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This article discusses cointegration tests for nonlinear cointegration in the presence of variance breaks in the errors. We build on approaches of Cavaliere and Taylor (2006, Journal of Time Series Analysis) for heteroskedastic cointegration tests and of Choi and Saikkonen (2010, Econometric Theory) for nonlinear cointegration tests. We propose a bootstrap test and prove its consistency. A Monte Carlo study shows the approach to have appealing finite sample properties and to work better than an approach using subresiduals. We provide an empirical application to the environmental Kuznets curves (EKC), finding that the cointegration tests do not reject the EKC hypothesis in most cases.

In many forecast evaluation applications, standard tests (e.g., Diebold and Mariano, 1995) as well as tests allowing for time-variation in relative forecast ability (e.g., Giacomini and Rossi, 2010) build on heteroskedasticity-and-autocorrelation consistent (HAC) covariance estimators. Yet, the finite-sample performance of these asymptotics is often poor. "Fixed-b" asymptotics (Kiefer and Vogelsang, 2005), used to account for long-run variance estimation, improve finitesample performance under homoskedasticity, but lose asymptotic pivotality under time-varying volatility. Moreover, loss of pivotality due to time-varying volatility is found in the standard HAC framework in certain cases as well. We prove a wild bootstrap implementation to restore asymptotically pivotal inference for the above and new CUSUM- and Cramér-von Mises based tests in a fairly general setup, allowing for estimation uncertainty from either a rolling window or a recursive approach when fixed-b asymptotics are adopted to achieve good finite-sample performance. We then investigate the (time-varying) performance of professional forecasters relative to naive no-change and model-based predictions in real-time. We exploit the Survey of Professional Forecasters (SPF) database and analyze nowcasts and forecasts at different horizons for output and inflation. We find that not accounting for time-varying volatility seriously affects outcomes of tests for equal forecast ability: wild bootstrap inference typically yields convincing evidence for advantages of the SPF, while tests using non-robust critical values provide remarkably less. Moreover, we find significant evidence for time-variation of relative forecast ability, the advantages of the SPF weakening considerably after the "Great Moderation"., Discussion Paper / SFB823;6/2021

This article discusses cointegration tests for nonlinear cointegration in the presence of variance breaks in the errors. We build on approaches of Cavaliere and Taylor (2006, Journal of Time Series Analysis) for heteroskedastic cointegration tests and of Choi and Saikkonen (2010, Econometric Theory) for nonlinear cointegration tests. We propose a bootstrap test and prove its consistency. A Monte Carlo study shows the approach to have appealing finite sample properties and to work better than an approach using subresiduals. We provide an empirical application to the environmental Kuznets curves (EKC), finding that the cointegration tests do not reject the EKC hypothesis in most cases., Discussion Paper / SFB823;26/2020

Web-based systems for assessment or homework are commonly used in many different domains. Several studies show that these systems can have positive effects on learning outcomes. Many research efforts also have made these systems quite flexible with respect to different item formats and exercise styles. However, there is still a lack of support for complex exercises in several domains at university level. Although there are systems that allow for quite sophisticated operations for generating exercise contents, there is less support for using similar operations for evaluating students' input and for feedback generation. This paper elaborates on filling this gap in the specific case of statistics. We present both the conceptional requirements for this specific domain as well as a fully implemented solution. Furthermore, we report on using this solution for formative and summative assessments in lectures with large numbers of participants at a big university.

The fixed-b asymptotic framework provides refinements in the use of heteroskedasticity and autocorrelation consistent variance estimators. The resulting limiting distributions of t-statistics are, however, not pivotal when the unconditional variance changes over time. Such time-varying volatility is an important issue for many financial and macroeconomic time series. To regain pivotal fixed-b inference under time-varying volatility, we discuss three alternative approaches. We (i) employ the wild bootstrap (Cavaliere and Taylor, 2008, ET), (ii) resort to time transformations (Cavaliere and Taylor, 2008, JTSA) and (iii) consider to select test statistics and asymptotics according to the outcome of a heteroscedasticity test, since small-b asymptotics deliver standard limiting distributions irrespective of the socalled variance profile of the series. We quantify the degree of size distortions from using the standard fixed-b approach assuming homoskedasticity and compare the effectiveness of the corrections via simulations. It turns out that the wild bootstrap approach is highly recommendable in terms of size and power. An application to testing for equal predictive ability using the Survey of Professional Forecasters illustrates the usefulness of the proposed corrections.

