Econometrics II
Tim Eijkenaar
February 2021
1
, 1 Limited dependent variables
We have already seen the linear regression model in previous courses:
yi = β0 + β1 x1i + β2 x2i + . . . + βk xki + ui .
In many models we unfortunately have to deal with a limited dependent variable (that is, a limited
yi ). In this section we will have a close look at this.
1.1 Binary dependent variables
Suppose we have the binary variable yi with yi = 1 if the salary of person i is above average and
yi = 0 if not. Of each person i we also know his or her number of years of education and we are
interested in the relationship between the number of year of education and salary. Based on our
knowledge of previous courses we would construct the following model
yi = β0 + β1 x1i + ui ,
but this is a very bad idea. A much better idea is to use the probability model which we will soon
construct. But first, let us rewrite the model to
yi = x0i β + ui
with β = [β0 β1 ]T and x0i = [1 x1i ]. For this model, we have (by strict exogeneity)
E(yi | xi ) = E(x0i β + ui | xi ) = E(x0i β | xi ) + E(ui | xi ) = x0i E(β | xi ) + 0 = x0i β
but also (by the definition of the expected value)
X
E(yi | xi ) = yi p(yi | xi ) = 0 · P(yi = 0 | xi ) + 1 · P(yi = 1 | xi ) = P(yi = 1 | xi )
yi ∈{0, 1}
hence
P(yi = 1 | xi ) = x0i β.
Note that the probability will always be in the [0, 1]-interval. Therefore, we know that x0i β ∈ [0, 1].
However, x0i β can be estimated outside this interval. For example, it could be that we obtain
x0i β̂ = −0.1 by least squares which is not in the [0, 1]-interval and therefore we know that this is
a very bad estimate. How do we solve this? We are going to assume that P(yi = 1 | xi ) = G(x0i β)
rather than P(yi = 1 | xi ) = x0i β where G(z) is strictly increasing with 0 ≤ G(z) ≤ 1 for each
z ∈ (−∞, +∞). It should be no surprise that the class of cumulative distribution functions satisfies
these properties. Two common used choices are the following cumulative distribution functions
Z z
1 ez 1 v2
G(z) = −z
= z
and G(z) = Φ(z) = φ(v)dv (with φ(v) = (2π)− 2 e− 2 ).
1+e 1+e −∞
These are the cumulative distribution functions of the logistic distribution and the standard normal
distribution, respectively. We refer to the model as ’probit model’ if the CDF of the standard normal
2
Tim Eijkenaar
February 2021
1
, 1 Limited dependent variables
We have already seen the linear regression model in previous courses:
yi = β0 + β1 x1i + β2 x2i + . . . + βk xki + ui .
In many models we unfortunately have to deal with a limited dependent variable (that is, a limited
yi ). In this section we will have a close look at this.
1.1 Binary dependent variables
Suppose we have the binary variable yi with yi = 1 if the salary of person i is above average and
yi = 0 if not. Of each person i we also know his or her number of years of education and we are
interested in the relationship between the number of year of education and salary. Based on our
knowledge of previous courses we would construct the following model
yi = β0 + β1 x1i + ui ,
but this is a very bad idea. A much better idea is to use the probability model which we will soon
construct. But first, let us rewrite the model to
yi = x0i β + ui
with β = [β0 β1 ]T and x0i = [1 x1i ]. For this model, we have (by strict exogeneity)
E(yi | xi ) = E(x0i β + ui | xi ) = E(x0i β | xi ) + E(ui | xi ) = x0i E(β | xi ) + 0 = x0i β
but also (by the definition of the expected value)
X
E(yi | xi ) = yi p(yi | xi ) = 0 · P(yi = 0 | xi ) + 1 · P(yi = 1 | xi ) = P(yi = 1 | xi )
yi ∈{0, 1}
hence
P(yi = 1 | xi ) = x0i β.
Note that the probability will always be in the [0, 1]-interval. Therefore, we know that x0i β ∈ [0, 1].
However, x0i β can be estimated outside this interval. For example, it could be that we obtain
x0i β̂ = −0.1 by least squares which is not in the [0, 1]-interval and therefore we know that this is
a very bad estimate. How do we solve this? We are going to assume that P(yi = 1 | xi ) = G(x0i β)
rather than P(yi = 1 | xi ) = x0i β where G(z) is strictly increasing with 0 ≤ G(z) ≤ 1 for each
z ∈ (−∞, +∞). It should be no surprise that the class of cumulative distribution functions satisfies
these properties. Two common used choices are the following cumulative distribution functions
Z z
1 ez 1 v2
G(z) = −z
= z
and G(z) = Φ(z) = φ(v)dv (with φ(v) = (2π)− 2 e− 2 ).
1+e 1+e −∞
These are the cumulative distribution functions of the logistic distribution and the standard normal
distribution, respectively. We refer to the model as ’probit model’ if the CDF of the standard normal
2
