Advanced Econometrics Takeshi Amemiya

Standard Tobit Model (Type 1 Tobit Model)

Tobin (1958) noted that the observed relationship between household ex­penditures on a durable good and household incomes looks like Figure 10.1,

Income

Figure 10.1 An example of censored data where each dot represents an observation for a particular household. An important characteristic of the data is that there are several observations where the expenditure is 0. This feature destroys the linearity assumption so that the least squares method is clearly inappropriate. Should we fit a nonlin­ear relationship? First, we must determine a statistical model that can generate the kind of data depicted in Figure 10.1. In doing so the first fact we should recognize is that we cannot use any continuous density to explain the condi­tional distribution of expenditure given income because a continuous density is inconsistent with the fact that there are several observations at 0. We shall develop an elementary utility maximization model to explain the phenome­non in question.

Define the symbols needed for the utility maximization model as follows:

у a household’s expenditure on a durable good y0 the price of the cheapest available durable good z all the other expenditures л: income

A household is assumed to maximize utility U(y, z) subject to the budget constraint у + z^x and the boundary constraint уШуоогу = 0. Suppose y* is the solution of the maximization subject toy + zSx but not the other constraint, and assume y* = /7, + P2x + u, where и may be interpreted as the collection of all the unobservable variables that affect the utility function. Then the solution to the original problem, denoted by y, can be defined by

У = У* if У*>Л

= 0 or y0 if y*Sy0.

If we assume that и is a random variable and that y0 varies with households but is assumed known, this model will generate data like Figure 10.1. We can write the likelihood function for n independent observations from the model

(10.1.1) as

(10.2.2)

where Ft and/) are the distribution and density function, respectively, ofyf, П0 means the product over those і for which yf ё уы, and means the product over those і for which yf > yoi. Note that the actual value of у when y* ^ y0 has no effect on the likelihood function. Therefore the second line of Eq. (10.2.1) may be changed to the statement “if y* S y0, one merely observes that fact.”

The model originally proposed by Tobin (1958) is essentially the same as the model given in the preceding paragraph except that he specifically as­sumed y* to be normally distributed and assumed y0to be the same for all households. We define the standard Tobit model (or Type 1 Tobit) as follows:

yf = xVJ+u„ i= 1,2,. . . ,n, Уі = У* if yf > 0 = 0 if yf s 0,

where {«,} are assumed to be i. i.d. drawings from N{0, a2). It is assumed that (у() and {x,} are observed for / = 1, 2,. . . , n but (yf) are unobserved if yf ё 0. Defining X to be the n X К matrix the rth row of which is x), we assume that {x,} are uniformly bounded and lim„_. n~l X'X is positive definite. We also assume that the parameter space of fi and a2 is compact. In the Tobit model we need to distinguish the vectors and matrices of positive observations from the vectors and matrices of all the observations; the latter appear with an underbar.

Note that yf > 0 and yf S 0 in (10.2.4) may be changed to yf > y0 and yf g y0 without essentially changing the model, whether y0 is known or un­known, because y0 can be absorbed into the constant term of the regression. If, however, y0i changes with і and is known for every i, the model is slightly changed because the resulting model would be essentially equivalent to the model defined by (10.2.3) and (10.2.4), where one of the elements of fi other than the constant term is known. The model where yoi changes with і and is unknown is not generally estimable.

The likelihood function of the standard Tobit model is given by

L = П П ~ П a 1Ф[(Уі - Х/Д)М

О 1

where Ф and фаге the distribution and density function, respectively, of the standard normal variable.

The Tobit model belongs to what is sometimes known as the censored regression model. In contrast, when we observe neither yt nor x, when yf ^ 0, the model is known as a truncated regression model. The likelihood function of the truncated version of the Tobit model can be written as

L = П Ф(х№-'а-'Ф[(Уі ~ (Ю.2.6)

1

Henceforth, the standard Tobit model refers to the model defined by (10.2.3) and (10.2.4), namely, a censored regression model, and the model the likeli­hood function of which is given by (10.2.6) will be called the truncated stan­dard Tobit model.