Reasons for Instability in Spatial Dependence Models Jesús Mur(*), Fernando López (**) and Ana Angulo(*) (*) Department of Economic Analysis University of Zaragoza Gran Vía, 2-4. (50005). Zaragoza. Spain. (**)Department of Quantitative Methods and Computing University of Cartagena. Paseo Alfonso XIII, Cartagena. Spain.

MOTIVATION It is interesting to elaborate a framework in which it is possible to deal simultaneously with the questions of heterogeneity and of spatial dependence. → For example, an outlier of the HL type, a ‘diamond in the rough’, may really be that, an outlying value in a stochastic distribution, but it may also be the logical result of a process of negative autocorrelation. Statistical anomaly Structural break

MOTIVATION However, the symptoms are confusing:  Anselin (1990, p.204) indicates, often ‘the misleading indication of heterogeneity is due to spatial autocorrelation’  Pace and Lesage (2004, p.31) state that there are frequent cases in which ‘spatial dependence arises due to inadequately modelled spatial heterogeneity’

A General Spatial Lag Model with INSTABILITY where Angulo, A., J. Mur and F. López (2008): An inquiry into the causes of instability in cross-sectional models: An application to outcomes in 2008 Spanish General Elections. 14 International Conference on Computing in Economics and Finance, Paris June 2008

Robust Multipliers for testing the hypothesis of stability

(a) (1) SLM ; (2) SLM with instability in  (3) SLM with instability in  (4) SLM with instability in  (5) SLM with instability in  and  (6) SLM with instability in  and  (7) SLM with instability in  and  SLM with instability in all the parameters ( ,  and  ).

An small step forward: This battery of tests, combined with some other technique, performs reasonable well under different circumstances. We are highly confident in detecting problems of instability in the parameters of a spatial model. Now the question is to identify the PATTERN of the spatial break: ¿How this instability takes place ? We will focus on the parameters of SPATIAL DEPENDENCE

We will combine two different approaches * LOCAL ESTIMATION Geographically Weighted Regression, GWR, (Brunsdon et al, 1998 and 1999, Fotheringham et al, 1999, Pace and Lesage, 2004, Leung et al., 2000 and 2003, Páez et al, 2002a and 2002b, Mur et al, 2007a and b) Locally Linear Weighted Regression, LWR, (Cleveland, 1979, Cleveland and Devlin, 1988, McMillen, 1996, McMillen and McDonald, 1997) PLUS * EXPANSION ALGORITHMS Linear expansion in the parameters (Casetti, 1972, Casetti and Poon, 1995, and Casetti, 1997 )

LOCAL ESTIMATION Replicate the ML estimation of the SAR process, but in a delimited zone surrounding each region. To do so, we will take the set of the m regions nearest to that of interest (in terms of the Geographically Weighted Regression, GWR, methodology, this is the bandwidth) With this restricted sample, we obtain the ML estimation AROUND each point. To sum up, we solve R ML estimations.

Example: Box Map of the log. of the income per capita. Year 2002.

Map of quantiles and histogram of the estimation of ρ(i) for m=20

EXPANSION ALGORITHM Let us suppose a linear model: Where the parameters evolve according to some contextual variables in a linear way: {z 1r, z 2r, z 3r, …., z kr } is the set of ‘contextual’ variables. The reduced form of this system of equations is the so-called final model:

Case to discuss: There exists instability only in the parameter of SPATIAL DEPENDENCE (Implicitly we assume stability in the other elements)

h(-) is an unknown function It is assumed that we have identified the ‘contextual’ variables: the z’s associated to the break. Solution: we will try to linearize the function by using a polynomial in the contextual variables:

Going back to the model specified We obtain a final useful expression to be used in a ML algorithm

Briefly and for N=1 (strictly linear) The score:The IM: Then we can obtain the tests, etc…..

Some examples Type 1. Plane with slope: Type 2. Elliptical paraboloid : Type 3. Hyperbolic paraboloid: R being the number observations, z 1 and z 2 the coordinates associated with point each of the grid (for regular square grid z 1, z 2 = 1,...,). The parameter  regulates the degree of instability that will range between 0 and 1. k =2.

Figure 1: Patterns of spatial instability in  (20x20)

Figure 2: The Local estimation algorithm

More on the local estimation algorithm Using a Linear Simple Model (only one ‘x’ variable in the RHS), with instabilities in the parameters

Regional distribution of the logarithm of the per capita income. REVISED. Year 2002.

The European regional income appears to be a SAR model: STABLE? z 1 and z 2 are the spatial coordinates of the regional centroid

Estimation results

Isopleths of Spatial Dependence. EU27 NUTS III Each point corresponds to a regional centroid.