Simultaneous Equations

The Problem

Price goes up — does quantity fall because demand fell, or rise because supply rose?

When variables are jointly determined, regressing one on the other gives inconsistent OLS estimates — not merely imprecise, but wrong in the limit. Simultaneity makes the regressor correlated with the error in its own equation. From Day 1, X^{'}\epsilon \neq 0

The way out is to be explicit about two levels of the model:

• the structural form — which equation is which (demand vs. supply, policy vs. response)
• the reduced form — what the data can actually tell us directly

The Structural Form

A system of G equations in G endogenous variables \mathbf{y}_t and K exogenous variables \mathbf{x}_t:

\mathbf{B}\,\mathbf{y}_t + \boldsymbol{\Gamma}\,\mathbf{x}_t = \boldsymbol{\varepsilon}_t

Each row is a separate economic relationship — a supply curve, a demand curve, a policy rule. By convention, normalise the diagonal of \mathbf{B} to -1 so that variable i stands alone on the left-hand side of equation i.

The Reduced Form

Premultiply by \mathbf{B}^{-1} (assuming it exists):

\mathbf{y}_t = \underbrace{-\mathbf{B}^{-1}\boldsymbol{\Gamma}}_{\boldsymbol{\Pi}}\,\mathbf{x}_t + \underbrace{\mathbf{B}^{-1}\boldsymbol{\varepsilon}_t}_{\mathbf{v}_t} \qquad\Longrightarrow\qquad \boxed{\mathbf{y}_t = \boldsymbol{\Pi}\,\mathbf{x}_t + \mathbf{v}_t}

Every endogenous variable is now expressed purely in terms of exogenous variables. Because the same regressors appear in every equation, \boldsymbol{\Pi} can be estimated by OLS applied equation by equation — consistently and without instruments.

The cost: the reduced-form errors \mathbf{v}_t = \mathbf{B}^{-1}\boldsymbol{\varepsilon}_t are a blend of all the structural disturbances and will generally be correlated across equations:

E[\mathbf{v}_t \mathbf{v}_t'] = \mathbf{B}^{-1}\boldsymbol{\Sigma}\left(\mathbf{B}^{-1}\right)'

A Concrete Example: Supply & Demand

Structural form (G = 2, endogenous p_t and q_t, exogenous \text{inc}_t and \text{cost}_t)

\begin{aligned} \text{Demand:}\quad & q_t = \alpha_1 p_t + \alpha_2\,\text{inc}_t + \varepsilon_t^d \\[4pt] \text{Supply:}\quad & q_t = \beta_1 p_t + \beta_2\,\text{cost}_t + \varepsilon_t^s \end{aligned}

Reduced form (solve for p_t and q_t)

\begin{aligned} p_t &= \pi_{11}\,\text{inc}_t + \pi_{12}\,\text{cost}_t + v_t^p \\[4pt] q_t &= \pi_{21}\,\text{inc}_t + \pi_{22}\,\text{cost}_t + v_t^q \end{aligned}

The \pi_{ij} are rational functions of (\alpha_1, \alpha_2, \beta_1, \beta_2) — ratios of the structural parameters. Recovering those parameters from \hat{\boldsymbol{\Pi}} is the identification problem.

The Identification Problem

We can estimate \hat{\boldsymbol{\Pi}} cleanly from data. But does it tell us anything about \mathbf{B} and \boldsymbol{\Gamma}?

\boldsymbol{\Pi} = -\mathbf{B}^{-1}\boldsymbol{\Gamma} \quad\Leftrightarrow\quad \mathbf{B}\,\boldsymbol{\Pi} + \boldsymbol{\Gamma} = \mathbf{0}

The reduced form \boldsymbol{\Pi} has G \times K free parameters. The structural pair (\mathbf{B}, \boldsymbol{\Gamma}) has up to G(G + K). Without further restrictions there are infinitely many structural models consistent with any given reduced form.

What rescues identification is prior knowledge about which variables belong in which equations: the exclusion restrictions.

The Order Condition

A variable excluded from equation i is a potential source of identifying information. The necessary condition is:

\boxed{\text{number of excluded variables} \geq G - 1}

where “excluded” means present in the system but absent from equation i.

Passing the order condition confirms you have enough excluded variables. It does not confirm they actually do the job — that is the rank condition’s business.

The Rank Condition

Let \mathbf{C}_i be the submatrix of structural coefficients on the variables excluded from equation i, drawn from the other G - 1 equations.

\boxed{\text{rank}(\mathbf{C}_i) = G - 1}

This is necessary and sufficient. The excluded variables need to genuinely shift the other equations — if their effects are collinear across those equations, or absent altogether, we cannot trace out equation i no matter how many are excluded.

