Vector Autoregressions

Recall: ADL Models

We have studied dynamic relationships using autoregressive distributed lag (ADL) models.

\[ y_t = \alpha + \sum_{j=1}^p \phi_j y_{t-j} + \sum_{k=0}^q \beta_k x_{t-k} + u_t. \]

The model captures:

• persistence (lags of \(y_t\))
• dynamic effects of \(x_t\) through its lags
• gradual adjustment over time

The Exogeneity Assumption

For consistent estimation and interpretation:

• the regressors must be weakly exogenous
• feedback from \(y_t\) to \(x_t\) is ruled out

A Simple Control-System Analogy

Consider a thermostat regulating room temperature:

• \(y_t\): room temperature
• \(x_t\): thermostat setting
• \(u_t\): unobserved shocks (weather, insulation, people entering)

The thermostat adjusts its setting in response to past temperature:

\[ x_t = \gamma (y_{t-1} - y^\ast), \]

where \(y^\ast\) is the target temperature.

Why Exogeneity Fails

Suppose the room temperature is generated by \[ y_t = \alpha y_{t-1} + \beta x_t + u_t. \]

Since \(x_t = \gamma(y_{t-1} - y^\ast)\) depends on \(y_{t-1}\), which itself depends on \(u_{t-1}\):

Feedback in Macro: Policy and Inflation

A simple bivariate policy–inflation system:

\[ \begin{aligned} i_t &= \rho\, i_{t-1} + \phi\, \pi_{t-1} + u_{it} \\ \pi_t &= \theta\, \pi_{t-1} - \lambda\, i_{t-1} + u_{\pi t}. \end{aligned} \]

• the policy rate responds to past inflation (Taylor-type rule)
• inflation responds to past policy rates (monetary transmission)
• each equation has its own shock

Large-Scale Macroeconometric Models

Macroeconomists historically tried to capture this kind of feedback through large-scale macroeconometric models.

These models:

• consist of many simultaneous equations
• specify behavioral relationships for each variable (e.g. consumption, investment, etc.)
• explicitly allow feedback across the system

Difficulties with Simultaneous-Equation Systems

To identify such a system, you must impose exclusion restrictions — assumptions that some variables do not enter certain equations.

• Exclusions are hard to justify. Why does \(r_t\) belong in investment but not in consumption? Intertemporal optimization puts it in both. In practice, exclusions reflect modeling convenience more than theory.

• Exclusions choose the transmission. Omitting \(r_t\) from \(C_t\) forces every effect of a rate shock on \(C\) to travel through \(Y_t\) — ruling out, by assumption, any direct consumer response to rates.

If we can’t defend the exclusions, we can’t defend the system.

Symmetric Dynamic System

The alternative: treat all variables symmetrically — each gets its own equation, and each equation can include every other variable.

The simplest case is a bivariate system:

\[ \begin{aligned} y_t &= b_{10} - b_{12} z_t + \gamma_{11} y_{t-1} + \gamma_{12} z_{t-1} + \varepsilon_{yt}, \\ z_t &= b_{20} - b_{21} y_t + \gamma_{21} y_{t-1} + \gamma_{22} z_{t-1} + \varepsilon_{zt}. \end{aligned} \]

Compact Vector Form

Let \[ x_t = \begin{bmatrix} y_t \\ z_t \end{bmatrix}, \qquad \varepsilon_t = \begin{bmatrix} \varepsilon_{yt} \\ \varepsilon_{zt} \end{bmatrix}. \]

The system can be written compactly as \[ B x_t = \Gamma_0 + \Gamma_1 x_{t-1} + \varepsilon_t, \] where

\[ B = \begin{bmatrix} 1 & b_{12} \\ b_{21} & 1 \end{bmatrix}, \qquad \Gamma_0 = \begin{bmatrix} b_{10} \\ b_{20} \end{bmatrix}, \qquad \Gamma_1 = \begin{bmatrix} \gamma_{11} & \gamma_{12} \\ \gamma_{21} & \gamma_{22} \end{bmatrix}. \]

Structural Restrictions and Reduced Form

The matrix \(B\) summarizes contemporaneous structural restrictions.

