Heteroscedastic Disturbances Research Articles

Short Guides to Static Panel Data Regression Model Estimator

This paper elaborates on the use of three regression model estimators under the classification of static panel data. In order to select the best regression model estimator for panel data analysis, the researcher should focus on conducting several tests to avoid bias in the estimation results. The Breusch Pagan Lagrangian Multiplier test (Breusch Pagan test) is a simple test carried out in the panel data analysis to check for heteroscedastic disturbances in the linear regression model in making the decision to choose whether Pooled Ordinary Least Square or Random Effects. If the Breusch Pagan test shows the rejection of the null hypothesis, it indicates that the Random Effect model is more appropriate than Pooled Ordinary Least Square because the data have a panel effect (unobserved heterogeneity). The panel effect should be checked using the Hausman test in order to identify whether the effects are correlated or uncorrelated with the regressors. In the case of rejection of the null hypothesis, the Fixed Effect estimator is a more appropriate or unbiased estimator to analyze the data.

Arbitrage Pricing with Heterogeneous Spatial Effects and Heteroscedastic Disturbances

Abstract We develop a heterogeneous spatial arbit and regression coefficients, and heteroscedastic variances, and further establish identification of parameters and asymptotic normality of the conditional QML estimators under some mild conditions. We apply the proposed model to study a real data set of 11 eurozone stock index returns and extend the Fama–French five-factor model to regional stock indices, in which heterogeneous spatial effects and heteroscedastic disturbances are highly significant and they both play very important roles in explaining distinct endogenous effects and distinct risks of the 11 eurozone stock markets. Our empirical results reveal unique characteristics of each of 11 eurozone stock markets and their inner connections.

Conditional Correlation Demand Systems

We address the estimation of singular demand systems with heteroscedastic disturbances. As in Serletis and Isakin (Econ Rev 36:1111–1122, 2017) and Serletis and Xu (Empir Econ, 2019, forthcoming) we assume that the covariance matrix of the errors of the demand system is time-varying, and contribute to the literature by considering the constant conditional correlation and dynamic conditional correlation parameterizations of the variance model. We derive a number of important practical results and also provide an empirical application to support our methodology.

Demand systems with heteroscedastic disturbances

We address the estimation of singular demand systems with heteroscedastic disturbances. We relax the homoscedasticity assumption and instead assume that the covariance matrix of the errors of the demand system is time varying. In doing so, we consider the VECH and BEKK parameterizations of the variance model. We analytically prove the invariance of the maximum likelihood estimator with respect to the choice of the good deleted from a singular demand system and also prove a number of important practical results regarding how to recover the mean and variance equation parameters (and their standard errors) of the full demand system from those of any subsystem obtained by deleting an arbitrary good.

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

ABSTRACTWe suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.

Asymptotic Efficiency of the Maximum Likelihood Estimator for the Box-Cox Transformation Model with Heteroscedastic Disturbances

This paper considers the asymptotic efficiency of the maximum likelihood estimator (MLE) for the Box-Cox transformation model with heteroscedastic disturbances. The MLE under the normality assumption (BC MLE) is a consistent and asymptotically efficient estimator if the “small ” condition is satisfied and the number of parameters is finite. However, the BC MLE cannot be asymptotically efficient and its rate of convergence is slower than ordinal order when the number of parameters goes to infinity. Anew consistent estimator of order is proposed. One important implication of this study is that estimation methods should be carefully chosen when the model contains many parameters in actual empirical studies.

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.

Improved beta modeling and forecasting: An unobserved component approach with conditional heteroscedastic disturbances

Recent research on time-varying systematic-risk (beta) modeling reveals significant advantages in utilizing daily financial data and unobserved-component models. This research proposes a state-space market model with conditional heteroscedastic errors, thus addressing the leptokurtosis of the unconditional distribution of the disturbances and reducing the influence of outliers in the estimation process. This approach outperforms the conventional models, providing better levels of in-sample goodness of fit and more accurate point- and interval-dynamic assets returns forecasts. The proposed model provides better levels of empirical, conditional, and unconditional coverage and independence of its interval returns forecasts and reaches lower loss-function scores. Therefore, our model allows improving financial strategies, such as stock pricing, determining the companies’ cost-of-equity, evaluating the performance of managed-investment and pension funds, making portfolio-rebalancing processes and computing the value at risk (VAR) of investment portfolios.

Inference on Treatment Effects after Selection among High-Dimensional Controls

We propose robust methods for inference about the effect of a treatment variable on a scalar outcome in the presence of very many regressors in a model with possibly non-Gaussian and heteroscedastic disturbances. We allow for the number of regressors to be larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by including a relatively small number of variables whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of regressors. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the “post-double-selection” method. The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus, our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We also present a generalization of our method to a fully heterogeneous model with a binary treatment variable. We illustrate the use of the developed methods with numerical simulations and an application that considers the effect of abortion on crime rates.