The fixed-b asymptotic framework provides refinements in the use of heteroskedasticity and autocorrelation consistent variance estimators. We show however that the fixed-b limiting distributions of t-statistics are not pivotal when the variance of the underlying data generating process changes over time. To regain pivotal fixed-b inference under such time heteroskedasticity, we discuss three alternative approaches. We employ (1) the wild bootstrap (Cavaliere and Taylor, 2008, ET), (2) resort to time transformations (Cavaliere and Taylor, 2008, JTSA) and (3) suggest to pick suitable the asymptotics according to the outcome of a heteroskedasticity test, since small-b asymptotics deliver standard limiting distributions irrespective of the so-called variance profile of the series. We quantify the degree of size distortions from using the standard fixed-b approach and compare the effectiveness of the corrections via simulations. We also provide an empirical application to excess returns.

While the limiting null distributions of cointegration tests are invariant to a certain amount of conditional heteroskedasticity as long as global homoskedasticity conditions are fulfilled, they are certainly affected when the innovations exhibit time-varying volatility. Worse yet, distortions from single units accumulate in panels, where one must anyway pay special attention to dependence among cross-sectional units, be it time-dependent or not. To obtain a panel cointegration test robust to both global heteroskedasticity and cross-unit dependence, we start by adapting the nonlinear instruments method proposed for the Dickey-Fuller test by Chang (J of Econometrics 110, 261--292) to an error-correction testing framework. We show that IV-based testing of the null of no error-correction in individual equations results in asymptotic standard normality of the test statistic as long as the t-type statistics are computed with White heteroskedasticity-consistent standard errors. Remarkably, the result holds even in the presence of endogenous regressors, irrespective of the number of integrated covariates, and for any variance profile. Furthermore, a test for the null of no cointegration---in effect, a joint test against no error correction in any equation of each unit---retains the nice properties of the univariate tests. In panels with fixed cross-sectional dimension, both types of test statistics from individual units are shown to be asymptotically independent even in the presence of correlation or cointegration across units, leading to a panel test statistic robust to cross-unit dependence and unconditional heteroskedasticity. The tests perform well in panels of usual dimensions with innovations exhibiting variance breaks and a factor structure.

We investigate the OLS-based estimator s2 of the disturbance variance in an error component linear panel regression model when the disturbances are homoskedastic, but spatially correlated. Although consistent (Song and Lee, Econ. Lett. 2008), s2 can be arbitrarily biased towards zero in finite samples., Discussion Paper / SFB 823;18/2010

This paper discusses two longstanding questions in growth econometrics which involve multiple hypothesis testing. In cross sectional GDP growth regressions many variables are simultaneously tested for significance. Similarly, when investigating pairwise convergence of output in panel data sets of n countries, n(n-1)/2 tests are performed. We propose to control the false discovery rate (FDR) so as not to erroneously declare variables significant in these multiple testing situations only because of the large number of tests performed. Doing so, we provide a simple new way to robustly select variables in economic growth models. We find that few other variables beyond the initial GDP level are needed to explain growth. We also show that convergence in panels of per capita output using a time series definition with the necessary condition of no unit root in the log per-capita output gap of two economies does not appear to hold.

The so-called Cauchy estimator uses the sign as instrument for the first lag in autoregressions, and the resulting t-type statistic has a standard normal distribution even in the unit root case. Thus, nonstandard asymptotics of the usual unit root tests such as the augmented Dickey-Fuller [ADF] test can be avoided. Moreover, the ADF test is affected by unconditional heteroskedasticity asymptotically; but the paper shows that, by using as instruments nonlinear transformations asymptotically homogenous of order 0 (of which the sign is one), asymptotic normality of the IV t-type statistic is maintained under unconditional heteroskedasticity when the series to be tested has no deterministic trends. No estimation of the so-called variance profile and no bootstrap application are required to this end, unlike for the ADF test. When adjusting the differences for deterministic components, however, the null distribution of the Cauchy test for a unit root becomes non-standard, reminiscent of the ADF test. In fact, an examination of the asymptotic distribution of the Cauchy test under a sequence of local alternatives reveals that it has power in the same 1/T neighborhoods as the the ADF test, irrespective of whether a deterministic trend is present in the data or not. The standard normality of the examined Cauchy test can be exploited to build a panel unit root test under cross-sectional dependence with an orthogonalization procedure. The panel test does not require any N asymptotics to establish the limiting distribution, but the paper's analysis of the joint N,T asymptotics for the panel statistic suggests that N should be smaller than T. To render the test applicable when the number of cross-sectional units is larger than the number of time observations, shrinkage estimators of the involved covariance matrix are used. The performance of the discussed procedures is found to be satisfactory in finite samples.

This paper suggests a combination procedure to exploit the imperfect correlation of cointegration tests to develop a more powerful meta test. To exemplify, we combine Engle and Granger (1987) and Johansen (1988) tests. Either of these un- derlying tests can be more powerful than the other one depending on the nature of the data-generating process. The new meta test is at least as powerful as the more powerful one of the underlying tests irrespective of the very nature of the data generating process. At the same time, our new meta test avoids the size distortion inherent in separately applying multiple tests for cointegration to the same data set.

