Description

- **Core Topic:** The chapter extends univariate time series concepts to the multivariate domain, focusing on Vector Autoregressive (VAR) and Vector Autoregressive Moving Average (VARMA) models.
- **White Noise Diagnostics:** It begins by simulating multivariate white noise and demonstrating that its autocorrelation, cross-correlation, and portmanteau test p-values behave as expected under the null hypothesis.
- **VARMA Properties:** A bivariate VARMA(1,1) model is analyzed to check for causality and invertibility by examining the determinants of its polynomial matrices.
- **VAR(2) Deep Dive:** A detailed case study of a 3-dimensional VAR(2) process covers stability checking via companion matrix eigenvalues, mean vector calculation, and VMA(∞) representation.
- **Granger Causality:** The chapter formally introduces Granger-causality tests, using the VAR(2) example to show how to test directional predictive relationships between variables.
- **Impulse Response:** Impulse response analysis is performed on the VAR(2) model to trace the effect of a one-time shock to one variable on the entire system over time.
- **Model Selection (Macro Data):** Using US macroeconomic data (GDP, consumption, investment), VAR(1) and VAR(2) models are compared, with portmanteau tests showing VAR(1) residuals are autocorrelated while VAR(2) is adequate.
- **Granger Network:** The US macro analysis reveals a Granger-causality network where GDP is the primary driver, with investment responding strongly to GDP shocks (accelerator effect).
- **Refinement by Restriction:** For UK-Canada-US GDP data and a GDP-unemployment system, overly complex VAR models are refined by setting highly insignificant coefficients to zero, improving parsimony while maintaining diagnostic adequacy.
- **Key Takeaway:** Proper multivariate modeling requires iterative steps: identification, estimation, diagnostic checking (portmanteau tests), and often refinement through zero restrictions to achieve a well-specified and interpretable model.