A Series on Time Series, Part IV: Copula Methods and Vector Autoregression

In our ongoing exploration of time series forecasting, we’ve delved into traditional methods like ARIMA and Exponential Smoothing, and more recently, into machine learning techniques such as decision trees, neural networks, and support vector machines. In this fourth installment, we take a closer look at copula methods and vector autoregression (VAR), both of which extend our toolkit for multivariate time series analysis. These advanced techniques enable us to capture the complex relationships between multiple variables over time, offering deeper insights for industries like real estate, finance, and energy.

Copula Methods: Modeling Dependencies in Time Series

As our datasets grow richer and more complex, traditional methods for time series forecasting may struggle to capture the relationships between multiple variables—especially when those relationships are non-linear. Copula methods allow us to model these dependencies in a more flexible way, focusing on how different variables move together without imposing restrictive assumptions about their marginal distributions.

Why Copulas?

A copula essentially decouples the marginal distributions of each variable from their joint behavior. This is particularly useful when dealing with multivariate time series, where the relationship between variables can vary across different time periods. For instance, in the real estate market, housing prices in different regions might show stronger correlations during times of economic stress, which would be difficult to model with traditional methods.

By applying copulas, we can analyze the dependencies between these variables without being constrained by their individual behaviors. This allows us to gain a more accurate understanding of how changes in one variable (e.g., interest rates) might influence others (e.g., home prices).

Applications of Copula Methods

• Risk Management: Financial institutions use copulas to model dependencies between asset prices, helping them estimate portfolio risk, especially under extreme market conditions.
• Real Estate Analytics: Copula methods can track how regional housing markets co-move, which helps in understanding market-wide risks and opportunities during housing booms or busts.

Vector Autoregression (VAR): Multivariate Time Series Forecasting

While copulas focus on dependency structures, vector autoregression (VAR) is a powerful tool for capturing the interdependencies between multiple time series. Unlike univariate methods like ARIMA, VAR models can simultaneously forecast multiple variables, each of which can influence the others over time. This is ideal for situations where variables—such as home prices, interest rates, and inflation—affect one another.

How VAR Works

VAR treats each variable in a system as a function of its own past values as well as the past values of all other variables. This makes it well-suited for dynamic economic systems, where feedback loops and lagged effects are common.

For example, in the housing market, interest rates can influence home prices, while home prices can, in turn, affect inflationary pressures. By using VAR, we can quantify these relationships and forecast how shifts in one variable might ripple through the system.

Key Applications of VAR

• Macroeconomic Forecasting: VAR is widely used to model interactions between economic variables such as inflation, GDP, and unemployment rates, providing insights into how these variables influence each other.
• Real Estate Market Forecasting: In real estate, VAR can be applied to model the interdependence between mortgage rates, housing demand, and regional home prices, helping investors and policymakers make informed decisions.

Testing and Validation in Multivariate Models

As with any time series model, validating copula methods and VAR requires careful testing to ensure that the models generalize well. Backtesting and walk-forward validation are commonly used to simulate how these models would perform in real-world scenarios. For VAR, it’s particularly important to account for lag structures, while for copulas, one must ensure that the dependency structures hold across different periods of volatility.

Conclusion

In this fourth part of our series, we’ve explored how copula methods and vector autoregression (VAR) expand our ability to analyze and forecast multivariate time series. Copulas offer flexibility in modeling non-linear dependencies, while VAR provides a framework for capturing dynamic, feedback-driven relationships between multiple variables. Together, these tools empower analysts and forecasters to make more informed predictions in complex domains like real estate, finance, and energy.

In the next installment, we will explore hybrid models, combining traditional time series techniques with machine learning approaches to offer a best-of-both-worlds strategy for forecasting accuracy.

Franklin Carroll

Franklin Carroll is Kukun's VP of Modelling and Analytics; After finishing his studies at the University of Chicago, he worked at Fannie Mae where he helped build Collateral Underwriter, roll out the Value Verify initiative, and specialized in collateral related and credit related policy.

Published October 1, 2024