Spatial Econometrics

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Spatial Econometrics

• Course Applied Econometrics
• Lecturer Zhigang Li

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Test for Spatial Correlation

• YXße
• Moran I (MI) Statistic (Case, 1991)
• Measure covariance in errors between joining
  districts relative to the variance in errors in a
  given district. The idea is similar to
  autocorrelation for time series data.
• Let ?ij1 if districts i and j share a join and
  0 otherwise. Let e be the average of residuals
  for observations in district i. J is the total
  number of joins and d is the total number of
  districts, then
• MI(SiSjei?ijej)/Sei2(d/2J)

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Spatial Econometric Model I Mixed Spatial
Autoregressive Model

• yfW1yXßu
• u?W2ue, eN(0,s2I)
• In the MSA model, y is a vector of n observations
  (for different locations, e.g. cities). X is a
  matrix of exogenous variables.
• f and ? are scalar (spatial) parameters.
• W1 and W2 are spatial weights matrices,
  typically with zero diagonal.

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Spatial Econometric Model II Mixed Spatial
Autoregressive Moving Average Model

• yfW1yXßu
• u?W2ee, eN(0,s2I)
• For the MSA model to identify, W1 needs to differ
  from W2. X needs to contain at least one
  exogenous variable in addition to the constant
  term (Anselin et al. 1996).
• These conditions are not needed for the MSAM
  model.

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Spatial Weight Matrix W1 and W2

• If W1 and W2 are zeroes, the spatial models
  degenerate into normal linear regression models.
• If W1 is nonzero while W2 is zero, the dependent
  variable (e.g. rental prices) is affected not
  only by exogenous variables X, but also by its
  spatial counterparts, e.g. the rental prices in
  neighbor cities.
• The spatial effect W1X is obviously endogenous
  so the estimate of the spatial correlation f is
  generally inconsistent.

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Spatial Weight Matrix W1 and W2 (Continued)

• If W1 is zero but W2 is not, OLS estimates of the
  model are consistent but the standard error
  estimates are inconsistent (like in the
  autoregressive error case).

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Estimation of W2 Models

• Large T, small N
• With spatial correlations in the error term,
  parameters of the model can still be consistently
  estimated so correct residuals can be obtained.
• With large T, the obtained residuals for
  different periods can then be used to estimate
  the spatial correlation structure ?W2 and then to
  correct the bias in standard error estimates.
• Small T, Large N
• With small T, we can not use the above approach
  to estimate an arbitrary W2. Instead, we have to
  assume the structure W2 to estimate ? and then to
  correct the bias in standard error estimates.

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What Do the (W1-) Spatial Econometric Models Do?

• The W1-model estimates (if consistent) indicate
  equilibrium spatial correlations between the
  dependent variable (e.g. prices) of
  geographically dispersed observations.
• A static model
• The mutual responses may not be symmetric (this
  depends on the spatial weight matrix).

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Estimation of W1 Models

• Estimation Methods
• Maximum Likelihood method (Ord 1975)
• Generalized Moments Apporach (Kelejian and Prucha
  1999)
• OLS (Lee 2002)

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Estimating W1 Models Using OLS I (Lee, 2002)

• OLS is applicable if the effect of each
  neighbor unit is small (i.e. if the endogeneity
  is little).
• YnXnß?WYnVn
• Here n is the number of cross-sectional units and
  V is a vector of i.i.d. homogeneous errors.
• OLS estimates are consistent if E((WY)V/n))
  converges to zero as n approaches infinity.
• This rules out the case where neighbor units
  are units with a contiguous border.
• A valid case for OLS may be one where each unit
  is influenced by many of its neighboring units.

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Estimating W1 Models Using OLS II (Lee, 2002)

• For example, Case (1991) looks at the household
  consumption of rice. Here neighbors refer to
  farmers who live in the same district. As n
  increases, the number of farmers in the same
  district also increase, so the effect of each
  farmer decreases (towards zero).
• For pure spatial autoregressive models (X is
  zero), OLS cant be consistent and ML or GMM
  methods are needed.
• For finite sample, the bias of ? and ß are also
  affected by the size of s2 (the variance of error
  term) and ? (the neighbor effect), and the
  collinearity between X and W(I-?W)-1Xß.
• Even when errors in V are spatially correlated,
  the consistency and asymptotic normality of
  estimates can still holds, but standard error
  estimates are biased and need to be corrected.

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MLE Method (Ord, 1975)

• Y?WYe
• Let BY-?WY, then eB.
• Given ?, B can be calculated.
• Since eNormal(0,s2I), we can calculate the
  likelihood of each observation.
• The MLE approach would capture the complex
  inter-relation between the observations.
• OLS does not fully account for this
  interrelation, so its estimate would generally be
  inconsistent.

