Point Optimal Tests Research Articles

Semiparametric testing with highly persistent predictors

We address the issue of semiparametric efficiency in the bivariate regression problem with a highly persistent predictor, where the joint distribution of the innovations is regarded an infinite-dimensional nuisance parameter. Using a structural representation of the limit experiment and exploiting invariance relationships therein, we construct invariant point-optimal tests for the regression coefficient of interest. This approach naturally leads to a family of feasible tests based on the component-wise ranks of the innovations that can gain considerable power relative to existing tests under non-Gaussian innovation distributions, while behaving equivalently under Gaussianity. When an i.i.d. assumption on the innovations is appropriate for the data at hand, our tests exploit the efficiency gains possible. Moreover, we show by simulation that our test remains well behaved under some forms of conditional heteroskedasticity.

Point optimal testing with roots that are functionally local to unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest that appear in practical work, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases and demonstrate how the power envelope changes in situations of practical interest. Against FLUR alternatives, conventional constant point optimal tests can be asymptotically infinitely deficient in power, with poor finite sample power performance particularly when the departure from unity occurs early in the sample period. New analytic explanation for this phenomenon is provided. Simulation results are reported and some implications for empirical practice are examined.

Distributions You Can Count On …But What’s the Point?

The Poisson regression model remains an important tool in the econometric analysis of count data. In a pioneering contribution to the econometric analysis of such models, Lung-Fei Lee presented a specification test for a Poisson model against a broad class of discrete distributions sometimes called the Katz family. Two members of this alternative class are the binomial and negative binomial distributions, which are commonly used with count data to allow for under- and over-dispersion, respectively. In this paper we explore the structure of other distributions within the class and their suitability as alternatives to the Poisson model. Potential difficulties with the Katz likelihood leads us to investigate a class of point optimal tests of the Poisson assumption against the alternative of over-dispersion in both the regression and intercept only cases. In a simulation study, we compare score tests of ‘Poisson-ness’ with various point optimal tests, based on the Katz family, and conclude that it is possible to choose a point optimal test which is better in the intercept only case, although the nuisance parameters arising in the regression case are problematic. One possible cause is poor choice of the point at which to optimize. Consequently, we explore the use of Hellinger distance to aid this choice. Ultimately we conclude that score tests remain the most practical approach to testing for over-dispersion in this context.

The role of information in nonstationary regression

ABSTRACTThe role of standard likelihood-based measures of information and efficiency is unclear when regressions involve nonstationary data. Typically the standardized score is not asymptotically Gaussian and the standardized Hessian has a stochastic, rather than deterministic limit. Here we consider a time series regression involving a deterministic covariate which can be evaporating, slowly evolving or nonstationary. It is shown that conditional information, or equivalently, profile Kullback–Leibler and Fisher information remain informative about both the accuracy, i.e. asymptotic variance, of profile maximum likelihood estimators, and the power of point optimal invariant tests for a unit root. Specifically, these information measures indicate fractional, rather than linear trends that may minimize inferential accuracy. Such is confirmed in a numerical experiment.

Properties of the Power Envelope for Tests Against Both Stationary and Explosive Alternatives: The Effect of Trends

This article details a precise analytic effect that inclusion of a linear trend has on the power of Neyman–Pearson point optimal unit root tests and thence the power envelope. Both stationary and explosive alternatives are considered. The envelope can be characterized by probabilities for two, related, sums of Chi‐square random variables. A stochastic expansion, in powers of the local‐to‐unity parameter, of the difference between these loses its leading term when a linear trend is included. This implies that the power envelope converges to size at a faster rate, which can then be exploited to prove that the power envelope must necessarily be lower. This effect is shown to be, analytically, greater asymptotically than in small samples and numerically far greater for explosive than for stationary alternatives. Only a linear trend has a specific rate effect on the power envelope, however other deterministic variables will have some effect. The methods of the article lead to a simple direct measure of this effect which is then informative about power, in practice.

Semiparametric Optimal Testing With Highly Persistent Predictors

We address the issue of semiparametric efficiency in the bivariate regression problem with a highly persistent predictor, where the joint distribution of the innovations is regarded an infinite-dimensional nuisance parameter. Using a structural representation of the limit experiment and exploiting invariance relationships therein, we construct invariant point-optimal tests for the regression coefficient of interest. This approach naturally leads to a family of feasible tests based on the component-wise ranks of the innovations that can gain considerable power relative to existing tests under non-Gaussian innovation distributions, while behaving equivalently under Gaussianity. When an i.i.d. assumption on the innovations is appropriate for the data at hand, our tests exploit the efficiency gains possible. Moreover, we show by simulation that our test remains well behaved under some forms of conditional heteroskedasticity.

Using point optimal test of a simple null hypothesis for testing a composite null hypothesis via maximized Monte Carlo approach

ABSTRACTKing’s Point Optimal (PO) test of a simple null hypothesis is useful in a number of ways, for example it can be used to trace the power envelope against which existing tests can be compared. However, this test cannot always be constructed when testing a composite null hypothesis. It is suggested in the literature that approximate PO (APO) tests can overcome this problem, but they also have some drawbacks. This paper investigates if King’s PO test can be used for testing a composite null in the presence of nuisance parameters via a maximized Monte Carlo (MMC) approach, with encouraging results.

