Cointegration Time Series Analysis Studylib Net

Sarah Lee AI generated o4-mini 6 min read · April 19, 2025 Time series data often exhibit common trends and shared long‑run relationships. Traditional regression of non‑stationary series can give spurious results, but cointegration unlocks the ability to model these relationships properly. In this article, we dive into the theory, step‑by‑step methods, and practical R/Python implementations for: Definition: Two or more non‑stationary series XtX_tXt​ and YtY_tYt​ are cointegrated if a linear combination, Zt=Yt−βXt,Z_t = Y_t - \beta X_t,Zt​=Yt​−βXt​,

is stationary (I(0)I(0)I(0)) even though XtX_tXt​ and YtY_tYt​ are I(1)I(1)I(1) (unit root processes). 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: This resource file contains information regarding lecture 20. 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. Sarah Lee AI generated o4-mini 5 min read · April 19, 2025 Dive into the fundamentals of cointegration relationships—learn theory, tests, and applications for robust time‑series analysis. Reference: Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.

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