Advanced Econometrics Takeshi Amemiya

Other Examples of Type 3 Tobit Models

Roberts, Maddala, and Enholm (1978) estimated two types of simultaneous equations Tobit models to explain how utility rates are determined. One of their models has a reduced form that is essentially Type 3 Tobit; the other is a simple extension of Type 3.

The structural equations of their first model are

У*і = x'ufiz + “2; (10.8.14)

and

yft = УУ*і + + Щ i> (10.8.15)

where y*t is the rate requested by the fth utility firm, y*t is the rate granted for the fth firm, x2i includes the embedded cost of capital and the last rate granted minus the current rate being earned, and x3/ includes only the last variable mentioned. It is assumed that y* and y% are observed only if

у*/ = г(х. + Vj > 0, (10.8.16)

where z, include the earnings characteristics of the fth firm. (Vvt is assumed to be unity.) The variable y* may be regarded as an index affecting a firm’deci - sion as to whether or not it requests a rate increase. The model (10.8.14) and

(10.8.15) can be labeled as P(yt < 0) • P(y, > 0, y2, y3) in our shorthand notation and therefore is a simple generalization of Type 3. The estimation method of Roberts, Maddala, and Enholm is that of Lee, Maddala, and Trost

(1980) and can be described as follows:

Step 1. Estimate a by the probit MLE.

Step 2. Estimate fi2 by Heckman’s two-step method.

Step 3. Replace y2i in the right-hand side of (10.8.15) by y2i obtained in step 2 and estimate у and fi3 by the least squares applied to (10.8.15) after adding the hazard rate term Ццу^Уи > 0).

The second model of Roberts, Maddala, and Enholm is the same as the first model except that (10.8.16) is replaced by

yl>R„ (10.8.17)

where R-i refers to the current rate being earned, an independent variable. Thus this model is essentially Type 3. (It would be exactly Type 3 if R, = 0.) The estimation method is as follows:

Step 1. Estimate fi2 by the Tobit MLE.

Step 2. Repeat step 3 described in the preceding paragraph.

Nakamura, Nakamura, and Cullen (1979) estimated essentially the same model as Heckman (1974) using Canadian data on married women. They used the WLS version of Heckman’s simultaneous equations two-step esti­mators, that is, they applied WLS to (10.8.10).

Hausman and Wise (1976, 1977, 1979) usedType3 and its generalizations to analyze the labor supply of participants in the negative income tax (NIT) experiments. Their models are truncated models because they used observa­tions on only those persons who participated in the experiments. The first model of Hausman and Wise (1977) is a minor variation of the standard T obit model, where earnings Y follow

Yi = Yf if Yf < L,, Yf a2), (10.8.18)

where Ц is a (known) poverty level that qualifies the fth person to participate in the NIT program. It varies systematically with family size. The model is estimated by LS and MLE. (The LS estimates were always found to be smaller in absolute value, confirming Greene’s result given in Section 10.4.2.) In the second model of Hausman and Wise (1977), earnings are split into wage and hours as Y = W • H, leading to the same equations as those of Heckman (Eqs.

10.8.2 and 10.8.7) except that the conditioning event is

log Wt - I - log Ht < log Lt (10.8.19)

instead of (10.8.8). Thus this model is a simple extension of Type 3 and belongs to the same class of models as the first model of Roberts, Maddala, and Enholm (1978), which we discussed earlier, except for the fact that the model of Hausman and Wise is truncated. The model of Hausman and Wise (1979) is also of this type. The model presented in their 1976 article is an extension of (10.8.18), where earnings observations are split into the preexperiment (sub­script 1) and experiment (subscript 2) periods as

Yu = Y*i and Y2i=Y% if Y^<Lt. (10.8.20)

Thus the model is essentially Type 3, except for a minor variation due to the fact that Ц varies with /.