Predictive Modeling of Asset Classes: Probability Density Function Shapes

 

This post discusses and suggests the use of suitable models based on the shape and properties of a Probability Density Function (PDF).  Data on the following ETFS (SPDR=SP500, IEF=10yr Treasury Yield, DBC=Commodities, DXY=USD, GLD=Gold) as at 4 June 2025. The chart above shows the shape of the PDF of each asset class. 

A summary of the properties and/or implications of each asset class’  PDF from the perspective of constructing predictive models. 

DXY

The DXY distribution is relatively symmetric and centered, with moderate tails. This suggests that standard linear models and time series models assuming normality may perform adequately. Outliers are less likely, so models will be less sensitive to extreme values, and regularization may not be as critical.

SP500

The SP500 distribution is sharply peaked and slightly left-skewed, indicating frequent small changes and occasional larger negative moves. Predictive models should account for potential negative outliers. Robust regression or models that can handle skewness, such as quantile regression or models with heavy-tailed error terms, may be beneficial.

GOLD

GOLD’s distribution is broad and slightly right-skewed, with heavier tails. This implies a higher probability of extreme positive returns. Predictive models should be robust to outliers and may benefit from using loss functions less sensitive to large deviations, such as Huber loss or quantile-based approaches.

10YR BOND

The 10YR BOND distribution is narrow and symmetric, indicating low volatility and a high concentration of values near the mean. Predictive models can assume normality and low variance, making simple linear models effective. However, the low variance may limit the predictive power of more complex models.

COMMODITIES

The COMMODITIES distribution is wide and exhibits heavy tails, suggesting frequent large deviations from the mean. Predictive models should be robust to outliers and may require transformation or regularization. Nonlinear models or those designed for fat-tailed distributions, such as tree-based methods or models with t-distributed errors, may perform better.

In summary, asset classes with symmetric, narrow distributions (like 10YR BOND) are well-suited for simple models, while those with skewness or heavy tails (like GOLD and COMMODITIES) require robust, flexible modeling approaches to handle outliers and non-normality.

Suitable Predictive models for data with significant Skewness and Kurtosis

Here are several models and approaches suitable for skewed data distributions, along with explanations and considerations for implementation:

Quantile Regression

Quantile regression models the conditional quantiles of the response variable, making it robust to skewness and outliers. It does not assume a symmetric error distribution and can provide a more complete view of the relationship between variables. When implementing, consider which quantiles (e.g., median, 90th percentile) are most relevant to your analysis.

Generalized Linear Models (GLM) with Non-Normal Families

GLMs allow you to specify error distributions that match the skewness of your data, such as Gamma, Poisson, or Inverse Gaussian. This flexibility helps model data with non-normal, skewed distributions. When using GLMs, select the family and link function that best matches your data’s characteristics.

Tree-Based Models (Random Forest, Gradient Boosting, XGBoost, LightGBM)

Tree-based models are non-parametric and do not assume any specific data distribution. They handle skewed and heavy-tailed data well, automatically capturing nonlinearities and interactions. For implementation, ensure proper tuning of hyperparameters and consider using robust evaluation metrics.

Support Vector Regression (SVR) with Nonlinear Kernels

SVR can model complex, skewed relationships using nonlinear kernels (e.g., RBF). It is less sensitive to the distribution of the target variable. When implementing, scale your features and tune kernel parameters for best performance.

Robust Regression (Huber Regression, Theil-Sen Estimator, RANSAC)

Robust regression methods reduce the influence of outliers and skewed data points. Huber regression, for example, uses a loss function that is quadratic for small errors and linear for large errors. When implementing, select the robustness parameter (e.g., epsilon in Huber) based on your data.

Transformation-Based Approaches (Box-Cox, Yeo-Johnson)

Transforming the target variable to reduce skewness (e.g., using Box-Cox or Yeo-Johnson transformations) can make the data more suitable for standard linear models. After modeling, invert the transformation to interpret predictions in the original scale.

Bayesian Regression with Skewed Priors

Bayesian models allow you to specify priors that reflect the skewness in your data, such as skew-normal or skew-t distributions. This approach can improve predictive accuracy and uncertainty estimation. When implementing, use probabilistic programming libraries and carefully choose priors.

Each of these models or approaches is designed to handle skewed data distributions, either by being robust to outliers, not assuming normality, or by explicitly modeling the skewness.