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L37 Error Correction Models

ECM

It is a type of TS model used to analyze the relationship between two or more non-stationary variables that share a stable, long-term equilibrium relationship (alt, cointegration exists).

Why ECM?

  • If two variables have a shared equilibrium, any deviations are temporary. The rise or fall will revert back to the mean (long-run equilibrium).
  • ECM has the capability of capturing the
    • short-term effects: immediate impact of changes in the related variables
    • long-term correction: adjusts for deviations from the equilibrium. Pulling variables back in line

Key Components

  1. Error Correction Term (ECT): Measures the gap between current value and long-term equilibrium.
    • Larger the ECT, larger the adjustment the model will make in the next period
  2. Short-term coefficients: without needing to consider equilibrium relationship, show immediate effect of changes in one variables on other.

Example

  • Inflation and Interest rates
  • Model short-term changes in interest rates based on recent changes
  • "Error correction process": high interest rate could slowly adjust downward if inflation is low

ECM Equation Structure

\[ \nabla Y_{t} = \alpha(\beta X_{t-1} - Y_{t-1}) + \gamma \nabla X_{t} + \epsilon_{t} \]
  • \(\beta X_{t-1} - Y_{t-1}\) is the ECT = deviation from long-term equilibrium at time \(t-1\)
    • \(\beta\) = "long term impact"
    • Degree of disequilibrium. High \(\implies\) Far from Equilibrium, requiring larger adjustments
  • \(\alpha\) = speed of adjustment coefficient (how fast \(Y_{t}\) in response to deviations in the equilibrium)
    • \(\alpha\) = "how slow or how quick"
    • Faster adjustment, vs slow correction process

  • \(\gamma\) = short term coefficient

    • \(\gamma\) = "short term impact"
    • independent of long-term relationship

  • If ECT is significant \(\implies\) \(Y_{t}\) adjusts to bring back equilibrium

  • changes in \(X\) have immediate effects on \(Y\) in the short run

Steps to Fit an ECM

  1. Test for Non-stationary: ADF → Ensure \(I(1)\)
  2. Test for Cointegration: Johansen cointegration → If yes, proceed with ECM
  3. Estimate ECM using cointegration equation (usually OLS)
    • Error correction term from cointegration equation.

Advantages & Disadvantages of ECM

  • Advantages
    • combine short- and long-term dynamics, more accurate models for cointegrated TS
    • effective in systems where relationships drift but maintain an equilibrium over time (SITUATION, common in economic in financial data)
  • Limitations
    • Requires cointegration (unsuitable otherwise)
    • If model is mis-specified, erroneous results follow

Practical Examples

  • Stock prices and dividends:
    • dividend-based valuations (long-term)
    • prices might diverge in the short term
  • (nominal) Interest rates and Inflation (Fisher effect)
    • central banks and investors
  • Oil Prices and Exchange Rates
    • Oil prices deviate from currency values
  • Electricity demand and temperature
    • Electricity demand has Short-term spikes due to extreme weather
    • returns to average level