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Enhancing Efficiency of Local Projections Estimation with Volatility Clustering in High-Frequency Data
The causal impact of interventions is central to applied economics. The local projections (LP) method, introduced by 10.1257/0002828053828518, has gained widespread use, typically applied with pre-identified structural shocks rather than identifying them during estimation. LP has been used to analyze US monetary policy's impact on global real exchange rates MA2024111891, assess tax progressivity's effects on US income inequality since 1970 JALLES2024111715, and explore crypto shocks' spillover into the US stock market MUSHOLOMBO2023111427.
LI2024105722 show that LPs exhibit a bias-variance trade off in impulse response estimation, finding that the least-squares LP estimator has lower bias than the least-squares vector autoregressive (VAR) estimator but suffers from higher variance, reducing efficiency. This inefficiency arises because LP error terms across horizons are uncorrelated, unlike VAR, where impulse response variance at future horizon depends on error terms from previous horizons. To improve efficiency, much research has focused on refining LP estimators, including the use of HAC and HAR standard errors stock2018identification. More recently, https://doi.org/10.3982/ECTA18756 show that incorporating lagged variables in LP regressions eliminates the need for standard error corrections.
This paper proposes LP models designed for high-frequency economic and financial data, which often exhibit volatility clustering. Our approach captures volatility clustering in the error term using a generalized autoregressive conditional heteroskedasticity (GARCH) process, addressing the serial correlation issue in LP models. We further extend the model by integrating local projection errors into the conditional covariance equation, incorporating exogenous terms (GARCH-X) and a heterogeneous structure (GARCH-HAR). By leveraging the serial dependence in LP error structures, Monte Carlo simulations demonstrate that our approach improves efficiency relative to standard LP across various forecast horizons, remains robust to persistent volatility, and yields greater gains as sample size increases.
The rest of the article is organized as follows. Section (ref) discusses the proposed LP models. Section (ref) presents results comparing different LP models using simulated datasets. Section (ref) concludes with a discussion of our approach and results. This article has an online Appendix containing additional technical details and empirical results.
This section describes the proposed LP models and how the impulse responses are generated from these models. We consider an $N\times1$ financial and economic variables $y_{t}=\left(y_{1t},...,y_{Nt}\right)^{\top}$ at time $t$. For ease of exposition, we assume $N=1$. The standard LP model for $h$-step projection is given by
where $c_{h}$ is the intercept term, $\beta_{h}$ is the impulse response at horizon $h$, and $e_{h,t}$ is the $h$-step projection error term, which is assumed to follow a normal distribution with zero mean and variance $\sigma^{2}$, and is correlated with the past errors. Eq. (ref) shows that the Local Projection approach estimates a separate regression for each time horizon $h$ to estimate impulse response. In general, the LP model yields higher impulse response variance than the VAR model LI2024105722 due to unmodeled serial correlation in LP error terms across different horizons.
In this paper, we extend the standard LP model in Eq. (ref) by modeling the local projection errors at $h$-step projection to follow a GARCH type model,
The model in Eqs. (ref) and (ref) is the LP-GARCH model.
The LP-GARCH model in Eqs. (ref) and (ref) does not account for serial correlation in the h-step projection error term $e_{h,t}$. Next, we introduce the LP-GARCHX model, where the conditional variance $\sigma_{h,t}^{2}$ depends on both the squared projection errors from the $h-1$ step, $e_{h-1,t}^{2}$, and the standard GARCH components, $\sigma_{h,t-1}^{2}$ and $e_{h,t-1}^{2}$. The LP-GARCHX is given by
Note that when $h=1$, the parameter $\alpha_{3,h}=0$. As $\sigma_{h,t}^{2}$ depends on $e_{h-1,t}$ from $h=2$ onwards, we need a set of estimates of $e_{h-1,t}$ to estimate $\sigma_{h,t}^{2}$. We can proceed with a recursive strategy for estimating LP equations starting from $h=1$ to the end of the forecast horizon. The steps are:
The final model, LP-GARCH-HAR, combines the GARCH framework with Heterogeneous Autoregressive (HAR) model of corsi2009simple to capture long-memory effects commonly observed in high-frequency financial time series. The LP-GARCH-HAR model is given by
where $1_{\left(h-1\right)>5}$ is an indicator function that equals $1$ when $h-1>5$, and $0$ otherwise,
and
The LP models are estimated using the Maximum Likelihood (ML) method, with optimization performed in Matlab. Section (ref) evaluates the efficiency of the proposed LP models through a Monte Carlo study.
This section evaluates the efficiency of different LP models in the Monte Carlo study. The Monte Carlo design is described in Section (ref). Section (ref) discusses the results.
