Exact permutation tests for non-nested non-linear regression models

This paper proposes exact distribution-free permutation tests for the specification of a non-linear regression model against one or more possibly non-nested alternatives. The new tests may be validly applied to a wide class of models, including models with endogenous regressors and lag structures. These tests build on the well-known J test developed by Davidson and MacKinnon [1981. Several tests for model specification in the presence of alternative hypotheses. Econometrica 49, 781–793] and their exactness holds under broader assumptions than those underlying the conventional J test. The J-type test statistics are used with a randomization or Monte Carlo resampling technique which yields an exact and computationally inexpensive inference procedure. A simulation experiment confirms the theoretical results and also shows the performance of the new procedure under violations of the maintained assumptions. The test procedure developed is illustrated by an application to inflation dynamics.     

Tests of non-nested regression models: Small sample adjustments and Monte Carlo evidence

It is known that the small sample significance levels of Cox-type tests of non-nested regression models can be much greater than the nominal level. Adjustments designed to overcome this problem are discussed and two tests are proposed. Monte Carlo evidence on the performance of the tests derived in this paper, the Davidson-MacKinnon J-test and the Fisher-McAleer test is presented. The F-test applied to the comprehensive model is also included in the simulation experiments.     

Class Analysis and Earnings Inequality: Nested and Non-Nested Comparisons of Two Class Models in Israel

This paper demonstrates how nesting and non-nesting analytical strategies provide different answers regarding the comparative utility of theoretical models. This paper demonstrates this incompatibility by testing the empirical efficacy of Goldthorpe's and Wright's class schemes in explaining earnings inequality in Israel. These models are non-nested, because while they partially overlap each other conceptually and empirically, neither can be written as a parametric restriction of the other. As they are non-nested, we cannot test each model against the other by using the conventional sociological approach to hypotheses testing. For the sake of demonstration, however, we show results obtained from the conventional Ordinary Least Squares regression models with conventional Baysian Information Coefficient statistic, serving as criterion for a decision rule. Wright's model was found to be more significant in explaining earnings variations in Israeli society. Yet when we used two models of non-nested specification tests (the Cox-Pesaran model and the J test) to examine each model's unique contribution, neither of these models were able to reject the rival hypothesis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Simple and powerful GMM over-identification tests with accurate size

Based on the series long run variance estimator, we propose a new class of over-identification tests that are robust to heteroscedasticity and autocorrelation of unknown forms. We show that when the number of terms used in the series long run variance estimator is fixed, the conventional J statistic, after a simple correction, is asymptotically F-distributed. We apply the idea of the F-approximation to the conventional kernel-based J tests. Simulations show that the J^∗ tests based on the finite sample corrected J statistic and the F-approximation have virtually no size distortion, and yet are as powerful as the standard J tests.     

Bootstrap J-Test for Panel Data Models with Spatially Dependent Error Components,a Spatial Lag and Additional Endogenous Variables

We develop a bootstrap J-test method for testing a panel model against one non-nested alternative when the competing specifications are estimated by Feasible Generalised Spatial Two Stage Least Squares/Generalised Method of Moments (FGS2SLS/GMM). Both models incorporate spatially correlated error components, thus accounting for spatial heterogeneity via random effects, and accommodate endogenous regressors other than the spatially lagged dependent variable. The proposed scheme is applied to a testing problem involving non-nested wage equations as motivated by the Wage Curve literature and the New Economic Geography theory. Results show that our bootstrap test is a reliable and effective procedure for correcting asymptotic reference critical values and distinguishing between the two rival hypotheses.     

Testing non-nested semiparametric models: an application to Engel curves specification

This paper proposes a test statistic for discriminating between two partly non-linear regression models whose parametric components are non-nested. The statistic has the form of a J-test based on a parameter which artificially nests the null and alternative hypotheses. We study in detail the realistic case where all regressors in the non-linear part are discrete and then no smoothing is required on estimating the non-parametric components. We also consider the general case where continuous and discrete regressors are present. The performance of the test in finite samples is discussed in the context of some Monte Carlo experiments. The test is well motivated for specification testing of Engel curves. We provide an application using data from the 1980 Spanish Expenditure Survey. © 1998 John Wiley & Sons, Ltd.     

