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We review the literature on the Autoregressive Distributed Lag (ARDL) model, from its origins in the analysis of autocorrelated trend stationary processes to its subsequent applications in the analysis of cointegrated non-stationary time series. We then survey several recent extensions of the ARDL model, including asymmetric and nonlinear generalisations of the ARDL model, the quantile ARDL model, the pooled mean group dynamic panel data model and the spatio-temporal ARDL model.

This article examines the null limit distribution of the quasi-likelihood ratio (QLR) statistic for testing linearity condition against the smooth transition autoregressive (STAR) model. We explicitly show that the QLR test statistic weakly converges to a functional of a multivariate Gaussian process under the null of linearity, which is done by resolving the issue of identification problem arises in two different ways under the null. In contrast with the Lagrange multiplier test that is widely employed for testing the linearity condition, the proposed QLR statistic has an omnibus power, and thus, it complements the existing testing procedure. We show the empirical relevance of our test by testing the neglected nonlinearity of the US fiscal multipliers and growth rates of US unemployment. These empirical examples demonstrate that the QLR test is useful for detecting the nonlinear structure among economic variables.

This study tests for the sandwich-form asymptotic covariance matrices entailed by conditionally heteroskedastic and/or autocorrelated regression errors or conditionally uncorrelated homoskedastic errors. In doing so, we enable the empirical researcher to estimate the asymptotic covariance matrix of the quasi-maximum likelihood estimator by supposing a possibly misspecified model for error distribution. Accordingly, we provide test methodologies by extending the approaches in Cho and White (in: Chang Y, Fomby T, Park JY (eds) Advances in econometrics: essays in honor of Peter CB Phillips. Emerald Group Publishing Limited, West Yorkshire, 2014) and Cho and Phillips (J Econ 202:45–56, 2018a) to detect the influence of heteroskedastic and/or autocorrelated regression errors on the asymptotic covariance matrix. In particular, we establish a sequential testing procedure to achieve our goal. We affirm the theory on our test statistics through simulation and apply the test statistics to energy price growth rate data for illustrative purposes; here, we also apply our test methodology to test the fully correct model hypothesis.

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two k dimensional symmetric positive-definite matrices A and B : the traces of A B − 1 and B A − 1 are equal to k if and only if A = B . Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.

We relax the differentiability condition of standard econometric models and instead assume that mod- els are directionally differentiable (Gdifferentiable). Using this new condition, we analyze the asymptotic distribution of the extremum estimator and show that its asymptotic distribution can be rep- resented as a function of a Gaussian stochastic process indexed by direction. Our analysis also treats the analysis of differentiable models as a special case. Furthermore, we refine the standard quasi-likelihood ratio, Wald, and Lagrange multiplier test statistics under the directional differentiability condition and examine their asymptotic behaviors. We also interrelate them with the asymptotic behaviors of the stan- dard test statistics obtained under the differentiability condition.

This study examined the extreme learning machine (ELM) applied to the Wald test statistic for the model specification of the conditional mean, which we call the WELM testing procedure. The omnibus test statistics available in the literature weakly converge to a Gaussian stochastic process under the null that the model is correct, and this makes their application inconvenient. By contrast, the WELM testing procedure is straightforwardly applicable when detecting model misspecification. We applied the WELM testing procedure to the sequential testing procedure formed by a set of polynomial models and estimate an approximate conditional expectation. We then conducted extensive Monte Carlo experiments to evaluate the performance of the sequential WELM testing procedure and verify that it consistently estimates the most parsimonious conditional mean when the set of polynomial models contains a correctly specified model. Otherwise, it consistently rejects all the models in the set.

Summary We provide a methodology for testing a polynomial model hypothesis by generalizing the approach and results of Baek, Cho, and Phillips (Journal of Econometrics, 2015, 187, 376–384; BCP), which test for neglected nonlinearity using power transforms of regressors against arbitrary nonlinearity. We use the BCP quasi-likelihood ratio test and deal with the new multifold identification problem that arises under the null of the polynomial model. The approach leads to convenient asymptotic theory for inference, has omnibus power against general nonlinear alternatives, and allows estimation of an unknown polynomial degree in a model by way of sequential testing, a technique that is useful in the application of sieve approximations. Simulations show good performance in the sequential test procedure in both identifying and estimating unknown polynomial order. The approach, which can be used empirically to test for misspecification, is applied to a Mincer (Journal of Political Economy, 1958, 66, 281–302; Schooling, Experience and Earnings, Columbia University Press, 1974) equation using data from Card (in Christofides, Grant, and Swidinsky (Eds.), Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp, University of Toronto Press, 1995, 201-222) and Bierens and Ginther (Empirical Economics, 2001, 26, 307–324). The results confirm that the standard Mincer log earnings equation is readily shown to be misspecified. The applications consider different datasets and examine the impact of nonlinear effects of experience and schooling on earnings, allowing for flexibility in the respective polynomial representations.

