Forecast Pooling, Ensemble Learning, and Combining Trading Strategies
Summary
The document distinguishes ensemble learning from pooling forecasts. Ensembles combine related estimates generated within a shared modeling process, while forecast pooling combines predictions that may come from different models. It also separates combining forecast values from dividing capital between strategies, which operates on realized return streams instead.
For forecasts of the same return and horizon, the answer describes a weighted combination and gives minimum-error-variance weights based on the covariance of out-of-sample forecast errors, under assumptions of approximate unbiasedness and weights summing to one. It recommends equal weighting as a starting point, evaluating alternatives with rolling or expanding walk-forward tests, and shrinking or constraining estimated weights because error covariance estimates can be noisy. Forecast correlation matters because models that make similar errors provide less diversification. Different forecast scales or biases may require adjustment or an intercept. Capital allocation should account for strategy-return covariance, turnover, costs, and risk limits. The answer emphasizes that complex weights must prove useful out of sample.
Key ideas
- Ensemble learning combines related estimates, while forecast pooling can combine distinct forecasting models.
- Forecast combination and allocation of capital across strategies are separate decisions.
- Forecast-error covariance determines the value of combining predictions and informs minimum-variance weights.
- Equal weighting is a useful baseline when estimated weights are noisy or history is limited.
- Walk-forward evaluation should account for biases, scale, turnover, transaction costs, and risk limits.
Tags
Full text
# What are the ensemble techniques to forecast returns?
# What are the ensemble techniques to forecast returns?
It was pointed in an other question that ensemble methods can help to reduce curve fitting. What are your experience with these and which one seems the most appropriate? If I had two forecasters that give reasonably good results. Would it be better to use both and invest half in each (diversification) or use one of the ensemble method?
## Answer by Dirk Eddelbuettel (score 7)
https://quant.stackexchange.com/a/270
Ensemble methods, or ensemble learning are a class of statistical methods that, loosely speaking, operate on many rather than a single instance of the data. Think bootstrapping, but then combine the estimates for an aggregate. The Wikipedia link has more.
Combining two forecasters is something else that is sometimes called pooling forecasts or, more generally, consensus forecast.
The main difference is that the pieces in an ensemble method are related---pooling is from the same class or instance of an estimate---whereas pooled forecast are aggregating over different forecast which may not have any commonality.
## Answer by Russlan Ramdowar (score 0)
https://quant.stackexchange.com/a/85718
With two forecasters, I would treat this as a forecast-combination problem first and a capital-allocation problem second.
Let $f_t=(f_{1,t},\ldots,f_{m,t})'$ be forecasts of the same return over the same horizon, and let the combined forecast be
$$\hat r_{t+1}=w'f_t.$$
If the forecasts are approximately unbiased and you impose $1'w=1$, the minimum forecast-error-variance weights are
$$w=\frac{\Sigma_e^{-1}1}{1'\Sigma_e^{-1}1},$$
where $\Sigma_e$ is the covariance matrix of the out-of-sample forecast errors. Correlation matters: two individually good forecasters add little diversification if they make the same mistakes.
In practice, estimating $\Sigma_e$ is noisy, so the theoretically optimal weights can overfit. A sensible workflow is:
- Start with equal weights.
- Estimate errors only through rolling or expanding walk-forward tests.
- Compare equal weights with constrained regression, inverse-error-variance weights, or Bayesian/model-averaging weights.
- Shrink any estimated weights back toward equal weights and cap extreme values.
- Evaluate both statistical accuracy and economic results after turnover, costs, and risk limits.
If the forecasts have different biases or scales, de-bias and standardize them first, or use a regression with an intercept; in that case the coefficients do not necessarily need to sum to one.
Investing half the capital in each strategy is a different layer. It combines PnL streams, so the relevant covariance is strategy-return covariance and equal risk is usually more meaningful than equal capital. If both forecasts trade the same book, pooling the forecasts first also makes position netting and transaction costs easier to control.
For two reasonably good forecasters and a limited history, equal-weight pooling is the baseline I would try to beat. Complexity should earn its place out of sample.
Disclosure: OpenAI Codex assisted with structuring and wording this answer; I reviewed and edited the formula and conclusions.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.