Tests for cointegration with two unknown regime shifts with an application to financial market integration

It is widely agreed in empirical studies that allowing for potential structural change in economic processes is an important issue. In existing literature, tests for cointegration between time series data allow for one regime shift. This paper extends three residual-based test statistics for cointegration to the cases that take into account two possible regime shifts. The timing of each shift is unknown a priori and it is determined endogenously. The distributions of the tests are non-standard. We generate new critical values via simulation methods. The size and power properties of these test statistics are evaluated through Monte Carlo simulations, which show the tests have small size distortions and very good power properties. The test methods introduced in this paper are applied to determine whether the financial markets in the US and the UK are integrated.      

The effect of the GARCH(1, 1) on autocorrelation tests in dynamic systems of equations

Using Monte Carlo methods, the properties of systemwise generalizations of the Breusch–Godfrey test for autocorrelated errors are studied when there are some kinds of GARCH effects among the errors. The analysis, regarding the size of the test, reveals that the GARCH have considerable effects of the properties of the test regarding the size, especially in large systems of equations. The corrected LR tests, however, have been shown to perform satisfactorily in small systems when the errors are white noise or they have low GARCH effects, whilst the commonly used TR2 test behaves badly even in single equations. All tests perform badly, however, when the number of equations increases and the GARCH effect is strong. As regards the power of the test, the GARCH was not found to have any significant effects on the power properties of the test.     

A guide to the computation of stationarity tests

In this paper we compare ways of computing stationarity tests. We show that whereas some of the procedures recommended lead to inconsistency of the tests, it is still possible to compute a test with good properties in finite sample in terms of empirical size and power. The guidance suggested in the paper is illustrated by testing for the purchasing power parity hypothesis in some developed countries.

Testing spatial regression models under nonregular conditions

In time series context, estimation and testing issues with autoregressive and moving average (ARMA) models are well understood. Similar issues in the context of spatial ARMA models for the disturbance of the regression, however, remain largely unexplored. In this paper, we discuss the problems of testing no spatial dependence in the disturbances against the alternative of spatial ARMA process incorporating the possible presence of spatial dependence in the dependent variable. The problems of conducting such a test are twofold. First, under the null hypothesis, the nuisance parameter is not identified, resulting in a singular information matrix (IM), which is a nonregular case in statistical inference. To take account of singular IM, we follow Davies (Biometrika 64(2):247–254, 1977; Biometrika 74(1):33–43, 1987) and propose a test procedure based on the supremum of the Rao score test statistic. Second, the possible presence of spatial lag dependence will have adverse effect on the performance of the test. Using the general test procedure of Bera and Yoon (Econom Theory 9:649–658, 1993) under local misspecification, we avoid the explicit estimation of the spatial autoregressive parameter. Thus our suggested tests are entirely based on ordinary least squares estimation. Tests suggested here can be viewed as a generalization of Anselin et al. (Reg Sci Urban Econ 26:77–104, 1996). We conduct Monte Carlo simulations to investigate the finite sample properties of the proposed tests. Simulation results show that our tests have good finite sample properties both in terms of size and power, compared to other tests in the literature. We also illustrate the applications of our tests through several data sets.     

Testing for panel unit roots in the presence of spatial dependency

In this article, the size and power properties of the Common-factor Im, Pesaran and Shin (CIPS), Wald (W), Likelihood Ratio (LR) and Lagrange Multiplier (LM) tests are investigated when the error term follows a spatial error model. In this study, the results from the Monte Carlo simulations, first, show that the CIPS test over-estimates the nominal size. Second, the simulation results show that the empirical size of the W test approaches the nominal size quickly, while the LR and LM tests underestimate the null hypothesis in both small and moderate sample sizes. Finally, the results also show that even though the LM and LR tests under-reject the true-null hypothesis they have higher power than the W test.     

