Post on 08-Jun-2020
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Asymmetric Information, AdverseSelection and Seller Revelation on
eBay Motors
Greg Lewis
University of Michigan
February 13, 2007
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Motivation
eBay MotorsOnline used car market$1 billion in revenues in 2005Pundits predicted it wouldn’t work
Market for Lemons (Akerlof 1970)Quality heterogeneityAsymmetric information (condition, history, parts)No credible disclosure technology ! adverseselection
Empirical questions:How much information is disclosed on eBayMotors?Is such disclosure selective?
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Motivation
eBay MotorsOnline used car market$1 billion in revenues in 2005Pundits predicted it wouldn’t work
Market for Lemons (Akerlof 1970)Quality heterogeneityAsymmetric information (condition, history, parts)No credible disclosure technology ! adverseselection
Empirical questions:How much information is disclosed on eBayMotors?Is such disclosure selective?
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Motivation
eBay MotorsOnline used car market$1 billion in revenues in 2005Pundits predicted it wouldn’t work
Market for Lemons (Akerlof 1970)Quality heterogeneityAsymmetric information (condition, history, parts)No credible disclosure technology ! adverseselection
Empirical questions:How much information is disclosed on eBayMotors?Is such disclosure selective?
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Line of Attack
Link price and informationObtain unique eBay datasetProvide quantitative measures of webpage contentRun hedonic regressionsFind large e!ect of amount of information on price
Analyze auction model with strategic sellerrevelation
Equilibrium with selective disclosureSellers reveal favorable private informationBidders view absence of information as bad signal
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Line of Attack
Link price and informationObtain unique eBay datasetProvide quantitative measures of webpage contentRun hedonic regressionsFind large e!ect of amount of information on price
Analyze auction model with strategic sellerrevelation
Equilibrium with selective disclosureSellers reveal favorable private informationBidders view absence of information as bad signal
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Line of Attack
Deduce model predictionsPrior mean valuation increasing in amount ofinformation disclosedInformation measures endogenous
Estimate model and test predictionsDevelop new common value auction estimatorRecover mean valuationsTest for endogeneity using semiparametric controlfunction
Quantify impact of disclosuresMagnitude of structural resultsCounterfactual simulation
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Line of Attack
Deduce model predictionsPrior mean valuation increasing in amount ofinformation disclosedInformation measures endogenous
Estimate model and test predictionsDevelop new common value auction estimatorRecover mean valuationsTest for endogeneity using semiparametric controlfunction
Quantify impact of disclosuresMagnitude of structural resultsCounterfactual simulation
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Line of Attack
Deduce model predictionsPrior mean valuation increasing in amount ofinformation disclosedInformation measures endogenous
Estimate model and test predictionsDevelop new common value auction estimatorRecover mean valuationsTest for endogeneity using semiparametric controlfunction
Quantify impact of disclosuresMagnitude of structural resultsCounterfactual simulation
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Findings
1 Strong evidence of selective disclosure by sellers2 Disclosures reduce information asymmetries
and potential for adverse selection
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Contribution
Adverse selection in used car marketsLittle empirical evidence (e.g. Bond (1982,1984),Genosove (1993))Quantify role of disclosure in limiting informationasymmetries
eBay LiteratureRole of reputation (Resnick/Zeckhauser (2002))Role of information dispersion (Pai-Ling Yin)Direct analysis of role of information on eBay
Empirical literature on disclosureJin (2005), Jin and Leslie (2003)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Contribution
Information disclosure in auctionsLinkage principle (Milgrom & Weber (1982))Costly revelation (Grossman (1981), Milgrom(1981), Jovanovic (1983))Provide measure of amount of disclosure, and showpositively correlated with value
New common values eBay estimatorBajari & Hortacsu (2003), Hong & Shum (2002)Robust to early and non-credible bidsComputationally lightControl function approach for endogeneity
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Outline of Talk
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
How eBay Motors works
Seller posts an auction webpageAuction has fixed duration, bid at any time
People “snipe”, bidding at last momentOver 10% of bids on Corvettes in last 10 minutes(Adams et al)Late Bidding Model
Potential bidders can communicate with seller
Close of auction, pay second highest bidPick up car and pay after close of auction
Ex-post verifiability of information
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
A Tale of Two Cars
Car 1 (good webpage)2 pages text, 28 photos37 bidsfinal price $6875, blue book = $5100
Car 2 (poor webpage)3 lines text, 3 photos4 bidsfinal price $1225, blue book = $4700
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
What is Information?
