Introduction to the Mathematical and Statistical Foundations of Econometrics

Likelihood Functions

There are many cases in econometrics in which the distribution of the data is neither absolutely continuous nor discrete. The Tobit model discussed in Section

8.3 is such a case. In these cases we cannot construct a likelihood function in the way I have done here, but still we can define a likelihood function indirectly, using the properties (8.4) and (8.7):

Definition 8.1: A sequence L n (в), n > і, of nonnegative random functions on a parameter space © is a sequence of likelihood functions if the following conditions hold:

(a) There exists an increasing sequence.^n, n > 0, of a-algebras such that for each в є © and n > і, Ln(в) is measurable ^n.

(b)

There exists a в0 є © such that for all в є ©, P (E [ L і(в )/L і(в0)^0] < і) = і, and, for n > 2,

(c) For all 6 = Є2 in &, P [Li(0i) = Lx{e2)^o] < i, and for n > 2,

p[Ln(ei)/L„_i(ei) = Ln(Є2)/Ln-i(e2)^"n-i] < i-1

The conditions in (c) exclude the case that Ln (Є) is constant on ©. Moreover, these conditions also guarantee that Є0 є © is unique:

Theorem 8.1: For all Є є ©{Є0} and n > i, E[ln(Ln(Є)/Ln(Є0))] < 0-

Proof: First, let n = i. I have already established that ln(L, i(e )/L^e0)) < Li(B)/Li(Bo) _ i if Ln(e)/Ln(Єо) = i. Thus, letting Y(e) = Ln(e)/Ln(Єо) _ ln(Ln(e)/Ln(e0)) _ i and X(e) = Ln(e)/Ln(e0), we have Y(e) > 0, and Y (e) > 0 if and only if X (e) = i. Now suppose that P (E [Y (e )&0] = 0) = i. Then P[Y(e) = 0&0] = i a. s. because Y(e) > 0; hence, P[X(e) = i&0] = i a. s. Condition (c) in Definition 8.i now excludes the possibility that e = e0; hence, P(E[ln(Li(e)/Li(e0))^0] < 0) = i if and only if e = e0. In its turn this result implies that

E [ln(^i(e )/^i(eo))] < 0 if e = Єо - (8.9)

By a similar argument it follows that, for n > 2,

E [ln(Ln (e )/L n_i(e)) _ ln(Ln (Єо)/L n-i(eo))] < 0 if Є = Єо-

(8.i0)

The theorem now follows from (8.9) and (8.i0). Q. E.D.

As we have seen for the case (8.i), if the support {z : f (ze) > 0} of f (ze) does not depend on Є, then the inequalities in condition (b) become equalities, with &n = a(Zn,---, Zi) for n > i, and &0 the trivial a-algebra. Therefore,

Definition 8.2: The sequence L n (Є), n > i, of likelihood functions has invari­ant support if, for all Є є ©, P(E[Li(e)/L/i(eo)&0] = i) = i, and, forn > 2,

As noted before, this is the most common case in econometrics.

See Chapter 3 for the definition of these conditional probabilities.

8.2. Examples 8.3.1. The Uniform Distribution

Let Zj, j = 1n be independent random drawings from the uniform [O,0o] distribution, where 00 > 0. The density function of Zj is f (zo) = 0—lI(0 < z < 00), and thus the likelihood function involved is

n

Ln(0) = 0"пПі(0 < Zj < 0). (8.11)

j=1

In this case Жп = a (Zn, Z1) for n > 1, and we may choose for У0 the trivial a-algebra {^, 0}. The conditions (b) in Definition 8.1 now read

= min(0, 0g)/0 < 1 for n > 2.

Moreover, the conditions (c) in Definition 8.1 read

P [0—11(0 < Z1 < 01) = 0—1 I(0 < Z1 < 02)]

= P(Z1 > max(0b 02)) < 1 if 01 = 02.

Hence, Theorem 8.1 applies. Indeed,

n ln(0o/0) + nE[ln(I(0 < Z1 < 0))] - E [ln(I(0 < Z1 < 0o))] n ln(0o/0) + nE[ln(I(0 < Z1 < 0))]

—<x if 0 < 0o,

n ln(0o/0) < 0 if 0 > 0o,

0 if 0 = 0o.