4: Matrices and Linear Regression
- Page ID
- 57716
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In the previous chapter, we were introduced to the classical linear model and estimating the parameters using ordinary linear regression. All of the work was done using a scalar representation of the data. When moving beyond simple linear regression, the estimators are more difficult to determine. The calculus remains almost as simple, but solving the system of equations becomes prohibitive.
The usual solution to solving complicated system of equations is to use a matrix representation of the problem. That is what this chapter does. Along the way, we discover more about linear models than we expected.
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As in the previous chapter, let \(x\) and \(y\) be numeric variables. The linear relationship between \(x\) and \(y\) can be summarized by a line that "best" fits the observed data. That is, we can (and will) summarize the relationship between \(x\) and \(y\) using a linear equation:
\begin{equation}y = \beta_0 + \beta_1 x \end{equation}
The above holds in the case of simple linear regression (SLR). So, what do we do when there are more independent variables? Here is that representation:
\begin{equation}
y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \cdots + \beta_k x_k \label{eq:lm2b-lobf}
\end{equation}
Here, \(\beta_0\) is the y-intercept (still). And, \(\beta_j\) is the effect of variable \(x_j\) on the dependent variable, assuming all the other variables remain constant. By the way, his is called the ceteris paribus assumption. If the independent variables are independent of each other, then this requirement it met. However, there is frequently some correlation among the independent variables. Read on to see what to do about this.
We say that the "line" given by equation \ref{eq:lm2b-lobf} best fits the observed data. However, when dealing with two independent variables, it is not a line but a plane; with three, a space; with four, a hyperplane; etc. Clearly, meaningfully representing an entire four-variable model is quite difficult.