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

ADVANCED ECONOMETRICS I

Theory (3/3)

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 Fénix: https://fenix.iscte-iul.pt/disciplinas/03089

Joaquim J.S. Ramalho

Ordered choices:

Values for the dependent variable: 𝑍 ∈ 0,1, … , 𝑁 − 1 Latent model: 𝑍

𝑗 ∗ = 𝑦𝑗 ′𝛾 + 𝑣𝑗

▪ 𝑦𝑗 cannot include an intercept

Individual behaviour observed only by intervals: 𝑍𝑗 = ൞ if 𝑍

𝑗 ∗ ≤ 𝛿0

𝑛 if 𝛿𝑛−1 < 𝑍

𝑗 ∗ ≤ 𝛿𝑛,

1 ≤ 𝑛 ≤ 𝑁 − 2 𝑁 − 1 if 𝑍

𝑗 ∗ > 𝛿𝑁−2

▪ Example:

– 𝑍

𝑗 ∗ is a latent measure of the health status

– 𝑍𝑗 is an observed health indicator: poor, satisfactory, good, excellent

Assumption: the 𝛿𝑘’s are not known

• 3. Discrete Choice Models

3.2. Models for Ordered Choices

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Probabilities:

Aim:

▪ Modelling the probability of observing 𝑍

𝑗 ∗ in a given interval

Each probability is based on the same 𝐻 ∙ functions used with binary choices, being given by:

𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗 = 𝑄𝑠 𝛿𝑛−1 < 𝑍 𝑗 ∗ ≤ 𝛿𝑛|𝑦𝑗

= 𝑄𝑠 𝑍

𝑗 ∗ ≤ 𝛿𝑛|𝑦𝑗 − 𝑄𝑠 𝑍 𝑗 ∗ < 𝛿𝑛−1|𝑦𝑗

= 𝑄𝑠 𝑦𝑗

′𝛾 + 𝑣𝑗 ≤ 𝛿𝑛|𝑦𝑗 − 𝑄𝑠 𝑦𝑗 ′𝛾 + 𝑣𝑗 < 𝛿𝑛−1|𝑦𝑗

= 𝑄𝑠 𝑣𝑗 ≤ 𝛿𝑛 − 𝑦𝑗

′𝛾|𝑦𝑗 − 𝑄𝑠 𝑣𝑗 < 𝛿𝑛−1 − 𝑦𝑗 ′𝛾|𝑦𝑗

= 𝐻 𝛿𝑛 − 𝑦𝑗

′𝛾 − 𝐻 𝛿𝑛−1 − 𝑦𝑗 ′𝛾

Hence, the general case is:

𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗 = ൞

𝐻 𝛿0 − 𝑦𝑗

′𝛾

if 𝑛 = 0 𝐻 𝛿𝑛 − 𝑦𝑗

′𝛾 − 𝐻 𝛿𝑛−1 − 𝑦𝑗 ′𝛾 if 1 ≤ 𝑛 ≤ 𝑁 − 2

1 − 𝐻 𝛿𝑁−2 − 𝑦𝑗

′𝛾

if 𝑛 = 𝑁 − 1

• 3. Discrete Choice Models

3.2. Models for Ordered Choices

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Estimation:

Parameters to be estimated:

▪ 𝛾 ▪ 𝛿0, … , 𝛿𝑁−2

Estimation method:

▪ Maximum likelihood

Most common models:

▪ Ordered logit ▪ Ordered probit

• 3. Discrete Choice Models

3.2. Models for Ordered Choices

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Stata

• logit Y 𝑌1 … 𝑌𝑙
• probit Y 𝑌1 … 𝑌𝑙

Joaquim J.S. Ramalho

Partial effects:

Each 𝑌𝑘 affects 𝑁 probabilities: ∆𝑌

𝑘 = 1 ⟹ ∆𝑄𝑠 𝑍 = 𝑛 𝑌

= ൞ −𝛾𝑘𝑕 𝛿0 − 𝑦𝑗

′𝛾

if 𝑛 = 0 𝛾𝑘 𝑕 𝛿𝑛 − 𝑦𝑗

′𝛾 − 𝑕 𝛿𝑛−1 − 𝑦𝑗 ′𝛾

if 1 ≤ 𝑛 ≤ 𝑁 − 2 𝛾𝑘𝑕 𝛿𝑁−2 − 𝑦𝑗

′𝛾

if 𝑛 = 𝑁 − 1 The sign of 𝛾𝑘 is informative about the direction of ∆𝑄𝑠 𝑍 = 0 𝑌 and ∆𝑄𝑠 𝑍 = 𝑁 − 1 𝑌 but not of the changes in the remaining probabilities

• 3. Discrete Choice Models

3.2. Models for Ordered Choices

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Multinomial choices:

Values for the dependent variable: 𝑍 ∈ 0,1, … , 𝑁 − 1 Latent model:

▪ Each individual has a given utility associated with each alternative:

𝑉𝑗𝑛 = 𝑦𝑗𝑛

′ 𝛾 + 𝑣𝑗𝑛

▪ The selected alternative is the one that maximizes utility: 𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗 = 𝑄𝑠 𝑉𝑗𝑛 = 𝑛𝑏𝑦 𝑉𝑗1, … 𝑉𝑗𝑁 |𝑦𝑗

Main models:

