Seasonal Unit Roots Research Articles

A novel suggestion on how to adequately treat stochastic non-stationary seasonality in tourism export forecasting

Using a quarterly data sample of 28 EU countries ranging from 2004Q1 to 2016Q4, real tourism exports are modeled and estimated as a function of relative tourism export prices and aggregate real GDP. For almost all countries, HEGY tests applied to first-differenced variables cannot reject the null hypothesis of the presence of seasonal unit roots. Seasonal differencing would be necessary to remove the seasonal patterns from the data, yet employing seasonally differenced data results in information loss, distorted elasticity estimates, and mediocre forecast accuracy. Our suggestion is therefore to perform estimation and forecasting for each quarter separately. This approach avoids the use of seasonal differencing, results in plausible, seasonally varying estimates of income elasticities, and yields statistically significantly superior forecast accuracy, also in comparison to two data-driven benchmarks for seasonal data (i.e., SARIMA and ETS).

DEVELOPMENT OF SEASONAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODEL WITH ERROR PROCESS

This research work investigated with error process. The auto covariance function maximum likelihood methods, iterative method and chi-squares test statistics is used to develop SARIMA model corrupted with error process that is used to estimate the true parameter of the SARIMA model. The forecast performance measurement and properties of error with different value are also investigated. Test of seasonal unit roots are also carried out in the work. The simulation, and real data of Zamfara state monthly Rain fall from 1998 to 2022 is used to validate the results with R - Statistical software version 4.1.1 and mini tab statistical software version 14. The result showed a significance p- value of 0.000, the proposed model provides a generalization and more flexible specification than the existing models of AR (1) error and ARMA (1,1) error in fitting time series processes in the presence of errors . Hence, the studies showed that, the finding is closely to the true parameter of the process and would be useful to researchers in the prediction and handling of natural calamities that disturb the otherwise stability of a system.

Determining seasonal unit roots with bridge estimator: Monte Carlo evidence and an application to convergence hypothesis

HEGY test is a commonly used procedure to test seasonal unit roots. However, the HEGY test is highly sensitive to the deterministic components of the model and the selected lag structure. Therefore, the test may give wrong results if the correct model is not used in terms of the lag structure and deterministic components. Moreover, there is a requirement in the HEGY test that unit roots at different frequencies should be tested separately. Lasso and Bridge estimators, which have been proposed recently with the arise of big data and artificial intelligence studies, are frequently used for model selection and parameter estimation for cross-sectional data. This study proposes a novel approach by adapting the Bridge estimator to the HEGY model and automatically selecting the correct model. Monte Carlo simulations have shown that this approach gives good results in terms of both choosing the correct model and automatically determining the frequencies with a seasonal unit root, with the help of the oracle feature of the Bridge estimator. Moreover, a real data example is provided to demonstrate the usage of our method and to provide a numerical comparison between the Bridge estimator and HEGY test to test the seasonal convergence hypothesis.

Investigating long range dependence in temperatures in Siberia

In this paper we examine monthly mean temperatures in 40 selected stations in Siberia for the time period January 1937–December 2020 using long range dependence techniques. In particular, we use a fractionally integrated model that incorporates a linear time trend along with a seasonal structure. Our results show first that long memory is present in all stations with significantly positive values for the differencing parameter, though, at the same time the seasonal component seems to be important in all cases. Performing seasonal unit root tests, the results support nonstationary seasonality and working with the seasonal differenced data, the results differ depending on the structure of the error term: if the errors are uncorrelated, long memory is present; however, allowing autocorrelation, this feature disappears in favor of a short memory pattern.

Non-parametric seasonal unit root tests under periodic non-stationary volatility.

This paper presents a new non-parametric seasonal unit root testing framework that is robust to periodic non-stationary volatility in innovation variance by making an extension to the fractional seasonal variance ratio unit root tests of Eroğlu et al. (Econ Lett 167:75–80, 2018). The setup allows for both periodic heteroskedasticity structure of Burridge and Taylar (J Econ 104(1):91–117, 2001) and non-stationary volatility structure of Cavaliere and Taylor (Econ Theory 24(1):43-71, 2008). We show that the limiting null distributions of the variance ratio tests depend on nuisance parameters derived from the underlying volatility process. Monte Carlo simulations show that the standard variance ratio tests can be substantially oversized in the presence of such effects. Consequently, we propose wild bootstrap implementations of the variance ratio tests. Wild bootstrap resampling schemes are shown to deliver asymptotically pivotal inference. The simulation evidence depicts that the proposed bootstrap tests perform well in practice and essentially correct the size problems observed in the standard fractional seasonal variance ratio tests, even under extreme patterns of heteroskedasticity.Supplementary InformationThe online version contains supplementary material available at 10.1007/s00180-022-01211-w.

