Parametric vs. Non-Parametric Survival Models for Trading

When applying survival analysis to trade duration, a key decision is the choice of model. The spectrum of survival models ranges from non-parametric to semi-parametric and fully parametric. Each class of models has its own set of assumptions, advantages, and disadvantages. For the quantitative trader, understanding these differences is important for selecting the most appropriate tool for the task at hand.

A Tale of Three Models

Let's compare the three main types of survival models in the context of financial data:

• Non-Parametric (Kaplan-Meier): The Kaplan-Meier estimator is a non-parametric model, meaning it makes no assumptions about the underlying distribution of survival times. It is a purely data-driven approach that provides a simple and intuitive way to visualize the survival function. However, it cannot be used to model the effects of covariates.

• Semi-Parametric (Cox Proportional Hazards): The Cox proportional hazards model is a semi-parametric model. It is parametric in the sense that it assumes a linear relationship between the covariates and the log-hazard, but it is non-parametric in the sense that it does not make any assumptions about the shape of the baseline hazard function. This flexibility makes the Cox model a popular choice for many applications.

• Fully Parametric: Fully parametric models, such as the Weibull, Exponential, and Log-Logistic models, assume a specific distribution for the survival times. For example, the Exponential model assumes that the hazard rate is constant over time, while the Weibull model allows the hazard rate to increase or decrease over time. The main advantage of parametric models is that they can be more efficient than the Cox model if the assumed distribution is correct. However, if the assumed distribution is incorrect, the results can be misleading.

Pros and Cons of Each Approach in the Context of Financial Data

• No assumptions about the distribution of survival times | - Cannot model the effects of covariates
• Can be inefficient for large datasets | | Semi-Parametric | - Flexible
• Can model the effects of covariates without assuming a specific distribution for the baseline hazard | - Assumes proportional hazards, which may not always hold
• Can be less efficient than parametric models if the distribution is known | | Fully Parametric | - More efficient than the Cox model if the assumed distribution is correct
• Can be used to extrapolate beyond the observed data | - The results can be misleading if the assumed distribution is incorrect
• Can be more difficult to fit than the Cox model |

How to Choose the Right Model for Your Trading Strategy

The choice of model will depend on the specific goals of your analysis and the characteristics of your data. Here are some general guidelines:

• For exploratory analysis: The Kaplan-Meier estimator is a good starting point for visualizing the survival function and comparing the survival distributions of different groups of trades.

• For predictive modeling: The Cox proportional hazards model is a good choice if you want to model the effects of covariates without making strong assumptions about the distribution of survival times.

• For a more detailed understanding of the hazard function: A parametric model can be a good choice if you have a strong reason to believe that the survival times follow a particular distribution. For example, if you believe that the hazard of a trade ending increases over time, you might choose a Weibull model with a shape parameter greater than 1.

Goodness-of-Fit Tests to Validate Model Assumptions

Regardless of which model you choose, it is important to validate its assumptions. For the Cox model, the key assumption is that the hazards are proportional. This can be tested using a variety of methods, such as the Schoenfeld residuals test. For parametric models, you can use goodness-of-fit tests, such as the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), to compare the fit of different distributions.