Point Optimal Invariant Tests Research Articles

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.

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). 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.

Some Properties of the Point Optimal Invariant Test for the Constancy of Parameters

In this paper we consider the time-varying parameter model. Since there is no uniformly most powerful test for the constancy of parameters, the locally best invariant (LBI) test has often been considered in the literature, including the study by Nabeya and Tanaka (1988). We show the existence of the limiting distribution of the point optimal invariant (POI) test statistic when we can derive the limiting distribution of the LBI test statistic. We prove that the limiting characteristic function of the POI test statistic can be expressed using that of the LBI test statistic.

Seasonal Moving‐average Unit Root Tests in the Presence of a Linear Trend

To distinguish stochastic from deterministic seasonality, test procedures are developed for a unit root in the integrated seasonal moving‐average (SMA) model when an underlying deterministic trend is present. Locally best invariant unbiase (LBIU) and point optimal invariant tests are considered. Their asymptotic distributions are developed and are found to differ from those for the no‐linear‐trend case. The limiting distribution of the LBIU statistic is expressed as a functional of Brownian motions. The procedures are extended to more general seasonal autoregressive moving‐average (ARMA) models, and to the inclusion of exogenous regressors. Finite‐sample distributions are also derived for the SMA(1) model. Simulations suggest that these distributions provide accurate approximations for more general ARMA models. A numerical example is included to illustrate the tests.

Tests for Seasonal Moving Average Unit Root in ARIMA Models

Procedures to test for a unit root in the seasonal moving average (MA) operator for seasonal autoregressive integrated moving average (ARIMA) models are examined. Such procedures are useful to distinguish deterministic from stochastic seasonal patterns. The simple (integrated) seasonal MA(1) model Z t ≡ Y t - Yt-s = (1 - θB s)ϵt, where {Y t} is the observed seasonal series and {ϵt} is a white noise sequence, is considered, and tests for H 0: θ = 1 versus H 1: θ < 1 are studied. The locally best invariant unbiased and point-optimal invariant tests are investigated. The finite-sample and asymptotic distributions of the test statistics under both the null hypothesis and local alternatives are derived. Quantiles of the null distributions are tabulated for quarterly and monthly data cases, and a limited study of the powers of the tests is reported. The test procedures are extended to more general seasonal ARMA models of the form Z t = (1 - θB s)Nt, where N t follows a general stationary and invertible ARMA model φ(B)Nt = θ*(B)ϵt. The generalizations suggested involve modifying the test procedures for the seasonal MA(1) case parametrically, based on the estimated ARMA structure for the noise process {N t}. Simulation results are presented to study the performance of the modified procedure, and numerical examples involving monthly and quarterly seasonal time series data are given.

Point Optimal Invariant Tests of a Unit Root in Models with Structural Change

This study proposes tests for the null hypothesis of a unit root against shifting trend stationary alternatives. In particular, models with structural change in the form of level or trend shifts are studied. Roots local-to-unity are addressed in models with structural change and asymptotic power functions of the tests are derived using local-to-unity asymptotic framework. These tests have good power properties against shifting trend stationary alternatives. Tests in the presence of multiple shifts are also examined. The asymptotic distributions of the tests are derived under exogenous and endogenous shift point assumptions. The asymptotic and finite sample percentiles of the tests are tabulated using Monte Carlo integration. Monte Carlo simulations reveal that tests have good power and size properties against various alternatives.

Testing for linearity in a semiparametric regression model

Abstract We propose a new exact test for nonlinearity in a regression model with one possibly nonlinear component. The test is based on the stochastic interpretation of spline smoothing given in Wahba (1978) and is a point optimal invariant test (as defined in King, 1988) based on this stochastic model. We show that the power of the exact test compares favourably with several other exact tests for nonlinearity proposed in the literature, and that its level and power are robust to long- and short-tailed symmetric error distributions. We also present an O(n) algorithm based on a state space approach to compute exact p-values for the point optimal invariant test, whereas previous implementations required O(n3) operations. An example is presented to illustrate the theory.

Testing for block effects and misspecification in regession models based on survey data*

Much applied work in econometrics and other social sciences involves regression analysis using survey data. King and Evans (1986) and Bhatti (1991) considered an equicorrelated error term model and proposed one-sided locally best invariant and point optimal invariant tests for this model, respectively. We consider more general situations and construct tests for equicorrelation over different blocks and even over different subblocks using King and Wu's (1990) locally most mean powerful invariant (LMMPI) test. Further, a procedure of diagnostic checking for model adequacy based on testing simultaneously for omitted regressors and equicorrelation in the errors is presented. Monte Carlo studies are used to investigate the performance of our tests. Finally, we apply these tests to Bangladesh data.

Nonnested testing for autocorrelation in the linear regression model

This paper investigates the construction and application of point-optimal invariant (POI) tests of joint AR(1)− AR(4) disturbances against joint MA(1)− MA(4) disturbances in the linear regression model. A Monte Carlo experiment is conducted to assess and compare the small sample performances of two asymptotic tests and two POI tests. Of the asymptotic tests, the Lagrange multiplier test is found to have distinctly better small sample properties than the prediction test. We also find that the extra computation required to perform a POI test is rewarded by a clear improvement in size and power properties in comparison to the asymptotic tests. The use of a POI test is illustrated with an application to a quarterly model of price inflation in the United Kingdom during 1947–1970. The paper concludes with some discussion of the problem of testing general AR( p) disturbances against MA( q) disturbances.

Testing for autoregressive disturbances in a time series regression with missing observations

Abstract This paper constructs tests for autoregressive disturbances in a time series regression with missing observations, where the disturbance terms are generated by (i) an AR(1) process and (ii) an AR( p ) process with a possible seasonal component. A point optimal invariant (POI) test is constructed for each problem. In addition, we show how to compute exact small sample p -values for the tests in O( n ) operations, and give a computationally efficient procedure for choosing the specific alternative against which to make POI test most powerful invariant.

Testing Moving Average Against Autoregressive Disturbances in the Li near-Reg ression Model

This article considers testing for first-order moving average against first-order autoregressive disturbances in the linear-regression model. Tests investigated include approximate point-optimal invariant (POI) tests, an asymptotic test of the second-order residual autocorrelation coefficient, and a Lagrange multiplier. (LM) test. A Monte Carlo experiment compares their small-sample performances. Of the asymptotic tests, the LM test has the most satisfactory sizes, but its rival has the better overall power. We find that the approximate POI tests have superior size and power properties in comparison to the asymptotic tests. An approximate POI test is applied to a random-walk model for Australian real interest rates.

Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors

Abstract Inference on the autocorrelation coefficient ϱ of a linear regression model with first-order autoregressive normal disturbances is studied. Both stationary and nonstationary processes are considered. Locally best and point-optimal invariant tests for any given value of ϱ are derived. Special cases of these tests include tests for independence and tests for unit-root hypotheses. The powers of alternative tests are compared numerically for a number of selected testing problems and for a range of design matrices. The results suggest that point-optimal tests are usually preferable to locally best tests, especially for testing values of ϱ greater than or equal to one.

Testing for fourth-order autocorrelation in regression disturbances when first-order autocorrrelation is present

This paper considers the problem of testing the null hypothesis of first-order autoregressive disturbances in the linear regression model against the alternative that the disturbances follow a joint first-order, simple fourth-order autoregressive process. The class of approximate point-optimal invariant tests are discussed and rules are given for choosing an appropriate member from this class of tests. The beneficial nature of these rules is illustrated by a limited empirical power comparison which shows the recommended test has good small-sample power properties.