LM tests of spatial dependence based on bootstrap critical values

To test the existence of spatial dependence in an econometric model, a convenient test is the Lagrange Multiplier (LM) test. However, evidence shows that, in finite samples, the LM test referring to asymptotic critical values may suffer from the problems of size distortion and low power, which become worse with a denser spatial weight matrix. In this paper, residual-based bootstrap methods are introduced for asymptotically refined approximations to the finite sample critical values of the LM statistics. Conditions for their validity are clearly laid out and formal justifications are given in general, and in detail under several popular spatial LM tests using Edgeworth expansions. Monte Carlo results show that when the conditions are not fully met, bootstrap may lead to unstable critical values that change significantly with the alternative, whereas when all conditions are met, bootstrap critical values are very stable, approximate much better the finite sample critical values than those based on asymptotics, and lead to significantly improved size and power. The methods are further demonstrated using more general spatial LM tests, in connection with local misspecification and unknown heteroskedasticity.     

Robust tests for heteroskedasticity in the one-way error components model

This paper constructs tests for heteroskedasticity in one-way error components models, in line with Baltagi et al. [Baltagi, B.H., Bresson, G., Pirotte, A., 2006. Joint LM test for homoskedasticity in a one-way error component model. Journal of Econometrics 134, 401–417]. Our tests have two additional robustness properties. First, standard tests for heteroskedasticity in the individual component are shown to be negatively affected by heteroskedasticity in the remainder component. We derive modified tests that are insensitive to heteroskedasticity in the component not being checked, and hence help identify the source of heteroskedasticity. Second, Gaussian-based LM tests are shown to reject too often in the presence of heavy-tailed (e.g. t$t$-Student) distributions. By using a conditional moment framework, we derive distribution-free tests that are robust to non-normalities. Our tests are computationally convenient since they are based on simple artificial regressions after pooled OLS estimation.     

Heteroskedasticity, autocorrelation, and spatial correlation robust inference in linear panel models with fixed-effects

This paper develops an asymptotic theory for test statistics in linear panel models that are robust to heteroskedasticity, autocorrelation and/or spatial correlation. Two classes of standard errors are analyzed. Both are based on nonparametric heteroskedasticity autocorrelation (HAC) covariance matrix estimators. The first class is based on averages of HAC estimators across individuals in the cross-section, i.e. “averages of HACs”. This class includes the well known cluster standard errors analyzed by Arellano (1987) as a special case. The second class is based on the HAC of cross-section averages and was proposed by Driscoll and Kraay (1998). The ”HAC of averages” standard errors are robust to heteroskedasticity, serial correlation and spatial correlation but weak dependence in the time dimension is required. The “averages of HACs” standard errors are robust to heteroskedasticity and serial correlation including the nonstationary case but they are not valid in the presence of spatial correlation. The main contribution of the paper is to develop a fixed-b asymptotic theory for statistics based on both classes of standard errors in models with individual and possibly time fixed-effects dummy variables. The asymptotics is carried out for large time sample sizes for both fixed and large cross-section sample sizes. Extensive simulations show that the fixed-b approximation is usually much better than the traditional normal or chi-square approximation especially for the Driscoll-Kraay standard errors. The use of fixed-b critical values will lead to more reliable inference in practice especially for tests of joint hypotheses.     

Evaluating GARCH models

In this paper, a unified framework for testing the adequacy of an estimated GARCH model is presented. Parametric Lagrange multiplier (LM) or LM type tests of no ARCH in standardized errors, linearity, and parameter constancy are proposed. The asymptotic null distributions of the tests are standard, which makes application easy. Versions of the tests that are robust against nonnormal errors are provided. The finite sample properties of the test statistics are investigated by simulation. The robust tests prove superior to the nonrobust ones when the errors are nonnormal. They also compare favourably in terms of power with misspecification tests previously proposed in the literature.     

Testing for heteroskedasticity and serial correlation in a random effects panel data model

This paper considers a panel data regression model with heteroskedastic as well as serially correlated disturbances, and derives a joint LM test for homoskedasticity and no first order serial correlation. The restricted model is the standard random individual error component model. It also derives a conditional LM test for homoskedasticity given serial correlation, as well as, a conditional LM test for no first order serial correlation given heteroskedasticity, all in the context of a random effects panel data model. Monte Carlo results show that these tests along with their likelihood ratio alternatives have good size and power under various forms of heteroskedasticity including exponential and quadratic functional forms.     