Research on the Impact of Sino-US Trade Structure on the Real Effective Exchange Rate of RMB

Since the beginning of 2018, Sino-US trade frictions have been escalating to the fields of science and technology, finance, and geography. Especially in the financial field, the United States has forcibly identified China as a “currency manipulator.” In order to analyze the impact of Sino-US trade on the RMB exchange rate, based on the Sino-US import and export trade data under the quarterly HS classification from 2003 to 2019 and the RMB real effective exchange rate, this article carries out the traditional time series test, seasonal unit root test, and cointegration test and further constructs the seasonal error correction model to explore the long-term and short-term dynamic impact of Sino-US import and export trade structure on RMB real effective exchange rate. The results shows that the upgrading and optimization of the overall trade structure between China and the United States will increase the appreciation pressure of RMB real effective exchange rate. There are seasonal and long-term trends between RMB real effective exchange rate and different types of import and export trade structures between China and the United States. Therefore, this article not only strongly refutes the “theory of RMB appreciation” and puts forward policy suggestions to effectively deal with the negative impact of Sino-US trade friction but also provides a research framework for global trade, especially the decoupling of trade structure and exchange rate between developing and developed countries.

Regulated seasonal unit root process

Abstract Unfortunately, time series problems do not appear in data singly. We focus on the joint occurrence of nonstationarity, seasonality and bounded data. Seasonal unit root tests and bounded unit root tests already exist in the literature, yet when all these issues are combined their performance needs improvement. That is why we offer a testing procedure for bounded seasonal unit root processes. The combination of these tests is not straightforward as the nonlinearity coming from bounds causes the limiting distribution of the proposed test statistic to be multivariate Brownian motion while the others have univariate distributions. The simulation exercises reveal that the existing tests, which ignores the presence of bounds or seasonality, suffer significant size problems. Our statistic removes the size distortions and also maintain satisfactory power performance.

Asymptotic Properties of Estimators for Seasonally Cointegrated State Space Models Obtained Using the CVA Subspace Method.

This paper investigates the asymptotic properties of estimators obtained from the so called CVA (canonical variate analysis) subspace algorithm proposed by Larimore (1983) in the case when the data is generated using a minimal state space system containing unit roots at the seasonal frequencies such that the yearly difference is a stationary vector autoregressive moving average (VARMA) process. The empirically most important special cases of such data generating processes are the I(1) case as well as the case of seasonally integrated quarterly or monthly data. However, increasingly also datasets with a higher sampling rate such as hourly, daily or weekly observations are available, for example for electricity consumption. In these cases the vector error correction representation (VECM) of the vector autoregressive (VAR) model is not very helpful as it demands the parameterization of one matrix per seasonal unit root. Even for weekly series this amounts to 52 matrices using yearly periodicity, for hourly data this is prohibitive. For such processes estimation using quasi-maximum likelihood maximization is extremely hard since the Gaussian likelihood typically has many local maxima while the parameter space often is high-dimensional. Additionally estimating a large number of models to test hypotheses on the cointegrating rank at the various unit roots becomes practically impossible for weekly data, for example. This paper shows that in this setting CVA provides consistent estimators of the transfer function generating the data, making it a valuable initial estimator for subsequent quasi-likelihood maximization. Furthermore, the paper proposes new tests for the cointegrating rank at the seasonal frequencies, which are easy to compute and numerically robust, making the method suitable for automatic modeling. A simulation study demonstrates by example that for processes of moderate to large dimension the new tests may outperform traditional tests based on long VAR approximations in sample sizes typically found in quarterly macroeconomic data. Further simulations show that the unit root tests are robust with respect to different distributions for the innovations as well as with respect to GARCH-type conditional heteroskedasticity. Moreover, an application to Kaggle data on hourly electricity consumption by different American providers demonstrates the usefulness of the method for applications. Therefore the CVA algorithm provides a very useful initial guess for subsequent quasi maximum likelihood estimation and also delivers relevant information on the cointegrating ranks at the different unit root frequencies. It is thus a useful tool for example in (but not limited to) automatic modeling applications where a large number of time series involving a substantial number of variables need to be modelled in parallel.

