Vector Autoregression (VAR)

Most real systems are not one lonely series drifting on its own; they are several series pushing on each other. Interest rates move inflation, inflation moves rates back. A stock's return and its trading volume shove each other around. A vector autoregression, or VAR, is the simplest honest way to model that back-and-forth: you let every variable depend on the recent history of itself and of all the others, and you estimate the whole tangle at once.

Think of it as stacking several AR models side by side and wiring them together. An ordinary AR(1) says "today is a multiple of yesterday plus noise." A VAR(1) says the same, but "today" is now a list of numbers, and each entry can lean on yesterday's value of every other entry.

The model

Collect your $k$ series into a single vector $Y_t$. A VAR with $p$ lags is

$$Y_t = c + A_1 Y_{t-1} + A_2 Y_{t-2} + \dots + A_p Y_{t-p} + \varepsilon_t .$$

Here $Y_t$ is the vector of all variables at time $t$; $c$ is a vector of constants; each $A_i$ is a $k \times k$ matrix of coefficients that says how strongly lag $i$ of every variable feeds into every variable today; and $\varepsilon_t$ is a vector of shocks (mean zero, no correlation across time, but the shocks at the same instant may be correlated with each other). The off-diagonal entries of the $A_i$ matrices are the whole point: they are the cross-effects, one series reaching into another.

Worked example: returns and volume

Suppose two demeaned daily series, returns $r_t$ and (standardized) volume $v_t$, follow a VAR(1):

$$\begin{aligned} r_t &= 0.10\, r_{t-1} - 0.05\, v_{t-1} + \varepsilon^r_t,\\ v_t &= 0.30\, r_{t-1} + 0.60\, v_{t-1} + \varepsilon^v_t. \end{aligned}$$

Yesterday's numbers were $r_{t-1} = 2.0$ (a big up day) and $v_{t-1} = 1.0$. The one-step-ahead forecasts are just the equations with the noise set to its average of zero:

$$\hat r_t = 0.10(2.0) - 0.05(1.0) = 0.15, \qquad \hat v_t = 0.30(2.0) + 0.60(1.0) = 1.20 .$$

Read the second equation: a big return day pushes tomorrow's volume up hard (coefficient $0.30$), and volume is sticky on its own ($0.60$). That cross-effect, return leading volume, is exactly what a separate AR model of volume would have thrown away.

Reading a VAR

You rarely stare at the raw coefficient matrices; they are hard to interpret. Three tools translate a VAR into answers:

• Granger causality asks whether one variable's past helps predict another beyond that other's own past, i.e. whether a block of off-diagonal coefficients is jointly zero.
• Impulse response functions trace what happens to every series over the next weeks if one shock hits today; the natural "what if" of a VAR.
• Forecast error variance decomposition splits each series' forecast uncertainty into how much comes from its own shocks versus the others'.

Where it goes wrong

• Parameter explosion. A $k$-variable VAR($p$) has $k^2 p$ slope coefficients. Three variables at four lags is already 36 numbers; add a fourth variable and you are fitting 64. Data runs out fast, so keep $k$ and $p$ small and lean on information criteria (AIC/BIC) to pick the lag length.
• Stationarity first. The math assumes the vector is stationary. Fitting a VAR to trending, unit-root series can produce nonsense, the multivariate cousin of spurious regression. If the series move together in the long run, they may be cointegrated, and you should use a vector error-correction model instead of differencing everything away.
• Ordering matters for structure. Turning a VAR into impulse responses requires assumptions about which shock hits which variable first; change the ordering and the story can change. That is the hard, contested part of the tool.

VAR is the workhorse for interacting time series, the multivariate step up from ARMA, the home of Granger causality, and, once you allow hidden drivers, a stepping stone to state-space models.