The rank condition implies the order condition, so rank is the definitive test. Order is just the quick preliminary count.

Rank & Order: The Supply/Demand Example

Coefficient matrix (normalised so q_t has coefficient -1 in both equations):

\begin{array}{c|cc|cc} & p_t & q_t & \text{inc}_t & \text{cost}_t \\ \hline \text{Demand} & \alpha_1 & -1 & \alpha_2 & 0 \\ \text{Supply} & \beta_1 & -1 & 0 & \beta_2 \end{array}

Demand excludes \text{cost}_t. Its coefficient in the supply equation is \beta_2:

\text{rank}([\,\beta_2\,]) = 1 = G-1 \quad\checkmark\quad (\text{provided } \beta_2 \neq 0)

Supply excludes \text{inc}_t. Its coefficient in the demand equation is \alpha_2:

\text{rank}([\,\alpha_2\,]) = 1 = G-1 \quad\checkmark\quad (\text{provided } \alpha_2 \neq 0)

Both equations are exactly identified.

The VAR as a Reduced Form

A Vector Autoregression of order p:

\mathbf{y}_t = \mathbf{A}_1\,\mathbf{y}_{t-1} + \mathbf{A}_2\,\mathbf{y}_{t-2} + \cdots + \mathbf{A}_p\,\mathbf{y}_{t-p} + \mathbf{u}_t, \qquad \mathbf{u}_t \sim WN(0,\,\boldsymbol{\Sigma}_u)

This is a reduced form. There are no contemporaneous endogenous variables on the right — only lags, which are predetermined. The \mathbf{A}_j matrices are reduced-form coefficients, not structural parameters. Because every equation shares the same regressors, OLS is consistent and fully efficient; SUR would give identical estimates.

The off-diagonal elements of \boldsymbol{\Sigma}_u are not evidence of direct causal links — they reflect structural shocks that get mixed together through \mathbf{B}_0^{-1}.

From Structural VAR to Reduced Form VAR

The underlying structural model:

\mathbf{B}_0\,\mathbf{y}_t = \mathbf{B}_1\,\mathbf{y}_{t-1} + \cdots + \mathbf{B}_p\,\mathbf{y}_{t-p} + \boldsymbol{\varepsilon}_t, \qquad \boldsymbol{\varepsilon}_t \sim WN(0,\,\mathbf{I})

Premultiply by \mathbf{B}_0^{-1} to recover the reduced form VAR:

\mathbf{y}_t = \underbrace{\mathbf{B}_0^{-1}\mathbf{B}_1}_{\mathbf{A}_1}\mathbf{y}_{t-1} + \cdots + \underbrace{\mathbf{B}_0^{-1}\mathbf{B}_p}_{\mathbf{A}_p}\mathbf{y}_{t-p} + \underbrace{\mathbf{B}_0^{-1}\boldsymbol{\varepsilon}_t}_{\mathbf{u}_t}

The covariance matrix ties the two levels together:

\boldsymbol{\Sigma}_u = \mathbf{B}_0^{-1}\left(\mathbf{B}_0^{-1}\right)'

Identification in the SVAR

We estimate \hat{\mathbf{A}}_j and \hat{\boldsymbol{\Sigma}}_u from the data. The question is whether \hat{\boldsymbol{\Sigma}}_u pins down \mathbf{B}_0.

\boldsymbol{\Sigma}_u = \mathbf{B}_0^{-1}\left(\mathbf{B}_0^{-1}\right)'

\boldsymbol{\Sigma}_u alone pins down \mathbf{B}_0 only up to right-multiplication by an orthogonal matrix. At least G(G-1)/2 restrictions are needed to resolve the ambiguity: three for a three-variable system, fifteen for six variables.

Imposing Identification

The restrictions on \mathbf{B}_0 are an economic choice, not a statistical one:

Recursive identification is point-identified; sign restrictions are set-identified — a range of \mathbf{B}_0 matrices is consistent with the data.

Summary

\underbrace{\mathbf{B}_0\,\mathbf{y}_t = \mathbf{B}_1\,\mathbf{y}_{t-1} + \cdots + \boldsymbol{\varepsilon}_t}_{\text{Structural VAR}} \quad\xrightarrow{\;\times\,\mathbf{B}_0^{-1}\;}\quad \underbrace{\mathbf{y}_t = \mathbf{A}_1\,\mathbf{y}_{t-1} + \cdots + \mathbf{u}_t}_{\text{Reduced Form VAR}}

The reduced form is always there. Whether you can get back to the structural form depends entirely on the restrictions you are prepared to impose and defend.