R Matrix Lab 1 — Covariance Geometry

Question

How can a proposed autocovariance sequence become a valid finite-dimensional distribution?

For a weakly stationary process,

The matrix must be symmetric and positive semidefinite because, for every $\mathbf a$,

Example 1 — AR(1) Toeplitz structure

For

stationarity requires $|\phi|<1$ and

The first column of $\Gamma_6$ is therefore

Read the output

• Equal diagonals confirm Toeplitz structure.
• Positive eigenvalues establish positive definiteness for this finite block.
• The Cholesky error checks the factorisation numerically.
• The quadratic-form check connects a matrix calculation to a familiar variance.

Change $\phi$ to $-0.6$. The first off-diagonal changes sign, but the matrix remains positive definite. Change it to $0.95$: the process is still stationary, but the condition number rises because neighbouring variables become nearly redundant.

Example 2 — Gaussian simulation by a matrix square root

Let $\mathbf z\sim N(\mathbf0,I_n)$ and choose $L$ with $LL^\top=\Gamma_n$. Then

has mean $\boldsymbol\mu$ and covariance $\Gamma_n$.

R returns $U=\operatorname{chol}(\Gamma_n)$ with $U^\top U=\Gamma_n$, so use $L=U^\top$.

The empirical error will not be exactly zero because only finitely many blocks were simulated. Increase replications by a factor of four: Monte Carlo error should shrink by roughly a factor of two.

Why pairwise correlations are not enough

Values satisfying $|\rho(h)|\le1$ do not automatically form a valid autocorrelation sequence. All finite Toeplitz blocks must be positive semidefinite. A proposed sequence can pass every pairwise bound and still produce a negative eigenvalue.

This is the matrix reason that an ACF cannot be invented one lag at a time.

Exercises

1. With $\phi=0.6$ and innovation variance 2, calculate $\gamma(0)$ and $\gamma(1)$ by hand.
2. Replace gamma by c(1, 0.9, -0.9) and inspect the eigenvalues of toeplitz(gamma). Is it a valid three-lag covariance block?
3. Show algebraically that $\operatorname{Var}(X_t-X_{t-1})=2[\gamma(0)-\gamma(1)]$.
4. Compare the condition numbers for $\phi=0$, $0.6$, $0.9$, and $0.99$. Explain the statistical meaning.
5. Graduate extension: prove that every principal submatrix of a positive semidefinite matrix is positive semidefinite.

Completion standard

You should be able to build $\Gamma_n$ from $\gamma(0),\ldots,\gamma(n-1)$, justify positive semidefiniteness with quadratic forms, and use Cholesky reconstruction as a reproducible numerical check.

Continue to Lab 2 — AR Recursions.