R Matrix Lab 2 — AR Recursions and Yule–Walker Equations

Convert AR models to companion form, compare polynomial roots with eigenvalues, propagate shocks, and recover coefficients from autocovariances.

Question

How do one scalar difference equation, one companion matrix, and one Toeplitz linear system describe the same AR process?

Consider

X_t=\phi_1X_{t-1}+\cdots+\phi_pX_{t-p}+\varepsilon_t.

Define the state

\mathbf s_t=(X_t,X_{t-1},\ldots,X_{t-p+1})^\top.

Then

\mathbf s_t=F\mathbf s_{t-1}+G\varepsilon_t,

where

F= \begin{bmatrix} \phi_1&\phi_2&\cdots&\phi_p\\ 1&0&\cdots&0\\ 0&1&\ddots&\vdots\\ \vdots&&\ddots&0 \end{bmatrix}, \qquad G=(1,0,\ldots,0)^\top.

Example 1 — Roots, eigenvalues, and shock propagation

For

X_t=0.75X_{t-1}-0.125X_{t-2}+\varepsilon_t,

the AR polynomial is

\phi(z)=1-0.75z+0.125z^2.

Its roots must lie outside the unit circle. The companion eigenvalues are their reciprocals and must lie inside.

Compare AR roots, companion eigenvalues, and impulse responses

The first output checks admissibility. The second shows dynamics: $F^hG$ is the state response to one innovation, and its first component is $\psi_h$ in the MA( $\infty$ ) representation.

Example 2 — Yule–Walker as a Toeplitz solve

For a stationary AR( $p$ ),

\Gamma_p\boldsymbol\phi=\boldsymbol\gamma_p,

with

\Gamma_p=\operatorname{Toeplitz}\{\gamma(0),\ldots,\gamma(p-1)\}, \qquad \boldsymbol\gamma_p=(\gamma(1),\ldots,\gamma(p))^\top.

The innovation variance follows from the lag-zero equation:

\sigma_\varepsilon^2 =\gamma(0)-\boldsymbol\phi^\top\boldsymbol\gamma_p.

Recover AR coefficients from theoretical autocovariances

This cell deliberately starts from population autocovariances. In estimation, replacing them by sample autocovariances introduces sampling error; the linear algebra remains the same.

One model, four equivalent views

View	Object	Stability or identification check
difference equation	$\phi(B)X_t=\varepsilon_t$	coefficients define recursion
polynomial	$\phi(z)$	roots outside unit circle
state equation	$\mathbf s_t=F\mathbf s_{t-1}+G\varepsilon_t$	eigenvalues of $F$ inside unit circle
moment equations	$\Gamma_p\boldsymbol\phi=\boldsymbol\gamma_p$	$\Gamma_p$ invertible

The views are equivalent under the model, but each makes a different task easier.

Exercises

Build the companion matrix for AR(3) coefficients $(0.5,-0.2,0.1)$ .
Verify numerically that each companion eigenvalue is the reciprocal of one AR-polynomial root.
Set $\phi=(1.2,-0.32)^\top$ . Check roots and eigenvalues before simulating.
For AR(1), reduce the matrix Yule–Walker equation to $\phi=\gamma(1)/\gamma(0)$ .
Set $\phi=(0.9,0.09)^\top$ . Inspect the spectral radius and the decay of $\psi_h$ . Explain why “coefficients sum to less than one” is not the general AR( $p$ ) stability test.
Graduate extension: solve $\operatorname{vec}(P)=(I-F\otimes F)^{-1}\operatorname{vec}(\sigma^2GG^\top)$ and compare $P_{11}$ with $\gamma(0)$ .

Checkpoints

The first row is $(0.5,-0.2,0.1)$ ; rows two and three shift the state with ones on the first subdiagonal.
Match the two unordered sets after taking reciprocals; allow floating-point tolerance.
A coefficient larger than one does not decide stability by itself; the polynomial roots or companion eigenvalues do.
$\Gamma_1=[\gamma(0)]$ and $\boldsymbol\gamma_1=[\gamma(1)]$ .
Stability depends on all roots jointly, not on one coefficient or one sum.

Completion standard

You should be able to move without ambiguity among lag-polynomial, companion-matrix, impulse-response, and Yule–Walker representations.

Continue to Lab 3 — Prediction and Likelihood.

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R Matrix Lab 1 — Covariance Geometry

Construct stationary Toeplitz covariance matrices, test positive definiteness, factor them, and simulate Gaussian finite blocks.

R Matrix Lab 3 — Prediction and Gaussian Likelihood

Solve best-linear-prediction equations, verify Schur-complement uncertainty, and compare conditional, moment, and exact Gaussian AR estimates.