GY460 Techniques of Spatial Analysis

1
GY460 Techniques of Spatial Analysis
Lecture 4 Techniques for dealing with spatial
sorting and selection(fixed effects,
diff-in-diff, matching and discontinuities etc.)

• Steve Gibbons

2
Introduction

• Sometimes we just want to eliminate problems
  induced by spatial sorting and heterogeneity
• i.e. differences between places which may lead to
  confounding factors and biased estimates of
  relationships of interest
• Selection (sorting) on observable and
  unobservable characteristics
• Examples
• Eliminating spatial factors from models of firm
  behaviour
• Eliminating geographical influences from models
  of school quality
• Various methods are available for dealing with
  this we have looked some of these already

3
Regression models with spatial effects
4
Data with discrete zones

• N observations in the data
• Grouped in to M zones (regions, districts,
  neighbourhoods)
• E.g.
• Cross-section data with gt1 cross-sectional
  observations in each neighbourhood
• Or panel data with more than one time period for
  each neighbourhood

5
Spatial variation in the mean

• Empirical model, with discrete neighbourhoods m
• yim for observation i in place m, depends on
• xim characteristics of observation i in place m
• ?im unobserved factors for observation i in
  place m
• um Unobserved factors common to all
  observations in place m
• X-sectional case i cross-sectional units, m
  places
• Panel data case i time units, mplaces

6
Random effects

• Empirical model, with discrete neighbourhoods m
• If um uncorrelated with xim, then OLS consistent
  just like spatial error model
• Error terms ? are correlated within spatial
  groups m
• But uncorrelated between spatial groups
• Use GLS or ML (assuming normality) for efficient
  estimates and unbiased s.e.s (multi-level
  modelling)

7
Fixed area effects dummy variables

• Empirical model, with discrete neighbourhoods m
• If um correlated with xim, then OLS inconsistent.
• Options
• Estimate the area fixed effects using OLS
• Least Squares Dummy variable model neighbourhood
  dummy variables

8
Fixed area effects within groups

• Or within-groups transformation difference the
  variables from the neighbourhood mean
• Where is the mean of y in group m
• Eliminates um
• Estimate by OLS
• Only uses deviation of variables from
  neighbourhood means so only within-neighbourhood
  variation counts
• LSDV and Within Groups (or (Fixed Effect)
  models are equivalent

9
Fixed area effects panel data

• Even better information with repeated
  observations on panel units (individuals, firms,
  regions etc.) over time
• Panel data
• Now all relationships of interest can be
  estimated from variation within panel units over
  time
• Use within-groups or first-differences over time,
  e.g.
• Q what does (vt-vt-1 ) represent? How could you
  control for it? Then, what variation in the data
  allows us to estimate ??? Hence what do we
  assume, if ? is to be estimated consistently?

10
Dynamic panel data models

• It would be useful to estimate this model e.g.
  to estimate the dependence of y on past values
  (or control for mean reversion)
• Q Can this within group model be estimated
  consistently by OLS?
• See Nickell (1981) Econometrica
• What about the first differenced model?
• Q Is there a useful IV here?

11
Dynamic panel data models

• In principle you could use instruments for
• This is the basis of the Arrelano Bond estimator
  (1991, Review of Economic Studies)
• They develop a GMM estimator which weights the
  instruments taking into account the
  first-differenced error structure e.g.
  implemented in xtabond in STATA
• Problems serial correlation in error terms?, if
  ? is close to zero the instruments will be very
  weak (since lagged values dont predict current
  values if ?0)
• Can also use
  as instruments for
• System GMM (Blundell and Bond 1998) xtabond2

12
Spatial panel data models

• These look attractive e.g. to eliminate sorting
  i.e. u_i
• But this still suffers from the simultaneity
  problems of the spatial y model requires
  maximum likelihood or instruments for
• Also difficult to defend that there is spatial
  correlation, but no time-dynamics
• So you have to estimate
• Have to deal with time dynamic y and spatial y!

13
Spatial panel data models

• Probably more useful to consider the reduced form
  e.g.

14
Difference in difference
15
Difference-in-difference

• Suppose we have places, firms individuals i
  observed over time.
• Treatment group D1 is exposed to some treatment
  x1,0 at time t1, whereas a control group D0 is
  not
• There is selection into treatment group
  (EfD?0) and common time effects g

16
Difference-in-difference

• The effect of the treatment can be estimated by a
  Difference in difference estimator
• Note that this is the same as youd get from OLS
  on

17
Difference-in-difference

• The DiD estimator is commonly used for evaluation
  of policy interventions
• DiD doesnt work if the treatment and control
  groups have different time trends
• If the composition of the treatment or control
  groups change before and after treatment e.g.

18
Matching
19
Matching estimators

• Matching tries to do something similar, when
  treatment and control group are not both observed
  pre and post policy
• Suppose we observe two groups
• Suppose the goal is to estimate the Average
  effect of the Treatment on the Treated (ATT)
• As we know, simple difference in means wont
  work
• i.e. because the treated and non-treated would
  have different Y in the absence of treatment

20
Matching estimators

• But suppose we have some observable
  characteristics Z for which
• i.e. mean pre-treatment Y for individuals with
  characteristics Z is the same, whether or not
  they are in the treatment group
• Called Conditional Independence Assumption CIA
• Allows for selection into treated and non-treated
  groups by Z (selection on observables), but not
  by unobservables.
• So if you can find individuals in group 0 who
  have the same Z as those in group 1 you can
  estimate from the
  individuals in group 0
• If Z is discrete this is straightforward..

