VAR and Granger causality

Variables that predict each other

A vector autoregression, VAR, models several series together. Each variable is regressed on the past values of all the variables, so for two series $x$ and $y$ and one lag, $y_t=c+A\,y_{t-1}+u_t$ with $y_t=(x_t,y_t)^\top$ and a $2\times2$ matrix $A$. Every equation is fitted by ordinary least squares, and in statsmodels VAR(data).fit(p) does all of them at once. The model is stable when every eigenvalue of $A$ lies inside the unit circle, which means shocks fade out instead of exploding. Forecasts follow by repeated multiplication. To choose the lag order $p$ you compare information criteria such as AIC and BIC, which reward fit and penalise extra coefficients.

Granger causality asks a narrow question: do past values of $y$ help predict $x$ once the past values of $x$ itself are known? You fit the equation for $x$ twice, once with the lags of $y$ (unrestricted, residual sum of squares $\mathrm{RSS}_u$) and once without them (restricted, $\mathrm{RSS}_r$), and compare with an F-statistic. A significant result says $y$ leads $x$ in time. It does not prove that $y$ causes $x$: a third variable, or investors who anticipate, can produce the same pattern.

On real monthly data, $258$ months from March 2005, take the SPY return in percent and the 10-year minus 2-year Treasury spread. AIC picks $3$ lags and BIC picks $1$, and a VAR(2) is stable, with the closest root at $1.02$ because the spread is very persistent (its own lags add to $0.98$). The Granger test of whether the spread helps predict returns gives $F=0.22$ ($p=0.81$), and whether returns help predict the spread gives $F=1.88$ ($p=0.15$). Neither direction is significant: knowing the yield-curve slope adds nothing to a monthly forecast of SPY.

Lab · what's graded

• One step ahead10 pts
• Two steps ahead15 pts
• The largest eigenvalue15 pts
• A Granger F-statistic15 pts
• Counting coefficients15 pts
• Reading a Granger test10 pts
• Prediction is not causation20 pts