ADVANCED ECONOMETRICS I Theory (2/3) Instructor: Joaquim J. S. - - PowerPoint PPT Presentation

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ADVANCED ECONOMETRICS I Theory (2/3) Instructor: Joaquim J. S. Ramalho E.mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: https://jjsramalho.wixsite.com/advecoi Fnix:

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ADVANCED ECONOMETRICS I

Theory (2/3)

Instructor: Joaquim J. S. Ramalho E.mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: https://jjsramalho.wixsite.com/advecoi FΓ©nix: https://fenix.iscte-iul.pt/disciplinas/03089

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Joaquim J.S. Ramalho

2.1. Model Estimation 2.1.1. Maximum Likelihood 2.1.2. Quasi-Maximum Likelihood Estimation 2.1.3. Generalized Method of Moments 2.2. Model Inference and Evaluation 2.3. Panel Data Models

  • 2. Nonlinear Regression Analysis

2020/2021 Advanced Econometrics I 2

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Joaquim J.S. Ramalho

Motivation:

Often, the dependent variable is discrete and/or bounded, in which case linear regression models cannot describe it appropriately Some continuous, bounded dependent variables may be transformed in such a way that linear regression models can still be used for their analysis; but in some cases such transformations are not available

  • 2. Nonlinear Regression Analysis

2020/2021 Advanced Econometrics I 3

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Joaquim J.S. Ramalho

Quantities of interest:

Linear models:

β–ͺ 𝐹 𝑍 π‘Œ

Nonlinear models:

β–ͺ 𝐹 𝑍 π‘Œ β–ͺ If using a probabilistic model: 𝑄𝑠 𝑍 π‘Œ β–ͺ In some models, there may be also interest on variants of the previous quantities:

– Example: when modelling a nonnegative outcome, 𝑍 β‰₯ 0, with lots of zeros, it may be interesting to estimate also:

Β» 𝑄𝑠 𝑍 = 0 π‘Œ Β» 𝐹 𝑍 π‘Œ, 𝑍 > 0

  • 2. Nonlinear Regression Analysis

2020/2021 Advanced Econometrics I 4

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Joaquim J.S. Ramalho

Partial Effects:

Linear models:

β–ͺ Model: 𝐹 𝑍 π‘Œ = π‘Œπ›Ύ β–ͺ Effects: βˆ†π‘Œ

π‘˜ = 1 ⟹ βˆ†πΉ 𝑍 π‘Œ = π›Ύπ‘˜

Nonlinear models:

β–ͺ Model:

– 𝐹 𝑍 π‘Œ = 𝐻 π‘Œπ›Ύ – 𝑄𝑠 𝑍 π‘Œ = 𝐺 π‘Œπ›Ύ

β–ͺ Effects: βˆ†π‘Œ

π‘˜ = 1 ⟹

– βˆ†πΉ 𝑍 π‘Œ = πœ–πΉ 𝑍|π‘Œ

πœ–π‘Œπ‘˜

=

πœ–π» π‘Œπ›Ύ πœ–π‘Œπ‘˜

= π›Ύπ‘˜

πœ–π» π‘Œπ›Ύ πœ–π‘Œπ›Ύ

= π›Ύπ‘˜π‘• 𝑦𝑗

′𝛾

– βˆ†π‘„π‘  𝑍 π‘Œ =

πœ–π‘„π‘  𝑍|π‘Œ πœ–π‘Œπ‘˜

=

πœ–πΊ π‘Œπ›Ύ πœ–π‘Œπ‘˜

= π›Ύπ‘˜

πœ–πΊ π‘Œπ›Ύ πœ–π‘Œπ›Ύ

= π›Ύπ‘˜π‘” 𝑦𝑗

′𝛾

  • 2. Nonlinear Regression Analysis

2020/2021 Advanced Econometrics I 5

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Joaquim J.S. Ramalho

Partial effects may be compared across different models, but the values of 𝛾 cannot However, because

πœ–π» π‘Œπ›Ύ πœ–π‘Œπ›Ύ

> 0 and

πœ–πΊ π‘Œπ›Ύ πœ–π‘Œπ›Ύ

> 0:

β–ͺ The sign of the partial effect is given by the sign of π›Ύπ‘˜ β–ͺ Testing the statistical significance of the partial effect is equivalent to test for 𝐼0: π›Ύπ‘˜ = 0

To calculate the magnitude of the partial effects, there are three main alternatives:

