Heterogeneous Peer E ects and Rank Concerns: Theory and ...uctpmt1/Tincani_supplementary...Online...

Online Supplementary Material for: “Heterogeneous Peer Effects

and Rank Concerns:

Theory and Evidence”

Michela M. Tincani1

November 12, 2014

1Tincani: University College London, 30 Gordon Street, London, WC1H0BE, UK, [email protected].

1 Intensity Attenuation Formula

To generate the intensity attenuation formula for the 2010 Maule earthquake, engineers visited

111 towns after the earthquake struck and evaluated the damage to at least 20 buildings per town

by direct observation. Each building was assigned a damage grade (DG). There are six damage

grades, ranging from no damage to collapse, and they are described in Table 1. The information

on damage to individual buildings in each sampled town is aggregated into a town-wide measure

of seismic intensity (on the MSK scale). Importantly, intensity on the MSK scale is independent of

the distribution of house types in a town.

To understand how the assignment of MSK intensity works formally, consider a simplified

example with only two building types and only two levels of damage. Assume that in town i there

are only two types of houses, a fraction α are adobe houses and a fraction 1 − α are reinforced

masonries. Assume also that each house can suffer only two types of damage grade (DG): high (H)

or low (L), with H,L ∈ R and H > L. Let Ii ∈ [0, 1] denote MSK intensity in town i. If a fraction

x of all adobe houses suffer damage H, then we assign MSK intensity Ii = x to town i. x is the

damage ratio for adobe houses. Reinforced masonry houses are more resistant than adobe houses.

Therefore, engineers know that when Ii = x only a fraction x − ∆m of all reinforced masonries

suffer damage H, where ∆m > 0 captures the additional resistance of reinforced masonries over

adobe houses. Hence, an alternative way to assign an MSK intensity to town i is to look at the

damage to reinforced masonries: if a fraction y of reinforced masonries suffers damage H, then

Ii = y+ ∆m. Notice that the assignment of the MSK intensity does not depend on the distribution

of house types, i.e. Ii is independent of α.1 MSK intensity, though based on observed damages

which vary by building type, is constructed in such a way that it signals intensity of the seismic

event in a town, and not overall damage to that town.

The mapping from damage grade in individual buildings to a town-wide measure of MSK

intensity can be found in Table 2 in Astroza et al. (2012), reporting damage grade and damage

ratios. Table 2 reports the damage grade and ratios for the two building types in my sample: old

traditional adobe constructions (11%) and unreinforced masonry houses (89%). These two building

1There is a redundancy due to the fact that ∆m is known. It is sufficient to look at damage in one type of house,because the damage to other types of houses can be inferred. In practice, engineers look simultaneously at damageto all types of houses.

1

Table 1: Description of damage grades to individual structures

Grade Description

G0 No damage

G1 Slight damage: fine cracks in plaster; falling of small pieces of plaster

G2 Moderate damage: fine cracks in walls; vertical cracks at wall intersections;horizontal cracks in chimneys, parapets and gables; spalling of fairly largepieces of plaster; falling of parts of chimneys; sliding of rood tiles

G3 Heavy damage (uninhabitable): large and deep diagonal cracks in most walls; large and deepvertical cracks at wall intersections; some walls lean out-of-plumb;falling of chimneys, parapets and gable walls; falling of rood tiles

G4 Very heavy damage (uninhabitable): partial or total collapse of a wall in the building;collapse of building partitions

G5 Collapse or destruction (uninhabitable): collapse of two or more walls in the building

types have similar earthquake resistance.

2 Estimation of reconstruction costs

To estimate reconstruction costs, I use the damage ratios proposed in Bommer et al. (2002) and

reported in Table 2.2 In their study, the authors develop a technique to estimate earthquake

restoration costs for Turkish catastrophe insurance. The damage ratios are the expected costs

to restore a building of a given damage grade, expressed in terms of a fraction of the cost of

reconstructing a completely collapsed house. The damage ratios are reported in Table 2. I used

the distribution of damage grade for each MSK-intensity to calculate expected damage cost. For

example, at MSK-intensity 7, 10 percent of adobe houses suffer damage grade 1, 35 percent damage

grade 2, 50 percent damage grade 3, and 5 percent damage grade 4. Using Table 3, the expected

damage ratio for MSK-intensity 7 is 0.10 ∗ 0.02 + 0.35 ∗ 0.02 + 0.50 ∗ 0.10 + 0.05 ∗ 0.50 = 0.084.

To translate this into USD, I multiply by the cost of reconstructing a rural adobe house in Chile

as reported in Comerio (2013) (USD 20, 000). These back-of-the-envelope calculations are for

illustrative purposes only, and are not used in the tests.

