Unit 4 — Multivariate and State-Space Models

Topics Covered

Syllabus topics: Multivariate time series: Vector ARMA models (VARMA), estimation, forecasting. State-space representation: formulation, Kalman filter basics, forecasting in state-space models.

The unit that is two units

These are two genuinely different frameworks that happen to share a page.

• Multivariate asks: what if several series influence each other?
• State-space asks: what if the thing I want to model is hidden, and I only see noisy evidence of it?

They meet at one point — a VAR can be written in state-space form — but learn them separately.

4.1 Vector autoregression

⚠️ Count the parameters before you fit one

k²p + k, plus k(k+1)/2 in the covariance matrix.

This is why VARs are limited to three or four variables in practice. The parameter count grows with the square of the number of series, and a macroeconomic dataset has 200 quarterly observations at best.

gdp[t]          = 0.6*gdp[t-1] + e
inflation[t]    = 0.3*inflation[t-1] + 0.5*gdp[t-1] + e
unemployment[t] = 0.5*unemp[t-1] - 0.4*gdp[t-1] + e

⚠️ The warning that belongs with every Granger result

It cannot distinguish a real cause from a common third driver that moves one series before the other, and it says nothing about what would happen if you intervened.

Ice cream sales Granger-cause drownings. Both are driven by summer.

Say "X Granger-causes Y" out loud. Never "X causes Y". Clive Granger himself was explicit about this, and examiners look for it.

VARMA, and why nobody uses it

The syllabus says VARMA. It exists — add a moving-average matrix polynomial — and it is almost never used, for one good reason: identification. Different VARMA parameterisations can produce identical dynamics, so the estimates are not unique without extra restrictions.

A VAR with enough lags approximates any VARMA, and it is identified. Say that if asked why practice is all VAR.

4.2 State-space models

   for each t:
     PREDICT:  x̂ₜ|ₜ₋₁ = T x̂ₜ₋₁       (where do I think the state is?)
               Pₜ|ₜ₋₁ = T Pₜ₋₁ Tᵀ + Q  (how sure am I?)

     observe yₜ

     UPDATE:   Kₜ = Pₜ|ₜ₋₁ Zᵀ (Z Pₜ|ₜ₋₁ Zᵀ + R)⁻¹      <- the KALMAN GAIN
               x̂ₜ = x̂ₜ|ₜ₋₁ + Kₜ(yₜ − Z x̂ₜ|ₜ₋₁)
               Pₜ  = (I − Kₜ Z) Pₜ|ₜ₋₁

What state-space does that ARIMA cannot

Practice problems

1. You have 8 quarterly macro series and 25 years of data. Can you fit a VAR(4)?

No. Count first:

• Observations: 25 × 4 = 100
• Coefficients: k²p + k = 64 × 4 + 8 = 264
• Plus a covariance matrix: k(k+1)/2 = 36

264 coefficients from 100 observations is not estimable — the system is badly under-determined, and even VAR(1) at 72 coefficients is hopeless.

What to do:

1. Reduce k. Pick the 3–4 series the question is actually about. This is the standard answer and usually the right one.

2. Reduce p. Let AIC choose; quarterly data often needs only 1–2.

3. Use a Bayesian VAR, whose priors shrink distant lags toward zero — the standard tool in macroeconomics for exactly this problem.

4. Use a factor model. Extract 2–3 principal components from the 8 series and fit a VAR to those.

2. Explain the difference between the Kalman filter and the Kalman smoother.

The filter is causal and real-time — it is what a navigation system runs, because it cannot use observations that have not happened.

The smoother is a backward pass after the forward one, and it is strictly more accurate at every interior point because it uses future observations too.

Which to use: the filter for live estimation and forecasting; the smoother for retrospective analysis — decomposing a historical series, or estimating the gap in the lab's missing-data demonstration.

3. Your VAR says X Granger-causes Y with p < 0.001. A colleague concludes X causes Y. What do you say?

That the test does not support that conclusion, and three specific reasons why.

1. Granger causality is about prediction, not mechanism. All it establishes is that X's past improves a forecast of Y beyond Y's own past.

2. A common driver produces exactly this pattern. If Z drives both, and reaches X before Y, then X Granger-causes Y with no causal link between them at all.

3. It is sensitive to what you left out. Add the omitted variable and the relationship can disappear — which is not true of a real causal effect.

And a fourth, if the series are non-stationary: Granger tests on non-stationary data produce spurious significance. Difference first, or use a VECM if the series are cointegrated.

What would support a causal claim: an intervention, a natural experiment, or an instrument — none of which a VAR provides.

4. Why is a local-level model equivalent to exponential smoothing?

The local-level model is:

$$y_t = \mu_t + \varepsilon_t, \qquad \mu_t = \mu_{t-1} + \eta_t$$

Its steady-state Kalman filter update is:

$$\hat\mu_t = \hat\mu_{t-1} + K(y_t - \hat\mu_{t-1}) = K y_t + (1-K)\hat\mu_{t-1}$$

which is simple exponential smoothing with α = K.

So they are the same model, and the equivalence is useful in both directions: it gives exponential smoothing a statistical foundation — prediction intervals, likelihood, AIC — that the ad-hoc version lacks, and it gives the state-space model an interpretation anyone can follow.

And it explains α. The smoothing constant is not arbitrary: it is the Kalman gain, determined by the ratio of state noise to measurement noise. A noisy sensor gives small α; a reliable one gives large α.

5. When is a state-space model worth the extra complexity over ARIMA?

Five situations, in rough order of how often they arise:

1. Missing data. ARIMA needs a complete series; the filter handles gaps natively, as the lab measured.

2. You need the components. If someone will ask "how much of that rise was trend and how much was seasonal?", ARIMA cannot answer and an unobserved-components model can.

3. Parameters that change over time. A relationship that drifts — a changing seasonal shape, a slope that shifts after an intervention — needs time-varying coefficients, which ARIMA does not have.

4. Real-time updating. One recursion per new observation instead of a refit.

5. Irregular spacing or mixed frequency. Monthly and quarterly series in one model is routine in state space and awkward otherwise.

When ARIMA is enough: a complete, regularly-spaced, single series where you want a forecast and nothing else. Which is most of the time — and "use the simpler model that answers the question" is a defensible exam position, provided you can name what you would gain by not doing so.

Exam questions from this unit

Two marks

1. Write the VAR(p) equation.
2. How many coefficients has a VAR(2) with 4 variables?
3. State the Granger causality null hypothesis.
4. Why is VARMA rarely used?
5. Write the two state-space equations.
6. What is the Kalman gain?
7. Give one difference between the Kalman filter and smoother.
8. How does a state-space model handle missing data?

Five marks

1. Explain VAR models and their parameter-count problem.
2. Explain Granger causality and its limitations.
3. Explain the state-space formulation and the role of each matrix.
4. Explain the Kalman filter's predict–update cycle.
5. Say when a state-space model is preferable to ARIMA, with reasons.

Ten marks

1. Explain multivariate time-series modelling in full — VAR, estimation, Granger causality, forecasting — and discuss when it beats a univariate model and when it does not.

2. Explain state-space models and the Kalman filter completely, with the equations, and relate them to models covered earlier in the course.

Mistakes that cost marks