A Test for Slope Heterogeneity in Fixed Effects Models

Typical panel data models make use of the assumption that the regression parameters are the same for each individual cross-sectional unit. We propose tests for slope heterogeneity in panel data models. Our tests are based on the conditional Gaussian likelihood function in order to avoid the incidental parameters problem induced by the inclusion of individual fixed effects for each cross-sectional unit. We derive the Conditional Lagrange Multiplier test that is valid in cases where N → ∞ and T is fixed. The test applies to both balanced and unbalanced panels. We expand the test to account for general heteroskedasticity where each cross-sectional unit has its own form of heteroskedasticity. The modification is possible if T is large enough to estimate regression coefficients for each cross-sectional unit by using the MINQUE unbiased estimator for regression variances under heteroskedasticity. All versions of the test have a standard Normal distribution under general assumptions on the error distribution as N → ∞. A Monte Carlo experiment shows that the test has very good size properties under all specifications considered, including heteroskedastic errors. In addition, power of our test is very good relative to existing tests, particularly when T is not large.     

Robust Lagrange multiplier test for detecting ARCH/GARCH effect using permutation and bootstrap

The Lagrange Multiplier (LM) test is one of the principal tools to detect ARCH and GARCH effects in financial data analysis. However, when the underlying data are non‐normal, which is often the case in practice, the asymptotic LM test, based on the χ²‐approximation of critical values, is known to perform poorly, particularly for small and moderate sample sizes. In this paper we propose to employ two re‐sampling techniques to find critical values of the LM test, namely permutation and bootstrap. We derive the properties of exactness and asymptotically correctness for the permutation and bootstrap LM tests, respectively. Our numerical studies indicate that the proposed re‐sampled algorithms significantly improve size and power of the LM test in both skewed and heavy‐tailed processes. We also illustrate our new approaches with an application to the analysis of the Euro/USD currency exchange rates and the German stock index. The Canadian Journal of Statistics 40: 405–426; 2012 © 2012 Statistical Society of Canada     

基于回归的时变偏度和时变峰度识别检验

资产收益率时变高阶矩建模的首要前提是资产收益率的偏度和峰度具有时变性,即资产收益率存在类似于异方差性的异偏度和异峰度特征。目前文献中的时变偏度和时变峰度识别检验存在适用性较差且检验功效较低等不足。本文提出基于回归的检验方法识别资产收益率偏度和峰度的时变性。该检验一方面利用概率积分变换缓解了拉格朗日乘数检验对资产收益率序列的高阶矩存在性的限制,另一方面考虑了检验统计量中参数估计的不确定性对其统计性质的影响,具有良好的渐近统计性质且适用性更广。蒙特卡洛模拟表明该检验具有良好的有限样本性质,具有合适的检验水平和较高的检验功效。最后,将基于回归的检验运用于上证综指和深圳成指收益率的时变建模研究中。     

On Testing Sample Selection Bias Under the Multicollinearity Problem

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993) is severe, (i) the t-test based on the Heckman-Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996) that the standard regression-based t-test (Heckman, 1979) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982), are robust to nonnormality but have very little power.     

Bootstrap methods for single structural change tests: power versus corrected size and empirical illustration

This paper assesses the performance of tests for a single structural change at unknown date when regressors are stationary, trending and when they have a break in mean. Size and power of the test procedures are compared in a simulation setup particularly aimed at autoregressive models using their limiting distribution and some bootstrap approximations. The comparisons are performed using graphical methods, namely P value discrepancy plots and size–power curves. The simulation study gives some interesting insights to the test procedures. Indeed, it documents that tests based on the conventional asymptotic distribution are oversized in small samples. The size correction is achieved by some bootstrap methods which appear to possess reasonable size properties. For the power study, the proposed bootstrap method improves on the asymptotic approximations of some tests for heteroskedastic regression errors especially when there is a mean-shift in the regressors. This result has not been found for the case of i.i.d. errors where the bootstrap tests have the same power properties as the tests based on the asymptotic approximations. We finally study the relationship between two monthly US interest rates. The results show that such relationship has been altered by a regime-shift located in May 1981.     

Robust Test for Spatial Error Model: Considering Changes of Spatial Layouts and Distribution Misspecification

This article suggests a robust LM (Lagrange Multiplier) test for spatial error model which not only reduces the influence of spatial lag dependence immensely, but also presents robust changes of spatial layouts and distribution misspecification. Monte Carlo simulation results imply that existing LM tests have serious size and power distortion with the presence of spatial lag dependence, group interaction or nonnormal distribution, but the robust LM test of this article shows well performance.     

Double-length regressions for linear and log-linear regressions with AR(1) disturbances

This paper derives Lagrange Multiplier tests based on double-length artificial regressions (DLR) for testing linear and log-linear regressions with AR(1) disturbances against Box-Cox alternatives These DLR tests are easier to compute than the corresponding likelihood ratio tests, and are easily generalized to test jointly for functional form and serial correlation. Two illustrative examples are given to show the importance of jointly testing for functional form and serial correlation.     