If \(B\) is invertible, the system implies a reduced-form representation: \[ x_t = A_0 + A_1 x_{t-1} + u_t, \qquad u_t = B^{-1}\varepsilon_t. \]

The reduced form is invariant to the choice of contemporaneous structural restrictions, and can be estimated by OLS.

Stability of the VAR(1)

Consider the VAR(1): \[ x_t = A_0 + A_1 x_{t-1} + u_t. \]

Iterating backward, \[ x_t = \Big(\sum_{i=0}^{k} A_1^i\Big) A_0 + \sum_{i=0}^{k} A_1^i u_{t-i} + A_1^{k+1} x_{t-k-1}. \]

Stability condition: all eigenvalues of \(A_1\) lie strictly inside the unit circle.

Stability Implies Stationarity

If all eigenvalues of \(A_1\) lie strictly inside the unit circle, then:

• \(A_1^{k+1} x_{t-k-1} \to 0\) as \(k \to \infty\) (the effect of initial conditions vanishes)
• The series \(\sum_{i=0}^{\infty} A_1^i u_{t-i}\) converges in mean square
• \(\sum_{i=0}^{\infty} A_1^i = (I-A_1)^{-1}\)

Generalization to VAR(\(p\))

Consider the VAR(\(p\)): \[ x_t = A_1 x_{t-1} + \cdots + A_p x_{t-p} + u_t. \]

Define the stacked state vector \[ X_t = \begin{bmatrix} x_t \\ x_{t-1} \\ \vdots \\ x_{t-p+1} \end{bmatrix}. \]

VAR(\(p\)) Companion Form

The system can be written in companion form

\[ X_t = \mathscr{A} X_{t-1} + \mathscr{U}_t, \]

where

\[ \mathscr{A} = \begin{bmatrix} A_1 & A_2 & \cdots & A_p \\ I & 0 & \cdots & 0 \\ & \ddots & \ddots & \vdots \\ 0 & & I & 0 \end{bmatrix}. \]

Stability of VAR(\(p\))

The VAR(\(p\)) is stable if all eigenvalues of the companion matrix \(\mathscr{A}\) lie strictly inside the unit circle:

\[ |\lambda_i(\mathscr{A})| < 1 \quad \text{for all } i. \]

Eigenvalues and Polynomial Roots

In ARMA models, stability was characterized via the roots of a polynomial. For VAR(\(p\)), the companion-matrix eigenvalues give the same thing.

Eigenvalues \(\lambda\) of \(\mathscr{A}\) solve \(\det(\lambda I - \mathscr{A}) = 0\). Expanding using the block structure, \[ \det(\lambda I - \mathscr{A}) = \det(\lambda^p I - A_1 \lambda^{p-1} - \cdots - A_p). \]

Factoring out \(\lambda^p\) and setting \(z = 1/\lambda\): \[ \det(I - A_1 z - \cdots - A_p z^p) = 0. \]

So the eigenvalues of \(\mathscr{A}\) are reciprocals of the roots of the characteristic polynomial: \(|\lambda|<1 \Leftrightarrow |z|>1\).

Estimation of VAR Models

A VAR(\(p\)) is estimated as a system of linear projections: \[ x_t = A_1 x_{t-1} + \cdots + A_p x_{t-p} + u_t. \]

Estimation is typically carried out by equation-by-equation ordinary least squares.

VAR Inference Conditions

If:

• the VAR is stable, so \(\{x_t\}\) is stationary
• the reduced-form specification is correct (innovations are orthogonal to lagged regressors)
• the dimension \((m,p)\) is fixed relative to sample size \(T\)

Is Equation-by-Equation OLS Efficient?

Equation-by-equation OLS estimates each VAR equation in isolation. VAR innovations are typically correlated across equations (\(\Sigma_u = \mathrm{Var}(u_t)\) not diagonal) — which usually makes GLS more efficient than OLS.

