Lecture 8 summary · March 5, 2027

Estimating a latent state as observations arrive

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The economic question

A latent state and its indicator follow separate equations. The Kalman gain weights an observation by its uncertainty relative to the predicted state.

Central relation

a_t=\phi m_{t-1},

V_t=\phi^2P_{t-1}+q,

K_t=\frac{V_t}{V_t+r},

m_t=a_t+K_t(y_t-a_t)

Individual problem

Predicted mean is two, predicted variance 0.6, observation variance 1.4, and new observation five. Find gain, updated mean, and variance.

Worked reasoning

Gain is 0.3, filtered mean 2.9, and filtered variance 0.42. A missing observation skips the update; prediction still advances.

A consequential failure

Smoothed historical states are used as if filtered at their origin, or a given-parameter likelihood is called parameter estimation.

Repair

Keep predictions, filters, and smoothers separate. Check limiting cases and identify given versus estimated parameters.