Classic Econometric Models. Part 1: Prices and Returns

Let's take a look at the fundamental models used in empirical finance.

Selecting an appropriate econometric model for time series forecasting presents a formidable challenge due to the wide array of available options. In this segment of the newsletter, I will introduce fundamental models commonly employed for this purpose, along with more recent, advanced alternatives.

This series of articles will primarily address several key categories of econometric models, encompassing point forecasting for stock prices, returns, volatility, as well as probabilistic models capable of predicting entire probability distributions. I will also discuss risk models, portfolio models, high-frequency (microstructure and transactional) models, as well as options pricing models and event study analysis.

In the first part of the series, the focus will be on models used for point forecasts of financial asset prices and their returns. It's important to note that the categorization utilized in this article is intended to improve clarity, as many of these models have versatile applications and can be used for both price and return forecasting.

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Point Price Forecasting

The primary interest of investors lies in projecting future stock prices, a task that poses significant challenges, as the literature has repeatedly demonstrated the near-impossibility of directly forecasting prices (Kendall & Hill, 1953; Fama 1965). Nevertheless, certain models may offer the capability to achieve this feat in specific instances where prices exhibit autoregressive behavior. One such model, renowned for its popularity and suitability as a starting point, is the Autoregressive Integrated Moving Average (ARIMA) model.

ARIMA - Popularized by Box and Jenkins (1970) as a generalization of the ARMA model (see next section), the ARIMA model is a versatile and widely-used tool in time series analysis and forecasting. It consists of three key components:

Autoregressive (AR) Term: This component captures the relationship between the current observation and its past values. It signifies the degree of correlation between the current value and previous values in the time series.

Integrated (I) Term: The integration term represents the number of differences required to make the time series stationary. Stationarity is essential for modeling, and this component ensures that the data is transformed accordingly.

Moving Average (MA) Term: The MA component accounts for the weighted average of past forecast errors, emphasizing recent errors more than distant ones.

ARIMA models are further categorized into different orders, typically denoted as ARIMA(p, d, q), where 'p' represents the order of the AR component, 'd' signifies the degree of differencing, and 'q' indicates the order of the MA component. Selecting appropriate values for these orders is a crucial aspect of ARIMA modeling.

ARIMA models have been proven effective in various fields, including finance, economics, and beyond. They serve as a foundational tool for time series forecasting, providing valuable insights into future price movements and trends and are often used as a benchmark for more sophisticated models. For more on ARIMA models, see (Tsay, 2010) from the references section.

Apart from ARIMA, we can also include models such as:

Regression Models - Regression models, including linear and nonlinear regression, are suitable for point price forecasting when distinct relationships exist between price and predictor variables. These models leverage historical data to predict future prices based on factors like past prices, trading volumes, or economic indicators (see Draper & Smith, 1998).

Exponential Smoothing Models - Exponential smoothing models assign exponentially decreasing weights to past data points, giving more importance to recent observations. These models are effective at capturing trends and seasonality in time series data, making them valuable for forecasting (see Hyndman et al., 2008).

VAR (Vector Autoregression) - VAR models extend autoregression to multiple time series variables. They capture interdependencies among multiple financial instruments, making them useful for analyzing relationships between stocks, bonds, and other assets in a portfolio. VAR modeling is commonly employed in macroeconomic and financial analysis (see Kilian & Lütkepohl, 2017; Luetkepohl, 2005; Juselius, 2007).

State Space Models: State space models, including the Kalman filter and particle filters, can be employed for forecasting point prices of assets, particularly in dynamic systems where the underlying data generation process is complex. (see Luetkepohl, 2005; Tsay, 2010).

These models, in addition to ARIMA, provide a versatile toolkit for forecasting of prices. The choice of model depends on the nature of the data and the specific forecasting objectives.

However, as mentioned, forecasting stock prices can be very difficult or nearly impossible. I will now move on to another section, which is stock returns forecasting.

Returns forecasting

While forecasting stock prices can be a daunting challenge, predicting stock returns offers an alternative approach. By focusing on returns, which consider both price changes and dividends, we can employ various quantitative models to gain insights into potential future investment performance. This shift in focus can provide valuable information for investors and traders seeking to make informed decisions in the financial markets.

ARMA (Autoregressive Moving Average) - ARMA stands as one of the most prevalent models for forecasting returns of financial instruments. While historically popular, it is increasingly employed as a benchmark due to the acknowledged challenge of achieving accurate predictions using this model. Its limitations arise from the assumption that past values alone can effectively predict future returns. In contrast to ARIMA (Autoregressive Integrated Moving Average), which includes differencing to make the time series stationary, ARMA focuses solely on autoregressive and moving average components, making it less capable of handling non-stationary data (see Tsay, 2010). 

VAR (Vector Autoregression) - VAR can be also used as a model for predicting returns of financial assets. Unlike univariate models that focus on a single time series, VAR considers multiple financial variables simultaneously, such as returns on various assets within a portfolio. By capturing the dynamic relationships and dependencies among these variables, VAR enables a more comprehensive understanding of how changes in one asset's returns can impact others. This makes it an invaluable tool for portfolio managers and investors seeking to optimize their strategies and assess the risk-return trade-offs across multiple assets. 

VECM (Vector Error Correction Model) - VECM extends the capabilities of VAR in predicting the returns. It accommodates the concept of cointegration, which implies a long-term equilibrium relationship among multiple financial variables. VECM is particularly useful when dealing with assets that exhibit both short-term fluctuations and long-term equilibrium tendencies. By correcting for short-term deviations from equilibrium, VECM provides a more nuanced perspective on return dynamics, making it a valuable tool for analyzing and predicting returns in situations where both short-term and long-term relationships are at play. This makes it especially relevant in scenarios where asset prices tend to revert to their long-term mean values.  For comprehensive literature on both VAR and VECM, see ( Kilian & Lütkepohl, 2017; Luetkepohl, 2005; Juselius, 2007).

Summary

In the above writeup, I have discussed popular econometric models commonly utilized for point price and returns forecasting. These models are fundamental tools in financial analysis, offering valuable information about future market movements. In the forthcoming parts of this series, I will explore other critical categories of models, including volatility, probabilistic, risk, and portfolio models. Additionally, I will address a few more specialized model categories. Stay tuned and let me know what you think so far in the comments!

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: forecasting and control. Holden-Day.

Draper, N. R., & Smith, H. (1998). Applied Regression Analysis (Third edition). Wiley-Interscience.

Fama, E. F. (1965). Random Walks in Stock Market Prices. Financial Analysts Journal, 21(5), 55–59.

Hyndman, R., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with Exponential Smoothing: The State Space Approach. Springer.

Juselius, K. (2007). The Cointegrated VAR Model: Methodology and Applications. Oxford University Press.

Kendall, M. G., & Hill, A. B. (1953). The Analysis of Economic Time-Series-Part I: Prices. Journal of the Royal Statistical Society. Series A (General), 116(1), 11–34.

Kilian, L., & Lütkepohl, H. (2017). Structural Vector Autoregressive Analysis. Cambridge University Press.

Luetkepohl, H. (2005). The New Introduction to Multiple Time Series Analysis. Springer.

Tsay, R. S. (2010). Analysis of Financial Time Series. John Wiley & Sons.