Cointegration Testing in Panels with Common Factors*

Panel unit‐root and no‐cointegration tests that rely on cross‐sectional independence of the panel unit experience severe size distortions when this assumption is violated, as has, for example, been shown by Banerjee, Marcellino and Osbat [Econometrics Journal (2004), Vol. 7, pp. 322–340; Empirical Economics (2005), Vol. 30, pp. 77–91] via Monte Carlo simulations. Several studies have recently addressed this issue for panel unit‐root tests using a common factor structure to model the cross‐sectional dependence, but not much work has been done yet for panel no‐cointegration tests. This paper proposes a model for panel no‐cointegration using an unobserved common factor structure, following the study by Bai and Ng [Econometrica (2004), Vol. 72, pp. 1127–1177] for panel unit roots. We distinguish two important cases: (i) the case when the non‐stationarity in the data is driven by a reduced number of common stochastic trends, and (ii) the case where we have common and idiosyncratic stochastic trends present in the data. We discuss the homogeneity restrictions on the cointegrating vectors resulting from the presence of common factor cointegration. Furthermore, we study the asymptotic behaviour of some existing residual‐based panel no‐cointegration tests, as suggested by Kao [Journal of Econometrics (1999), Vol. 90, pp. 1–44] and Pedroni [Econometric Theory (2004a), Vol. 20, pp. 597–625]. Under the data‐generating processes (DGP) used, the test statistics are no longer asymptotically normal, and convergence occurs at rate T rather than as for independent panels. We then examine the possibilities of testing for various forms of no‐cointegration by extracting the common factors and individual components from the observed data directly and then testing for no‐cointegration using residual‐based panel tests applied to the defactored data.     

Testing for the cointegration rank when some cointegrating directions are changing

We develop some tests for characterizing the cointegration space of a cointegrated vector autoregressive model when its long-run parameters are modified by a structural break at a known date. We first consider the case in which the break does not affect the loading factors and second the more general one in which all long-run parameters change. For each configuration, we design procedures to test for the cointegration rank as for the number of directions which are changing between the two regimes. For the simplest case, the cointegration rank test is also extended to the case of an unknown date of shift.     

Testing for Panel Cointegration with Multiple Structural Breaks*

This paper proposes a Lagrange multiplier (LM) test for the null hypothesis of cointegration that allows for the possibility of multiple structural breaks in both the level and trend of a cointegrated panel regression. The test is general enough to allow for endogenous regressors, serial correlation and an unknown number of breaks that may be located at different dates for different individuals. We derive the limiting distribution of the test and conduct a small Monte Carlo study to investigate its finite sample properties. In our empirical application to the solvency of the current account, we find evidence of cointegration between saving and investment once a level break is accommodated.     

A Wald test for the cointegration rank in nonstationary fractional systems

This paper develops new methods for determining the cointegration rank in a nonstationary fractionally integrated system, extending univariate optimal methods for testing the degree of integration. We propose a simple Wald test based on the singular value decomposition of the unrestricted estimate of the long run multiplier matrix. When the “strength” of the cointegrating relationship is less than 1/2, the test statistic has a standard asymptotic distribution, like Lagrange Multiplier tests exploiting local properties. We consider the behavior of our test under estimation of short run parameters and local alternatives. We compare our procedure with other cointegration tests based on different principles and find that the new method has better properties in a range of situations by using information on the alternative obtained through a preliminary estimate of the cointegration strength.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

Improved likelihood ratio tests for cointegration rank in the VAR model

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but as usual the asymptotic results do not require normally distributed innovations. Our tests differ from existing tests in two respects. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott et al. (1996). Second, our tests incorporate a “sign” restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.     

Testing for Error Correction in Panel Data*

This paper proposes new error correction‐based cointegration tests for panel data. The limiting distributions of the tests are derived and critical values provided. Our simulation results suggest that the tests have good small‐sample properties with small size distortions and high power relative to other popular residual‐based panel cointegration tests. In our empirical application, we present evidence suggesting that international healthcare expenditures and GDP are cointegrated once the possibility of an invalid common factor restriction has been accounted for.     

