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represented by in the model. So, in this example, the model would be expressed
as shown in Equation 6.2 .
6.2
In the linear model, the represent the unknown p parameters. The estimates
for these unknown parameters are chosen so that, on average, the model provides
a reasonable estimate of a person's income based on age and education. In other
words, the fitted model should minimize the overall error between the linear model
and the actual observations. Ordinary Least Squares (OLS) is a common technique
to estimate the parameters.
To illustrate how OLS works, suppose there is only one input variable, x, for
an outcome variable y. Furthermore, n observations of (x,y) are obtained and
plotted in Figure 6.1 .
Figure 6.1 Scatterplot of y versus x
The goal is to find the line that best approximates the relationship between the
outcome variable and the input variables. With OLS, the objective is to find the
line through these points that minimizes the sum of the squares of the difference
between each point and the line in the vertical direction. In other words, find the
values of
and
such that the summation shown in Equation 6.3 is minimized.
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