Finite Distributed Lag Models

Core Idea

A finite distributed lag (FDL) model lets a variable z affect y with a delay by including current and past values of z as regressors: y_t = α₀ + δ₀z_t + δ₁z_{t−1} + ... + δ_q z_{t−q} + u_t. The coefficient δ₀ is the impact propensity, the immediate effect of a one-unit rise in z. The sequence δ₀, δ₁, ..., δ_q is the lag distribution, tracing the effect of a temporary change over time. The sum δ₀ + δ₁ + ... + δ_q is the long-run propensity (LRP), the eventual effect of a permanent one-unit increase in z.

How It's Best Learned

Take a two-lag model with given coefficients and trace by hand what happens to y after z rises by one unit for a single period and after it rises by one unit permanently. Plot both paths. The temporary path shows the lag distribution; the permanent path climbs to the LRP. Then show how to estimate the LRP and its standard error directly by rewriting the regression.

Common Misconceptions

• The coefficient on the current value of z measures the full effect of z on y. (It is only the impact effect; with lags, the effect of a permanent change builds up to the sum of all lag coefficients.)
• Insignificant individual lag coefficients mean the lags do not matter. (Lags of z are usually highly correlated, so individual coefficients are imprecise even when the LRP is estimated precisely and is significant.)
• A temporary and a permanent change in z have the same long-run effect. (A temporary change has no effect after q periods, while a permanent change raises y by the LRP.)

Explainer

In cross-section regression, x affects y at the same moment. Over time, effects often take a while. A rise in the minimum wage may change employment over several quarters. A tax change may affect births with a delay of a year or more, since decisions about having children respond slowly. A finite distributed lag model captures this by including several past values of the explanatory variable.

With two lags the model is y_t = α₀ + δ₀z_t + δ₁z_{t−1} + δ₂z_{t−2} + u_t. To read the coefficients, run two thought experiments.

First, a temporary change. Suppose z is constant at c, rises to c + 1 in period 0, and returns to c in period 1. In period 0, y rises by δ₀. In period 1, the bump in z appears as z_{t−1}, so y is δ₁ above its old level. In period 2, it appears as z_{t−2}, so y is δ₂ above. From period 3 on, y is back to normal. The sequence δ₀, δ₁, δ₂, plotted against the lag, is the lag distribution. δ₀ is called the impact propensity.

Second, a permanent change. Suppose z rises from c to c + 1 in period 0 and stays there. In period 0, y rises by δ₀. In period 1 both z_t and z_{t−1} are higher, so y is δ₀ + δ₁ above its old level. From period 2 on, y is δ₀ + δ₁ + δ₂ above. That sum is the long-run propensity, often the number policy makers care most about.

Estimation uses OLS, and the usual time-series assumptions apply. One practical issue is multicollinearity. Consecutive values of z tend to move together, so the regression has trouble separating δ₀ from δ₁ from δ₂, and individual standard errors are large. The sum is often estimated well anyway. To get its standard error, define θ = δ₀ + δ₁ + δ₂ and substitute δ₀ = θ − δ₁ − δ₂. The model becomes y_t = α₀ + θz_t + δ₁(z_{t−1} − z_t) + δ₂(z_{t−2} − z_t) + u_t, and the coefficient on z_t estimates θ with its own standard error.

Choosing the number of lags involves a tradeoff. More lags capture slow effects but use up degrees of freedom and add collinearity. F-tests on the last lags and information criteria help. Models that also include lagged y are a natural next step.