Granger Causality

"Does money supply cause inflation?" is a question about mechanism, and mechanism is hard. Clive Granger asked a humbler, sharper question that a computer can actually answer: does knowing the past of x help me forecast y better than y's own past alone? If yes, we say x Granger-causes y. It is an operational, predictive notion — all about forecasting, and deliberately silent about true cause. That modest reframing was influential enough to earn Granger a share of the 2003 Nobel Prize in economics.

The definition: does the past of x add forecasting power?

Compare two forecasts of y_t. The first uses only the history of y itself; the second uses the history of both y and x.

x Granger-causes y if the past of x improves the forecast of y beyond what y's own past achieves — i.e. the forecast error variance strictly falls when lagged x is added.
It is a statement about predictability, not physical causation.
Feedback is allowed: x may Granger-cause y and y Granger-cause x at the same time (mutual Granger causality).

The natural home for the test is the vector autoregression. In the equation for y_t, the past of x shows up as the coefficients on x_{t-1}, \dots, x_{t-p}. Granger non-causality is simply the claim that all of those coefficients are zero — and that is an ordinary hypothesis you can test.

Predictability made visible

In the plot below, series x turns first and series y echoes each turn a few steps later. Stand at any time knowing only y's own past and you are guessing; add the recent path of x and the next moves of y become far easier to call. That extra forecasting power — nothing more, nothing less — is what a Granger test detects.

Worked example: the restricted-vs-unrestricted F-test

The test is a textbook application of hypothesis testing. Fit two nested regressions for y_t using p lags:

\begin{aligned} \text{unrestricted:}\quad & y_t = c + \sum_{i=1}^{p} a_i\, y_{t-i} + \sum_{j=1}^{p} b_j\, x_{t-j} + \varepsilon_t, \\ \text{restricted:}\quad & y_t = c + \sum_{i=1}^{p} a_i\, y_{t-i} + u_t. \end{aligned}

The null hypothesis is Granger non-causality, H_0: b_1 = b_2 = \dots = b_p = 0 — the p lags of x are jointly useless. Let the two residual sums of squares be \mathrm{RSS}_r (restricted) and \mathrm{RSS}_u (unrestricted). The F-statistic is

F = \frac{(\mathrm{RSS}_r - \mathrm{RSS}_u)/p}{\mathrm{RSS}_u/(n - 2p - 1)} \;\sim\; F_{p,\; n-2p-1} \text{ under } H_0.

Put in numbers: p = 2 lags, n = 100 observations, \mathrm{RSS}_r = 120, \mathrm{RSS}_u = 100. Then

F = \frac{(120 - 100)/2}{100/(100 - 5)} = \frac{10}{1.053} \approx 9.5.

The 5% critical value of F_{2,\,95} is about 3.1. Since 9.5 > 3.1 we reject H_0: adding the lags of x cut the residual variance by far more than chance would, so x Granger-causes y. Testing the other direction just swaps the roles of x and y; find both significant and you have feedback.

The word "causality" oversells what the test delivers. Granger causality is predictive precedence in a particular information set — and any of three things can make it lie:

A common driver. If a hidden third variable z drives both series but reaches x first, then x will Granger-cause y despite having no effect on it whatsoever — the rooster "Granger-causes" the sunrise.
An omitted variable. Leave the true driver out of the model and the remaining series soak up its predictive content, manufacturing spurious Granger causality. Enlarge the information set and it can vanish or even reverse.
Anticipation. A forward-looking series can Granger-cause its own cause: because markets price in the future, stock prices Granger-cause future dividends — clearly not because prices make the dividends.

So read a positive result as "x carries usable advance information about y, given what else is in the model" — never as proof that wiggling x would move y.

Yes — and it is common. Granger causality is not a one-way street: two series can each improve the forecast of the other, a situation called feedback or bidirectional Granger causality. Prices and trading volume, or two competing firms' advertising spends, often show it. Feedback is exactly why the VAR framework is the right setting: you test the x-lags in the y equation and, separately, the y-lags in the x equation, and any of the four patterns — neither, one way, the other way, or both — can come out. The subject Granger built his test to referee was the endless macroeconomic quarrel over whether money "causes" output; the honest answer his method gives is usually "each helps predict the other," which is more accurate than either side wanted to hear.