268 The Basics of financial economeTrics
method is an approach to estimation that is essentially based on changing
the observed variables when the original ones cannot be estimated with any
of the above methods. The instrumental variables method is also an instance
of the generalized method of moments.
As will be explained in this chapter, there are variants of the methods
that are used when assumptions of a model fail. The choice between differ-
ent estimation methods depends on the data and the modeling assumptions
we make. More specifically, when the assumptions of the general linear
regression model that were described in Chapter 4 hold, the ordinary least
squares method—a type of least squareds method—and the maximum
likelihood method are appropriate. An additional consideration in select-
ing an estimation method is the computational cost that might favor one
or another method.
Least-Squares Estimation Method
The least-squares (LS) estimation method is a best-fit technique adapted
to a statistical environment. As a data-fitting technique, LS methods can
always be used. When LS methods are applied to linear regressions they are
called ordinary least squares (OLS). OLS methods require that the standard
assumptions of regressions are satisfied. As we will see, when some assump-
tions of standard regression are violated, two alternative methods described
in this chapter—weighted least squares or generalized least squares—are
applicable.
Let’s begin with the basic task of fitting data. Suppose we observe a set of
data and we want to find the straight line that best approximates these points.
A sensible criterion in this case is to compute the distance of each point from
a generic straight line, compute the sum of the squares of these distances, and
choose the line that minimizes this sum—in short, the LS method.
To illustrate, suppose that we are given the set of 10 data points listed
in Table 13.1. We can think of these data as 10 simultaneous observations
of two variables. Figure 13.1 represents the scatterplot of the data, which is
a figure with a point corresponding to each coordinate.
Suppose we want to draw the optimal straight line that minimizes the
sum of squared distances. A straight line is represented by all points that
satisfy a linear relationship
y = a + bxAfter choosing a and b, a straight line will take the values yi = a + bi, i =
1,... , 10 in correspondence with our sample data. For example, Figure 13.2