Basics

Let us define a pair of companies as follows: two competing companies that are equally efficient in running an identical business. For example, PepsiCo, Inc and The Coca-Cola companies are likely to be such a pair. Each part of a pair returns the same profit/loss per $1 of investments. Mathematically this can be expressed as follows:
 dY - adX=0,   (1)
where dY and dX stand for the change in the stock price of the pair components, Y and X, per time dt. The solution of (1) gives the relationship, Y= aX + b, which should hold true if both parts of the pair are fairly priced. Let us introduce  a synthetical underlying, R(t) = Y(t) - aX(t). Since market forces are expected ensuring  <R(t) - b>=0  ( <...> stands for averaging over time), R is an ideal vehicle for mean reverting /statistical arbitrage strategies.

Pair trading strategies look to profit from a temporary divergence in R. 

For example, when  Y moves up while aX moves up slower than Y, or X doesn't move up, or X moves down then R can become larger than its stationary value b. A trader expects this divergence to revert back to the mean. In the above example, the trader ought to sell the outperformer, Y, and to buy the underperformer, X. The bet that R would converge to b relies only on relative changes in X and Y. It means that pair trading is not sensitive to market uptrend, downtrend, or sideways movements.

Why Trade Pairs?

Hedging is an intrinsic property of pair trading. Thus, if properly diversified, pair trading is not prone to market crashes.

What Pairs are suitable for a retail trader?

Creating a market-neutral portfolio of stock pairs requires significant investments of capital, analytical and computational resources which is usually beyond the limits of retail trading. It doesn't mean that a retail trader should avoid pair trading at all. In retail, it is reasonable to trade pairs made of index ETFs/futures, currency ETFs/futures and commodity ETF/futures where the causal relationship between X and Y is of a general economical/financial character while  X & Y are diversified by the very design.


How to generate Y = aX +b.

In the public domain, there are several quantitative methods (correlation, distance, stochastic, stochastic differential residual, and cointegration) describing how to select a pair and how to generate the corresponding residual series.