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.     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

Asymmetry and nonstationarity for a seasonal time series model

Tests for symmetry and seasonal unit roots are developed for an extended model of Hylleberg et al. (1990. Seasonal integration and cointegration. Journal Econometrics 44, 215–238.) which can represent both partial seasonal unit roots and threshold effects. Methods based on ordinary least squares (OLS) estimation and instrumental variable (IV) estimation are proposed and compared. For adjusting mean functions, ordinary mean adjustment and recursive mean adjustment are both considered. Several tests are constructed from various combination of estimation schemes and mean adjustment schemes. Among the tests, the tests based on IV-estimation are recommended because they have very simple limiting null distributions and have finite sample power properties comparable to those based on the OLSE. The recommended tests are applied to a US unemployment rate data set and find evidences for both nonstationarities associated with zero frequency and threshold effects.     

Cross-sectional dependence robust block bootstrap panel unit root tests

In this paper we consider the issue of unit root testing in cross-sectionally dependent panels. We consider panels that may be characterized by various forms of cross-sectional dependence including (but not exclusive to) the popular common factor framework. We consider block bootstrap versions of the group-mean (Im et al., 2003) and the pooled (Levin et al., 2002) unit root coefficient DF tests for panel data, originally proposed for a setting of no cross-sectional dependence beyond a common time effect. The tests, suited for testing for unit roots in the observed data, can be easily implemented as no specification or estimation of the dependence structure is required. Asymptotic properties of the tests are derived for T going to infinity and N finite. Asymptotic validity of the bootstrap tests is established in very general settings, including the presence of common factors and cointegration across units. Properties under the alternative hypothesis are also considered. In a Monte Carlo simulation, the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests. The power properties of the bootstrap tests appear to be similar to those of the asymptotic tests.     

Testing for Seasonal Unit Roots in Monthly Panels of Time Series*

We consider the problem of testing for seasonal unit roots in monthly panel data. To this aim, we generalize the quarterly cross‐sectionally augmented Hylleberg–Engle–Granger–Yoo (CHEGY) test to the monthly case. This parametric test is contrasted with a new non‐parametric test, which is the panel counterpart to the univariate record unit–root seasonal (RURS) test that relies on counting extrema in time series. All methods are applied to an empirical data set on tourism in Austrian provinces. The power properties of the tests are evaluated in simulation experiments that are tuned to the tourism data.     

Taking a new contour: A novel approach to panel unit root tests

The paper introduces a novel approach to testing for unit roots in panels, which takes a new contour that is drawn along the line given by the equi-squared-sum instead of the traditional one given by the equi-sample-size. We show in the paper that the distributions of the unit root tests are asymptotically normal along the new contour under both the null and the local-to-unity alternatives. Subsequently, we demonstrate that this startling finding may be exploited constructively to invent tools and methodologies for effective inferences in panel unit root models. Simulations show that our approach works quite well in finite samples.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.     

Inference in a nearly integrated autoregressive model with nonnormal innovations

Robust tests and estimators based on nonnormal quasi-likelihood functions are developed for autoregressive models with near unit root. Asymptotic power functions and power envelopes are derived for point-optimal tests of a unit root when the likelihood is correctly specified. The shapes of these power functions are found to be sensitive to the extent of nonnormality in the innovations. Power loss resulting from using least-squares unit-root tests in the presence of thick-tailed innovations appears to be greater than in stationary models.     

Panel Unit‐root Tests for Cross‐sectionally Correlated Panels: A Monte Carlo Comparison*

This paper deals with the finite‐sample performance of a set of unit‐root tests for cross‐correlated panels. Most of the available macroeconomic time series cover short time periods. The lack of information, in terms of time observations, implies that univariate tests are not powerful enough to reject the null of a unit‐root while panel tests, by exploiting the large number of cross‐sectional units, have been shown to be a promising way of increasing the power of unit‐root tests. We investigate the finite sample properties of recently proposed panel unit‐root tests for cross‐sectionally correlated panels. Specifically, the size and power of Choi's [Econometric Theory and Practice: Frontiers of Analysis and Applied Research: Essays in Honor of Peter C. B. Phillips, Cambridge University Press, Cambridge (2001)], Bai and Ng's [Econometrica (2004), Vol. 72, p. 1127], Moon and Perron's [Journal of Econometrics (2004), Vol. 122, p. 81], and Phillips and Sul's [Econometrics Journal (2003), Vol. 6, p. 217] tests are analysed by a Monte Carlo simulation study. In synthesis, Moon and Perron's tests show good size and power for different values of T and N, and model specifications. Focusing on Bai and Ng's procedure, the simulation study highlights that the pooled Dickey–Fuller generalized least squares test provides higher power than the pooled augmented Dickey–Fuller test for the analysis of non‐stationary properties of the idiosyncratic components. Choi's tests are strongly oversized when the common factor influences the cross‐sectional units heterogeneously.     