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.     

Heteroskedasticity-consistent covariance matrix estimators for spatial autoregressive models

In the presence of heteroskedasticity, conventional test statistics based on the ordinary least squares (OLS) estimator lead to incorrect inference results for the linear regression model. Given that heteroskedasticity is common in cross-sectional data, the test statistics based on various forms of heteroskedasticity-consistent covariance matrices (HCCMs) have been developed in the literature. In contrast to the standard linear regression model, heteroskedasticity is a more serious problem for spatial econometric models, generally causing inconsistent extremum estimators of model coefficients. This paper investigates the finite sample properties of the heteroskedasticity-robust generalized method of moments estimator (RGMME) for a spatial econometric model with an unknown form of heteroskedasticity. In particular, it develops various HCCM-type corrections to improve the finite sample properties of the RGMME and the conventional Wald test. The Monte Carlo results indicate that the HCCM-type corrections can produce more accurate results for inference on model parameters and the impact effects estimates in small samples.     

Testing for serial correlation,spatial autocorrelation and random effects using panel data

This paper considers a spatial panel data regression model with serial correlation on each spatial unit over time as well as spatial dependence between the spatial units at each point in time. In addition, the model allows for heterogeneity across the spatial units using random effects. The paper then derives several Lagrange multiplier tests for this panel data regression model including a joint test for serial correlation, spatial autocorrelation and random effects. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin and Bera [1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York] and in the panel data context by Baltagi et al. [2003. Testing panel data regression models with spatial error correlation. Journal of Econometrics 117, 123–150]. The second is the LM tests for the error component panel data model with serial correlation derived by Baltagi and Li [1995. Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics 68, 133–151]. Hence, the joint LM test derived in this paper encompasses those derived in both strands of earlier works. In fact, in the context of our general model, the earlier LM tests become marginal LM tests that ignore either serial correlation over time or spatial error correlation. The paper then derives conditional LM and LR tests that do not ignore these correlations and contrast them with their marginal LM and LR counterparts. The small sample performance of these tests is investigated using Monte Carlo experiments. As expected, ignoring any correlation when it is significant can lead to misleading inference.     

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.     

New robust tests for the parameters of the Weibull distribution for complete and censored data

Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown is examined. In simulation studies the behavior in finite sample size and in contaminated data is analyzed and the new method is compared to existing ones. Here it is shown that the new tests based on likelihood depth give quite good results compared to standard methods and are robust against contamination. They are also robust in right-censored data in contrast to existing methods like the method of medians.     

Heteroskedasticity and Neglected Parameter Heterogeneity

The paper studies the consequences of neglecting parameter heterogeneity for the linear regression model and cross-sectional data. Monte-Carlo experiments are used to illustrate that neglected parameter heterogeneity typically leads to (a) regression coefficients that are economically meaningless and (b)significant test statistics for heteroskedasticity and, possibly non-normality. The paper concludes that evidence for heteroskedasticity should not routinely lead to the use of White's well-known heteroskedasticity-consistent variance covariance matrix estimator. If heteroskedasticity is caused by neglected parameter heterogeneity or other causes of heteroskedasticity, such as wrong functional form, White's estimator will not serve any useful purpose.     

Real‐time detection of regimes of predictability in the US equity premium

We propose new real‐time monitoring procedures for the emergence of end‐of‐sample predictive regimes using sequential implementations of standard (heteroskedasticity‐robust) regression t‐statistics for predictability applied over relatively short time periods. The procedures we develop can also be used for detecting historical regimes of temporary predictability. Our proposed methods are robust to both the degree of persistence and endogeneity of the regressors in the predictive regression and to certain forms of heteroskedasticity in the shocks. We discuss how the monitoring procedures can be designed such that their false positive rate can be set by the practitioner at the start of the monitoring period using detection rules based on information obtained from the data in a training period. We use these new monitoring procedures to investigate the presence of regime changes in the predictability of the US equity premium at the 1‐month horizon by traditional macroeconomic and financial variables, and by binary technical analysis indicators. Our results suggest that the 1‐month‐ahead equity premium has temporarily been predictable, displaying so‐called “pockets of predictability,” and that these episodes of predictability could have been detected in real time by practitioners using our proposed methodology.     