Least squares model averaging by Mallows criterion

This paper is in response to a recent paper by Hansen (2007) who proposed an optimal model average estimator with weights selected by minimizing a Mallows criterion. The main contribution of Hansen’s paper is a demonstration that the Mallows criterion is asymptotically equivalent to the squared error, so the model average estimator that minimizes the Mallows criterion also minimizes the squared error in large samples. We are concerned with two assumptions that accompany Hansen’s approach. The first is the assumption that the approximating models are strictly nested in a way that depends on the ordering of regressors. Often there is no clear basis for the ordering and the approach does not permit non-nested models which are more realistic from a practical viewpoint. Second, for the optimality result to hold the model weights are required to lie within a special discrete set. In fact, Hansen noted both difficulties and called for extensions of the proof techniques. We provide an alternative proof which shows that the result on the optimality of the Mallows criterion in fact holds for continuous model weights and under a non-nested set-up that allows any linear combination of regressors in the approximating models that make up the model average estimator. These results provide a stronger theoretical basis for the use of the Mallows criterion in model averaging by strengthening existing findings.     

Robust trend inference with series variance estimator and testing-optimal smoothing parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed-K asymptotics are second-order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).     

Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non‐nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non‐nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well‐behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.     

Testing the specification of multivariate models in the presence of alternative hypotheses

In this paper we propose a non-nested hypothesis test for testing the specification of a multivariate econometric model in the presence of an alternative model which purports to explain the same phenomenon. We demonstrate that the new test statistic tends to minus the same random variable as the CPD test statistic introduced by Pesaran and Deaton (1978), provided that the truth is ‘close’ to the null hypothesis. Since the new test is simpler to compute than the multivariate CPD test, it would seem to be the procedure of choice.     

Testing for a change in persistence in the presence of non-stationary volatility

In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0)$I\left(0\right)$ to I(1)$I\left(1\right)$ behaviour or vice versa, of, inter alia, [Kim, J., 2000. Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116]. We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical illustration using US price inflation data is provided.     

A consistent nonparametric test for nonlinear causality—Specification in time series regression

Since the pioneering work by Granger (1969), many authors have proposed tests of causality between economic time series. Most of them are concerned only with “linear causality in mean”, or if a series linearly affects the (conditional) mean of the other series. It is no doubt of primary interest, but dependence between series may be nonlinear, and/or not only through the conditional mean. Indeed conditional heteroskedastic models are widely studied recently. The purpose of this paper is to propose a nonparametric test for possibly nonlinear causality. Taking into account that dependence in higher order moments are becoming an important issue especially in financial time series, we also consider a test for causality up to the Kth conditional moment. Statistically, we can also view this test as a nonparametric omitted variable test in time series regression. A desirable property of the test is that it has nontrivial power against T^1/2-local alternatives, where T is the sample size. Also, we can form a test statistic accordingly if we have some knowledge on the alternative hypothesis. Furthermore, we show that the test statistic includes most of the omitted variable test statistics as special cases asymptotically. The null asymptotic distribution is not normal, but we can easily calculate the critical regions by simulation. Monte Carlo experiments show that the proposed test has good size and power properties.     

A consistent characteristic function-based test for conditional independence

Y is conditionally independent of Z given X   if Pr{f(y|X,Z)=f(y|X)}=1$\{f(y|X,Z)=f(y|X)\}=1$ for all y on its support, where f(·|·)$f(·|·)$ denotes the conditional density of Y   given (X,Z)$(X,Z)$ or X.$X.$ This paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal. We extend the test of Su and White (2005. A Hellinger-metric nonparametric test for conditional independence. Discussion Paper, Department of Economics, UCSD) in two directions: (1) our test is less sensitive to the choice of bandwidth sequences; (2) our test has power against deviations on the full support of the density of (X,Y,Z$X,Y,Z$). We establish asymptotic normality for our test statistic under weak data dependence conditions. Simulation results suggest that the test is well behaved in finite samples. Applications to stock market data indicate that our test can reveal some interesting nonlinear dependence that a traditional linear Granger causality test fails to detect.     

