What are the long-run effects of flooding on home values? We use new differencein-difference methods for setups with local spillovers to analyze the effects of 2012 hurricane Sandy on New York’s housing market. We show that, more than a decade after the storm, and long after damage was repaired, flooded properties continue to sell at a substantial discount. Our estimates also suggest that the discount is inversely related to the severity of flooding, consistent with the stronger informational signal conveyed by flooding of less exposed properties. Last, we estimate large local spillover effects that persistently lowered the values of nearby non-flooded properties. Spillover effects reduce homeowners’ incentives to invest in flood-mitigation actions, as such individual efforts cannot fully prevent value depreciation.

We investigate likelihood-based estimations of unbalanced panel data models with entity and time fixed effects that allow for cross-sectional dependence specified through matrix exponential terms. We consider a hybrid approach to deal with the fixed effects that can create the incidental parameters problem. We first eliminate the time fixed effects from the model using an orthogonal transformation. We then concentrate out the entity fixed effects from the quasi log-likelihood function of the transformed model. We introduce an M-estimator that utilises analytically adjusted score functions of the concentrated quasi log-likelihood function. We show that the suggested M-estimator is consistent and has an asymptotic normal distribution irrespective of whether the number of time periods is large or small. For consistent estimation of the variance-covariance matrix of the M-estimator, we propose an analytical bias correction approach involving the sample counterpart and plug-in methods. Through an extensive Monte Carlo study, we show that the suggested M-estimator has good finite sample properties. Finally, we use our model specification to study spatial correlation in crime rates and total factor productivity.

The estimation of matrix exponential spatial panel data models with entity and time fixed effects is considered under both homoskedastic and heteroskedastic error terms. Under the assumption of homoskedasticity, a quasi-maximum likelihood estimator (QMLE) formulated from the partial likelihood functions is proposed. In the case of heteroskedasticity, an M-estimator derived from the adjusted quasi score functions obtained from the partial likelihood functions is suggested. The large sample properties of the proposed estimators are established under certain assumptions. Through Monte Carlo simulations, the proposed estimators are shown to exhibit good finite sample properties. In an empirical application, the relationship between carbon emissions and economic activity is investigated using state-level data from the United States.

In this study, we suggest an imputation approach for estimating treatment effect parameters when untreated potential outcomes follow a panel data model that has both interactive fixed effects (IFE) and additive two-way fixed effects. In settings with common treatment timing and staggered treatment adoption, we consider a hybrid approach involving classical and Bayesian methods. First, using the classical random sampling approach across units, we show that treatment effect parameters are identified in our setting under a selection on observables and unobservables assumption. We then suggest an efficient Gibbs sampler for estimating the treatment effect parameters using our suggested imputation approach. We consider two Bayesian methods for selecting the number of factors in the postulated model for untreated potential outcomes. We provide simulation evidence showing that our imputation approach performs satisfactorily. In an empirical application, we use our approach to study the causal effect of police presence on crime.

In this paper, we provide a comprehensive review of the literature on estimation, inference, and model selection approaches for cross-sectional matrix exponential spatial models. We first discuss the properties of the matrix exponential specification in modeling cross-sectional dependence in comparison to the spatial autoregressive specification. We then provide a survey of the existing estimation and inference methods for cross-sectional matrix exponential spatial models. We carefully discuss summary measures for the marginal effects of regressors, detail the matrix-vector product method for efficient computation of matrix exponential terms, and then explore model selection approaches. Our aim is not only to summarize the main findings from the spatial econometric literature but also to make them more accessible to applied researchers. Additionally, we contribute to the literature by presenting several new results. We propose an M-estimation approach for models with heteroskedastic error terms and demonstrate that the resulting M-estimator is consistent and asymptotically normally distributed. Moreover, we provide additional results for model selection exercises. Finally, in a Monte Carlo study, we evaluate the finite sample properties of various estimators from the literature alongside the M-estimator.

