a chapter titled "Computationally Intensive Methods for Integration in Econometrics"

Economic models are often used to study a single decision or outcome. The outcome variable may be fully observed, continuous, and unrestricted (for example, log consumption); fully observed and continuous but restricted to an interval (fraction of expenditure devoted to a certain category of goods); continuous but censored (earnings subject to known withholding limits for social insurance); a mixture of discrete and continuous outcomes (earnings of full-time high school students); categorical (income from survey data known only to be in a designated interval); or discrete (dichotomous choice, such as labor force participation). Depending on the model and data, other kinds of outcomes may be observed as well. In all of these models, it is useful to conceive first of a latent outcome (denoted˜y t , for observation t), and then a corresponding set-valued observed outcome, denoted y t. For example, in the case of a continuous outcome censored from above at c, y t = ˜ y t if˜y t ≤ c , and y t = c, ∞ () if˜y t > c. In the case of a dichotomous outcome, one observes y t = −∞, 0 (] if˜y t ≤ 0 and y t = 0, ∞ This section treats the linear model˜y t = ′ β x t + u t , with observed outcomes of the form y t = c t ,d t [ ], y t = c t , d t [), or y t = c t , d t (], it being understood that c t ≤ d t and that c t and d t are extended real numbers. The disturbances u t t = 1,K,T () are independent and identically distributed conditional on x t t = 1,K,T (). The disturbance u t has a normal mixture distribution. We make this assumption because the normal mixture density can approximate any density arbitrarily well (Ferguson, 1983), and because it leads to practical methods for inference. It avoids the well-known problems that arise if the