空间计量经济学导论

• Independence of observations is a basic assumption of most statistical modeling procedures. • Why is independence important? – The formulas used to fit the line are only correct if we have independence – Wrong intercept and slope estimates mean all our conclusions are wrong!
so that tract 1 is a neighbor to 2, and 2 is a neighbor to both 1 and 3, while tract 3 is a neighbor to 2. Then our weight matrix takes the form:
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Back to our students
Table 2: Seating location of students (in a row) Seats Occupied Study Time Exam Score John Steve Mary Devon Billy Denise
0 70
15 70
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Spatial weight matrices
Example #1: Let three census tracts be located in a line: Table 3: Location of 3 Census tracts in Space Tract #1 Tract #2 Tract #3
80
100
Score = α + β · Study Time (in minutes)
• Focus in on the slope =
∆Score ∆Study time
=β
• Intercept = α = score with no study time
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ARE SAMPLE OBSERVATIONS INDEPENDENT?
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Spatial Dependence
• Cross-sectional spatial data samples. • Spatial data samples represents observations that are associated with points or regions, for example homes, counties, states, or census tracts. • Observations are regions (or points) • One observation depends on others, e.g. Does Devon’s score of 85 given 5 minutes of study time depend on neighboring students Mary and Billy’s scores/study time. • Suppose we let observations i = 1 and j = 2 represent neighbors (perhaps regions with borders that touch), then a data generating process might take the form shown in (1).
90 100
5 85
60 90
30 80
• Is independence valid here? • Is knowledge of who each student is sitting next to important?
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Definition of Spillovers
• A (spatial) spillover arises when a causal relationship between characteristics/actions of entity/agent (Xi) located at position i in space exerts a significant influence on the outcomes/decisions/actions (Yj ) of an agent/entity located at position j • Formally, ∂Yj /∂Xi ̸= 0 • If locations j are neighbors to location i, we have a local spatial spillover • If locations j include not only neighbors to i, but neighbors to neighbors of i, neighbors to neighbors to neighbors, and so on, we have a global spillover
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RELATING EXAM SCORES TO STUDY TIME
• We suppose exam scores are linearly related to study time
Score = α + β · Study Time (in minutes)
Table 1: Sample of exam scores and study time Student Exam Scores Study Times John y1 = 70 x1 = 0 Devon y2 = 85 x2 = 5 Steve y3 = 70 x3 = 15 Denise y4 = 80 x4 = 30 Billy y5 = 90 x5 = 60 Mary y6 = 100 x6 = 90
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Importance of Spillovers
• Costs vs. direct benefits + indirect (spillover) benefits analysis • Examples include: – disaster aid to increase probability of firm A reopening may spillover and increase probability of neighboring firms B and C re-opening. LeSage, James, R. Kelley Pace, Nina Lam, Richard Campanella, and Xingjian Liu (2011),“New Orleans business recovery in the aftermath of Hurricane Katrina,” Journal of the Royal Statistical Society, Series A. – home mortgage adjustment program that decreases loan-to-value ratio to reduce the probability of homeowner A defaulting may decrease probability of neighboring homeowners B and C defaulting on their mortgages. Zhu, Shuang and R. Kelley Pace (2011) “Modeling Spatially Interdependent Mortgage Decisions,” Journal of Real Estate Finance and Economics Vol. 44, Nos. 1/2, 2012
Introduction to Spatial Econometrics
James P. LeSage Fields Endowed Chair for Urban and Regional Economics McCoy College of Business Administration Department of Finance and Economics Texas State University San Marcos, Texas 78666 jlesage@spatial-econometrics.com March, 2014
yi yj ε i , εj
= = ∼
αiyj + Xiβ + εi αj yi + Xj β + εj N (0, σ )
2
(1)
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The spatial autoregressive process
• Ord (1975 JASA) proposed a parsimonious parameterization for the dependence relations (which built on early work by Whittle (1954). Applied to the dependence relations between the observations on variable y , we have expression (2).
n ∑ j =1
yi εi
= ∼ ∑n
ρ
Wij yj + xiβ + εi
2
(2)
N (0, σ )
i = 1, . . . , n
• The term: j =1 Wij yj is called a spatial lag, since it represents a linear combination of values of the variable y constructed from observations/regions that neighbor observation i. • This is accomplished by placing elements Wij in the ∑n n × n spatial weight matrix W , such that j =1 Wij yj results in a scalar that represents a linear combination of values taken by neighboring observations.
2
Linear relationship
110 105 100 95 Exam Scores 90 85 80 75 70 65 60 0 20 minutes more study time 5.75 points higher score
20
40 60 Study time (in minutes)
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WHY DO WE LOVE INDEPENDENCE?
• It makes things simple ← ← . . . ←
y1 y2
x1 x2
yn
xn
• Student 1’s score (y1) depends only on their own study time (x1) • House i’s selling price (yi) depends only on its own characteristics (xi) • Teen 1’s behavior (y1) depends only on their own characteristics (x1) • Land use of property/parcel (yi) depends only on its own characteristics (xi)
Overview
• • • • • •
A description of spatial dependence Spatial autoregressive processes Spatial regression models Interpreting spatial regression model estimates Applied examples Motivations for spatial regression models