Distinction between:Seller’s Private InformationBidder’s Private InformationPublic Information(seller chooses to post on webpage)
I refer to information as the content of auctionwebpages (public)I refer to the amount of information as aquantitative measure of that content
Bytes of text in descriptionNumber of photos
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Data
Collect bids, item, seller characteristics fromover 40000 auctions
Download eBay auction webpages over a 6 monthperiod ! unique panel datasetModels chosen include classic cars, reliable cars,pickupsUse text parsing program to obtain variables
Drop auctions with < 2 bids, re-listings,missing data, proprietary software
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Characteristics of Dropped Auctions
Table: Characteristics of Dropped Auctions
n " 2 n < 2Age 19.51 18.52Mileage 123050 107853Text 1680 1701Photos 12.1 10.8Starting Bid 2640 10071
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Trading Activity
Sellers listing exactly one car”Private Sellers” = 83.4% of sellersAccount for 54.3% of listings
Sellers listing more than one car”Dealers” = 16.6% of sellersAccount for 45.7% of listings
Repeat buyers are a small fraction (2.2%)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Variation in Information Measures
Table: Variation in Information Measures
Private Sellers DealersText 1434 2248
(1479) (2765)Photos 11 13.8
(6.3) (7.1)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Hedonic Regressions
Hedonic regression (OLS):
log(p) = Z! + " (1)
wherep is final auction priceZ is a vector of covariatesInclude title, transmission, year and model fixede!ects
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Table: Hedonic Regressions
(1) (2)Log Text Size 0.0775*** 0.0886***Photos 0.0212*** 0.0221***Number of Options 0.0239***Log Feedback -0.0193***% Negative Feedback -0.0029***Total Listings 0.0011Warranty 0.7979***Warranty*Logtext -0.0780***Warranty*Photos -0.0165***
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Table: Hedonics by Subsample
Full Private Dealers DealersLog Text Size 0.0804*** 0.0998*** 0.0330*** 0.1584***Photos 0.0221*** 0.0223*** 0.0195*** 0.0219***Model FE yes yes yes yesYear FE yes yes yes yesSeller FE no no no yes
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Table: Hedonics for 1984-2006 Mustangs
Estimated Coe"cientLog Text Size 0.0907***Photos 0.0179***Year FE yes
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Text coe!cients by year
5 10 15 20Age
!0.05
0.05
0.1
0.15
0.2
Estimated Coefficient
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Photo coe!cients by year
5 10 15 20Age
0.01
0.02
0.03
Estimated Coefficient
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
eBay Motors
Potential Explanations
Seller E!ectFixed e!ects results suggest no
Advertising E!ectWarranty result suggests genuine informationTest later
Strategic E!ect (Winner’s Curse)
Selective Disclosure
Auction model with disclosure nests last three
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Modeling Choices
Symmetric pure common value auction modelJustified by tests (Athey/Haile)Consistent with strategic Winner’s Curse storyParticipants in an auction have similar preferences
Multidimensional private information for sellersMeasure amount of information revealed
Credible but costly revelationCredibility from institutional featuresOpportunity cost of posting text, photos
Bajari & Hortacsu (2003) late bidding model
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Formalism
N bidders, realization n common knowledge
Common Value V # FBidder signals X1 · · ·Xn c.i.i.d # G |v
Private communications
Seller signals S1 · · · Sm c.i.i.d. # H |vCar features, history, conditionSignals may be of di!erential importance
V , X1 · · ·XN , S1 · · · Sm a"liated
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Formalism
Seller can reveal signal Si publicly at cost ci
Amount of information I = # signals revealedEach revelation must be written up / displayedEqual weighting of signals