Bootstrapping the Box–Pierce Q test: A robust test of uncorrelatedness

This paper describes a test of the null hypothesis that the first K autocorrelations of a covariance stationary time series are zero in the presence of statistical dependence. The test is based on the Box–Pierce Q statistic with bootstrap-based P-values. The bootstrap is implemented using a double blocks-of-blocks procedure with prewhitening. The finite sample performance of the bootstrap Q   test is investigated by simulation. In our experiments, the performance is satisfactory for samples of n=500$n=500$. At this sample size, the differences between the empirical and nominal rejection probabilities are essentially eliminated.     

What model for entry in first-price auctions? A nonparametric approach

We develop a selective entry model for first-price auctions that nests two polar models often estimated in the empirical literature on auctions, Levin and Smith (1994), and Samuelson (1985). The selective entry model features a pro-competitive selection effect. The selection effect is shown to be nonparametrically identifiable, and a nonparametric test for its presence is proposed. This test can be used to discriminate between the two polar models.     

An unbiased model comparison test using cross-validation

Social scientists often consider multiple empirical models of the same process. When these models are parametric and non-nested, the null hypothesis that two models fit the data equally well is commonly tested using methods introduced by Vuong (Econometrica 57(2):307–333, 1989) and Clarke (Am J Political Sci 45(3):724–744, 2001; J Confl Resolut 47(1):72–93, 2003; Political Anal 15(3):347–363, 2007). The objective of each is to compare the Kullback–Leibler Divergence (KLD) of the two models from the true model that generated the data. Here we show that both of these tests are based upon a biased estimator of the KLD, the individual log-likelihood contributions, and that the Clarke test is not proven to be consistent for the difference in KLDs. As a solution, we derive a test based upon cross-validated log-likelihood contributions, which represent an unbiased KLD estimate. We demonstrate the CVDM test’s superior performance via simulation, then apply it to two empirical examples from political science. We find that the test’s selection can diverge from those of the Vuong and Clarke tests and that this can ultimately lead to differences in substantive conclusions.     

Local indirect least squares and average marginal effects in nonseparable structural systems

We study the scope of local indirect least squares (LILS) methods for nonparametrically estimating average marginal effects of an endogenous cause X on a response Y in triangular structural systems that need not exhibit linearity, separability, or monotonicity in scalar unobservables. One main finding is negative: in the fully nonseparable case, LILS methods cannot recover the average marginal effect. LILS methods can nevertheless test the hypothesis of no effect in the general nonseparable case. We provide new nonparametric asymptotic theory, treating both the traditional case of observed exogenous instruments Z and the case where one observes only error-laden proxies for Z.     

Inference on inequality from household survey data

I develop a theory of asymptotic inference for the Lorenz curve and the Gini coefficient for testing economic inequality when the data come from stratified and clustered household surveys with large number of clusters per stratum. Using the asymptotic framework of Bhattacharya [Asymptotic Inference from multi-stage surveys. Journal of Econometrics 126(1), 145–171], I derive a weak convergence result for the continuously-indexed Lorenz process even when the underlying density is not uniformly bounded away from zero. I provide analytical formulae for the asymptotic covariance functions that are corrected for both stratification and clustering and develop consistent tests for Lorenz dominance. Inference on the Gini coefficient follows as a corollary. The methods are applied to per capita household expenditure data from the complexly designed Indian national sample survey to test for changes in inequality before and after the reforms of the early 1990s. Ignoring the survey design is seen to produce qualitatively different results, especially in the urban sector where the population sorts more completely into rich and poor neighborhoods.     

Data-based ranking of realised volatility estimators

This paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process. I provide conditions under which the relative accuracy of competing estimators can be consistently estimated (as T→∞), and show that forecast evaluation tests may be adapted to the problem of ranking these estimators. The proposed methods avoid making specific assumptions about microstructure noise, and facilitate comparisons of estimators that would be difficult using methods from the extant literature, such as those based on different sampling schemes. An application to high frequency IBM data between 1996 and 2007 illustrates the new methods.