▪ Multinomial Logit: 𝑉𝑗𝑛 ~ 𝐻𝑣𝑛𝑐𝑓𝑚 and 𝑉𝑗𝑛 independent ∀𝑛 ▪ Multinomial Probit: 𝑉𝑗𝑛 ~ 𝑂𝑝𝑠𝑛𝑏𝑚 ▪ Nested Logit ▪ Random Parameters Logit

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Explanatory variables:

𝑦𝑗𝑛 may include:

▪ 𝑦𝑗𝑛: variables that are different across individuals and alternatives ▪ 𝑦𝑛: variables that differ across alternatives but not individuals ▪ 𝑦𝑗: variables that differ across individuals but not alternatives

Example:

▪ 𝑍

𝑗 - selected means of transport to go to work

▪ 𝑦𝑗𝑛 - time that each individual 𝑗 takes in going to work when using transport 𝑛 ▪ 𝑦𝑛 - price of transport 𝑛 ▪ 𝑦𝑗 - age of individual 𝑗

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Multinomial logit:

𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗𝑛 = 𝐻𝑛 𝑦𝑗𝑛 ′ 𝛾 + 𝑦𝑗 ′𝛾𝑛 =

𝑓𝑦𝑗𝑛

′ 𝛾+𝑦𝑗 ′𝛾𝑛

σ𝑘=0

𝑁−1 𝑓𝑦𝑗𝑘

′ 𝛾+𝑦𝑗 ′𝛾𝑘

𝛾𝑛 has to be normalized, that is for one alternative (base

• utcome) its value is set to zero

𝛾 cannot include a constant term Independence of Irrelevant Alternatives (IIA) – the odds ratio between two alternatives does not depend on the remaining alternatives:

𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗𝑛

𝑄𝑠 𝑍

𝑗 = 𝑚|𝑦𝑗𝑛

= 𝑓𝑦𝑗𝑛

′ 𝛾+𝑦𝑗 ′𝛾𝑛

𝑓𝑦𝑗𝑚

′ 𝛾+𝑦𝑗 ′𝛾𝑚

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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When all explanatory variables are of the type 𝑦𝑗𝑛 and 𝑦𝑛, the choice between alternatives 𝑛 and 𝑚 is fully explained by diferences in the alternative characteristics: 𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗𝑛

𝑄𝑠 𝑍

𝑗 = 𝑚|𝑦𝑗𝑚

= 𝑓𝑦𝑗𝑛

′ 𝛾

𝑓𝑦𝑗𝑚

′ 𝛾 = 𝑓 𝑦𝑗𝑛 ′ −𝑦𝑗𝑚 ′ 𝛾

▪ Is this case, the model is often called ‘conditional logit’

When all explanatory variables are of the type 𝑦𝑗, the choice between alternatives 𝑛 e 𝑚 is fully explained by diferences between 𝛾𝑛 e 𝛾𝑚: 𝑄𝑠 𝑍

𝑗 = 𝑛|𝑦𝑗

𝑄𝑠 𝑍

𝑗 = 𝑚|𝑦𝑗

= 𝑓𝑦𝑗

′𝛾𝑛

𝑓𝑦𝑗

′𝛾𝑚 = 𝑓𝑦𝑗 ′ 𝛾𝑛−𝛾𝑚

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Stata asclogit Y 𝑌1𝑛 …, case(id) alternatives(varname) casevars(𝑌𝑗 …) basealternative(name) Stata mlogit Y 𝑌1 … 𝑌𝑙, baseoutcome(0)

Joaquim J.S. Ramalho

Estimation:

▪ Maximum likelihood based on the following log-likelihood function: 𝑀𝑀 = ෍

𝑗=1 𝑂

𝑒𝑗𝑛𝑚𝑝𝑕 𝐻𝑛 𝑦𝑗𝑛

′ 𝛾 + 𝑦𝑗 ′𝛾𝑛

▪ 𝑒𝑗𝑛 = 1 if individual 𝑗 chooses alternative 𝑛

Partial effects:

▪ ∆𝑌𝑗𝑘 = 1 ⟹

– ∆𝑄𝑠 𝑍

𝑗 = 𝑛 𝑌 = 𝛾𝑘𝐻𝑛 ∙ 𝑒𝑗𝑛 − 𝐻𝑛 ∙

– 𝛾𝑘 gives the sign of the partial effect

▪ ∆𝑌𝑗 = 1 ⟹

– ∆𝑄𝑠 𝑍

𝑗 = 𝑛 𝑌 = 𝐻𝑛 ∙

𝛾𝑘 − ҧ 𝛾 , onde ҧ 𝛾 = σ𝑛=1

𝑁−1 𝛾𝑛𝐻𝑛 ∙

– 𝛾𝑘 gives the sign of the partial effect relative to the base alternative, not the sign of the overall effect

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Testing IIA

▪ Hausman test comparing:

– Full multinomial logit model – Multinomial logit model excluding one or more alternatives

▪ If multinomial logit is the correct model, then both models produce consistent estimators (null hypothesis) ▪ If multinomial logit is not the correct model, then the results generated by both models will be different (alternative hypothesis)

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Stata mlogit Y 𝑌1 … 𝑌𝑙, baseoutcome(0) (ou asclogit…) estimates store Mod1 mlogit Y 𝑌1 … 𝑌𝑙 if Y != 3, baseoutcome(0) (ou asclogit…) estimates store Mod2 hausman Mod1 Mod2

