Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Bias corrections for two-step fixed effects panel data estimators

This paper introduces large-T bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These models include systems of equations with limited dependent variables and unobserved individual effects, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time varying heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control variable from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control variable accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two-step nature of the procedure. As these two-step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large-T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.     

Forecasting in dynamic models with stochastic regressors

When a dependent variable y is related to present and past values of an exogenous variable x in a dynamic regression (distributed lag) model, and when x must be forecast in order to forecast y, necessary and sufficient conditions are derived in order for optimal forecasts of y to possess lower mean square error as a result of including x in the model, relative to forecasting y solely from its own past. The contribution to this forecast MSE reduction of non-invertibility in the lag distribution is assessed. Examples from econometrics and engineering are provided to illustrate the results.     

A Meta‐Analysis of β‐Convergence: the Legendary 2%

Abstract.  The topic of convergence is at the heart of a wide‐ranging debate in the growth literature, and empirical studies of convergence differ widely in their theoretical backgrounds, empirical specifications, and in their treatment of cross‐sectional heterogeneity. Despite these differences, a rate of convergence of about 2% has been found under a variety of different conditions, resulting in the widespread belief that the rate of convergence is a natural constant. We use meta‐analysis to investigate whether there is substance to the 'myth' of the 2% convergence rate and to assess several unresolved issues of interpretation and estimation. Our data set contains approximately 600 estimates taken from a random sample of empirical growth studies published in peer‐reviewed journals. The results indicate that it is misleading to speak of a natural convergence rate since estimates of different growth regressions come from different populations, and we find that correcting for the bias resulting from unobserved heterogeneity in technology levels leads to higher estimates of the rate of convergence. We also find that correcting for endogeneity of the explanatory variables has a substantial effect on the estimates and that measures of financial and fiscal development are important determinants of long‐run differences in per capita income levels. We show that although the odds of a study being published is not uniform for studies with different p ‐values, publication bias has no significant effect on the conclusions of the analysis.     

Spatial Vector Autoregressions

Abstract

A spatial vector autoregressive model (SpVAR) is defined as a VAR which includes spatial as well as temporal lags among a vector of stationary state variables. SpVARs may contain disturbances that are spatially as well as temporally correlated. Although the structural parameters are not fully identified in SpVARs, contemporaneous spatial lag coefficients may be identified by weakly exogenous state variables. Dynamic spatial panel data econometrics is used to estimate SpVARs. The incidental parameter problem is handled by bias correction rather than more popular alternatives such as generalised methods of moments (GMM). The interaction between temporal and spatial stationarity is discussed. The impulse responses for SpVARs are derived, which naturally depend upon the temporal and spatial dynamics of the model. We provide an empirical illustration using annual spatial panel data for Israel. The estimated SpVAR is used to calculate impulse responses between variables, over time, and across space. Finally, weakly exogenous instrumental variables are used to identify contemporaneous spatial lag coefficients.     

Estimation and recent behavior of urban population and employment density gradients

The monocentric model of urban structure predicts that urban population density declines with distance from the central business district. Using the negative exponential function to approximate the decline, Mills (E. S. Mills, “Studies in the Structure of the Urban Economy,” Johns Hopkins Press, Baltimore, Md. (1972)) estimated population and employment density gradients from 1948 to 1963 for a sample of 18 SMSAs. This paper updates Mills' estimates and examines recent patterns in population and employment suburbanization. The updated series estimated here is obtained using a “corrected” version of Mills' method. The original procedure incorporated a bias which Mills noted and later corrected (E. Mills and K. Ohta, in “Asia's New Giant” (Patrick and Rosovsky, Eds.), The Brookings Institution, Washington, D.C. (1976)). The comparability of the series begun by Mills and extended here is not interrupted, however, because, as Mills suspected, the bias is indeed small provided SMSA data are used. On the other hand, Urbanized Area definitions of metropolitan areas cause the original and corrected versions to yield significantly different results. This finding has implications for the appropriate choice of data for urban studies.     