Model extremely generalMultidimensional real-valued signalsArbitrary revelation costsNecessitates strong assumption later
Two-stage gameSeller makes disclosuresBidders participate in auction
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Late Bidding Model
eBay format is English auction with re-entry
Sniping ! late observation of bids impossibleTwo Bidding Rounds:
Early-bidding, [0, # $ $), bidders observe eachothers bids, exit non-bindingLate-bidding, [# $ $, # ], sealed-bid auction
Late bids unrecorded if pt > bt
bidder i ’s true valuation unrecordedmay misinterpret earlier bid by i as valuation
Can show bid zero in early stage, play secondstage as sealed bid auction
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Modeling Disclosure
Seller chooses which signals to revealSeller’s private information realization sReporting policy R(s) : Rm ! {0, 1}m
Reported Signals sRDefine expected value with public signals
w(x , y , s; n) = E [V |X = x , Y = y , S = s, N = n] (2)
Make additivity assumption
w(x , y , s; n) =m!
j=1
fj(sj) + g(x , y , n) (3)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Proposition (Costly Revelation)
For c > 0, there exists a sequential equilibrium in which
1 The reporting policy R is characterized by cuto!st = t1 · · · tm % R &' such that:
R(si ) =
"1 if si " ti0 otherwise
where ti = inf {t : E [!i (t, t !; n)|t ! < t] " ci}.2 Bidders bid zero during the first stage of the auction, and bid
vR(x , x , sR ; n) = E [w(x , x , sR ; n)|{si < ti}i"U ]
during the second stage, where sR is the vector of reported signalsand U = {i : R(si ) = 0} is the set of unreported signals.
3 The final auction price p! = vR(x (n#1:n), x (n#1:n), sR ; n).
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Key Result
Proposition (Expected Monotonicity)
The interim prior mean E [V |I ] is strictly increasing in theamount of information I .
New theory resultProof shows that equilibrium strategies inducea"liation between I and VHinges on additivity and conditional independenceof signalsImplies that if seller’s private information is“separable” in some sense then amount ofdisclosure is positively correlated with value
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Theory
Predictions
Theory predicts:1 Interim prior mean E [V |I ] increasing in I2 Endogeneity of information measure I
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Motivation
Why do I need a structural econometric model?
1 To recover the relationship between latentvalues and covariates
Object of demand estimation2 To carefully test prediction
Prediction is E [V |I ] increasing in I , not E [p|I ]Winner’s Curse - better informed bidders bid moreSo E [p|I ] could be increasing for strategic reasons
3 To perform the counterfactual simulation
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Parameterization
Parameterize interim priorBidders observe covariates z , have interim prior F |zLet interim prior F |z be Log Normal (µ, %)Let conditional signals G |v be Log Normal (v , r)Let µ = &z , % = #(!z)Model parameters ' = (&, !, r)
Include number of photos and bytes of text asregressors
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Estimation Strategy
Inherent Di"cultiesLatent v and bidder signals x1 · · · xn
Bid function depends non-linearly on latentvariables
Approach the problem by pseudo-maximumlikelihood (PML)
Dependent variable is price
p = v(x (n!1:n), x (n!1:n), n)
Posit log-normal distribution for p with mean andvariance E [log p|zj , '; n] and $[log p|zj , '; n]Maximize pseudo-likelihood function over '
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
PML Asymptotics
Estimator consistent and asymptotically normal(Gourieroux, Monfort and Trognon (1984))
Idea is:Pseudo-likelihood from quadratic exponentialfamily + moments correctly specified( Limiting PL maximized at '0
Standard M-estimation conditions to get '̂n ! '0
Global identification required - numerical checks
