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)

Подпись: P Подпись: E Подпись: L n (в )/іп-і(в) L n (во)/Ln-l(во) Подпись: ^n-1 Подпись: і Подпись: і.

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,

Подпись: P Подпись: E Подпись: L n (e)/L n_i(e) Ln (ЄО)/L n-i(eo) Подпись: =i Подпись: = i-

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.

Подпись: iSee 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

Подпись: E [7.1(0 )/7.1(0C)^G] Г 7n(0)/Ln-1(0) E — - Ln (0G)/L n-1(0o) image634

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,

Подпись: E [ln( 7. n (0)/7. n (0o))]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.

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