Cointegration Explains Why Pairs Spreads Use Price Levels
Summary
The document asks why a pairs-trading spread is formed from stock price levels when the hedge ratio is estimated using price changes. The answer identifies cointegration: two nonstationary price series may share a long-run relationship such that a particular linear combination of their levels is stationary or otherwise less integrated. That combination, rather than the changes themselves, is the candidate spread for a mean-reversion strategy.
The response cautions that regressing unrelated nonstationary series can produce spurious results, while cointegration provides the relevant statistical condition. It describes the setting where individual prices are integrated of order one and their spread is less integrated, and points to formal cointegration tests. The explanation does not establish that any specific pair is cointegrated or profitable; a trader must test the relationship and assess whether it persists in the chosen sample and trading period.
Key ideas
- A price-level spread can be useful when the underlying price series are cointegrated.
- The hedge ratio defines a linear combination of price levels whose integration order is lower.
- Regressing unrelated nonstationary price series risks a spurious relationship.
- Cointegration tests can assess a candidate pair, but do not guarantee persistence or profitability.
Tags
Full text
# Why the spread is calculated on raw prices instead on the price changes? # Why the spread is calculated on raw prices instead on the price changes? I'm reading Quantitative Trading With R written by Harry Georgakopoulos. In chapter 6 he exposes a basic quantitative strategy based on setting up a stock spread and buy when it is below a lower threshold and sell when it is above a upper threshold. Here is where I have my question: ``` x<- x[time_range] #x is the time series of the price of the stock A y<- y[time_range] #y is the time series of the price of the stock B dx<- diff(x) #stock A price changes dy<- diff(y) #stock B price changes r<- prcomp( ~ dx + dy) #linear regression using total least squares beta<- r $ rotation[2,1]/r $ rotation[1,1] #slope of the regression line spread<- y-beta*x ``` If we are calculating the regression line between the price changes of the stocks (`dx` and `dy`), why we calculate the spread on their raw prices (x and y)? Maybe is an obvious thing, but for me it would be more intuitive to calculate the spread on the price changes. Thank you. ## Answer by skoestlmeier (score 3) https://quant.stackexchange.com/a/43488 #### That is the concept of Cointegration Regressing two non-stationary variables results in spurious regression. However, if these two variables are cointegrated, spurious regression no longer arises. As stated on p. 120, > We call these resulting betas a cointegrating vector. Cointegration is a statistical property of time series, mainly proposed by Engle/Granger (1987). Assume two time-series $Y_t$ and $X_t$, which are integrated of order $d(Y_t, X_t \sim I(d) )$. If there exists a $\beta$ such that $$Y_t - \beta \cdot X_t = u_t$$ ,where $u_t$ is integrated of order less than $d$ (say $d-b$, than $Y_t$ and $X_t$ are cointegrated of order $d-b$. Especially if $Y_t$ and $X_t$ are of order 1 (which should be the case for absolute stock prices) and are cointegrated, you do not have to difference the data or use stock return data. Statistical tests for cointegration are clearly and detailed described in Hayashi (2000), Econometrics and implemented in the R-package urca.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.