Inference on Treatment Effects after Selection Amongst High-Dimensional Controls

We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances where the number of controls may be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the “post-double-selection” method. Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties.The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We illustrate the use of the developed methods with numerical simulations and an application to the effect of abortion on crime rates.

Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain

We develop results for the use of LASSO and Post-LASSO methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p, that apply even when p is much larger than the sample size, n. We rigorously develop asymptotic distribution and inference theory for the resulting IV estimators and provide conditions under which these estimators are asymptotically oracle-efficient. In simulation experiments, the LASSO-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument-robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the LASSO based IV estimator substantially reduces estimated standard errors allowing one to draw much more precise conclusions about the economic effects of these decisions. Optimal instruments are conditional expectations; and in developing the IV results, we also establish a series of new results for LASSO and Post-LASSO estimators of non-parametric conditional expectation functions which are of independent theoretical and practical interest. Specifically, we develop the asymptotic theory for these estimators that allows for non-Gaussian, heteroscedastic disturbances, which is important for econometric applications. By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for these estimators that are as sharp as in the homoscedastic Gaussian case under the weak condition that log p = o(n1=3). Moreover, as a practical innovation, we provide a fully data-driven method for choosing the user-specified penalty that must be provided in obtaining LASSO and Post-LASSO estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.

Estimation of higher‐order spatial autoregressive cross‐section models with heteroscedastic disturbances

AbstractThis paper generalizes the two‐step approach to estimating a first‐order spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) in a cross‐section with heteroscedastic innovations by Kelejian and Prucha to the case of spatial autoregressive models with spatial autoregressive disturbances of arbitrary (finite) order (SARAR(R,S)). We derive a generalized moments (GM) estimation procedure of the spatial regressive parameters of the disturbance process and a generalized two‐stage least squares estimator for the regression parameters of the model, prove consistency of proposed estimators thereof, and establish their (joint) asymptotic distribution. Monte Carlo evidence suggests that the estimation procedure performs reasonably well in small samples and that – apart from being of interest in itself – a proper specification of the spatial regressive disturbance process is also crucial for obtaining consistent estimates of the variance‐covariance matrix used in the generalized least squares estimation.ResumenEste artículo generaliza el enfoque de dos pasos para estimar un modelo autorregresivo espacial de primer orden con perturbaciones autorregresivas espaciales (SARAR(1,1)) en una sección transversal con innovaciones heterocedásticas por Kelejian y Prucha al caso de modelos autorregresivos espaciales con perturbaciones autorregresivas espaciales de orden (finito) arbitrario (SARAR(R,S)). Inferimos un procedimiento de estimación de momentos generalizados (GM) de los parámetros regresivos espaciales del proceso de perturbación y un estimador generalizado de mínimos cuadrados bietápicos para los parámetros de regresión del modelo, demostramos la uniformidad de los mismos estimadores propuestos, y establecemos su distribución asintótica (conjunta). Las pruebas obtenidas mediante Monte Carlo sugieren que el desempeño del procedimiento de estimación es razonablemente bueno en muestras pequeñas y también que –aparte de ser interesante de por sí– es crucial el especificar adecuadamente el proceso de perturbación regresiva espacial si queremos obtener estimaciones uniformes de la matriz de varianza‐covarianza utilizada en la estimación generalizada de mínimos cuadrados.

Has the Volatility of U.S. Inflation Changed and How?

The local level model with stochastic volatility, recently proposed for U.S. Inflation by Stock and Watson (“Why Has U.S. Inflation Become Harder to Forecast?”, Journal of Money, Credit and Banking, Supplement to Vol. 39, No. 1, February 2007), provides a simple yet sufficiently rich framework for characterizing the evolution of the main stylized facts concerning the U.S. inflation. The model decomposes inflation into a permanent component, evolving as a random walk, and a transitory component. The volatility of the disturbances driving both components is allowed to vary over time. The paper provides a full Bayesian analysis of this model and readdresses some of the main issues that were raised by the literature concerning the evolution of persistence and predictability and the extent and timing of the great moderation. The assessment of various nested models of inflation volatility and systematic model selection provide strong evidence in favor of a model with heteroscedastic disturbances in the permanent component, whereas the transitory component has time invariant size. The main evidence is that the great moderation is over, and that volatility, persistence and predictability of inflation underwent a turning point around 1995. During the last decade, volatility and persistence have been increasing and predictability has been going down.