For the reduced-form VAR, though, GLS coincides with OLS — because every equation shares the same regressors, GLS’s usual advantage disappears.

The next slides make this precise.

SUR Representation

Each VAR equation, stacked over \(t = 1, \ldots, T\), gives \(y_i = Z\beta_i + u_i\), with \(Z \in \mathbb{R}^{T \times mp}\) the lagged-regressor matrix and \(\beta_i \in \mathbb{R}^{mp}\) equation \(i\)’s coefficient vector.

Stacking across the \(m\) equations:

\[ \underbrace{\begin{bmatrix} y_1 \\ \vdots \\ y_m \end{bmatrix}}_{y \,\in\, \mathbb{R}^{mT}} = \underbrace{\begin{bmatrix} Z & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & Z \end{bmatrix}}_{X \,\in\, \mathbb{R}^{mT \times m^2 p}} \underbrace{\begin{bmatrix} \beta_1 \\ \vdots \\ \beta_m \end{bmatrix}}_{\beta \,\in\, \mathbb{R}^{m^2 p}} + \underbrace{\begin{bmatrix} u_1 \\ \vdots \\ u_m \end{bmatrix}}_{u \,\in\, \mathbb{R}^{mT}}. \]

Each equation shares the same regressor matrix \(Z\).

Kronecker Product

The block-diagonal \(X\) above has a name: the Kronecker product.

For matrices \(A\) (\(p \times q\)) and \(B\), \(A \otimes B\) replaces each scalar entry \(a_{ij}\) of \(A\) with the block \(a_{ij}B\):

\[ A \otimes B = \begin{bmatrix} a_{11} B & \cdots & a_{1q} B \\ \vdots & & \vdots \\ a_{p1} B & \cdots & a_{pq} B \end{bmatrix}. \]

SUR in Kronecker Form

Since \(I_m\) has 1s on the diagonal, the block-diagonal \(X\) is just \(I_m \otimes Z\):

\[ X = I_m \otimes Z. \]

Because \(u_t\) is i.i.d. across time with \(\mathrm{Var}(u_t) = \Sigma_u\) (an \(m \times m\) contemporaneous covariance), the stacked error has

\[ \mathrm{Var}(u) = \Sigma_u \otimes I_T \;\in\; \mathbb{R}^{mT \times mT}. \]

So the SUR system is compactly \(y = (I_m \otimes Z)\beta + u\) with \(\mathrm{Var}(u) = \Sigma_u \otimes I_T\).

Useful Kronecker properties (we’ll need these):

• \((A \otimes B)' = A' \otimes B'\)
• \((A \otimes B)(C \otimes D) = (AC) \otimes (BD)\)
• \((A \otimes B)^{-1} = A^{-1} \otimes B^{-1}\)

GLS Setup

We want the efficient estimator for this stacked system. Work generically first: \(y = X\beta + u\) with \(\mathrm{Var}(u) = \Omega\) positive-definite (later we take \(\Omega = \Sigma_u \otimes I_T\)).

OLS minimizes the unweighted sum of squared residuals:

\[ S_{\mathrm{OLS}}(\beta) = (y - X\beta)'(y - X\beta). \]

Equal weighting is optimal when \(\Omega = \sigma^2 I\), but suboptimal when \(\Omega\) is general:

• observations with larger variance carry less information
• correlated observations carry overlapping information

Weighted Least Squares

For general \(\Omega\), minimize a weighted sum of squared residuals instead:

\[ S_{\mathrm{GLS}}(\beta) = (y - X\beta)' \Omega^{-1} (y - X\beta). \]

First-order condition (derivative in \(\beta\) set to zero):

\[ -2\, X' \Omega^{-1} (y - X\beta) = 0. \]

Solving the normal equations yields the GLS estimator:

\[ \widehat\beta_{\mathrm{GLS}} = (X'\Omega^{-1}X)^{-1} X'\Omega^{-1} y. \]

Why \(\Omega^{-1}\)? The Pre-Whitening View

The \(\Omega^{-1}\) weighting isn’t arbitrary — it comes from transforming the model so errors become i.i.d.

