Let g Mt be the annual growth in the money supply and let...

Let $g M_{t}$ be the annual growth in the money supply and let unem, be the unemployment rate. Assuming that unem_ follows a stable AR(1) process, explain in detail how you would test whether $g M$ Granger causes unem.

Key Concepts

Stationarity

Stationarity is a key requirement in time series analysis, meaning that a time series’ statistical properties such as mean, variance, and autocorrelation structure remain constant over time. Testing for stationarity ensures that the relationships inferred from Granger causality tests are valid and not driven by spurious trends or non-constant variance.

Hypothesis Testing in Time Series

In the context of Granger causality, hypothesis testing involves assessing whether the coefficients on the lagged values of the predictor variable are jointly statistically different from zero. This is often done using an F-test or a Wald test to evaluate the null hypothesis that past values of the predictor variable do not have predictive power for the target variable.

Lag Selection

Lag selection refers to the process of determining the appropriate number of past periods (lags) to include in a time series model. Adequate lag selection is crucial in Granger causality tests to ensure that the model captures the dynamics of the series without overfitting or underfitting, typically using information criteria such as AIC or BIC.

Autoregressive (AR) Process

An autoregressive process is a time series model in which the current value of a variable is expressed as a function of its previous values (lags) and a stochastic error term. The AR(1) process, specifically, indicates that only the immediately preceding value (one lag) is used in modeling the current outcome.

Granger Causality

Granger causality is a statistical concept used to determine whether past values of one time series contain information that helps predict another time series beyond what is contained in its own past values. It involves testing the null hypothesis that lagged values of one variable do not provide statistically significant information about future values of another variable.

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