Statistica economica - BES 1

University of Perugia

Degree program in Finance and Quantitative Methods for Economics

Business and Economic Statistics

Specification issues in binary choice models

October 2025

Academic Year 2025-2026

David Aristei – Business and Economic Statistics Academic Year 2025-2026

2/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Specification issues in binary choice models

In general, estimator: a maximum likelihood estimator;

• 1) Is a consistent estimator;
• 2) Is asymptotically efficient;
• 3) Follows an asymptotically normal distribution.

These properties hold provided that the likelihood function is specified correctly.

This means that we must be sure about the entire distribution that we impose upon the data, as any deviation will cause the loss of all these properties and, most of all, lead to inconsistent estimators.

David Aristei – Business and Economic Statistics Academic Year 2025-2026

3/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Specification issues in binary choice models

In binary choice models, misspecification problems typically arise when the probability of y_i = 1 as a function of x_i P(y_i = 1 | x_i) = F(x_i′β) is “misspecified”.

Such misspecifications are usually motivated from the latent variable model and reflect violations in the assumptions about the error term ε_i.

In particular, specification problems in binary choice models mainly relate to:

• Heteroscedasticity of the error terms;
• Non-normality of ε_i (in the probit case);
• Endogeneity of the regressors (i.e. some explanatory variables are correlated with ε_i).

It is therefore necessary (even if not so frequent in empirical applications…) to test for the presence of specification issues and, where necessary, to correct for them.

David Aristei – Business and Economic Statistics Academic Year 2025-2026

4/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Heteroscedasticity in binary choice models

The Maximum Likelihood estimator of the parameters of a binary choice is no longer consistent and asymptotically efficient when the error term ε_i is heteroscedastic.

It is therefore essential to assess the validity of the homoscedasticity hypothesis, upon which we have based the definition of both logit and probit models.

We now relax the homoscedasticity assumption and assume that the variance of ε_i depends upon a set of J exogenous variables w_i′ = [w_i1, ..., w_iJ] (usually a subset of the regressors in x_i), not including the constant: Var(ε_i) = k ⋅ h(w_i′θ)

where h(⋅) is a continuously differentiable function, such that h(0) = 1 and h′(⋅) ≠ 0, and k is a positive constant.

David Aristei – Business and Economic Statistics Academic Year 2025-2026

5/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Heteroscedasticity in binary choice models

The extended heteroscedastic binary choice model, in which we allow the variance of the errors ε_i to vary with the values of the variables in w_i, can be written as:

y_i = 1(x_i′β + ε_i > 0), ε_i ∼ IID(0, k ⋅ h(w_i′θ))

which can be equivalently rewritten as:

y_i = 1((x_i′β / h(w_i′θ)) + (ε_i / h(w_i′θ)) > 0), ε_i ∼ IID(0, k)

where the scaled errors ε_i / h(w_i′θ) have constant variance equal to k.

In this extended binary choice model, the conditional probability of success is thus given by:

p_i = P(y_i = 1 | x_i, w_i) = F((x_i′β) / h(w_i′θ))

and depends not only on regressors in x_i but also on variables in w_i.

David Aristei – Business and Economic Statistics Academic Year 2025-2026

6/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Heteroscedasticity in binary choice models

The log-likelihood function of the heteroscedastic binary choice model is given by:

⎡ ⎤⎛ ⎞ ⎛ ⎞N Nx x′ ′β β⎟ ⎟∑ ∑⎜⎜ ⎜⎜log L(β, θ) = y_i log F ⎢ ⎥i i + (1 − y_i)log 1 − F⎟⎟ ⎟⎟⎜ ⎜⎢ ⎥h(w_i′θ) h(w_i′θ)⎝ ⎠ ⎝ ⎠′ ′θ) θ)⎣ ⎦i i i i =1 =1

⎡ ⎤⎡ ⎡ ⎤ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞N NN N N Nx xx x x x′ ′β β′ ′ ′ ′β β β β⎟ ⎟⎟ ⎟ ⎟ ⎟⎟∑ ∑∑ ∑ ∑ ∑⎜ ⎜⎜⎜ ⎜ ⎜⎜ ⎜log L(β, y log F (1 y )log 1 Flog L(β, log L(β, y log F y log F (1 y )log(1 1 y F)log 1 FP(y , D | x , z ) [(2y 1)( x ), (2D 1) z (2y 1)(2D 1)ρ]⎢ ⎥⎢ ⎢ ⎥ ⎥′ ′i ii i i iθ) = + − −θ) = θ) = + −+ −− −= Φ − β + γD α, − −⎟ ⎟⎟ ⎟ ⎟⎜⎜ ⎜ ⎜⎟ ⎟⎟ ⎟ ⎟ ⎟⎜ ⎜⎢ ⎥⎜ ⎜ ⎜ ⎜⎢ ⎢ ⎥ ⎥h(w_i h(w_ih(w_i h(w_i h(w_i h(w_i′θ) θ)′ ′ ′ ′θ) θ) θ) θ)⎣ ⎦⎣ ⎣ ⎦ ⎦i ii i i ii ii i i i=1 =1=1 =1 =1 =1 θ

and can be maximized (using iterative methods) over β and θ to obtain consistent estimates of model parameters.

To complete the formulation of the model, we have to make specific assumptions on the distribution of the error terms ε_i (i.e. by specifying the CDF F(⋅): e.g., standard normal or standard logistic) and on the form of heteroscedasticity (i.e. by specifying the function h(⋅)).

David Aristei – Business and Economic Statistics Academic Year 2025-2026

7/28

University of Perugia – Degree program in Finance and Quantitative Methods for Economics

Heteroscedasticity in binary choice models

A heteroscedastic probit model can be obtained by assuming k = 1, F(⋅) = Φ(⋅) and h(⋅) = [exp(⋅)]² (with h(0) = 1 and h′(⋅) ≠ 0), which corresponds to assuming multiplicative heteroscedasticity (i.e. the variance is an exponential function of w_i):

Var(ε_i) = k ⋅ h(w_i′θ) = [exp(w_i′θ)]²

so that p_i = Φ((x_i′β) / exp(w_i′θ)) and the log-likelihood function of the model can be written as:

N NN N⎡ ⎤N N ⎛ ⎞⎡ ⎤ N N⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞x xxx x x′ ′β β′ β′ ′β β ′ β∑ ∑∑ ∑log L(β, y log (1 y )log 1∑ ∑ ⎟log L(β, y log (1 y )log∑ ∑log L(β, y log (1 y )log 1 P(y , D | x , z ) [(2y 1)( x ), (2D⎟⎟ ⎟⎟⎜⎜log L(β, y log F (1 y )lo⎟⎟⎟ ⎟⎟ ′⎜⎜ ⎜⎜i i⎜⎜θ) = Φ + − − Φi⎜⎜ ⎜⎜ ⎢ ⎥i i θ) = Φ + −= Φ − β + γDθ) = Φ + − − Φ i⎢ ⎥θ) = + −⎟⎟ ⎟exp(w exp(w⎝ ⎠ ⎝ ⎠exp(w⎝ ⎠exp(w exp(w ⎜⎝ ⎠ ⎝ ⎠′ ′θ) θ)⎢ ⎥i i ′θ)i i i i 2 i i ii ih(w′ ′θ) θ)⎢ ⎥i i i i⎝ ⎠′θ)⎣ ⎦⎣ ⎦i iii i ii i i ii ii i =1 =1=1 =1=1 =1=1 =1

Accordingly, a heteroscedastic logit model is obtained by assuming k = π² / 3, F(⋅) = Λ(⋅) and h(⋅) = [exp(⋅)]², so that p_i =