Reduced-Form VAR Specification and Interpretation

Why Specify a VAR?

When several time series may influence one another over time, modeling each series separately can obscure their joint dynamics. A vector autoregression, or VAR, represents each series as a function of lags of every series in the system. The specification makes the timing and direction of the modeled lagged associations explicit: a past value of one variable can help describe the current conditional mean of another.

The focus here is the reduced-form VAR: its equations contain lagged values of the system’s variables, but not contemporaneous values of the other variables on the right-hand side. Its coefficients describe conditional associations, not automatically causal effects. The lag order $p$ is treated as given; choosing it and estimating the system are separate tasks.

Definition and Assumptions

Definition: A reduced-form VAR($p$) for a $k$-dimensional vector $y_t$ is $$ y_t=c+A_1y_{t-1}+A_2y_{t-2}+\cdots+A_py_{t-p}+u_t. $$ Here $y_t$ is a $k\times1$ vector of jointly modeled series, $c$ is a $k\times1$ intercept vector, each $A_j$ is a $k\times k$ lag coefficient matrix, and $u_t$ is a $k\times1$ vector of reduced-form innovations.

The equation is a specification of conditional means when the innovations have mean zero given the available past. Let $\mathcal{F}_{t-1}$ denote information from the modeled series through time $t-1$. A standard innovation assumption is $E(u_t\mid\mathcal{F}_{t-1})=0$, with a time-invariant covariance matrix $E(u_tu_t')=\Sigma$. Thus the innovations are serially uncorrelated under this setup, although components of $u_t$ at the same time may be correlated. Normality is not required for the basic specification or coefficient interpretations below.

For a baseline model, suppose the variables and lag order have been defined and the system includes a constant but no additional deterministic or exogenous regressors. These choices are part of specifying the model; the constant-only setup is not appropriate for every application. The notation $A_j[r,s]$ will mean the entry in row $r$, column $s$ of $A_j$.

Reading the Lag Matrices

Rows correspond to the current outcome equations; columns correspond to the lagged variables used as predictors. In particular, $A_j[r,s]$ multiplies the value of series $s$ at time $t-j$ in the equation for series $r$ at time $t$. A diagonal entry is an own-lag coefficient: it connects a series to its own past. An off-diagonal entry is a cross-lag coefficient: it connects one series’ past to another series’ current conditional mean.

More precisely, holding all other included lagged values fixed, increasing $y_{s,t-j}$ by one unit changes the conditional mean of $y_{r,t}$ by $A_j[r,s]$ units. This is a statement about the model’s conditional mean, not about a change in the realized outcome: the innovation $u_t$ can make the realized value differ from that mean. Nor does the coefficient alone establish a causal effect; unmodeled factors, measurement choices, and the assumptions behind a causal design still matter.

Worked Example: Separating Own-Lag and Cross-Lag Effects

Suppose $y_t=(x_t,z_t)'$ follows the hypothetical VAR(1) $$ \begin{pmatrix}x_t\\z_t\end{pmatrix} = \begin{pmatrix}1\\-2\end{pmatrix} + \begin{pmatrix}0.6&0.25\\-0.10&0.5\end{pmatrix} \begin{pmatrix}x_{t-1}\\z_{t-1}\end{pmatrix} +u_t. $$ Assume $x_t$ is measured in index points and $z_t$ in percentage points. Interpret the four lag coefficients.

Expanding the rows gives $$ x_t=1+0.6x_{t-1}+0.25z_{t-1}+u_{x,t}, \qquad z_t=-2-0.10x_{t-1}+0.5z_{t-1}+u_{z,t}. $$

In the $x_t$ equation, $0.6$ is an own-lag coefficient: holding $z_{t-1}$ fixed, a one-index-point increase in $x_{t-1}$ is associated with a $0.6$-index-point increase in the conditional mean of $x_t$. The coefficient $0.25$ is a cross-lag coefficient: a one-percentage-point increase in $z_{t-1}$, holding $x_{t-1}$ fixed, is associated with a $0.25$-index-point increase in the conditional mean of $x_t$.

