10.3 Cointegration and error correction models

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

• 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 long-run equilibrium 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 speed of adjustment 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: $y_t = \beta_0 + \beta_1 x_t + u_t$
  2. Test the residuals $u_t$ for stationarity using a unit root test like the Augmented Dickey-Fuller test
     • If the residuals are stationary, the series are cointegrated (income and expenditure)
• Johansen test: 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 short-run dynamics and long-run equilibrium relationships
• General form of an ECM for two cointegrated variables: $\Delta y_t = \alpha_0 + \alpha_1 \Delta x_t + \alpha_2 (y_{t-1} - \beta_0 - \beta_1 x_{t-1}) + \varepsilon_t$
  • $\Delta y_t$ and $\Delta x_t$ capture short-run dynamics (changes in stock prices)
  • $(y_{t-1} - \beta_0 - \beta_1 x_{t-1})$ is the error correction term, 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 ($\alpha_1$): 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 ($\beta_1$): Represent the long-run equilibrium relationship between the variables (long-run relationship between price and quantity demanded)
• Adjustment parameter ($\alpha_2$):
  • 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

• Forecasting 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)