From an Economic Question to a Local Projection

Lecture 01

Tyler Sotomayor · Columbia University

Korea, 1950

  • 1950Q3: military news worth 0.600 of a year’s trend GDP
  • GDP: 2.75 percent below trend in 1950Q2, 5.12 percent above it in 1953Q1, a rise of 7.87 points
  • The same quarters: tax increases, the 1951 Treasury–Fed Accord, wage and price controls

One realized path, with everything mixed in. We want what happened because of the news: a path nobody observed.

Two paths, one gap

Two twelve-period outcome paths coincide through period 4, separate by exactly one unit at period 5, and the gap then halves each period, while the realized path swings well above and below its period-4 value.

The gap y_t-y^{c}_t: 1, 0.5, 0.25 at t=5,6,7. The change since y_4: -0.731, -0.197, 1.970.

Two clocks

Calendar time t dates the intervention and the regression row. Horizon h counts the periods after it.

Intervention switched off Dates of the gaps Horizons Gaps
at t=5 5, 6, 7 0, 1, 2 1, 0.5, 0.25
at t=10 10, 11, 12 0, 1, 2 1, 0.5, 0.25

Same gaps at different dates: the rule treats an intervention the same way whenever it arrives, so the response depends on h, not on t. The impulse-response function is h\mapsto\theta_h.

The 7.87-point rise was a calendar-time fact about 1950–1953. The policymaker asked a horizon question, and one regression will estimate one point on its curve.

Persistence sets the shape

Iterate y_t=\rho\,y_{t-1}+\theta_0 s_t+v_t forward: \theta_h=\theta_0\rho^{h}.

Two geometric decay curves start from the same value of 1 at horizon 0; the curve for rho equal to 0.5 is close to zero by horizon 5, while the curve for rho equal to 0.9 is still near 0.3 at horizon 12.

The intervention sets only the first step; persistence does the rest. An AR(1) response can only decay.

The forward equation is a regression

\begin{aligned} y_{t+h}&=\theta_0\rho^{h}\,s_t+\rho^{h+1}\,y_{t-1}+u_{t,h}\\ u_{t,h}&=\textstyle\sum_{j=0}^{h}\rho^{h-j}v_{t+j}+\theta_0\sum_{j=1}^{h}\rho^{h-j}s_{t+j} \end{aligned}

u_{t,h} holds only terms dated t or later, and not s_t.

Interventions are unpredictable and independent of the disturbances (A2, A3), so u_{t,h} is uncorrelated with s_t and y_{t-1} and \beta_h=\theta_h. Least squares did none of that work.

One regression per horizon: a local projection. Without y_{t-1}, the same \beta_h but a noisier estimate (sd of \hat\beta_0 0.218, not 0.073).

Rows by hand

Three stacked panels for horizons 0, 1, and 2, each showing twelve dated rows; the usable rows start at row 2 in every panel and end at rows 12, 11, and 10, and each usable row is linked to the outcome h periods later.

T_h=T-h-p: 11, 10, 9 rows. Slopes 0.617, -0.983, 2.050 against \theta_h of 1, 0.5, 0.25.

One episode

With a single intervention row \tau:

\hat\beta_h=y_{\tau+h}-\bar y_{\text{others}}

The hand table with only the t=5 intervention, at h=0:

\hat\beta_0=-1.632=\underbrace{1}_{\theta_0} +\underbrace{0.5\,y_4+v_5-\bar y_{\text{others}}}_{\text{everything else}\;=\;-2.632}

Korea is one such row. More episodes help only if what else happened is unrelated to the news.

One draw against the truth

Estimated responses from one simulated sample over horizons 0 to 12 lie below the true geometric response at every horizon and fall below zero after horizon 9, while a wide simulated 90 percent band surrounds the true response.

One T=200 sample: all estimates below \theta_h=0.9^h.

A caption names the intervention, the units of s and y, the horizon unit, and the counterfactual.

Inside the residual

\begin{aligned} u_{t,1}&=\rho\,v_t+v_{t+1}+\theta_0s_{t+1}\\ u_{t+1,1}&=\rho\,v_{t+1}+v_{t+2}+\theta_0s_{t+2} \end{aligned}

Adjacent rows share v_{t+1}: correlation \rho/(2+\rho^2). At horizon h, residuals up to h rows apart are correlated.

Across horizons, u_{t,h+1}=\rho\,u_{t,h}+v_{t+h+1}+\theta_0s_{t+h+1}.

Lecture 5 shows that inference depends on the autocorrelation of s_tu_{t,h}, which is zero here because s_t is independent of all the other terms and y_{t-1} is a control.

First contact: military news and GDP

Estimated response of GDP relative to trend over horizons 0 to 20 quarters rises from about 0.08 at impact to a peak of about 0.40 at quarter 10 and declines to about 0.08 by quarter 20.

y_{t+h} on (1,s_t,y_{t-1}), 1890Q1–2015Q4, T_h=504-h.

News worth 1 percent of trend GDP is followed ten quarters later by GDP 0.40 percent of trend GDP above what the regression otherwise predicts.

What has not yet been argued

Quarter News, fraction of trend GDP Share of variation Cumulative share
1941Q4 0.6919 0.2612 0.2612
1950Q3 0.6001 0.1958 0.4570
1942Q3 0.4328 0.1009 0.5579
1950Q4 0.4022 0.0869 0.6448

Shares of \sum_t(s_t-\bar s)^2 in the h=8 sample of 496 quarters. Dropping 1950Q3 alone moves \hat\beta_8 from 0.362 to 0.434.

Still to argue: units (Lecture 2); controls, since Ramey and Zubairy’s set lowers \hat\beta_{10} to 0.294 (Lecture 3); identification (Lectures 3–4); uncertainty (Lectures 5–6); episode dependence (Lecture 13).

The Shock-to-Response Explorer

Open the Explorer; predict before touching a control.

  1. Two paths: switch off the t=5 intervention.
  2. Rows: move h and watch rows leave the sample.
  3. One draw: toggle y_{t-1} and shorten T.
  4. Military news: drop 1941Q4 or 1950Q3 at h=8.

Practicum 01 reruns these rows in Stata and reproduces the levels curve of Jordà–Taylor Figure 1a at a reduced draw count.

Exercises, hints, and worked solutions

  1. Start with the Lecture 01 exercises: six by hand, four in Stata, one of them on the military-news data.
  2. Open Hint only when you need a direction.
  3. Compare your work with the Detailed solution. Every Stata solution ends in assertions you can run.

The solutions companion also provides a printable PDF with all 10 hints and solutions expanded.

Next lecture

  • Every regression here used the level y_{t+h}, with y_{t-1} on the right-hand side.
  • The opening’s 7.87 points was a change, y_{t+h}-y_{t-1}, and a change since the date before is not a causal response.

If we regress y_{t+h}-y_{t-1} instead of y_{t+h}, is the answer different, or the same answer in different clothes?

Lecture 2: levels, differences, cumulative responses, and units, on the same military-news data.