\(\Omega\) is symmetric positive-definite, so it has a symmetric square root \(\Omega^{1/2}\) with \(\Omega^{1/2}\Omega^{1/2} = \Omega\). Premultiply by \(\Omega^{-1/2}\):

\[ \Omega^{-1/2} y = \Omega^{-1/2} X \beta + \Omega^{-1/2} u, \qquad \mathrm{Var}(\Omega^{-1/2} u) = I. \]

Applying OLS to the transformed system gives the same formula, \((X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y\) — pre-whitening and weighted LS are the same estimator.

Applying GLS to the SUR System

Start with \(X = I_m \otimes Z\) and \(\Omega = \Sigma_u \otimes I_T\).

Transpose and inverse: \[ X' = I_m \otimes Z', \qquad \Omega^{-1} = \Sigma_u^{-1} \otimes I_T. \]

Mixed-product gives \[ X'\Omega^{-1} = (I_m \otimes Z')(\Sigma_u^{-1} \otimes I_T) = \Sigma_u^{-1} \otimes Z', \]

\[ X'\Omega^{-1}X = (\Sigma_u^{-1} \otimes Z')(I_m \otimes Z) = \Sigma_u^{-1} \otimes (Z'Z). \]

Inverse property: \[ (X'\Omega^{-1}X)^{-1} = \Sigma_u \otimes (Z'Z)^{-1}. \]

Why OLS Is Efficient

Combining the pieces in the GLS formula:

\[ \widehat\beta_{\mathrm{GLS}} = (X'\Omega^{-1}X)^{-1} X'\Omega^{-1} y = \big(\Sigma_u \otimes (Z'Z)^{-1}\big)\big(\Sigma_u^{-1} \otimes Z'\big) y = \big(I_m \otimes (Z'Z)^{-1}Z'\big) y. \]

The block-diagonal form means \((Z'Z)^{-1}Z'\) is applied equation by equation — which is equation-by-equation OLS. So

\[ \widehat\beta_{\mathrm{GLS}} = \widehat\beta_{\mathrm{OLS}}. \]

Dimensionality in VAR Models

The reduced-form VAR(\(p\)) \(x_t = \nu + A_1 x_{t-1} + \cdots + A_p x_{t-p} + u_t\) contains

\[ m^2 p + m \]

parameters (\(m\) = number of variables).

Rule of Thumb for Sample Size

Each equation in a VAR(\(p\)) involves approximately \(mp\) slope coefficients.

Empirical practice typically requires \[ T \gtrsim 5\text{–}10 \times mp \] for reliable estimation.

Lag-Length Selection

Because dimensionality grows with \(p\), lag length must be chosen carefully. Standard criteria trade off fit against parameterization:

• Akaike Information Criterion (AIC)
• Bayesian Information Criterion (BIC)
• Hannan–Quinn Criterion (HQ)

In typical macroeconomic systems:

• BIC tends to select the most parsimonious lag length
• AIC tends to select longer lags
• HQ lies between the two

Forecasting in a VAR

Forecasts are computed recursively from the VAR coefficients:

\[ \widehat{x}_{t+h|t} = A_1 \widehat{x}_{t+h-1|t} + \cdots + A_p \widehat{x}_{t+h-p|t}, \qquad h \ge 1, \]

with \(\widehat{x}_{s|t} = x_s\) for \(s \le t\).

Forecast Error Variance

For a VAR(1) with \(\mathrm{Var}(u_t) = \Sigma_u\), the FEV satisfies

\[ \Omega_1 = \Sigma_u, \qquad \Omega_h = A\,\Omega_{h-1}A' + \Sigma_u. \]

Under stability, \(\Omega_h\) grows with \(h\) and converges to the unconditional variance of \(x_t\).

Example: Forecasting in a Stable VAR (Simulated Data)

We simulate a stable bivariate VAR(1) process and illustrate multi-step forecasts and forecast confidence intervals.

