Heteroskedastic Disturbances Research Articles

Asymptotically efficient root estimators for spatial autoregressive models with spatial autoregressive disturbances

This paper considers closed-form root estimators for spatial autoregressive models with spatial autoregressive disturbances (SARAR model). We first derive a simple consistent closed-form estimator. Then we construct feasible moment conditions that are quadratic in the spatial lag and spatial error dependence parameters separately, which generate root estimators with closed forms. We consider both the cases with homoskedastic and unknown heteroskedastic disturbances. In the homoskedastic case, the root estimator can be asymptotically as efficient as the quasi-maximum likelihood estimator (QMLE); in the heteroskedastic case, it can be asymptotically as efficient as a method of moments estimator (MME) that sets adjusted quasi-maximum likelihood scores to zero, where the adjusted scores have zero means at the true parameters. The root estimators and their associated standard errors can avoid the computation of any matrix determinants or inverses, so they are computationally simple without iterations, especially valuable for big data where the sample size is large.

Testing for mean reversion in Bitcoin returns with Gibbs-sampling-augmented randomization

In the present paper, we attempt to verify whether the Bitcoin log-returns are mean reverted in the presence of heteroskedastic disturbances driven by a mixture distribution. To tackle this problem, we use the autoregression test of mean reversion based on the Gibbs-sampling-augmented randomization methodology. In general, our results indicated that Bitcoin is mean averting for different returns horizons, model specifications and for sub-sample periods, which show the explosive characteristic of the Bitcoin in the period of analysis from 2010 to 2019.

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

ABSTRACTThis paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.

Testing-Based Forward Model Selection

This paper defines and studies a variable selection procedure called Testing-Based Forward Model Selection. The procedure inductively selects covariates which increase predictive accuracy into a working statistical regression model until a stopping criterion is met. The stopping criteria and selection criteria are defined using statistical hypothesis tests. The paper explicitly describes a testing procedure in the context of high-dimensional linear regression with heteroskedastic disturbances. Finally, a simulation study examines finite sample performance of the proposed procedure and shows that it behaves favorably in high-dimensional sparse settings in terms of prediction error and size of selected model.

Outer-product-of-gradients tests for spatial autoregressive models

For Lagrangian multiplier (LM) tests of restrictions on parameters in spatial autoregressive (SAR) models with (SARAR models) or without SAR disturbances, their outer-product-of-gradient (OPG) variants can be simple and robust to unknown heteroskedasticity. However, for certain tests, asymptotic distributions of test statistics might depend on the constrained maximum likelihood or quasi maximum likelihood (QML) estimators, so their OPG variants would not be valid. To overcome such a hurdle, we propose to use C(α)-type score vectors to obtain valid OPG variants. Such OPG tests can be systematically constructed for SARAR models with homoskedastic and heteroskedastic disturbances, which might not be normally distributed. They also have the advantage that any n-consistent estimator can be used in place of a restricted QML estimate. In particular, OPG tests based on generalized method of moments (GMM) estimates are computationally simple and powerful compared to LM tests. Corresponding OPG tests based on C(α)-type gradient vectors in the GMM framework are also investigated.

Dynamic spatial autoregressive models with autoregressive and heteroskedastic disturbances

SummaryWe propose a new class of models specifically tailored for spatiotemporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, that is, SARAR(1, 1), by exploiting the recent advancements in score‐driven (SD) models typically used in time series econometrics. In particular, we allow for time‐varying spatial autoregressive coefficients as well as time‐varying regressor coefficients and cross‐sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite‐sample properties of the maximum likelihood estimator for the new class of models as well as its flexibility in explaining a misspecified dynamic spatial dependence process. The new proposed class of models is found to be economically preferred by rational investors through an application to portfolio optimization.

Estimation of the Heteroskedastic Canonical Contagion Model with Instrumental Variables.

Knowledge of contagion among economies is a relevant issue in economics. The canonical model of contagion is an alternative in this case. Given the existence of endogenous variables in the model, instrumental variables can be used to decrease the bias of the OLS estimator. In the presence of heteroskedastic disturbances this paper proposes the use of conditional volatilities as instruments. Simulation is used to show that the homoscedastic and heteroskedastic estimators which use them as instruments have small bias. These estimators are preferable in comparison with the OLS estimator and their asymptotic distribution can be used to construct confidence intervals.