Detecting periods of exuberance: A look at the role of aggregation with an application to house prices

The recently developed SADF and GSADF unit root tests of Phillips and Yu (2011) and Phillips et al. (2015a,b) have become popular in the literature for detecting exuberance in asset prices. In this paper, we examine through simulation experiments the effect of cross-sectional aggregation on the power properties of these tests. The simulation design considered is based on simulated data and actual housing data for both U.S. metropolitan areas and international housing markets and thus allows us to draw conclusions for different levels of aggregation. Our findings suggest that aggregation lowers the power of both the SADF and GSADF tests. The effect, however, is much larger for the SADF test. We also provide evidence that tests based on panel data techniques, namely the panel GSADF test recently proposed by Pavlidis et al. (2016), can perform substantially better than univariate tests applied to aggregated series. Furthermore, we also illustrate the date-stamping procedure under the univariate/panel GSADF procedure uncovering novel evidence on the role of interest rates and policy uncertainty as factors explaining episodes of widespread mildly explosive dynamics in housing markets.     

South African stock return predictability in the context data mining: The role of financial variables and international stock returns

In this paper, we examine the predictive ability, both in-sample and the out-of-sample, for South African stock returns using a number of financial variables, based on monthly data with an in-sample period covering 1990:01 to 1996:12 and the out-of-sample period of 1997:01 to 2010:04. We use the t-statistic corresponding to the slope coefficient in a predictive regression model for in-sample predictions, while for the out-of-sample, the MSE-F and the ENC-NEW tests statistics with good power properties were utilised. To guard against data mining, a bootstrap procedure was employed for calculating the critical values of both the in-sample and out-of-sample test statistics. Furthermore, we use a procedure that combines in-sample general-to-specific model selection with out-of-sample tests of predictive ability to further analyse the predictive power of each financial variable. Our results show that, for the in-sample test statistic, only the stock returns for our major trading partners have predictive power at certain short and long run horizons. For the out-of-sample tests, the Treasury bill rate and the term spread together with the stock returns for our major trading partners show predictive power both at short and long run horizons. When accounting for data mining, the maximal out-of-sample test statistics become insignificant from 6-months onward suggesting that the evidence of the out-of-sample predictability at longer horizons is due to data mining. The general-to-specific model shows that valuation ratios contain very useful information that explains the behaviour of stock returns, despite their inability to predict stock return at any horizon. The model also highlights the role of multiple variables in predicting stock returns at medium- to long run horizons.     

A robust test for multivariate normality

The size of the Jarque-Bera test for multivariate normality can be severely distorted in small samples. An alternative test procedure, that turns out to have good size and power properties, is suggested.     

Structural break,nonlinearity and asymmetry: a re-examination of PPP proposition

In this study, we examine the validity of the PPP proposition for 28 European countries. For this purpose, we propose a new unit root test procedure that allows for both gradual structural breaks and asymmetric nonlinear adjustment towards the equilibrium level. Small-sample properties of the new tests are examined through Monte-Carlo simulations. The simulation results suggest that the new tests have satisfactory size and power properties. We then apply these new tests along with other unit root tests to examine stationarity properties of real exchange rate series of the sample countries. Our tests reject the null of unit root in more cases when compared to alternative tests. Overall, we find that the PPP proposition holds in majority of the European countries examined in this article.     

MACROECONOMETRIC MODEL EVALUATION,WITH SPECIAL REFERENCE TO THE NIF88 MODEL*

This paper has argued that a mixture of procedures is required for the evaluation of a macroeconomic model. Both individual equations and overall model properties are involved, while both formal tests and an informal understanding of the model are needed. The procedures advocated reflect the need for evaluation to be informative and manageable. The bulk of the statistical testing is best conducted at the single equation stage. Evaluation of single equations involves:

• (i) employing a battery of diagnostic tests;
• (ii) specification and stability tests should be designed, where possible, to enhance the power of the test conducted, including by making use of external information;
• (iii) procedures designed to detect influence of particular observations can provide
• (iv) comparison with other information, both from other empirical investigations and from theoretical priors, is necessary to ensure sound results.

     

A panel stationarity test with gradual structural shifts: Re-investigate the international commodity price shocks

This paper proposes a simple panel stationarity test which takes into account structural shifts and cross-section dependency. Structural shifts are modelled as gradual/smooth process with a Fourier approximation. The so-called Fourier panel stationarity test has a standard normal distribution. The Monte Carlo simulations indicate that (i) if the error terms are i.i.d, the test shows good size and power properties even in small samples; and (ii) if the error terms are serially correlated, the test has reasonable size and high power. We re-examine the behavior of the international commodity prices and find out an evidence on the persistence of shocks.     