Xiao (2009) develops a novel estimation technique for quantile cointegrated time series by extending Phillips and Hansen’s (1990) semiparametric approach and Saikkonen’s (1991) parametrically augmented approach. This paper extends Pesaran and Shin’s (1998) autoregressive distributed-lag approach into quantile regression by jointly analyzing short-run dynamics and long-run cointegrating relationships across a range of quantiles. We derive the asymptotic theory and provide a general package in which the model can be estimated and tested within and across quantiles. We further affirm our theoretical results by Monte Carlo simulations. The main utilities of this analysis are demonstrated through the empirical application to the dividend policy in the US.

We study Kolmogorov–Smirnov goodness-of-fit tests for evaluating distributional hypotheses where unknown parameters need to be fitted. Following the work of Pollard (1980), our approach uses a Cramér–von Mises minimum distance estimator for parameter estimation. The asymptotic null distribution of the resulting test statistic is represented by invariance principle arguments as a functional of a Brownian bridge in a simple regression format for which asymptotic critical values are readily delivered by simulations. Asymptotic power is examined under fixed and local alternatives and finite sample performance of the test is evaluated in simulations. The test is applied to measure top income shares using Korean income tax return data over 2007–2012. When the data relate to estimating the upper 0.1% or higher income shares, the conventional assumption of a Pareto tail distribution cannot be rejected. But the Pareto tail hypothesis is rejected for estimating the top 1.0% or 0.5% income shares at the 5% significance level. A supplement containing proofs and data descriptions is available online.

We introduce a statistic testing for neglected nonlinearity using extreme learning machines and call it ELMNN test. The ELMNN test is very convenient and can be widely applied because it is obtained as a by-product of estimating linear models. For the proposed test statistic, we provide a set of regularity conditions under which it asymptotically follows a chi-squared distribution under the null. We conduct Monte Carlo experiments and examine how it behaves when the sample size is finite. Our experiment shows that the test exhibits the properties desired by our theory.

We provide a methodology for testing a polynomial model hypothesis by extending the approach and results of Baek, Cho, and Phillips (2015; Journal of Econometrics; BCP) that tests for neglected nonlinearity using power transforms of regressors against arbitrary nonlinearity. We examine and generalize the BCP quasi-likelihood ratio test dealing with the multifold identification problem that arises under the null of the polynomial model. The approach leads to convenient asymptotic theory for inference, has omnibus power against general nonlinear alternatives, and allows estimation of an unknown polynomial degree in a model by way of sequential testing, a technique that is useful in the application of sieve approximations. Simulations show good performance in the sequential test procedure in identifying and estimating unknown polynomial order. The approach, which can be used empirically to test for misspecification, is applied to a Mincer (1958, 1974) equation using data from Card (1995). The results confirm that Mincer’s log earnings equation is easily shown to be misspecified by including nonlinear effects of experience and schooling on earnings, with some flexibility required in the respective polynomial degrees.

We use the quasi-likelihood ratio statistic to test the hypothesis that a power transformation is ineffective. The null is not standard because a parameter is unidentified, and we can also let the power coefficient be zero for the null.

Testing the correct model specification hypothesis for artificial neural network (ANN) models of the conditional mean is not standard. The traditional Wald, Lagrange multiplier, and quasi-likelihood ratio statistics weakly converge to functions of Gaussian processes, rather than to convenient chi-squared distributions. Also, their large-sample null distributions are problem dependent, limiting applicability. We overcome this challenge by applying functional regression methods of Cho et al. [8] to extreme learning machines (ELM). The Wald ELM (WELM) test statistic proposed here is easy to compute and has a large-sample standard chi-squared distribution under the null hypothesis of correct specification. We provide associated theory for time-series data and affirm our theory with some Monte Carlo experiments.

