Why HAR Realized Variance Forecasts Do Not Simply Explode

Summary

The document raises a concern about the heterogeneous autoregressive model for realized variance, which regresses current realized variance on daily, weekly, and monthly lagged measures. Positive coefficients do not mean the model repeatedly multiplies yesterday’s realized variance by a single factor: the predictors represent averages over different horizons, and the model is a forecasting regression rather than a recursion that replaces observed variance with its own prediction indefinitely.

Realized variance is calculated from intraday squared returns for each period, and those observations are used to estimate the HAR relationship. The model’s residual captures the difference between observed realized variance and its fitted value; it is not a separate variance input that must always be added to forecasts. The question provides coefficient estimates but no data or model diagnostics, so it does not establish whether a particular implementation is correct. The discussion centers on interpreting the specification and the role of observed realized variance, rather than on forecast evaluation or HARQ estimation.

Key ideas

• HAR uses realized variance measured over daily, weekly, and monthly horizons as regression predictors.
• Positive coefficients do not imply that the model recursively replaces every realized observation with a forecast.
• Intraday squared returns provide the realized variance observations used to fit and update the model.
• The residual represents forecast error and can be negative without making observed realized variance negative.
• Model behavior depends on the precise construction of horizon averages and the forecasting procedure.

Tags

# What is wrong with my HAR model - constantly increasing?


# What is wrong with my HAR model - constantly increasing?












I am trying to code the HAR (and eventually the HARQ) from Bollerslev et al 2016 and Corsi 2009.

$$ RV_t = \beta_0 +\beta_1 RV^d_{t-1} + \beta_2 RV^W_{t-1}+\beta_3 RV^M_{t-1}+u_t $$

Bollerslev estimates the parameters as: $\beta_0 = 0.1123, \beta_1 = 0.2273, \beta_2 = 0.4903$, and $\beta_3= 0.1864 $

With these all being positive my $RV_t$ is just an increasing amount of $RV_{t-1}$ and approaching infinity. What have I misunderstood? I tried including the $u_t$ but this sometimes led to negative realised variance values so I assumed it was wrong.

Aside: In a lot of these realised variance papers they discuss that $RV_t = \sum^M_{i=1} r_{t,i}^2$ as the squared returns for M periods in day $t$. Then they go on to propose the HAR/HARQ etc model which is calculating $RV_t$. Are we only using the squared returns calculation for the $RV_0$ and from then on using the proposed model?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.