Bayesian estimation and model selection for spatial Durbin error model with finite distributed lags

In this paper we investigate a spatial Durbin error model with finite distributed lags and consider the Bayesian MCMC estimation of the model with a smoothness prior. We study also the corresponding Bayesian model selection procedure for the spatial Durbin error model, the spatial autoregressive model and the matrix exponential spatial specification model. We derive expressions of the marginal likelihood of the three models, which greatly simplify the model selection procedure. Simulation results suggest that the Bayesian estimates of high order spatial distributed lag coefficients are more precise than the maximum likelihood estimates. When the data is generated with a general declining pattern or a unimodal pattern for lag coefficients, the spatial Durbin error model can better capture the pattern than the SAR and the MESS models in most cases. We apply the procedure to study the effect of right to work (RTW) laws on manufacturing employment.     

On lags and chaos in economic dynamic models

This paper discusses the role of lags in dynamic economic models. Applications of discrete dynamical systems to economics are considered and the shortcomings of their (implicit) treatment of lags are criticized. An abstract, probabilistic view of lags is then provided, within which fixed delay lags are shown to be a special case. A basic equivalence is proved between: (i) a system with an indefinitely large number of agents, reacting to inputs with randomly gamma distributed, discrete lags, and (ii) a system with one single ‘representative‘ agent reacting to inputs with a continuous, multiple exponential lag. Finally, the paper analyses a single-loop feedback system cuopling a multiple exponential lag and a non-linearity of the one-hump type. The dynamic behavior of the system is studied by means of analytical and numerical methods, and the conditions for periodic and chaotic solutions are investigated.     

Union status of young men in Britain: a decade of change

Previous empirical studies of individual union status in Britain have been cross‐sectional. In contrast, we use longitudinal data from the National Child Development Study, to estimate the determinants of male trade union membership over the period 1981–1991. As suggested by union theories, we find that it is important to control for unobserved individual heterogeneity, and our preferred model allows for correlation of individual heterogeneity with observable variables. Our estimates reveal that the observed decline in very large workplaces, and the contraction of the public sector, explain about one third of the predicted decline in union membership over the period. Copyright © 2000 John Wiley & Sons, Ltd.     

The nonrandom selection of don't knows in binary and ordinal responses: Corrections with the bivariate probit model with sample selection

On social surveysdon't knows are a common answer to attitudinal questions, which often have binary or ordinal response categories.Don't knows can be nonrandomly selected according to certain demographic or socioeconomic characteristics of the respondent. To model the sample selection and correct for its bias, this paper discusses two types of bivariate models —binary-probit and the ordinal probit model with sample selection. The difference between parameter estimates and predicted probabilities from the analysis modelling the sample selection bias ofdon't knows and those from the analysis not modellingdon't knows is emphasized. Two empirical examples using the 1989 General Social Survey data demonstrate the necessity to correct for the bias in the nonrandom selection ofdon't knows for binary and ordinal attitudinal response variables. A replication of the analyses using the 1990 and 1991 General Social Survey data helps demonstrate the reliability of the sample selection bias ofdon't knows.     

Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

Agglomeration elasticities and firm heterogeneity

This paper examines three key issues encountered when estimating the relationship between agglomeration and multi factor productivity (‘agglomeration elasticities’): the sorting of heterogeneous firms, the convexity of agglomeration effects, and the challenges of identifying the impact of persistent spatial differences in effective density. We use a firm-level panel containing production data together with detailed information on the geographic location of employment, covering a high proportion of the New Zealand economy. We are able to control for heterogeneity along firm, region, and industry dimensions, and to estimate separate agglomeration elasticities across industries and regions. Sorting leads to upward biased elasticity estimates but using firm fixed effects can lead to downward bias due to the highly persistent nature of agglomeration variables. Our preferred estimates control for sorting across regions and industries. Overall, we find a positive agglomeration elasticity of 0.066. Within industries and, to a lesser extent within regions, there is pronounced variation in the strength of agglomeration effects, and evidence of decreasing returns to agglomeration. High density areas attract firms that benefit most from agglomeration.     