The problem of annual inflation rate indicator

AbstractIn economic practice, the annual inflation rate is commonly used. As a one‐sided moving average of the annualized inflation rate, it is approximately 6 months behind the annualized and monthly inflation rates and the CPI. The annual rate of inflation is, in fact, a smoothing transformation that removes the seasonal component from the annualized inflation rate, assuming the presence of all seasonal unit roots. In reality, however, the CPIs, and thus the annualized inflation rates, do not contain most of seasonal unit roots, resulting in spurious cycles of annual inflation rates, which pose difficulties in assessing and interpreting their development dynamics. Their practical applications in econometric analyses are limited because the unit root tests too often do not reject the zero hypothesis of the presence of unit roots when annual inflation rates are actually stationary. Parameter estimates of the annual inflation rate models are biased and inaccurate, allowing only incorrect short‐term forecasts. The above facts make use of the annual inflation rate indicator in real economic situations questionable.

Periodic autoregressive models for time series with integrated seasonality

A comprehensive seasonally integrated periodic autoregressive model is suggested which is shown to be flexible enough to include both the stochastic seasonal integrated and random trigonometric polynomial-based models. The demonstration of the equivalence between the two approaches is the objective of two theorems that are stated and proved in some details. A nice advantage of our model building procedure is that it is able to provide the user not only with a detailed model for data description and forecasting purpose but in addition with a hint at the presence of seasonal unit roots. The model which is illustrated in the present paper may be considered parsimonious, i.e. the number of estimated parameters, for a given goodness-of-fit criterion, is taken as low as possible, in two ways. First, by imposing unit roots and seasonal unit roots so that some estimated parameters are replaced by a differencing operator with fixed coefficients, and, second, by adopting a subset periodic autoregressive model, so that some parameters do not need to be estimated as they are constrained to equal zero. The effectiveness of our model is highlighted by an extensive simulation experiment that supports our claim that the model building procedure described here may be of good use as well for checking the existence of seasonal unit roots. Applications to real-world time series data sets are reported, and promising results are obtained that allow us to suggest that the seasonally integrated periodic autoregressive model may be safely used for modelling a wide range of seasonal time series data. In addition, well-known widely used tests, such as HEGY and the Taylor variance ratio test, are shown to provide us with results that generally agree with our findings.

Bootstrap seasonal unit root test under periodic variation

Both seasonal unit roots and periodic variation can be prevalent in seasonal data. In the testing of seasonal unit roots under periodic variation, the validity of the existing methods, such as the HEGY test, remains unknown. The behavior of the augmented HEGY test and the unaugmented HEGY test under periodic variation is analyzed. It turns out that the asymptotic null distributions of the HEGY statistics testing the single roots at 1 or −1 when there is periodic variation are identical to the asymptotic null distributions when there is no periodic variation. On the other hand, the asymptotic null distributions of the statistics testing any coexistence of roots at 1, −1,i, or −i when there is periodic variation are non-standard and are different from the asymptotic null distributions when there is no periodic variation. Therefore, when periodic variation exists, HEGY tests are not directly applicable to the joint tests for any concurrence of seasonal unit roots. As a remedy, bootstrap is proposed; in particular, the augmented HEGY test with seasonal independent and identically distributed (iid) bootstrap and the unaugmented HEGY test with seasonal block bootstrap are implemented. The consistency of these bootstrap procedures is established. The finite-sample behavior of these bootstrap tests is illustrated via simulation and prevails over their competitors’. Finally, these bootstrap tests are applied to detect the seasonal unit roots in various economic time series.