Joaquim J.S. Ramalho

Multinomial probit:

Not affected by the IIA property Very complex, requiring the computation of 𝑁 − 1 integrals The version implemented in Stata assumes independent errors, which eliminates the only advantage of multinomial probit over multinomial logit

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Stata mprobit Y 𝑌1 … 𝑌𝑙, baseoutcome(0)

Joaquim J.S. Ramalho

Nested logit:

Not affected by the IIA property, grouping the choices in several sets in such a way that:

▪ Within each group, alternatives may be correlated ▪ Between groups, alternatives are independent

Results from a sequential decision process – example for a two-level process:

▪ Level 1 – defining J groups, ▪ Level 2 – defining 𝑁

𝑘 choices in each group

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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Financing Own funds Bank debt Bank 1 Stock market Bank N Lisbon Frankfurt

Stata nlogit …

Joaquim J.S. Ramalho

Random parameters logit:

Latent model: 𝑉𝑗𝑛 = 𝑦𝑗𝑛

′ 𝛾𝑗 + 𝑣𝑗𝑛

Most common assumption: 𝛾𝑗 ~ 𝑂 𝛾, Σ𝛾 Not affected by the IIA property If Σ𝛾 = 0, it reduces to the Multinomial Logit model; hence, comparing the two models allows the IIA property to be tested

• 3. Discrete Choice Models

3.3. Models for Multinomial Choices

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4.1. Models for Nonnegative Outcomes 4.2. Models for Fractional Responses 4.3. Models for Discrete-Continuous Responses

• 4. Models for Continuous Limited Dependent Variables

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Nonnegative outcomes can be:

▪ Continuous: 𝑍 ϵ 0, +∞

– Examples: prices, wages,…

▪ Discrete (counts): 𝑍 ϵ 0,1,2,3, …

– Examples: patents applied for by a firm in a year, times someone is arrested in a year,...

Linear regression models are not the most suitable option because:

▪ May generate negative predictions for the dependent variable ▪ At least close to the lower bound of 𝑍, it does not make sense to assume constant partial effects

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Log-linear regression model:

ln 𝑍

𝑗 = 𝛾0 + 𝛾1𝑦1𝑗 + ⋯ + 𝛾𝑙𝑦𝑙𝑗 + 𝑣𝑗

With this transformation, the dependent variable becomes unbounded: 𝑍 ∈ ]0, +∞[⟹ ln 𝑍 ∈ ] − ∞, +∞[ Assumption: 𝐹 𝑣𝑗|𝑦 = 0 However, two new problems arise:

▪ The log-linear model is not defined for 𝑍 = 0; adding a small constant value to 𝑍 or dropping zeros are not in general good solutions ▪ Prediction is more interesting in the original scale, ෡ 𝑍

𝑗, and not in the

logarithmic scale, ෣ ln 𝑍

𝑗 ; the log-linear model gives the latter directly

but retransforming it to the original scale requires additional assumptions and calculations and/or the application of relatively complex methods (see the next slide to understand the problem)

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Assumed model: ln 𝑍

𝑗 = 𝛾0 + 𝛾1𝑦1𝑗 + ⋯ + 𝛾𝑙𝑦𝑙𝑗 + 𝑣𝑗

▪ Consistent estimation requires 𝐹 𝑣𝑗|𝑦 = 0 ▪ Under 𝐹 𝑣𝑗|𝑦 = 0: 𝐹 ln 𝑍

𝑗 |𝑦 = 𝛾0 + 𝛾1𝑦1𝑗 + ⋯ + 𝛾𝑙𝑦𝑙𝑗

෣ ln 𝑍

𝑗 = መ

𝛾0 + መ 𝛾1𝑦1𝑗 + ⋯ + መ 𝛾𝑙𝑦𝑙𝑗

Prediction of 𝑍

𝑗:

▪ If ln 𝑍

𝑗 = 𝛾0 + 𝛾1𝑦1𝑗 + ⋯ + 𝛾𝑙𝑦𝑙𝑗 + 𝑣𝑗, then:

𝑍

𝑗 = 𝑓𝛾0+𝛾1𝑦1𝑗+⋯+𝛾𝑙𝑦𝑙𝑗+𝑣𝑗

and 𝐹 𝑍

𝑗|𝑦 = 𝑓𝛾0+𝛾1𝑦1𝑗+⋯+𝛾𝑙𝑦𝑙𝑗𝐹 𝑓𝑣𝑗|𝑦

▪ Consistent prediction of 𝑍

𝑗 would require assuming 𝐹 𝑓𝑣𝑗|𝑦 = 1;

however, the assumption made, 𝐹 𝑣𝑗|𝑦 = 0, implies that, in general, 𝐹 𝑓𝑣𝑗|𝑦 ≠ 1 ▪ Alternatively, we need to get a consistent estimate of 𝐹 𝑓𝑣𝑗|𝑦 , which requires additional assumptions

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Exponential regression model:

𝑍 = exp 𝑦′𝛾 + 𝑣 𝐹 𝑍|𝑌 = exp 𝑦′𝛾 Assumption: 𝐹 𝑓𝑣|𝑦 = 1 Advantages:

▪ ෡ 𝑍

𝑗 is always nonnegative

▪ Predictions are obtained directly in the original scale, without requiring any retransformations