Point Optimal Testing with Roots That Are Functionally Local to Unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases. It is shown that against FLUR alternatives, conventional constant point optimal tests can have extremely low power, particularly when the departure from unity occurs early in the sample period. Simulation results are reported and some implications for empirical practice are examined.

Powerful Unit Root Tests Free of Nuisance Parameters

We propose a variance ratio‐type unit root test where the nuisance parameter cancels asymptotically under both the null of a unit root and a local‐to‐unity alternative. Critical values and asymptotic power curves can be computed using standard numerical techniques. Our test exhibits higher power compared with tests that share the virtue of being free of tuning parameters. In fact, the local asymptotic power curves of our procedure get close to the power functions of the point optimal test, where the latter suffers from the drawback of having to correct for a nuisance parameter consistently.

Parametric and Semi‐Parametric Efficient Tests for Parameter Instability

This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time‐varying parameters are asymptotically equivalent. Our finding is then extended to a semi‐parametric set‐up in which the error distribution is treated as an unknown infinite‐dimensional nuisance parameter. We find that semi‐parametric tests can be adaptive without further restrictive conditions on the error distribution.

Confidence sets for the break date based on optimal tests

This study proposes constructing a confidence set for the date of a one-time structural change using a point optimal test. Following Elliott and Muller (2007), we first construct a test for the break date that maximizes the weighted average of the power function. The confidence set is then obtained by inverting the test statistic. We carefully choose the weights and show by Monte Carlo simulations that the confidence set based on our method has a relatively accurate coverage rate, while the length of our confidence set is significantly shorter than the lengths proposed in the literature.

Point optimal testing: A survey of the post 1987 literature

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King [80] suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal tests for multi-dimensional testing.

Special Issue on ``Estimation, Testing and Forecasting in Econometrics''

One of us (King) joined Monash University as a young lecturer in econometrics in 1979. At about that time, there was a noticeable leap in the availability of computer power that allowed us to contemplate a likelihood based approach to econometrics. For example, the Faculty of Economics and Politics had taken delivery of its own VAX11/780 mini computer which allowed serious statistical and econometric computations to be made in double precision if required. Econometricians no longer had their CPU time rationed. We finally had the computational power and ability to run maximum likelihood estimation for most models and consequently conduct likelihood ratio (LR) tests or if more convenient, Lagrange multiplier (LM) or Wald tests. Did these advances mean that the theory of econometrics was “sorted”? It didn’t take long for cracks to appear. One testing problem that generated a disproportionately large literature in econometrics is that of testing for first-order autoregressive disturbances in the linear regression model, see King [5] for a survey. Here the LR test was found to be a very poor performer [1]. Unit root testing is another area where the classical tests (LR, LM and Wald) were found to perform poorly. In general, problems were caused by the small sample sizes and difficulties in dealing with nuisance parameters that can cause biases in estimated parameters that can also affect the performance of tests and forecasts. The last 35 years has seen all sorts of challenges overcome in an unending search for improvement in model based estimation, testing and forecasting. Many of these developments have become feasible because of the advances in computer hardware and software during that time. Also the ready availability of simulation methods have helped assess the small sample performance of proposed methods, both absolutely and comparatively. This special issue makes a contribution to this literature with the aim of bringing some of these advances to the notice of statisticians and others interested in model based statistical inference. The main ideas behind point optimal testing plus a strategy for the general application of such tests were outlined by King [6]. This issue opens with a paper that surveys the post 1987 literature on point optimal testing with particular emphasis on dealing with nuisance parameters. That period has witnessed the development of asymptotic point optimal testing led by the pioneering work of Elliott, Rothenberg and Stock [2] and, more recently, Muller [10] with a range of applications involving unit root testing, testing for cointegration and breaks or variation in regression coefficients. The review closes with an outline of a new class of point optimal tests for multi-dimensional testing, an area with few applications to date. Jahar Bhowmik contributes a paper that demonstrates the t-test of a classical regression coefficient is uniformly most powerful for a wider class of invariant tests. It is widely known that the F-test is uniformlymost powerfulwithin the class of tests that are invariant to four separate sets of transformations of the data. Because the t-test statistic is the square root of the classical F-test statistic, research papers and standard texts (see for example [3,8,13]) typically state that the t-test is uniformly most powerful for the same four sets of transformations as the F-test. Bhowmik

Unit root tests for cross-sectionally dependent panels: The influence of observed factors

This paper considers a heterogeneous panel unit root model with cross-sectional dependence generated by a factor structure—the factor common to all units being an observed covariate. The model is shown to be Locally Asymptotically Mixed Normal (LAMN), with the random part of the limiting Fisher information due to information generated by the covariate. Because of the LAMN structure, no asymptotically uniformly most powerful test exists; we investigate the asymptotic power properties of the locally optimal test, the best point optimal test, and the conditionally optimal t-test. Although one might expect the best point optimal test to be superior, the performance of the computationally simpler t-test is comparable.

