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Why GARCH Volatility Forecasts Can Rise or Fall with Horizon

Article Quant Q&A · Author: user3384794
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Summary

The document explains why a multi-step GARCH volatility forecast may rise or fall as the forecast horizon extends. For a GARCH(1,1) model, the conditional expectation of future variance can be written recursively using the model’s persistence, the latest conditional variance, and its long-run variance component. The cited explanation notes that, under stationarity, the influence of the latest variance decays with horizon while the intercept contribution accumulates.

These opposing effects mean the forecast need not move monotonically: it can initially fall toward its long-run level and later rise, depending on the starting variance and parameters. The question concerns an eGARCH forecast in R, but the response gives only a qualitative statement that a related, more complicated pattern applies there. It does not diagnose the specific fitted model or confirm the software output, so users should check their parameter estimates and model specification before generalizing the explanation.

Key ideas

  • A GARCH forecast recursively computes expected future variance from current information.
  • The latest variance’s influence decays with horizon when the model is stationary.
  • The intercept contribution accumulates, so forecasts are not necessarily monotonic.
  • The response does not diagnose the specific eGARCH fit or R output.

Tags

StatisticsVolatilityMachine learning
Full text
# Forecasting using GARCH in R


# Forecasting using GARCH in R












I am using the predict and ugarchforecast functions in R. When I fit my models and try to forecast, I get either only increasing or decreasing values for sigma, does anyone know why?

Thank you

Example:

eGARCHfit2 = ugarchspec(variance.model=list(model="eGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0), include.mean=TRUE), distribution.model="norm") eGARCH2 <- ugarchfit(brentlog2, spec=eGARCHfit2)

ugarchforecast(eGARCH2, data =brentlog2, n.ahead = 21) * GARCH Model Forecast * ------------------------------------ * Model: eGARCH * Horizon: 21 Roll Steps: 0 Out of Sample: 0 0-roll

forecast [T0=1976-06-26 01:00:00]: Series Sigma T+1 0.0002619 0.008350 T+2 0.0002619 0.008387 T+3 0.0002619 0.008423 T+4 0.0002619 0.008459 T+5 0.0002619 0.008496 T+6 0.0002619 0.008532 T+7 0.0002619 0.008569 T+8 0.0002619 0.008605 T+9 0.0002619 0.008642 T+10 0.0002619 0.008678

The first value is the mean which is always constant and the second one is sigma which is always increasing as you can see.

## Answer by Stefan Voigt (score 1)

https://quant.stackexchange.com/a/18175

This should follow from the properties of the forecast - for example the GARCH(1,1) forecast for $h$ steps is computing the conditional expectation of $\sigma^2_{t+h}$ based on the information set-up in $t$. This can be computed recursively by

$$ V(\varepsilon_{t+h}|F_t)=\omega+\alpha\varepsilon_{t+h-1|F_t}+\beta\sigma^2_{t+h-1|F_t}\\ =\omega\sum\limits_{i=0}^{h-2}(\alpha+\beta)^i+(\alpha+\beta)^{h-1}\sigma^2_{t+1} $$ something similar but more complicated should hold for the EGARCH model. Stationarity for $\varepsilon_t^2$ hold if $|\alpha+\beta|<1$. If we assume stationarity the second term of the formula above should decrease with $h$. The first term is an increasing function in $h$ for $\omega>0$ which is a standard assumption to ensure the positiveness of the conditional variance. Therefore it seems to me like there is no monotonicity in $h$ but one saddle point at which the forecast starts to increase.

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