2Bommer, J., R. Spence, M. Erdik, S. Tabuchi, N. Aydinoglu, E. Booth, D. del Re, and O. Peterken (2002):“Development of an earthquake loss model for Turkisj catastrophe insurance,” Journal of Seismology, 6(3), 431-446.

2

Table 2: Assumed damage ratios (Bommer at al. 2002)

Damage Grade Damage Ratio

G0 0%G1 or G2 2%G3 10%G4 50%G5 100%

Table 3: Damage grade and damage ratios for the two types of constructions in my sample

MSK Intensity Adobe Unreinforced masonryDG N (%) DG N (%)

V G1 5 G0 100G0 95

VI G2 5 G1 5G1 50 G0 95G0 45

VII G4 5 G2 50G3 50 G1 35G2 35 G0 15G1 10

VIII G5 5 G4 5G4 50 G3 50G3 35 G2 35G2 10 G1 10

IX G5 50 G5 5G4 35 G4 50G3 15 G3 35

G2 10

X G5 75 G5 50G4 25 G4 35

G3 15

XI G5 100 G5 75G4 25

XII G5 100 G5 100

3

Figure 1: Damage difference between neighboring towns. The graph shows the damage gradedistribution for the least and most affected town within the 4 mile wide land strip closest to theearthquake rupture. The graph was generated using MSK-intensities computed with the intensityattenuation formula. Even towns close to each other were affected differently by the earthquake, dueto differences in intensity of shaking and in soil type. For example, the proportion of unreinforcedmasonry building that collapsed in the most affected town is more than three times as large as theproportion collapsing in the least affected town.

4

Figure 2: Towns sampled in Astroza, Rui and Astroza (2012). The number in parenthesis is thetown identifier used in the isoseismal map. Towns are classified according to their MSK intensity,as determined by direct observation by structural engineers. Source: Astroza, Rui and Astroza(2012).

5

Table 4: Difference-in-differences evaluation of the effect of seismic intensity on test scores, withoutthe regressor (1− Pi)Ii, dependent variable Math test score in eighth grade

(1) (2) (3) (4)Municipal Voucher Unsubsidized All Schools

Math test score in fourth grade 0.635*** 0.655*** 0.646*** 0.645***(0.00234) (0.00231) (0.00623) (0.00159)

Household income (CLP) 7.68e-08*** 4.98e-08*** 6.44e-08*** 6.03e-08***(1.03e-08) (5.91e-09) (8.46e-09) (4.35e-09)

Father’s education (yrs) 0.00634*** 0.00755*** 0.0115*** 0.00715***(0.000741) (0.000724) (0.00183) (0.000497)

Mother’s education (yrs) 0.00806*** 0.00784*** 0.0162*** 0.00859***(0.000765) (0.000768) (0.00169) (0.000515)

Female -0.0999*** -0.110*** -0.0881*** -0.104***(0.00410) (0.00385) (0.00914) (0.00268)

Household lives in 0.110** 0.00622 -0.0245 0.0381+earthquake region (E) (0.0389) (0.0312) (0.0675) (0.0229)

Cohort 2007-2011, 0.0500*** 0.0397*** 0.0133 0.0423***affected by earthquake (P) (0.00739) (0.00783) (0.0233) (0.00523)

P*Earthquake Intensity -0.00472*** -0.0110*** -0.00727+ -0.00836***(0.00143) (0.00148) (0.00419) (0.000995)

Constant -0.373*** -0.0997*** 0.0579 -0.184***(0.0288) (0.0263) (0.0677) (0.0185)

School Fixed Effects yes yes yes yes

Observations 97658 117011 20501 235170

Standard errors in parentheses

+ p < 0.10, * p < 0.05, ** p < 0.01, *** p < 0.001

6

Table 5: Difference-in-differences evaluation of the effect of seismic intensity on test scores, withoutthe regressor (1− Pi)Ii, dependent variable Spanish test score in eighth grade


Spanish test score in fourth grade 0.656*** 0.642*** 0.595*** 0.645***(0.00249) (0.00240) (0.00651) (0.00167)

Household income (CLP) 3.61e-08** 2.86e-08*** 1.51e-08 2.51e-08***(1.10e-08) (6.38e-09) (9.61e-09) (4.70e-09)



Female 0.134*** 0.121*** 0.100*** 0.125***(0.00439) (0.00415) (0.0104) (0.00290)

Household lives in 0.0711+ 0.0431 0.0506 0.0546*earthquake region (E) (0.0416) (0.0336) (0.0765) (0.0247)