• Data-generating process is stationary by construction
• Forecasts are produced recursively from the VAR
• Confidence intervals reflect uncertainty from future shocks

Nonstationary Variables: Why Not Just Difference?

When \(x_t\) is \(I(1)\), differencing restores stationarity — but it removes long-run relationships between variables.

A VAR in differences cannot capture equilibrium comovement (Sims, Stock, and Watson, 1990).

From VAR Levels to a Differenced Form

Start from a VAR(\(p\)) in levels: \[ x_t = A_1 x_{t-1} + \cdots + A_p x_{t-p} + u_t. \]

Subtract \(x_{t-1}\) from both sides: \[ \Delta x_t = \Big(A_1 + \cdots + A_p - I\Big)x_{t-1} + \sum_{i=1}^{p-1} \Big(-A_{i+1}-\cdots-A_p\Big)\Delta x_{t-i} + u_t. \]

Rewriting as a VECM

Define \[ \Pi := \sum_{i=1}^p A_i - I, \qquad \Gamma_i := -\sum_{j=i+1}^p A_j. \]

Then the VAR can always be written as \[ \Delta x_t = \Pi x_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta x_{t-i} + u_t. \]

Cointegration Hypothesis

Suppose \(x_t\) is \(I(1)\) and cointegrated with rank \(r<m\).

That is, there exists \(\beta \in \mathbb{R}^{m\times r}\) such that \[ \beta' x_t \ \text{is stationary}. \]

Why Π Must Have Reduced Rank

In the VECM form, \[ \Delta x_t = \Pi x_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta x_{t-i} + u_t, \] the left-hand side is stationary, as are \(\Delta x_{t-i}\) and \(u_t\). Hence \(\Pi x_{t-1}\) must be stationary.

The Vector Error-Correction Model

Substituting \(\Pi=\alpha\beta'\) gives the VECM:

\[ \Delta x_t = \alpha \beta' x_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta x_{t-i} + u_t. \]

• short-run dynamics: \(\Gamma_i \Delta x_{t-i}\)
• long-run equilibrium errors: \(\beta' x_{t-1}\)
• adjustment to equilibrium: \(\alpha \in \mathbb{R}^{m \times r}\)

A Bivariate VECM Example

Let \[ x_t = \begin{bmatrix} y_t \\ z_t \end{bmatrix}, \qquad y_t,\; z_t \text{ are } I(1). \]

Suppose there exists a scalar \(\theta\) such that \[ \beta' x_t = y_t - \theta z_t \quad \text{is stationary}, \qquad \beta'=(1,\,-\theta). \]

Then \(y_t\) and \(z_t\) are cointegrated (rank \(r=1\)).

Bivariate VECM: Error-Correction Equations

Define the equilibrium error \(e_{t-1} \equiv y_{t-1} - \theta z_{t-1}\). The VECM \[ \Delta x_t = \alpha e_{t-1} + \sum_{i=1}^{p-1}\Gamma_i \Delta x_{t-i} + u_t, \qquad \alpha \in \mathbb{R}^{2\times 1}, \]

Estimation of the VECM Parameters

Consider the VECM \[ \Delta x_t = \Pi x_{t-1} + \sum_{i=1}^{p-1}\Gamma_i \Delta x_{t-i} + u_t, \qquad \Pi=\alpha\beta', \qquad \mathrm{rank}(\Pi)=r. \]

What needs to be estimated:

• short-run parameters \(\Gamma_i\)
• long-run matrix \(\Pi\), subject to rank restriction \(\mathrm{rank}(\Pi)=r<m\)

Why OLS Fails for the VECM

• OLS treats \(\Pi\) as unrestricted
• unrestricted OLS produces a full-rank estimate of \(\Pi\)

Johansen: Partial Out Short-Run Dynamics

Start from the VECM: \[ \Delta x_t=\Pi x_{t-1}+\sum_{i=1}^{p-1}\Gamma_i \Delta x_{t-i}+u_t. \]

Idea: partial out short-run dynamics so the remaining relation is purely long-run.