The data generating process (DGP) follows a first-order autoregressive (AR(1)) model with a time-varying conditional variance governed by a generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) process:
with the initial conditional variance
We set the true parameter values to $\beta_0=0$, $\gamma=1$, $\alpha_1=0.5$, $\alpha_2=0.3,0.4,0.48$, and $\beta_{1} \in \{0.6, 0.8, 0.9, 0.95 \}$. For each true model, we generate $R=500$ datasets with lengths $T = 500, 1000, 2000, 5000$. The standard errors, derived from the specified true model, serve as benchmarks for evaluating four LP variants: (1) standard LP, (2) LP-GARCH, (3) LP-GARCHX, and (4) LP-GARCH-HAR (see Section (ref)). We compare the impulse response standard errors from these LP models against those from the true models to assess their relative accuracy. The standard errors for the estimates of $\beta_{h}$ for $h=1,...,24$ obtained from different LP models and the true model are calculated as the standard deviation of the estimates $\beta^{(r)}_{h}$ for $r=1,...,R$ and $h=1,...,24$. For robustness in the standard errors, we examine different levels of persistence in the mean process by varying $\beta_1$ and in the GARCH process by adjusting $\alpha_2$.
Figure (ref) reports the standard errors of impulse responses for $h=1,…,24$ steps ahead across four LP models and the AR(1)-GARCH(1,1) model, considering different sample sizes with $\beta_1=0.95$, $\alpha_1=0.5$, and $\alpha_2=0.4$. For brevity, results for $\beta_1=0.6, 0.8, 0.9$ are provided in the online Appendix (Figures (ref) to (ref) in Section (ref)). Overall, the LP-GARCH model yields smaller standard errors than the standard LP model. Additionally, the LP-GARCHX and LP-GARCH-HAR models exhibit similar standard errors, consistently outperforming the LP-GARCH model in terms of efficiency.
To assess the relative accuracy of standard errors across the four LP models compared to the AR(1)-GARCH(1,1) data-generating process (DGP), Figure (ref) plots the differences in standard errors of the estimated impulse responses. The gap between the LP models and the true model narrows as sample size increases, indicating that LP estimator efficiency improves with larger samples. Corresponding results for $\beta_1=0.6,0.8,0.9$ are provided in the online Appendix (Figures (ref) to (ref) in Section (ref)).
Table (ref) presents the mean standard errors relative to the true model (AR(1)-GARCH(1,1)) across the 24-step forecast horizon for the four LP models. It shows that mean standard errors decrease as the sample size increases, and that the proposed LP-GARCH, LP-GARCHX, and LP-GARCH-HAR models consistently produce smaller standard errors compared to the standard LP model. This suggests that our proposed methods enhance the efficiency of LP estimators.
Sections (ref) and (ref) of the online Appendix present results for $\alpha_2=0.3$ and $\alpha_2=0.48$, respectively. Table (ref), along with Tables (ref) and (ref) in the online Appendix, shows that for standard LP models, mean standard errors relative to the true model are larger when $\alpha_2=0.48$ (i.e., a more persistent volatility process) and slightly smaller or unchanged when $\alpha_2=0.3$ (i.e., a less persistent volatility process). In contrast, for LP-GARCH, LP-GARCHX, and LP-GARCH-HAR models, the mean standard errors remain stable across different volatility persistence governed by the various $\alpha_2$ values.
In summary, the simulation study suggests that: (1) The proposed LP models, LP-GARCHX and LP-GARCH-HAR exhibit smaller standard errors than the LP-GARCH and standard LP models; (2) The gap measuring the differences in standard errors of the estimated impulse responses between the LP models and the true model decreases as sample size increases, indicating that LP estimator efficiency improves with larger samples; (3) For the standard LP model, the mean standard errors relative to the true model tend to be larger when $\alpha_2=0.48$ and smaller when $\alpha_2=0.3$. In contrast, for the LP-GARCH, LP-GARCHX, and LP-GARCH-HAR models, the mean standard errors relative to the true model remain consistent regardless of persistence in the GARCH process (i.e., variations in $\alpha_2$).
In conclusion, this paper enhances the LP method by addressing its inefficiency in high-frequency economic and financial data exhibiting volatility clustering. By incorporating a GARCH process and extending the model with GARCH-X and GARCH-HAR structures, we capture the serial correlation in the local projection errors. Monte Carlo simulations demonstrate that exploiting serial dependence in LP error structures improves efficiency across forecast horizons, remains robust to persistent volatility, and yields greater gains as sample size increases. These findings refine LP estimation, broadening its applicability in analyzing high-frequency economic and financial data.