Category Advanced Econometrics Takeshi Amemiya

Model of Kenny, Lee, Maddala, and Trost

Kenny et al. (1979) tried to explain earnings differentials between those who went to college and those who did not. We shall explain their model using the variables appearing in (10.9.1). In their model, у f refers to the desired years of college education, yf to the earnings of those who go to college, and yf to the earnings of those who do not go to college. A small degree of simultaneity is introduced into the model by letting yf appear in the right-hand side of the yf equation. Kenny and his coauthors used the MLE. They noted that the MLE iterations did not converge when started from the LS estimates but did con­verge very fast when started from Heckman’s two-step estimates (simulta­neous equations version).

Read More

Two Error Components Model with a Serially Correlated Error

In the subsection we shall discuss the 2ECM defined by Eqs. (6.6.18) and

(6.6.19) in which e follows an AR(1) process, that is,

Подпись: (6.6.39)in = УЧ,-1 + 4,

where {<!;„} are i. i.d. with zero mean and variance cj. As in the Balestra-Ner – love model, the specification of єю will be important if T is small. Lillard and Willis (1978) used model (6.6.39) to explain the log of earnings by the inde­pendent variables such as race, education, and labor force experience. They assumed stationarity for (ей), which is equivalent to assuming Еею = 0, Уєю = a/{ 1 — у2), and the independence of єю from £(1, £a,. . . . Thus, in the Lillard-Willis model, Euu’ = (a2^TVT+Y) ® L,, where Г is like (5.2.9) wither2 = cr^andp = y...

Read More

Discriminant Analysis

The purpose of discriminant analysis is to measure the characteristics of an individual or an object and, on the basis of the measurements, to classify the individual or the object into one of two possible groups. For example, accept or reject a college applicant on the basis of examination scores, or determine whether a particular skull belongs to a man or an anthropoid on the basis of its’ measurements.

We can state the problem statistically as follows: Supposing that the vector of random variables x* is generated according to either a density g, or g0, we are to classify a given observation on x*, denoted xf, into the group character­ized by either g, or g0. It is useful to define yt = 1 if xf is generated by g, and у і = 0 if it is generated by g0...

Read More

Probit Maximum Likelihood Estimator

The Tobit likelihood function (10.2.5) can be trivially rewritten as

т = П [І – Ф(х’,0/о) П Ф(x’,0/a) (10.4.1)

0 1

П Ф(х’іР/о)~1а~1Ф[(Уі ~ *іР)/о].

і

Then the first two products of the right-hand side of (10.4.1) constitute the likelihood function of a probit model, and the last product is the likelihood function of the truncated Tobit model as given in (10.2.6). The probit ML estimator ofa = fifa, denoted a, is obtained by maximizing the logarithm of the first two products.

Note that we can only estimate the ratio fi/a by this method and not flora separately. Because the estimator ignores a part of the likelihood function that involves fi and a, it is not fully efficient...

Read More

Duration Models

11.1.3 Stationary Models—Basic Theory

We shall first explain a continuous-time Markov model as the limit of a discrete-time Markov model where the time distance between two adjacent time periods approaches 0. Paralleling (11.1.2), we define

Aj*(/) At = Prob[ith person is in state к at time t + At (11.2.1)

given that he or she was in state j at time /].

In Sections 11.2.1 through 11.2.4 we shall deal only with stationary (possibly heterogeneous) models so that we have Aj*(f) = Aj* for all t. •

Let us consider a particular individual and omit the subscript і to simplify the notation. Suppose this person stayed in state j in period (0, t) and then moved to state к in period (/, t + At). Then, assuming t/At is an integer, the probability of this event (called A) is

P( A) = (1 – Xj Aty/A...

Read More

Asymptotic Distribution of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator

The LIML and 2SLS estimators of a have the same asymptotic distribution. In this subsection we shall derive it without assuming the normality of the observations.

We shall derive the asymptotic distribution of 2SLS. From (7.3.4) we have

Подпись: (7.3.5)yfT(a2s – a) = (Г-*ZJ PZ^-‘r-^Z; Pu,.

The limit distribution of VT(d2s — a) is derived by showing that plim T~lZ PZ, exists and is nonsingular and that the limit distribution of T~i/2Z[ Pu, is normal.

First, consider the probability limit of T~lZ PZ,. Substitute (ХП, + V,, X,) for Zx in T~ lZ PZ,. Then any term involving У, converges to 0 in probability. For example,

plim T~lX[ X(X’X)-lX’Vl = plim T~lX X(T~lX, XrlT-lX, Vl
= plim Г-’Х; X(plim T-‘X’X)-1 plim Г^Х’У, = 0.

image503

The second equality follows from Theorem 3.2...

Read More