Using singular spectrum analysis for inference on seasonal time series with seasonal unit roots

The problem of optimal linear filtering, smoothing and trend extraction for m-period differences of processes with a unit root is studied. Such processes arise naturally in economics and finance, in the form of rates of change (price inflation, economic growth, financial returns) and finding an appropriate smoother is thus of immediate practical interest. The filter and resulting smoother are based on the methodology of singular spectrum analysis (SSA). An explicit representation for the asymptotic decomposition of the covariance matrix is obtained. The structure of the impulse and frequency response functions indicates that the optimal filter has a 'permanent' and a 'transitory component', with the corresponding smoother being the sum of two such components. Moreover, a particular form for the extrapolation coefficients that can be used in out-of-sample prediction is proposed. In addition, an explicit representation for the filtering weights in the context of SSA for an arbitrary covariance matrix is derived. This result allows one to examine the specific effects of smoothing in any situation. The theoretical results are illustrated using different datasets, namely US inflation and real GDP growth.

Using singular spectrum analysis for inference on seasonal time series with seasonal unit roots

The problem of optimal linear filtering, smoothing and trend extraction for m-period differences of processes with a unit root is studied. Such processes arise naturally in economics and finance, in the form of rates of change (price inflation, economic growth, financial returns) and finding an appropriate smoother is thus of immediate practical interest. The filter and resulting smoother are based on the methodology of singular spectrum analysis (SSA). An explicit representation for the asymptotic decomposition of the covariance matrix is obtained. The structure of the impulse and frequency response functions indicates that the optimal filter has a 'permanent' and a 'transitory component', with the corresponding smoother being the sum of two such components. Moreover, a particular form for the extrapolation coefficients that can be used in out-of-sample prediction is proposed. In addition, an explicit representation for the filtering weights in the context of SSA for an arbitrary covariance matrix is derived. This result allows one to examine the specific effects of smoothing in any situation. The theoretical results are illustrated using different datasets, namely US inflation and real GDP growth.

Artificial neural networks in modelling seasonal tourism demand - case study of Croatia

The purpose of this paper is to design an artificial neural network in the attempt to define the data generating process of the number of German tourist arrivals in Croatia considering the strong seasonal character of empirical data. The presence of seasonal unit roots in tourism demand determinants is analysed using the approach developed by Hylleberg, Engle, Granger and Yoo – Hegy test. The study is based on seasonality analysis and Artificial Neural Networks approach in building a model which intend to describe the behaviour of the German tourist flows to Croatia. Different neural network architectures were trained and tested, and after the modelling phase, the forecasting accuracy and model performances were analysed. Model performance and forecasting accuracy evaluation was tested using the mean absolute percentage error. Based on the augmented HEGY test procedure it can be concluded the German tourist arrivals to the Republic of Croatia have nonstationary behaviour associated with the zero frequency and seasonal frequency. Taking this into consideration, in the analysis of the phenomenon it is necessary to consider its seasonal character. Given the importance of the tourism for Croatian economic development, the research results could be useful, for both, researchers and practitioners, in the process of planning and routing the future Croatian hotel industry development and improvement of business performances.

Sea Surface Temperatures: Seasonal Persistence and Trends

AbstractThis paper addresses analysis of the global monthly sea surface temperatures using a reconstructed dataset that goes back to 1884. We use fractional integration methods to examine features such as persistence, seasonality, and time trends in the data. The results show that seasonality is a relevant issue, finding evidence of seasonal unit roots. With the seasonal component removed, persistence is also very significant, and, when looking at the data month by month, evidence of significant linear trends is detected in all cases. According to these results, monthly sea surface temperatures increase by between 0.07° and 0.11°C every 100 years.

The Elasticity of Substitution in Demand for Non-Tradable Goods in Bolivia

This paper uses a CES function to estimate the constant elasticity of substitution in consumption for non-tradables relative to tradables in a dependent economy framework. The methodology for generating data on real consumption of tradable and non-tradable goods, real prices of tradable and non-tradable goods and real absorption is based on the Bolivian Input-Output Matrix, producing quarterly data for the period 1990. 1 to 2002. 4. The data identify Bolivia as a country highly open to trade, with an average ratio of 55 percent in the value of exports and imports relative to GDP, non-tradable production accounting for 52 percent of GDP, and differences in the behavior of the internal and external real exchange rates. The HEGY test is used to identify and separate out seasonal unit roots in the data. A cointegration relationship was found between real absorption, the non-tradable to tradable consumption ratio and the non-tradable to tradable price ratio, suggesting inelasticity of substitution.

