Granger Causality

Granger Causality tests whether past values of one time series improve prediction of another series beyond what the target’s own past explains, typically via VAR or regression models. It indicates predictive directionality, not true causal mechanism, and depends on model specification and stationarity. In AI/ML, it helps select lagged features and infer temporal influence, e.g., whether search trends help forecast product demand.

Imagine you’re trying to predict tomorrow’s ice cream sales. If knowing today’s temperature helps you make a better prediction than using past sales alone, you might say temperature “drives” sales. Granger causality is a similar idea for data that changes over time: one time series “Granger-causes” another if its past values improve predictions of the other series. It doesn’t prove true cause-and-effect like a controlled experiment; it’s about better forecasting using history. In AI and machine learning, it’s often used to test whether one signal contains useful predictive information about another.

import numpy as np  # numerical arrays and random number generation
import pandas as pd  # time-indexed tabular data
from statsmodels.tsa.stattools import grangercausalitytests  # Granger causality test routine

# --- Create a realistic "marketing spend -> sales" monthly dataset ---
rng = np.random.default_rng(7)  # reproducible randomness
n = 120  # 10 years of monthly observations
idx = pd.date_range("2015-01-01", periods=n, freq="MS")  # monthly start dates

marketing = np.zeros(n)  # marketing spend time series
sales = np.zeros(n)      # sales time series

for t in range(1, n):
    marketing[t] = 0.7 * marketing[t - 1] + rng.normal(0, 1.0)  # persistent spend with noise
    sales[t] = 0.6 * sales[t - 1] + 0.5 * marketing[t - 1] + rng.normal(0, 1.0)  # lagged marketing impacts sales

df = pd.DataFrame({"sales": sales, "marketing": marketing}, index=idx)  # align both series on the same time axis

# --- Test whether past marketing helps predict sales (beyond past sales itself) ---
max_lag = 6  # test up to 6 months of lagged effects
# *** KEY LINE ***
res = grangercausalitytests(df[["sales", "marketing"]], maxlag=max_lag, verbose=False)  # tests "marketing -> sales"

# Extract p-values (ssr F-test) across lags into a compact table for inspection
pvals = {lag: out[0]["ssr_ftest"][1] for lag, out in res.items()}  # p-value is element [1]
pvals = pd.Series(pvals, name="p_value").to_frame().assign(lag=lambda x: x.index)
print(pvals.sort_index())  # smaller p-values at some lags indicate evidence of Granger-causality

# Optional: reverse direction to check "sales -> marketing" as a sanity check
# *** KEY LINE ***
res_rev = grangercausalitytests(df[["marketing", "sales"]], maxlag=max_lag, verbose=False)  # tests "sales -> marketing"
pvals_rev = {lag: out[0]["ssr_ftest"][1] for lag, out in res_rev.items()}
print(pd.Series(pvals_rev, name="p_value").sort_index())

• The key output is the lag-by-lag p-value table extracted from statsmodels.tsa.stattools.grangercausalitytests; small p-values indicate that adding lagged values of the second series improves prediction of the first.
• maxlag controls how many past periods are tested; in ML workflows it’s often chosen via domain knowledge (e.g., business cycles) or validated with a holdout period.
• Common mistake: passing columns in the wrong order—grangercausalitytests expects [target, candidate_cause] in the DataFrame slice.
• Typical use in AI/ML: feature engineering for forecasting (include lagged drivers) and directional screening before fitting multivariate models (e.g., VAR or dynamic regression).

library(lmtest)  # provides grangertest() for Granger causality testing

# --- Create a realistic "marketing spend -> sales" monthly dataset ---
set.seed(7)  # reproducibility
n <- 120  # 10 years of monthly observations
dates <- seq.Date(from = as.Date("2015-01-01"), by = "month", length.out = n)  # monthly index

marketing <- numeric(n)  # marketing spend time series
sales <- numeric(n)      # sales time series

for (t in 2:n) {
  marketing[t] <- 0.7 * marketing[t - 1] + rnorm(1, 0, 1.0)  # persistent spend with noise
  sales[t] <- 0.6 * sales[t - 1] + 0.5 * marketing[t - 1] + rnorm(1, 0, 1.0)  # lagged marketing impacts sales
}

df <- data.frame(date = dates, sales = sales, marketing = marketing)  # keep series aligned by date

# --- Test whether past marketing helps predict sales (beyond past sales itself) ---
lag_k <- 6  # test a 6-month lag structure
# *** KEY LINE ***
gc_ms <- grangertest(sales ~ marketing, order = lag_k, data = df)  # tests "marketing -> sales" at lag_k
print(gc_ms)  # inspect the F-test and p-value for the chosen lag order

# Optional: reverse direction to check "sales -> marketing" as a sanity check
# *** KEY LINE ***
gc_sm <- grangertest(marketing ~ sales, order = lag_k, data = df)  # tests "sales -> marketing" at lag_k
print(gc_sm)

• The printed result from lmtest::grangertest includes an F-test and p-value for the specified lag order; it summarizes whether lagged predictors add explanatory power.
• order sets the number of lags included; unlike the Python example that reports multiple lags, this call tests a single chosen lag length.
• Common mistake: interpreting significance as true causation; in practice, you still need to check stationarity, lag selection, and potential confounders before using the result operationally.
• Typical use in AI/ML: decide whether to include lagged exogenous drivers in forecasting pipelines (e.g., dynamic regression / ARIMAX-style features).