Draft:Local Projections

Basic local projection regression

Let ${\displaystyle y_{t}}$ be an outcome of interest (e.g., output, inflation, employment), and let ${\displaystyle s_{t}}$ be a shock, policy variable, or other treatment measure at time ${\displaystyle t}$. For each horizon ${\displaystyle h}$, a baseline local projection estimates: $${\displaystyle y_{t+h}=\alpha _{h}+\beta _{h}s_{t}+\gamma _{h}'X_{t}+u_{t,h},\qquad h=0,1,\dots ,H.}$$

Here:

• ${\displaystyle \beta _{h}}$ is interpreted as the response of ${\displaystyle y}$ at horizon ${\displaystyle h}$ to a one-unit change in ${\displaystyle s_{t}}$, conditional on controls.
• ${\displaystyle X_{t}}$ is a vector of predetermined controls, often including lags of ${\displaystyle y}$ and lags of ${\displaystyle s}$ (and possibly other variables).
• ${\displaystyle u_{t,h}}$ is the horizon-specific error term.

Estimating the regression separately for each horizon produces the sequence ${\displaystyle \{{\hat {\beta }}_{h}\}_{h=0}^{H}}$. Plotting ${\displaystyle {\hat {\beta }}_{h}}$ against ${\displaystyle h}$ yields an estimated impulse response function.

Interpretation as a linear projection

Each ${\displaystyle \beta _{h}}$ is the coefficient of the best linear predictor (population linear projection): $${\displaystyle (\alpha _{h},\beta _{h},\gamma _{h})=\arg \min _{a,b,c}\;\mathbb {E} \!\left[\left(y_{t+h}-a-bs_{t}-c'X_{t}\right)^{2}\right].}$$ Under conditions such as ${\displaystyle \mathbb {E} [u_{t,h}\mid s_{t},X_{t}]=0}$, ${\displaystyle \beta _{h}}$ is often interpreted as a conditional causal response at horizon ${\displaystyle h}$.^[1]

Impulse responses as conditional expectations

A common reduced-form definition of an impulse response at horizon ${\displaystyle h}$ is a change in a conditional expectation of ${\displaystyle y_{t+h}}$ following a change in the shock or intervention ${\displaystyle s_{t}}$, holding fixed an information set ${\displaystyle {\mathcal {F}}_{t-1}}$: $${\displaystyle \mathrm {IRF} (h)={\frac {\partial }{\partial \delta }}\,\mathbb {E} \!\left[y_{t+h}\mid s_{t}=\delta ,{\mathcal {F}}_{t-1}\right]{\Big |}_{\delta =0}.}$$ In linear settings, if the controls ${\displaystyle X_{t}}$ span (or adequately approximate) the relevant information set ${\displaystyle {\mathcal {F}}_{t-1}}$, this object coincides with the coefficient in the corresponding linear projection of ${\displaystyle y_{t+h}}$ on ${\displaystyle s_{t}}$ and ${\displaystyle X_{t}}$.^[1][3]

Cumulative responses and multipliers

Some applications report cumulative effects: $${\displaystyle \mathrm {CIRF} (h)=\sum _{j=0}^{h}\beta _{j},}$$ or ratios of cumulative responses (e.g., fiscal multipliers), depending on the shock and outcomes studied.

Exogenous or externally measured shocks

If ${\displaystyle s_{t}}$ is constructed to be plausibly exogenous (e.g., a policy “surprise” measured from high-frequency data or a narrative shock), then conditioning on appropriate controls may justify treating ${\displaystyle s_{t}}$ as conditionally exogenous.

Instrumental variables (LP-IV)

When ${\displaystyle s_{t}}$ is endogenous, LPs can be combined with two-stage least squares using an instrument ${\displaystyle z_{t}}$. A common approach estimates horizon-specific IV regressions with ${\displaystyle s_{t}}$ instrumented by ${\displaystyle z_{t}}$. This strategy is widely used in macroeconomics to estimate dynamic causal effects using external instruments (“proxy” identification).^[4]

Connection to VAR impulse responses

In linear settings, LP impulse responses are closely related to VAR-based impulse responses. Plagborg-Møller and Wolf (2021) show that, under appropriate conditions and with a sufficiently rich set of controls, linear LPs and VARs target the same population impulse responses; differences arise from finite-sample performance and specification choices.^[3]

Distributed-lag and direct forecasting interpretation

Each horizon-specific LP is related to a distributed lag model and to “direct” multi-step forecasting regressions, in contrast to iterating one-step-ahead forecasts from a fitted model.

State dependence and nonlinear local projections

LPs are frequently generalized to allow responses to vary across regimes or states by interacting the shock with a state indicator ${\displaystyle q_{t}}$: $${\displaystyle y_{t+h}=\alpha _{h}+\beta _{h,0}s_{t}+\beta _{h,1}(s_{t}\cdot q_{t})+\gamma _{h}'X_{t}+u_{t,h}.}$$ This framework is used to study asymmetries and regime dependence in macroeconomic responses.^[6]

Panel local projections

With panel data (units ${\displaystyle i}$ over time ${\displaystyle t}$), LPs often include unit and time fixed effects: $${\displaystyle y_{i,t+h}=\alpha _{h}+\beta _{h}s_{i,t}+\gamma _{h}'X_{i,t}+\mu _{i}+\tau _{t}+u_{i,t,h}.}$$ Standard errors are often clustered by unit (and sometimes two-way clustered).

Event studies and difference-in-differences

LP-style regressions are used to estimate dynamic treatment effects in difference-in-differences and event study designs, especially with staggered treatment timing.^[7]

Smooth and regularized local projections

To reduce sampling variability across horizons, some methods impose smoothness or regularization on ${\displaystyle \{\beta _{h}\}}$. “Smooth local projections” use basis expansions and penalties to trade bias for variance at longer horizons.^[8]