Stationarity of seasonal patterns in weekly agricultural prices

Weekly series of agricultural prices usually exhibit seasonal variations and the stationarity of these variations should be taken into account to analyse price relationships. However, unit root tests at seasonal frequencies are unlikely to have good power properties. Furthermore, movements in actual price series are often not as expected when unit roots are present. Therefore, stationarity tests at seasonal frequencies also need to be applied. In this paper, a procedure to test for the null hypothesis of stationarity at seasonal frequencies was extended to the weekly case. Once critical values were obtained by simulation exercises, unit root and stationarity tests were applied to weekly retail prices of different agricultural commodities in Spain. The most relevant finding was that many unit roots that seasonal unit root tests failed to reject did not seem to be present from the results of seasonal stationarity tests, whereas seasonal unit root tests led to the rejection of some unit roots that seemed to be present according to the results of seasonal stationarity tests. In conclusion, unit root tests should be complemented with stationarity tests before making decisions about the behaviour of seasonal patterns.

Robust tests for deterministic seasonality and seasonal mean shifts

We develop tests for the presence of deterministic seasonal behaviour and seasonal mean shifts in a seasonally observed univariate time series. These tests are designed to be asymptotically robust to the order of integration of the series at both the zero and seasonal frequencies. Motivated by the approach of Hylleberg, Engle, Granger and Yoo [1990, Journal of Econometrics vol. 44, pp. 215-238], we base our approach on linear filters of the data which remove any potential unit roots at the frequencies not associated with the deterministic component(s) under test. Test statistics are constructed using the filtered data such that they have well defined limiting null distributions regardless of whether the data are either integrated or stationary at the frequency associated with the deterministic component(s) under test. In the same manner as Vogelsang [1998, Econometrica vol. 66, pp. 123-148], Bunzel and Vogelsang [2005, Journal of Business and Economic Statistics vol. 23, pp. 381-394] and Sayginsoy and Vogelsang [2011, Econometric Theory vol. 27, pp. 992-1025], we scale these statistics by a function of an auxiliary seasonal unit root statistic. This allows us to construct tests which are asymptotically robust to the order of integration of the data at both the zero and seasonal frequencies. Monte Carlo evidence suggests that our proposed tests have good finite sample size and power properties. An empirical application to U.K. GDP indicates the presence of seasonal mean shifts in the data.

TOURISM DEMAND AND EXOGENOUS EXCHANGE RATE IN CAMBODIA: A STOCHASTIC SEASONAL ARIMAX APPROACH

Time series seasonally displays toward either an autoregressive or moving average process where persist in seasonality fluctuation. The paper examines the relationship of tourism demand with controlling an exogenous exchange rate using seasonal ARIMA model. The empirical results reveal that HEGY test for seasonal unit roots with lower and upper panel indicates the statistical significance which explains the failure of rejection of having unit roots at different frequencies. The estimated outcomes from tourism demand model specify that per capita income and exchange rate have the power in explaining tourism demand measured as tourist arrivals. In particular to forecasting model and due to the lower statistical value of RMSE and MAE displays that the SARIMAX (4, 1, 1) – (1, 1, 1)12 model is the best accuracy model to perform the long run ex-ante forecasting of tourism demand. This suggests tourism policy maker to pay more reflection in formulating the policy toward the exogenous factors in line with the uncertainty and unobserved seasonality.

Quantifying the impact of Ramadan on global raw sugar prices

PurposeThe purpose of this study is to quantify the impact of Ramadan on both the level and the growth of global raw sugar price.Design/methodology/approachThe study uses a dummy and a fractional variable to capture Ramadan to overcome the asynchronicity of time between Ramadan fasting (which is based on the Islamic lunar calendar) and the movement in prices (which follows the Gregorian solar calendar). To capture the seasonality of sugar production, the data on sugar price span 34 years so that the Islamic calendar makes a complete cycle of the Gregorian calendar. The empirical model is estimated using both autoregressive integrated moving average model and unobserved components model.FindingsThe results show that monthly raw sugar prices in the global market increases by roughly 6.06 per cent (or $17.78 per metric ton) every year ahead of Ramadan.Practical implicationsThe study illustrates the implications of the results for the consumption of imported sugar in Bangladesh.Originality/valueThe study uses a broader set of Ramadan indicators in its empirical models and checks the robustness of its baseline model using the unobserved components model. It also performs seasonal unit root tests on the global raw sugar prices.