Grokking Linear Regression Analysis in Finance – Part III

See Part I and Part II for an overview of the linear models and the concept of “regression”.

Multiple linear regression

Let’s now say we believe there are multiple factors that tell us something about KO‘s returns. They could be SPY‘s returns, its competitor PepsiCo’s (NASDAQ : PEP) returns, and the US Dollar index (ICE : DX) returns. We denote these variables with the letter XX and add subscripts for each of them. We use the notation Xi,1,Xi,2 and Xi,3 to refer to the ith observation of SPY, PEP and DX returns respectively.

Like before, let’s put them all in an equation format to make things explicit.

β012 and β3 are the model parameters in equation 2.

Here, we have a multiple linear regression model to describe the relation between YY (the returns on KO) and Xi;i = 1, 2, 3 (the returns on SPY, PEP, and DX respectively).

We call it multiple, since there is more than one explanatory variable (three, in this case); and we call it linear, since the coefficients are linear.

When we go from one to two explanatory variables, we can visualize it as a 2-D plane (which is the generalization of a line) in three dimensions.

For ex. Y = 3 − 2X1 + 4X2 can be plotted as shown below.

As we add more features, we move to n-dimensional planes (called hyperplanes) in (n + 1) dimensions which are much harder to visualize (anything above three dimensions is). Nevertheless, they would still be linear in their coefficients and hence the name.

The objective of multiple linear regression is to find the “best” possible values for β012 and β3 such that the formula can “accurately” calculate the value of Yi.

In our example here, we have three X′s.

Multiple regression allows for any number of X′s (as long as they are less than the number of observations).

Stay tuned for the next installment in which Vivek will discuss linear regression of a non-linear relationship.

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