Statistical Properties of Polynomial Regression Channels
The Black Book of Day Trading Strategies
1,000 complete strategies · 31 chapters · Full trade plans
A deep understanding of the statistical properties of Polynomial Regression Channels (PRC) is what separates the novice from the expert quantitative analyst. These properties provide insights into the reliability of the channel and can be used to develop more sophisticated and robust trading strategies.
Distribution of Residuals
The residuals, or the differences between the actual prices and the fitted regression values, are a key component of the PRC. The standard deviation of the residuals is used to determine the channel width. It is often assumed that the residuals are normally distributed, but this is not always the case in financial markets. Financial asset returns are known to exhibit fat tails and skewness. A thorough analysis of the residuals should include tests for normality, such as the Jarque-Bera test.
Jarque-Bera Test Statistic:
ight)$$Where (m) is the number of observations, (S) is the sample skewness, and (K) is the sample kurtosis. A high JB statistic indicates that the data is not normally distributed.
Stationarity of Channel Width
The width of the PRC is determined by the standard deviation of the residuals. For the channel to be a reliable indicator of volatility, the channel width should be relatively stable, or stationary, over time. A non-stationary channel width would imply that the volatility of the asset is changing in a way that is not being captured by the model. Tests for stationarity, such as the Augmented Dickey-Fuller (ADF) test, can be applied to the time series of the channel width.
Autocorrelation of Residuals
Another important statistical property to examine is the autocorrelation of the residuals. If the residuals are autocorrelated, it means that there is still information in the price series that is not being captured by the polynomial regression model. This could be an indication that a higher-degree polynomial is needed, or that other variables should be included in the model.
| Lag | Autocorrelation | p-value |
|---|---|---|
| 1 | 0.05 | 0.32 |
| 2 | -0.02 | 0.78 |
| 3 | 0.08 | 0.15 |
| 4 | 0.01 | 0.92 |
Trade Example:
If a trader observes that the residuals of a PRC are not normally distributed and exhibit fat tails, they might adjust their trading strategy to be more conservative. For example, instead of placing a take-profit order at the regression line, they might place it at a level closer to the entry price to account for the possibility of large, adverse price movements.
Conclusion
The statistical properties of Polynomial Regression Channels provide a wealth of information that can be used to improve the performance and robustness of trading strategies. A quantitative analyst who takes the time to understand these properties will have a significant edge over those who simply apply the PRC as a black box. The next article in this series will focus on specific trading strategies that can be implemented using PRCs.