Multiple (linear) regression

Question

I am looking for some inputs on a pair trading strategy that I am trying to improve with some semi-fundamental input.

The basic idea is to use multiple linear regression to estimate the price of a stock ($Y$) based on fundamentals:

$$ Y(t) = \beta_0 + \beta_1*x_1(t) + \beta_2*x_2(t) + \cdots $$

Just to give you an idea of a case:

• $Y$ = Stock price, e.g. Starbucks
• $x_1$ = Market, e.g. S&P500
• $x_2$ = Coffe price
• $x_3$ = Competitor something..
• $x_4$ = Forex something..

My idea would be to only trade in the direction of the residual at the last date of the regression, meaning that if starbucks is at 55 and the regression gives a "fundamental" price at 60 I would only go long. And, since this is a part of a larger pair trading strategy I would short something else and include all the commonly used pair trading parameters.

After some extensive googling I have not been able to find any similar approaches. Anybody who has seen anything similar somewhere? Does it make any sense? Or is this just way off?

Answer 1

I personally would not do that!

Your regression model has been fitted to approximate $Y(t)$ (the reality) as much as possible.

If I understand you well, you say:

• at the previous period $Y(t)=55$ (Starbucks traded at 55 USD)
• the last period's estimate from the regression is $\hat{Y}(t)=60$
• Since $\hat{Y}(t)-Y(t) > 0$, you want to invest.

This does not make sense to me because you are trusting more the model than the actual process $Y$ and the model has been optimized (fitted) on $Y$.

When you computed the regression parameters, you found the best linear relationship between $Y$ and your different dependent variables $X$. Yet, your model does not perfectly fit the data (it has residuals: 60-55=5). So in a sense, the model hasn't been able to completely understand Starbuck's price process. And yet, you are willing to trade in the direction of the residual, which is the error of the regression.

Besides, beware of the dependent variables $X$ you use. Multiple linear regression requires them to be uncorrelated. This would generate problem if you look at the statistical significance of the estimated parameters $\beta_1, ... \beta_k$

Finally, under the assumptions of the multiple regression, the residuals (which you trade on) are supposed to be normally distributed with mean 0! So this means that you are in fact trading on an indicator that is just random.