Testing Integration of Macroeconomic Time Series in Transitional Socialist Economies. A Modification of Perron Test

The presence of structural breaks reduces the power of integration tests. A number of methods were suggested to improve the statistical properties of integration tests in the presence of structural breaks. The most known are Perron tests, which allow to test for the level of integration of time series with one structural break. Perron tests allow for two types of structural breaks: additive outlier an innovative outlier. These tests are, however, not very useful in testing the level of integration of macroeconomic time series in countries in transition from centrally-planned to market economy. In such case one should expect two structural breaks to affect the time series: one at the beginning and one at the end of the transformation process. Test that allows for two additive outlier type structural breaks in time series is developed in this paper. This test has superior power as compared to standard Dickey-Fuller and Perron tests. This paper provides asymptotic distribution as well as finite sample properties of proposed test. Therefore practitioners receive a reliable tool for analyzing macroeconomic processes in transitional economies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Diagnostic testing for cointegration

We develop a sequence of tests for specifying the cointegrating rank of, possibly fractional, multiple time series. Memory parameters of observables are treated as unknown, as are those of possible cointegrating errors. The individual test statistics have standard null asymptotics and are related to Hausman specification test statistics: when the memory parameter is common to several series, an estimate of this parameter based on the assumption of no cointegration achieves an efficiency improvement over estimates based on individual series, whereas if the series are cointegrated the former estimate is generally inconsistent. However, a computationally simpler but asymptotically equivalent approach, which avoids explicit computation of the “efficient” estimate, is instead pursued here. Two versions of it are initially proposed, followed by one that robustifies to possible inequality between memory parameters of observables. Throughout, a semiparametric approach is pursued, modelling serial dependence only at frequencies near the origin, with the goal of validity under broad circumstances and computational convenience. The main development is in terms of stationary series, but an extension to non-stationary ones is also described. The algorithm for estimating cointegrating rank entails carrying out such tests based on potentially all subsets of two or more of the series, though outcomes of previous tests may render some or all subsequent ones unnecessary. A Monte Carlo study of finite sample performance is included.     

Another look at Swedish business cycles, 1861–1988

The linearity of nine long Swedish macroeconomic time series, whose business cycle properties were discussed by Englund, Persson, and Svensson (1992), is tested and rejected for all but two. Non-linear (STAR) models are estimated, and their properties are investigated. Business cycle frequency variation does not seem to be constant over time for all series; it is difficult to find a ‘Swedish business cycle’. Pairwise Granger non-causality tests are adapted to the STAR case, and non-causality is tested. The results point at strong temporal interactions and indicate that the functional form (linear or STAR) strongly affects the outcome of these tests. Copyright © 1999 John Wiley & Sons, Ltd.     

Size and power of tests of stationarity in highly autocorrelated time series

Tests of stationarity are routinely applied to highly autocorrelated time series. Following Kwiatkowski et al. (J. Econom. 54 (1992) 159), standard stationarity tests employ a rescaling by an estimator of the long-run variance of the (potentially) stationary series. This paper analytically investigates the size and power properties of such tests when the series are strongly autocorrelated in a local-to-unity asymptotic framework. It is shown that the behavior of the tests strongly depends on the long-run variance estimator employed, but is in general highly undesirable. Either the tests fail to control size even for strongly mean reverting series, or they are inconsistent against an integrated process and discriminate only poorly between stationary and integrated processes compared to optimal statistics.     

Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

Testing conditional independence via empirical likelihood

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.     

Tests for a Break in Level when the Order of Integration is Unknown*

In this paper, we consider tests for a break in the level of a series at an unknown point in time. It is often the case that uncertainty exists concerning the order of integration of the series; consequently, we focus on tests that are applicable when the order of integration is not known. The size and power of existing tests are analysed, and a modification to one of the established sets of tests is proposed which offers improved performance in certain circumstances.     

