Heteroskedastic Disturbances Research Articles

Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

One important goal of this study is to develop 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 generalized moments (GM) estimator suggested in Kelejian and Prucha (1998,1999) for the spatial autoregressive parameter in the disturbance process. We prove the consistency of our estimator; unlike in our earlier paper we also determine its asymptotic distribution, and discuss issues of efficiency. We then define instrumental variable (IV) estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GM estimator under reasonable conditions. Much of the theory is kept general to cover a wide range of settings. We note the estimation theory developed by Kelejian and Prucha (1998, 1999) for GM and IV estimators and by Lee (2004) for the quasi-maximum likelihood estimator under the assumption of homoskedastic innovations does not carry over to the case of heteroskedastic innovations. The paper also provides a critical discussion of the usual specification of the parameter space.

GM Estimation of Higher-Order Spatial Autoregressive Processes in Cross-Section Models with Heteroskedastic Disturbances

This paper generalizes the approach to estimating a first-order spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) in a cross-section with heteroskedastic innovations by Kelejian and Prucha (2008) to the case of spatial autoregressive models with spatial autoregressive disturbances of arbitrary (finite) order (SARAR(R,S)). We derive the moment conditions and the optimal weighting matrix for a generalized moments (GM) estimation procedure of the spatial regressive parameters of the disturbance process and define a generalized two-stages least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators, derive their (joint) asymptotic distribution, and provide Monte Carlo evidence on their small sample performance.

Blacks and the family cap: pregnancy, abortion, and spillovers

While reducing out-of-wedlock childbearing is a central goal of welfare reform, most policymakers prefer achieving this objective via a reduction in nonmarital pregnancy rates rather than through an increase in the incidence of abortion. Using aggregate state-level data from 1984 to 1998, I estimate fixed effects models that allow for autocorrelated and heteroskedastic disturbances to examine the association between the family cap and nonmarital birth, pregnancy, and abortion rates. I find robust evidence that the family cap is associated with a reduction in nonmarital birth rates, particularly among black women. This reduction is driven by a reduction in nonmarital pregnancy rates rather than through an increase in abortion or marriage rates. These findings suggest that that the stigmatizing effect of the family cap may influence the nonmarital pregnancy decisions of black women.

On the predictive power of the term structure of interest rates for future inflation changes in the presence of political instability: the Turkish economy

This study investigates whether the term structure contains useful information about future inflation for Turkey during 1990–2000, a period of high inflation, high budget deficits, and political instability. Constant parameter and time varying parameter models are rejected by the data. The relationship between term structure of interest rates and inflation changes is found to be explained by a time-varying-parameter model with Markov-switching heteroskedastic disturbances. Thus, the term structure of interest rates is limited as a guide for monetary policy in an economy subject to regime changes such as that of Turkey. Stability can be achieved only by reducing inflation through circumscribing substantial government budget deficits and the political instability underlying them.

State-Space Models with Regime-Switching: Classical and Gibbs Sampling Approaches with Applications

Both state-space models and Markov switching models have been highly productive paths for empirical research in macroeconomics and finance. This book presents recent advances in econometric methods that make feasible the estimation of models that have both features. One approach, in the classical framework, approximates the likelihood function; the other, in the Bayesian framework, uses Gibbs-sampling to simulate posterior distributions from data. The authors present numerous applications of these approaches in detail: decomposition of time series into trend and cycle, a new index of coincident economic indicators, approaches to modeling monetary policy uncertainty, Friedman's plucking model of recessions, the detection of turning points in the business cycle and the question of whether booms and recessions are duration-dependent, state-space models with heteroskedastic disturbances, fads and crashes in financial markets, long-run real exchange rates, and mean reversion in asset returns.

