Non-causality in Bivariate Binary Panel Data



or, in compact vectorial form:

Y* = Bx + ε = V + ε

Y = [⅛∙>o}]

where Y*,Y,υ and ε are (n × 1) vectors of elements Yj*,Yj, βjX and ej∙, whereas
B = [β1, ∙ ∙ ∙ , βn'. Notice that no loss of generality is involved in assuming that
the same vector
x enters all equations, since some of the coefficients in B may
be zero.

The probability of Yj can be easily calculated by adapting the results of the
previous section:

Pr (yj = Уз I <≈) = Φ ((2 ∙ yj - 1) ∙ βjχ∙, 0, σ2)

i.e., the marginal distribution of Yj is a simple univariate probit model; this
result can be very useful in applications, since it shows that the distribution of
Yj depends only on the parameters of the corresponding latent regression Y.*.

However, in this case, it is more interesting to express the probability of a
vector
у = [τ∕ι,... ,yn]' whose elements are only O’s or l’s; in order to do so,
define the diagonal matrix
Dy

Dy = 2 ∙ diag (y) - In

Then the probability of у is

Pr (У = у I x) = Pr (Dy (Bx + ɛ) > 0)

= Pr (DyBx + Dyε > 0)

= Pr (—Dyε < DyBx)

Now suppose that ε is a random Gaussian vector, so that

ε ~ N (0, Σ)

Then -Dyε is itself a random Gaussian vector

-Dyε ~ N (θ,DyΣD'y)

and the probability of observing у is:

Pr (У = у I x) = Φn (pyBx; 0, DyΣD'y)

This formula allows to express the probability of у in compact form, involving
only an «-dimensional integral parameterized as a function of
y. Here, DyYD1y
is a positive definite matrix: in fact, consider dy, a column vector equal to the
main diagonal of
Dy; then, by Equation (2.11) in Styan (1973)

DyVD1y = (dyd!y) * Σ
30



More intriguing information

1. Does Presenting Patients’ BMI Increase Documentation of Obesity?
2. Delivering job search services in rural labour markets: the role of ICT
3. Does adult education at upper secondary level influence annual wage earnings?
4. A dynamic approach to the tendency of industries to cluster
5. ‘I’m so much more myself now, coming back to work’ - working class mothers, paid work and childcare.
6. Two-Part Tax Controls for Forest Density and Rotation Time
7. Cancer-related electronic support groups as navigation-aids: Overcoming geographic barriers
8. Tariff Escalation and Invasive Species Risk
9. Evidence on the Determinants of Foreign Direct Investment: The Case of Three European Regions
10. SOCIOECONOMIC TRENDS CHANGING RURAL AMERICA