β–ͺ Calculate the partial effects for each individual in the sample and then

  • btain the mean of those effects

β–ͺ Replace x by its sample means β–ͺ Replace x by specific values

  • 2. Nonlinear Regression Analysis

2020/2021 Advanced Econometrics I 6

Stata

(after estimating the model)

margins, dydx(π‘€π‘π‘ π‘šπ‘—π‘‘π‘’) at(…) margins, dydx(π‘€π‘π‘ π‘šπ‘—π‘‘π‘’) atmeans margins, dydx(π‘€π‘π‘ π‘šπ‘—π‘‘π‘’)

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Joaquim J.S. Ramalho

Estimation:

Most common estimation methods:

β–ͺ Maximum Likelihood (ML): more efficient β–ͺ Quasi-Maximum Likelihood (QML): more robust

In both cases it is necessary to specify:

β–ͺ The 𝐻 function in 𝐹 𝑍 π‘Œ = 𝐻 π‘Œπ›Ύ β–ͺ The 𝐺 function in 𝑄𝑠 𝑍 π‘Œ = 𝐺 π‘Œπ›Ύ

Main assumptions:

β–ͺ ML:

– Correct specification of both 𝐻 and 𝐺

β–ͺ QML:

– Correct specification of 𝐻 – 𝐺 does not need to be correctly specified but needs to belong to the linear exponential family (e.g. Normal, Bernoulli, Poisson, Exponencial, Gama, etc.)

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 7

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ML / QML estimation - Statistics:

Distribution function - 𝐺 𝑧 : gives the probability of the random variable 𝑍 taking a value less than or equal to 𝑧: 𝐺 𝑧 = 𝑄𝑠 𝑍 ≀ 𝑧 Density function - 𝑔 𝑧 :

β–ͺ Derivative of the distribution function: 𝑔 𝑧 =

πœ–πΊ 𝑧 πœ–π‘§

𝐺 𝑧 = Χ¬

βˆ’βˆž 𝑧 𝑔 𝑧 𝑒𝑍

β–ͺ In the continuous case, describes the relative likelihood for the random variable 𝑍 being equal to 𝑧 (not the absolute likelihood) β–ͺ In the discrete case gives the probability of the random variable 𝑍 being equal to 𝑧

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 8

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Likelihood function:

β–ͺ In individual terms, it is the same as the density function β–ͺ Usually, it is calculated for the full sample, giving the likelihood of

  • bserving that sample under the assumption that the density function

𝑔 𝑧 describes appropriately the population behaviour β–ͺ Assuming independence across individuals and the same distribution for all of them, it is calculated as: 𝑀 𝑧 = ς𝑗=1

𝑂

𝑔 𝑧𝑗 , 0 ≀ 𝑀 𝑧 ≀ 1

Usually:

β–ͺ 𝐺 𝑧 , 𝑔 𝑧 and 𝑀 𝑧 depend also on 1 or 2 parameters β–ͺ One of the parameters represents 𝐹 𝑧

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 9

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  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

𝒁 Function π’ˆ 𝒛 ] βˆ’ ∞, +∞[ Normal 𝜈, 𝜏2 1 2𝜌𝜏2 1/2 exp βˆ’ 𝑧 βˆ’ 𝜈 2 2𝜏2 ]0, +∞[ Exponential 𝜈 1 𝜈 𝑓

βˆ’π‘§ 𝜈

]0,1[ Beta 𝜈, πœ€ Ξ“ πœ€ Ξ“ πœˆπœ€ Ξ“ 1 βˆ’ 𝜈 πœ€ π‘§πœˆπœ€βˆ’1 1 βˆ’ 𝑧

1βˆ’πœˆ πœ€βˆ’1

{0,1} Bernoulli 𝜈 πœˆπ‘§ 1 βˆ’ 𝜈 1βˆ’π‘§ {0,1,2, … } Poisson 𝜈 πœˆπ‘§π‘“βˆ’πœˆ 𝑧!

