1.5.2 State-Space Models

Basic Prerequisites: Econometrics & Time Series (stationarity, ARMA, unit roots) and Linear Algebra.

---

1. Intuition & Practical Objective

A state-space model draws a clean line between what is happening and what we see. The state $s_t$ is the full minimal description of the system at time $t$ - the latent level, the true beta, the hidden volatility, the current phase of a cycle. The observation $y_t$ is a noisy, possibly incomplete, possibly indirect view of that state. The model is two equations: how the state evolves, and how observations are generated from it.

Everything you already know is a special case. An ARMA model, a regression, a trend-plus-seasonal decomposition, the “signal + noise” of page 01, a dynamic hedge ratio - all of them are state-space models wearing different clothes. That is the practical payoff: one estimation engine (the Kalman filter) fits all of them, handles missing data for free, delivers the likelihood for free, and separates the part you care about (the state) from the noise you don’t.

The two structural features that make the state-space view worth the notation:

• Latent structure. The state can include things you never observe directly - a random slope, a seasonality term, a slowly-varying hedge ratio. You cannot regress on an unobserved variable; you can filter it.
• Handling of missing/irregular data. A missing observation simply drops out of the update (the filter just predicts); a series sampled at irregular intervals is handled by the same recursion. Rolling OLS cannot do this without ad-hoc patching.

---

2. Mathematical Ground Truth & Derivations

2.1 The general linear-Gaussian form (Tsay 11.26–11.27)

$$s_{t+1}=d_t+T_t{s}_t+R_t{\eta }_t,{\eta }_t\sim N(0,Q_t)$$ $$y_t=c_t+Z_t{s}_t+e_t,e_t\sim N(0,H_t).$$

• $s_t\in {\mathbb{R}}^m$ is the state vector; $T_t$ the transition matrix (what the state does when left alone); $Q_t$ the process-noise covariance ($R_t$ selects which state components are disturbed).
• $y_t\in {\mathbb{R}}^p$ is the observation; $Z_t$ the observation/design matrix (how the state is measured); $H_t$ the measurement-noise covariance.
• $d_t,c_t$ are known (possibly time-varying) intercepts. All matrices may be time-varying and are known or parameterized; the state is random.

The model is linear (observations are linear in the state) and Gaussian (all noise is normal). These two assumptions are exactly what make the Kalman filter the optimal (minimum-MSE) estimator - and exactly what its failure modes violate (pages 05–06).

2.2 The local-level model: the smallest useful state-space model

$$y_t={\mu }_t+e_t,e_t\sim N(0,{\sigma }_e^2);{\mu }_{t+1}={\mu }_t+{\eta }_t,{\eta }_t\sim N(0,{\sigma }_{\eta }^2).$$

State $s_t={\mu }_t$ ($m=1$), $T=1$, $Z=1$, $R=1$, $H={\sigma }_e^2$, $Q={\sigma }_{\eta }^2$. This is the pure “random-walk signal observed through noise” model of page 01. Despite its size it drives realized-volatility filtering, quote smoothing, and - via the equivalence below - the ARIMA(0,1,1) used throughout volatility modeling.

2.3 ARMA and ARIMA are state-space models

ARMA(1,1) $x_t=\varphi x_{t-1}+\theta a_{t-1}+a_t$ has the companion state $s_t=(x_t,\theta a_t{)}^{\top }$ with

$$T=\left(\begin{array}{cc}\varphi & 1\\ 0& 0\end{array}\right),R=\left(\begin{array}{c}1\\ \theta \end{array}\right),Z=\left(\begin{array}{cc}1& 0\end{array}\right),Q={\sigma }_a^2,H=0.$$

Local level $\iff$ ARIMA(0,1,1) (Tsay Eqs. 11.4–11.5). Differencing the local-level model gives an MA(1):

$$(1-B)y_t=(1-\theta B)a_t,(1+{\theta }^2){\sigma }_a^2=2{\sigma }_e^2+{\sigma }_{\eta }^2,\theta {\sigma }_a^2={\sigma }_e^2.$$

So the same two variances $({\sigma }_e^2,{\sigma }_{\eta }^2)$ can be written as a state-space model (with a latent level) or as an ARIMA(0,1,1) (with only observables). The state-space form is the one that separates the signal ${\mu }_t$ from the noise - ARIMA gives you the reduced-form autocovariances but throws the latent level away.

General regression with ARMA errors - the standard “signal plus correlated noise” model in econometrics - is likewise a state-space model: put the regression coefficients and the ARMA error state in $s_t$, set $Z_t=$ (regressors, 1).

---

3. Computational Implementation - the two representations, checked against each other

Part A verifies the local-level $\leftrightarrow$ ARIMA(0,1,1) algebra on Tsay’s own verified Alcoa numbers; Part B confirms an ARMA(1,1) simulated directly and simulated through its companion state-space form are the identical series.

Two things are verified. First, the local-level/ARIMA algebra reproduces Tsay’s published Alcoa decomposition - and the punchline (${\sigma }_e=0.48\gg {\sigma }_{\eta }=0.074$, a 6.5× noise-to-signal ratio) is the state-space statement that most of what you see in a realized-vol series is measurement noise, not real moves. Second, the ARMA and its companion state-space form generate the bit-identical series (difference $0.00\times 10^0$) - the representations are the same object.

---

4. Failure Modes & First-Principles Breakdowns

1. A state-space model is only as good as its transition matrix. $T$ is the model of how the world evolves. If you set $T=1$ for the beta of a name that genuinely re-prices its factor exposures each quarter, no filter can recover the truth - the structure is wrong (this is misspecification, page 05).
2. Non-identifiability between $Q$ and $R$. For a local-level model estimated from observables alone, $Q$ and $R$ are separately identified only through the autocovariances of $y_t$. With short samples or near-unit-root behavior the likelihood is flat, and wildly different $({\sigma }_e,{\sigma }_{\eta })$ pairs fit equally well. The state-space view adds a latent level - but that level’s precision is exactly what the data limits.
3. State-augmentation explodes dimension. Cramming every regressor and its lags into $s_t$ grows $Q,T$ as $O(m^2)$ and destabilizes numerical conditioning; prefer the minimal state and let the filter reject the rest.
4. Differencing is not the same as state-space. ARIMA throws the latent level away; if your goal is to filter (denoise, nowcast) rather than to forecast autocovariances, the state-space form is the right one - they are not interchangeable tools even though they share the same likelihood.

---

5. References

• Tsay, Analysis of Financial Time Series (3rd ed.)
• Tsay
• Durbin & Koopman, Time Series Analysis by State Space Methods
• Harvey, Forecasting, Structural Time Series Models and the Kalman Filter
• Hamilton, J. D.: Time Series Analysis

---

6. Connected Graph Bridges

• Back: 01 · From Zero
• Continue: 03 · The Kalman Filter · Index Hub
• Base: Econometrics & Time Series · Linear Algebra
• Sibling: Fundamental Multi-Factor Models (the static-beta benchmark this generalizes)