# Regressing price on volume

What are the basic pitfalls of regressing stock price on volume of shares traded? This is a time series dataset.

The model I'm using is:

$$\ln(price)=b_0+b_1t+b_2\ln(volume)+\epsilon$$

As you can see I include a time trend $t$.

The R2 and p-value appears to come out good, but I'm more accustomed to cross-sectional analysis so don't really know what I'm doing. I'd appreciate it if someone with more time series knowledge could let me know if this is totally naive.

Thanks

• Can you tell us what you are looking for? – Zarbouzou Mar 30 '12 at 22:51

The standard time series "pitfall" is the dreaded unit root or, more generally, non-stationary processes. For example, suppose price and volume are given by:

$ln(price) = a + b*t +\epsilon_1$

$ln(volume) = c + d*t +\epsilon_2$

Running a regression will give you an exceptionally good fit (in terms of $R^2$ and t-values), but is fundamentally a worthless equation to have estimated. These are referred to as "spurious regressions." Econometrically, the standard approach is to first-difference your data until both are stationary prior to running any regressions.

In addition to insuring that both processes are stationary, you may also have to worry that the two variables are cointegrated, which has its own associated procedures. Read Johansen (1988) for everything you ever wanted to know about dealing with cointegration, and more.

• You can add that your parameter estimates will not follow the classical student distribution. You will end up finding them significant while there might be absolutly no link. I have experimented that by simulating two indépendent random walks and running a regression. Standard test gave a significant coeficients at the 1% level about 80% of the time. – Zarbouzou Mar 30 '12 at 11:01
• Nice answer. Also, the price of a stock doesn't mean much on its own. Typically you would have to scale it by something like earnings or assets to give it a meaning. This is why financial economists invariably run regressions with returns. – itzy Mar 31 '12 at 23:31

As ever with these models, the only thing that counts is how successful it is in predicting data outside the calibration set. It doesn't matter whether your explanatory variables are trading volume, air pressure or skirt length: if it increases explanatory power significantly, include it. If not, exclude it.