Lagged Variables and Granger Causality

In time series analysis, a lagged variable refers to a value of a variable from a previous time period that is used to predict or explain the current value. By shifting past observations forward, analysts can identify patterns, trends, and relationships over time, which can enhance forecasting accuracy.

In the notation Y_{t-1}, the subscript "t-1" signifies that the value of Y is being referenced from one time period prior to the current time t. This indicates a lagged value, commonly used in time series analysis to assess how past values influence the present.

Granger causality assesses the predictive power of one time series over another by determining if past values of X provide additional information for forecasting Y, beyond what is available from Y's own past. If X's past values significantly improve predictions of Y, then X is said to Granger-cause Y.

Including lagged dependent variables in a regression model helps to account for autocorrelation, where past values influence current outcomes. This approach captures dynamic relationships over time, allowing the model to better reflect the underlying processes and improve predictive accuracy by incorporating historical data into the analysis.

This model is called an Autoregressive Distributed Lag (ARDL) model because it incorporates both autoregressive components (Y_{t-1}) and distributed lag of explanatory variables (X_t). ARDL is used for analyzing the relationship between a dependent variable and its lagged values alongside other independent variables over time, making it suitable for various time series data.

In Granger causality testing, the null hypothesis posits that past values of variable X do not provide any information about future values of variable Y. This means that any observed correlation between X and Y is not indicative of a causal relationship, allowing researchers to test whether X truly influences Y over time.

Akaike Information Criterion (AIC) and Schwarz Criterion are statistical tools used to determine the optimal number of lags in a VAR model. They balance the model's goodness of fit against its complexity, helping to avoid overfitting while ensuring that the model captures essential dynamics in the data.

Granger causality indicates a predictive relationship between variables, but it does not confirm true causation. Just because one variable can predict another does not imply that changes in the first variable cause changes in the second. Economic causality requires deeper analysis of underlying mechanisms and factors, which Granger causality alone cannot provide.

In a bivariate VAR(1) model, Granger causality testing involves estimating the impact of each variable on the other. This requires two equations: one that regresses X on past values of X and Y, and another that regresses Y on past values of X and Y. Hence, two equations are estimated.

A statistically significant lag of X in the Y equation indicates that past values of X can help predict future values of Y. This relationship implies that changes in X precede and influence changes in Y, thus demonstrating that X Granger-causes Y.

For Granger causality tests to be valid, the time series data must be stationary, which is indicated by being integrated of order zero (I(0)). Alternatively, if the series are non-stationary, they must be cointegrated, meaning they share a long-term equilibrium relationship. This ensures that any causal inference made is reliable and meaningful.

In a distributed lag model, the long-run multiplier reflects the total impact of a change in an explanatory variable over time. It is calculated by summing the coefficients of the lagged values of that variable, as these coefficients represent the effect of past values on the current outcome, thus determining the overall long-term effect.