# Extracting Signal from Noisy Data

Consider a scenario in which Y_t represents the % change in price and we want to use X_t to predict Y_t. We assume that X_t is information we get before Y_t is revealed.

Suppose that in reality Y_t = 0.01*X_t + e_t (beta = 0.01), where e_t ~ N(0,1). Moreover, X_t ~ N(0,1). Suppose we have 1000 points of historical data to test on. This generates the distribution of beta when simulating the estimation 1000 times: A few questions:

1) Is this approach a feasible money making strategy in finance? First, the assumptions I have made are already really optimistic (Y_t,X_t are iid over 1000 samples and the true relationship is linear). However, because of the noise the variance of the estimated parameter is very high, and 40% of the parameters are the wrong sign. Adding more rhs variables would presumably just make this worse.

Even presuming we got beta = 0.01, the signal is so small that we will have a high error rate of falsely rejecting our signal when the strategy does poorly.

2) What are possible solutions? If we had enough data (HFT?) we could precisely estimate small signals. What if we are looking at lower frequency time series (daily data?). Is another solution to look for a Z_t that has a much higher beta? However, given that markets are relatively efficient, why would a large beta continue to exist? Do we need to add more low-signal variables into the RHS of the regression (this would just increase the variance though)?

Another approach to get a higher "beta" seems to be stat arb? If Y_t and X_t are contemporaneous stock prices of cointegrated price series, we could expect beta to be large enough to estimate more precisely (relative to its magnitude). However, I have read (although I clearly don't have first hand experience) that stat arb profits have been declining since everybody knows the idea nowadays.

## 2 Answers

Yes, these are the fundamental building blocks for a money making strategy.

To partially solve the issues you mention (small/low positive means/profits with large standard errors), you can investigate on many assets simultaneously. The idea is to take the advantage of Central Limit Theorem.

Assuming the signal for each asset are i.i.d., and each signal $i=1,2,.., N$ can make you money $r_t^{(i)} \sim \mathbb{N}(\mu, \sigma^2)$ if you fully invest in that signal, where $\mu$ is quite small when compared with $\sigma$. Central Limit Theorem tells us that $$\sum_{i=1}^N \frac{1}{N} r_t^{(i)} \sim \mathbb{N}(\mu, \sigma^2/N)$$ so now your portfolio profit doesn't hurt but has a much smaller standard error $\sigma^2/\sqrt{N}$. Taking $N$ to a large value, say 1000, and by taking leverage strategically, you can make your portfolio profit attractive while reduce the volatility to be a manageable level.

You estimate a model $$Y_t = \beta X_t + \epsilon_t.$$ which is just time-series regression.

Concerning your question 1): One usually looks at the beta of a single security w.r.t. a stock index (see the CAPM). High beta (above 1)will indicate that the stock would rise and fall more than the market.

Other approaches where one estimates a beta is in hedging. If you have a security $X$ and a security $Y$ and you wonder how many pices of $X$ you need to hedge $Y$ (i.e. minimize $|\beta X-Y|^2$) then you can take the beta.

Directly making money with the regression coefficient beta: I would really say no. If beta is too small: forget the model at all (what would one expect?).

For 2): estiamting a beta is really basic. If there were profits to make (even in HFT) then one of the large companies already does this.

• Thanks Richard. I was trying to give a simplified example of a case in which information known beforehand (X_t) could predict the future (Y_t). The simplest way they could be related would be something like the linear model I described. You say to just forget the model if the beta is small, does that mean there are cases in which signal X_t is a strong predictor of the future? I just thought real signals would be relatively small since the markets are so competitive. – applicative_x Jun 18 '15 at 13:13
• Yes ... in modern markets you can not expect large sources of arbitrage (small $\beta$). But if $\beta$ is small then speaking the the terms of your model the error dominates and you are likely to lose money. – Ric Jun 18 '15 at 13:34