This paper proposes a new panel unit root test based on Simes' [Biometrika 1986, An Improved Bonferroni Procedure for Multiple Tests of Significance'] classical intersection test. The test is robust to general patterns of cross-sectional dependence and yet straightforward to implement, only requiring p-values of time series unit root tests of the series in the panel, and no resampling. Monte Carlo experiments show good size and power properties relative to existing panel unit root tests. Unlike previously suggested tests, the new test allows to identify the units in the panel for which the alternative of stationarity can be said to hold. We provide two empirical applications to panels of real gross domestic product (GDP) and real exchange rate data.

We propose new tests for panel cointegration by extending the panel unit root tests of Choi [2001] and Maddala and Wu [1999] to the panel cointegration case. The tests are flexible, intuitively appealing and relatively easy to compute. We investigate the finite sample behavior in a simulation study. Several variants of the tests compare favorably in terms of both size and power with other widely used panel cointegration tests.

This paper examines whether, in addition to standard unit root and cointegration tests, panel approaches also produce test statistics behaving erratically when applied to tests for PPP. We show that if appropriate tests (which are robust to cross-sectional dependence and more powerful than single time series tests) are used, any evidence of erratic behaviour disappears, and strong empirical support is found for PPP. It appears therefore that recent advances in panel data econometrics might enable us to settle the PPP debate. JEL Classification: C12, C23, F31

We use meta analytic combination procedures to develop new tests for panel cointegration. The main idea consists in combining p-values from time series cointegration tests on the different units of the panel. The tests are robust to heterogeneity as well as to cross-sectional dependence between the different units of the panel. To achieve the latter, we employ a sieve bootstrap procedure with joint resampling of the residuals of the different units. A simulation study shows that the suggested bootstrap tests can have substantially smaller error-in-rejection probabilities than tests ignoring the presence of cross-sectional dependence while preserving high power. We apply the tests to a panel of Post-Bretton Woods data to test for weak Purchasing Power Parity (PPP).

Time series cointegration tests, even in the presence of large sample sizes, often yield conflicting conclusions (“mixed signals”) as measured by, inter alia, a low correlation of empirical p-values [see Gregory et al., 2004, Journal of Applied Econometrics]. Using their methodology, we present evidence suggesting that the problem of mixed signals persists for popular panel cointegration tests. As expected, there is weaker correlation between residual and system-based tests than between tests of the same group.

Meta-analytic panel unit root tests such as Fisher’s Chi^2 test, which consist of pooling the p-values of time series unit root tests, are widely applied in practice. Recently, several Monte Carlo studies have found these tests’ Error-in-Rejection Probabilities (or, synonymously, size distortion) to increase with the number of series in the panel. We investigate this puzzling finding by modelling the finite sample p-value distribution of the time series tests with local deviations from the asymptotic p-value distribution. We find that the size distortions of the panel tests can be explained as the cumulative effect of small size distortions in the time series tests.

We use recent advances in multiple testing to identify the countries for which Purchasing Power Parity (PPP) held over the last century. The approach controls the multiplicity problem inherent in simultaneously testing for PPP on several time series, thereby avoiding spurious rejections. It has higher power than traditional multiple testing techniques by exploiting the dependence structure between the countries with a bootstrap approach. We use a sieve bootstrap approach to account for nonstationarity under the null hypothesis. Our empirical results show that, plausibly, controlling for multiplicity in this way leads to a number of rejections of the null of no PPP that is intermediate between that of traditional multiple testing techniques and that which results if one tests the null on each single time series at some level alpha. JEL Classification: C12, C23, F31

We investigate the OLS-based estimator s^2 of the disturbance variance in the standard linear regression model with cross section data when the disturbances are homoskedastic, but spatially correlated. For the most popular model of spatially autoregressive disturbances, we show that s^2 can be severely biased in finite samples, but is asymptotically unbiased and consistent for most types of spatial weighting matrices as sample size increases.

We analyse whether tests of PPP exhibit erratic behaviour (as previously reported by Caporale et al., 2003) even when (possibly unwarranted) homogeneity and proportionality restrictions are not imposed, and trivariate cointegration (stage-three) tests between the nominal exchange rate, domestic and foreign price levels are carried out (instead of stationarity tests on the real exchange rate, as in stage-two tests). We examine the US dollar real exchange rate vis-à-vis 21 other currencies over a period of more than a century, and find that stage-three tests produce similar results to those for stage-two tests, namely the former also behave erratically. This confirms that neither of these traditional approaches to testing for PPP can solve the issue of PPP.