We provide a family of tests for the IID hypothesis based on generalized runs, powerful against unspecified alternatives, providing a useful complement to tests designed for specific alternatives, such as serial correlation, GARCH, or structural breaks. Our tests have appealing computational simplicity in that they do not require kernel density estimation, with the associated challenge of bandwidth selection. Simulations show levels close to nominal asymptotic levels. Our tests have power against both dependent and heterogeneous alternatives, as both theory and simulations demonstrate.

Tests for regression neglected nonlinearity based on artificial neural networks (ANNs) have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold. This implies that the asymptotic behavior of general ANN-based tests for neglected nonlinearity is still an open question. Here we analyze a convenient ANN-based quasi-likelihood ratio statistic for testing neglected nonlinearity, paying careful attention to both components of the null. We derive the asymptotic null distribution under each component separately and analyze their interaction. Somewhat remarkably, it turns out that the previously known asymptotic null distribution for the type 1 case still applies, but under somewhat stronger conditions than previously recognized. We present Monte Carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new, stronger regularity conditions are violated.

We examine the use of the likelihood ratio (LR) statistic to test for unobserved heterogeneity in duration models, based on mixtures of exponential or Weibull distributions. We consider both the uncensored and censored duration cases. The asymptotic null distribution of the LR test statistic is not the standard chi-square, as the standard regularity conditions do not hold. Instead, there is a nuisance parameter identified only under the alternative, and a null parameter value on the boundary of the parameter space, as in Cho and White (2007a) . We accommodate these and provide methods delivering consistent asymptotic critical values. We conduct a number of Monte Carlo simulations, comparing the level and power of the LR test statistic to an information matrix (IM) test due to Chesher (1984) and Lagrange multiplier (LM) tests of Kiefer (1985) and Sharma (1987) . Our simulations show that the LR test statistic generally outperforms the IM and LM tests. We also revisit the work of van den Berg and Ridder (1998) on unemployment durations and of Ghysels et al. (2004) on interarrival times between stock trades, and, as it turns out, affirm their original informal inferences.

Least absolute deviations (LAD) estimation of linear time series models is considered under conditional heteroskedasticity and serial correlation. The limit theory of the LAD estimator is obtained without assuming the finite density condition for the errors that is required in standard LAD asymptotics. The results are particularly useful in application of LAD estimation to financial time series data.

We analyze use of a quasi-likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a Markov regime-switching context. This test exploits mixture properties implied by the regime-switching process, but ignores certain implied serial correlation properties. When formulated in the natural way, the setting is nonstandard, involving nuisance parameters on the boundary of the parameter space, nuisance parameters identified only under the alternative, or approximations using derivatives higher than second order. We exploit recent advances by Andrews (2001) and contribute to the literature by extending the scope of mixture models, obtaining asymptotic null distributions different from those in the literature. We further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime-switching tests. We compare the size and power of our statistics to other useful tests for regime switching via Monte Carlo methods and find relatively good performance. We apply our methods to reexamine the classic cartel study of Porter (1983) and reaffirm Porter's findings.

We study Kolmogorov-Smirnov goodness of fit tests for evaluating distributional hypotheses where unknown parameters need to be fitted. Following work of Pollard (1979), our approach uses a Cramer-von Mises minimum distance estimator for parameter estimation. The asymptotic null distribution of the resulting test statistic is represented by invariance principle arguments as a functional of a Brownian bridge in a simple regression format for which asymptotic critical values are readily delivered by simulations. Asymptotic power is examined under fixed and local alternatives and finite sample performance of the test is evaluated in simulations. The test is applied to measure top income shares using Korean income tax return data over 2007 to 2012. When the data relate to the upper 0.1% or higher tail of the income distribution, the conventional assumption of a Pareto tail distribution cannot be rejected. But the Pareto tail hypothesis is rejected for the top 1.0% or 0.5% incomes at the 5% significance level.

We give an apparently new characterization of the equality of two positive definite matrices A and B; and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive definite matrices. Our primary focus is on testing the information matrix equality (e.g., White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small asn = 50: Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they don’t, they nevertheless perform respectably.

We revisit the twofold identification problem discussed by Cho, Ishida, and White (2011), which arises when testing for neglected nonlinearity by artificial neural networks. We do not use the so-called “no-zero” condition and employ a sixth-order expansion to obtain the asymptotic null distribution of the quasi-likelihood ratio (QLR) test. In particular, we avoid restricting the number of explanatory variables in the activation function by using the distance and direction method discussed in Cho and White (2012). We find that the QLR test statistic can still be used to handle the twofold identification problem appropriately under the set of mild regularity conditions provided here, so that the asymptotic null distribution can be obtained in a manner similar to that in Cho, Ishida, and White ( 2011). This also implies that the weighted bootstrap in Hansen (1996) can be successfully exploited when testing the linearity hypothesis using the QLR test.

