Market error for single stock APT

[Malcs]

Hi there, we are currently working on an APT project to check the influence of macroeconomic variables on the returns of a single stock (SOL).

So we first regressed the returns of the market index (which excludes SOL from the index) against the macroeconomic variables. We then removed insignificant macroeconomic variables from this market regression.

Once complete, we used the market error term (MER) in this regression for our SOL regression.

Our SOL regression then takes SOL's returns minus the risk free rate on the left hand side of the equation.

On the right hand side of this equation, we have the constant plus the first beta coefficient which is multiplied by the MER minus the risk free rate, then the rest of our beta coefficients that are associated with the macroeconomic variables that were found to be significant in our first regression of the market model.

Is this the correct procedure to follow?
What does the market error term represent?
And does the market error term minus the risk free rate represent the market proxy?
And why do we add the first beta coefficient multiplied by the market error term minus the risk free rate?

[trubador]

This is a well studied theory and is therefore very rich in terms of resources. Aside from many textbooks written on the subject, you can find good references on the internet. For instance, you can start from here:
Arbitrage Pricing Theory
Beta coefficient