Testing fractional integration with monthly data

Seasonal roots can help to explain the seasonal fluctuations in macroeconomic time series. In this paper we concentrate on monthly data and look at different versions of Robinson’s (1994) tests for testing unit roots and other fractionally integrated hypotheses when the root is located at zero and/or at the seasonal frequencies. A Monte Carlo experiment is carried out to check the power of these tests against different fractional alternatives, and an empirical application, using Spanish monthly data for the consumer price index, is also carried out in the article.     

Unit and Fractional Roots at the Long Run and the Seasonal Frequencies in Macroeconomic Time Series

In this article, we show that macroeconomic time series may contain unit and fractional roots at both, at zero and at zero and at the seasonal frequencies. The importance of the root at the long run or zero frequency requires in many cases to consider this root at both, separately in an independent polynomial, and also included in the seasonal one. Several Monte Carlo experiments are conducted to examine cases when the root at the zero frequency is not appropriately considered. An empirical application based on the tests of Robinson, Peter M. “Efficient Tests of Nonstationary Hypotheses,” Journal of the American Statistical Association, 89, 1994, pp. 1420–37 is also carried out at the end of the article.The author gratefully acknowledges financial support from the Government of Navarra (“Ayudas de Formación e Investigación y Desarrollo”).     

Seasonal fractional components in macroeconomic time series

Seasonal fractional models are shown in this article to be alternative credible ways of modelling the seasonal component in macroeconomic time series. A testing procedure that allows one to test different orders of integration at zero and at each of the seasonal frequencies is described. This procedure is then applied to the Italian consumption and income series, the results being very sensitive to the way of modelling the I(0) disturbances.     

The seasonal unit root structure in New Zealand Macroeconomic variables

ADF unit root tests are generally applied to macroeconomic data prior to testing theoritical models to ensure that all relevant variables are integrated of the same order. Not only is it important to test that these variables are integrated of the same order but also that a cointegrating relationship exists; failure to do so raise the specture of false inference associated with the spurious regression problem. The seasonal nature of quarterly data adds a further proplem which has generally been overcome by seasonally adjusting the data using procedure such as the census X-11 rather than suppressing it, have attempted to determine whether the seasonal component in each variable exhibits stochastic non-stationary. This paper analysisunit roots in a seasonal setting and compares the recently developed tests for seasonal unit roots as well as the standard augmented Dickey-Fuller zerop frequency unit root tests. Of the variables tested relatively few paper to be integrated at the seasonal frequenciues and, as other studies suggest,determinstic seasonal effects are typically more important than stochastic ones.     

Seasonal Cointegration and Long‐Run Neutrality of Money in the USA

Using the notion of seasonal cointegration and a monetarist model, this paper re‐examines the long‐run monetary neutrality hypothesis, based on the seasonally unadjusted quarterly data of the US over the period 1959Q1–2004Q4. The results indicate that money is cointegrated with price at all possible frequencies while real output is cointegrated with price only at an annual frequency. The cointegration between money and price at the zero frequency, and non‐cointegration between real output and money at all possible frequencies, suggests that money affects nominal but not real variables in the long run.     

Long run and cyclical strong dependence in macroeconomic time series: Nelson and Plosser revisited

This paper deals with the presence of long range dependence at the long run and the cyclical frequencies in macroeconomic time series. We use a procedure that allows us to test unit roots with fractional orders of integration in raw time series. The tests are applied to an extended version of Nelson and Plosser’s (Nelson CR, Plosser CI (1982) J Monetary Econ 10:139–162) dataset, and the results show that, though the classic unit root hypothesis cannot be rejected in most of the series, fractional degrees of integration at both the zero and the cyclical frequencies are plausible alternatives in some cases. Additionally, the root at the zero frequency seems to be more important than the cyclical one for all series, implying that shocks affecting the long run are more persistent than those affecting the cyclical part. The results are consistent with the empirical fact observed in many macroeconomic series that the long-term evolution is non-stationary, while the cyclical component is stationary.

