State Space, Mixed Frequency, and Nowcasting

Use Kalman prediction and update matrices for latent financial states, macroeconomic nowcasts, missing releases, and ragged edges.

State Space, Mixed Frequency, and Nowcasting

One architecture for unobserved states

A linear Gaussian state-space model is

\boldsymbol\alpha_t =T_t\boldsymbol\alpha_{t-1}+R_t\boldsymbol\eta_t, \qquad \boldsymbol\eta_t\sim(0,Q_t),

\mathbf y_t =Z_t\boldsymbol\alpha_t+\boldsymbol\varepsilon_t, \qquad \boldsymbol\varepsilon_t\sim(0,H_t).

The state $\boldsymbol\alpha_t$ is not directly observed. The data $\mathbf y_t$ are noisy, incomplete measurements of it.

Finance state	Possible measurements
latent efficient price	transaction prices from several venues
time-varying beta	asset and factor returns
latent volatility	returns, realised measures, option-implied measures
yield-curve factors	yields at multiple maturities

Economics state	Possible measurements
current economic activity	employment, surveys, production, quarterly GDP
output gap	output, inflation, unemployment
common inflation trend	sectoral price indices and expectations
potential growth	noisy and revised national accounts

The equations are shared. The measurement calendar and substantive interpretation differ.

Prediction and update

Given filtered state $(\mathbf a_{t-1|t-1},P_{t-1|t-1})$ :

Predict

\mathbf a_{t|t-1}=T_t\mathbf a_{t-1|t-1},

P_{t|t-1}=T_tP_{t-1|t-1}T_t^\top+R_tQ_tR_t^\top.

Measure

\mathbf v_t=\mathbf y_t-Z_t\mathbf a_{t|t-1}, \qquad F_t=Z_tP_{t|t-1}Z_t^\top+H_t.

Update

K_t=P_{t|t-1}Z_t^\top F_t^{-1},

\mathbf a_{t|t}=\mathbf a_{t|t-1}+K_t\mathbf v_t, \qquad P_{t|t}=P_{t|t-1}-K_tF_tK_t^\top.

The Kalman gain weights new information by relative uncertainty. A noisy release receives less weight; a precise surprise receives more.

Missing values are a measurement design

If a series is unavailable at time $t$ , remove its row from $\mathbf y_t$ and $Z_t$ and the matching rows and columns from $H_t$ . The state transition still runs.

This matters in two different ways:

Finance: markets close at different times, instruments become illiquid, and high-frequency measures are asynchronous.
Economics: releases arrive on different dates, quarterly outcomes are absent in most months, and the latest panel has a ragged edge.

Imputing future releases before filtering would destroy the information-set interpretation.

Mixed-frequency measurement

Let monthly latent activity be $x_t$ and define the state

\boldsymbol\alpha_t=(x_t,x_{t-1},x_{t-2})^\top.

A monthly indicator measures

m_t= \begin{bmatrix}1&0&0\end{bmatrix}\boldsymbol\alpha_t+\varepsilon_{m,t},

while a quarterly average released every third month measures

q_t= \begin{bmatrix}1/3&1/3&1/3\end{bmatrix} \boldsymbol\alpha_t+\varepsilon_{q,t}.

The frequency mismatch is encoded in $Z_t$ rather than erased by filling the quarterly series forward.

R laboratory: a ragged-edge nowcast

Kalman nowcast with monthly and delayed quarterly measurements

At month 33 the quarterly release usually reduces uncertainty. At month 36 it is missing, so the filter nowcasts from the monthly indicator and propagated state rather than pretending that GDP has already been observed.

Real-time macroeconomic nowcasting

Giannone, Reichlin, and Small show how a dynamic-factor framework can update the current-quarter assessment as new releases arrive. A credible nowcast records:

the release calendar and vintage;
the state or factor specification estimated at that date;
the news component $\mathbf v_t$ of each release;
the Kalman weight translating news into the nowcast revision;
later data revisions separately from model error.

Evaluating on the latest revised panel answers “How well can final values be reconstructed?” It does not answer “What could have been known in real time?”

Finance interpretation

For a time-varying beta,

r_{i,t}=\alpha_t+\beta_t r_{m,t}+\varepsilon_t, \qquad \beta_t=\beta_{t-1}+\eta_t.

The filter balances noisy current returns against gradual beta movement. The same warning applies: a filtered beta is a model-dependent latent estimate, not a directly observed characteristic. Trading use additionally requires an execution rule and out-of-sample risk assessment.

Smoothing is not forecasting

Filtering estimates $\boldsymbol\alpha_t$ using data through $t$ .
Forecasting estimates a future state using data through $t$ .
Smoothing estimates a past state using later data.

Smoothing is valuable for historical decomposition but leaks future information if presented as a real-time signal.

Practice

What happens to $K_t$ as measurement variance $H_t$ becomes very large?
Why does carrying the last quarterly GDP value forward create a false measurement?
When is a smoothed recession state inappropriate?

Answers

The gain on that measurement approaches zero; the filter trusts the state prediction.
It says the old value is a new observation with artificial precision.
When evaluating what an analyst could classify or forecast at the historical date.

Next: Forecast Evaluation for Decisions.

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