Likelihood based testing for no fractional cointegration

We consider two likelihood ratio tests, the so-called maximum eigenvalue and trace tests, for the null of no cointegration when fractional cointegration is allowed under the alternative, which is a first step to generalize the so-called Johansen’s procedure to the fractional cointegration case. The standard cointegration analysis only considers the assumption that deviations from equilibrium can be integrated of order zero, which is very restrictive in many cases and may imply an important loss of power in the fractional case. We consider the alternative hypotheses with equilibrium deviations that can be mean reverting with order of integration possibly greater than zero. Moreover, the degree of fractional cointegration is not assumed to be known, and the asymptotic null distribution of both tests is found when considering an interval of possible values. The power of the proposed tests under fractional alternatives and size accuracy provided by the asymptotic distribution in finite samples are investigated.     

Testing for a Change in Persistence in the Presence of a Volatility Shift*

We consider the impact of a break in the innovation volatility process on ratio‐based persistence change tests. We demonstrate that the ratio statistics used do not have pivotal limiting null distributions and that the associated tests display a considerable degree of size distortion with size approaching unity in some cases. In practice, therefore, on the basis of these tests the practitioner will face difficulty in discriminating between persistence change processes and processes which display a simple volatility break. A wild bootstrap‐based solution to the identified inference problem is proposed and is shown to work well in practice.     

Multiple Testing for No Cointegration under Nonstationary Volatility

With cointegration tests often being oversized under time‐varying error variance, it is possible, if not likely, to confuse error variance non‐stationarity with cointegration. This paper takes an instrumental variable (IV) approach to establish individual‐unit test statistics for no cointegration that are robust to variance non‐stationarity. The sign of a fitted departure from long‐run equilibrium is used as an instrument when estimating an error‐correction model. The resulting IV‐based test is shown to follow a chi‐square limiting null distribution irrespective of the variance pattern of the data‐generating process. In spite of this, the test proposed here has, unlike previous work relying on instrumental variables, competitive local power against sequences of local alternatives in 1/T‐neighbourhoods of the null. The standard limiting null distribution motivates, using the single‐unit tests in a multiple testing approach for cointegration in multi‐country data sets by combining P‐values from individual units. Simulations suggest good performance of the single‐unit and multiple testing procedures under various plausible designs of cross‐sectional correlation and cross‐unit cointegration in the data. An application to the equilibrium relationship between short‐ and long‐term interest rates illustrates the dramatic differences between results of robust and non‐robust tests.     

Forecasting with equilibrium-correction models during structural breaks

When location shifts occur, cointegration-based equilibrium-correction models (EqCMs) face forecasting problems. We consider alleviating such forecast failure by updating, intercept corrections, differencing, and estimating the future progress of an ‘internal’ break. Updating leads to a loss of cointegration when an EqCM suffers an equilibrium-mean shift, but helps when collinearities are changed by an ‘external’ break with the EqCM staying constant. Both mechanistic corrections help compared to retaining a pre-break estimated model, but an estimated model of the break process could outperform. We apply the approaches to EqCMs for UK M1, compared with updating a learning function as the break evolves.     

Confidence sets for the date of a single break in linear time series regressions

This paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression. We establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude. Given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon, we proceed to develop an appropriate alternative. We suggest constructing confidence sets by inverting a sequence of tests. Each of the tests maintains a specific break date under the null hypothesis, and rejects when a break occurs elsewhere. By inverting a certain variant of a locally best invariant test, we ensure that the asymptotic critical value does not depend on the maintained break date. A valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number.     

Alternative bootstrap procedures for testing cointegration in fractionally integrated processes

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The investigation focuses on (a) choice of statistic, (b) use of bias correction techniques, and (c) designing the simulation of the null hypothesis. Three residual-based tests are considered, two of the null hypothesis of non-cointegration, the third of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments to throw light on the relative roles of issues (a)–(c) in test performance.     