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Computation
Moments complicated, for example:
v(x , x , n;µ,!, r) =
#eq"((x $ q)/r)2"((x $ q)/r)n#1"((q $ µ)/!)dq#"((x $ q)/r)2"((x $ q)/r)n#1"((q $ µ)/!)dq
Standard tricksPre-compute moments on gridUse scale properties of log normal distribution
New trick exploits PML approachScale properties ! $&(!, r) computable by WLSNested loop, inner loop &, outer loop (!, r)Avoids curse of dimensionality
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Advantages of Estimator
Previous approachesBajari/Hortacsu (2003) - Bayesian MCMCHong/Shum (2002) - Quantile Estimation
Advantages:Robust estimation (only use second highest bid)Much less computationally intensive in highdimensions (minutes vs days)Can use non-linear control function for endogeneity
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Endogeneity of Information
May be concerned about endogeneity ofinformation
Omitted quality index (µ = &z + (, % = !z "
Information measure I correlated with (
Does this cause an endogeneity problem?Depends on true moment specificationNo advertising e!ect ( I redundant given (Control function approach to test for advertisinge!ect
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Control Function Approach
Suppose I have an instrument W
Specify policy function: I = g(z ), W ) + h(()
Kernel Estimation of residualsm = I $ E [I |z ), W ]
h monotone, so * control function f (m)
Idea: Blundell/Powell(2003), Pinkse (2000),Newey et al (1999)
Asymptotics: Andrews (1994a, 1994b)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Control Function Asymptotics
M-estimation with plugged-in $mSpecification correct( maximizer of limiting PL function = '0
Need stochastic equicontinuity of criterionfunction
Infinite-dimensional first stage nuisance parameterNeed criterion function “smooth” in function space
Kernel estimationBandwidth by cross validation (no oversmoothing)“Smooth” Trimming (underweight low density obs)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Estimation
Instrument
Exploit panel data
Instrument W = amount of information It$1
for previous car sold
Find It$1 and I strongly correlated in dataNeed It$1 independent of (t
Need ( randomly assigned over time within sellersUnobserved car characteristics idiosyncraticIt!1 related to (t!1, independent of (t
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Table: Structural Demand Estimation
(1) (2)µLog Text 0.0537*** 0.0592***Photos 0.0185*** 0.0179***Log Feedback -0.0195***Percentage Negative Feedback -0.0027**%Age 0.0127*** 0.0126***Log of Text Size -0.0070 -0.0081Photos -0.0178*** -0.0182***Mean % 0.8472*** 0.8442***r 1.2099* 1.2095*
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Structural Results I
Model prediction µ = E [V |I ] + I confirmedWinner’s Curse e!ect
Estimates show 1 unit increase in % ( 10% , pText insignificant, photos do decrease uncertaintyEach photo adds about $7 by decreasing Winner’sCurse
Sources of informationForming posterior, bidder weights private signaland public signalWeighting depends roughly on r/(r + %)Estimate 59% public signal, 41% private signal
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Table: Structural Demand Estimation by Car Group
Classic Cars Reliable Cars PickupsµLog of Text Size 0.1157*** 0.0395*** 0.0596***Photos 0.202*** 0.0062*** 0.0172***%Log of Text Size -0.0107 -0.0047 -0.0021Photos -0.0052*** -0.0034 -0.0072Mean % 0.8512 0.6567 0.8065r 1.1996 1.0918 1.2658
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Table: Structural Demand Estimation with Controls
No Control With ControlµLog of Text Size 0.0623*** 0.0444***Control 0.0508***Control Squared -0.0179*%Log of Text Size -0.0450*** -0.0447***Mean % 0.8324 0.8319r 1.2162 1.2160Log Likelihood -1591.6473 -1581.7041
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Structural Results II
Coe"cient on text size falls when control addedEvidence of endogeneity
Use likelihood ratio testLR test statistic - 19.88Chi-squared statistic )2