In the $z_t$ equation, $-0.10$ is a cross-lag coefficient: a one-index-point increase in $x_{t-1}$, holding $z_{t-1}$ fixed, corresponds to a $0.10$-percentage-point decrease in the conditional mean of $z_t$. The $0.5$ coefficient is an own-lag association: a one-percentage-point increase in $z_{t-1}$, holding $x_{t-1}$ fixed, corresponds to a $0.5$-percentage-point increase in the conditional mean of $z_t$. None of these interpretations is a causal claim.

Justifying the Conditional-Mean Interpretation

The coefficient interpretation follows directly by conditioning on the lagged information and using the innovation assumption. Since the lagged vectors are known given $\mathcal{F}_{t-1}$, they can be treated as fixed when taking the conditional expectation:

\begin{aligned} E(y_t\mid\mathcal{F}_{t-1}) &=E\left(c+\sum_{j=1}^{p}A_jy_{t-j}+u_t\mid\mathcal{F}_{t-1}\right)\\ &=c+\sum_{j=1}^{p}A_jy_{t-j} +E(u_t\mid\mathcal{F}_{t-1})\\ &=c+\sum_{j=1}^{p}A_jy_{t-j}. \end{aligned}

For a particular entry $A_j[r,s]$, increase only $y_{s,t-j}$ by $\delta$, holding the other lagged values fixed. The conditional mean vector changes by $A_j e_s\delta$, where $e_s$ is the vector with a one in position $s$ and zeros elsewhere. Its $r$th component changes by $A_j[r,s]\delta$. This verifies both the units-based interpretation and the need to hold the other included lagged values fixed.

Computing a Conditional Mean

For specified lag values, the conditional mean is obtained by multiplying each lag vector by its coefficient matrix and adding the intercept. This calculation does not require observing the current innovation; it gives the model’s expected current vector conditional on the stated past.

Worked Example: A VAR(2) Conditional Mean

Consider the hypothetical VAR(2) $$ y_t= \begin{pmatrix}1\\-2\end{pmatrix} + \begin{pmatrix}0.5&0.2\\-0.1&0.6\end{pmatrix}y_{t-1} + \begin{pmatrix}-0.1&0\\0.3&0.1\end{pmatrix}y_{t-2} +u_t. $$ Suppose $y_{t-1}=(4,10)'$ and $y_{t-2}=(2,8)'$, where the first series is in index points and the second is in percentage points. Assume the innovation has conditional mean zero.

The first lag contribution is $$ \begin{pmatrix}0.5&0.2\\-0.1&0.6\end{pmatrix} \begin{pmatrix}4\\10\end{pmatrix} = \begin{pmatrix}0.5(4)+0.2(10)\\-0.1(4)+0.6(10)\end{pmatrix} = \begin{pmatrix}4\\5.6\end{pmatrix}. $$ The second lag contribution is $$ \begin{pmatrix}-0.1&0\\0.3&0.1\end{pmatrix} \begin{pmatrix}2\\8\end{pmatrix} = \begin{pmatrix}-0.1(2)+0(8)\\0.3(2)+0.1(8)\end{pmatrix} = \begin{pmatrix}-0.2\\1.4\end{pmatrix}. $$

Adding the intercept gives the conditional mean: $$ E(y_t\mid\mathcal{F}_{t-1}) = \begin{pmatrix}1\\-2\end{pmatrix} + \begin{pmatrix}4\\5.6\end{pmatrix} + \begin{pmatrix}-0.2\\1.4\end{pmatrix} = \begin{pmatrix}4.8\\5.0\end{pmatrix}. $$ Thus the model’s conditional mean is 4.8 index points for the first series and 5.0 percentage points for the second. A realized $y_t$ need not equal this vector; its difference from the conditional mean is $u_t$.

What the Intercept and Innovation Covariance Mean

The intercept $c_r$ is the contribution to the conditional mean in equation $r$ when every included lagged variable equals zero. That may be a useful reference point, but zero can be outside the relevant range of a series, so the intercept need not have a direct substantive interpretation. It is also not generally the unconditional mean. If a finite, time-invariant mean $\mu=E(y_t)$ exists, taking expectations in the VAR gives $$ \mu=c+\left(\sum_{j=1}^{p}A_j\right)\mu. $$ When $I-\sum_{j=1}^{p}A_j$ is invertible, this implies $\mu=(I-\sum_{j=1}^{p}A_j)^{-1}c$. This identity is conditional on the existence of that mean; it is not a claim that the system has a stable or stationary solution.

