Information criteria

Information Criteria: A Guide for Model Selection

Information criteria are a set of statistical tools used to evaluate and compare different statistical models, with the goal of identifying the model that best balances goodness of fit with model complexity. While seemingly abstract, understanding these criteria is crucial for anyone building predictive models – and in the world of crypto futures trading, building accurate predictive models is paramount. This article will provide a comprehensive introduction to information criteria, focusing on their application in a statistical context and highlighting why they are relevant to traders and analysts.

What are Statistical Models and Why Do We Need to Compare Them?

Before diving into information criteria, let's establish the context. A statistical model is a mathematical representation of a real-world process. In trading, models can range from simple moving averages to complex machine learning algorithms designed to predict future price movements. These models rely on historical trading volume analysis and other data.

The challenge arises because numerous models can potentially fit the same data. A highly complex model might perfectly capture every nuance of the historical data (high goodness of fit), but it could be overfitting – meaning it performs poorly on new, unseen data. A simpler model might not fit the historical data as well, but it could generalize better to future data, offering more reliable predictions.

The core problem is finding the optimal balance between these two extremes: a model complex enough to capture the underlying patterns, but not so complex that it's overly sensitive to noise. This is where information criteria come in. They provide a quantitative way to compare models and penalize complexity.

The Core Principle: Balancing Goodness of Fit and Complexity

All information criteria share a common principle: they aim to estimate the relative information lost when a given model is used to represent the process that generated the data. Think of it like compressing a file. A perfect compression (high fit) might require a complex algorithm (high complexity). A simpler compression might lose some information (lower fit) but be more efficient.

The key is that information criteria don’t just measure how well a model fits the data; they also penalize models for having more parameters. The penalty reflects the risk of overfitting. A model with more parameters has more "freedom" to fit the training data, but this freedom comes at the cost of potentially poor performance on new data.

Common Information Criteria

Several information criteria are commonly used. Here, we’ll focus on the three most prevalent:

**Akaike Information Criterion (AIC)**: Developed by Hirotatsu Akaike, AIC estimates the relative amount of information lost by a given model. It’s widely used and relatively simple to calculate.
**Bayesian Information Criterion (BIC)**: Also known as the Schwarz Information Criterion (SIC), BIC is an alternative to AIC. It tends to penalize complexity more strongly than AIC, favoring simpler models.
**Hannan-Quinn Information Criterion (HQIC)**: A compromise between AIC and BIC, HQIC offers a penalty for complexity that falls between the two.

Understanding the Formulas

Let’s look at the formulas for each criterion. Don't be intimidated; the concepts are more important than memorizing the equations.

**AIC = 2k - 2ln(L)**
**BIC = kln(n) - 2ln(L)**
**HQIC = 2kln(ln(n)) - 2ln(L)**

Where:

**k** is the number of parameters in the model. This includes all coefficients and variables used in the model. For example, a simple linear regression model (y = ax + b) has two parameters: 'a' (slope) and 'b' (intercept).
**n** is the number of data points used to fit the model (sample size). In trading, this would be the number of historical data points used to train your model – for example, the number of daily closing prices used to fit a time series analysis.
**L** is the maximized value of the likelihood function for the model. The likelihood function represents how well the model fits the observed data. A higher value of L indicates a better fit. The natural logarithm (ln) of L is used to simplify calculations and improve numerical stability.

Interpreting the Results

The core principle for interpreting information criteria is simple: **lower values are better**. The model with the lowest AIC, BIC, or HQIC is considered the most preferred model, given the data.

However, it’s crucial to understand *why* a lower value is better. A lower value indicates a better balance between goodness of fit and model complexity. The criterion is effectively saying, “This model explains the data well without being unnecessarily complicated.”

It's important to note that these criteria provide *relative* rankings. They tell you which model is better *among the set of models you've compared*. They don't tell you if any of the models are actually "good" in an absolute sense. A model with a low AIC might still be a poor predictor if the underlying assumptions of the model are violated.

Example: Comparing Moving Average Strategies

Imagine you’re developing a trading strategy based on moving averages. You've tested three different models:

**Model 1:** A simple 10-day moving average crossover strategy. (k = 2 parameters: the short-period MA and the long-period MA)
**Model 2:** A more complex strategy using two moving averages and Relative Strength Index (RSI). (k = 5 parameters)
**Model 3:** A very complex strategy incorporating three moving averages, RSI, and MACD. (k = 8 parameters)

You fit each model to 1000 days of historical price data (n = 1000) and obtain the following results:

| Model | AIC | BIC | HQIC | |-------|--------|--------|--------| | 1 | 1020 | 1035 | 1028 | | 2 | 1015 | 1045 | 1030 | | 3 | 1010 | 1060 | 1038 |

Based on AIC, Model 3 has the lowest value. However, BIC penalizes complexity more heavily. According to BIC, Model 1 is the preferred model. HQIC falls in between.

This example illustrates a key point: the choice of information criterion can influence the selected model. AIC might favor more complex models, while BIC tends to favor simpler ones. The appropriate criterion depends on the specific application and the goals of the modeling exercise. If you believe that overfitting is a significant risk, BIC might be a better choice. If you prioritize capturing as much information as possible, AIC might be more appropriate.