Sources of Monetary Growth Uncertainty and Economic Activity: The Time- Varying-Parameter Model with Heteroskedastic Disturbances

A large body of research suggests that uncertainty is an important factor affecting economic activity. Most earlier research, however, fails to consider the possibility that uncertainty may affect the value of new information and economic activity differently, depending upon its source. The author first examines the importance of these different sources for modeling U.S. monetary growth uncertainty. He then empirically relates different sources of monetary growth uncertainty to economic activity measured by real GNP. The results suggest that different sources of uncertainty have different impacts on economic activity. The results tend to confirm predictions arising in the literature on irreversible investment and uncertainty. Copyright 1993 by MIT Press.

Robustness issues in structural equation modeling: a review of recent developments

In structural equation modeling the statistician needs assumptions inorder (1) to guarantee that the estimates are consistent for the parameters of interest, and (2) to evaluate precision of the estimates and significance level of test statistics. With respect to purpose (1), the typical type of analyses (ML and WLS) are robust against violation of distributional assumptions; i.e., estimates remain consistent or any type of WLS analysis and distribution of z. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vector z of observable follows a multivariate normal distribution. In relation to purpose (2), distributional misspecification may have consequences for efficiency, as well as power of test statistics (see Satorra, 1989a); that is, some estimation methods may bemore precise than others for a given specific distribution of z. For instance, ADF-WLS is asymptotically optimal under a variety of distributions of z, while the asymptotic optimality of NT-WLS may be lost when the data is non-normal Violation of a distributional assumption may have consequences for purpose (2). However, recent theory, such as the one described in Sections 7 and 8, showes that asymptotic variances of estimates and asympttic null distributions of test statistics derived under the normality assumption may be correct even when z is non-normal provided certain model conditions hold (the conditions of Theorem 1). That is, in a specific application with z non-normally distributed, the assumption that z is normal play the role of a “working device” that facilitates calculation of the correct distribution of statistics of interest. This corresponds to what in Section 7 and 8 has been called asymptotic robustness. For most of the models considered in practice, replacing the assyumption uncorrelation for the assumption of independence implised reaching the properties of asymptotic robustness; in that case, in order to evaluate the asymptotic behavior of statistics of interest, a NT form for Γ produces correct results even for non-normal data. This robustness result applies regardless of the type of fitting criterion used. Distinction between “uncorrelation’ and ‘independence’ becomes crucial when dealing with the asymptotic robustness issue. Statistical independence among variables of the model guarantee that the distribution of statistics of interest are asymptotically distribution-free of the non-normal variables; thus a NT form for Γ applies. As an example of where such distinction is apparent, consider a simple regression model with a heteroskedastic disturbance term. Here the disturbance term is uncorrelated with the regressor, but the variance varies with the value of the regressor. For a study showing that ADF-WLS protects against heteroskedasticity of erros, while ML wil generally fail, see Mooijaart and Satorra (1987). In regresion analysis the usual method for detecting heteroskedasticity is by looking at residual plots. Presumably, alsi in structural equation modeling, the need to distinguish between uncorrelation and independence will force the researcher to go back to the row data in order to do a similar type of “residuals’ inspection. In concluding, an importance consideration is to compute sampling variability for estimates and test statistics using appropriate formulae, without requiring that the estimation procedure be the ‘best’ in some sense. We have seen that such computations can be carried out correctly using the wrong assumptions with respect to the distribution of the vector of observable variables, provided some additional model conditions hold. Roughly speaking, such additional model conditions amount to strengthen the usual assumption of uncorrelation among some random constituents of the model to the assumption of stochastic independecen.

Adaptive estimation of regression models via moment restrictions

This paper considers adaptive estimation of regression models by means of generalized method of moments estimators. Two models are considered, one with an i.i.d. disturbance that is independent of the regressors and one with a conditionally symmetric but (possibly) heteroskedastic disturbance. For both cases the paper develops linearized estimators that are asymptotically efficient if the number and variety of moment conditions are allowed to grow at an appropriate rate with the sample size. In the general symmetric case no other adaptive estimator has yet been proposed. Also, results of a small Monte Carlo study indicate that in the independence case the small sample performance of the generalized method of moments estimator can be quite good vis-a-vis other estimators previously proposed.

