Common mistake 1 with Time-varying coefficients (Kalman)

Reading the smoothed path as a real-time signal. It is the single most common misuse of a Kalman filter in finance - backtesting a rule on smoothed states makes it look prescient because the states knew the future.

Common mistake 2 with Time-varying coefficients (Kalman)

Concluding drift from a rolling window. A rolling regression ALWAYS wiggles: sampling noise alone produces a moving line, and this app can show it - on genuinely constant data q comes back at about 1e-17 and the path is flat, where a 60-observation window shows a spread of 0.22.

Common mistake 3 with Time-varying coefficients (Kalman)

Testing q = 0 with a chi-square. The null sits on the boundary of the parameter space, so the usual critical value is conservative and a p-value would look more precise than it is.

Common mistake 4 with Time-varying coefficients (Kalman)

Letting the coefficients absorb a misspecification. If the errors are still autocorrelated, the dynamics are missing, and drifting coefficients will happily mop that up while explaining nothing.

EconStatsCraft › Time-varying coefficients (Kalman)

Time-varying coefficients (Kalman)

A regression whose coefficients are allowed to move, estimated by a Kalman filter. It answers what a rolling window is usually asked for, without the arbitrary window: one estimated signal-to-noise ratio instead, and it reports zero movement when there is none - which a rolling window never does.

What it needs

Dependent variable
Regressors - Each gets its own path over time.

The intuition

A regression assumes the relationship is the same in 1995 and 2025. Often it is not - the pass-through from oil prices to inflation, the slope of the Phillips curve, the beta of a stock all move. A time-varying parameter model lets the coefficients follow a random walk and estimates their path, so instead of one number you get a line with a confidence band, and the question becomes whether it moves.

When to use it

When you suspect a relationship changed but cannot say when: monetary pass-through, a stock’s beta, the response of yields to inflation, the effectiveness of an instrument. Also the honest alternative to a rolling regression, which is what most people reach for and which cannot distinguish genuine drift from sampling noise.

Where the estimator comes from

The observation equation is y_t=x_t'\beta_t+\varepsilon_t with \varepsilon_t\sim N(0,\sigma^2), and the coefficients evolve as \beta_t=\beta_{t-1}+\eta_t with \eta_t\sim N(0,Q). Putting \beta_t in the state vector makes this a linear Gaussian state-space model, so the Kalman filter delivers the likelihood and the filtered path \beta_{t\mid t}, and the smoother delivers \beta_{t\mid T}. The ratio of Q to \sigma^2 controls how fast coefficients are allowed to move: Q=0 collapses the model exactly to OLS, and large Q lets the coefficients chase every observation. Estimating Q rather than fixing it is what stops the model from assuming its own conclusion.

The model

\begin{aligned} y_t &= x_t'\beta_t + \varepsilon_t, \quad \varepsilon_t \sim N(0,\sigma^2) \\ \beta_t &= \beta_{t-1} + \eta_t, \quad \eta_t \sim N(0, q\sigma^2 I) \end{aligned}

What it assumes

That the coefficients move as random walks - gradual drift with no tendency to return. That fits a slowly changing structure and fits a one-off regime break less well, though it will still locate the break as a rapid move.
That the observation errors are independent given the state. The one-step-ahead errors are tested for leftover autocorrelation; if they fail, the missing ingredient is usually a lag of y rather than more coefficient movement.
One signal-to-noise ratio q shared by every coefficient. Where one coefficient is genuinely stable and another moves fast, this compromises between them.
The variation being attributed to the coefficients is not really an omitted variable. A coefficient that drifts is observationally close to a model missing a regressor that itself trends, and the data cannot separate them.

Statistical properties

Q = 0 is nested and testable, so "do the coefficients actually move" is a real question rather than an assumption. The test is non-standard because the null is on a variance boundary.
Recursive least squares and CUSUM are the non-model alternatives: cheaper, but they detect instability without estimating the path.
Filtered coefficients are what was knowable at the time; smoothed coefficients use the whole sample. Only the filtered path is legitimate for a real-time claim.
A random walk for coefficients implies they can wander without bound, which is often implausible over long samples. A stationary AR(1) alternative keeps them tethered to a mean.
With several time-varying coefficients the state dimension grows and identification weakens quickly. Letting one coefficient vary while the rest stay fixed is usually the more informative specification.
Wide bands are the normal outcome, not a defect. A coefficient path whose band always contains the OLS estimate is evidence FOR stability, and worth reporting as such.

How to read the output

Read the SMOOTHED path for history and the FILTERED path for anything you would have acted on. The smoothed estimate at 2015 has seen 2024, and the gap between the two lines is the value of hindsight.
Where the band comfortably contains the horizontal OLS line all the way across, that coefficient is not moving and a constant-coefficient regression is adequate.
q is the whole story in one number: the ratio of how fast the coefficients move to how noisy the data is. q estimated at essentially zero means the likelihood preferred a constant-coefficient model, and that is a result rather than a failure.
The bands widen at both ends of the sample, because the filter has least information there. Drift at the very start or the very end deserves the most scepticism and usually gets the least.

A worked reading

A rolling-window regression and a TVP model on the same data usually tell a similar story, and the TVP version tells it better in one specific way: the window length in a rolling regression is an arbitrary choice that determines how much movement you see, whereas the TVP model estimates the equivalent quantity from the data as Q. If Q comes back at zero the model is saying, in its own terms, that the rolling-window movement you were about to interpret was sampling noise from a short window.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Estimated Q at zero	No evidence of variation	OLS is adequate. This is a real finding
Coefficient path extremely jagged	Q too large relative to noise	Check it was estimated rather than fixed
Bands so wide they contain everything	Little information about variation	Report stability rather than a story about the path
A jump at one date	A structural break, not smooth drift	A break model fits that better
Smoothed path telling a clean story	It used the future	Show the filtered path for real-time claims
Several coefficients varying at once	Weak identification	Free one at a time

Common mistakes

Reading the smoothed path as a real-time signal. It is the single most common misuse of a Kalman filter in finance - backtesting a rule on smoothed states makes it look prescient because the states knew the future.
Concluding drift from a rolling window. A rolling regression ALWAYS wiggles: sampling noise alone produces a moving line, and this app can show it - on genuinely constant data q comes back at about 1e-17 and the path is flat, where a 60-observation window shows a spread of 0.22.
Testing q = 0 with a chi-square. The null sits on the boundary of the parameter space, so the usual critical value is conservative and a p-value would look more precise than it is.
Letting the coefficients absorb a misspecification. If the errors are still autocorrelated, the dynamics are missing, and drifting coefficients will happily mop that up while explaining nothing.

Where to read this properly

Cooley and Prescott (1976), "Estimation in the Presence of Stochastic Parameter Variation", Econometrica 44(1).
Durbin and Koopman, Time Series Analysis by State Space Methods (2012), ch. 3.
Primiceri (2005), "Time Varying Structural Vector Autoregressions", Review of Economic Studies 72(3).
Stock and Watson (1996), "Evidence on Structural Instability in Macroeconomic Time Series Relations", JBES 14(1).

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Time-varying coefficients (Kalman)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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