The panel data unit root test suggested by Levin and Lin (LL) has been widely used in several applications, notably in papers on tests of the purchasing power parity hypothesis. This test is based on a very restrictive hypothesis which is rarely ever of interest in practice. The Im–Pesaran–Shin (IPS) test relaxes the restrictive assumption of the LL test. This paper argues that although the IPS test has been offered as a generalization of the LL test, it is best viewed as a test for summarizing the evidence from a number of independent tests of the sample hypothesis. This problem has a long statistical history going back to R. A. Fisher. This paper suggests the Fisher test as a panel data unit root test, compares it with the LL and IPS tests, and the Bonferroni bounds test which is valid for correlated tests. Overall, the evidence points to the Fisher test with bootstrap-based critical values as the preferred choice. We also suggest the use of the Fisher test for testing stationarity as the null and also in testing for cointegration in panel data.

/pdf/a-comparative-study-of-unit-root-tests-with-panel-data-and-a-3mrtl7499l.pdf

A Comparative Study of Unit Root Tests with Panel Data and a New Simple Test

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

The asymptotic power envelope is derived for point-optimal tests of a unit root in the autoregressive representation of a Gaussian time series under various trend specifications. We propose a family of tests whose asymptotic power functions are tangent to the power envelope at one point and are never far below the envelope. When the series has no deterministic component, some previously proposed tests are shown to be asymptotically equivalent to members of this family. When the series has an unknown mean or linear trend, commonly used tests are found to be dominated by members of the family of point-optimal invariant tests. We propose a modified version of the Dickey-Fuller t test which has substantially improved power when an unknown mean or trend is present. A Monte Carlo experiment indicates that the modified test works well in small samples.

/pdf/efficient-tests-for-an-autoregressive-unit-root-15aexwgsnj.pdf

Efficient Tests for an Autoregressive Unit Root

Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme of the text is the use of artificial regressions for estimation, reference, and specification testing of nonlinear models, including diagnostic tests for parameter constancy, serial correlation, heteroscedasticity, and other types of mis-specification. Explaining how estimates can be obtained and tests can be carried out, the authors go beyond a mere algebraic description to one that can be easily translated into the commands of a standard econometric software package. Covering an unprecedented range of problems with a consistent emphasis on those that arise in applied work, this accessible and coherent guide to the most vital topics in econometrics today is indispensable for advanced students of econometrics and students of statistics interested in regression and related topics. It will also suit practising econometricians who want to update their skills. Flexibly designed to accommodate a variety of course levels, it offers both complete coverage of the basic material and separate chapters on areas of specialized interest.

Estimation and Inference in Econometrics

General characterizations of valid confidence sets and tests in problems which involve locally almost unidentified (LAU) parameters are provided and applied to several econo- metric models. Two types of inference problems are studied: (i) inference about parame- ters which are not identifiable on certain subsets of the parameter space, and (ii) inference about parameter transformations with discontinuities. When a LAU parameter or parametric function has an unbounded range, it is shown under general regularity conditions that any valid confidence set with level 1 - a for this parameter must be unbounded with probability close to 1 - a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model: no valid confidence set which is almost surely bounded does exist. These properties hold even if "identifying restrictions" are imposed. Similar results also obtain for parameters with bounded ranges. Consequently, a confidence set which does not satisfy this characterization has zero coverage probability (level). This will be the case in particular for Wald-type confidence intervals based on asymptotic standard errors. Furthermore, Wald-type statistics for testing given values of a LAU parameter cannot be pivotal functions (i.e., they have distributions which depend on unknown nuisance param- eters) and even cannot be usefully bounded over the space of the nuisance parameters. These results are applied to several econometric problems: inference in simultaneous equations (instrumental variables (IV) regressions), linear regressions with autoregressive errors, inference about long-run multipliers and cointegrating vectors. For example, it is shown that standard "asymptotically justified" confidence intervals based on IV estimators (such as two-stage least squares) and the associated "standard errors" have zero coverage probability, and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions, a result of interest for interpreting IV regressions with "weak instruments." Furthermore, expansion methods (e.g., Edgeworth expansions) and bootstrap techniques cannot solve these difficulties. Finally, in a number of cases where Wald-type methods are fundamentally flawed (e.g., IV regressions with poor instruments), it is observed that likelihood-based methods (e.g., likelihood-ratio tests and confidence sets) combined with projection techniques can easily yield valid tests and confidence sets.

/pdf/some-impossibility-theorems-in-econometrics-with-48xphcbjg3.pdf

Some impossibility theorems in econometrics with applications to structural and dynamic models

Causality in the sense of Granger is typically defined in terms of predictibility of a vector of variables one period ahead. Recently, Liitkepohl (1993) proposed to define noncausality between two variables in terms of nonpredictibility at any number of periods ahead. When more than two vectors are considered (i.e., when the information set contains auxiliary variables), these two notions are not equivalent. In this paper, we first generalize the notion of causality by considering causality at a given (arbitrary) horizon h. Then we derive necessary and sufficient conditions for noncausality between vectors of variables (inside a larger vector) up to any given horizon h, where h can be infinite. In particular, for general possibly nonstationary processes with finite second moments, relatively simple exhaustivity and separation conditions, which are sufficient for noncausality at all horizons, are provided. To deal with cases where such conditions do not apply, we consider a more specific, although still very wide, class of vector autoregressive processes (possibly of infinite order, stationary or nonstationary), which include multivariate ARIMA processes, and we derive general parametric characterizations of noncausality at various horizons for this class (including a causality chain characterization). We also observe that the coefficients of lagged variables in forecasts at various horizons h ≥ 1 can be interpreted as generalized impulse response coefficients which yield a complete picture of linear causality properties, in contrast with usual response coefficients which can be quite misleading in this respect.

Short run and long run causality in time series: Theory

Short run and long run causality in time series

We discuss statistical inference problems associated with identification and testability in econometrics. We consider inference in non-parametric models and weakly identified structural models (weak instruments). We point out that many ill-defined statistical problems, such as non-testable hypotheses, occur in these areas and are typically associated with asymptotic approximations. In non-parametric models, such problems include testing moments and inference under heteroscedasticity or serial dependence of unknown form. For weakly identified structural models, difficulties are typically associated with improper pivots, and we review recent developments aimed at proposing more reliable procedures, including alternative proposed statistics, bounds, projection, split-sampling, conditioning, Monte Carlo tests.

Identification, Weak Instruments, and Statistical Inference in Econometrics

http://www.cirano.qc.ca/pdf/publication/2005s-02.pdf

Jean-Marie Dufour

Papers

Some impossibility theorems in econometrics with applications to structural and dynamic models

Short run and long run causality in time series: Theory

Short run and long run causality in time series

Identification, Weak Instruments, and Statistical Inference in Econometrics

Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics