Glossary

10.3 Cointegration and error correction models

3 min read•july 22, 2024

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.

Cointegration

Cointegration in time series

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  • Statistical property of two or more non-stationary time series that exhibit time-varying means, variances, or both (GDP and consumption)
  • Cointegrated time series share a common stochastic trend and have a relationship
    • Deviations from this equilibrium are stationary and mean-reverting (price of a stock and its futures contract)
  • Cointegrated series move together in the long run, despite short-run deviations, preventing them from drifting too far apart (interest rates and inflation)
  • Allows for the estimation of long-run equilibrium parameters and the to equilibrium

Tests for cointegrating relationships

  • Engle-Granger test: A two-step residual-based test for cointegration
    1. Estimate the long-run equilibrium relationship using OLS regression: yt=β0+β1xt+uty_t = \beta_0 + \beta_1 x_t + u_t
    2. Test the residuals utu_t for using a unit root test like the Augmented Dickey-Fuller test
      • If the residuals are stationary, the series are cointegrated (income and expenditure)
  • : A maximum likelihood-based test for cointegration in a vector autoregressive (VAR) framework
    • Allows for testing multiple cointegrating relationships among several variables (GDP, consumption, and investment)
    • Based on the rank of the matrix of long-run coefficients in the VAR model, which determines the number of cointegrating relationships
    • Uses the trace statistic and the maximum eigenvalue statistic

Error Correction Models (ECMs)

Error correction models

  • Incorporate both and long-run equilibrium relationships
  • General form of an ECM for two cointegrated variables: Δyt=α0+α1Δxt+α2(yt−1−β0−β1xt−1)+εt\Delta y_t = \alpha_0 + \alpha_1 \Delta x_t + \alpha_2 (y_{t-1} - \beta_0 - \beta_1 x_{t-1}) + \varepsilon_t
    • Δyt\Delta y_t and Δxt\Delta x_t capture short-run dynamics (changes in stock prices)
    • (yt−1−β0−β1xt−1)(y_{t-1} - \beta_0 - \beta_1 x_{t-1}) is the , representing the deviation from long-run equilibrium (spread between a stock and its futures contract)
  • Estimating an ECM:
    1. Estimate the long-run equilibrium relationship using OLS regression
    2. Estimate the ECM using OLS, including the lagged residuals from the long-run relationship as the error correction term

Interpretation of ECM parameters

  • Short-run coefficients (α1\alpha_1): Represent the immediate impact of changes in the explanatory variable on the dependent variable (effect of a change in income on consumption)
  • Long-run coefficients (β1\beta_1): Represent the long-run equilibrium relationship between the variables (long-run relationship between price and quantity demanded)
  • Adjustment parameter (α2\alpha_2):
    • Represents the speed at which the dependent variable adjusts to deviations from the long-run equilibrium
    • A negative and statistically significant adjustment parameter indicates the presence of a stable long-run relationship (adjustment of stock prices to their fundamental values)
    • The larger the absolute value of the adjustment parameter, the faster the adjustment to equilibrium

Applications of ECMs

  • with ECMs:
    • Provide more accurate forecasts than models that ignore cointegration (forecasting exchange rates)
    • The error correction term helps to keep the forecasts in line with the long-run equilibrium
  • Policy analysis with ECMs:
    • Assess the short-run and long-run effects of policy changes or shocks (impact of a tax cut on consumption and GDP)
    • The adjustment parameter indicates the speed at which the system returns to equilibrium after a shock
    • Impulse response functions derived from ECMs analyze the dynamic effects of shocks on the variables in the system (response of inflation to a monetary policy shock)

Key Terms to Review (15)

