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.NET barcode Recall from 1 that the union of -algebras is not necessarily a -algebra. in .NET Build Code 128C in .NET Recall from 1 that the union of -algebras is not necessarily a -algebra.

Recall from 1 that the union of -algebras is not necessarily a -algebra. generate, create none none on none projectsvb.net barcode Maximum Likelihood Theory UPC-13 8.3.3.

Probit and Logit Models Again, let Z j = (Y j , X T )T , j = 1, . . .

, n be independent random vectors, but j now Y j takes only two values, 0 and 1, with conditional Bernoulli probabilities. T P(Y j = 1 0 , X j ) = F 0 + 0 X j , T P(Y j = 0 0 , X j ) = 1 none none F 0 + 0 X j ,. (8.14). T where F is a gi none none ven distribution function and 0 = ( 0 , 0 )T . For example, let the sample be a survey of households, where Y j indicates home ownership and X j is a vector of household characteristics such as marital status, number of children living at home, and income. If F is the logistic distribution function, F(x) = 1/[1 + exp( x)], then model (8.

14) is called the Logit model; if F is the distribution function of the standard normal distribution, then model (8.14) is called the Probit model. In this case the conditional likelihood function is n.

n L c ( ) =. Y j F + T X j + (1 Y j ) 1 F + T X j , (8.15). where = ( , T )T . Also in this case , the marginal distribution of X j does not affect the functional form of the ML estimator as a function of the data. The -algebras involved are the same as in the regression case, namely, 0 = ({X j } ) and, for n 1, n = ({Y j }n ) 0 . Moreover, note that j=1 j=1.

1 1 E[ L c ( )/ L c ( 0 ). 0 ] =. y F + T X1 + ( none for none 1 y) 1 F + T X 1 = 1,. and similarly n L c ( )/ L c ( ) n 1 n 1 = c ( )/ L c ( ) Ln 0 0 n 1 1. yF + T X n + (1 y) 1 F + T X n = 1;. hence, the condit none for none ions (b) of De nition 8.1 and the conditions of De nition 8.2 apply.

Also the conditions (c) in De nition 8.1 apply, but again it is rather tedious to verify this..

The Mathematical and Statistical Foundations of Econometrics 8.3.4.

The Tobit none none Model Let Z j = (Y j , X T )T , j = 1, . . .

, n be independent random vectors such that j Y j = max(Y j , 0), where. T Y j = 0 + 0 X j + U j 2 with U j X j N 0, 0 .. (8.16). The random variables Y j are only observed if they are positive. Note that T P[Y j = 0 X j ] = P 0 + 0 X j + U j 0 X j T = P U j > 0 + 0 X j X j = 1 x T 0 + 0 X j / 0 ,. where (x) =. exp( u 2 /2)/ 2 du. This is a Probit model. Because model (8.16) was proposed by Tobin (1958) and involves a Probit model for the case Y j = 0, it is called the Tobit model.

For example, let the sample be a survey of households, where Yj is the amount of money household j spends on tobacco products and X j is a vector of household characteristics. But there are households in which nobody smokes, and thus for these households Y j = 0. In this case the setup of the conditional likelihood function is not as straightforward as in the previous examples because the conditional distribution of Y j given X j is neither absolutely continuous nor discrete.

Therefore, in this case it is easier to derive the likelihood function indirectly from De nition 8.1 as follows. First note that the conditional distribution function of Y j , given X j and Y j > 0, is P[Y j y.

X j , Y j > 0] = = = P[0 < Y j y X j ] P[Y j > 0 X j ]. T T P 0 0 X j < U j y 0 0 X j X j P[Y j > 0 X j ] T y 0 0 X j / 0 T 0 0 X j / 0. T 0 + 0 X j / 0. I (y > 0);. hence, the conditional density function of Y j , given X j and Y j > 0, is h(y 0 , X j , Y j &g t; 0) = where (x) = 0. T y 0 0 X j none none / 0 T 0 + 0 X j / 0. I (y > 0),. exp( x 2 /2) . 2 .
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