# Multiple (linear) regression

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?

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Sounds like you want principal component analysis and a basket calculator. You should read-up on arbitrage pricing theory. – chrisaycock Feb 5 '13 at 23:25
Why is there always people voting to close without giving any rational? I find that very disingenuous. Or are questions of new users generally voted to be closed? If the question does not fit the Q&A why not commenting on that? – Matt Wolf Feb 6 '13 at 9:48
@Freddy: I didn't vote to close, but this is close to "Could someone help me develop a trading strategy?" and therefore some may deem it's off-topic according to the FAQ. – Joshua Ulrich Feb 6 '13 at 14:09
I see your point but as long as the question is relevant to quant space one always has the choice to ignore the question right? Or vote to close with comment. That's at least my take on it. – Matt Wolf Feb 6 '13 at 15:00
@Freddy: I agree with you; just playing devil's advocate. – Joshua Ulrich Feb 6 '13 at 17:06

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.

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I guess what he wanted to say is that he is setting up a function to estimate the stock price based on fundamental data and if the stock price does not converge to what he believes is the fundamental value of the stock then he will either long or short the stock. In that sense I do not see why PCA is that far off? I would identify the driving components of the stock, periodically re-calibrate, and then look to calculate the deviations between the stock price and future model prices. – Matt Wolf Feb 6 '13 at 5:13
We only speak about the last price - this is misleading. If the model fits the whole process well, then the model is ok, no matter what it is relative to the last price - this is just one price of many. If the model is good and we can forecast the covariates then a trade could be based on it. – Richard Feb 6 '13 at 9:10
@Freddy I agree that CPA would be useful if he wants to enhance the model. Anyway, I'm assuming he wants to use his model (he asks whether it's good or not). If you create a model trying to estimate $S_{t+1}$ using $X$ at time $t$, then yes you can trade that; it makes sense. $S_t < \hat{S}_{t+1}$ would be a buy signal. Otherwise he's trading on something that was expected to happen according to the model, but didn't. – SRKX Feb 6 '13 at 9:32
@Richard yes, but here he computes the estimate "backwards" and he trades because the model's expected did not materialize. – SRKX Feb 6 '13 at 9:46
Of course I trust my model more than the market price at any given point in time, otherwise I should not be in the business of trading. If we believe markets are strongly efficient we should neither buy nor sell because prices reflect the true value. I hope we agree that this is far from being the case. – Matt Wolf Feb 6 '13 at 11:51