Granger Causality

Granger causality is a statistical notion of causation based on prediction. A time series X is said to Granger-cause another series Y if past values of X contain information that helps to forecast Y, over and above the information already contained in the past values of Y itself. The idea rests on a simple principle: a cause must come before its effect.

The concept was introduced by the economist Clive Granger in a 1969 paper, and he later shared the Nobel Memorial Prize in Economic Sciences. It is widely used in economics and finance to study lead-lag relationships: whether money growth leads inflation, whether futures prices lead cash prices, or whether a sentiment measure leads returns. Despite its name, it establishes predictive precedence only and not causation in the everyday sense.

The standard test compares two regressions. The restricted model explains Y by its own lags. The unrestricted model adds lags of X:

The null hypothesis is that all the coefficients $b_j$ are zero, meaning X does not Granger-cause Y. It is tested with an F-statistic:

• $RSS_r$ is the residual sum of squares of the restricted model, without lags of X.
• $RSS_u$ is the residual sum of squares of the unrestricted model.
• $q$ is the number of lags of X being tested.
• $n$ is the number of observations and $k$ the number of parameters in the unrestricted model.

A large F means that adding the history of X reduced the forecast errors by more than chance would explain. The test can be run in both directions. X may lead Y, Y may lead X, each may lead the other, which is called feedback, or neither.

An analyst asks whether daily changes in an index future help predict daily returns on a basket of small company shares. With 200 observations and two lags of each variable, the unrestricted model has five parameters. The restricted regression has a residual sum of squares of 110, and the unrestricted one has 100.

The statistic is F = ((110 − 100) ÷ 2) ÷ (100 ÷ 195) = 5 ÷ 0.513 = 9.75. The 5 percent critical value for an F distribution with 2 and 195 degrees of freedom is about 3.04. Since 9.75 is well above it, the null hypothesis is rejected: in this sample, past futures returns Granger-cause the small company returns. The corresponding p-value would be far below 0.05.

The series must satisfy stationarity. Applying the test to trending or random-walk series, such as price levels, gives unreliable results, so prices are usually converted to returns or differences first, after a unit root check such as the augmented Dickey-Fuller test. Where two non-stationary series share a long-run relationship, known as cointegration, the test is carried out within an error correction model, and Granger showed that causality must then run in at least one direction.

The number of lags has to be chosen, usually by an information criterion, and the outcome can change with that choice. The test is normally run inside a vector autoregression, a system in which each variable is regressed on lags of all the variables, an extension of autoregression to several series.

Lead-lag effects are a long-standing subject of market research. Studies have found that index futures tend to lead the underlying cash index by minutes, because futures are cheaper and faster to trade when news arrives. Returns on large company shares have been found to lead those on small ones, as information reaches thinly traded stocks more slowly. Option markets, credit default swaps and bond yields have each been tested as leading indicators for share prices.

Researchers also test whether text-based measures from sentiment analysis, search volumes or order flow Granger-cause returns or volatility. Central banks and economists apply the test to macroeconomic series as part of time series forecasting. For a trader, a finding that X Granger-causes Y suggests that X could serve as a signal for Y, subject to the cautions below.

• It is not true causation. If a third factor drives both series and affects X sooner, X will appear to cause Y. Forward-looking behavior can even reverse the apparent direction: share prices fall before a recession because investors anticipate it, not because they cause it.
• It is linear and backward-looking. The standard test detects only linear effects of past values. It misses nonlinear links and effects that occur within the same period.
• Data problems create false leads. If one asset trades less often than another, its recorded price is stale, and the more active asset will seem to lead it even when there is no real delay. Time stamps recorded at different moments have the same effect.
• Searches produce chance findings. Testing many pairs of series and many lag lengths guarantees some significant results, and standard hypothesis testing thresholds then overstate the evidence.

A statistically significant lead also need not be profitable. The effect may be smaller than trading costs, may last for less time than it takes to act, and may fade once it is widely known. Relationships found in one period often weaken in the next, so results need confirmation on later data.