests have been exhaustively studied for real-valued time series model though, similar results for integer-valued autoregressive (INAR) models are not well documented, nevertheless. In view of this, we investigate the asymptotic behaviour of the Box-Pierce and Ljung-Box portmanteau tests in an INAR model. It turns out that these tests are chi-squared distributed asymptotically under mild conditions regardless of the process being stable or nearly unstable.

Center-outward Rank- and Sign-based VARMA Portmanteau Tests: Chitturi, Hosking, and Li–McLeod revisited

The pseudo-Gaussian portmanteau tests of Chitturi, Hosking, and Li and McLeod for VARMA models are revisited from a Le Cam perspective, providing a precise and more rigorous description of the asymptotic behavior of the multivariate portmanteau test statistic, which depends on the dimension d of the observations, the number m of lags involved, and the length n of the observation period. Then, based on the concepts of center-outward ranks and signs recently developed (Hallin, del Barrio, Cuesta-Albertos, and Matrán, Annals of Statistics 49, 1139–1165, 2021), a class of multivariate rank- and sign-based portmanteau test statistics is proposed which, under the null hypothesis and under a broad family of innovation densities, can be approximated by an asymptotically chi-square variable. The asymptotic properties of these tests are derived; simulations demonstrate their advantages over their classical pseudo-Gaussian counterpart.

A portmanteau-type test for detecting serial correlation in locally stationary functional time series

The portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.

On portmanteau-type tests for nonlinear multivariate time series

A general framework to devise portmanteau-type test statistics for a general class of multivariate nonlinear time series models with vector martingale difference errors is formulated. Based on this framework a suite of individual and mixed multivariate test statistics is considered. Two applications are developed: single- and multiple-lag test statistics. In each case, the resulting portmanteau test statistic is based on multivariate residuals and multivariate squared residuals. Moreover, single- and multiple-lag mixed multivariate portmanteau-type tests are introduced. These test statistics are designed to detect different forms of inadequacies in the model residuals jointly. All proposed tests take uncertainty due to model estimation properly into account. The asymptotic null distribution of each test statistic follows from the asymptotic distribution of the general portmanteau-type test statistic in a natural way. Some considerations are given to the empirical size and power of six test statistics via a simulation study. All tests have satisfactory size and power properties in finite samples. To demonstrate their practical use, the proposed test statistics are applied to the residuals of a vector bivariate nonlinear threshold model fitted to U.S. interest rates.

An Automatic Portmanteau Test For Nonlinear Dependence

A data-driven version of a portmanteau test for detecting nonlinear types of statistical dependence is considered. An attractive feature of the proposed test is that it properly controls the type I error without being sensitive with respect to the number of autocorrelations used. In addition, the automatic test is found to have higher power in simulations when compared to the standard portmanteau test, for both raw data and residuals.

FORECASTING CONSUMER PRICE INDEX AND EXCHANGE RATE USING ARIMA MODELS: EMPIRICAL EVIDENCE FROM NIGERIA

Considering the high level of uncertainty in the foreign exchange market and the adverse effects of inflation in Nigeria, the need to utilize current data to work on appropriate models capable of predicting the future values of exchange rates and CPI has become necessary to guide monetary policy makers. This paper applies the techniques of Autoregressive Integrated Moving Average models to forecast the CPI and exchange rates of Nigeria using the dataset of monthly CPI and Exchange rates of naira against US Dollar from January 2010 to August 2022 obtained from National Bureau of Statistics. After the original series were appropriately differenced to attain stationarity, autocorrelation function (ACF) and partial autocorrelation function (PACF) were used to select a number of tentative models for parameter estimation. Based on selection criteria such as AIC, SBIC, HQC, R2 and Durbin Watson Statistics, ARIMA(1, 2, 0) was chosen as the best model for forecasting Nigeria’s monthly Inflation (CPI) and ARIMA(1, 1, 1) was selected as the most ideal model for forecasting monthly foreign exchange rates of Nigeria. The portmanteau tests carried out show that the residuals from both models are white noise which further confirms the adequacy of the fitted models. The results reveal that both inflation and Exchange rates of Naira against the Dollar will continue to rise. However, the rise in exchange rates for the short time is relatively steady. The findings from this study will furnish the monetary and policy makers with necessary information needed to reverse the expected trend

Empirical likelihood-based portmanteau tests for autoregressive moving average models with possible infinite variance innovations

Empirical likelihood-based portmanteau tests for autoregressive moving average models with possible infinite variance innovations

Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

This work considers the problem of modified Portmanteau tests for testing the adequacy of FARIMA models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak FARIMA). We first study the joint distribution of the least squares estimator and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) modified Portmanteau statistics for weak FARIMA models. We also propose another method based on a self-normalization approach to test the adequacy of FARIMA models. Finally some simulation studies are presented to corroborate our theoretical work. An application to the Standard & Poor’s 500 and Nikkei returns also illustrates the practical relevance of our theoretical results.