21
Matching estimators

• So we can estimate
• The naïve estimate of the effect of the treatment
  is 190-125 65

22
Matching estimators

• For the treated, Y0 is unobserved but can be
  estimated by re-weighting (under the CIA
  assumption)
• So the ATT is 190-180 10

23
Matching estimators

• But what if (as is usual) Z is not discrete?
  Propensity score matching does this reweighting
  using an estimate of the probabilty that
  individual with characteristics z is in the
  treatment group
• (Rosenbaum and Rubin (1983) Biometrika)
• Requires a first stage estimate of Pr(D1 Z)
  e.g. from a probit or logit regression on Z
• Then the treatment effect for an individual i in
  the treated group can be estimated as
• Where the weights depend on the difference
  between the propensity score for individual i and
  the untreated controls j, and

24
Matching estimators

• In practice Matching estimators behave like
  kitchen sink regressions you are just
  controlling for as many observable
  characteristics as possible (Z)
• However, you are controlling for these Z in a
  very non-linear way like having lots of control
  variables and their interactions in an OLS
  regression
• Matching estimators allow for heterogenous
  treatment effects
• You can re-weight in other ways, e.g. to estimate
  the effect of the treatment on the population, or
  on the un-treated
• No solution to selection on unobservables which
  is surely the main issue!
• Requires common support no overlap between Z
  in the treated and untreated groups ? you cant
  match.

25
Discontinuity designs
26
Discontinuity designs

• Regression discontinuity method tries to identify
  causal effects from abrupt changes
• Requires a discontinuity induced by institutional
  rules, policy etc.
• e.g. majority voting
• Class size rules e.g. Maimonides rule
• Geographical administrative boundaries
• Assumption is that assignment to treatment is
  determined by some covariate X when it reaches a
  value d
• The outcome is otherwise only related to X by a
  smooth function e.g. EyX m(X)

27
Discontinuity designs
28
Discontinuity designs

• So
• Idea is to estimate the average effect of the
  treatment at the discontinuity point
• We could control for a m(x) parametrically
  (polynomial series etc.)
• Or restrict the sample to observations for which
  x is close to c i.e.

29
Boundary discontinuities
School quality in district B
ve quality-price relationship across boundary
Price, homeowner characteristics
Price, homeowner characteristics
School quality in district A
Unobserved local amenity
30
Discontinuity designs

• In principle, X is identical for treatment and
  controls exactly at the discontinuity
• But practical applications require non-zero
  differences between X and discontinuity
• E.g. can rarely find a large enough sample of
  housing transactions exactly on the boundary
• Trade off between adequate sample size and
  elimination of biases due to m(x)
• We looked at practical spatial examples e.g.
  Black (1999), Duranton et al (2006)
• See also Gibbons, S., Machin, S and Silva, O.
  (2009), Valuing School Quality Using Boundary
  Discontinuity Regressions, SERC DP0018
  http//www.spatialeconomics.ac.uk/textonly/SERC/pu
  blications/download/sercdp0018.pdf

31
Applications to spatial policy evaluation

• Research designs can incorporate elements of all
  these methods e.g. match treatment and control
  groups using propensity score matching, then
  implement dif in dif
• Machin, S., McNally, S., Meghir,C. (2007),
  Resources and Standards in Urban Schools, IZA
  DP2653 http//ftp.iza.org/dp2653.pdf
• Busso, M. and P. Kline (2006) Do Local Economic
  Development Programs Work, Evidence from Federal
  Empowerment Zone Program, http//www.econ.berkeley
  .edu/pkline/papers/Busso-Kline20EZ20(web).pdf
• Romero, R. and M. Noble (2008) Evaluating
  Englands New Deal for Communities Programme
  Using the Difference in Difference method,
  Journal of Economic Geography 8(6) 1-20

32
The partial linear model
33
Continuous space

• A general model with spatial heterogeneity
• Si is an index of the location of observation i
• Model continuous unobserved variation over space
• m(.) is supposed to represent large-scale
  predictable variation over space e.g. land
  values
• ? random shocks sales price of specific houses
• We discussed these issues in the lecture on
  smoothing
• Could do it parametrically e.g. polynomial series
  or Cheshire and Sheppard (1995) see earlier
  lectures

34
Partial linear model

• Suppose
• If we know ?, function m(.) is just the expected
  (mean) value of y-xb given the location s1, s2
• Refer to the lecture on smoothing this can be
  inferred from values of y in neighbouring
  locations once we know ?
• Spatial weighting again
• Kernel weighting, nearest neighbours etc..

35
Semi-parametric spatial models

• Must get estimates of beta first? How?
• e.g. see Robinson (1988), Econometric, Root-n
  consistent Semiparametric Regression
• Estimate averages of y and all x at each point in
  the data, non-parametrically
• Estimate the betas by OLS on
• Note analogy to the within-groups model
• Can then estimate

36
Applications to housing analysis

• Clapp, J. M., H.-J. Kim, and A. E. Gelfand
  (2002) "Predicting Spatial Patterns of House
  Prices Using Lpr and Bayesian Smoothing," Real
  Estate Economics, 30, (4), 505-532
• Use of non-parametric methods to construct house
  price indices
• Gibbons, S., and S. Machin (2003) "Valuing
  English Primary Schools," Journal of Urban
  Economics, 53, (2).
• Use of the semi-parametric model for eliminating
  larger-scale neighbourhood effects on school
  performance

37
Conclusions

• Underlying issue we have considered is selection
  or sorting e.g. people, firms etc of different
  types sort into different locations and this can
  lead to biased estimates of causal relationships
• Selection can be on unobservables, or observables
• We considered various techniques for dealing with
  these problems
• Other solutions random assignment, IV we have
  or will consider elsewhere.