Local Projections and Covariance for Dependent Data

Conceptual map

II-11.01The local-projection contract
II-11.02Shock units
II-11.03Serial dependence
II-11.04Spatial and clustered dependence
II-11.05Meaning of robustness

1. A local projection is horizon specific

A coefficient and its covariance are defined by the response horizon, outcome transformation, shock units, controls, usable dates, and dependence model.

For horizon $h\ge0$ , consider $y_{t+h}=\alpha _{h}+\beta _{h}s_{t}+\gamma _{h}^{\top}x_{t}+u_{t,h}$ . Stack regressors as $z_{t}$ and define the OLS score $g_{t,h}=z_{t}u_{t,h}$ . The usable sample has $T_{h}$ observations after lags, leads, and missing values are removed.

Proposition 1 · HAC sandwich

Under stationarity or an appropriate triangular-array analogue, finite long-run score variance, a central limit theorem, and nonsingular $Q_{h}=\mathbb{E}[z_{t}z_{t}^{\top}]$ , the asymptotic covariance is

\[Q_{h}^{-1}S_{h}Q_{h}^{-1}/T_{h}, \quad S_{h}=\sum _{\ell =-\infty }^{\infty }E[g_{t,h}g_{t-\ell ,h}^{\top }].\](1)

Proof sketch. The OLS normal equations give $\sqrt{T_{h}}(\hat{\theta}_{h}-\theta _{h})=\hat{Q}_{h}^{-1}T_{h}^{-1/2}\sum g_{t,h}$ . A dependent-data central limit theorem assigns the score sum covariance $S_{h}$ , and the sample regressor moment converges to $Q_{h}$ . Slutsky's theorem yields (1). ∎

A Bartlett HAC estimate truncates at bandwidth $L<T_{h}$ and weights lag $\ell$ by $1-\lvert \ell\rvert/(L+1)$ . Its validity requires a bandwidth sequence and dependence conditions matched to the data.

2. Shock scaling transforms coefficients and standard errors together

Replace the shock by $s_{t}^{*}=a s_{t}$ with $a\ne0$ , leaving the remaining regressor span unchanged. Then $\beta _{h}^{*}=\beta _{h}/a$ and $\operatorname{se}(\beta _{h}^{*})=\operatorname{se}(\beta _{h})/|a|$ . The absolute $$t$$ -statistic is invariant.

For $\beta=0.4$ and standard error 0.1, multiplying the shock by 100 gives coefficient 0.004 and standard error 0.001; both $$t$$ -statistics equal four. A plotted response to a 25-basis-point shock must multiply the coefficient by the shock size in the same units.

3. Dependence designs imply different score sums

Time-series HAC aggregates score autocovariances by lag. Cluster covariance aggregates scores within each assignment or sampling cluster before forming outer products. Spatial HAC weights cross-sectional score pairs by distance. Two-way clustering uses an inclusion-exclusion construction over declared cluster dimensions. These estimators coincide only under special covariance structures.

Overlapping local-projection outcomes induce serial correlation through shared future observations even if one-step innovations are uncorrelated. Cumulative responses across horizons require the full cross-horizon covariance; summing horizon-specific standard errors is invalid.

4. Horizon and calendar boundaries change the sample

Failure case · Robustness table built from changing units

Changing annualization, shock normalization, outcome denominator, calendar aggregation, sample endpoint, or cumulative-response definition can move a coefficient without changing the underlying fitted relation. A robustness comparison must hold the response contract fixed or transform every result to common units.

At large $$h$$ , $T_{h}$ shrinks and a fixed bandwidth can approach the sample length. Missing leads can induce composition changes. Near-unit-root outcomes, generated shocks, and pre-estimated factors add terms beyond the elementary sandwich in (1). A singular regressor moment matrix makes the stated OLS coefficient unidentified.

5. Implementation, exercises, and sources

Store one protocol row per horizon: dependent variable, transformation, shock series and scale, controls and lags, date range, usable count, covariance estimator, bandwidth or cluster variables, small-sample correction, and cumulative-response rule. Verify symmetry and PSD behavior of each covariance matrix.

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Exercises

Reproduce the shock-rescaling example.
Write the finite Bartlett estimator corresponding to (1).
For horizons zero through four, enumerate which dates enter each regression and cumulative response.

Partial solutions

1. Divide the coefficient by 100 and its standard error by 100; their ratio remains four. 2. Sum the lag-zero score covariance and weighted symmetric lag pairs through $$L$$ , then premultiply and postmultiply by the sample regressor-moment inverse. 3. Horizon $$h$$ loses the final $$h$$ outcome dates before any lag-related trimming.

Òscar Jordà (2005), “Estimation and Inference of Impulse Responses by Local Projections,” American Economic Review 95, 161–182.Horizon-specific local projections.
Whitney K. Newey and Kenneth D. West (1987), “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica 55, 703–708.Bartlett HAC covariance.
Bruce E. Hansen, Econometrics, chapters on least squares and covariance estimation.Sandwich and cluster formulas.

6. Audit checkpoint

State horizon, outcome units, shock normalization, controls, date support, effective sample, overlap, generated regressors, dependence design, bandwidth or clusters, boundary correction, cross-horizon covariance, cumulative rule, and numerical conditioning.

7. Scope boundary

The chapter covers linear horizon-specific projections and covariance bookkeeping. Identification of structural shocks and nonlinear time-series models require separate arguments.