Module 5 — Systems, Seasonality, and Cointegration

Core question

What changes when variables forecast one another, seasonal positions have their own dynamics, or non-stationary levels remain tied by a stable long-run relation?

Learning outcomes

You will be able to:

• formulate and check the stability of a VAR;
• solve a stationary VAR covariance with a Kronecker-product equation;
• distinguish Granger predictability from causal effect;
• compute and qualify an impulse response;
• choose among seasonal indicators, seasonal differencing, and SARIMA terms;
• recognise cointegration and interpret an error-correction model;
• explain why forecasts across products or time aggregates must be reconciled.

1. VAR: every variable can use every lag

For a $k$-variable VAR($p$),

A VAR(1) is stationary when every eigenvalue of $A_1$ has modulus below one. For VAR($p$), use the equivalent companion matrix.

Parameter growth is the first constraint

With an intercept, a VAR has $k(kp+1)$ regression coefficients. For $k=5$ and $p=4$, that is 105 coefficients before the 15 distinct innovation-covariance terms. A rich system estimated from 80 quarters is not automatically informative; shrinkage, fewer variables/lags, or theory-based restrictions may be necessary.

Stacked regression form

For VAR(1), collect rows $\mathbf y_2^\top,\ldots,\mathbf y_n^\top$ in $Y$ and rows $(1,\mathbf y_{t-1}^\top)$ in $\mathcal X$. Then

Equation-by-equation OLS and multivariate least squares give the same coefficient estimates when every equation has the same regressors. The innovation covariance still matters for joint inference.

2. A two-variable shock calculation

Let demeaned electricity demand and price follow

Starting from a one-unit shock to the first innovation, $\boldsymbol\psi_0=(1,0)'$, propagation is $\boldsymbol\psi_h=A^h\boldsymbol\psi_0$:

If contemporaneous innovations are correlated, “a shock to variable 1 while variable 2 is unchanged” is not identified by the reduced-form covariance alone. Cholesky ordering, long-run restrictions, sign restrictions, external instruments, or a structural model add different assumptions and may produce different paths.

3. Stationary covariance is a Lyapunov equation

For VAR(1),

the unconditional covariance $\Gamma_0$ satisfies

Using $\operatorname{vec}(AXB)=(B^\top\otimes A)\operatorname{vec}(X)$,

The eigenvalues check stability, the covariance eigenvalues check validity, and the residual norm checks the Lyapunov equation. These are separate claims.

4. Granger predictability is deliberately modest

$x$ Granger-causes $y$ when lags of $x$ improve forecasts of $y$ after conditioning on the included history. Operationally, compare

with

and test $b_1=\cdots=b_p=0$.

Omitted common causes, measurement timing, anticipation, aggregation, and regime changes can all alter the result.

5. Seasonality is not one phenomenon

For monthly data, $s=12$. Choose the smallest assumption that matches the mechanism:

A multiplicative SARIMA model is written

Reading the notation

SARIMA$(1,1,1)(0,1,1)_{12}$ means:

• one ordinary difference and one annual seasonal difference;
• one non-seasonal AR and MA term;
• one seasonal MA term linking shocks 12 months apart.

It does not mean the order is appropriate. Compare it with seasonal naive and inspect whether double differencing created strong negative autocorrelation.

6. Cointegration preserves a long-run relation

Suppose $x_t$ is a random walk and

where $u_t$ is stationary. Both levels are non-stationary, but

is stationary. The pair is cointegrated. Differencing both series independently would hide the equilibrium error $y_{t-1}-2x_{t-1}$.

A two-variable error-correction equation may be

If $\alpha=-0.3$ and the previous equilibrium error is 10, the correction contributes $-3$ to the next change in $y$. The sign tells how deviations are repaired; cointegration alone does not tell which variable adjusts or identify a causal mechanism.

7. Hierarchical forecasts must add up

Campus demand may be forecast for buildings, faculties, and the whole university; monthly forecasts must also aggregate to quarters and years. Independently fitted forecasts are usually incoherent:

Reconciliation projects or conditions base forecasts so aggregation constraints hold. Recent work on mixed-type probabilistic reconciliation extends this idea when a hierarchy combines counts and continuous quantities. Coherence is a logical requirement; accuracy gains remain empirical and must be backtested at every level.

8. Model-choice map

Practice

1. Multiply the worked VAR matrix by $(0.38,0.11)'$ to verify the horizon-3 response.
2. For $k=8,p=3$, count VAR regression coefficients with intercepts.
3. Give a predictive but non-causal explanation for why temperature lags improve electricity-demand forecasts.
4. Explain why seasonal indicators and seasonal differencing encode different counterfactuals.
5. If $y_{t-1}-2x_{t-1}=-5$ and $\alpha=-0.2$, what is the error-correction contribution to $\Delta y_t$?
6. For the R example, substitute the printed $\Gamma_0$ into $A\Gamma_0A^\top+\Sigma$ and verify one matrix entry by hand.

Takeaway

Multivariate and seasonal models are block-matrix extensions of the univariate theory, but identification becomes harder as dimension grows. The 2024/25 VARMA review is a graduate bridge from these classical equations to current identification and estimation research. Continue to Spectral Analysis to study covariance by frequency rather than lag.