Dynamic Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

We propose a new class of models specifi cally tailored for spatio-temporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, i.e. SARAR(1,1), by exploiting the recent advancements in Score Driven (SD) models typically used in time series econometrics. In particular, we allow for time-varying spatial autoregressive coefficients as well as time-varying regressor coefficients and cross-sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite sample properties of the Maximum Likelihood estimator for the new class of models as well as its exibility in explaining several dynamic spatial dependence processes. The new proposed class of models are found to be economically preferred by rational investors through an application in portfolio optimization.

ON SIZE AND POWER OF HETEROSKEDASTICITY AND AUTOCORRELATION ROBUST TESTS

Testing restrictions on regression coefficients in linear models often requires correcting the conventional F-test for potential heteroskedasticity or autocorrelation amongst the disturbances, leading to so-called heteroskedasticity and autocorrelation robust test procedures. These procedures have been developed with the purpose of attenuating size distortions and power deficiencies present for the uncorrected F-test. We develop a general theory to establish positive as well as negative finite-sample results concerning the size and power properties of a large class of heteroskedasticity and autocorrelation robust tests. Using these results we show that nonparametrically as well as parametrically corrected F-type tests in time series regression models with stationary disturbances have either size equal to one or nuisance-infimal power equal to zero under very weak assumptions on the covariance model and under generic conditions on the design matrix. In addition we suggest an adjustment procedure based on artificial regressors. This adjustment resolves the problem in many cases in that the so-adjusted tests do not suffer from size distortions. At the same time their power function is bounded away from zero. As a second application we discuss the case of heteroskedastic disturbances.

Assessing the suitability of the minimum capture size and protection regimes in the gooseneck barnacle shellfishery

The suitability of a total-length-based, minimum capture-size and different protection regimes was investigated for the gooseneck barnacle Pollicipes pollicipes shellfishery in N Spain. For this analysis, individuals that were collected from 10 sites under different fishery protection regimes (permanently open, seasonally closed, and permanently closed) were used. First, we applied a non-parametric regression model to explore the relationship between the capitulum Rostro-Tergum (RT) size and the Total Length (TL). Important heteroskedastic disturbances were detected for this relationship, demonstrating a high variability of TL with respect to RT. This result substantiates the unsuitability of a TL-based minimum size by means of a mathematical model. Due to these disturbances, an alternative growth-based minimum capture size of 26.3 mm RT (23 mm RC) was estimated using the first derivative of a Kernel-based non-parametric regression model for the relationship between RT and dry weight. For this purpose, data from the permanently protected area were used to avoid bias due to the fishery. Second, the size-frequency distribution similarity was computed using a MDS analysis for the studied sites to evaluate the effectiveness of the protection regimes. The results of this analysis indicated a positive effect of the permanent protection, while the effect of the seasonal closure was not detected. This result needs to be interpreted with caution because the current harvesting based on a potentially unsuitable minimum capture size may dampen the efficacy of the seasonal protection regime.

Time variation in the relative importance of permanent and transitory components in the U.S. housing market

This paper uses an unobserved component model with heteroskedastic disturbances based on Harvey et al. (1992) to measure the time-varying importance of permanent and transitory components in the U.S. house prices. Our findings show that the cyclical component in the U.S. housing market is highly persistent and house prices were more than 20% above the trend at the peak of the housing boom in 2006. Our results also suggest that there was a big increase in the relative importance of the transitory shock variance at the peak of the housing crisis.

Fixed Effects and Random Effects Estimation of Higher-Order Spatial Autoregressive Models with Spatial Autoregressive and Heteroskedastic Disturbances

This paper develops a unified framework for fixed and random effects estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroskedasticity 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 a random effects and a fixed effects 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 heteroskedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman-test of the spatial random against the spatial fixed effects model.

A jump diffusion model for spot electricity prices and market price of risk

We construct a jump-diffusion model with seasonality, mean-reversion, time-dependent jump intensity and heteroskedastic disturbance for electricity spot prices, while keeping the analytical tractability of futures prices. We find that the jump component plays a considerably larger role than the diffusion component in the variance of spot prices. Moreover, the jump intensity is much higher during summer and winter. We also explore the seasonal market price of risk (MPR) with different maturities, from one month to five months. Our results show that the diffusion risk and the jump risk are priced quite differently.