This short paper explores the estimation of a dynamic spatiotemporal autoregressive conditional heteroscedasticity (ARCH) model. The log-volatility term in this model can depend on (i) the spatial lag of the log-squared outcome variable, (ii) the time-lag of the log-squared outcome variable, (iii) the spatiotemporal lag of the log-squared outcome variable, (iv) exogenous variables, and (v) the unobserved heterogeneity across regions and time, i.e., the regional and time fixed effects. We examine the small- and large-sample properties of two quasi-maximum likelihood estimators and a generalised method of moments estimator for this model. We first summarize the theoretical properties of these estimators and then compare their finite sample properties through Monte Carlo simulations.

In this article, we propose a spatio-temporal model to investigate the dynamics of contagion in the credit event risks of sovereigns. More specifically, we examine how changes in the credit default swap (CDS) spreads of a sovereign are influenced by the CDS spreads of other sovereigns over time. Our model incorporates spatial, temporal, and spatio-temporal lags of CDS spreads while accounting for unobserved heterogeneity across sovereigns and time periods. We consider several candidates for the underlying contagion network matrix using cross-border domestic bank exposures, geographical distances between sovereigns, and pairwise correlations of CDS spreads. We propose an efficient Bayesian algorithm for estimation and a simple method to address nested and non-nested model selection problems. Using a quarterly dataset of fourteen sovereigns from 2009 to 2022, we find evidence of contagion in CDS spreads which was relatively stronger during the period 2009–2012.

Geo-referenced data are characterized by an inherent spatial dependence due to the geographical proximity. In this paper, we introduce a dynamic spatiotemporal autoregressive conditional heteroscedasticity (ARCH) process to describe the effects of (i) the log-squared time-lagged outcome variable, i.e., the temporal effect, (ii) the spatial lag of the log-squared outcome variable, i.e., the spatial effect, and (iii) the spatial lag of the log-squared time-lagged outcome variable, i.e., the spatiotemporal effect, on the volatility of an outcome variable. Furthermore, our suggested process allows for the fixed effects over time and space to account for the unobserved heterogeneity. For this dynamic spatiotemporal ARCH model, we derive a generalized method of moments (GMM) estimator based on the linear and quadratic moment conditions of a specific transformation. We show the consistency and asymptotic normality of the GMM estimator, and determine the best set of moment functions. We investigate the finite-sample properties of the proposed GMM estimator in a series of Monte-Carlo simulations with different model specifications and error distributions. Our simulation results show that our suggested GMM estimator has good finite sample properties. In an empirical application, we use monthly log-returns of the average condominium prices of each postcode of Berlin from 1995 to 2015 (190 spatial units, 240 time points) to demonstrate the use of our suggested model. Our estimation results show that the temporal, spatial and spatiotemporal lags of the log-squared returns have statistically significant effects on the volatility of the log-returns.

We introduce a dynamic spatiotemporal volatility model that extends traditional approaches by incorporating spatial, temporal, and spatiotemporal spillover effects, along with volatility-specific observed and latent factors. The model offers a more general network interpretation, making it applicable for studying various types of network spillovers. The primary innovation lies in incorporating volatility-specific latent factors into the dynamic spatiotemporal volatility model. Using Bayesian estimation via the Markov Chain Monte Carlo (MCMC) method, the model offers a robust framework for analyzing the spatial, temporal, and spatiotemporal effects of a log-squared outcome variable on its volatility. We recommend using the deviance information criterion (DIC) and a regularized Bayesian MCMC method to select the number of relevant factors in the model. The model's flexibility is demonstrated through two applications: a spatiotemporal model applied to the U.S. housing market and another applied to financial stock market networks, both highlighting the model's ability to capture varying degrees of interconnectedness. In both applications, we find strong spatial/network interactions with relatively stronger spillover effects in the stock market.

Spatial and spatiotemporal volatility models are a class of models designed to capture spatial dependence in the volatility of spatial and spatiotemporal data. Spatial dependence in the volatility may arise due to spatial spillovers among locations; that is, if two locations are in close proximity, they can exhibit similar volatilities. In this paper, we aim to provide a comprehensive review of the recent literature on spatial and spatiotemporal volatility models. We first briefly review time series volatility models and their multivariate extensions to motivate their spatial and spatiotemporal counterparts. We then review various spatial and spatiotemporal volatility specifications proposed in the literature along with their underlying motivations and estimation strategies. Through this analysis, we effectively compare all models and provide practical recommendations for their appropriate usage. We highlight possible extensions and conclude by outlining directions for future research.