Johansen: Partial Out (Residualized Variables)

Regress \(\Delta x_t\) and \(x_{t-1}\) on \(\{\Delta x_{t-1},\dots,\Delta x_{t-p+1}\}\) and keep residuals: \[ \begin{aligned} R_{0t} &= \Delta x_t - \mathbb{E}[\Delta x_t \mid \Delta x_{t-1},\dots,\Delta x_{t-p+1}],\\ R_{1t} &= x_{t-1} - \mathbb{E}[x_{t-1} \mid \Delta x_{t-1},\dots,\Delta x_{t-p+1}]. \end{aligned} \]

Johansen: Gaussian MLE Setup

Assume the VECM errors are Gaussian: \[ u_t \sim N(0,\Sigma_u), \quad t=1,\dots,T. \]

By FWL, the partialled-out long-run regression has the same error: \[ R_{0t} = \Pi R_{1t} + u_t, \qquad u_t \sim N(0,\Sigma_u). \]

Johansen: Log-Likelihood

The (conditional) log-likelihood is

\[ \ell(\Pi,\Sigma_u) = -\frac{T}{2}\log|\Sigma_u| -\frac{1}{2}\sum_{t=1}^T (R_{0t}-\Pi R_{1t})'\Sigma_u^{-1}(R_{0t}-\Pi R_{1t}) + \text{const}. \]

Profile Out \(\Sigma_u\)

For fixed \(\Pi\), maximizing the Gaussian log-likelihood over \(\Sigma_u\) gives \[ \widehat\Sigma_u(\Pi) = \tfrac{1}{T}\sum_{t=1}^T (R_{0t} - \Pi R_{1t})(R_{0t} - \Pi R_{1t})'. \]

Imposing Π = αβ’; Profile Out α

Expand the residual covariance: \[ \widehat\Sigma_u(\Pi) = S_{00} - \Pi S_{10} - S_{01}\Pi' + \Pi S_{11}\Pi', \qquad S_{ij} = \tfrac{1}{T}\textstyle\sum_t R_{it}R_{jt}'. \]

From β to a Generalized Eigenvalue Problem

Two determinant identities — \(|A-B|=|A|\,|I-A^{-1}B|\) and Sylvester’s \(|I+UV|=|I+VU|\) — plus the normalization \(\beta'S_{11}\beta = I_r\), reduce the problem to \[ \max_{\beta}\;\bigl|\beta' A \beta\bigr| \quad\text{s.t.}\quad \beta'S_{11}\beta = I_r, \qquad A := S_{10}S_{00}^{-1}S_{01}. \]

CCA in Variational Form

The remark just said \(\widehat\lambda_i = \widehat\rho_i^{\,2}\) — eigenvalues are squared canonical correlations. Make this operational.

What Does a Large \(\widehat\lambda_i\) Mean?

Cointegration as the Escape Hatch

A non-vanishing \(\widehat\lambda_i\) requires a \(\beta_i\) such that \(\beta_i'x_{t-1}\) is stationary. Then \(\beta_i' S_{11}\,\beta_i = O_p(1)\) rather than \(O_p(T)\), and \(\widehat\lambda_i\) stays bounded away from 0.

\[ \widehat\lambda_i \;\not\xrightarrow{p}\; 0 \;\;\Longleftrightarrow\;\; \beta_i'x_{t-1}\sim I(0) \;\;\Longleftrightarrow\;\; \beta_i \text{ cointegrates } x_t. \]

How Eigenvalues Encode rank(Π)

Each non-zero \(\lambda_i\) of \(M := \Sigma_{11}^{-1}\Sigma_{10}\Sigma_{00}^{-1}\Sigma_{01}\) flags a cointegrating direction. To get a count, chain three facts:

Eigenvalues and Cointegrating Rank

A single \(\widehat\lambda_i\) describes one direction. Their collective pattern encodes the cointegrating rank \(r\):

From Eigenvalues to a Test

In population, the rank dictionary cleanly separates non-vanishing from vanishing eigenvalues. At finite \(T\), sampling noise keeps every \(\widehat\lambda_i > 0\) — we need a hypothesis test.

