Linear Probability Models and Big Data: Prediction, Inference and Selection Bias

Suneel ChatlaGalit Shmueli

Institute of Service ScienceNational Tsing Hua University Taiwan

Outline Introduction to binary outcome models

Motivation : Rare use of LPM

Study goals

o Estimation and inference

o Classification

o Selection bias

Simulation study

eBay data – in paper

Conclusions2

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Binary outcome models

Logit

Probit

LPM

OLS Regression:

Standard normal cdf

The purpose of binary-outcome regression models?

Inference

and estimation

Selection Bias

Prediction (Classificatio

n)

5

Summary of IS literature (MISQ,JAIS,ISR and MS: 2000~2016)

• Inference and estimation60

• Selection bias31• Classification and

prediction5

Only 8 used LPM 3 are from this year alone

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”Implementing a campaign fixed effects model with Multinomial logit is challenging due to incidental parameter problem so we opt to employ LPM …” – Burtch et al. (2016)”The LPM is simple for both estimation and inference. LPM is fast and it allows for a reasonable accurate approximation of true preferences.” – Schlereth & Skiera (2016)

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Statisticians don’t like LPMEconometricians love LPM

Researchers rarely use LPM

WHY?

Criticisms

Non normal error

Non constant error varianceUnbounded predictions

Functional form

Logit

✔

✔

✔

✔✖

Probit

✔

✔

✔

✔✖

LPM

✖

✖

✖

✖

Comparison of three models in terms their theoretical properties

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Advantages

Convergence issues

Incidental parameters

Easier interpretation

Computational speed

Logit

✖

✖

✔

✔✖

Probit

✖

✖

✖

✔✖

LPM

✔

✔

✔

✔

Comparison in terms of practical issues

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The Questions that Matter to Researchers?

Logit Probit LPMInference & Estimation

Classification

Selection Bias

10

Inference and

estimation• Consistency

• Marginal effects

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Latent Framework

𝑍={ 1, 𝑖𝑓 𝒀>00 , h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒

Latent continuous (not observable)

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Inference and

estimation

𝑙𝑜𝑔𝑖𝑠 (0,1) • Logit model

𝑁 (0,1)• Probit model

1)• Linear

probability model

The MLE’s of both logit and probit are consistent.�̂� 𝑝→𝛽

LPM estimates are proportionally and directionally consistent (Billinger, 2012) .

�̂�𝑙𝑝𝑚𝑝→𝑘𝛽

n

𝑘 𝛽

𝛽�̂�

�̂�𝑙𝑝𝑚

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Inference and

estimation

Marginal effects for interpreting effect size For LPM

ME for = =

For logit model

ME for = =

For probit model

ME for = = ()

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Easy Interpretation

No direct Interpretation

Inference and

estimation

Simulation study• Sample sizes {50,500,50000}• Error distribution {Logistic, Normal, Uniform}

• 100 Bootstrap samples

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Inference and

estimation

Comparison of Standard Models

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True Logit Probit LPMIntercept 0 0 0 0.5

1 0.99 1 0.47-1 -1 -1.01 -0.430.5 0.5 0.5 0.21-0.5 -0.5 -0.5 -0.21

k=0.4

Inference and

estimation

1.02-1.070.52-0.52

Non-significance results are identical

coefficient significance results are identical

Comparison of significance

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Inference and

estimation

Comparison of marginal effects

distributions of marginal effects are identical

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Inference and

estimation

Classification and

prediction• Predictions

beyond [0,1]

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Is trimming appropriate?

Replace with 0.99, 0.999

Replace with 0.001, 0.0001

Logi

t Pre

dict

ions

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Classification and

prediction

Classification

21

Classification accuracies are

identical

Classification and

prediction

Selection Bias

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Quasi-experiments

Like randomized experimental designs that test causal hypotheses but lack random assignment

Treatment Assignment

● Assigned by experimenter

● Self selection

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Selection Bias

Selection BiasTwo-Stage (2SLS) Methods

Stage 1:Selection model (T)

AdjustmentStage 2:Outcome model (Y)

𝐸 [𝑇∨𝑋 ]=Φ (𝑋 𝛾) 𝐼𝑀𝑅=𝜙 (𝑋𝛾)Φ (𝑋 𝛾)

(Heckman, 1977)

𝐸 [𝑇∨𝑋 ]=𝑋 𝛾 𝜆=𝑋𝛾−1(Olsen, 1980)

Probit

LPM24

Selection Adjustment

Olsen is simpler

Selection BiasOutcome model coefficients (bootstrap)

Both Heckman and Olsen’s methods perform similar to the MLE

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Selection Bias

Bottom lineInference and

Estimation• Use LPM with

large sample; otherwise logit/probit is preferable

• With small-sample LPM use robust standard errors

Classification

• Use LPM if goal is classification or ranking

• Trim predicted probabilities

• If probabilities are needed, then logit/probit is preferable

Selection Bias

• Use LPM if the sample is large

• If both selection and outcome models have the same predictors, LPM suffers from multicollinearity

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Thank you!

Suneel Chatla, Galit Shmueli, (2016), An Extensive Examination of Linear Regression Models with a Binary Outcome Variable, Journal of the Association for Information Systems (Accepted).

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