State-Space Representation

Definition

A state-space representation $(A,B,C,D)$ expresses the equilibrium of a linear (or linearized) dynamic stochastic economic model as a system in which unobserved states $x_t$ evolve via a first-order recursion driven by economic shocks $w_t$ , and observables $y_t$ are a linear combination of states and shocks. It is the canonical form connecting dynamic stochastic general equilibrium (DSGE) theory to empirical time-series objects including vector autoregressions (VARs).

Key Ideas

Any linear/linearized DSGE model can be cast in $(A,B,C,D)$ form by collecting the equilibrium decision rules.
The same $(A,B,C,D)$ implies an innovations representation $(A,K,C,\Sigma)$ via the Kalman filter, which in turn implies a Wold moving average (MA) and an infinite-order VAR for the observables.
Different $(A,B,C,D)$ tuples can represent the same impulse response function; the minimum-state realization is canonical.
The innovations representation contains all first- and second-moment implications of the model for $\{y_t\}$ and equals the model's conditional likelihood function.

How It Works

Economic Model State-Space $(A,B,C,D)$

$x_{t+1} = A x_t + B w_t \tag{1}$ $y_t = C x_t + D w_t \tag{2}$

where $w_t \sim \mathcal{N}(0, I_m)$ is a vector of $m$ orthogonal economic shocks, $x_t \in \mathbb{R}^n$ is the (possibly unobserved) state, and $y_t \in \mathbb{R}^k$ is the vector of observables. Matrix dimensions: $A$ is $n\times n$ , $B$ is $n\times m$ , $C$ is $k\times n$ , $D$ is $k\times m$ .

The impulse response of the economic model is directly readable from $(A,B,C,D)$ :

$y_t = \mu_y + [C(I - AL)^{-1}BL + D]\,w_t, \qquad d_0 = D,\quad d_j = CA^{j-1}B,\; j \geq 1 \tag{5}$

Innovations Representation $(A,K,C,\Sigma)$

The Kalman filter applied to $(A,B,C,D)$ produces a second state-space system — the innovations representation — that has the same $A$ and $C$ but replaces $(B,D,w_t)$ with $(KG, G, \varepsilon_t)$ :

$\hat{x}_{t+1} = A\hat{x}_t + KG\varepsilon_t \tag{10}$ $y_t = C\hat{x}_t + G\varepsilon_t \tag{11}$

where $\hat{x}_t = \mathbb{E}[x_t \mid y_{t-1}]$ , and $G\varepsilon_t = a_t = y_t - \mathbb{E}[y_t \mid y_{t-1}]$ are the one-step innovations.

The steady-state Kalman gain $K$ and state estimation error covariance $\Sigma = \mathbb{E}(x_t - \hat{x}_t)(x_t - \hat{x}_t)'$ satisfy:

$\Sigma = A\Sigma A' + BB' - (A\Sigma C' + BD')(C\Sigma C' + DD')^{-1}(A\Sigma C' + BD')' \tag{12}$

$K = (A\Sigma C' + BD')(C\Sigma C' + DD')^{-1} \tag{13}$

The innovation covariance is:

$GG' = C\Sigma C' + DD' \tag{14}$

The $k \times k$ matrix $G$ is determined only up to postmultiplication by an orthogonal matrix; a specific choice of $G$ corresponds to a structural identification scheme (see Structural Identification).

Wold MA Representation

From the innovations representation, the Wold moving average is:

$y_t = [G + C(I - AL)^{-1}KGL]\,\varepsilon_t, \qquad c_0 = G,\quad c_j = CA^{j-1}KG,\; j \geq 1 \tag{15}$

VAR Representation

Applying a partitioned inverse to (15), when the eigenvalues of $A - KC$ are strictly less than 1 in modulus:

$y_t = C[I - (A-KC)L]^{-1}K\,y_{t-1} + G\varepsilon_t, \qquad A_j^{\text{VAR}} = C(A-KC)^{j-1}K \tag{16-17}$

This is an infinite-order VAR. The dominant eigenvalue of $A - KC$ controls the rate of decay of the VAR coefficients. If any eigenvalue of $A - KC$ equals 1 in modulus, the infinite-order VAR representation does not exist (though finite-order approximations still converge in mean square).

Vector Autoregressive Moving Average (VARMA) Representation (Square Case)

When $k = n$ and $C$ is invertible, premultiplying (16) by $[I - (A-KC)L]C^{-1}$ yields a first-order VARMA:

$y_t = C[(A-KC)C^{-1} + K]\,y_{t-1} + G\varepsilon_t - C(A-KC)C^{-1}G\varepsilon_{t-1} \tag{19}$

This shows that the pure AR representation is generically of infinite order; the finite-order approximation requires sufficiently many lags.

Dynamic Factor Model (Engle-Watson 1981)

The single-factor state-space model for $k$ observable series $y_t$ and one latent factor $x_t$ is:

$x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + v_t, \quad v_t \sim \mathcal{N}(0, Q)$ $y_t = \alpha x_t + \beta z_t + e_t, \quad e_t \sim \mathcal{N}(0, R)$

where $z_t$ is a vector of observed covariates, $\alpha$ is a $k\times 1$ vector of factor loadings (one element normalized to 1 for identification), and $R$ is diagonal (sector errors uncorrelated conditional on the factor). This is the $(A,B,C,D)$ system with $A = [\phi_1, \phi_2; 1, 0]$ , $C = [\alpha, 0]$ in companion form, and the innovations representation is the Kalman filter prediction-error decomposition.