The Log-Likelihood Ratio

For nested models with maximized likelihoods \(L_0 \le L_1\) (restricted vs. unrestricted), \[ \mathrm{LR} \;=\; -2\log\frac{L_0}{L_1} \;=\; -2(\ell_0 - \ell_1) \;\ge\; 0. \]

Large \(\mathrm{LR}\) → restricted model fits much worse → reject \(H_0\).

Johansen’s LR Rank Test

For each \(r\), test \(H_0:\mathrm{rank}(\Pi)\le r\) vs. \(H_1:\mathrm{rank}(\Pi) > r\). Substituting the profile log-likelihood \(\ell(\Pi) = -\tfrac{T}{2}\log|\widehat\Sigma_u(\Pi)| + \text{const}\), \[ \mathrm{LR}(r) \;=\; -2(\ell_r - \ell_m) \;=\; -T\log\frac{|\widehat\Sigma_m|}{|\widehat\Sigma_r|}, \] where \(\widehat\Sigma_m, \widehat\Sigma_r\) are the residual covariances under the unrestricted and rank-\(r\) models.

LR in Eigenvalue Form

The determinant ratio reduces to a product over the Johansen eigenvalues: \[ \frac{|\widehat\Sigma_m|}{|\widehat\Sigma_r|} \;=\; \prod_{i=r+1}^m (1 - \widehat\lambda_i), \qquad \mathrm{LR}(r) \;=\; -T\sum_{i=r+1}^m \log(1 - \widehat\lambda_i). \]

Two tests are built from this: the trace test and the maximum eigenvalue test.

Trace Test (Joint LR Test)

Compares the rank-\(r\) model to the unrestricted model: \[ \mathrm{LR}_{\text{trace}}(r) = -T\sum_{i=r+1}^m \log(1-\hat\lambda_i). \]

This tests \[ H_0:\ \mathrm{rank}(\Pi)\le r \quad \text{vs.} \quad H_1:\ \mathrm{rank}(\Pi)>r. \]

The null requires every \(\hat\lambda_{r+1},\dots,\hat\lambda_m\) to be zero.

Maximum Eigenvalue Test

Compares the rank-\(r\) model to the rank-\((r+1)\) model: \[ \mathrm{LR}_{\max}(r,r+1) = -T\log(1-\hat\lambda_{r+1}). \]

This tests \[ H_0:\ \mathrm{rank}(\Pi)=r \quad \text{vs.} \quad H_1:\ \mathrm{rank}(\Pi)=r+1. \]

Connecting Back to the Eigenvalue Magnitude

For small \(\widehat\lambda_i\) (the relevant range under the null of no further cointegration), Taylor expansion gives \[ -\log(1-\widehat\lambda_i) \;=\; \widehat\lambda_i + \tfrac{1}{2}\widehat\lambda_i^2 + O(\widehat\lambda_i^3). \]

Trace vs. Max-Eigen: Practical Guidance

Why they differ.

• Trace sums contributions across \(\widehat\lambda_{r+1}, \dots, \widehat\lambda_m\) — diffuse evidence across many small eigenvalues accumulates → tends to pick larger \(r\).
• Max-eigen uses only \(\widehat\lambda_{r+1}\) — needs one clearly large next eigenvalue → tends to pick smaller \(r\).

Engle–Granger vs. Johansen

Both estimate \(\beta\) identified only up to rotation. The difference is when the normalization enters.