Testing of Fractional Cointegration in Macroeconomic Time Series*

We propose in this article a two‐step testing procedure of fractional cointegration in macroeconomic time series. It is based on Robinson's (Journal of the American Statistical Association, Vol. 89, p. 1420) univariate tests and is similar in spirit to the one proposed by Engle & Granger (Econometrica, Vol. 55, p. 251), testing initially the order of integration of the individual series and then, testing the degree of integration of the residuals from the cointegrating relationship. Finite‐sample critical values of the new tests are computed and Monte Carlo experiments are conducted to examine the size and the power properties of the tests in finite samples. An empirical application, using the same datasets as in Engle & Granger (Econometrica, Vol. 55, p. 251) and Campbell & Shiller (Journal of Political Economy, Vol. 95, p. 1062), is also carried out at the end of the article.     

Unit root tests for panel MTAR model with cross-sectionally dependent error

Unit root tests are constructed for dynamic panels whose component series are momentum threshold autoregressive processes. Gaussian null asymptotics are established for the proposed tests. A Monte–Carlo experiment is conducted to compare finite sample properties of the proposed tests. The tests are illustrated by a real data set.     

Out-of-sample tests of forecasting accuracy: an analysis and review

In evaluations of forecasting accuracy, including forecasting competitions, researchers have paid attention to the selection of time series and to the appropriateness of forecast-error measures. However, they have not formally analyzed choices in the implementation of out-of-sample tests, making it difficult to replicate and compare forecasting accuracy studies. In this paper, I (1) explain the structure of out-of-sample tests, (2) provide guidelines for implementing these tests, and (3) evaluate the adequacy of out-of-sample tests in forecasting software. The issues examined include series-splitting rules, fixed versus rolling origins, updating versus recalibration of model coefficients, fixed versus rolling windows, single versus multiple test periods, diversification through multiple time series, and design characteristics of forecasting competitions. For individual time series, the efficiency and reliability of out-of-sample tests can be improved by employing rolling-origin evaluations, recalibrating coefficients, and using multiple test periods. The results of forecasting competitions would be more generalizable if based upon precisely described groups of time series, in which the series are homogeneous within group and heterogeneous between groups. Few forecasting software programs adequately implement out-of-sample evaluations, especially general statistical packages and spreadsheet add-ins.     

COMMON SEASONAL FEATURES: GLOBAL UNEMPLOYMENT

Seasonal patterns in economic time series are generally examined from a univariate point of view. Using extensions of the unit root literature, important classes of seasonal processes are deterministic, stationary stochastic or mean reverting, and unit root stochastic. Time series tests have been developed for each of these. This paper examines seasonality in a multivariate context. Systems of economic variables can have trends, cycles and unit roots as well as the various types of seasonality. Restrictions such as cointegration and common cycles are here applied also to examine multivariate seasonal behaviour of economic variables. If each of a collection of series has a certain type of seasonality but a linear combination of these series can be found without seasonality, then the seasonal is said to be ‘common’. New tests are developed to determine if seasonal characteristics are common to a set of time series. These tests can be employed in the presence of various other time series structures. The analysis is applied to OECD data on unemployment for the period 1975.1 to 1993.4, and it is found that four diverse countries (Australia, Canada, Japan and USA) not only have common trends in their unemployment, but also have common deterministic seasonal features and a common cycle/stochastic seasonal feature. Such a collection of characteristics were not found in other groups of OECD countries.     

Joint and marginal specification tests for conditional mean and variance models

This article proposes a class of joint and marginal spectral diagnostic tests for parametric conditional means and variances of linear and nonlinear time series models. The use of joint and marginal tests is motivated from the fact that marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new tests are based on a generalized spectral approach and do not need to choose a lag order depending on the sample size or to smooth the data. Moreover, the proposed tests are robust to higher order dependence of unknown form, in particular to conditional skewness and kurtosis. It turns out that the asymptotic null distributions of the new tests depend on the data generating process. Hence, we implement the tests with the assistance of a wild bootstrap procedure. A simulation study compares the finite sample performance of the proposed and competing tests, and shows that our tests can play a valuable role in time series modeling. Finally, an application to the S&P 500 highlights the merits of our approach.     

Correlation testing in time series,spatial and cross-sectional data

We provide a general class of tests for correlation in time series, spatial, spatio-temporal and cross-sectional data. We motivate our focus by reviewing how computational and theoretical difficulties of point estimation mount, as one moves from regularly-spaced time series data, through forms of irregular spacing, and to spatial data of various kinds. A broad class of computationally simple tests is justified. These specialize to Lagrange multiplier tests against parametric departures of various kinds. Their forms are illustrated in case of several models for describing correlation in various kinds of data. The initial focus assumes homoscedasticity, but we also robustify the tests to nonparametric heteroscedasticity.     