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Generalized Moments (GM) Approach (Kelejian and
Prucha, 1999)

• YXße where e?Weu
• Moments conditions
• E(uu/N)s2
• E(uWWu/N)s2Tr(WW)/N
• E(uWu)0
• Procedure
• First estimate the model with OLS to obtain
  consistent estimates of ß and then calculate
  residuals.
• Use residuals, the left-hand-side of the moments
  conditions above can be replaced by practical
  moment conditions using the residuals.
• Solve the three-equation system for parameters ?,
  ?2, s2.

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Estimation Softwares (Florax and Vlist, 2003)

• SpaceStat (Anselin 2000)
• Spatial econometrics toolkit (for Matlab) by
  James Lesage (www.spatial-econometrics.com)

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Note

• The spatial econometric models discussion do not
  identify the causal effects between different
  districts. Instead, they only measure some form
  of equilibrium relationship. (See Charles Manski
  1993 Review of Economic Studies paper for a
  discussion of the reflection problem.)

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Wage Spillovers in Public Sector Contract
Negotiations (Babcock et al. 2005)

• Theory of Wage Spillovers
• Collective bargaining using wage comparables
  (or the wages of reference groups).
• It is a commonplace (in the US) for the two sides
  of public sector unionized labor market to refer
  to outcomes negotiated in comparable
  municipalities during public sector contract
  negotiations.

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Empirical Strategy I

• Yaift?WYdSWYXße
• e?WGeu
• W is the spatial reference matrix indicating
  the actual spatial pattern of references by
  schools wage negotiators. WG, in contrast, is
  just some reduced-form geographic structure.
• Y are the current stage negotiated wage. Y are
  the wages in contracts negotiated within the last
  three years and are still in use. Therefore, this
  model is not the traditional static spatial
  econometric model and makes more sense for causal
  inference.
• The model permits the amount of spillover to vary
  with community support for union activity, S.

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Empirical Strategy II

• Data 364 Pennsylvania school districts during
  the mid-1980s.
• Y Salaries of teachers with bachelors degree
  plus 15 years of experience.
• S (support for union) is approximated by the
  percentage of Democrats voters, the percentage of
  prolabor votes by the state senator, and the
  percentage of unionized private sector workers.
• W (the reference matrix) is first estimated using
  survey information on the reference process of 70
  schools (Appendix B).
• The model of wage spillover is estimated using
  3-stage IV regression (Kelejian and Prucha).

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Findings

• Union strength is positively related to wages.
• The salaries of districts referred to by the
  union have the largest impact on the negotiated
  outcome (elasticity of about .87)

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Spatial Correlation in Contracts(Pinkse and
Slade, 1998)

• Spatial error correlation in a Discrete-Choice
  framework
• yXß0u, u?0Wue, eN(0, I)
• y1 if ygt0, 0 if ylt0
• y Contract type (low vs. high powered)
• X is the observed attributes of the gas stations
  (hours, service bays, car wash, )
• Estimation Method A GMM approach.

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Spatial Weights Matrix

• Six Alternative Spatial Matrixes
• Distance
• On the same street
• Distance on the same street
• Nearest neighbor on the same street
• Nearest neighbor
• Common boundary
• Nearest neighbor seems most plausible (also
  common boundary) by significance of ?0 table.4.

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Interaction among Local Governments The Case of
Growth Controls (Brueckner, 1998)

• Theoretical Model of Competing in Growth Control
• Consumers are mobile across cities, and people
  prefer small to large cities.
• Local governments may have the incentive to
  compete in growth controls.

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Measures of Growth Control

• Examples of Growth Control Approaches
• Population growth limitations
• Housing permit limitations
• Commercial building height limitations
• In total nine growth control approaches are
  considered. The measure of growth control is the
  number of control approaches adopted by a local
  government.

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Spatial Weights

• wij1
• wij1/dij
• wijPj
• wijPj/dij

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Spatial Spillovers between University Research
and Innovations (Anselin, 1997)

• Knowledge Production Function (State level)
• kßk1rßk2ußk3uc ek
• rßr1ußr2z1er
• ußu1rßu2z2eu
• K is a proxy for knowledge output (k is lnK,
  similar for others). R is industry RD. U is
  university research.
• C is a geographic coincidence index, indicating
  how closely are university and industry research
  located spatially.

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Data

• Geographic Coincidence Index
• Correlation between university research and
  industrial RD for the MSAs in the state.
• Proportion of counties where industry research
  and university research are co-located.
• Gravity with distance
• How much university research is carried out in
  counties within a given distance band of a
  industry RD county
• Innovation
• Data obtained from new product announcements in
  trade and technical publications. 4200
  innovations are identified by the address of the
  innovating establishment.