P*Earthquake Intensity -0.00602*** -0.00766*** -0.0156** -0.00793***(0.00154) (0.00160) (0.00476) (0.00108)

Constant -0.428*** -0.274*** -0.167* -0.331***(0.0309) (0.0283) (0.0768) (0.0200)


Observations 97057 116446 20389 233892


+ p < 0.10, * p < 0.05, ** p < 0.01, *** p < 0.001

7

Table 6: Difference-in-differences evaluation of the effect of seismic intensity on test scores, control-ling for earthquake intensity in pre-earthquake cohort ((1 − Pi)Ii), dependent variable Math testscore in eighth grade


Math test score in fourth grade 0.635*** 0.655*** 0.646*** 0.645***(0.00234) (0.00231) (0.00623) (0.00159)

Household income (CLP) 7.69e-08*** 4.98e-08*** 6.44e-08*** 6.03e-08***(1.03e-08) (5.91e-09) (8.46e-09) (4.35e-09)



Female -0.1000*** -0.110*** -0.0881*** -0.104***(0.00410) (0.00385) (0.00914) (0.00268)

Household lives in 0.275* -0.0924 -0.0827 0.0243earthquake region (E) (0.125) (0.0969) (0.230) (0.0726)


(1-P)*Earthquake Intensity -0.0279 0.0173 0.0100 0.00238(0.0202) (0.0161) (0.0378) (0.0119)

P*Earthquake Intensity -0.0326 0.00622 0.00266 -0.00599(0.0202) (0.0161) (0.0378) (0.0119)

Effect of Earthquake Intensity -0.0046316** -0.0110978*** -0.0073453+ -0.0083705***

θpost − θpre (0.0014325) (0.0014855) (0.0042001) (0.0009965)

Constant -0.369*** -0.103*** 0.0565 -0.184***(0.0289) (0.0265) (0.0679) (0.0186)


Observations 97658 117011 20501 235170


+ p < 0.10, * p < 0.05, ** p < 0.01, *** p < 0.001

8

Table 7: Difference-in-differences evaluation of the effect of seismic intensity on test scores, con-trolling for earthquake intensity in pre-earthquake cohort ((1− Pi)Ii), dependent variable Spanishtest score in eighth grade


Spanish test score in fourth grade 0.656*** 0.642*** 0.595*** 0.645***(0.00249) (0.00240) (0.00651) (0.00167)

Household income (CLP) 3.61e-08** 2.87e-08*** 1.51e-08 2.52e-08***(1.10e-08) (6.38e-09) (9.61e-09) (4.70e-09)



Female 0.134*** 0.121*** 0.100*** 0.125***(0.00439) (0.00415) (0.0104) (0.00290)

Household lives 0.0699 -0.317** -0.147 -0.171*in earthquake region (E) (0.133) (0.104) (0.258) (0.0780)


(1-P)*Earthquake Intensity 0.000206 0.0633*** 0.0340 0.0391**(0.0214) (0.0173) (0.0425) (0.0128)

P*Earthquake Intensity -0.00581 0.0553** 0.0181 0.0310*(0.0214) (0.0173) (0.0424) (0.0128)

Effect of Earthquake Intensity -0.0060204*** -0.0080081*** -0.0158481*** -0.0081014***

θpost − θpre (0.0015408) (0.0016039) (0.004772) (0.0010798)

Constant -0.428*** -0.287*** -0.171* -0.338***(0.0310) (0.0285) (0.0770) (0.0201)


Observations 97057 116446 20389 233892


+ p < 0.10, * p < 0.05, ** p < 0.01, *** p < 0.001

9

Figure 3: Examples of estimated m(c) functions in two classroom categories. The left panel is fromthe pre-earthquake cohort, the right panel is from the post-earthquake cohort.

10

Table 8: Values of test statistics and critical values for test of monotonicity at the α = 0.10significance level, by classroom category. 8 randomly selected categories.

Mathematics SpanishClassroom Category Test statistic Critical Value Test statistic Critical Value

Pre-earthquake classrooms1 2.2874460E-02 4.60e+19 4.0707965E-03 1.05e+192 6.2840671E-04 1.08e+19 1.6759724E-03 7.75e+183 3.6209350E-04 4.98e+18 1.0020613E-03 1.79e+194 3.9056635E-03 1.92e+19 2.2328943E-03 1.97e+19Post-earthquake classrooms5 1.0598215E-03 6.01e+18 3.8213478E-04 1.63e+196 2.7184933E-03 1.41e+19 1.1514544E-03 1.32e+197 4.1919011E-03 4.22e+19 1.3525186E-03 1.19e+198 1.7282768E-03 1.22e+19 3.4069275E-03 1.42e+19

In al classroom categories, the test statistic is below the critical value. Therefore, the null

hypothesis that m(c) is increasing is not rejected.