Model selection for nonlinear time series

We investigate the finite-sample performance of model selection criteria for local linear regression by simulation. Similarly to linear regression, the penalization term depends on the number of parameters of the model. In the context of nonparametric regression, we use a suitable quantity to account for the Equivalent Number of Parameters as previously suggested in the literature. We consider the following criteria: Rice T, FPE, AIC, Corrected AIC and GCV. To make results comparable with other data-driven selection criteria we consider also Leave-Out CV. We show that the properties of the penalization schemes are very different for some linear and nonlinear models. Finally, we set up a goodness-of-fit test for linearity based on bootstrap methods. The test has correct size and very high power against the alternatives investigated. Application of the methods proposed to macroeconomic and financial time series shows that there is evidence of nonlinearity.First version received: September 2002/Final version received : October 2003I would like to thank Cees Diks, Cars Hommes and an anonymous referee for useful comments that significantly improved the paper.     

More powerful threshold cointegration tests

Threshold cointegration tests have made a big splash in the literature by allowing for asymmetric adjustment in linear cointegration tests. This paper contributes to this literature by proposing new tests to improve the power of the conventional threshold cointegration tests. The new tests intuitively resolve one of the possible reasons that attribute to the low power of existing threshold cointegration tests and are easy to implement since they do not require any additional information outside of the system. Our simulation results show that the proposed tests improve the power of the existing threshold cointegration tests, especially as the signal-to-noise ratio increases, in contrast to other considered procedures. The efficiency gains are achieved regardless of sample size, the number of cointegrated variables, and the types of threshold specifications. The newly developed tests are applied to examine long-run purchasing power parity in the Pacific nations. In contrast to conventional cointegration tests, the proposed tests found long-run PPP holds in 5 out of 7 countries with appropriate asymmetric adjustments.     

The export-output growth nexus in Japan: a bootstrap rolling window approach

The purpose of this article is to examine the export–output nexus in Japan by taking into account the time variation in the causal link with bootstrap Granger non-causality test and rolling estimation. The data used cover the seasonally adjusted real export and real Gross Domestic Product (GDP) for the 1957:1–2009:1 period. Standard Granger causality tests indicate no causality between export and real GDP series. On the contrary, full sample-modified Granger causality tests based on bootstrap, which are applicable irrespective of integration–cointegration properties of the data, indicate a bi-directional causal link between exports and real GDP. Accordingly, export growth should be an important factor behind Japan’s high-economic growth in the last three decades. Using parameter stability tests, we show that these results are not uniform for different sample periods and results vary due to structural changes. Using bootstrap rolling window estimation, we find that there is a positive bi-directional predictive power from the mid 1970s to the late-1980s between the series, while from the late 1990s to 2009 there is a positive predictive power only from export growth to output growth.     

Time-Series Based Empirical Assessment of Random Urban Growth: New Evidence from France

Modern urban growth literature frequently uses unit-root tests in order to check the empirical relevance of Gibrat’s law of random growth. The contradictory nature of the test results provided by this literature is most likely linked to the low power of unit-root tests. To address this problem, we apply unit-root testing to a large-sized sample of high-quality French census data covering an exceptionally long time span of more than two centuries. We add subsequent cointegration tests in order to detect the possible presence of cointegrated random growth, which may reflect the fact that cities with a similar economic structure react fairly similarly to exogenous growth shocks. According to the test results, the random growth hypothesis cannot be rejected for a very large majority of the tested French cities; on the other hand, the null hypothesis of absence of cointegration cannot be rejected in more than 95% of the cases. Our findings therefore provide empirical support for non-cointegrated random growth.

     

Bootstrap and Pretest J-Type Non-Nested Tests for Orthogonal Regression Models: Some Monte Carlo Evidence

This paper compares the size and power of two J-type tests for weakly correlated or nearly orthogonal non-nested regression models: a bootstrap and a pretest test. The latter seems to outperform the former in terms of its size characteristics, especially when the alternative model has more non-nested regressors and the orthogonality between the two sets of regressors is severe. The bootstrap test does better in terms of power.