Testing network autocorrelation without replicates.

In this paper, we propose a portmanteau test for whether a graph-structured network dataset without replicates exhibits autocorrelation across units connected by edges. Specifically, the well known Ljung-Box test for serial autocorrelation of time series data is generalized to the network setting using a specially derived central limit theorem for a weakly stationary random field. The asymptotic distribution of the test statistic under the null hypothesis of no autocorrelation is shown to be chi-squared, yielding a simple and easy-to-implement procedure for testing graph-structured autocorrelation, including spatial and spatial-temporal autocorrelation as special cases. Numerical simulations are carried out to demonstrate and confirm the derived asymptotic results. Convergence is found to occur quickly depending on the number of lags included in the test statistic, and a significant increase in statistical power is also observed relative to some recently proposed permutation tests. An example application is presented by fitting spatial autoregressive models to the distribution of COVID-19 cases across counties in New York state.

Forecasting wildfires in major forest types of India

Severity of wildfires witnessed in different parts of the world in the recent times has posed a significant challenge to fire control authorities. Even when the different fire early warning systems have been developed to provide the quickest warnings about the possible wildfire location, severity, and danger, often it is difficult to deploy the resources quickly to contain the wildfire at a short notice. Response time is further delayed when the terrain is complex. Early warning systems based on physics-based models, such as WRF-FIRE/SFIRE, are computationally intensive and require high performance computing resources and significant data related to fuel properties and climate to generate forecasts at short intervals of time (i.e., hourly basis). It is therefore that when the objective is to develop monthly and yearly forecasts, time series models seem to be useful as they require lesser computation power and limited data (as compared to physics-based models). Long duration forecasts are useful in preparing an efficient fire management plan for optimal deployment of resources in the event of forest fire. The present research is aimed at forecasting the number of fires in different forest types of India on a monthly basis using “Autoregressive Integrated Moving Average” time series models (both univariate and with regressors) at 25 km × 25 km spatial resolution (grid) and developing the fire susceptibility maps using Geographical Information System. The performance of models was validated based on the autocorrelation function (ACF), partial ACF, cumulative periodogram, and Portmanteau (L-Jung Box) test. Both the univariate- and regressor-based models performed equally well; however, the univariate model was preferred due to parsimony. The R software package was used to run and test the model. The forecasted active fire counts were tested against the original 3 years monthly forecasts from 2015 to 2017. The variation in coefficient of determination from 0.94 (for year 1 forecast) to 0.64 (when all the 3-year forecasts were considered together) was observed for tropical dry deciduous forests. These values varied from 0.98 to 0.89 for tropical moist deciduous forest and from 0.97 to 0.88 for the tropical evergreen forests. The forecasted active fire counts were used to estimate the future forest fire frequency ratio, which has been used as an indicator of forest fire susceptibility.

Bootstrapping a powerful mixed portmanteau test for time series

A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. The new portmanteau statistic is calculated from the log of the determinant of a matrix comprised of the autocorrelations and cross-correlations of the residuals and squared residuals of a fitted time series. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-square distributed random variables and can be approximated by a gamma distribution. A bootstrapping approach is shown to be robust when distributional assumptions are relaxed. The efficacy of the statistic is studied against linear and nonlinear dependency structures of some stationary time series models. It is shown that the new test can provide higher power than other tests in many situations. We demonstrate the advantages of the proposed test by investigating linear and nonlinear effects in an economic series and two environmental time series.

BRC-GARCH-X model: the empirical evidence in stock returns

A covariate-driven random coefficient generalized conditional heteroscedasticity (GARCH) time series model with the form of the buffered autoregression (BRC-GARCH-X) for modeling financial time series data is considered. As an extension of the classical two-regime threshold process, the buffered autoregression enjoys a more flexible regime-switching mechanism. Furthermore, the main feature of this model is that the threshold variable for regime-switching is formulated as a weighted average of important auxiliary variables. The estimator for regression parameters is obtained by the quasi-maximum exponential likelihood (QMEL) estimator and the corresponding asymptotic properties are established. Moreover, a mixed portmanteau test is developed for diagnostic checking. And a reasonable method for selecting search ranges for thresholds is also proposed and simulation studies are considered. As an application, we bring attention to some features of of stock returns of SP500 which shows that our model is feasible.