Most popular density functions:

10 Advanced Econometrics I 2020/2021

In all cases: 𝐹 𝑧 = 𝜈

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Joaquim J.S. Ramalho

Econometrics:

All the analysis is conditional on a set of explanatory variables The parameter 𝜈 (= 𝐹 𝑧 in Statistics) is replaced by the function assumed for 𝐹 𝑧|π‘Œ , for example π‘Œπ›Ύ (linear regression model) It is assumed that the likelihood function is known up to the set of parameters 𝛾 (and, in case the original function has 2 parameters, the other parameter) Density function to be considered: 𝑔 𝑧𝑗|𝑦𝑗; 𝛾

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 11

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Estimation:

Given that:

β–ͺ The density function 𝑔 βˆ™ is known, except for 𝛾 β–ͺ The probability that the sample values were in fact generated by the chosen density 𝑔 βˆ™ is measured by the likelihood function

Then:

β–ͺ We should choose for 𝛾 the value that maximizes 𝑀 𝑧𝑗|𝑦𝑗; 𝛾 β–ͺ Optimization problem: max𝛾𝑀 𝑧|π‘Œ; 𝛾 = ΰ·‘

𝑗=1 𝑂

𝑔 𝑧𝑗|𝑦𝑗; 𝛾 β–ͺ Actually, it is more common to maximize 𝑀𝑀 π‘Œπ›Ύ = ln 𝑀 𝑧|π‘Œ; 𝛾 : max𝛾𝑀𝑀 π‘Œπ›Ύ = ෍

𝑗=1 𝑂

ln 𝑔 𝑧𝑗|𝑦𝑗; 𝛾

– It is easier to maximize – It produces the same estimates for 𝛾

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 12

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Properties of ML / QML Estimators:

Asymptotic properties of ML estimators:

β–ͺ Consistency β–ͺ Efficiency β–ͺ Normality

Asymptotic properties of QML estimators:

β–ͺ Consistency β–ͺ Normality β–ͺ Efficiency is lost; variance calculated in a robust way β–ͺ Not possible to predict 𝑄𝑠 𝑍 π‘Œ and associated partial effects

Finite sample properties for both estimators:

β–ͺ Unknown

  • 2. Nonlinear Regression Analysis

2.1. Model Estimation

2020/2021 Advanced Econometrics I 13

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Alternative forms for estimating parameter variances:

Standard / Efficient ⟢ only available for ML Robust ⟢ only makes sense for QML Cluster-robust ⟢ panel data Bootstrap

Classical tests:

Likelihood Ratio (LR) ⟢ only available for ML Wald Score/LM

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 14

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Test for the joint significance of a set of parameters:

Competing models:

β–ͺ Restricted (smaller) model, based on 𝑀𝑆 𝛾0 + 𝛾1𝑦1 + β‹― + 𝛾𝑕𝑦𝑕 β–ͺ Full (larger) model , based on 𝑀𝐺࡫ ΰ΅― 𝛾0 + 𝛾1𝑦1 + β‹― + 𝛾𝑕𝑦𝑕 + 𝛾𝑕+1𝑦𝑕+1 + β‹― + 𝛾𝑙𝑦𝑙

Hypotheses:

𝐼0: 𝛾𝑕+1 = β‹― = 𝛾𝑙= 0 (restricted model) 𝐼1: No 𝐼0 (full model)

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 15

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LR test:

𝑀𝑆 = 2 𝑀𝑀𝐺 π‘Œπ›ΎπΊ βˆ’ 𝑀𝑀𝑆 π‘Œπ›Ύπ‘† ~πœ“π‘™βˆ’π‘•

2

β–ͺ Available in most econometric packages β–ͺ Easy calculation β–ͺ Both the competing models need to be estimated

Wald test: 𝑋 = መ 𝛾𝐸

β€² Var መ

𝛾𝐸

βˆ’1 መ

𝛾𝐸~πœ“π‘™βˆ’π‘•

2 where መ 𝛾𝐸 = መ 𝛾𝑕+1, … , መ 𝛾𝑙 is estimated based on 𝑀𝑀𝐺 π‘Œπ›ΎπΊ

β–ͺ When 𝐼0: π›Ύπ‘˜ = 0, 𝑋 simplifies to: 𝑒 = መ π›Ύπ‘˜ ො 𝜏ෑ

π›Ύπ‘˜

~π’ͺ 0,1 β–ͺ Available in most econometric packages β–ͺ Only the full model needs to be estimated

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 16

Stata

(estimate one model)

estimates store Model1

(estimate the other model)

estimates store Model2 lrtest π‘π‘π‘’π‘“π‘š1 π‘π‘π‘’π‘“π‘š2 Stata

(after estimating the full model)

test π‘Œπ‘•+1 β‹― π‘Œπ‘™

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Score/LM test: Score = πœ–π‘€π‘€πΊ π‘Œ መ 𝛾𝑁 πœ–π›Ύ π‘Šπ‘π‘ 

