Tanizaki (1993) Kalman Filter Model with Qualitative Dependent Variables

Summary

Tanizaki (1993) extends the standard Kalman filter to handle binary (qualitative) dependent variables, bridging state-space time-series models and cross-sectional binary choice models. The key idea is to replace the continuous observation equation with a binary choice model $y_t = F(x_t'\beta_t) + e_t$ where $F(\cdot)$ is a Cumulative Distribution Function (CDF), then linearize $F$ around the current state estimate via first-order Taylor expansion to produce a pseudo-linear measurement equation that the standard Kalman filter recursion can process. Parameters $R$ and $\Phi$ are estimated without assuming a distribution for the discrete state variable using Expectation-Maximization (EM)-like moment equations on smoothed states. Applied to money excess demand estimation (Jan 1959–May 1989, $n=364$ ), the time-varying parameter model strongly dominates the fixed-parameter probit/logit on both log-likelihood and predictive fit.

Key Claims

The problem: Standard Kalman filter requires a continuous observation equation $y_t = x_t'\beta_t + u_t$ ; when $y_t$ is binary the standard algorithm breaks down because $u_t$ is no longer normally distributed.
State-space-plus-binary-choice formulation: The unobservable latent variable is $y_t^* = x_t'\beta_t - \varepsilon_t$ , with $y_t = \mathbf{1}[y_t^* \geq 0]$ , so $\Pr(y_t = 1) = F(x_t'\beta_t) \equiv F_t$ ; this is rewritten as $y_t = F(x_t'\beta_t) + e_t$ where $e_t$ has mean zero and variance $F_t(1 - F_t)$ (eq. 18).
Linearization: $F(x_t\beta_t)$ is approximated around the current filtered estimate $\beta_{t|t}$ by $F(x_t\beta_{t|t}) + x_t f_{t|t}(\beta_t - \beta_{t|t})$ , yielding a pseudo-linear observation equation $\tilde{y}_t = x_t f_{t|t} \beta_t + e_t$ where $\tilde{y}_t = y_t - F_{t|t} + x_t f_{t|t}\beta_{t|t}$ (eq. 20). This enables the standard mixed-estimation Kalman gain/covariance/state update (eqs. 21–23) — but iteration at each $t$ is required because $f_{t|t}$ and $F_{t|t}$ depend on $\beta_{t|t}$ .
Smoothing: Shumway-Stoffer (1982) backward recursions yield smoothed states $\beta_{t|n}$ , $P_{t|n}$ , and cross-covariances $P_{t,t-1|n}$ (eqs. 25–28).
Distribution-free parameter estimation: Since $y_t$ is qualitative, $\beta_t$ follows a discrete distribution whose exact form is difficult to derive; $R$ is updated via an EM-like moment formula using smoothed states (eq. 24); $\Phi$ is estimated by minimizing the sum of weighted squared state residuals (eq. 31) — neither step assumes a parametric distribution for $\beta_t$ .
Log-likelihood: Evaluated at smoothed states (eq. 32) as $\sum_t [y_t \log F(x_t\beta_{t|n}) + (1-y_t)\log(1-F(x_t\beta_{t|n}))]$ ; used for convergence monitoring.
Three CDF choices: Normal (probit), logistic (logit), Cauchy — qualitatively similar filtered-path shapes; normal distribution achieves highest log L and correlation $\rho$ .
Application — money excess demand: Observable proxy: $y_t = 1$ iff the interest rate rises next period ( $r_{t+1} \geq r_t$ ), used as a proxy for excess money demand since actual money demand $M_t^D$ is unobservable; state vector $(\alpha_t, \beta_t)'$ follows a random walk ( $\Phi = I$ ); $\beta_t$ is the real-interest-rate elasticity of excess demand.
Empirical results: Fixed-parameter probit/logit/Cauchy: $\log L \approx -248$ , $\rho \approx 0.11$ ; time-varying models: $\log L \approx -222$ – $-234$ , $\rho \approx 0.33$ – $0.48$ — strong improvement; $\hat\beta_t$ shows peaks around 1975 (oil shock) and 1984 (high U.S. rates), indicating time-varying responsiveness of excess demand to the real interest rate.
Limitation — local approximation: The first-order Taylor linearization is accurate only near the current estimate; when $F(\cdot)$ is strongly nonlinear at $\beta_{t|t}$ , the approximation can fail; no formal asymptotic theory is derived in the paper.

Concepts Introduced or Extended

Entities Mentioned

Hisashi Tanizaki

Quotes

"The time-varying parameter problem is easily solvable with the Kalman filter. The traditional methods break down if the dependent variable is dichotomous."

"Since $y_t$ is a qualitative variable, $\beta_t$ must be distributed as a random variable of discrete type. Therefore we need to consider an estimation method which does not depend on the distribution of $\beta_t$ ."

My Take

A useful bridge between the state-space/Time-Varying Parameter (TVP) literature and the discrete-choice literature. The core trick — linearize the CDF around the current state estimate and iterate — is the same idea underlying the extended Kalman filter for nonlinear state-space models. The distribution-free estimation of $R$ and $\Phi$ is an honest response to the genuinely awkward status of a discrete latent state. However, the paper provides no asymptotic theory, and the approximation quality depends heavily on the quality of the initial estimate; modern particle-filter approaches would handle the nonlinearity more rigorously. The money demand application is illustrative but narrow: the mapping from interest-rate changes to excess demand is mechanical and the economic identification is thin.