Long memory at the long run and at the cyclical frequencies: modelling real wages in England, 1260–1994

This paper examines historical data on daily real wages in England for the time period 1260–1994 by means of new statistical techniques suitable for modelling long memory both at the long run and the cyclical frequencies. Specifically, it uses a procedure due to Robinson ((1994) Efficient tests of nonstationary hypotheses. J Am Stat Assoc 89:1420–1437) which is based, for the cyclical component, on Gegenbauer processes. We test for the presence of unit (and fractional) roots at both the zero and the cyclical frequencies, and find that the root at the zero frequency plays a much more important role than the cyclical one, though the latter frequency also has a component of long memory behaviour. It also appears that the trending (zero frequency) component is nonstationary while the cyclical one is stationary, with shocks having permanent effects on the former, but transitory effects on the latter. Similar conclusions are reached when allowing for a break in 1875 (the beginning of the Second Industrial Revolution).

Testing for seasonal integration and cointegration: The Austrian consumption income relationship

This paper analyzes the nature of seasonal fluctuations in quarterly observations for Austrian consumption and income data. We begin with univariate tests of the order of integration and then move on to tests of cointegration. Seasonally adjusted as well as raw data are used in these tests. in univariate tests, the outcome for seasonally adjusted and raw data is in line. The unit roots at the zero frequency found in the seasonally adjusted series are also present in the raw data. In bivariate tests, the results for seasonally adjusted and raw data differ. While we find cointegration at the zero frequency between consumption and income for seasonally adjusted series, this hypothesis is generally rejected for the raw data.     

GLS detrending-based unit root tests in nonlinear STAR and SETAR models

We extend GLS detrending procedure to testing for unit roots against STAR and SETAR alternatives. Monte Carlo simulations and applications to DM/Yen real exchange rates demonstrate that GLS detrending-based nonlinear unit root tests are more powerful than OLS detrending-based counterparts.     

Bayesian analysis of periodic unit roots in the presence of a break

A Bayesian testing approach for a periodic unit root in quarterly and monthly data is presented. Further a Bayesian test is introduced to test for unit roots at (non)seasonal spectral frequencies. All procedures admit one structural break in the periodic trend function, where the occurrence of a break and the associated timing are treated as additional model parameters. A Bayesian model averaging (BMA) approach is proposed and power functions of the tests are computed. Overall the results indicate that the BMA periodic unit root test exhibits favourable test properties even in small samples. In an empirical application the presented testing procedures are used to test for (non)seasonal forms of unemployment persistence among OECD countries.     

Seasonal cointegration,common seasonals,and forecasting seasonal series

Seasonal cointegration generalizes the idea of cointegration to processes with unit roots at frequencies different from 0. Here, common seasonals, also a dual notion of common trends, is adopted for the seasonal case. The features are demonstrated in exemplary models for German and U.K. data. An evaluation of the predictive value of accounting for seasonal cointegration shows that seasonal cointegration may be difficult to exploit to improve predictive accuracy even in cases where seasonal non-cointegration is clearly rejected on statistical grounds. The findings from the real-world examples are corroborated by Monte Carlo simulation.Part of this paper was written while the author was visiting professor at the University of California San Diego.     

A Nonparametric Test for I(0)

There is frequently interest in testing that a scalar or vector time series is I(0), possibly after first-differencing or other detrending, while the I(0) assumption is also taken for granted in autocorrelation-consistent variance estimation. We propose a test for I(0) against fractional alternatives. The test is nonparametric, and indeed makes no assumptions on spectral behaviour away from zero frequency. It seems likely to have good efficiency against fractional alternatives, relative to other nonparametric tests. The test is given large sample justification, subjected to a Monte Carlo analysis of finite sample behaviour, and applied to various empirical data series.     

TESTS OF UNIT ROOTS AND COINTEGRATION HYPOTHESES IN ECONOMETRIC MODELS*

A number of statistical procedures for testing the unit roots hypotheses and the cointegration hypothesis have been proposed by statisticians and econometricians. This paper unifies many of the previous studies on unit roots tests and cointegration tests in the framework of a multivariate regression model and develops some new test statistics. We give a convenient quadratic representation of the limiting distributions of test statistics using stochastic integrals with respect to Brownian motions. The test procedures in this paper include the statistics for testing the unit root, the double unit roots, the seasonal unit roots, and the cointegrating relations for special cases. We also discuss some useful generalizations of unit roots tests and cointegration tests for empirical studies.     