𝐺 መ

𝛾𝑁

βˆ’1 πœ–π‘€π‘€πΊ π‘Œ መ

𝛾𝑁 πœ–π›Ύ ~πœ“π‘™βˆ’π‘•

2

where መ 𝛾𝑁 = መ 𝛾0, … , መ 𝛾𝑕, 0, … 0 , with መ 𝛾0, … , መ 𝛾𝑕 estimated based on 𝑀𝑀𝑆 π‘Œπ›Ύπ‘† β–ͺ Only the restricted model needs to be estimated, which may be an advantage when the full model is complex and hard to estimate β–ͺ Rarely available in econometric packages, requiring programming

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 17

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Specification tests:

For 𝐹 𝑍 π‘Œ :

β–ͺ RESET test β–ͺ Chow test

For 𝑄𝑠 𝑍 π‘Œ :

β–ͺ Information Matrix text, usually very hard to implement β–ͺ More common: tests designed specifically to particular models

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 18

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RESET test:

Implementation:

β–ͺ Estimate the original model: 𝑄𝑠 𝑍|π‘Œ = 𝐺 𝛾0 + 𝛾1𝑦1 + β‹― 𝛾𝑙𝑦𝑙 β–ͺ Generate the variables π‘Œ መ 𝛾

2, π‘Œ መ

𝛾

3, π‘Œ መ

𝛾

4, …

β–ͺ Add the generated variables to the original model and estimate the following auxiliary model: 𝑄𝑠 𝑍|π‘Œ = 𝐺 𝛾0 + 𝛾1𝑦1 + β‹― + 𝛾𝑙𝑦𝑙 + Ξ³1 π‘Œ መ 𝛾

2 + Ξ³2 π‘Œ መ

𝛾

3 + Ξ³3 π‘Œ መ

𝛾

4 + β‹―

β–ͺ Apply a LR / Wald test for the significance of the added variables: 𝐼0: Ξ³1 = Ξ³2 = Ξ³3 = β‹― = 0 (suitable model functional form) 𝐼1: No 𝐼0 (unsuitable model functional form)

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 19

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Chow Test for Structural Breaks:

Context:

β–ͺ Two groups of individuals / firms / ...: 𝐻𝐡, 𝐻𝐢 β–ͺ It is suspected that the behaviour of the two groups in which regards the dependent variable may have different determinants

Implementation:

β–ͺ Generate the dummy variable 𝐸 = α‰Š1 if the individual belongs to 𝐻𝐡 0 if the individual belongs to 𝐻𝐢 β–ͺ Estimate the original model β€˜duplicated’: 𝑄𝑠 𝑍|π‘Œ = 𝐺 πœ„0 + πœ„1π‘Œ1 + β‹― + πœ„π‘™π‘Œπ‘™ + Ξ³0𝐸 + Ξ³1πΈπ‘Œ1 + β‹― + Ξ³π‘™πΈπ‘Œπ‘™ β–ͺ Apply a LR / Wald test for the significance of the variables where 𝐸 is present: 𝐼0: Ξ³0 = β‹― = γ𝑙 = 0 (no structural break) 𝐼1: NΓ£o 𝐼0 (with a structural break)

  • 2. Nonlinear Regression Analysis

2.2. Model Inference and Evaluation

2020/2021 Advanced Econometrics I 20

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Base nonlinear model for panel data:

Individual effects model: 𝐹 𝑍

𝑗𝑒 𝑦𝑗𝑒, 𝛽𝑗 = 𝐻 𝛽𝑗 + 𝑦𝑗𝑒 β€² 𝛾

𝑄𝑠 𝑍

𝑗𝑒 𝑦𝑗𝑒, 𝛽𝑗 = 𝐺 𝛽𝑗 + 𝑦𝑗𝑒 β€² 𝛾

Unlike the linear case:

β–ͺ Assuming 𝐹 𝛽𝑗|𝑦𝑗𝑒 = 0 is not enough to get consistent estimators β–ͺ In general, methods based on subtracting time averages or first- diferences do not eliminate fixed effects β–ͺ Inconsistent estimation of 𝛽𝑗 leads to inconsistent estimation of 𝛾 (incidental parameters problem)