Modelling financial time series with threshold nonlinearity in returns and trading volume

AbstractThis paper investigates the effect of past returns and trading volumes on the temporal behaviour of international market returns. We propose a class of nonlinear threshold time‐series models with generalized autoregressive conditional heteroscedastic disturbances. Using Bayesian approach, an implementation of Markov chain Monte Carlo procedure is used to obtain estimates of unknown parameters. The proposed family of models incorporates changes in log of volumes in the sense of regime changes and asymmetric effects on the volatility functions. The results show that when differences of log volumes are involved in the system of log return and volatility models, an optimum selection can be achieved. In all the five markets considered, both mean and variance equations involve volumes in the best models selected. Our best models produce higher posterior‐odds ratios than that in Gerlach et al.'s (Phys. A Statist. Mech. Appl. 2006; 360:422–444) models, indicating that our return–volume partition of regimes can offer extra gain in explaining return‐volatility term structure. Copyright © 2007 John Wiley & Sons, Ltd.

A regime switching approach to the Feldstein–Horioka puzzle: Evidence from some European countries

The domestic savings-investment correlation is re-examined using data for some European countries in order to obtain some information about the degree of capital mobility in the sense of the Feldstein and Horioka [Feldstein, M., & Horioka, C. Y. (1980). Domestic saving and international capital flows. Economic Journal 90, 314–329]. Applying a Markov-switching model with heteroscedastic disturbances, we find that the correlation coefficients are unstable due to the policy regime changes consistent with the Lucas [Lucas, R. (1976). Econometric policy evaluation: A critique. In K. Brunner, & A. H. Meltzer (Eds.), The Phillips curve and labor markets. Amsterdam: North-Holland] critique.

Most mean powerful test for testing heteroscedastic disturbances in the linear regression model

This paper investigates the application of the most mean powerful test to the problem of testing for heteroscedastic disturbances in the linear regression model. The most mean powerful test was introduced by Begum and King [5] and is based on the generalized Neyman-Pearson lemma. This test provides an optimal test of certain composite hypotheses. Previous applications have only involved testing problems whose null hypotheses, after reduction through invariance arguments, are one-dimensional (see, Begum and King [5]) and two dimensional (see, Begum and King [6]). This is the first application involving the problem of testing heteroscedastic disturbances in the linear regression model. A Monte Carlo simulation experiment was conducted to assess the small sample performances of the test with encouraging results.

The relationship between inflation and different sources of inflation uncertainty in Turkey

The relationship between inflation and different sources of inflation uncertainty is investigated in Turkey during the period of 1995:3–2000:12. A time-varying parameter model of inflation with heteroscedastic disturbances is used to generate different sources of inflation uncertainty and then Granger methods are employed to test for causality between inflation and different sources of inflation uncertainty. The findings show that there is a causative influence of inflation on its uncertainty arising due to time-varying parameters of the inflation model. No evidence is found on the correlation between inflation and inflation uncertainty arising due to heteroscedasticity in the disturbance terms.

Testing for heteroscedasticity in regression models

A new test for heteroscedasticity in regression models is presented based on the Goldfeld-Quandt methodology. Its appeal derives from the fact that no further regressions are required, enabling widespread use across all types of regression models. The distribution of the test is computed using the Imhof method and its power is assessed by performing a Monte Carlo simulation. We compare our results with those of Griffiths & Surekha (1986) and show that our test is more powerful than the wide range of tests they examined. We introduce an estimation procedure using a neural network to correct the heteroscedastic disturbances.

Exact inference for the linear model with groupwise heteroscedastic spherical disturbances

Exact nonparametric inference on a single coefficient in a linear regression model, as considered by Bekker (Working Paper, Department of Economics, University of Groningen, 1997), is elaborated for the case of spherically distributed heteroscedastic disturbances. Instead of approximate inference based on feasible weighted least squares, exact inference is formulated based on partial rotational invariance of the distribution of the vector of disturbances. Thus, classical exact inference based on t-statistics is generalized to exact inference that remains valid in a groupwise heteroscedastic context. The approach is applied to a basic two-sample problem, and to the random- and fixed-effects models for panel data.

Testing for Unit Roots in Short Dynamic Panels with Serially Correlated and Heteroscedastic Disturbance Terms

In this paper we introduce fixed-T unit root tests for panel data models with serially correlated and heteroscedastic disturbance terms. The tests are based on pooled least squares estimators for the autoregressive coefficient of the AR(1) panel model adjusted for their inconsistency. The proposed test statistics have normal limiting distributions when the cross-section dimension of the panel grows large, provided a condition involving the 4+delta-th order moments of the first differences of the data is satisfied. Monte Carlo evidence suggests that the tests have empirical size close to the nominal level and considerable power, even for MA(1) disturbance terms which exhibit strong negative autocorrelation.