Residual based tests for cointegration in dependent panels

In the paper, we propose residual based tests for cointegration in general panels with cross-sectional dependency, endogeneity and various heterogeneities. The residuals are obtained from the usual least squares estimation of the postulated cointegrating relationships from each individual unit, and the nonlinear IV panel unit root testing procedure is applied to the panels of the fitted residuals using as instruments the nonlinear transformations of the adaptively   fitted lagged residuals. The t$t$-ratio, based on the nonlinear IV estimator, is then constructed to test for unit root in the fitted residuals for each cross-section. We show that such nonlinear IV t$t$-ratios are asymptotically normal and cross-sectionally independent under the null hypothesis of no cointegration. The average or the minimum of the IVt$t$-ratios can, therefore, be used to test for the null of a fully non-cointegrated panel against the alternative of a mixed panel, i.e., a panel with only some cointegrated units. We also consider the maximum of the IV t$t$-ratios to test for a mixed panel against a fully cointegrated panel. The critical values of the minimum, maximum as well as the average tests are easily obtained from the standard normal distribution function. Our simulation results indicate that the residual based tests for cointegration perform quite well in finite samples.     

Testing for structural stability in the whole sample

The paper examines a Lagrange Multiplier type test for the constancy of the parameter in general models with dependent data without imposing any artificial choice of the possible location of the break. In order to prove the asymptotic behaviour of the test, we extend a strong approximation result for partial sums of a sequence of random variables. We also present a Monte-Carlo experiment to examine the finite sample performance of the test and how it compares with tests which assume some knowledge of the possible location of the break.     

Unit root testing under a local break in trend

Recent approaches to testing for a unit root when uncertainty exists over the presence and timing of a trend break employ break detection methods, so that a with-break unit root test is used only if a break is detected by some auxiliary statistic. While these methods achieve near asymptotic efficiency in both fixed trend break and no trend break environments, in finite samples pronounced “valleys” in the power functions of the tests (when mapped as functions of the break magnitude) are observed, with power initially high for very small breaks, then decreasing as the break magnitude increases, before increasing again. In response to this problem, we propose two practical solutions, based either on the use of a with-break unit root test but with adaptive critical values, or on a union of rejections principle taken across with-break and without-break unit root tests. These new procedures are shown to offer improved reliability in terms of finite sample power. We also develop local limiting distribution theory for both the extant and the newly proposed unit root statistics, treating the trend break magnitude as local-to-zero. We show that this framework allows the asymptotic analysis to closely approximate the finite sample power valley phenomenon, thereby providing useful analytical insights.     

Panel Stationarity Test with Structural Breaks*

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.     

Testing cointegration relationship in a semiparametric varying coefficient model

In this paper, we develop two cointegration tests for two varying coefficient cointegration regression models, respectively. Our test statistics are residual based. We derive the asymptotic distributions of test statistics under the null hypothesis of cointegration and show that they are consistent against the alternative hypotheses. We also propose a wild bootstrap procedure companioned with the continuous moving block bootstrap method proposed in  Paparoditis and Politis (2001) and  Phillips (2010) to rectify severe distortions found in simulations when the sample size is small. We apply the proposed test statistic to examine the purchasing power parity (PPP) hypothesis between the US and Canada. In contrast to the existing results from linear cointegration tests, our varying coefficient cointegration test does not reject that PPP holds between the US and Canada.     

Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.     

Micro versus macro cointegration in heterogeneous panels

We consider the issue of cross-sectional aggregation in nonstationary and heterogeneous panels where each unit cointegrates. We derive asymptotic properties of the aggregate estimate, and necessary and sufficient conditions for cointegration to hold in the aggregate relationship. We then analyze the case when cointegration does not carry through the aggregation process, and we investigate whether the violation of the formal conditions for perfect aggregation can still lead to an aggregate equation that is observationally equivalent to a cointegrated relationship. We derive a measure of the degree of noncointegration of the aggregate relationship and we explore its asymptotic properties. We propose a valid bootstrap approximation of the test. A Monte Carlo exercise evaluates size and power properties of the bootstrap test.