  • 2. Nonlinear Regression Analysis

2.3. Panel Data Models

2020/2021 Advanced Econometrics I 21

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Main estimators:

Pooled estimator:

β–ͺ Based on the estimation of the model 𝐹 𝑍

𝑗𝑒 𝑦𝑗𝑒 = 𝐻 𝑦𝑗𝑒 β€² 𝛾 and

𝑄𝑠 𝑍

𝑗𝑒 𝑦𝑗𝑒 = 𝐺 𝑦𝑗𝑒 β€² 𝛾 , being consistent only under the assumption of

no individual effects β–ͺ Even with random effects this estimator will be, in general, inconsistent

Pooled estimator with individual effects:

β–ͺ Adds dummies for each individual, allowing estimation of the 𝛽𝑗

β€²s

β–ͺ Consistent only if π‘ˆ β†’ ∞

  • 2. Nonlinear Regression Analysis

2.3. Panel Data Models

2020/2021 Advanced Econometrics I 22

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Fixed effects estimator:

β–ͺ Assumes 𝐹 𝛽𝑗|𝑦𝑗𝑒 β‰  0 β–ͺ Long panels

– Use the pooled estimator with individual effects

β–ͺ Short panels:

– In a few cases:

Β» It is possible to drop the 𝛽𝑗

β€²s from the model to be estimated using methods

defined on a case-by-case basis (may also be used with long panels) Β» In general, prediction and quantification of partial effects are not possible

– In most cases, no fixed effects estimator is available

  • 2. Nonlinear Regression Analysis

2.3. Panel Data Models

2020/2021 Advanced Econometrics I 23

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Random effects estimator:

β–ͺ Most popular panel data estimator for probabilistic models β–ͺ It is necessary to:

– Correctly specify 𝑔 𝑧𝑗|𝑦𝑗, 𝛽𝑗; 𝛾 – Assume that 𝛽𝑗 follows some distribution 𝑔 𝛽𝑗; πœƒ

β–ͺ Density function for maximum likelihood estimation:

𝑔 𝑧𝑗|𝑦𝑗; 𝛾, πœƒ = ΰΆ± 𝑔 𝑧𝑗|𝑦𝑗, 𝛽𝑗; 𝛾 𝑔 𝛽𝑗; πœƒ 𝑒𝛽𝑗

– In general, this expression cannot be simplified – Because of the integral, it requires numerical methods

β–ͺ QML estimation not available β–ͺ In general, prediction and quantification of partial effects are not possible

  • 2. Nonlinear Regression Analysis

2.3. Panel Data Models

2020/2021 Advanced Econometrics I 24

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Joaquim J.S. Ramalho

3.1. Models for Binary Choices 3.2. Models for Ordered Choices 3.3. Models for Multinomial Choices

  • 3. Discrete Choice Models

2020/2021 Advanced Econometrics I 25

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Models for:

Binary choices:

β–ͺ 𝑍 ∈ 0,1 β–ͺ Ex.: be (or not) successful in a mortgage application

Multinomial choices

β–ͺ 𝑍 ∈ 0,1, … , 𝑁 βˆ’ 1 β–ͺ Ex.: choosing a brand

Ordered choices

β–ͺ 𝑍 ∈ 0,1, … , 𝑁 βˆ’ 1 β–ͺ Ex.: firms getting a specific investment rating

  • 3. Discrete Choice Models

2020/2021 Advanced Econometrics I 26

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Common structure:

𝑁 choices Aim - explaning the probability of observing 𝑍

𝑗 = 𝑧𝑗 given π‘Œπ‘— =

𝑦𝑗:

𝑄𝑠 𝑍

𝑗 = 𝑧𝑗 π‘Œπ‘— = 𝑦𝑗 = 𝐺 𝑦𝑗 ′𝛾

Since σ𝑧 𝑄𝑠 𝑍

𝑗 = 𝑧𝑗 𝑦𝑗 = 1:

β–ͺ Only 𝑁 βˆ’ 1 choices are modelled, the probability of the other being

  • btained by difference

β–ͺ The sum of the partial effects has to be null, σ𝑧 βˆ†π‘„π‘  𝑍

𝑗 = 𝑧𝑗 𝑦𝑗 = 0,

with one of them being obtained by difference

  • 3. Discrete Choice Models

2020/2021 Advanced Econometrics I 27

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Joaquim J.S. Ramalho

Binary choices:

Dependent variable only takes on the values 0 and 1 Bernoulli density function: 𝑔 𝑧𝑗|𝑦𝑗 = πœˆπ‘—

𝑧𝑗 1 βˆ’ πœˆπ‘— 1βˆ’π‘§π‘—

πœˆπ‘— = 𝐹 𝑍

𝑗 𝑦𝑗 = 𝐻 𝑦𝑗 ′𝛾 ,

where 0 < 𝐻 βˆ™ < 1 Note that 𝐹 𝑍

𝑗 𝑦𝑗 = 𝑄𝑠 𝑍 𝑗 = 1 𝑦𝑗 , since:

𝐹 𝑍

𝑗 𝑦𝑗 = 1 Γ— 𝑄𝑠 𝑍 𝑗 = 1 𝑦𝑗 + 0 Γ— 𝑄𝑠 𝑍 𝑗 = 0 𝑦𝑗

= 𝑄𝑠 𝑍

𝑗 = 1 𝑦𝑗 ;

Therefore, one may choose for 𝐻 βˆ™ a distribution function, which, by definition, is bounded by 0 and 1

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2020/2021 Advanced Econometrics I 28

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Estimation:

QML not possible:

β–ͺ The correct specification of 𝐹 𝑍

𝑗 π‘Œπ‘— implies automatically the correct

specification of 𝑄𝑠 𝑍

𝑗 = 1 π‘Œπ‘—

Estimation by ML based on:

𝑀𝑀 = ෍

𝑗=1 𝑂

𝑧𝑗ln 𝐻 𝑦𝑗

′𝛾

+ 1 βˆ’ 𝑧𝑗 ln 1 βˆ’ 𝐻 𝑦𝑗

′𝛾

According to the specification of 𝐻, different the resultant model – examples:

β–ͺ Probit: 𝐻 𝑦𝑗

′𝛾 = Ξ¦ 𝑦𝑗 ′𝛾 = Χ¬ βˆ’βˆž 𝑦𝛾 1 2𝜌 π‘“βˆ’

𝑦𝑗 ′𝛾 2 2

𝑒𝑦𝛾 β–ͺ Logit: 𝐻 𝑦𝑗

′𝛾 = Ξ› 𝑦𝑗 ′𝛾 = 𝑓𝑦𝑗

′𝛾

1+𝑓𝑦𝑗

′𝛾

β–ͺ Cloglog: 𝐻 𝑦𝑗

′𝛾 = 1 βˆ’ π‘“βˆ’π‘“π‘¦π‘—

′𝛾

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2020/2021 Advanced Econometrics I 29

Stata logit Y π‘Œ1 … π‘Œπ‘™ probit Y π‘Œ1 … π‘Œπ‘™ cloglog Y π‘Œ1 … π‘Œπ‘™

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Joaquim J.S. Ramalho

  • 3. Discrete Choice Models

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2020/2021 Advanced Econometrics I 30

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Partial effects:

βˆ†π‘Œ

π‘˜ = 1 ⟹ βˆ†πΉ 𝑍 π‘Œ = βˆ†π‘„π‘  𝑍 = 1 π‘Œ = πœ–π» π‘Œπ›Ύ πœ–π‘Œπ‘˜

= π›Ύπ‘˜

πœ–π» π‘Œπ›Ύ πœ– π‘Œπ›Ύ

= π›Ύπ‘˜π‘• 𝑦𝑗

′𝛾 , with 𝑕 𝑦𝑗 ′𝛾 given by:

β–ͺ Logit: 𝑕 𝑦𝑗

′𝛾 = πœ‡ 𝑦𝑗 ′𝛾 = Ξ› 𝑦𝑗 ′𝛾 1 βˆ’ Ξ› 𝑦𝑗 ′𝛾

β–ͺ Probit: 𝑕 𝑦𝑗

′𝛾 = 𝜚 𝑦𝑗 ′𝛾 = 1 2𝜌 π‘“βˆ’

𝑦𝑗 ′𝛾 2 2

β–ͺ Cloglog: 𝑕 𝑦𝑗

′𝛾 = 1 βˆ’ 𝐻 𝑦𝑗 ′𝛾 𝑓𝑦𝑗

′𝛾

  • 3. Discrete Choice Models

3.1. Models for Binary Choices

2020/2021 Advanced Econometrics I 31

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  • 3. Discrete Choice Models