2,0.999 = 13.81Reject null of no endogeneity
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Table: Demand Estimation by Car Group with Control
Classic Cars Reliable Cars PickupsµLog of Text Size 0.0660*** -0.0216 0.0108Control 0.0749*** 0.1157*** 0.0641**Control Squared -0.0293* -0.0180 -0.0003%Log of Text Size -0.0322 -0.0423 -0.0548**Mean % 0.8753 0.6064 0.7784r 1.2241 1.2379 1.3442
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Results
Results Summary
Validate model predictionsE [V |I ] increasing in information measures II endogenous
Selective disclosure accounts for most of theobserved relationship between price and I
Little evidence of an advertising e!ect
Value of information highest for classic cars,then pickups, then reliable cars
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Simulation
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Simulation
Simulation Idea
Counterfactual where sellers can’t discloseinformation on webpage
Get at importance of public disclosures for marketBidders now have “coarse priors”(e.g. E [log V |z ] vs E [log V |z , I ])Prior underestimates value of “peaches” ;overestimate “lemons”Still receive unbiased private signals
For random sample of 1000 datapoints:Structural estimates + !E [I |z ] ! coarse priorsCompute expected counterfactual prices pc
Compute baseline prices pb, for a car of “averagequality”
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Simulation
Simulation results
!15 !10 !5 5 10 15
Rel. Value !%"
!15
!10
!5
5
10
15
Rel. Price !%"
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Simulation
Simulation results
Fit straight lines:Actual slope = 1.01 (mechanically true)Counterfactual slope = 0.61Implies for an $11000 peach with characteristicsworth $10000, get about $400 less undercounterfactual ! adverse selection
Can also simulate value/price ratio - 1.17Kelly blue book retail/private party price between1.15 and 1.25Think of retail price as value ! correct ballpark
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Outline
1 eBay Motors
2 Theory
3 Estimation
4 Results
5 Simulation
6 Conclusions
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Conclusions and Future Research
ConclusionsStrong evidence of selective disclosure by sellersDisclosures reduce information asymmetries andpotential for adverse selection
ExtensionsE"ciency implications?Examine fraction of non-traded cars in data and incounterfactual
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Test for Common Values
Implement Athey/Haile (2002) test ofsymmetric PV vs CV environmentPV + Exchangeability implies:
2
nPr(B(n#2:n) < b)+
n $ 2
nPr(B(n#1:n) < b) = Pr(B(n#2:n#1) < b)
Test equality of distributions using modifiedKolmogorov-Smirnov statistic and subsampling
Haile/Hong/Shum (working paper)
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Table: Test for Common Values
Critical ValuesSample Size S0.9 S0.95 S0.99
30 % 48.592 50.184 52.59540 % 49.41 50.393 53.07Test Statistic 51.664
Reject private values framework at 5% level
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Robustness: Starting Bids
Table: Full Data vs Reserves < 20% of final price
Full Low ReserveµLog Text 0.0537*** 0.0546***Photos 0.0185*** 0.0183***%Age 0.0127*** 0.0123***Log of Text Size -0.0070 -0.0038Photos -0.0178*** -0.0057*Mean % 0.8472*** 0.8235***r 1.2099* 1.2212
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Robustness: Estimation Technique
Table: PML vs NLS
PML NLSµLog Text 0.0537*** 0.0485***Photos 0.0185*** 0.0154***%Age 0.0127*** 0.0071***Log of Text Size -0.0070 -0.0111*Photos -0.0178*** -0.0202***Mean % 0.8472*** 0.6480r 1.2099* 1.3503
Asymmetric Information, Adverse Selection and Seller Revelation on eBay Motors
Conclusions
Goodness of Fit Tests
Table: Full vs Restricted Model
Full Model No Covariates for %µLog Text 0.0537*** 0.0807***Photos 0.0185*** 0.0255***%Age 0.0127***Log of Text Size -0.0070Photos -0.0178***Mean % 0.8472*** 0.8592***r 1.2099* 1.1069***Log Likelihood -2478.5 -3857.3