Testing for parameter constancy in the time series direction in panel data models

We propose tests for parameter constancy in the time series direction in panel data models. We construct a locally best invariant test based on Tanaka [Time series analysis: nonstationary and noninvertible distribution theory. New York: Wiley; 1996] and an asymptotically point optimal test based on Elliott and Müller [Efficient tests for general persistent time variation in regression coefficients. Rev Econ Stud. 2006;73:907–940]. We derive the limiting distributions of the test statistics as T→∞ while N is fixed, and calculate the critical values by applying numerical integration and response surface regression. Simulation results show that the proposed tests perform well if we apply them appropriately.

Unit Root Tests for Cross-Sectionally Dependent Panels: The Influence of Observed Factors

This paper considers a heterogeneous panel unit root model with cross-sectional dependence generated by a factor structure – the factor common to all units being an observed covariate. The model is shown to be Locally Asymptotically Mixed Normal (LAMN), with the random part of the limiting Fisher information due to information generated by the covariate. Because of the LAMN structure, no asymptotically uniformly most powerful test exists; we investigate the asymptotic power properties of the locally optimal test, the best point optimal test, and the conditionally optimal t-test. Although one might expect the best point optimal test to be superior, the performance of the computationally simpler t-test is comparable.

On the asymptotic distribution of the Dickey Fuller-GLS test statistic

In a very influential paper, Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836] show that no uniformly most powerful test for the unit root testing problem exits, derive the relevant power envelope and characterize a family of point-optimal tests. As a by-product, they also propose a ‘generalized least squares (GLS) detrended’ version of the conventional Dickey–Fuller test, denoted DF-GLS, that has since then become very popular among practitioners, much more so than the point-optimal tests. In view of this, it is quite strange to find that, while conjectured in Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836], so far there seems to be no formal proof of the asymptotic distribution of the DF-GLS test statistic. By providing three separate proofs, the current paper not only substantiates the required result, but also provides insight regarding the pros and cons of different methods of proof.

Testing linear regression model with AR(1) errors against a first-order dynamic linear regression model with white noise errors: A point optimal testing approach

We know very little about the performance of point optimal (PO) and approximate point optimal (APO) tests in the presence of unavoidable nuisance parameters. Because marginal likelihood based tests are said to perform well in the presence of unavoidable nuisance parameters, this paper compares the performance of marginal likelihood based APO tests and classical tests using a testing problem which has been largely overlooked by econometric practitioners, namely testing for a static linear regression model with AR(1) errors against a dynamic linear regression model with white noise errors. It is well known that the classical tests are specifically designed for nested testing, they are applied to test for the significance of the dynamic coefficient of a dynamic linear regression model with AR(1) errors.A testing procedure is proposed, where the size and power comparisons used are based on near-exact non-similar critical values of tests obtained using the simulated annealing (SA) algorithm, as the near-exact non-similar critical values control the sizes of the tests well overall.Among marginal likelihood based classical tests, the likelihood ratio (LR) test and Lagrange multiplier (LM) test seem to perform well under the alternative hypothesis, particularly when the dynamic parameter is large and the sample size is reasonably big. The Wald (W) test is the worst performer overall. This concurs with previous observations that the W test performs poorly in small samples. Compared to the classical approach, APO tests appear to have good power properties, particularly in the neighborhood of the chosen parameter point under the alternative hypothesis. This finding may advance the use of PO and APO tests.

Bootstrap Point Optimal Unit Root Tests

In this article, we investigate and compare the behaviour of some bootstrap unit root tests in finite ARMA models with a constant and/or a trend and use them to obtain asymptotic results for the point optimal (hereafter PO) test, in terms of both size and power. We demonstrate the asymptotic validity of bootstrapping the PO test. We provide a feasible method for obtaining approximate critical values for the PO unit root test. Through simulations, we investigate how effective the bootstrap is in different sample sizes, correlative coefficients and close unity autoregressive roots in two different models. Our main objective is to show that the bootstrap PO test can be used in regression models with AR and MA errors and trending regressors. The results reported here provide an analytical investigation of the use of the bootstrap for PO tests with dependent data.The main contribution of this article has two features. First, we choose the PO test and make this powerful but unfeasible procedure both powerful and feasible, by plugging in a consistent estimation of the coefficient structure, and we show that the bootstrap PO test provides asymptotically valid critical values. Second, through simulation, our numerical results suggest that the bootstrap PO test performs well in having the correct size properties and retaining good power in the parametric (and semi-parametric) bootstrap procedure.

The Law of One Price in Six Central and Eastern European Economies

This study examines the validity of the purchasing power parity (PPP) for six Central and East European countries (CEEC) during the 1991–2010 period. Because this rather short-time period is usually not sufficient to produce reliable findings using conventional univariate unit root tests, I use the covariate point optimal test (CPT) developed by Elliott and Jansson, which enhances the power of the test by exploiting information in stationary covariates. CPT tests provide support for the PPP hypothesis in four out of the six CEEC economies. Evidence of PPP is especially strong in the Czech Republic, Poland and Slovak Republic.