5. Multiple regression: when a single factor is not enough

Level 4: Correlation, regression and the myth of causality

The market rises 1%. Good news for the airline, in principle.

But oil has risen 2% the same day, and that eats up almost all the gain: the model estimates that the share rises by barely 0.12%.

With a single factor you would not have seen that result coming.

A single factor leaves out too much

The daily return of a share rarely responds to a single factor.

Think of an airline. It moves with the market in general, but also with:

• The price of fuel, which directly affects its costs.
• Travel demand indicators.
• Company-specific news.

Explaining its behavior with a single variable, as we did in module 3, leaves out relevant and available information.

Multiple regression extends the same least squares logic to several explanatory variables at once.

How each coefficient is read

The equation, in words, comes out like this:

estimated return of the share = a base value, plus a coefficient multiplied by the market return, plus another coefficient multiplied by the movement in the oil price, plus another coefficient multiplied by a travel demand indicator, plus the residual that remains unexplained.

Each coefficient is interpreted as the marginal effect of that variable, holding all the others constant.

An example of that reading: if the oil coefficient is -0.4, it means that, for the same behavior of the market and of demand, a 1% rise in the oil price is associated on average with a 0.4% fall in the share.

The full calculation

A complete numerical example. Suppose the model:

return = 0.02% + 0.9 × market - 0.4 × oil + 0.3 × demand.

And a day on which the market rises 1%, oil rises 2% and demand does not move. The estimated return is:

0.02% + 0.9×1% - 0.4×2% + 0.3×0%, which gives 0.02% + 0.9% - 0.8%, that is, roughly 0.12%.

Each term contributes its part. The result combines the positive effect of the market with the negative effect of dearer fuel.

The risk of multicollinearity

Adding factors is not free of risks of its own. The most important is multicollinearity.

It occurs when two explanatory variables are themselves correlated with each other. For example, a consumer spending index and a travel demand index, which tend to move together over an economic cycle.

In that case, the model struggles to divide the explanatory credit between the two.

The typical symptom is that the coefficients become unstable or take on counterintuitive signs. A variable you know should have a positive effect turns up with a negative coefficient.

It is not that the economic relationship has reversed. It is that the model cannot cleanly separate the contribution of each factor when they move almost in unison.

Where this is really used

This kind of multifactor model has a real and well established use in the industry.

Fama-French style models are widely used to explain share returns. They start from the CAPM market factor, which you will see formally in the next module, and add further factors: the size of the company, or its relative valuation.

The reason is precise: a single market factor leaves a great deal unexplained in the variation of returns across different shares.

Adding factors is not free

It is worth bearing in mind, to close this module, that adding factors to a multiple model is not free from a statistical point of view.

Every new variable, whether or not it makes economic sense, tends to raise the model's R² simply through the mechanics of fitting.

That is why the choice of which factors to include must rest on a defensible economic reason for each one, not on testing variables until the fit improves.

It is a problem taken up in depth in the last module of this level.