The excess co-movement of commodity prices reconsidered

This paper provides an empirical reconsideration of evidence for excess co-movement of commodity prices within the framework of univariate and multivariate GARCH(1, 1) models. Alternative formulations of zero excess co-movement are provided, and corresponding score and likelihood ratio tests are developed. Monthly time series data for two sample periods, 1960–85 and 1974–92, on up to nine commodities are used. In contrast to earlier work, only weak evidence of excess co-movement is found.     

Testing normality: a GMM approach

In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (Proceedings of the Sixth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 2, pp. 583–602) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (Econometrica 63 (1995) 767) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopt is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a heteroskedastic-autocorrelation-consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (Econom. Lett. 6 (1980) 255) and OPG regression tests of Davidson and MacKinnon (Estimation and Inference in Econometrics, Oxford University Press, Oxford). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, two applications to GARCH and realized volatility models are presented.     

Convergence of Prices and Rates of Inflation*

We consider how unit‐root and stationarity tests can be used to study the convergence of prices and rates of inflation. We show how the joint use of these tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whether convergence occurs in relative prices or just in rates of inflation. Special attention is paid to the issue of whether a mean should be extracted in carrying out tests in first differences and whether there is an advantage to adopting a (Dickey–Fuller) unit‐root test based on deviations from the last observation. The asymptotic distribution of this last test statistic is given and Monte Carlo simulation experiments show that the test yields considerable power gains for highly persistent autoregressive processes with ‘relatively large’ initial conditions. The tests are applied to the monthly series of the consumer price index in the Italian regional capitals over the period 1970–2003.     

同期相关面板数据退势单位根检验的小样本性质

本文基于SUR回归将时间序列的两种单位根检验（ADF—GLS检验）推广到面板数据,得到了同期相关面板数据退势单位根检验,称为SUR—ADF—GLS检验。通过蒙特卡洛试验研究发现,SUR—ADF—GLS检验具有良好的小样本性质。并且,SUR—ADF—GLS检验关于面板数据的同期相关性结构存在着较强的“依存性”。     

Seasonal Unit Root Tests with Structural Breaks in Deterministic Seasonality

The seasonal root tests of Hylleberg et al (1990) are extended using the sequential approach of Zivot and Andrews (1992). This paper presents Monte Carlo evidence to support a sequential approach to estimation and critical values are estimated. It is demonstrated that non-stationary data with structurally unstable deterministic seasonality can lead to low power in standard tests for seasonal roots. The sequential tests are applied to US agricultural price data and macroeconomic data and compared with the standard tests. Seasonal roots are rejected in all series.     

Tests for Multiple Breaks in the Trend with Stationary or Integrated Shocks

In this paper, we propose new tests of the presence of multiple breaks in the trend of a univariate time‐series where the number and dates of the breaks are unknown and that are valid in the presence of stationary or unit root shocks. These tests can also be used to sequentially estimate the number of breaks. The behaviour of the proposed tests is studied through Monte Carlo experiments.     

Autocovariance functions of series and of their transforms

We derive a method to link exactly the autocovariance functions of two arbitrary instantaneous transformations of a time series. For example, this is useful when one wants to derive the autocovariance of the logarithm of a series from the known autocovariance of the original series and, more generally, when one wishes to describe the time-series effects of applying a nonlinear transformation to a process whose properties are known. As an illustration, we provide two corollaries and three examples. The first corollary is on the commonly used logarithmic transformation, and is applied to a geometric auto-regressive (AR) process, as well as to a positive moving-average (MA) process. The second corollary is on the tan⁻¹(·) transformation which will turn possibly unstable series into stable ones. As an illustration, we obtain the autocovariance function of the tan⁻¹(·) of an arithmetic AR process. This filter, while always producing a bounded process, preserves the stability/instability distinction of the original series, a feature that can be turned to an advantage in the design of tests. We then present a probabilistic interpretation of the main features of the new autocovariance function. We also provide a mathematical lemma on a general integral which is of independent interest.