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Lagged variables
Values of a time series observed at previous periods, used as explanatory variables in Granger causality models. The optimal number of lags is crucial and determined by information criteria such as AIC or BIC.
Vector autoregression
Econometric model where each variable is explained by its own past values and the past values of other variables in the system. VAR models constitute the reference framework for Granger causality analysis between multiple time series.
Engle-Granger test
Two-step procedure for testing cointegration between two time series, first estimating the long-term relationship then testing the stationarity of the residuals. This test is fundamental for determining whether long-term causality exists between variables.
Impulse response function
Graphical representation of the effect of a shock to one variable on the future evolution of all variables in the VAR system. Impulse response functions allow analysis of the direction and intensity of causal relationships over time.
Variance decomposition
Analysis that determines the proportion of the forecast error variance of one variable explained by shocks to itself and to other variables in the system. This technique quantifies the relative importance of causal relationships between variables.
Akaike information criterion
Statistical metric used to select the optimal number of lags in a VAR model or Granger causality test. AIC balances model fit and parsimony, penalizing the addition of unnecessary variables.
Bayesian information criterion
Model selection criterion similar to AIC but with stricter penalty for complexity, tending to favor more parsimonious models. BIC is particularly useful in large samples to avoid overfitting.
Bivariate causality test
Granger causality analysis involving only two time series, where each variable is regressed on its own lags and the lags of the other variable. This test can be biased if omitted variables influence the causal relationship.
Toda-Yamamoto Test
A robust causality testing procedure that is invariant to the integration order of time series, eliminating the need for pre-tests of stationarity or cointegration. This method uses an augmented VAR including the maximum order of integration of the series plus one additional lag.
Error Correction Model
Extension of the VAR model for cointegrated series, incorporating an error correction term to represent adjustment towards long-term equilibrium. The VECM allows distinguishing between short-term causality and long-term causality among cointegrated variables.
Non-causality Hypothesis
Null hypothesis in Granger causality tests, stating that past values of a variable X do not add any significant predictive information to the forecast of a variable Y. Rejection of this hypothesis suggests the existence of a causal relationship from X to Y.
First Difference Regression
Transformation of non-stationary time series into first differences to achieve stationarity before applying causality tests. This approach eliminates deterministic trends but may lose information about long-term relationships between variables.
Instantaneous Causality Test
Test verifying whether contemporaneous variations of one variable are correlated with those of another variable, beyond lagged effects. Instantaneous causality suggests the existence of omitted variables or simultaneity in the relationships between variables.