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Journal of Business & Economic Statistics data

1 Introduction The most commonly applied econometric tool is least-squares estimation, also known as regression. As we will see, least-squares is a tool to estimate an approximate conditional mean of one variable (the dependent variable) given another set of variables (the regressors, conditioning variables, or covariates). In this chapter we abstract from estimation, and focus on the probabilistic foundation of the conditional expectation model and its projection approximation. 2.2 The Distribution of Wages Suppose that we are interested in wage rates in the United States. Since wage rates vary across workers, we cannot describe wage rates by a single number. Instead, we can describe wages using a probability distribution. Formally, we view the wage of an individual worker as a random variable wage with the probability distribution F(u) = P(wage  u): When we say that a personís wage is random we mean that we do not know their wage before it is measured, and we treat observed wage rates as realizations from the distribution F: Treating unobserved wages as random variables and observed wages as realizations is a powerful mathematical abstraction which allows us to use the tools of mathematical probability. A useful thought experiment is to imagine dialing a telephone number selected at random, and then asking the person who responds to tell us their wage rate. (Assume for simplicity that all workers have equal access to telephones, and that the person who answers your call will respond honestly.) In this thought experiment, the wage of the person you have called is a single draw from the distribution F of wages in the population. By making many such phone calls we can learn the distribution F of the entire population. When a distribution function F is di§erentiable we deÖne the probability density function f(u) = d duF(u): The density contains the same information as the distribution function, but the density is typically easier to visually interpret.