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 non-nested conditional moment restrictions using unconditional empirical likelihood

We propose non-nested hypothesis tests for conditional moment restriction models based on the method of generalized empirical likelihood (GEL). By utilizing the implied GEL probabilities from a sequence of unconditional moment restrictions that contains equivalent information of the conditional moment restrictions, we construct Kolmogorov–Smirnov and Cramér–von Mises type moment encompassing tests. Advantages of our tests over Otsu and Whang’s (2011) tests are: (i) they are free from smoothing parameters, (ii) they can be applied to weakly dependent data, and (iii) they allow non-smooth moment functions. We derive the null distributions, validity of a bootstrap procedure, and local and global power properties of our tests. The simulation results show that our tests have reasonable size and power performance in finite samples.     

Generalizing weak instrument robust IV statistics towards multiple parameters,unrestricted covariance matrices and identification statistics

We generalize the weak instrument robust score or Lagrange multiplier and likelihood ratio instrumental variables (IV) statistics towards multiple parameters and a general covariance matrix so they can be used in the generalized method of moments (GMM). The GMM extension of Moreira's [2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio statistic towards GMM preserves its expression except that it becomes conditional on a statistic that tests the rank of a matrix. We analyze the spurious power decline of Kleibergen's [2002. Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803, 2005. Testing parameters in GMM without assuming that they are identified. Econometrica 73, 1103–1124] score statistic and show that an independent misspecification pre-test overcomes it. We construct identification statistics that reflect if the confidence sets of the parameters are bounded. A power study and the possible shapes of confidence sets illustrate the analysis.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.     

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.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.     

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.     

Performance of conditional Wald tests in IV regression with weak instruments

We compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression. Following Moreira [2003, A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048], all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the (unknown) concentration parameter, so these conditional tests are asymptotically valid under weak instrument asymptotics. Four of the tests are based on k-class Wald statistics (two-stage least squares, LIML, Fuller's [Some properties of a modification of the limited information estimator. Econometrica 45, 939–953], and bias-adjusted TSLS); the fifth is Moreira's (2003) conditional likelihood ratio (CLR) test. The heretofore unstudied conditional Wald (CW) tests are found to perform poorly, compared to the CLR test: in many cases, the CW tests have almost no power against a wide range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for the computation of the asymptotic conditional p-value of the CLR test.     

Estimation for spatial dynamic panel data with fixed effects: The case of spatial cointegration

Yu et al. (2008) establish asymptotic properties of quasi-maximum likelihood estimators for a stable spatial dynamic panel model with fixed effects when both the number of individuals n and the number of time periods T are large. This paper investigates unstable cases where there are unit roots generated by temporal and spatial correlations. We focus on the spatial cointegration model where some eigenvalues of the data generating process are equal to 1 and the outcomes of spatial units are cointegrated as in a vector autoregressive system. The asymptotics of the QML estimators are developed by reparameterization, and bias correction for the estimators is proposed. We also consider the 2SLS and GMM estimations when T could be small.     

The Flexible Fourier Form and Local Generalised Least Squares De‐trended Unit Root Tests*

In two recent papers Enders and Lee (2009) and Becker, Enders and Lee (2006) provide Lagrange multiplier and ordinary least squares de‐trended unit root tests, and stationarity tests, respectively, which incorporate a Fourier approximation element in the deterministic component. Such an approach can prove useful in providing robustness against a variety of breaks in the deterministic trend function of unknown form and number. In this article, we generalize the unit root testing procedure based on local generalized least squares (GLS) de‐trending proposed by Elliott, Rothenberg and Stock (1996) to allow for a Fourier approximation to the unknown deterministic component in the same way. We show that the resulting unit root tests possess good finite sample size and power properties and the test statistics have stable non‐standard distributions, despite the curious result that their limiting null distributions exhibit asymptotic rank deficiency.     

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.