3.1. Models for Binary Choices

2020/2021 Advanced Econometrics I 32

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Selection criteria:

To select the most suitable model, in addition to the RESET test, it is common to calculate the percentage of correct predictions of each model:

β–ͺ ΰ·  𝑍

𝑗 = ࡝1 if

ΰ·£ 𝑄𝑠 𝑍

𝑗 = 1|𝑦𝑗 β‰₯ 0.5

0 if ΰ·£ 𝑄𝑠 𝑍

𝑗 = 1|𝑦𝑗 < 0.5

β–ͺ % correct predictions: π‘œ11 + π‘œ00 /π‘œ β–ͺ % 1’s correctly predicted: π‘œ11/π‘œ1 β–ͺ % 0’s correctly predicted: π‘œ00/π‘œ0

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2020/2021 Advanced Econometrics I 33

𝑍

𝑗 = 1

𝑍

𝑗 = 𝟏

Total ΰ·  𝑍

𝑗 = 1

π‘œ11 ΰ·  𝑍

𝑗 = 0

π‘œ00 Total π‘œ1 π‘œ0 π‘œ

Stata

(after estimating the model)

estat classification

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Joaquim J.S. Ramalho

Alternative motivation:

In Economics, using the utility concept to explain the optimal choices of agents is very common The satisfaction experienced by the consumer of a good cannot be measured accurately; however, the decision to buy

  • r not the good can be observed

Strategy:

β–ͺ Linear regression model to explain the difference in utilities (𝑍

𝑗 βˆ—) of the

two goods, using as dependent variable a continuous latent variable 𝑍

𝑗 βˆ— = 𝑦𝑗 ′𝛾 + 𝑣𝑗

β–ͺ Binary regression model to explain the probability of choosing a good:

– Instead of 𝑍

𝑗 βˆ—, one observes 𝑍𝑗 = ࡝0 se 𝑍 𝑗 βˆ— ≀ 0

1 se 𝑍

𝑗 βˆ— > 0

– Model: 𝑄𝑠 𝑍

𝑗 = 1|𝑦𝑗 = 𝑄𝑠 𝑍 𝑗 βˆ— > 0|𝑦𝑗 = 𝑄𝑠 𝑦𝑗 ′𝛾 + 𝑣𝑗 > 0|𝑦𝑗 =

𝑄𝑠 𝑣𝑗 > βˆ’π‘¦π‘—

′𝛾|𝑦𝑗 = 𝑄𝑠 𝑣𝑗 < 𝑦𝑗 ′𝛾|𝑦𝑗 = 𝐻 𝑦𝑗 ′𝛾

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2020/2021 Advanced Econometrics I 34

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Panel data:

Base model – individual effects model: 𝑄𝑠 𝑍

𝑗𝑒 = 1 𝑦𝑗𝑒, 𝛽𝑗 = 𝐹 𝑍 𝑗𝑒 𝑦𝑗𝑒 = 𝐻 𝛽𝑗 + 𝑦𝑗𝑒 β€² 𝛾

Estimators:

β–ͺ Pooled (omits 𝛽𝑗; consistent only if 𝛽𝑗 = 𝛽) β–ͺ Pooled with individual effects (consistent only if π‘ˆ ⟢ ∞) β–ͺ Random effects (assumes 𝛽𝑗~𝑂 0, πœπ›½

2 )

β–ͺ Fixed effects logit

  • 3. Discrete Choice Models

3.1. Models for Binary Choices

2020/2021 Advanced Econometrics I 35

Stata Pooled: same commands as for cross-sectional data Random effects: xtprobit, xtlogit, xtcloglog

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Fixed effects logit model:

It can be shown that the 𝛽𝑗’s may be eliminated from a logit model if the analysis is conditional on individuals for whom σ𝑒=1

π‘ˆ

𝑍

𝑗𝑒 β‰  0 and σ𝑒=1 π‘ˆ

𝑍

𝑗𝑒 β‰  π‘ˆ:

β–ͺ All individuals whom display the value of 1 for the dependent variable in all time periods are dropped from the sample β–ͺ The same occurs with individuals displaying always the value of 0 for the dependent variable β–ͺ Only individuals that change their states at least once over time are relevant for estimation

This method only works for the logit model

  • 3. Discrete Choice Models

3.1. Models for Binary Choices

2020/2021 Advanced Econometrics I 36

Stata xtlogit Y π‘Œ1 … π‘Œπ‘™, fe