固定效应模型基于个体LM的新可混合性检验

对固定效应模型,本文基于拉格朗日乘数（LM）原理提出了一种新的可混合性检验。不同于已有的LM型可混合性检验,这里使用每个截面个体的LM统计量构建可混合性检验统计量。数理分析表明,本文所提的方法有着渐进正态性,对于扰动项的异方差和非正态均稳健,且与PY检验（Pesaran&Yamagata,2008）渐近等价。Monte Carlo模拟实验表明,相对于PY检验及另外两种LM型的可混合性检验,对于不同大小的 ,本文提出的方法有着良好的水平表现和更优越的检验势。     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

SIMPLE LM TESTS FOR THE UNBALANCED NESTED ERROR COMPONENT REGRESSION MODEL

This paper derives several Lagrange Multiplier tests for the unbalanced nested error component model. Economic data with a natural nested grouping include firms grouped by industry; or students grouped by schools. The LM tests derived include the joint test for both effects as well as the test for one effect conditional on the presence of the other. The paper also derives the standardized versions of these tests, their asymptotic locally mean most powerful version as well as their robust to local misspecification version. Monte Carlo experiments are conducted to study the performance of these LM tests.     

Bias correction for the regression-based LM fractional integration test

This paper examines the finite-sample behavior of the Lagrange Multiplier (LM) test for fractional integration proposed by Breitung and Hassler (J. Econom. 110:167–185, 2002). We find by extensive Monte Carlo simulations that size distortions can be quite large in small samples. These are caused by a finite-sample bias towards the alternative. Analytic expressions for this bias are derived, based on which the test can easily be corrected.     

Pooling multivariate data under W, LR and LM tests

Two independent random samples are drawn from two multivariate normal populations with mean vectors μ1 and μ2 and a common variance-covariance matrix Σ. Ahmed and Saleh (1990) considered preliminary test maximum likelihood estimator (PMLTE) for estimating μ1 based on the Hotelling's T _N ², when it is suspected that μ1=μ2. In this paper, the PTMLE based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. Using the quadratic risk function, the conditions of superiority of the proposed estimator for departure parameter are derived. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the PTMLE based on W test produces the highest minimum guaranteed efficiencies compared to UMLE among the three test procedures.     

UNIFORM ASYMPTOTIC EXPANSIONS APPLIED TO SEQUENTIAL t - and t2-TESTS

Using some uniform asymptotic expansions for parabolic cylinder functions recently developed by Olver (1959), various integrals associated with the sequential t- and t² -tests are evaluated asymptotically in terms of the sample size. Then the continuation region inequalities for these tests are inverted and expressed in terms of well known test criteria. It should be pointed out that the inversion of the continuation regions in terms of the well known statistics yields forms for the sequential tests that are more easily applicable by the practitioner than the forms yielded by the method of Rushton. Furthermore, using these inequalities and the asymptotic normality of the test criteria, finite sure termination of sequential t- and t²-test procedures readily follow. Based on simulation studies, power comparisons of the two approximations are also made.     

Testing for Serial Correlation in Fixed-Effects Panel Data Models

In this article, we propose various tests for serial correlation in fixed-effects panel data regression models with a small number of time periods. First, a simplified version of the test suggested by Wooldridge (2002) and Drukker (2003) is considered. The second test is based on the Lagrange Multiplier (LM) statistic suggested by Baltagi and Li (1995), and the third test is a modification of the classical Durbin–Watson statistic. Under the null hypothesis of no serial correlation, all tests possess a standard normal limiting distribution as N tends to infinity and T is fixed. Analyzing the local power of the tests, we find that the LM statistic has superior power properties. Furthermore, a generalization to test for autocorrelation up to some given lag order and a test statistic that is robust against time dependent heteroskedasticity are proposed.     

A monte carlo study of the wald,lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.     

Modified first-difference estimator in a panel data model with unobservable factors both in the errors and the regressors when the time dimension is small

Panel data models with factor structures in both the errors and the regressors have received considerable attention recently. In these models, the errors and the regressors are correlated and the standard estimators are inconsistent. This paper shows that, for such models, a modified first-difference estimator (in which the time and the cross-sectional dimensions are interchanged) is consistent as the cross-sectional dimension grows but the time dimension is small. Although the estimator has a non standard asymptotic distribution, t and F tests have standard asymptotic distribution under the null hypothesis.     

Misspecification tests for periodic long memory GARCH models

Distributional theory for Quasi-Maximum Likelihood estimators in long memory conditional heteroskedastic models is not formally defined, even asymptotically. Because of that, this paper analyses the real size and power of the likelihood ratio and the Lagrange multiplier misspecification tests when periodic long memory GARCH models are involved. The performance of these tests is studied by means of Monte Carlo simulations with respect to the class of generalized long memory GARCH models. For this class of models, analytical derivatives are developed. An application to the USD/JPY exchange rate is also provided.     

Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Robust inference for predictability in smooth transition predictive regressions

This article provides a novel test for predictability within a nonlinear smooth transition predictive regression (STPR) model where inference is complicated due not only to the presence of persistent, local to unit root, predictors, and endogeneity but also the presence of unidentified parameters under the null of no predictability. In order to circumvent the unidentified parameters problem, t? statistic for the predictor in the STPR model is optimized over the Cartesian product of the spaces for the transition and threshold parameters; and to address the di?culties due to persistent and endogenous predictors, the instrumental variable (IVX) method originally developed in the linear cointegration testing framework is adopted within the STPR model. Limit distribution of this statistic (i.e., sup?t^IVX test) is shown to be nuisance parameter-free and robust to the local to unit root and endogenous regressors. Simulations show that sup?t^IVX has good size and power properties. An application to stock return predictability reveals presence of asymmetric regime-dependence and variability in the strength and size of predictability across asset-related (e.g., dividend/price ratio) vs. other (e.g., default yield spread) predictors.