Time Seriesintermediate

Static and Finite Distributed Lag Models

A static model relates $y_t$ to $z_t$ in the same period, assuming the effect is immediate. A finite distributed lag (FDL) model lets the effect play out over several periods by including lags of the regressor, as in $y_t = \alpha_0 + \delta_0 z_t + \delta_1 z_{t-1} + \dots + \delta_q z_{t-q} + u_t$ . The coefficient $\delta_0$ is the impact propensity (the immediate effect of a one-unit change in $z$ ), while the sum $\delta_0 + \delta_1 + \dots + \delta_q$ is the long-run propensity (LRP), the total effect once the change has fully worked through. FDL models are how econometrics captures dynamics without leaving the linear regression framework.

Why it matters

Many economic responses are not instant. A cut in interest rates today nudges investment a little now and more over the following quarters. The FDL model strings together the contemporaneous variable and its lags so each period’s contribution gets its own coefficient. Add those coefficients up and you get the long-run propensity, the answer to "if $z$ rises permanently by one unit, where does $y$ eventually settle?"

Formulas

Finite distributed lag model

y_t = \alpha_0 + \delta_0 z_t + \delta_1 z_{t-1} + \dots + \delta_q z_{t-q} + u_t

Order q lags of the regressor enter alongside its current value.

Long-run propensity

\text{LRP} = \delta_0 + \delta_1 + \dots + \delta_q

Total effect of a permanent one-unit increase in z; the impact propensity is just the first term.

Worked examples

Scenario

You study how a change in the personal exemption affects the general fertility rate, allowing effects to spread over two years.

Solution

Estimate an FDL of order 2: `regress gfr pe L.pe L2.pe, robust`. The coefficient on `pe` is the impact propensity, and the LRP is the sum of the three coefficients. Test the LRP with `lincom pe + L.pe + L2.pe` to get a point estimate and standard error for the total effect.

Note`lincom` gives a valid standard error for the summed coefficients, which you cannot read off the individual rows.

Scenario

A static model `regress consumption income, robust` assumes spending reacts fully within the same period.

Solution

If consumption actually adjusts with a delay, the static specification is too restrictive. Adding `L.income` and `L2.income` lets the data reveal whether the response is spread over time, turning the static model into an FDL.

NoteA static model is the special case of an FDL with no lags included.

Common mistakes

✗The impact propensity is the full effect of a change in $z$ . It is only the immediate effect; the long-run propensity sums the current and lagged coefficients.
✗The long-run propensity is just the largest lag coefficient. The LRP is the sum of all the distributed-lag coefficients, not any single one.
✗Static models are wrong and FDL models are right. A static model is fine when the response really is contemporaneous; the FDL generalizes it for delayed effects.
✗You can read the standard error of the LRP straight from the regression table. You must combine the coefficients, for example with Stata’s `lincom`, because the LRP standard error depends on their covariances.

Revision bullets

•Static model assumes an immediate effect of $z$ on $y$
•FDL adds lags so effects spread over time
•Impact propensity is $\delta_0$ , the contemporaneous effect
•Long-run propensity is the sum $\delta_0 + \dots + \delta_q$
•Use `lincom` for a valid standard error on the LRP

Quick check

In the FDL model $y_t = \alpha_0 + \delta_0 z_t + \delta_1 z_{t-1} + \delta_2 z_{t-2} + u_t$ , the long-run propensity is:

A researcher reports impact and lag coefficients 0.3, 0.2, 0.1. What does the impact propensity tell us?

Connected topics

Logs & Elastic.TS Data Trends TS.1-TS.6 Serial Corr.

Sources

Wooldridge (2019), §10.2
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Defines static and finite distributed lag models and the impact and long-run propensities.
Wooldridge (2019), Example 10.4
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Fertility-rate FDL example used to illustrate estimating and testing the long-run propensity.

How to cite this page

Dr. Phil's Quant Lab. (2026). Static and Finite Distributed Lag Models. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-static-fdl

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