Specification Testing of Markov Switching Models*

This paper proposes a set of formal tests to address the goodness‐of‐fit of Markov switching models. These formal tests are constructed as tests of model consistency and of both parametric and non‐parametric encompassing. The formal tests are then combined with informal tests using simulation in combination with non‐parametric density and conditional mean estimation. The informal tests are shown to be useful in shedding light on the failure (or success) of the encompassing tests. Several examples are provided.     

On Critical Values of Tests against a Change in Persistence*

Kim, Belaire‐Franch and Amador [Journal of Econometrics (2002) Vol. 109, pp. 389–392] and Busetti and Taylor [Journal of Econometrics (2004) Vol. 123, pp. 33–66] present different percentiles for the same mean score test statistic. We find that the difference by a factor 0.6 is due to systematically different sample analogues. Furthermore, we clarify which sample versions of the mean‐exponential test statistic should be correctly used with which set of critical values. At the same time, we correct some of the limiting distributions found in the literature.     

Simulation‐based Finite Sample Linearity Test against Smooth Transition Models*

In this paper, we use Monte Carlo (MC) testing techniques for testing linearity against smooth transition models. The MC approach allows us to introduce a new test that differs in two respects from the tests existing in the literature. First, the test is exact in the sense that the probability of rejecting the null when it is true is always less than or equal to the nominal size of the test. Secondly, the test is not based on an auxiliary regression obtained by replacing the model under the alternative by approximations based on a Taylor expansion. We also apply MC testing methods for size correcting the test proposed by Luukkonen, Saikkonen and Teräsvirta (Biometrika, Vol. 75, 1988, p. 491). The results show that the power loss implied by the auxiliary regression‐based test is non‐existent compared with a supremum‐based test but is more substantial when compared with the three other tests under consideration.     

Fluctuation Tests for a Change in Persistence*

In this paper, we develop a set of new persistence change tests which are similar in spirit to those of Kim [Journal of Econometrics (2000) Vol. 95, pp. 97–116], Kim et al. [Journal of Econometrics (2002) Vol. 109, pp. 389–392] and Busetti and Taylor [Journal of Econometrics (2004) Vol. 123, pp. 33–66]. While the exisiting tests are based on ratios of sub‐sample Kwiatkowski et al. [Journal of Econometrics (1992) Vol. 54, pp. 158–179]‐type statistics, our proposed tests are based on the corresponding functions of sub‐sample implementations of the well‐known maximal recursive‐estimates and re‐scaled range fluctuation statistics. Our statistics are used to test the null hypothesis that a time series displays constant trend stationarity [I(0)] behaviour against the alternative of a change in persistence either from trend stationarity to difference stationarity [I(1)], or vice versa. Representations for the limiting null distributions of the new statistics are derived and both finite‐sample and asymptotic critical values are provided. The consistency of the tests against persistence change processes is also demonstrated. Numerical evidence suggests that our proposed tests provide a useful complement to the extant persistence change tests. An application of the tests to US inflation rate data is provided.     

Regression‐based Tests for a Change in Persistence*

We show that the minimal forward (reverse) recursive unit tests of Banerjee, Lumsdaine and Stock [Journal of Business and Economics Statistics (1992) Vol. 10, pp. 271–288] are consistent against the alternative of a change in persistence from I(0) to I(1) [I(1) to I(0)]. However, these statistics are also shown to diverge for series which are I(0) throughout. Consequently, a rejection by these tests does not necessarily imply a change in persistence. We propose a further test, based on the ratio of these statistics, which is consistent against changes either from I(0) to I(1), or vice versa, yet does not over‐reject against constant I(0) series. Consistent breakpoint estimators are proposed.     

Selecting a Nonlinear Time Series Model using Weighted Tests of Equal Forecast Accuracy*

Nonlinear time series models have become fashionable tools to describe and forecast a variety of economic time series. A closer look at reported empirical studies, however, reveals that these models apparently fit well in‐sample, but rarely show a substantial improvement in out‐of‐sample forecasts, at least over linear models. One of the many possible reasons for this finding is the use of inappropriate model selection criteria and forecast evaluation criteria. In this paper we therefore propose a novel criterion, which we believe does more justice to the very nature of nonlinear models. Simulations show that this criterion outperforms those criteria currently in use, in the sense that the true nonlinear model is more often found to perform better in out‐of‐sample forecasting than a benchmark linear model. An empirical illustration for US GDP emphasizes its relevance.     

Panel LM Unit‐root Tests with Level Shifts*

This paper proposes a new panel unit‐root test based on the Lagrangian multiplier (LM) principle. We show that the asymptotic distribution of the new panel LM test is not affected by the presence of structural shifts. This result holds under a mild condition that N/T→k, where k is any finite constant. Our simulation study shows that the panel LM unit‐root test is not only robust to the presence of structural shifts, but is more powerful than the popular Im, Pesaran and Shin (IPS) test. We apply our new test to the purchasing power parity (PPP) hypothesis and find strong evidence for PPP.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

Bootstrap HAC Tests for Ordinary Least Squares Regression*

There is a need for tests that are derived from the ordinary least squares (OLS) estimators of regression coefficients and are useful in the presence of unspecified forms of heteroskedasticity and autocorrelation. A method that uses the moving block bootstrap and quasi‐estimators in order to derive a consistent estimator of the asymptotic covariance matrix for the OLS estimators and robust significance tests is proposed. The method is shown to be asymptotically valid and Monte Carlo evidence indicates that it is capable of providing good control of significance levels in finite samples and good power compared with two other bootstrap tests.     

Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

Levels,Differences and ECMs – Principles for Improved Econometric Forecasting*

Unit‐root testing can be a preliminary step in model development, an intermediate step, or an end in itself. Some researchers have questioned the value of any unit‐root and cointegration testing, arguing that restrictions based on theory are at least as effective. Such confusion is unsatisfactory. Needed is a set of principles that limit and define the role of the tacit knowledge of the model builders. In a forecasting context, we enumerate the various possible model selection strategies and, based on simulation and empirical evidence, recommend using these tests to improve the specification of an initial general vector autoregression model.     

Testing for Heteroskedasticity and Predictive Failure in Linear Regression Models*

It is argued that, when researchers wish to carry out a Chow test of the significance of prediction errors, it is necessary to assume homoskedasticity because standard results on heteroskedasticity‐robust tests are not available. The effects of heteroskedasticity on the Chow prediction error test are examined. The implementation of tests for heteroskedasticity is discussed, with the case in which the regressors include dummy variables for prediction error tests receiving special attention. Monte Carlo results are reported.     

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.     

Tests for a Break in Level when the Order of Integration is Unknown*

In this paper, we consider tests for a break in the level of a series at an unknown point in time. It is often the case that uncertainty exists concerning the order of integration of the series; consequently, we focus on tests that are applicable when the order of integration is not known. The size and power of existing tests are analysed, and a modification to one of the established sets of tests is proposed which offers improved performance in certain circumstances.     

Controlling the Overall Significance Level of a Battery of Least Squares Diagnostic Tests*

Double bootstrap methods are used to control the overall significance level of a battery of diagnostic tests applied to a regression model estimated by ordinary least squares. Monte Carlo evidence on the finite sample performance of the bootstrap methods is reported and discussed.     

Testing Steady‐State Restrictions of Linear Rational Expectations Models when Data are Highly Persistent*

Steady‐state restrictions are commonly imposed on highly persistent variables to achieve stationarity prior to confronting rational expectations models with data. However, the resulting steady‐state deviations are often surprisingly persistent indicating that some aspects of the underlying theory may be empirically problematic. This paper discusses how to formulate steady‐state restrictions in rational expectations models with latent forcing variables and test their validity using cointegration techniques. The approach is illustrated by testing steady‐state restrictions for alternative specifications of the New Keynesian model and shown to be able to discriminate between different assumptions on the sources of the permanent shocks.     

Testing for Misspecification in Binary Response Models with Competing Distributions*

In empirical studies, the probit and logit models are often used without checks for their competing distributional specifications. It is also rare for econometric tests to be focused on this issue. Santos Silva [Journal of Applied Econometrics (2001 ), Vol. 16, pp. 577–597] is an important recent exception. By using the conditional moment test principle, we discuss a wide class of non‐nested tests that can easily be applied to detect the competing distributions for the binary response models. This class of tests includes the test of Santos Silva (2001 ) for the same task as a particular example and provides other useful alternatives. We also compare the performance of these tests by a Monte Carlo simulation.     

Diagnostic Tests of Cross‐section Independence for Limited Dependent Variable Panel Data Models*

This article considers the problem of testing for cross‐section independence in limited dependent variable panel data models. It derives a Lagrangian multiplier (LM) test and shows that in terms of generalized residuals of Gourieroux et al. (1987) it reduces to the LM test of Breusch and Pagan (1980) . Because of the tendency of the LM test to over‐reject in panels with large N (cross‐section dimension), we also consider the application of the cross‐section dependence test (CD) proposed by Pesaran (2004) . In Monte Carlo experiments it emerges that for most combinations of N and T the CD test is correctly sized, whereas the validity of the LM test requires T (time series dimension) to be quite large relative to N. We illustrate the cross‐sectional independence tests with an application to a probit panel data model of roll‐call votes in the US Congress and find that the votes display a significant degree of cross‐section dependence.     

Hedonic Price Models for Dynamic Markets*

The price of a product depends on its characteristics and will vary in dynamic markets. The model describes a processing firm that bids in an auction for a heterogeneous and perishable input. The reduced form of this model is estimated as an expanded random parameter model that combines a nonlinear hedonic bid function and inverse input demand functions for characteristics. The model was estimated by using 289,405 transactions from the Icelandic fish auctions. Total catch and gut ratio were the main determinants of marginal prices of characteristics, while the price of cod mainly depended on size, gutting and storage.     

Exact Skewness–Kurtosis Tests for Multivariate Normality and Goodness‐of‐Fit in Multivariate Regressions with Application to Asset Pricing Models*

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross‐equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared with a simulation‐based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal and stable error distributions. In the Gaussian case, finite‐sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi‐stage Monte Carlo test methods. For non‐Gaussian distribution families involving nuisance parameters, confidence sets are derived for the nuisance parameters and the error distribution. The tests are applied to an asset pricing model with observable risk‐free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over 5‐year subperiods from 1926 to 1995.