Lesson M08.L02: Static and Finite Distributed Lag Models

Field	Detail
Module	08 — Time Series Basics
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Distinguish static from FDL models; interpret impact effects, lagged effects, and the long-run propensity; apply FDL reasoning to monetary policy transmission.
Provenance	Wooldridge, Introductory Econometrics, Ch 10.2

Explanation

1. Static Model

A static model assumes that the effect of $z_t$ on $y_t$ is entirely contemporaneous — there are no lagged effects:

\[y_t = \beta_0 + \beta_1 z_t + u_t\]

When appropriate: - The adjustment from $z$ to $y$ is instantaneous (financial markets: asset prices react within the period to information). - You have no reason to believe past values of $z$ matter independently.

When inappropriate: - Economic processes take time: investment responds to interest rates over quarters; wages adjust to inflation with a lag; exchange rate changes pass through to import prices gradually.

In most macroeconomic applications, the static model is a restrictive special case of the FDL model.

2. Finite Distributed Lag (FDL) Model

A finite distributed lag model of order $q$ allows $q$ past values of $z$ to independently affect $y$:

\[\boxed{y_t = \alpha + \delta_0 z_t + \delta_1 z_{t-1} + \delta_2 z_{t-2} + \cdots + \delta_q z_{t-q} + u_t}\]

The coefficients $\{\delta_0, \delta_1, \ldots, \delta_q\}$ are called the lag distribution or distributed lag weights.

Interpretation of each coefficient:

Suppose $z$ increases by one unit in period $t$ and remains at the new level forever (a permanent change). Then:

In period $t$: $y$ changes by $\delta_0$ (the impact effect or impact multiplier).
In period $t+1$: additional change is $\delta_1$ (beyond what already happened).
In period $t+j$: additional change is $\delta_j$.

Alternatively, if $z$ increases by one unit in period $t$ only (a temporary change, a pulse), then:

Period $t$: $y$ changes by $\delta_0$.
Period $t+1$: $y$ changes by $\delta_1$.
Period $t+j$: $y$ changes by $\delta_j$.
Period $t+q+1$: effect is zero (the pulse has worked through all lags).

3. Impact Effect

\[\delta_0 = \frac{\partial y_t}{\partial z_t}\]

The impact effect is the immediate response of $y$ to a one-unit change in $z$ in the same period, holding all other lags constant. It is the coefficient on the contemporaneous value of the regressor.

4. Lagged Effects

\[\delta_j = \frac{\partial y_{t+j}}{\partial z_t}, \quad j = 1, 2, \ldots, q\]

The $j$-period lagged effect measures how much $y$ changes $j$ periods after a one-unit change in $z$. These capture the delayed transmission of shocks through the economy.

Lag distribution shape: The pattern of $\{\delta_j\}$ over $j$ traces out the impulse response function — how $y$ responds over time to a one-time shock in $z$.

5. Long-Run Propensity (LRP)

The long-run propensity (or long-run multiplier) is the total cumulative effect on $y$ after all lag effects have worked through:

\[\boxed{LRP = \sum_{j=0}^{q} \delta_j = \delta_0 + \delta_1 + \cdots + \delta_q}\]

Economic interpretation: The LRP tells us the long-run equilibrium change in $y$ for a sustained one-unit increase in $z$. It is the sum of all immediate and delayed effects.

Graphically: The LRP is the area under the lag distribution plot (coefficients vs lag length).

6. Monetary Policy Example: Money Supply and Inflation

Quantity theory motivation. In the long run, sustained growth in the money supply ($m_t$, %) drives inflation ($\pi_t$, %), but the transmission operates with a lag: - Banks receive reserves, expand credit. - Firms and households spend more. - Producers raise prices. - Wage negotiations adjust. - Import prices react.

A simple FDL model:

\[\pi_t = \alpha + \delta_0 \Delta m_t + \delta_1 \Delta m_{t-1} + \delta_2 \Delta m_{t-2} + \cdots + \delta_q \Delta m_{t-q} + u_t\]

Friedman's hypothesis: in the long run, a sustained 1% increase in money growth raises inflation by approximately 1%, so $LRP \approx 1$. But the impact effect $\delta_0$ may be small — prices don't respond immediately.

7. Australian Example: RBA Rate Changes and Inflation

The RBA (Reserve Bank of Australia) adjusts the cash rate to control inflation. A classic stylised fact of monetary policy is the "long and variable lags" (Friedman, 1961):

Rate changes take roughly 12–18 months to fully affect inflation in Australia.
Early effects (0–3 months): mainly exchange rate channel — AUD appreciates, import prices fall.
Medium-term (3–12 months): credit channel — higher rates reduce borrowing, cool demand.
Long-term (12–18+ months): aggregate demand falls, labour market softens, wage growth slows, inflation falls.