Partial effects: ∆𝑌

𝑘 = 1 ⟹ ∆𝐹 𝑍 𝑌 = 𝛾𝑘exp 𝑦′𝛾

▪ The sign of the effect is given by the sign of 𝛾𝑘 ▪ 𝛾𝑘 can be interpreted as a semi-elasticity (see the next slide for a proof)

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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∆𝑌

𝑘 = 1 ⟹ ∆𝐹 𝑍 𝑌 = 𝛾𝑘exp 𝑦′𝛾

⟹ ∆𝐹 𝑍 𝑌 = 𝛾𝑘𝐹 𝑍 𝑌 ⟹ ∆𝐹 𝑍 𝑌 𝐹 𝑍|𝑌 = 𝛾𝑘 ⟹ 100 ∆𝐹 𝑍 𝑌 𝐹 𝑍|𝑌 = 100𝛾𝑘 ⟹ %∆𝐹 𝑍 𝑌 = 100𝛾𝑘%

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Assumptions and estimation methods according to the type of nonnegative outcome:

▪ Continuous response:

– Assumption: only 𝐹 𝑍|𝑌 ; estimation: QML

▪ Count data - two alternatives:

– Assumption: only 𝐹 𝑍|𝑌 ; estimation: QML – Assumption: 𝐹 𝑍|𝑌 and 𝑄𝑠 𝑍 = 𝑘|𝑌 ; estimation: ML

Three main distribution functions are used as basis for QML and/or ML estimation:

▪ Poisson ▪ Negative Binomial 1 ▪ Negative Binomial 2

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Poisson regression model:

𝑍

𝑗 ~ 𝑄𝑝𝑗𝑡𝑡𝑝𝑜 𝜇𝑗 ⟹ 𝑄𝑠 𝑍 𝑗 = 𝑧|𝑦𝑗 = 𝑓−𝜇𝑗𝜇𝑗 𝑧

𝑧! where 𝜇𝑗 = 𝐹 𝑍|𝑌 = exp 𝑦′𝛾 Estimation methods: ML (only count data) or QML, since the Poisson distribution belongs to the linear exponential family By definition, 𝐹 𝑍|𝑌 = 𝑊𝑏𝑠 𝑍|𝑌 (equidispersion), which may be a strong assumption is some empirical applications

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

2020/2021 Advanced Econometrics I 22

Stata ML: poisson Y 𝑌1 … 𝑌𝑙 QML: poisson Y 𝑌1 … 𝑌𝑙, robust

Joaquim J.S. Ramalho

Negative binomial regression models:

Two variants, both allowing for overdispersion (𝜀 > 0):

▪ NEGBIN1: 𝑊𝑏𝑠 𝑍|𝑌 = 1 + 𝜀 𝐹 𝑍|𝑌 - ML estimation ▪ NEGBIN2: 𝑊𝑏𝑠 𝑍|𝑌 = 1 + 𝜀𝐹 𝑍|𝑌 𝐹 𝑍|𝑌 - it belongs to the linear exponential family, enabling estimation by both ML (only count data) and QML

Overdispersion test:

𝐼0: 𝜀 = 0 (Poisson model) 𝐼1: 𝜀 ≠ 0 (Negative Binomial 1 or 2 model)

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Stata NEGBIN1: nbreg Y 𝑌1 … 𝑌𝑙, dispersion(constant) NEGBIN2 (ML): nbreg Y 𝑌1 … 𝑌𝑙, dispersion(mean) NEGBIN2 (QML): nbreg Y 𝑌1 … 𝑌𝑙, dispersion(mean) robust

Joaquim J.S. Ramalho

Base panel data model:

Continuous / count data: 𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢, 𝛽𝑗 = exp 𝛿𝑗 + 𝑦𝑗𝑢 ′ 𝛾 = 𝛽𝑗exp 𝑦𝑗𝑢 ′ 𝛾

Count data: 𝑄𝑠 𝑍

𝑗𝑢 = 𝑧|𝑦𝑗𝑢, 𝛽𝑗 = 𝑓−𝜇𝑗𝑢𝜇𝑗𝑢 𝑧

𝑧! 𝜇𝑗 = 𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢, 𝛽𝑗 = 𝛽𝑗exp 𝑦𝑗𝑢 ′ 𝛾

Pooled estimator:

Based on the cross-sectional assumption 𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢 =

exp 𝑦𝑗𝑢

′ 𝛾

Produces consistent estimators only if 𝐹 𝛽𝑗 𝑦𝑗𝑢 = 1

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Stata poisson Y 𝑌1 … 𝑌𝑙, vce(cluster clustvar)

Joaquim J.S. Ramalho

Random Effects Poisson Estimator:

Assumptions:

▪ 𝑍

𝑗𝑢 ~ 𝑄𝑝𝑗𝑡𝑡𝑝𝑜 𝜇𝑗𝑢

▪ 𝜇𝑗 = 𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢, 𝛽𝑗 = 𝛽𝑗exp 𝑦𝑗𝑢 ′ 𝛾

▪ log 𝛽𝑗 = 𝛿𝑗 ~ 𝐻𝑏𝑛𝑛𝑏 1, 𝜃

Resulting model:

▪ NEGBIN2-type model ▪ Estimation method: ML ▪ 𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢 = exp 𝑦𝑗𝑢 ′ 𝛾 , which implies that the Pooled estimator is

consistent under random effects of this type

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Stata xtpoisson Y 𝑌1 … 𝑌𝑙, re vce(robust)

Joaquim J.S. Ramalho

Fixed Effects Estimators:

Fixed effects Poisson estimator (three equivalent versions):

▪ Pooled estimator with individual effects ▪ Estimator conditional on σ𝑢=1

𝑈

𝑍

𝑗𝑢, with σ𝑢=1 𝑈

𝑍

𝑗𝑢 ≠ 0

▪ Quasi mean-differenced GMM estimator (Hausman, Hall and Griliches, 1984)

Quasi-differences GMM estimator:

▪ Chamberlain (1992) ▪ Wooldridge (1997)

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

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Fixed effects Poisson estimator:

May be derived using the three equivalent versions Pooled estimator with individual effects:

▪ Adds individual dummies, associated to the 𝛿𝑗

′s

▪ As in linear models, 𝛾 is consistently estimated even in short panels (no incidental parameters problem)

The quasi mean-differenced GMM estimator is based on the following moment condition: 𝐹 ቤ 𝑍

𝑗𝑢 − 𝜇𝑗𝑢

ҧ 𝜇𝑗 ത 𝑍

𝑗 𝑦𝑗𝑢

= 0, where 𝜇𝑗𝑢 = 𝑓𝑦𝑞 𝑦𝑗𝑢

′ 𝛾

Requires strictly exogenous explanatory variables

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

2020/2021 Advanced Econometrics I 27

Stata xtpoisson Y 𝑌1 … 𝑌𝑙, fe vce(robust)

Joaquim J.S. Ramalho

Quasi-differences GMM estimator :

Chamberlain (1992): 𝐹 ቤ 𝜇𝑗𝑢 𝜇𝑗,𝑢−1 𝑍

𝑗𝑢 − 𝑍 𝑗,𝑢−1 𝑦𝑗𝑢

= 0 Wooldridge (1997):

𝐹 ቤ 𝑍

𝑗𝑢

𝜇𝑗𝑢 − 𝑍

𝑗,𝑢−1

𝜇𝑗,𝑢−1 𝑦𝑗𝑢 = 0

In both cases the explanatory variables do not need to be strictly exogenous, so these estimators are particularly useful in dynamic models

• 4. Models for Continuous Limited Dependent Variables

4.1. Models for Nonnegative Outcomes

2020/2021 Advanced Econometrics I 28

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Fractional outcomes:

𝑍 ϵ 0,1

Base specification:

𝐹 𝑍 𝑌 = 𝐻 𝑦′𝛾 where the 𝐻 ∙ function must respect the restriction 0 ≤ 𝐻 ∙ ≤ 1

Main models:

Fractional regression model: assumes only 𝐹 𝑍|𝑌 Beta regression model: assumes also 𝑄𝑠 𝑍|𝑌 Transformation regression models (assume only 𝐹 𝑍|𝑌 ):

▪ Linear transformation ▪ Exponential transformation

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Fractional regression models:

Very similar to binary regression models

▪ Main models: Logit, Probit, Cloglog ▪ Partial effects calculated using the same expressions ▪ Estimation also based on the Bernoulli function, but only by QML

• 4. Models for Continuous Limited Dependent Variables

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Stata glm Y 𝑌1 … 𝑌𝑙, family(binomial) link(logit) robust glm Y 𝑌1 … 𝑌𝑙, family(binomial) link(probit) robust glm Y 𝑌1 … 𝑌𝑙, family(binomial) link(cloglog) robust

Joaquim J.S. Ramalho

Beta regression model:

Assumes also 𝐹 𝑍 𝑌 = 𝐻 𝑦′𝛾 , using the same functions for 𝐻 ∙ Additional assumption: 𝑍

𝑗 ~ 𝐶𝑓𝑢𝑏, with mean given by 𝐻 𝑦′𝛾

and precision parameter 𝜚 Estimation only by ML: more efficient, less robust Only available when 𝑍 ϵ 0,1

• 4. Models for Continuous Limited Dependent Variables

4.2. Models for Fractional Responses

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Linear transformation:

𝑍

𝑗 = 𝐻 𝑦𝑗 ′𝛾 + 𝑣𝑗

𝐼 𝑍

𝑗 = 𝑦𝑗 ′𝛾 + 𝑣𝑗

Alternative specifications:

▪ Logit: 𝐼 𝑍

𝑗 = ln 𝑍𝑗 1−𝑍𝑗

▪ Probit: 𝐼 𝑍

𝑗 = Φ−1 𝑍 𝑗

▪ Cloglog: 𝐼 𝑍

𝑗 = ln −ln 1 − 𝑍 𝑗

Advantages:

▪ Estimation: OLS ▪ Easy to deal with panel data and endogenous variables

Limitations:

▪ 𝐼 𝑍

𝑗 is not defined for 𝑍 𝑗 = 0 and 𝑍 𝑗 = 1

▪ Prediction in the original scale requires additional assumptions and calculations and/or the application of relatively complex methods

• 4. Models for Continuous Limited Dependent Variables

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2020/2021 Advanced Econometrics I 32