STATE DEPENDENCE AND HETEROGENEITY IN HEALTH USING A BIAS‐CORRECTED FIXED‐EFFECTS ESTIMATOR

This paper estimates a dynamic ordered probit model of self‐assessed health with two fixed effects: one in the linear index equation and one in the cut‐points. This robustly controls for heterogeneity in unobserved health status and in reporting behavior, although we cannot separate both sources of heterogeneity. We find important state dependence effects, and small but significant effects of income and other socioeconomic variables. Having dynamics and flexibly accounting for unobserved heterogeneity matters for those estimates. We also contribute to the bias correction literature in nonlinear panel models by comparing and applying two of the existing proposals to our model. Copyright © 2012 John Wiley & Sons, Ltd.     

Raising the bar (9)

ABSTRACT

This editorial summarizes the papers published in issue 13(4) so as to raise the bar in applied spatial economic research and highlight new trends. The first paper develops an economic geography model with trade costs in all sectors and different shares of unskilled labour in all locations. The second paper translates an economic geography model into a dynamic spatial econometric model and then estimates the unknown parameters to test for congestion spillover effects among Chinese cities. The following paper also investigates spillover effects, but of sovereign and banking risks across countries. The fourth paper empirically examines if a higher market potential results in higher average productivity and lower productivity dispersion of Italian retailers. The fifth paper demonstrates that modelling more than one spatial lag in the independent variables, using different specifications of the spatial weight matrix, can be used as a tool to correct for an omitted variable bias. The final paper develops a test for the existence of non-parametric non-linearities in a linear spatial econometric model.     

Evaluating Californian under‐age drunk driving laws: endogenous policy lags

When evaluating the effectiveness of a policy, most studies assume that the policy effect begins with the date of enactment. However, there often exists an endogenous policy lag, due to information acquisition and the cost of adjustment. Meanwhile, the policy impact may be a gradual implementation from one level to another, instead of a one‐time shift behavior. To account for these issues when evaluating the Californian under‐age drunk driving laws, this paper adopts two econometric techniques: the multiple structural change methodology and the smooth transition method. The methods detect two effective policy changes and also reveal the existence of policy lags. In addition, ignoring these lags leads to severely biased estimates of policy effects. A long transition period is identified for the first under‐age drunk driving policy, while an abrupt transition is found for the other. In summary, the paper shows that the two econometric techniques complement each other and will be useful for policy evaluation. Copyright © 2011 John Wiley & Sons, Ltd.     

Selection correction and sensitivity analysis for ordered treatment effect on count response

In estimating the effect of an ordered treatment τ on a count response y with an observational data where τ is self‐selected (not randomized), observed variables x and unobserved variables ε can be unbalanced across the control group (τ = 0) and the treatment groups (τ = 1, …, J). While the imbalance in x causes ‘overt bias’ which can be removed by controlling for x, the imbalance in ε causes ‘covert (hidden or selection) bias’ which cannot be easily removed. This paper makes three contributions. First, a proper counter‐factual causal framework for ordered treatment effect on count response is set up. Second, with no plausible instrument available for τ, a selection correction approach is proposed for the hidden bias. Third, a nonparametric sensitivity analysis is proposed where the treatment effect is nonparametrically estimated under no hidden bias first, and then a sensitivity analysis is conducted to see how sensitive the nonparametric estimate is to the assumption of no hidden bias. The analytic framework is applied to data from the Health and Retirement Study: the treatment is ordered exercise levels in five categories and the response is doctor office visits per year. The selection correction approach yields very large effects, which are however ruled out by the nonparametric sensitivity analysis. This finding suggests a good deal of caution in using selection correction approaches. Copyright © 2004 John Wiley & Sons, Ltd.     