A quarterly FDL of inflation on the cash rate change ($\Delta r_t$) with $q = 6$ lags (18 months) would capture this:

\[\pi_t = \alpha + \sum_{j=0}^{6} \delta_j \Delta r_{t-j} + u_t\]

The LRP = $\sum_{j=0}^6 \delta_j$ should be negative (higher rates → lower inflation) and its magnitude reveals the total disinflationary power of a 1 pp rate hike sustained over 18 months.

Worked Example

Given FDL coefficients from a quarterly model of Australian unemployment ($u_t$, %) on GDP growth ($g_t$, %):

\[u_t = 5.8 - 0.2\, g_t - 0.4\, g_{t-1} - 0.3\, g_{t-2} - 0.1\, g_{t-3} + \text{error}\]

Q1. What is the impact effect of a 1 pp increase in GDP growth on unemployment?

\[\delta_0 = -0.2\]

A 1 percentage point rise in GDP growth this quarter is associated with a 0.2 pp fall in unemployment this quarter.

Q2. What is the effect 2 quarters later?

\[\delta_2 = -0.3\]

Two quarters after the 1 pp GDP growth increase, unemployment falls by an additional 0.3 pp (cumulative change from the pulse: $\delta_0 + \delta_1 + \delta_2 = -0.2 - 0.4 - 0.3 = -0.9$ pp).

Q3. Compute the LRP.

\[LRP = -0.2 + (-0.4) + (-0.3) + (-0.1) = -1.0\]

Interpretation: A sustained 1 pp increase in GDP growth is associated with a 1 percentage point permanent fall in the unemployment rate over 3 quarters. This is close to an Okun's Law relationship — a rule of thumb widely used in macroeconomics.

Q4. Where is the largest single-period effect?

The lag with the largest $|\delta_j|$ is $j=1$ ($\delta_1 = -0.4$): the biggest unemployment response occurs one quarter after the GDP growth shock, not immediately.

Common Misconception

"The long-run effect equals the sum of all lagged coefficients — regardless of how many lags I've included."

This requires an important caveat: the LRP formula $\sum_{j=0}^q \delta_j$ is only valid if you have included enough lags to capture the full distributed lag response.

If the true model has $q^* = 8$ lags but you estimate with $q = 3$, your "LRP" is $\sum_{j=0}^3 \delta_j$, which underestimates the true long-run effect. The remaining effects (lags 4–8) are omitted, causing omitted variable bias in all estimated coefficients and a biased LRP.

How to choose $q$? - Use economic theory (e.g. monetary policy lags are roughly 4–6 quarters in Australia). - Start with a long lag length and use AIC/BIC to select the optimal $q$. - Verify that the $\delta_j$'s become small and statistically insignificant at long lags (indicating the distributed lag has been captured).

Practice Prompts

LRP and impact effect. A researcher estimates the following quarterly FDL model for Australian housing starts ($h_t$) on the mortgage rate ($r_t$):

$$h_t = 120 - 5.0\, r_t - 8.0\, r_{t-1} - 6.0\, r_{t-2} - 3.0\, r_{t-3} - 1.0\, r_{t-4} + u_t$$

(a) Interpret the impact effect $\delta_0 = -5.0$. (b) Compute the LRP. Interpret it in economic terms. (c) At which lag does the largest effect on housing starts occur?

Lag selection. You estimate two FDL models of inflation on money growth: one with $q = 4$ lags (AIC = −3.42, BIC = −3.18) and one with $q = 8$ lags (AIC = −3.51, BIC = −3.09). (a) Which model does AIC prefer? BIC? (b) Explain intuitively why AIC and BIC can disagree. (c) If the true lag length is 8, what is the consequence of selecting $q = 4$ for the estimated LRP?
Policy interpretation. The RBA increases the cash rate by 50 basis points (0.5 pp) in a single meeting. Using the lag distribution below, compute the cumulative effect on quarterly inflation after 1, 2, and 4 quarters. Plot (sketch) the impulse response.

Lag $j$	$\delta_j$
0	−0.03
1	−0.08
2	−0.10
3	−0.09
4	−0.07

(Multiply each $\delta_j$ by the shock size of 0.5 pp.)

Further Resources

Wooldridge, Introductory Econometrics (7e), §10.2 — Finite Distributed Lag Models.
Friedman, M. (1961). "The lag in effect of monetary policy." Journal of Political Economy, 69(5), 447–466. — Original "long and variable lags" reference.
RBA (2020). "Monetary Policy Transmission in Australia." RBA Bulletin, March 2020. — Australian policy transmission lags evidence.
Stock, J.H. & Watson, M.W. (2020). Introduction to Econometrics (4e), §14.2 — FDL and distributed lag models.