The significance of roll calls in voting bodies: A model and statistical estimation

In the long history of legislative roll call analyses, there continues to exist a particularly troubling problem: There is no satisfactory method for measuring the relative importance or significance of individual roll calls. A measure of roll call significance would be intersting in and of itself, but many have realized that it could also substantially improve empirical research. The consequence of this situation is that hundreds of researchers risk heteroskedastic disturbances (resulting in inefficient estimates and biased standard errors and test statistics), are unable to appropriately choose the roll calls most suited to their theory (resulting in analyses that may not correctly test their theory), and often use methods that create more problems than they solve (resulting in selection bias, unrealistic weighting schemes, or relatively subjective measures). This article introduces a new method designed to meet these problems. Based on an application of Box-Tiao intervention analysis, the method extracts from observed voting participation scores the “revealed preferences” of legislators as a measure of roll call significance. Applying this method to roll calls from the U.S. Senate demonstrates the success of the method and suggests its utility in applied research.

A temporal cross section specification of the demand for gasoline using a random coefficient regression model

This paper uses a random coefficient regression approach to estimate the demand for gasoline by pooling cross-sectional (state level) and time series data. The analysis proceeds by estimating two alternative models, namely a stock adjustment model and a flow adjustment model. The two models are estimated using state level data on gasoline consumption, gasoline price, income and the stock of automobiles. The random coefficient specification of each demand model is estimated assuming heteroskedastic disturbances across states, autocorrelated disturbances over time and variable intercept and slope coefficients across states. The resultant price and income elasticities are compared and inferences concerning the ability of the flow adjustment model to approximate the underlying demand function are made.

Empirical Criminology

Statistical results based on aggregate data are, in part, artifacts of discre tionary research activity. In demonstrating this, the author examines the rela tionships between theoretical constructs and empirical concepts and between empirical concepts and statistical proxies, as well as the sensitivity of statistical results to the selection of control variables, the structure of the explanatory model (single v. multiple equation systems), and the mode of esti mation (ordinary least squares v. a simultaneous equation estimation tech nique). Further, aggregation can produce heteroskedastic disturbances, heter oskedasticity produces biased statistical results, and there is no general pro cedure for eliminating aggregation bias. Improper correction for bias can pro duce misleading results. The author proposes that, since empirical research is far from pure, it behooves empiricists to develop a formal metastatistical methodology.

On the Sensitivity of Simultaneous-Equations Estimators to the Stochastic Assumptions of the Models

Abstract The paper reports the results of several sampling experiments investigating the sensitivity of various simultaneous-equations estimators to errors in the exogenous variables, stochastic coefficients, heteroskedastic disturbances and auto correlated disturbances. The estimates studied are direct least squares, two-stage least squares, Nagar's unbiased k-class estimator, limited-information maximum likelihood, three-stage least squares, and full-information maximum likelihood. Stochastic coefficients altered the ranking of the estimators and increased their dispersions greatly. The other conditions did not produce great changes in the rankings of the estimators, the central tendencies of their distributions or the usefulness of their standard errors.

On the Sensitivity of Simultaneous-Equations Estimators to the Stochastic Assumptions of the Models

Abstract The paper reports the results of several sampling experiments investigating the sensitivity of various simultaneous-equations estimators to errors in the exogenous variables, stochastic coefficients, heteroskedastic disturbances and auto correlated disturbances. The estimates studied are direct least squares, two-stage least squares, Nagar's unbiased k-class estimator, limited-information maximum likelihood, three-stage least squares, and full-information maximum likelihood. Stochastic coefficients altered the ranking of the estimators and increased their dispersions greatly. The other conditions did not produce great changes in the rankings of the estimators, the central tendencies of their distributions or the usefulness of their standard errors.