     

Bootstrap testing for cointegration of international commodity prices

This paper investigates cointegration with respect to nine commodity groups traded on international markets. Nonparametric bootstrapping is utilized in the testing procedure. Of the 21 pairs of price series, investigated here, for 13 the no-cointegration null hypothesis is rejected in favour for the cointegration of the series. In addition to five out of the remaining eight cases that were not cointegrated, a plausible explanation is the prevailing trade policy. Thus a great majority of the institutionally nonregulated cases turn out to get empirical support for being cointegrated. An important statistical finding is that the augmented Dickey-Fuller test for cointegration (CRADF) generally yields p-values that are close to the p-values obtained by the bootstrap testing. But once they differ substantially, it is usually an indication of irregular periods (e.g. structural changes) in the series. The paper conducts also a Monte Carlo simulation experiment to investigate the power and size properties of the tests. Generally the results indicate that the test procedures have pretty low power in small samples. Bootstrapping improves the testing somewhat by leading consistently to a bit more powerful inference.     

TESTING FOR STATIONARITY IN HETEROGENEOUS PANEL DATA IN THE CASE OF MODEL MISSPECIFICATION

This paper investigates the performance of the tests proposed by Hadri and by Hadri and Larsson for testing for stationarity in heterogeneous panel data under model misspecification. The panel tests are based on the well known KPSS test (cf. Kwiatkowski et al.) which considers two models: stationarity around a deterministic level and stationarity around a deterministic trend. There is no study, as far as we know, on the statistical properties of the test when the wrong model is used. We also consider the case of the simultaneous presence of the two types of models in a panel. We employ two asymptotics: joint asymptotic, T, N →∞ simultaneously, and T fixed and N allowed to grow indefinitely. We use Monte Carlo experiments to investigate the effects of misspecification in sample sizes usually used in practice. The results indicate that the assumption that T is fixed rather than asymptotic leads to tests that have less size distortions, particularly for relatively small T with large N panels (micro‐panels) than the tests derived under the joint asymptotics. We also find that choosing a deterministic trend when a deterministic level is true does not significantly affect the properties of the test. But, choosing a deterministic level when a deterministic trend is true leads to extreme over‐rejections. Therefore, when unsure about which model has generated the data, it is suggested to use the model with a trend. We also propose a new statistic for testing for stationarity in mixed panel data where the mixture is known. The performance of this new test is very good for both cases of T asymptotic and T fixed. The statistic for T asymptotic is slightly undersized when T is very small (≤10).     

Combination of “combinations of <Emphasis Type="Italic">p</Emphasis> values”

We investigate the impact of an uncertain number of false individual null hypotheses on commonly used p value combination methods. Under such uncertainty, these methods perform quite differently and often yield conflicting results. Consequently, we develop a combination of “combinations of p values” (CCP) test aimed at maintaining good power properties across such uncertainty. The CCP test is based on a simple union–intersection principle that exploits the weak correspondence between two underlying p value combination methods. Monte Carlo simulations show that the CCP test controls size and closely tracks the power of the best individual methods. We empirically apply the CCP test to explore the stationarity in real exchange rates and the information rigidity in inflation and output growth forecasts.     

Nonparametric Tests of Differences in Medians: Comparison of the Wilcoxon–Mann–Whitney and Robust Rank-Order Tests

The nonparametric Wilcoxon–Mann–Whitney test is commonly used by experimental economists for detecting differences in central tendency between two samples. This test is only theoretically appropriate under certain assumptions concerning the population distributions from which the samples are drawn, and is often used in cases where it is unclear whether these assumptions hold, and even when they clearly do not hold. Fligner and Pollicello's (1981, Journal of the American Statistical Association. 76, 162–168) robust rank-order test is a modification of the Wilcoxon–Mann–Whitney test, designed to be appropriate in more situations than Wilcoxon–Mann–Whitney. This paper uses simulations to compare the performance of the two tests under a variety of distributional assumptions. The results are mixed. The robust rank-order test tends to yield too many false positive results for medium-sized samples, but this liberalness is relatively invariant across distributional assumptions, and seems to be due to a deficiency of the normal approximation to its test statistic's distribution, rather than the test itself. The performance of the Wilcoxon–Mann–Whitney test varies hugely, depending on the distributional assumptions; in some cases, it is conservative, in others, extremely liberal. The tests have roughly similar power. Overall, the robust rank-order test performs better than Wilcoxon–Mann–Whitney, though when critical values for the robust rank-order test are not available, so that the normal approximation must be used, their relative performance depends on the underlying distributions, the sample sizes, and the level of significance used.