Enhancing Efficiency of Local Projections Estimation with Volatility Clustering in High-Frequency Data

Abstract: This paper advances the local projections (LP) method by addressing its inefficiency in high-frequency economic and financial data with volatility clustering. We incorporate a generalized autoregressive conditional heteroskedasticity (GARCH) process to resolve serial correlation issues and extend the model with GARCH-X and GARCH-HAR structures. Monte Carlo simulations show 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. Our findings contribute to refining LP estimation, enhancing its applicability in analyzing economic interventions and financial market dynamics.

• The paper enhances LP estimation by integrating a GARCH process to address serial correlation in high-frequency data.
• It introduces LP-GARCHX and LP-GARCH-HAR models that incorporate exogenous inputs and heterogeneous structures to capture volatility persistence.
• Monte Carlo simulations show that these models reduce standard errors of impulse responses compared to the standard LP approach.

Enhancing Efficiency of Local Projections Estimation with Volatility Clustering in High-Frequency Data

This paper introduces enhancements to the Local Projections (LP) method, specifically tailored for high-frequency economic and financial data characterized by volatility clustering. The core innovation involves integrating a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) process into the LP framework. This approach aims to address the serial correlation present in LP error terms, thereby improving the efficiency of impulse response estimation. The authors further extend the model by incorporating exogenous terms (GARCH-X) and a heterogeneous structure (GARCH-HAR) into the conditional covariance equation.

Model Specification

The paper begins by outlining the standard LP model, highlighting its limitations in capturing serial correlation in error terms across different horizons, which leads to higher impulse response variance compared to VAR models [LI2024105722]. To address this, the authors propose the LP-GARCH model, where the local projection errors at each step h follow a GARCH-type process:

$$y_{t+h} = c_h + \beta_h y_t + e_{h,t}, \; e_{h,t} \sim N(0, \sigma_{h,t}^2)$$

$$\sigma_{h,t}^2 = \gamma_h + \alpha_{1,h} \sigma_{h,t-1}^2 + \alpha_{2,h} e_{h,t-1}^2$$

To further account for serial correlation, the LP-GARCHX model is introduced. In this model, the conditional variance depends on both the squared projection errors from the h-1 step ($e_{h-1,t}^2$) and the standard GARCH components:

$$y_{t+h} = c_h + \beta_h y_t + e_{h,t}, \; e_{h,t} \sim N(0, \sigma_{h,t}^2)$$

$$\sigma_{h,t}^2 = \gamma_h + \alpha_{1,h} \sigma_{h,t-1}^2 + \alpha_{2,h} e_{h,t-1}^2 + \alpha_{3,h} e_{h-1,t}^2$$

The estimation of the LP-GARCHX model employs a recursive strategy, starting from h=1 to the end of the forecast horizon, to estimate the conditional variance.

Finally, the LP-GARCH-HAR model incorporates the Heterogeneous Autoregressive (HAR) model to capture long-memory effects often observed in high-frequency financial time series [corsi2009simple]:

$$y_{t+h} = c_h + \beta_h y_t + e_{h,t}, \; e_{h,t} \sim N(0, \sigma_{h,t}^2)$$

$$\sigma_{h,t}^2 = \gamma_h + \alpha_{1,h} \sigma_{h,t-1}^2 + \alpha_{2,h} e_{h,t-1}^2 + \alpha_{3,h} e_{h-1,t}^2 + \alpha_{4,h} \tilde{e}_{h-1,t}^2 + 1_{(h-1)>5} \alpha_{5,h} \overline{e}_{h-5,t}^2$$

where $$\tilde{e}_{h-1,t}^2 = \frac{1}{h-1} \sum_{i=1}^{h-1} e_{i,t}^2$$ and $$\overline{e}_{h-5,t}^2 = \frac{1}{5} \sum_{i=1}^5 e_{i,t}^2$$. The LP models are estimated using the Maximum Likelihood (ML) method.

Simulation Study and Results

The authors conduct a Monte Carlo simulation to evaluate the efficiency of the proposed LP models. The data generating process (DGP) is an AR(1) model with a GARCH(1,1) process for the conditional variance. The simulation compares the standard errors of impulse responses from four LP variants: standard LP, LP-GARCH, LP-GARCHX, and LP-GARCH-HAR, against those from the true AR(1)-GARCH(1,1) model.

The simulation results indicate that the LP-GARCH, LP-GARCHX, and LP-GARCH-HAR models generally yield smaller standard errors than the standard LP model, with LP-GARCHX and LP-GARCH-HAR exhibiting similar performance and consistently outperforming LP-GARCH. The differences in standard errors between the LP models and the true model decrease as the sample size increases, suggesting improved efficiency with larger samples. The mean standard errors relative to the true model are smaller for the proposed LP-GARCH, LP-GARCHX, and LP-GARCH-HAR models compared to the standard LP model, particularly when volatility persistence is high.

Conclusion

This paper makes a contribution by integrating GARCH processes into the LP framework to address volatility clustering in high-frequency data. The simulation results support the claim that the proposed LP-GARCHX and LP-GARCH-HAR models enhance the efficiency of LP estimators, especially in the presence of persistent volatility and with larger sample sizes. These models offer a refined approach for analyzing economic interventions and financial market dynamics in high-frequency settings.