In this paper, we propose an integrated modified harmonic mean estimator (IHME) for nested and non-nested model selection problems in spatial panel data models with entity and time fixed effects. We formulate the IHME based on the integrated likelihood functions obtained by analytically integrating out the high-dimensional entity and time fixed effects from the complete likelihood functions. To investigate the finite sample properties of the IHME, we design a comprehensive simulation study that allows for both nested and non-nested model selection exercises in some popular spatial panel data models. Our simulation results show that the IHME has excellent finite sample performance and outperforms some competing estimators in terms of precision. We provide an empirical application on the US house price changes to show the usefulness of the proposed IHME in a model selection exercise.

A robust test statistic for testing homoskedasticity in spatial panel data models that have entity and time fixed effects is introduced in a quasi maximum likelihood estimation setting. A size-correction approach is introduced to ensure that the score functions have an asymptotic distribution centered around zero in the local presence of certain nuisance parameters. The outer-product-of-martingale-difference (OPMD) method is used to formulate an estimator for the asymptotic variance of the score functions. The OPMD estimator and the adjusted score functions are used to formulate a computationally simple robust test statistic. The suggested test statistic does not require knowing the presence of spatial dependence in the outcome variable and/or the disturbance terms. The asymptotic distribution of the test statistic is established under the null and local alternative hypotheses. Through Monte Carlo simulations, the finite sample size and power properties of the proposed test statistic are investigated. Finally, two empirical applications are provided to illustrate the practical use of the proposed test statistic.

A dynamic spatiotemporal stochastic volatility (SV) model is introduced, incorporating explicit terms accounting for spatial, temporal, and spatiotemporal spillover effects. Alongside these features, the model encompasses time-invariant site-specific factors, allowing for differentiation in volatility levels across locations. The statistical properties of an outcome variable within this model framework are examined, revealing the induction of spatial dependence in the outcome variable. Additionally, a Bayesian estimation procedure employing the Markov Chain Monte Carlo (MCMC) approach, complemented by a suitable data transformation, is presented. Simulation experiments are conducted to assess the performance of the proposed Bayesian estimator. Subsequently, the model is applied in the domain of environmental risk modeling, addressing the scarcity of empirical studies in this field. The significance of climate variation studies is emphasized, illustrated by an analysis of local air quality in Northern Italy during 2021, which underscores pronounced spatial and temporal clusters and increased uncertainties/risks during the winter season compared to the summer season.

ABSTRACT In this paper we consider a high-order spatial generalized autoregressive conditional heteroskedasticity (GARCH) model to account for the volatility clustering patterns observed over space. The model consists of a log-volatility equation that includes the high-order spatial lags of the log-volatility term and the squared outcome variable. We use a transformation approach to turn the model into a mixture of normals model, and then introduce a Bayesian Markov chain Monte Carlo (MCMC) estimation approach coupled with a data-augmentation technique. Our simulation results show that the Bayesian estimator has good finite sample properties. We apply a first-order version of the spatial GARCH model to US house price returns at the metropolitan statistical area level over the period 2006Q1–2013Q4 and show that there is significant variation in the log-volatility estimates over space in each period.

In this article, we consider a matrix exponential unbalanced panel data model that allows for (i) spillover effects using matrix exponential terms, (ii) unobserved heterogeneity across entities and time, and (iii) potential heteroscedasticity in the error terms across entities and time. We adopt a likelihood based direct estimation approach in which we jointly estimate the common parameters and fixed effects. To ensure that our estimator has the standard large sample properties, we show how the score functions should be suitably adjusted under both homoscedasticity and heteroscedasticity. We define our suggested estimator as the root of the adjusted score functions, and therefore our approach can be called the M-estimation approach. For inference, we suggest an analytical bias correction approach involving the sample counterpart and plug-in methods to consistently estimate the variance-covariance matrix of the suggested M-estimator. Through an extensive Monte Carlo study, we show that the suggested M-estimator has good finite sample properties. In an empirical application, we use our model to investigate the third country effects on the U.S. outward foreign direct investment (FDI) stock at the industry level.