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.     

Statistical inference on regression with spatial dependence

Central limit theorems are developed for instrumental variables estimates of linear and semiparametric partly linear regression models for spatial data. General forms of spatial dependence and heterogeneity in explanatory variables and unobservable disturbances are permitted. We discuss estimation of the variance matrix, including estimates that are robust to disturbance heteroscedasticity and/or dependence. A Monte Carlo study of finite-sample performance is included. In an empirical example, the estimates and robust and non-robust standard errors are computed from Indian regional data, following tests for spatial correlation in disturbances, and nonparametric regression fitting. Some final comments discuss modifications and extensions.     

Testing panel data regression models with spatial error correlation

This paper derives several lagrange multiplier (LM) tests for the panel data regression model with spatial error correlation. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin (Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Dordrecht; Rao's score test in spatial econometrics, J. Statist. Plann. Inference 97 (2001) 113) and Anselin et al. (Regional Sci. Urban Econom. 26 (1996) 77), and the second is the LM tests for the error component panel data model discussed in Breusch and Pagan (Rev. Econom. Stud. 47(1980) 239) and Baltagi et al. (J. Econometrics 54 (1992) 95). The idea is to allow for both spatial error correlation as well as random region effects in the panel data regression model and to test for their joint significance. Additionally, this paper derives conditional LM tests, which test for random regional effects given the presence of spatial error correlation. Also, spatial error correlation given the presence of random regional effects. These conditional LM tests are an alternative to the one-directional LM tests that test for random regional effects ignoring the presence of spatial error correlation or the one-directional LM tests for spatial error correlation ignoring the presence of random regional effects. We argue that these joint and conditional LM tests guard against possible misspecification. Extensive Monte Carlo experiments are conducted to study the performance of these LM tests as well as the corresponding likelihood ratio tests.     

Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients

Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients. Spatial Economic Analysis. The spatial model with space-varying coefficients proposed by Sun et al. in 2014 has proved to be useful in detecting the location effects of the impacts of covariates as well as spatial interaction in empirical analysis. However, Sun et al.’s estimator is inconsistent when heteroskedasticity is present – a circumstance that is more realistic in certain applications. In this study, we propose a kind of semi-parametric generalized method of moments (GMM) estimator that is not only heteroskedasticity robust but also takes a closed form written explicitly in terms of observed data. We derive the asymptotic distributions of our estimators. Moreover, the results of Monte Carlo experiments show that the proposed estimators perform well in finite samples.     

带未知异方差广义空间模型的有效估计

空间单元大小以及其它的经济特征上的差异,常常会导致空间异方差问题。本文给出了广义空间模型异方差问题的三种不同估计方法。第一种方法是将异方差形式参数化,来克服自由度的不足,使用ML估计进行实现。而针对异方差形式未知时,分别采用了基于2SLS的迭代GMM估计和更加直接的MCMC抽样方法加以解决,特别是MCMC方法表现得更加优美。蒙特卡罗模拟表明,给定异方差形式条件下, ML估计通过异方差参数化的方法依然可以获得较好的估计效果。而异方差形式未知的情况下,另外两种方法随着样本数的增大时也可以与ML的估计结果趋于一致。     

Convenient specification tests for logit and probit models

We propose several Lagrange multiplier tests of logit and probit models, which may be inexpen- sively computed by means of artificial linear regressions. These maybe used to test for various forms of model inadequacy, including the omission of specified variables and heteroskedasticity of known form. We perform a number of sampling experiments, in which we compare the small-sam- ple properties of these tests and of likelihood ratio tests. One of the LM tests turns out to have better small-sample properties than any of the others. We then investigate the power of the tests against local alternatives, and conduct a further series of sampling experiments to compare the power of various tests.     

Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix

This paper considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimator introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The finite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation.     

Powerful tests for structural changes in volatility

Detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence. We propose modified CUSUM and LM tests that are built on a robust estimator of the long-run variance of squared series. We establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative. The theory allows smooth and abrupt structural changes that can be small. The smoothing parameter is automatically selected such that the proposed test has good finite-sample size and meanwhile achieves decent power gain.