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
Estimate the long-run equilibrium relationship using OLS regression: yt=β0+β1xt+ut
Test the residuals ut 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
Δyt and Δxt capture short-run dynamics (changes in stock prices)
(yt−1−β0−β1xt−1) is the , representing the deviation from long-run equilibrium (spread between a stock and its futures contract)
Estimating an ECM:
Estimate the long-run equilibrium relationship using OLS regression
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): 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): Represent the long-run equilibrium relationship between the variables (long-run relationship between price and quantity demanded)
Adjustment parameter (α2):
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)