Engle–Granger (single-equation)

• OLS on \(y_t = \theta' z_t + e_t\), then ECM with \(\widehat e_{t-1}\)
• normalization is baked into estimation — picking \(y\) as LHS fixes its coefficient at \(1\)
• different LHS choices give different finite-sample \(\widehat\theta\) (asymmetric)
• cannot handle \(r > 1\)

Johansen (system VECM)

• system estimator for the cointegration space \(\mathrm{span}(\beta)\)
• normalization is post-estimation — rotate eigenvectors after fitting; rank tests are invariant
• handles \(r = 0, 1, \dots, m-1\); LR rank tests built in

From Rank to VECM Structure

With \(r\) fixed by the rank tests, both VECM components come from the eigenproblem: \[ \widehat\beta_r = \text{top-}r\ \text{eigenvectors (normalized: } \widehat\beta_r'\,S_{11}\,\widehat\beta_r = I_r\text{)}, \qquad \widehat\alpha_r = S_{01}\,\widehat\beta_r. \]

The estimated VECM \[ \Delta x_t = \widehat\alpha\,\widehat\beta'\,x_{t-1} + \cdots + u_t \] has two structural objects to interrogate:

• \(\beta\) — long-run equilibrium relations,
• \(\alpha\) — adjustment coefficients (which variables respond to disequilibrium).

Testing Adjustment in a VECM: Restrictions on \(\alpha\)

In the VECM \[ \Delta x_t = \alpha\beta' x_{t-1} + \cdots + u_t, \] \(\alpha \in \mathbb{R}^{m\times r}\) collects adjustment coefficients: row \(i\) describes how \(\Delta x_{i,t}\) responds to the \(r\) equilibrium errors.

Linear hypotheses on \(\alpha\) take the form \[ H_0:\ R\,\mathrm{vec}(\alpha) = q, \qquad R \in \mathbb{R}^{k \times mr},\ q \in \mathbb{R}^{k}, \] where \(\mathrm{vec}(\alpha) \in \mathbb{R}^{mr}\) stacks the columns of \(\alpha\), and \(k\) = # independent restrictions = degrees of freedom of the LR/Wald \(\chi^2\) test.

Conditional on the cointegration rank \(r\), \(\alpha\) is the coefficient on the stationary regressor \(\widehat\beta_r' x_{t-1},\) so standard regression / MLE asymptotics apply.

Examples of Restrictions on \(\alpha\)

• No adjustment for variable \(i\) (weak exogeneity for \(\beta\)): \[ H_0:\ \alpha_{i\cdot}=0 \quad (r\ \text{restrictions}) \]

• Only relation 1 adjusts variable \(i\): \[ H_0:\ \alpha_{i2}=\cdots=\alpha_{ir}=0 \]

• Same adjustment for variables \(i\) and \(j\): \[ H_0:\ \alpha_{i\cdot}=\alpha_{j\cdot} \]

Example: A Small Open Economy

\(x_t = (y_t,\ p_t,\ p_t^{\ast})'\) — domestic output, domestic price, foreign price.

Question. Do prices satisfy PPP in the long run, and which side adjusts to disequilibrium — domestic or foreign?

Small Open Economy: A Meaningful Restriction

Small open economy assumption: foreign prices do not adjust to domestic disequilibrium: \[ H_0:\ \alpha_{p^\ast}=0. \]

Interpretation:

• deviations from PPP are corrected through domestic adjustment
• \(p_t^{\ast}\) is weakly exogenous for the cointegrating relation \(\beta\)

Diagnostics

Once a VAR or VECM is estimated, check that the specification is adequate:

• residual autocorrelation: Ljung–Box or portmanteau test on multivariate residuals
• stability: all eigenvalues of the companion matrix inside the unit circle
• distributional checks: normality or at least symmetry of residuals for inference

Failure of these checks typically points back to an under-specified lag length or an omitted structural break.

Summary: What VARs and VECMs Do

We studied VARs and VECMs as reduced-form representations of multivariate time series.

They allow us to:

• model joint dynamics without imposing exogeneity
• estimate long-run relations via cointegration
• produce forecasts and forecast uncertainty

Summary: What They Do Not Do

The shocks \(u_t\) in a VAR or VECM are:

• statistical innovations
• generally correlated across equations
• not economically interpretable by construction