GMM Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

We consider a spatial econometric model containing a spatial lag in the dependent variable and the disturbance term with an unknown form of heteroskedasticity in innovations. We first prove that the maximum likelihood (ML) estimator for spatial autoregressive models is generally inconsistent when heteroskedasticity is not taken into account in the estimation. We show that the necessary condition for the consistency of the ML estimator of spatial autoregressive parameters depends on the structure of the spatial weight matrices. Then, we extend the robust generalized method of moment (GMM) estimation approach in Lin and Lee (2010) for the spatial model allowing for a spatial lag not only in the dependent variable but also in the disturbance term. We show the consistency of the robust GMM estimator and determine its asymptotic distribution. Finally, through a comprehensive Monte Carlo simulation, we compare finite sample properties of the robust GMM estimator with other estimators proposed in the literature.

Improved instrumental variables estimation of simultaneous equations under conditionally heteroskedastic disturbances

AbstractIn this paper we develop estimation techniques and a specification test for the validity of instrumental variables allowing for conditionally heteroskedastic disturbances. We propose modified two‐stage least squares (2SLS) and modified 3SLS procedures where the conditional heteroskedasticity is taken into account, which are natural extensions of the traditional 2SLS and 3SLS estimators and which achieve a lower variance. We recommend the use of these modified 2SLS and 3SLS procedures in practice instead of alternative estimators like limited‐information maximum likelihood/full‐information maximum likelihood, where the non‐existence of moments leads to extreme values, and also for ease of computation. It is shown theoretically and with simulation that in some cases 2SLS, 3SLS and our modified 2SLS and 3SLS procedures can have very severe biases (including the weak instruments case), and we present bias correction procedures to apply in practice along the lines of Flores‐Lagunes (2007). Our new estimation procedures can also be used to extend the test for weak instruments of Stock and Yogo (2005) and to allow for conditional heteroskedasticity. Finally, we show the usefulness of our estimation procedures with an application to the demand and supply of fish. Copyright © 2010 John Wiley & Sons, Ltd.

Efficient Maximum Likelihood Estimation of Spatial Autoregressive Models with Normal But Heteroskedastic Disturbances

The likelihood functions for spatial autoregressive models with normal but heteroskedastic disturbances have been derived [Anselin (1988, ch.6)], but there is no implementation of maximum likelihood estimation for these likelihood functions in general cases with heteroskedastic disturbances. Therefore, less efficient IV-based methods must be applied if disturbance terms might be heteroskedastic. In the present paper, we develop a new computer program for maximum likelihood estimation of heteroskedastic models that also utilizes multiple spatial weight matrices and confirm the efficiency of the obtained estimator in cases with heteroskedastic disturbances through Monte Carlo simulations.

GMM estimation of spatial autoregressive models with unknown heteroskedasticity

In the presence of heteroskedastic disturbances, the MLE for the SAR models without taking into account the heteroskedasticity is generally inconsistent. The 2SLS estimates can have large variances and biases for cases where regressors do not have strong effects. In contrast, GMM estimators obtained from certain moment conditions can be robust. Asymptotically valid inferences can be drawn with consistently estimated covariance matrices. Efficiency can be improved by constructing the optimal weighted estimation. The approaches are applied to the study of county teenage pregnancy rates. The empirical results show a strong spatial convergence among county teenage pregnancy rates.

Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances

This study develops a methodology of inference for a widely used Cliff–Ord type spatial model containing spatial lags in the dependent variable, exogenous variables, and the disturbance terms, while allowing for unknown heteroskedasticity in the innovations. We first generalize the GMM estimator suggested in Kelejian and Prucha (1998, 1999) for the spatial autoregressive parameter in the disturbance process. We also define IV estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GMM estimator. Much of the theory is kept general to cover a wide range of settings.

The wild bootstrap, tamed at last

The wild bootstrap is studied in the context of regression models with heteroskedastic disturbances. We show that, in one very specific case, perfect bootstrap inference is possible, and a substantial reduction in the error in the rejection probability of a bootstrap test is available much more generally. However, the version of the wild bootstrap with this desirable property is without the skewness correction afforded by the currently most popular version of the wild bootstrap. Simulation experiments show that this does not prevent the preferred version from having the smallest error in rejection probability in small and medium-sized samples.

A variance ratio test of the behaviour of Chinese stock indices

This study utilizes tests based on ranks and signs suggested by Wright (2000) together with the traditional variance ratio test to examine the behaviour of some Chinese stock indices. The results have shown that the null hypothesis of martingale difference behaviour of the Chinese index returns series examined is rejected for the whole samples. However, when the heteroskedastic stochastic disturbance term is used, the finding supports the random walk hypothesis for B shares by end of 1996, the Chinese stock market has become more efficient.