Adjustment Coefficient: The adjustment coefficient is a parameter used in error correction models to indicate the speed at which a dependent variable returns to equilibrium after a disturbance. It reflects how quickly the model adjusts to deviations from the long-term relationship established through cointegration. A higher adjustment coefficient suggests that the variable responds more rapidly to changes, making it crucial for understanding dynamic relationships between time series data.
Clive W.J. Granger: Clive W.J. Granger was a prominent economist known for his groundbreaking work in time series analysis, particularly in the development of cointegration and error correction models. His contributions significantly advanced the understanding of the long-run relationships between non-stationary time series variables, providing tools for analyzing economic data that exhibit trends and seasonal patterns.
Cointegration rank: Cointegration rank refers to the number of independent cointegrating relationships present in a set of non-stationary time series variables. It plays a crucial role in understanding the long-term equilibrium relationships between these variables and is essential for correctly specifying error correction models. Identifying the cointegration rank helps to determine how many combinations of the variables can maintain a stable relationship over time, influencing both the estimation and interpretation of economic models.
Engle-Granger Two-Step Approach: The Engle-Granger Two-Step Approach is a method used to test for cointegration between two or more time series variables. This approach involves first estimating the long-run relationship between the variables using ordinary least squares (OLS) and then testing the residuals from this regression for stationarity, typically using the Augmented Dickey-Fuller (ADF) test. This method is fundamental in understanding error correction models as it identifies whether a stable long-term relationship exists, which is crucial for further analysis.
Error correction term: The error correction term is a component in econometric models that indicates the deviation from a long-term equilibrium relationship between variables. It captures the speed at which variables return to this equilibrium after a short-term disturbance, providing crucial insights into dynamic adjustments in time series data. This term is especially significant in the context of cointegration, where it helps to correct deviations, ensuring that the model reflects both short-term dynamics and long-term relationships.
Forecasting: Forecasting is the process of making predictions about future events based on historical data and analysis. It involves identifying patterns and trends in time series data to estimate future values, which is crucial for planning and decision-making in various fields.
Johansen Test: The Johansen Test is a statistical method used to determine the presence and number of cointegration relationships among multiple time series. This test is particularly useful because it can handle more than two time series at once, allowing researchers to understand the long-run relationships between them. Its significance lies in its ability to identify whether a linear combination of non-stationary time series can produce a stationary series, which is crucial for building error correction models.
Long-run equilibrium: Long-run equilibrium refers to a state in which all economic forces are balanced, leading to stable prices and quantities in the market. In this state, supply equals demand, and there are no incentives for firms or consumers to change their behavior. This concept is particularly important when considering how variables interact over time, especially in the context of cointegration and error correction models.
Non-stationarity: Non-stationarity refers to a time series that exhibits changes in its statistical properties over time, such as mean, variance, or autocorrelation. This concept is crucial as many statistical methods assume that the underlying data is stationary. Recognizing non-stationarity is vital for making accurate predictions and understanding the relationships between variables in time-dependent data.
Residual diagnostics: Residual diagnostics refers to the process of analyzing the residuals, or differences between observed and predicted values, from a statistical model to assess the model's fit and underlying assumptions. This analysis helps identify potential issues such as autocorrelation, heteroscedasticity, or non-normality of errors, which can impact the validity of conclusions drawn from the model. Understanding these diagnostics is crucial when dealing with cointegration and error correction models, as they help ensure that the relationships between non-stationary time series are accurately captured and modeled.
Robert F. Engle: Robert F. Engle is an influential American economist best known for his work on the analysis of time series data, particularly in the context of volatility modeling. He developed the Autoregressive Conditional Heteroskedasticity (ARCH) model, which captures changing variance over time, a key concept for understanding financial time series and economic phenomena.
Short-run dynamics: Short-run dynamics refer to the behavior of a time series in the immediate period following a shock or change, often characterized by temporary adjustments before returning to a long-term equilibrium. This concept is essential for understanding how variables interact and adjust over time, especially in the context of cointegration and error correction models, where short-term fluctuations can provide insights into long-term relationships between integrated time series.
Speed of adjustment: The speed of adjustment refers to the rate at which a variable returns to its long-term equilibrium after experiencing a shock or disturbance. This concept is crucial in understanding how quickly an economy or system can adapt to changes, which is particularly relevant in the context of cointegration and error correction models, as these frameworks assess the relationship between non-stationary time series and how they correct towards equilibrium.
Stationarity: Stationarity refers to a property of a time series where its statistical characteristics, such as mean, variance, and autocorrelation, remain constant over time. This concept is crucial for many time series analysis techniques, as non-stationary data can lead to unreliable estimates and misleading inferences.
System Error Correction Model: A system error correction model is a statistical approach used to analyze the relationship between multiple time series that are cointegrated, focusing on the short-term dynamics while correcting for long-term equilibrium relationships. This model incorporates both the levels of the variables and their changes, allowing for a better understanding of how deviations from the long-term equilibrium can be adjusted over time. By using an error correction term, the model captures how quickly a system returns to equilibrium after a shock or disturbance.
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