Forecasting Realized Volatility with HAR Models: In-Sample Estimation, Out-of-Sample Forecasting & VaR Backtesting

Overview

An advanced econometrics project examining the accuracy of the Heterogeneous Autoregressive (HAR) family of realized volatility models and their downstream application to Value-at-Risk (VaR) risk management, using two decades of daily realized measures and returns for the SPDR S&P 500 ETF (SPY).

What I did

• •Extracted a defined sample period of daily realized variance, realized quarticity, signed semivariances, VIX², and returns for SPY, and structured the dataset for modelling in Python
• •Tested whether VIX² was an unbiased estimator of realized variance via linear regression, including individual and joint hypothesis tests on the intercept and slope at 5% and 1% significance, repeated under square-root and logarithmic transformations
• •Computed summary statistics and tested the normality of the realized variance and VIX² series (and their transformations) using the Jarque-Bera test
• •Estimated four HAR-family in-sample models (HAR, SHAR, HAR-Q, HAR-IV) across raw, square-root, and logarithmic functional forms, reporting Newey-West robust standard errors (10 lags) and adjusted R² for each
• •Interpreted the statistical significance, sign, and explanatory power of each coefficient, and evaluated the trade-offs between the different functional-form transformations
• •Selected the two best-performing in-sample models and generated out-of-sample forecasts using a 1,500-day rolling window, evaluated via the Mincer-Zarnowitz test
• •Constructed a model-averaging forecast combining the selected models and benchmarked it against the individual models using MSE and QLIKE loss functions, explaining why MAE and HMSE were unsuitable metrics for this setting
• •Tested whether model averaging significantly outperformed the individual forecasts using the Diebold-Mariano test
• •Built Value-at-Risk measures at the 1% and 5% significance levels from both the individual and model-averaged volatility forecasts, and plotted returns against each VaR series
• •Backtested the VaR estimates using the Proportion of Failures (PoF) and Traffic Light tests to assess whether violation rates matched theoretical expectations
• •Delivered a 10-minute individual presentation of the methodology and findings alongside the written report and Python code

Skills & tools