Distributed lags and intermediate good imports

In this paper we draw upon some recent developments in the theory of dynamic economics and econometrics to characterize the time series behavior of intermediate good imports. It is shown that adjustment costs, delivery lags, and inventory holding costs considerations lead to distributed lag import demand functions that are not subject to the Nerlove (1972) and Lucas (1976) critiques of ad hoc and fixed distributed lag modelling.     

ALCOHOL MARKETING,ADOLESCENT DRINKING AND PUBLICATION BIAS IN LONGITUDINAL STUDIES: A CRITICAL SURVEY USING META‐ANALYSIS

Abstract This paper presents a meta‐analysis of prospective cohort (longitudinal) studies of alcohol marketing and adolescent drinking, which accounts for publication bias. The paper provides a summary of 12 primary studies of the marketing–drinking relationship. Each primary study surveyed a sample of youth to determine baseline drinking status and marketing exposure, and re‐surveyed the youth to determine subsequent drinking outcomes. Logistic analyses provide estimates of the odds ratio for effects of baseline marketing variables on adolescent drinking at follow‐up. Using meta‐regression analysis, two samples are examined in this paper: 23 effect‐size estimates for drinking onset (initiation); and 40 estimates for other drinking behaviours (frequency, amount, bingeing). Marketing variables include ads in mass media, promotion portrayals, brand recognition and subjective evaluations by survey respondents. Publication bias is assessed using funnel plots that account for ‘missing’ studies, bivariate regressions and multivariate meta‐regressions that account for primary study heterogeneity, heteroskedasticity, data dependencies, publication bias and truncated samples. The empirical results are consistent with publication bias, omitted variable bias in some studies, and lack of a genuine effect, especially for mass media. The paper also discusses ‘dissemination bias’ in the use of research results by primary investigators and health policy interest groups.     

Estimating dynamic panel data discrete choice models with fixed effects

This paper considers the estimation of dynamic binary choice panel data models with fixed effects. It is shown that the modified maximum likelihood estimator (MMLE) used in this paper reduces the order of the bias in the maximum likelihood estimator from O(T^-1) to O(T^-2), without increasing the asymptotic variance. No orthogonal reparametrization is needed. Monte Carlo simulations are used to evaluate its performance in finite samples where T is not large. In probit and logit models containing lags of the endogenous variable and exogenous variables, the estimator is found to have a small bias in a panel with eight periods. A distinctive advantage of the MMLE is its general applicability. Estimation and relevance of different policy parameters of interest in this kind of models are also addressed.     

Testing a linear dynamic panel data model against nonlinear alternatives

The most popular econometric models in the panel data literature are the class of linear panel data models with unobserved individual- and/or time-specific effects. The consistency of parameter estimators and the validity of their economic interpretations as marginal effects depend crucially on the correct functional form specification of the linear panel data model. In this paper, a new class of residual-based tests is proposed for checking the validity of dynamic panel data models with both large cross-sectional units and time series dimensions. The individual and time effects can be fixed or random, and panel data can be balanced or unbalanced. The tests can detect a wide range of model misspecifications in the conditional mean of a dynamic panel data model, including functional form and lag misspecification. They check a large number of lags so that they can capture misspecification at any lag order asymptotically. No common alternative is assumed, thus allowing for heterogeneity in the degrees and directions of functional form misspecification across individuals. Thanks to the use of panel data with large $N$ and $T$, the proposed nonparametric tests have an asymptotic normal distribution under the null hypothesis without requiring the smoothing parameters to grow with the sample sizes. This suggests better nonparametric asymptotic approximation for the panel data than for time series or cross sectional data. This is confirmed in a simulation study. We apply the new tests to test linear specification of cross-country growth equations and found significant nonlinearities in mean for OECD countries’ growth equation for annual and quintannual panel data.