Stock Market Efficiency and COVID-19 with Multiple Structural Breaks: Evidence from India

The objective of the study is to investigate the influence of the coronavirus pandemic (endogenous crisis) on the stock market efficiency of India during the multiple break periods. The empirical analysis is performed using conditional heteroscedasticity and a small sample robust wild bootstrap automatic variance ratio test and automatic portmanteau test on a daily stock return data of two benchmark indices, that is, NIFTY and SENSEX. The empirical results demonstrate that the stock return of two indices deviates from market efficiency during some periods of the analysis, notably during the nationwide lockdown and peak periods of coronavirus cases in India. These findings indicate that changing stock market behaviour becomes more speculative and earns abnormal profits. To the best of the author’s knowledge, this study provides the first evidence of investigating the variations in the stock market efficiency of India in response to this endogenous crisis.

Time-frequency analysis of venous pulsatile flow and sigmoid sinus vascular wall displacement causing pulsatile tinnitus using time series and magnitude squared coherence methods

Displacement of the sigmoid sinus vascular wall and intrasinus blood flow motion is known to cause pulsatile tinnitus (PT) in patients with sigmoid plate dehiscence. To investigate the source of sound that causes PT, this study sought to determine the correlation between in vivo displacement of the vascular wall and intrasinus flow hemodynamics. A confocal laser displacement sensor and color-coded Doppler ultrasound system were implemented on a participant diagnosed with venous PT. The displacement and Doppler velocity data were analyzed using time series and magnitude squared coherence methods. The median and peak displacement values of the vascular wall were 6.7 [Formula: see text]m and 11.7 [Formula: see text]m, respectively. The major frequency range of vascular displacement was below 4 Hz. The portmanteau test was rejected for different values of m. The cross-correlation of the two processes was not zero, indicating the existence of cross-correlation. The peak magnitude squared coherence of the two gauged signals was 0.3 at 7.143 Hz. The displacement of the sigmoid sinus vascular wall was temporally correlated with the intrasinus flow, although the major frequency range of the induced vascular displacement fell below the human hearing threshold. Therefore, the hydroacoustic and vibroacoustic sounds induced by the fluid–structure interaction between the vascular and osseous structures overweighed the displacement of the vascular wall at the focal dehiscence area in PT production.

Portmanteau test for a class of multivariate asymmetric power GARCH model

We establish the asymptotic behaviour of the sum of squared residuals autocovariances and autocorrelations for the class of multi‐variate power transformed asymmetric models. We then derive a portmanteau test. We establish the asymptotic distribution of the proposed statistics. These asymptotic results are illustrated by Monte Carlo experiments. An application to a bivariate real financial data is also proposed.

A study on prediction model estimation of financial assets (exchange rate, KOSPI index, interest rate)

In this study, financial assets such as exchange rate, KOSPI index, and interest rate (3-year government bond) were predicted using the vector error correction model used in various financial markets. For this purpose, time series data from February 2000 to January 2021 provided by the Bank of Korea were used. To estimate the prediction model, the stability of the time series variables was confirmed by the ADF test, the causal relationship between the time series variables by the Granger causality test, and the cointegration relationship between the time series variables by the cointegration test. In addition, the prediction model was estimated by identifying the model based on the minimum information criterion based on AICC statistics, and the validity of the model was confirmed by the significance test of the cross-correlation matrix and the multivariate Portmanteau test. As the result of forecasting with the estimated prediction model, the exchange rate was predicted to rise steadily, the KOSPI index was predicted to fall, but remain in the mid-3,000 range, and the interest rate (3-year government bond) was predicted to decline.

The Puzzle of Informational Efficiency & Macroeconomic Variables: A Comparative Evidence in Shari’ah Stock Markets of Pakistan viz-a-viz Malaysia

The present study examines the weak-form informational efficiency of the Shari’ah stock market of Pakistan and Malaysia. For this purpose, the study uses the daily index price data of the KSE Meezan Index (Pakistan) & FTSE Malaysia EMAS Shariah Index (Malaysia) for the time span starting from 1st July 2009 to 30th June 2019. By employing a more robust test namely Automatic Portmanteau Test (AQ Test) through testing the Martingale Difference Hypothesis, this study finds that KSE Meezan Index (KMI-30) was informationally efficient (weak-form) whereas FTSE Malaysia EMAS Shariah Index was not weak-form of efficient. Further, this research investigates the long-run relationship viz-a-viz short-run relationship among the returns of both Islamic stock markets and three macroeconomic variables (Exchange Rate, Inflation Rate, Interest Rate) of both said countries. In this connection, this study employs the Philips-Perron Test for stationary analysis, Johansen co-integration test, Vector Error Correction Model on the monthly time series data for the same said sample period. The statistical findings show that returns of both Shari’ah stock markets have a long-run association with said macroeconomic variables and have the co-integration or long-term association with long-run causality running from macroeconomic variables to the return of the Islamic stock market in both sample countries but the short-run association viz-a-viz short-run causality is not found.