A simple unit root test against asymmetric STAR nonlinearity with an application to real exchange rates in Nordic countries

Existing tests of the unit root hypothesis against the alternative hypothesis of exponential smooth transition autoregressive (ESTAR) nonlinearity implicitly assume symmetry under the alternative. This paper proposes a simple unit root test against the alternative of symmetric or asymmetric ESTAR nonlinearity. In the event that the unit root hypothesis is rejected, a simple test of symmetric versus asymmetric ESTAR nonlinearity is also proposed. The asymptotic distributions of the test statistics are straightforward to establish and finite-sample performance is studied with Monte Carlo simulations. An empirical application involving the real exchange rates of four Nordic countries against the U.S. dollar illustrates the usefulness of the new tests.     

Measuring persistence in Croatian tourism: evidence from the Adriatic region

This study examines the degree of persistence in foreign tourist arrivals and overnight stays for seven Croatian coastal counties over the period January 1998 to December 2013 using fractional integration techniques. Our findings reveal that the respective regional tourism indicators exhibit seasonal unit roots which require seasonal first differences to render the respective time series stationary. With respect to the long-run evolution of the respective time series, both the parametric and semi-parametric fractional integration approaches show the degree of persistence is greater than zero, but significantly less than one for the majority of the coastal counties. Impulse response analysis reveals indeed shocks to the deseasonalized time series, either foreign tourist arrivals or foreign tourist overnight stays, appear short-lived with the exception of Istria and Primorje-Gorski kotar counties. Policy implications of the results are also discussed.     

Some cautions on the use of nonlinear panel unit root tests: Evidence from a modified series-specific non-linear panel unit-root test

The purpose of this paper is to examine the relevance of applying nonlinear panel unit root test to examine the non-linear mean reversion behaviors of real exchange rates. We find that nonlinear panel unit root test may achieve lower power performance as compared to its alternative of linear panel unit test when the data generating process does not contain significant non-linear components. This finding post cautions to researchers in modeling and testing real exchanges behavior. We also develop a modified series-specific nonlinear panel unit root test and find evidence in favor of purchasing power parity hypothesis for China's four ASEAN trading partners in the period of February 1997 to August 2009.     

Breaking Trend Functions in Real Exchange Rates: Evidence from Seventeen OECD Countries

This paper tests for unit roots in dollar-based and DM-based real exchange rates using quarterly data (from 1957:i to 1995:iv) for seventeen OECD countries. The results show that the unit root hypothesis cannot be rejected even if allowance is made for the possibility of a one-time change in the mean of the series at an unknown point in time. This is evidence against the hypothesis of absolute long-run purchasing power parity over this period.     

Testing of unit roots and other fractionally integrated hypotheses in the presence of structural breaks

Tests of unit roots and other nonstationary hypotheses that were proposed by Robinson (1994) are applied in this article to the Nelson and Plosser's (1982) series. The tests can be expressed in a way allowing for structural breaks under both the null and the alternative hypotheses. When applying the tests to the same dataset as in Perron (1989), we observe that our results might be consistent with those in Perron (1989) when testing the nulls of trend-stationarity or a unit-root. However, we also observe that fractionally integrated hypotheses may be plausible alternatives in the context of structural breaks at a known period of time. Final version received: August 2000/Final version accepted: August 2001 RID="*" ID="*"  The author gratefully acknowledges the financial support from the European TMR grant No. ERBFMRX-CT-98-0213. Comments of two anonymous referees are also acknowledged.     

Long- versus medium-run identification in fractionally integrated VAR models

We state that long-run restrictions that identify structural shocks in VAR models with unit roots lose their original interpretation if the fractional integration order of the affected variable is below one. For such fractionally integrated models we consider a medium-run approach that employs restrictions on variance contributions over finite horizons. We show for alternative identification schemes that letting the horizon tend to infinity is equivalent to imposing the restriction of Blanchard and Quah (1989) introduced for the unit-root case.     

Efficient Tests for a Unit Root When the Initial Observation is Drawn From Its Unconditional Distribution

Researchers desire powerful tests for unit roots. This paper derives the family of asymptotically most powerful tests for unit roots when the initial condition is drawn from its unconditional distribution under the alternative. This enables both the examination of previously proposed statistics and the construction of powerful tests against this alternative model.