Example for logit: 𝑍

𝑗 =

𝑓𝑦𝑗

′𝛾+𝑣𝑗

1 + 𝑓𝑦𝑗

′𝛾+𝑣𝑗

𝑍

𝑗 + 𝑍 𝑗𝑓𝑦𝑗

′𝛾+𝑣𝑗 = 𝑓𝑦𝑗 ′𝛾+𝑣𝑗

𝑍

𝑗 = 𝑓𝑦𝑗

′𝛾+𝑣𝑗 − 𝑍

𝑗𝑓𝑦𝑗

′𝛾+𝑣𝑗

𝑍

𝑗 = 1 − 𝑍 𝑗 𝑓𝑦𝑗

′𝛾+𝑣𝑗

𝑍

𝑗

1 − 𝑍

𝑗

= 𝑓𝑦𝑗

′𝛾+𝑣𝑗

ln 𝑍

𝑗

1 − 𝑍

𝑗

= 𝑦𝑗

′𝛾 + 𝑣𝑗

Joaquim J.S. Ramalho

Exponential transformation:

𝑍

𝑗 = 𝐻 𝑦𝑗 ′𝛾 + 𝑣𝑗 = 𝐻1 exp 𝑦𝑗 ′𝛾 + 𝑣𝑗

𝐼1 𝑍

𝑗 = exp 𝑦𝑗 ′𝛾 + 𝑣𝑗

Alternative specifications:

▪ Logit: 𝐼1 𝑍

𝑗 = 𝑍𝑗 1−𝑍𝑗

▪ Cloglog: 𝐼1 𝑍

𝑗 = −ln 1 − 𝑍 𝑗

Advantages:

▪ Estimation: same methods as those used for nonnegative responses ▪ Easy to deal with panel data and endogenous variables

Limitations:

▪ Not aplicable to the probit model ▪ 𝐼 𝑍

𝑗 is not defined for 𝑍 𝑗 = 1 (but it is for 𝑍 𝑗 = 0)

▪ Prediction in the original scale requires additional assumptions and calculations and/or the application of relatively complex methods

• 4. Models for Continuous Limited Dependent Variables

4.2. Models for Fractional Responses

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Multivariate fractional outcomes:

𝑍

𝑗𝑛 ϵ 0,1 , 𝑛 = 0, … , 𝑁 − 1

σ𝑛=0

𝑁−1 𝑍 𝑗𝑛 = 1

Base specification:

𝐹 𝑍

𝑗𝑛 𝑌𝑗 = 𝐻𝑛 𝑦′𝛾

The 𝐻𝑛 ∙ function must respect the restrictions 0 ≤ 𝐻𝑛 ∙ ≤ 1 and σ𝑛=0

𝑁−1 𝐻𝑛 = 1

Main models:

Multivariate fractional regression model Dirichlet regression model

• 4. Models for Continuous Limited Dependent Variables

4.2. Models for Fractional Responses

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Multivariate fractional regression model:

Very similar to multinomial choice models

▪ Main models: Logit Multinomial, Nested Logit, Random Parameters Logit, … ▪ Partial effects calculated using the same expressions

QML estimation based on the multivariate Bernoulli function

Dirichlet regression model:

Assumes the same specifications for 𝐻𝑛 ∙ Additional assumption: 𝑍

𝑗 ~ 𝐸𝑗𝑠𝑗𝑑ℎ𝑚𝑓𝑢, with means given by

𝐻𝑛 𝑦′𝛾 and precision parameter 𝜚 Estimation only by ML: more efficient, less robust Only available when 𝑍

𝑗𝑛 ϵ 0,1

• 4. Models for Continuous Limited Dependent Variables

4.2. Models for Fractional Responses

2020/2021 Advanced Econometrics I 35

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Panel data - base specification:

𝐹 𝑍

𝑗𝑢 𝑦𝑗𝑢, 𝛽𝑗 = 𝐻 𝛽𝑗 + 𝑦𝑗𝑢 ′ 𝛾

Estimators:

Pooled estimator (requires 𝛽𝑗 = 𝛽 for consistency) Pooled with individual effects (requires 𝑈 ⟶ ∞ for consistency) Random effects (assumes 𝛽𝑗~𝑂 0, 𝜏𝛽

2 )

Fixed effects (based on linear or exponential transformations)

• 4. Models for Continuous Limited Dependent Variables

4.2. Models for Fractional Responses

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Joaquim J.S. Ramalho

Tobit Model Two-Part Model Sample Selection Model

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

2020/2021 Advanced Econometrics I 37

Joaquim J.S. Ramalho

Motivation:

Sometimes, the dependent variable has both discrete and continuous values; typically:

▪ Discrete value: for many individuals, 𝑍

𝑗 = 0

▪ Continuous component: for the remaining individuals, 𝑍

𝑗 may take on

some positive value, which may be bounded (fractional outcome) or not (nonnegative outcome)

Examples:

▪ Expenditures on durable goods, alcohol,,... ▪ Work hours

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Joaquim J.S. Ramalho

Alternative models:

Tobit model: a single model explains all values Two-part model: uses two independent models for explaining separately the zeros and the positive values Sample selection model: uses two different, but interdependent, models for explaining the zeros and the positive values

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

2020/2021 Advanced Econometrics I 39

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Tobit model - specification:

Latent model: 𝑍

𝑗 ∗ = 𝑦𝑗 ′𝛾 + 𝑣𝑗, −∞ < 𝑍 𝑗 ∗ < +∞

Instead of 𝑍

𝑗 ∗, it is observed:

𝑍

𝑗 = ൝0 if 𝑍 𝑗 ∗ ≤ 0

𝑍

𝑗 ∗ if 𝑍 𝑗 ∗ > 0

Assumption: 𝑣𝑗 ~ 𝑂 0, 𝜏2

▪ 𝑄𝑠 𝑍

𝑗 = 0|𝑦𝑗 = 𝑄𝑠 𝑍 𝑗 ∗ ≤ 0|𝑦𝑗 = 𝑄𝑠 𝑦𝑗 ′𝛾 + 𝑣𝑗 ≤ 0|𝑦𝑗 = 𝑄𝑠(

) 𝑣𝑗 ≤ −𝑦𝑗

′𝛾|𝑦𝑗 = 𝑄𝑠

ฬ

𝑣𝑗 𝜏2 ≤ − 𝑦𝑗

′𝛾

𝜏2 𝑦𝑗

= Φ −

𝑦𝑗

′𝛾

𝜏2

= 1 − Φ

𝑦𝑗

′𝛾

𝜏2

▪ Hence: 𝑔 𝑧𝑗|𝑦𝑗 = 1 − Φ

𝑦𝑗

′𝛾

𝜏2

if Y = 0

1 2𝜌𝜏2 𝑓−

𝑧𝑗−𝑦𝑗 ′𝛾 2 2𝜏2

if 𝑍 > 0

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

2020/2021 Advanced Econometrics I 40

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Estimation:

Method: ML Parameters to be estimated: 𝛾 and 𝜏 Log-likelihood function:

𝑀𝑀 = ෍ 1 − 𝑒𝑗 𝑚𝑝𝑕 1 − Φ 𝑦𝑗

′𝛾

𝜏2 + 𝑒𝑗𝑚𝑝𝑕 1 2𝜌𝜏2 𝑓− 𝑧𝑗−𝑦𝑗

′𝛾 2

2𝜏2

where 𝑒𝑗 = ቊ0 if 𝑍

𝑗 = 0

1 if 𝑍

𝑗 > 0

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

2020/2021 Advanced Econometrics I 41

Stata tobit Y 𝑌1 … 𝑌𝑙, ll(0)

Joaquim J.S. Ramalho

Quantities of interest:

Conditional mean given that 𝑍

𝑗 is positive:

𝐹 𝑍

𝑗|𝑦𝑗, 𝑍 𝑗 > 0 = 𝑦𝑗 ′𝛾 + 𝜏𝜇 𝑦𝑗 ′𝛾

𝜏

where 𝜇

𝑦𝑗

′𝛾

𝜏

=

𝜚

𝑦𝑗 ′𝛾 𝜏

Φ

𝑦𝑗 ′𝛾 𝜏

is the Mills ratio

Probability of observing positive values for 𝑍

𝑗:

Pr 𝑍

𝑗 > 0|𝑦𝑗 = Φ 𝑦𝑗 ′𝛾

𝜏 Overall conditional mean:

𝐹 𝑍

𝑗|𝑦𝑗 = 𝑄𝑠 𝑍 𝑗 = 0|𝑦𝑗 𝐹 𝑍 𝑗|𝑦𝑗, 𝑍 𝑗 = 0 + Pr 𝑍 𝑗 > 0|𝑦𝑗 𝐹 𝑍 𝑗|𝑦𝑗, 𝑍 𝑗 > 0

= Φ 𝑦𝑗

′𝛾

𝜏 𝑦𝑗

′𝛾 + 𝜏𝜚 𝑦𝑗 ′𝛾

𝜏

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Partial effects:

∆𝑌

𝑘 = 1 ⟹

▪ ∆𝐹 𝑍

𝑗|𝑦𝑗, 𝑍 𝑗 > 0 = 𝛾𝑘 1 − 𝜇 𝑦𝑗

′𝛾

𝜏 𝑦′𝑗𝛾 𝜏

+ 𝜇

𝑦𝑗

′𝛾

𝜏

▪ ∆𝑄𝑠 𝑍

𝑗 > 0|𝑦𝑗 = 𝛾𝑘 𝜏 𝜚 𝑦𝑗

′𝛾

𝜏

▪ ∆𝐹 𝑍

𝑗|𝑦𝑗

= 𝛾𝑘Φ

𝑦𝑗

′𝛾

𝜏

The three effects have the same sign

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Two-part model specification:

First part – binary regression model: 𝑄𝑠 𝑒𝑗 = 1|𝑦𝑗 = 𝐻1 𝑦𝑗

′𝛾

▪ 𝑒𝑗 = ቊ0 se 𝑍

𝑗 = 0

1 se 𝑍

𝑗 > 0

Second part – exponential or fractional regression model 𝐹 𝑍

𝑗|𝑦𝑗, 𝑒𝑗 = 1 = 𝐻2 𝑦𝑗 ′𝜄

Overall conditional mean: 𝐹 𝑍

𝑗|𝑦𝑗

= 𝑄𝑠 𝑍

𝑗 = 0|𝑦𝑗 𝐹 𝑍 𝑗|𝑦𝑗, 𝑍 𝑗 = 0 + Pr 𝑍 𝑗 > 0|𝑦𝑗 𝐹 𝑍 𝑗|𝑦𝑗, 𝑍 𝑗 > 0

= 𝐻1 𝑦𝑗

′𝛾 𝐻2 𝑦𝑗 ′𝜄

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Estimation:

Each part of the model is estimated separately:

▪ In each part, use the standard methods for the type of data being analyzed ▪ In the first part of the model, use the full sample ▪ In the second part of the model, use the subsample for which 𝑍

𝑗 > 0

▪ One may use different explanatory variables in each part of the model

Partial effects:

∆𝑄𝑠 𝑒𝑗 = 1|𝑦𝑗 ∆𝐹 𝑍

𝑗|𝑦𝑗, 𝑒𝑗 = 1

∆𝐹 𝑍

𝑗|𝑦𝑗

= ∆𝑄𝑠 𝑒𝑗 = 1|𝑦𝑗 𝐹 𝑍

𝑗|𝑦𝑗, 𝑒𝑗 = 1 + 𝑄𝑠(

) 𝑒𝑗 = 1|𝑦𝑗 ∆𝐹 𝑍

𝑗|𝑦𝑗, 𝑒𝑗 = 1

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Sample selection model - latent variable:

𝑍

2𝑗 ∗ : main variable

𝑍

1𝑗 ∗ : variable that determines whether 𝑍 2𝑗 ∗ is observed or not

Two equations:

Participation equation (e.g. to work or not): 𝑍

1𝑗 = ൝0 if 𝑍 1𝑗 ∗ ≤ 0

1 if 𝑍

1𝑗 ∗ > 0

Outcome equation (e.g. how much to work): 𝑍

2𝑗 = ൝−

if 𝑍

1𝑗 ∗ ≤ 0

𝑍

2𝑗 ∗ if 𝑍 1𝑗 ∗ > 0

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Latent linear models:

൝𝑍

1𝑗 ∗ = 𝑦1𝑗 ′ 𝛾1 + 𝑣1𝑗

𝑍

2𝑗 ∗ = 𝑦2𝑗 ′ 𝛾2 + 𝑣2𝑗

Assumptions:

The error terms of the two equations are assumed to be correlated, having a bivariate normal distribution: 𝑣1𝑗 𝑣2𝑗 ~𝑂 0 , 1 𝜏12 𝜏12 𝜏2

2

Only when 𝜏12 = 0 the two equations will be independent (the selection mechanism is exogenous or ignorable):

▪ In this case, the second equation may be estimated by OLS using only the observed data

• 4. Models for Continuous Limited Dependent Variables

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Quantities of interest:

Conditional mean of the main latent variable: 𝐹 𝑍

2𝑗 ∗ |𝑦𝑗 = 𝑦2𝑗 ′ 𝛾2

Conditional mean of the main observed dependent variable: 𝐹 𝑍

2𝑗|𝑦𝑗, 𝑍 1𝑗 = 1 = 𝑦2𝑗 ′ 𝛾2 + 𝜏12𝜇 𝑦1𝑗 ′ 𝛾1

Probability of observing positive values: 𝑄𝑠 𝑍

2𝑗 > 0|𝑦𝑗 = 𝑄𝑠 𝑍 1𝑗 = 1|𝑦𝑗 = Φ 𝑦1𝑗 ′ 𝛾1

• 4. Models for Continuous Limited Dependent Variables

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Parameters to be estimated: 𝛾, 𝜏12, 𝜏2 Estimation methods:

ML Heckman’s two-step method

ML:

Based on the following log-likelihood function:

𝑀𝑀 = ෍ 1 − 𝑒𝑗 Pr 𝑍

1𝑗 = 0|𝑦1𝑗 + 𝑒𝑗 𝑔 𝑍 1𝑗 = 1|𝑍 2𝑗 + 𝑔 𝑍 2𝑗

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

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Stata heckman 𝑍

2 𝑌1 … 𝑌𝑙, select(𝑍 1 𝑌1 … 𝑌𝑙)

Joaquim J.S. Ramalho

Heckman’s two-step method:

Based on 𝐹 𝑍

2𝑗|𝑦𝑗, 𝑍 1𝑗 = 1 = 𝑦2𝑗 ′ 𝛾2 + 𝜏12𝜇 𝑦1𝑗 ′ 𝛾1

First step: estimate the probit model 𝑄𝑠 𝑍

1𝑗 = 1|𝑦𝑗 =

Φ 𝑦1𝑗

′ 𝛾1 and get 𝜇 𝑦1𝑗 ′ መ

𝛾1 =

𝜚 𝑦1𝑗

′ ෡

𝛾1 Φ 𝑦1𝑗

′ ෡

𝛾1

Second step: regress 𝑍

2𝑗 on 𝑦2𝑗 and 𝜇 𝑦1𝑗 ′ መ

𝛾1 using only individuals fully observed and OLS, and correct the variances t test for H0: 𝜏12 = 0 (exogenous selection mechanism) If the same regressors are used in both steps, multicolinearity may arise; to avoid it, it is usual to exclude from 𝑦2𝑗 some of the variables included in 𝑦1𝑗

• 4. Models for Continuous Limited Dependent Variables

4.3. Models for Discrete-Continuous Responses

2020/2021 Advanced Econometrics I 50

Stata heckman 𝑍

2 𝑌1 … 𝑌𝑙, twostep select(𝑍 1 𝑌1 … 𝑌𝑙)