HAR models the predictive relationship between next day's realized variance and current realized variance. The model equation is:

Question

BlackTom AI · Accepted Answer

Question restatement: The HAR model describes the predictive relationship between next day’s realized variance and current realized variance. The model equation is:

RV_{t+1} = a + b1 RV_t^{D} + b2 RV_t^{W} + b3 RV_t^{M} + ε_t

Answer options analysis:

Option 1:
RV_{t} = a + b1 RV_{t} D + b2 RV_{t} W + b3 RV_{t} M + ε_t

- This option mirrors the standard HAR specification structure, where the dependent variable is the next-day realized variance, and the regressors are current realized variance measured at daily (D), weekly (W), and monthly (M) horizons. The inclusion of an intercept a, coefficients b1, b2, b3, and the error term ε_t aligns with the typical formulation. The only potential concern is whether the left-hand side should be RV_{t+1} rather than RV_t, but the intended meaning is clear: the relation forecasts next day’s variance from multi-horizon aggregates. This option captures the essential HAR framework components and proper horizon labeling.

Option 2:
RV_{t+1} = a + b1 RV_t^{D} + b2 RV_t^{W} + b3 RV_t^{M} + ε_t + 1

- Here, there is an extra constant term "+ 1" at the end, which is not part of the standard HAR specification and would bias the error structure unless there’s a specific reason to include a deterministic unit term. The addition of this constant does not reflect standard calibration of a regression model for variance forecasting, making this option incorrect.

Option 3:
RV_{t} = a_{t} + b1 RV_t^{D} + b2 RV_t^{W} + b3 RV_t^{M} + ε_t

- This option changes the dependent variable to RV_t instead of RV_{t+1}, which means predicting the current day’s variance from lagged variances. It reverses the temporal direction that HAR is designed for (forecasting the next day’s variance). Additionally, the intercept is written as a_t, implying time-varying intercepts, which is inconsistent with the standard fixed-parameter HAR setup. This mis-specification undermines the forecasting purpose.

Option 4:
RV_{t+1} = a_t + b1 RV_t^{D} + b2 RV_t^{W} + b3 RV_t^{M} + ε_t + 1

- This option combines multiple deviations from the standard model: the intercept is time-varying a_t rather than a constant a, there is an extra constant term "+ 1" at the end, and the dependent variable is correctly RV_{t+1}. The mixture of time-varying intercept and an added constant is not characteristic of HAR, and the extra +1 term again mis-specifies the regression. This makes it incorrect.

Additional notes on conceptual correctness:
- The HAR model uses realized variance measurements at multiple horizons (D, W, M) as regressors to capture different persistence levels; this is consistent with the idea that variance exhibits long memory and multi-scale behavior. The core idea is to explain RV_{t+1} as a linear combination of current RV measured across several aggregations, plus an innovation term. Options that deviate from this structure (wrong dependent variable, extra constants, time-varying intercepts) fail to represent the standard HAR specification.

In summary, Option 1 best matches the conventional HAR specification: RV_{t+1} = a + b1 RV_t^{D} + b2 RV_t^{W} + b3 RV_t^{M} + ε_t, with the appropriate multi-horizon terms and the correct forward-looking dependent variable. The other options introduce extraneous constants, temporal direction issues, or parameter structure that conflict with the HAR model’s intended form.

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HAR models the predictive relationship between next day's realized variance and current realized variance. The model equation is:

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