Figure 4: Books Read from Month 3-4 to Month 8-9 After the Earthquake as a Function of Earth-quake Intensity. Source: SIMCE student questionnaire, 10th grade 2010, public school students.Notice that 5 on the horizontal axis groups earthquake intensities 1 to 5. Regions unaffected bythe earthquake were not used to produce this graph.

11

Table 9: Interaction between seismic intensity and homework load. Dependent variable Math testscore in the 8th grade.

(1) (2) (3)

Math test score 0.642*** 0.640*** 0.640***in 4th grade (0.00344) (0.00347) (0.00347)

Household income 7.51e-08*** 7.25e-08*** 7.25e-08***(CLP) (1.49e-08) (1.50e-08) (1.50e-08)

Father’s education 0.00596*** 0.00572*** 0.00572***(yrs) (0.00110) (0.00111) (0.00111)

Mother’s education 0.00891*** 0.00893*** 0.00893***(yrs) (0.00113) (0.00114) (0.00114)

Female -0.0970*** -0.0968*** -0.0968***(0.00592) (0.00597) (0.00597)

Seismic Intensity 0.0119 0.0110 0.0116θ (0.00761) (0.00768) (0.00786)

Seismic Intensity* -0.00546+ -0.00548 -0.00685High homework load θInt (0.00330) (0.00334) (0.00524)

Additional Impact −0.0173614∗ −0.0164933+ −0.018438+

θInt − θ (0.0088227) (0.008914) (0.0105999)

Class size 0.0119*** 0.0120***(0.00169) (0.00170)

Math teacher 0.000769 0.000764experience (0.000687) (0.000688)

High homework workload 0.00884in Math class (0.0261)

Constant -0.316*** -0.679*** -0.683***(0.0342) (0.0622) (0.0636)

Observations 47196 46394 46394


+ p < 0.10, * p < 0.05, ** p < 0.01, *** p < 0.001

12

3 Testing Monotonicity of m(c), details

Consider the i.i.d. sample {ci,−yi}1≤i≤nl , where nl is the size of the lth classroom category.3 Letci and cj be a pair of observations for c. The test function within each category l is defined as:

b(s) = b({ci,−yi}, s) =1

2

∑1≤i,j≤n

(−yi + yj)sign(cj − ci)Q(ci, cj , s)

where I dropped the l subscript for convenience, and where Q(ci, cj , s) is a weighting functionindexed by s ∈ S. To each s corresponds a choice of point c and bandwidth h for the followingspecification of the weighting function:

Q(c1, c2, (c, h)) = K

(c1 − ch

)K

(c2 − ch

)where K(u) = 0.75(1 − u2) if −1 < u < 1, and = 0 otherwise, and where h = 12n

− 15

l .4 I let c take

on 100 values, which identify equally spaced points going from the smallest to the largest observedvalue of ci in the population. As a result, there are 100 weighting functions for each classroomcategory l.

Conditional on {ci}, the variance of b(s) is given by:

V (s) = V ({ci}, {σi}, s) =∑

1≤i≤nσ2i

∑1≤j≤n

sign(cj − ci)Q(ci, cj , s)

2 (1)where σi =

(E[�2i |ci]

) 12 and �i = −yi − (−m(ci)). Following Chetverikov (2013), I use the residual

�̂i = −yi − (−m(ci)) as an estimator for σi, and obtain the estimated conditional variance of b(s)by substituting σ2i with σ̂

2i in equation 1. The test statistic is given by:

T = T ({ci,−yi}, {σ̂i}, S) = maxs∈S

b({ci,−yi}, s)√V̂ ({ci}, {σ̂i}, s)

.

Large values of T indicate that the null hypothesis that −m is increasing is violated.To simulate the critical values, I adopt the plug-in approach. The goal is to obtain a test of

level α. Let {ξi} be a sequence of B independent N(0, 1) random variables that are independentof the data. Let −y∗i,b = σ̂iξi,b for each b = 1, B and i = 1, n, where σ̂i = �̂i. For each b = 1, B,calculate the value T ∗b of the test statistic using the sample {ci,−y∗i,b}ni=1. The plug-in critical valuec1−α is the (1− α) sample quantile of {T ∗b }Bb=1.

3yi is replaced by −yi, and m will be replaced by −m, because this procedure tests that −m is increasing, whichis equivalent to testing that m is decreasing.

4This is the value for h recommended in Ghosal, Sen, and Van Der Vaart (2000).

13

Intensity Attenuation Formula Estimation of reconstruction costsTesting Monotonicity of m(c), details

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