In this paper, we develop a new version of Rao's score (RS) statistic for testing a non-linear hypothesis under both distributional and local parametric misspecification. Our suggested test statistic is constructed through a size correction approach so that it becomes robust to both types of misspecification. We establish the asymptotic properties of the robust test statistic and provide several examples to illustrate its implementation. We also investigate the finite sample properties of our test along with some other well-known tests through simulations. Our simulation results demonstrate that the new test statistic has good finite sample properties in terms of empirical size and power.

In this study, we suggest using information criteria for nested and non-nested model selection problems for the matrix exponential spatial specifications (MESS) under both homoskedasticity and heteroskedasticity. To this end, we consider the deviance information criterion, the Akaike information criterion and the Bayesian information criterion in a Bayesian setting. In the heteroskedastic case, we assume that the error terms have a scale mixture of normal distributions, where the scale mixture variables are latent variables that lead to different distributions. We demonstrate how the integrated likelihood function can be obtained analytically by integrating out the scale mixture variables from the complete-data likelihood function, and how this integrated likelihood function can be used to formulate the information criteria. We investigate the finite sample performance of these criteria in selecting the true model in a simulation study. The results show that these criteria perform satisfactorily and can be useful for selecting the correct model in specification search exercises. Finally, we apply the proposed information criteria to a spatially augmented growth model and a carbon emission model to show their usefulness for both nested and non-nested model selection problems.

The matrix exponential spatial models exhibit similarities to the conventional spatial autoregressive model in spatial econometrics but offer analytical, computational, and interpretive advantages. This paper provides a comprehensive review of the literature on the estimation, inference, and model selection approaches for the cross-sectional matrix exponential spatial models. We discuss summary measures for the marginal effects of regressors and detail the matrix-vector product method for efficient estimation. Our aim is not only to summarize the main findings from the spatial econometric literature but also to make them more accessible to applied researchers. Additionally, we contribute to the literature by introducing some new results. We propose an M-estimation approach for models with heteroskedastic error terms and demonstrate that the resulting M-estimator is consistent and has an asymptotic normal distribution. We also consider some new results for model selection exercises. In a Monte Carlo study, we examine the finite sample properties of various estimators from the literature alongside the M-estimator.

In this paper, we provide a general result under some high level assumptions that shows how to account for the parameter uncertainty problem in test statistics formulated with the quasi maximum likelihood (QML) estimator. We use our general result to develop various test statistics for testing skewness, kurtosis and normality for time series data. We show that the asymptotic distributions of our test statistics coincide with the asymptotic distributions of some tests suggested in the literature. Thus, our general result provides a unified approach for test statistics formulated with the QML estimator for time series data.

In this paper, we focus on a model specification problem in spatial econometric models when an empiricist needs to choose from a pool of candidates for the spatial weights matrix. We propose a model selection (MS) procedure for the matrix exponential spatial specification (MESS), when the true spatial weights matrix may not be in the set of candidate spatial weights matrices. We show that the selection estimator is asymptotically optimal in the sense that asymptotically it is as efficient as the infeasible estimator that uses the best candidate spatial weights matrix. The proposed selection procedure is also consistent in the sense that when the data generating process involves spatial effects, it chooses the true spatial weights matrix with probability approaching one in large samples. We also propose a model averaging (MA) estimator that compromises across a set of candidate models. We show that it is asymptotically optimal. We further flesh out how to extend the proposed selection and averaging schemes to higher order specifications and to the MESS with heteroscedasticity. Our Monte Carlo simulation results indicate that the MS and MA estimators perform well in finite samples. We also illustrate the usefulness of the proposed MS and MA schemes in a spatially augmented economic growth model.