AR, MA, VAR, Granger causality and cointegration

Let e_t be a purely random (white-noise) process. Three families of stationary models:

• Autoregressive, AR(p): the series depends on its own past, x_t = α₁x_t−1 + … + α_px_t−p + e_t. The current value carries the whole past, through the lags.
• Moving average, MA(q): the series is a weighted sum of the current and past shocks, x_t = β₀e_t + β₁e_t−1 + … + β_qe_t−q. A shock matters for q periods and then disappears. (Not to be confused with the moving average used to smooth data.)
• ARMA(p, q): both. A low-order ARMA often describes a series with far fewer parameters than a long AR or MA.

An AR(p) is stationary when the roots of its characteristic equation lie outside the unit circle (for AR(1): |φ| < 1; for AR(2): φ₁ + φ₂ < 1, φ₂ − φ₁ < 1 and |φ₂| < 1). A finite MA is always stationary. An ARIMA(p, d, q) model fits an ARMA(p, q) to a series after differencing it d times, which is how a unit-root series is modelled. The mean of x_t = c + φx_t−1 + e_t is c ÷ (1 − φ).

Which model fits a series? The autocorrelation function (ACF) plots the correlation of the series with itself at each lag. The partial autocorrelation function (PACF) plots the correlation at lag k after removing the effect of the shorter lags. Their shapes are the fingerprints:

Try each of the five in the lab and compare the bars with the table, then lengthen the series to sharpen the pattern. The dashed lines are approximate 95% bands (±1.96 ÷ √n): a bar outside is significantly different from zero, and one in twenty bars will be outside by chance.

The practical method, due to Box and Jenkins, has four steps: identify a tentative model from the ACF and PACF (after differencing if there is a unit root); estimate its parameters; diagnose by checking that the residuals are white noise, for example with the Ljung-Box statistic Q = n(n + 2) Σ r_k² ÷ (n − k), chi-square under the null of no autocorrelation up to the chosen lag; and forecast. When several models fit, an information criterion (AIC or BIC, lesson 4) chooses between them.

A vector autoregression (VAR) treats several series symmetrically: each variable is regressed on lags of itself and of all the others, with no variable assumed exogenous in advance. For two variables x and y, a VAR(1) is

Granger causality. x Granger-causes y if past values of x help predict y beyond what past values of y already give. In the VAR it is an F test that all the lags of x are jointly zero in the equation for y: if rejected, x Granger-causes y. Testing both ways: lags of Y significant in X’s equation, and lags of X insignificant in Y’s, means Y Granger-causes X and X does not Granger-cause Y (the opening question). Two-way significance is bidirectional causality (feedback), and none is no Granger causality.

The name is misleading. Granger causality is about predictive content and timing, not about causes. A variable that moves first because people anticipate something (stock prices before earnings, Christmas cards before Christmas) predicts without causing. It also depends on the lag length and on which other variables are in the model.

Real data. For India’s annual growth rates of private consumption and GDP (65 observations, 1961 to 2025), a one-lag Granger test of whether GDP growth helps predict consumption growth gives F = 0.05 (p = 0.82), and whether consumption growth helps predict GDP growth F = 0.6 (p = 0.44). Neither is significant: in annual data the two move together within the same year, and the previous year’s growth adds nothing. That is no statement about whether consumption drives output or the other way round.

Lesson 15 warned that regressing one I(1) series on another is spurious. There is one exception, and it is the most important idea in applied macroeconometrics. Two I(1) series are cointegrated if some linear combination of them is stationary: each wanders, but they wander together, and the gap between them keeps returning to a fixed level. Consumption and income, prices at home and abroad adjusted for the exchange rate, short and long interest rates: theory says such pairs are tied together in the long run. The test is a two-step recipe, due to Engle and Granger:

1. Check that both series are I(1) (the Dickey-Fuller test of lesson 15).
2. Regress one on the other in levels, y_t = a + βx_t + u_t, and test the residuals for a unit root. If they are stationary, the series are cointegrated, and β is the long-run relationship (the estimator is, remarkably, superconsistent).

The residual test uses its own critical values, more negative than Dickey-Fuller’s, because the regression has already chosen the line that makes the residuals look as stationary as it can. The lab computes them by simulation. Try the two settings: with a cointegrated pair the residual hugs zero and the statistic is far below the cutoff; with two unrelated random walks it wanders and the test rightly fails to reject.

Cointegration implies an error-correction model (ECM): the short-run changes of y respond to the last period’s deviation from the long-run relationship, which pulls y back towards it:

Real data. India’s log consumption on log GDP, 1960 to 2025 (World Bank, read 2026-10-24): the long-run slope is 0.88 and the Engle-Granger statistic of the residual is −1.21, against a simulated 5% critical value of −3.43 for independent walks of this length. We cannot reject no cointegration. The error-correction coefficient is −0.072 (t = −1.4), a small and insignificant speed of adjustment. This is a sober result: theory says consumption and income are tied together, but 65 annual observations are few for a test that has low power against a gap as persistent as this one (the residual’s autocorrelation is about 0.94, lesson 10), and nothing here shows the pair is cointegrated. The honest summary is “no evidence of cointegration in these data”, not “proof that they are unrelated”.

The questions matching tests with their purposes (Ljung-Box, Engle-Granger, Granger) are collected in the test atlas at the end of the module. For the unit-root tests that come before cointegration, see lesson 15.