• 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!
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)
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:
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
3
ARE SAMPLE OBSERVATIONS INDEPENDENT?
7
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
10
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
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.
1
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
5
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
8
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).
yi yj ε i , εj
= = ∼
αiyj + Xiβ + εi αj yi + Xj β + εj N (0, σ )
2
(1)
9
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).
90 100
5 85
60 90
30 80
• Is independence valid here? • Is knowledge of who each student is sitting next to important?
6
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