We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three different ways, each producing its own identification problem. We call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More specifically, the QLR statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided. Test power is analyzed and an empirical application to crop-yield distributions is provided. The paper also considers generalizations of the Box-Cox transformation, which are associated with the QLR test statistic.

We provide an altrnative proof that the Ordinary Least Squares estimator is the (conditionally) best linear unbiased estimator.

We provide a three line proof that the Ordinary Least Squares estimator is the (conditionally) best

We illustrate the need to use higher-order (specifically sixth-order) expansions in order to properly determine the asymptotic distribution of a standard artificial neural network test for neglected nonlinearity. The test statistic is a quasi-likelihood ratio (QLR) statistic designed to test whether the mean square prediction error improves by including an additional hidden unit with an activation function violating the no-zero condition in Cho, Ishida, and White ( 2011 ). This statistic is also shown to be asymptotically equivalent under the null to the Lagrange multiplier (LM) statistic of Luukkonen, Saikkonen, and Teräsvirta ( 1988 ) and Teräsvirta ( 1994 ). In addition, we compare the power properties of our QLR test to one satisfying the no-zero condition and find that the latter is not consistent for detecting a DGP with neglected nonlinearity violating an analogous no-zero condition, whereas our QLR test is consistent.

Statistics are developed to test for the presence of an asymptotic discontinuity (or infinite density or peakedness) in a probability density at the median. The approach makes use of work by Knight (1998) on L_1 estimation asymptotics in conjunction with non-parametric kernel density estimation methods. The size and power of the tests are assessed, and conditions under which the tests have good performance are explored in simulations. The new methods are applied to stock returns of leading companies across major U.S. industry groups. The results confirm the presence of infinite density at the median as a new significant empirical evidence for stock return distributions.

Least absolute deviations (LAD) estimation of linear time-series models is considered under conditional heteroskedasticity and serial correlation. The limit theory of the LAD estimator is obtained without assuming the finite density condition for the errors that is required in standard LAD asymptotics. The results are particularly useful in application of LAD estimation to financial time-series data.

In this paper, we study functional regression and its properties in testing the hypothesis of a constant zero mean function or an unknown constant non-zero mean function. As we show, the associated Wald test statistics have standard chi-square limiting null distributions, standard non-central chi-square distributions for local alternatives converging to zero at root-n rate, and are consistent against global alternatives. These properties permit computationally convenient tests for hypotheses involving nuisance parameters. In particular, we develop new alternatives to tests for mixture distributions and for regression misspecification, both of which involve nuisance parameters identified only under the alternative. In Monte Carlo studies, we find that our tests have well behaved levels. We find that the new procedures may sacrifice only exploit the covariance structure of the Gaussian processes underlying our statistics. Further, functional regression tests can have power better than existing methods that do not exploit this covariance structure, like the specification testing procedures of Bierens (1982, 1990) or Stinchcombe and White (1998).

We provide a familiy of tests for the IID hypothesis based on generalized runs, powerful against unspecified alternatives, providing a useful complement to test designed for specific alternatives, such as serial correlation, GARCH, or structural breaks. Our tests have appealing computational simplicity in that they do not require kernel density estimation, with the associated challenge of bandwidth selection. Simulations show levels close to nominal asymptotic levels.Our tests have power against both dependent and heterogeneous alternatives, as both theory and simulations demonstrate.

We provide a new test for equality of covariance matrices that leads to a convenient mechanism for testing specification using the information matrix equality. The test relies on a new characterization of equality between two k dimensional positive-definite matrices A and B: the traces of AB^{–1} and BA^{–1} are equal to k if and only if A = B. Using this criterion, we introduce a class of omnibus test statistics for equality of covariance matrices and examine their null, local, and global approximations under some mild regularity conditions. Monte Carlo experiments are conducted to explore the performance characteristics of the test criteria and provide comparisons with existing tests under the null hypothesis and local and global alternatives. The tests are applied to the classic empirical models for voting turnout investigated by Wolfinger and Rosenstone (1980) and Nagler (1991, 1994). Our tests show that all classic models for the 1984 presidential voting turnout are misspecified in the sense that the information matrix equality fails.