The covariance matrix $\Sigma$ describes the contemporaneous covariance of the reduced-form innovations. Its off-diagonal entries can be nonzero: shocks to different equations may move together in the same period. This covariance is distinct from the lag coefficient matrices. The VAR contains no contemporaneous $y_t$ terms on the right-hand side, so neither the lag coefficients nor $\Sigma$ alone identify a structural, causal contemporaneous relationship.

Units and Rescaling

Coefficient values depend on measurement units. To see the transformation exactly, let $y_t^*=Dy_t$, where $D$ is a diagonal matrix of nonzero scale factors. Multiplying the VAR by $D$ and substituting $y_t=D^{-1}y_t^*$ gives $$ y_t^*=Dc+\sum_{j=1}^{p}\left(DA_jD^{-1}\right)y_{t-j}^*+Du_t. $$ Therefore the transformed lag matrix is $A_j^*=DA_jD^{-1}$. For diagonal $D$, its $(r,s)$ entry is $d_r A_j[r,s]/d_s$. A rescaling generally changes the numerical values of nonzero off-diagonal coefficients in its row and column. The diagonal coefficient for the rescaled series is unchanged, and any zero coefficient remains zero. These facts prevent the mistaken conclusion that every coefficient in a rescaled row or column must change.

Worked Example: Rescaling One Series

Use the first lag matrix from the bivariate example, $$ A_1=\begin{pmatrix}0.5&0.2\\-0.1&0.6\end{pmatrix}, $$ and rescale only the first series by a factor of 2. Then $D=\begin{pmatrix}2&0\\0&1\end{pmatrix}$, so $$ A_1^*=DA_1D^{-1} = \begin{pmatrix}2&0\\0&1\end{pmatrix} \begin{pmatrix}0.5&0.2\\-0.1&0.6\end{pmatrix} \begin{pmatrix}0.5&0\\0&1\end{pmatrix} = \begin{pmatrix}0.5&0.4\\-0.05&0.6\end{pmatrix}. $$

The first-series own-lag coefficient remains $0.5$, as does the second-series own-lag coefficient $0.6$. The cross-lag coefficient from the second series to the first changes from $0.2$ to $0.4$; the coefficient from the first series to the second changes from $-0.1$ to $-0.05$. A zero off-diagonal coefficient, if present, would remain zero under this diagonal rescaling. The transformed intercept is $Dc$, and the transformed innovation covariance is $D\Sigma D'$.

Common Mistakes and Why Specification Matters

Reading rows as predictors. Rows identify the current outcome equation; columns identify lagged predictors. Check the row and column before interpreting a matrix entry.
Calling a cross-lag coefficient causal. It is a conditional association in the specified model. A causal interpretation requires additional assumptions and identification, not just a nonzero lag coefficient.
Confusing the intercept with the mean. The intercept is the conditional-mean contribution when all included lags are zero. The unconditional mean, when it exists, also depends on the lag matrices.
Assuming innovations in different equations are uncorrelated. The reduced-form covariance matrix may have nonzero off-diagonal entries, even though the innovations have conditional mean zero given the past.
Assuming all coefficients change when units change. Under diagonal rescaling, the series’ diagonal coefficient is unchanged and zero entries remain zero; nonzero off-diagonal entries generally change according to their row and column scale factors.

A clear specification records the variables, their ordering, their units, the lag order, and the deterministic terms included. These details make the matrix entries interpretable and make comparisons across specifications meaningful. The model is a compact description of conditional dynamics; it is not, by itself, evidence that one series causes another to move.

Check Your Understanding

Answer each question using the definitions and interpretations above.

In a three-variable VAR, what does entry $A_2[1,3]$ describe, and which units should appear in its interpretation?
Why can the innovations in two equations be contemporaneously correlated without either equation containing the other current-period variable on its right-hand side?
For the VAR(2) numerical example, explain what the zero in row 1, column 2 of $A_2$ means.
Under $y_t^*=Dy_t$ with diagonal $D$, what happens to a diagonal lag coefficient and to a coefficient that is exactly zero?
Why is it generally incorrect to interpret the intercept vector as the unconditional mean vector?