Unit Root and Cointegration Tests for Spatially Dependent Panel Data

  • Michael Beenstock
  • Daniel Felsenstein
Part of the Advances in Spatial Science book series (ADVSPATIAL)


The main purpose is to introduce the spatial vector error correction model (SpVECM), which is the spatial panel counterpart to VECMs in time series. Unlike SpVAR models, SpVECMs test structural hypotheses involving nonstationary spatial panel data.

We begin by developing a panel unit root test for spatially dependent panel data. A unit root is induced when the sum of the AR and SAR coefficients equals 1. This means that spatial panel data are more likely to be nonstationary than independent panel data or strongly dependent panel data. SAR coefficients induces contagion and nonstationarity. Monte Carlo simulation methods are used to calculate the distribution of the sum of AR and SAR coefficients under the null hypothesis that they sum to unity. Critical values for this sum are obtained to reject the null hypothesis. These critical values depend on the specification of W. We provide empirical illustrations of these spatial unit root rests for spatial panel data for Israel.

If spatial panel data are nonstationary, the parameter estimates of spatial panel data models may be spurious or nonsense. If the AR and SAR coefficients of the residual errors sum to less than 1, the variables are spatially panel cointegrated, and the parameter estimates are genuine rather than spurious or nonsense. We use Monte Carlo methods to calculate these critical values, which depend on the numbers of variables, panel units and time periods, and the specification of W.

We show that least squares estimates of spatially cointegrated parameter vectors are super-consistent with respect to T. This includes the estimates of SAR coefficients for spatial lagged dependent variables. Consequently, recourse to maximum likelihood, IV/GMM methods is not necessary when the data are nonstationary.

Cointegration is “local” if nonspatial variables are specified in the cointegrating vector. It is “spatial” when spatial variables alone are specified, and it is “global” when both types of variables are specified. Since the parameter estimates in cointegrating vectors have non-standard distributions, a semiparametric bootstrap procedure is proposed for calculating their confidence intervals by resampling from the empirical distribution functions of the spatiotemporal DGPs for the variables and the residuals.

We conclude with a discussion of spatial error correction models generated by the estimated cointegrating vectors. The spatial dynamics in error correction models may differ from the spatial dynamics in their cointegrating vectors. Also, error correction may be “local”, “spatial” or “global”.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michael Beenstock
    • 1
  • Daniel Felsenstein
    • 2
  1. 1.Department of EconomicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Department of GeographyHebrew University of JerusalemJerusalemIsrael

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