Cointegration And Error Correction Models Chapter 6 Time Series

Leo Migdal

-Dec 14, 2025, 2:29 AM

cointegration and error correction models chapter 6 time series

Published online by Cambridge University Press: 05 December 2014 The study of equilibrium relationships is at the heart of time series analysis. Because cointegration provides one way to study equilibrium relationships, it is a cornerstone of current time series analysis. The original idea behind cointegraton is that two series may be in equilibrium in the long run, but in the short run the two series deviate from that equilibrium. Clarke, Stewart, and Whiteley (1998, 562) explain that “cointegrated series are in a dynamic equilibrium in the sense that they tend to move together in the long run. Shocks that persist over a single period are ‘reequilibrated’ or adjusted by this cointegrating relationship.” Thus cointegration suggests a long-run relationship between two or more series that may move in quite different ways in...

Put a bit more formally, cointegration says that a specific combination of two non stationary series may be stationary. We then say these two series or variables are cointegrated, and the vector that defines the stationary linear combination is called the cointegrating vector. Recall from the previous chapter that a time series is stationary when its mean and variance do not vary over or depend on time. Lin and Brannigan(2003, 153) point out that “many times series variables in the social sciences and historical studies are nonstationary since the variables typically measure the changing properties of social events over, for example,... These variables display time varying means, variances, and sometimes autocovariances.” To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your...

Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle. Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Many statistical procedures are well-defined only when the processes of interest are stationary.

As a result, especially when one wants to investigate the joint dynamics of different variables, one often begins by making the data stationary (by, e.g., taking first differences or removing deterministic trends). However, doing so may remove information from the data. Heuristically, removing trends amounts to filtering out the long-run variations of the series. However, it may be the case that the different variables interact in the short run and in the long run. For instance, the left plot of Figure 6.1 suggests that the trends of \(x_t\) and \(y_t\) are positively correlated. However, the right plot shows that, for low values of \(h\), the correlation between \(x_t - x_{t-h}\) and \(y_t - y_{t-h}\) is negative.

This is notably the case for \(h=1\), which means that the first differences of the two variables (i.e., \(\Delta x_t\) and \(\Delta y_t\)) are negatively correlated. Hence, focusing on the first differences would lead the researcher to think that the relationship between \(x_t\) and \(y_t\) is a negative one (while it is only the case when one focuses on the... Figure 6.1: Situation where the conditional and uncondional correlation between \(x_t\) and \(y_t\) do not have the same sign. Definition 6.1 (Integrated variables) A univariate process \(\{y_t\}\) is said to be \(I(d)\) if its \(d^{th}\) difference is stationary (but not its \((d-1)^{th}\) difference). If we regress an \(I(1)\) variable \(y_t\) on another independent \(I(1)\) variable \(x_t\), the usual (OLS-based) t-tests on regression coefficients often (misleadingly) show statistically significant coefficients (we then speak of spurious regressions, see Section... A solution is to regress \(\Delta y_t\) (that is \(I(0)\)) on \(\Delta x_t\) and then inference will be correct.

However, as stated above, the economic interpretation of the regression then changes, as doing so amounts to focusing on the high-frequency movements of the variables. Explore the theory and practical applications of cointegration and error correction models in time-series analysis, focusing on financial markets, testing procedures, and modeling strategies. Ever chatted with a friend about how two assets seem to “move together” in the market? Sometimes, they follow a common path because they’re driven by similar fundamentals—maybe they’re both large-cap tech stocks, or perhaps they are two types of government bonds. But here’s the catch: just because prices look similar doesn’t necessarily mean they share a long-term stable relationship. That’s where cointegration comes in.

Cointegration is a concept that captures long-run equilibrium relationships among time-series variables, even when each individual series itself might wander around (i.e., be non-stationary). In finance, this is crucial for understanding everything from pairs trading strategies to interest-rate dynamics. If two series, say Yₜ and Xₜ, are both integrated of order 1 (I(1))—in other words, they both contain a unit root—yet a certain linear combination of them (a₁Yₜ + a₂Xₜ) is stationary (I(0)),... This tells us there’s a stable, long-term link between them, even if they drift up or down in the short run. Before diving into cointegration, it’s important to recall the basics of integrated processes and stationarity (see also “12.3 Unit Roots, Stationarity, and Forecasting” in this book). A stationary process is one whose statistical properties—including mean and variance—do not depend on time.

Many real-world financial time series like prices and exchange rates are not stationary; they often exhibit trends or random walk behavior. When a series must be differenced (subtracted from its own lag) once to become stationary, we say it’s I(1). This concept forms the foundation for cointegration analyses. If Yₜ and Xₜ are each I(1) processes, but a linear combination such as: An Error Correction Model (ECM) is a powerful econometric tool used to model the relationship between non-stationary time series variables that are cointegrated. Cointegration implies that while individual time series may be non-stationary, a linear combination of them is stationary, indicating a long-run equilibrium relationship.

ECMs are particularly useful for capturing both short-term dynamics and long-term equilibrium adjustments between variables. An Error Correction Model (ECM) is specifically designed to handle non-stationary data by addressing both short-term dynamics and long-term equilibrium relationships between time series variables. The term "error correction" refers to the mechanism by which deviations from the long-run equilibrium are corrected over time. In an ECM, the error correction term represents the extent to which the previous period's disequilibrium influences the current period's adjustments. This allows the model to capture both short-term fluctuations and the speed at which the variables return to equilibrium. An Error Correction Model (ECM) is specifically designed to handle non-stationary data by addressing both short-term dynamics and long-term equilibrium relationships between time series variables.

Non-stationary data are time series that have properties such as mean, variance, and autocorrelation that change over time. When dealing with non-stationary data, traditional regression models can lead to spurious results. However, if two or more non-stationary series are cointegrated, it means they share a common stochastic trend and move together in the long run, despite being non-stationary individually. Cointegration is a powerful concept in time series analysis, revealing long-term relationships between non-stationary variables. It's like finding a hidden connection between two seemingly unrelated trends, allowing us to make sense of complex economic systems. Error Correction Models (ECMs) take cointegration a step further, showing how variables adjust to maintain their long-term relationship.

They're like relationship counselors for data, helping us understand how economic factors interact and recover from short-term disruptions. Applied Econometric Time Series, 4th Edition SAS-RATS Version of Instructor's Manual* Copyright © 2000-2025 by John Wiley & Sons, Inc., or related companies. All right reserved. | Privacy Policy

Cointegration And Error Correction Models Chapter 6 Time Series

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