Maximum likelihood (ML) via scoring algorithm (Eq. 3.3–3.12 of Engle-Watson). Let $\theta$ denote all parameters. The scoring update is:

$\theta^{k+1} = \theta^k + \lambda_k J_k^{-1} \frac{\partial L}{\partial \theta}\bigg|_{\theta^k}$

where the gradient $\partial L/\partial \theta$ and the information matrix $J$ are both computed from first derivatives only, using $K$ additional Kalman filter passes. Second derivatives are never required. This makes ML feasible for moderate-dimensional factor models before automatic differentiation existed.

Lagrange multiplier (LM) / score diagnostics. Serial-correlation tests based on the score vector are computable under the null model (without estimating the alternative), a practical advantage over Wald or likelihood-ratio tests that require fitting an expanded model.

Application. Los Angeles metropolitan wages, five sectors (Construction, Retail Trade, Durable Mfg, Nondurable Mfg, Wholesale Trade), 1947–1975. Model B — autoregressive (AR)(2) metro factor ( $\hat\phi_1 = 1.606$ , $\hat\phi_2 = -0.619$ , sum $\approx 0.987$ , near unit root) plus AR(1) sector errors — passes the LM test ( $\text{stat} = 9.1 < \chi^2_6$ ) and outperforms ordinary least squares (OLS) in out-of-sample sum of squared errors (SSE). Factor loadings decline from non-traded (Construction = 1.0) to traded sectors (Wholesale Trade = 0.302), confirming economic interpretability.

Finite-Order VAR Approximation

For practical use, the $n$ th-order VAR coefficients $A_j^{(n)}$ can be computed directly from the autocovariances of $y_t$ via the Yule-Walker equations:

$\operatorname{cy}(j) = CA^j \operatorname{cx}(0)C' + CA^{j-1}BD', \quad j \geq 1$

where $\operatorname{cx}(0)$ solves the discrete Sylvester equation $\operatorname{cx}(0) = A\,\operatorname{cx}(0)A' + BB'$ .

Kalman Filter with Qualitative Dependent Variables (Tanizaki 1993)

Standard Kalman filter requires a continuous observation equation $y_t = x_t'\beta_t + u_t$ with Gaussian $u_t$ . When $y_t$ is binary, the error distribution is no longer normal and the standard algorithm breaks down. Tanizaki (1993) replaces the observation equation with a binary choice model:

$y_t = F(x_t'\beta_t) + e_t, \qquad \operatorname{E}[e_t]=0, \quad \operatorname{Var}(e_t) = F_t(1-F_t)$

where $F(\cdot)$ is a cumulative distribution function (CDF) (normal/probit, logistic/logit, or Cauchy). The CDF is linearized around the current filtered state $\beta_{t|t}$ via first-order Taylor expansion:

$F(x_t\beta_t) \approx F(x_t\beta_{t|t}) + x_t f_{t|t}(\beta_t - \beta_{t|t})$

yielding the pseudo-linear observation equation $\tilde{y}_t = x_t f_{t|t}\beta_t + e_t$ , where $\tilde{y}_t = y_t - F_{t|t} + x_t f_{t|t}\beta_{t|t}$ . This enables the standard Kalman gain/covariance/state update, but iteration at each $t$ is required because $f_{t|t}$ and $F_{t|t}$ themselves depend on $\beta_{t|t}$ .

Algorithm: Forward pass (iterated linearized Kalman filter) → backward pass (Shumway-Stoffer 1982 smoother, yielding $\beta_{t|n}$ , $P_{t|n}$ , cross-covariances $P_{t,t-1|n}$ ) → update $R$ via expectation-maximization (EM)-like moment formula and $\Phi$ by minimizing weighted squared state residuals → repeat until log-likelihood converges.

Distribution-free estimation: Since $y_t$ is qualitative, $\beta_t$ follows a discrete distribution whose exact form is intractable. $R$ and $\Phi$ are therefore estimated without assuming any parametric distribution for $\beta_t$ .

Application. Money excess demand, Jan 1959–May 1989 ( $n=364$ ); $y_t=1$ iff the interest rate rises next period. Time-varying models: $\log L \approx -222$ – $-234$ , $\rho \approx 0.33$ – $0.48$ vs. fixed-parameter probit/logit: $\log L \approx -248$ , $\rho \approx 0.11$ . Estimated $\hat\beta_t$ peaks around 1975 (oil shock) and 1984 (high U.S. rates), confirming time-varying responsiveness. See Tanizaki (1993).

Why It Matters

The $(A,B,C,D)$ formalism is the bridge between DSGE theory and empirical VAR analysis. It enables:

Deriving the VAR implied by a theoretical model analytically.
Computing the exact identification matrix $G = D$ (when invertibility holds).
Evaluating whether a finite-order VAR is a good approximation.
Testing DSGE models against unrestricted VARs via Marginal Data Density comparison.
Extracting unobserved common factors from multivariate panel data (Engle-Watson 1981); the scoring algorithm for ML remains applicable in non-conjugate factor models where closed-form posteriors are unavailable.

Open Questions

Non-square cases ( $k \neq m$ ) are substantially harder; no analogous clean criterion exists.
The minimum-state realization is not always easy to compute for large DSGE systems.