parameters allow for the characterization of each state with various features such as higher
or lower volatility, slower or faster reversion to the long-run mean, strong or insignificant
leverage effects.
The regime-switching models described below the behavior of implied volatility function
of past returns and other conditioning variables. The following model (1) tests for regimes
of volatility expectations depending on the long-run mean and leverage effects
vt = wi +βirt-1 +ζt (1)
where the error terms are distributed asζt ~ i.i.d. N (0,σζ2) . This model defines regimes in
terms of higher and lower implied volatility judging from the magnitude of drifts and the
sign and significance of slope coefficients. A negative coefficient suggests that the implied
volatility index tends to increase during bearish markets relationship, providing a measure
following Whaley (2000) of investors’ fear and anxiety. It is also possible to test for mean
reversion in expectations of future volatility by accounting for information contained in past
observations of implied volatility.
vt = wi +δivt-1 +βirt-1 +ζt (2)
According to model equation (2), the dynamics of implied volatility can be driven by
past levels of implied volatility as well as past returns. An extension of model (1) to include
the squared returns provides a more appropriate test for asymmetric effects of news on
implied volatility as model (3) accounts for both the sign and magnitude of shocks to the
return-generating process.
vt = wi + βirt-1 +γirt2-1 +ζt (3)
It is also possible to integrate models (2) Model (3) to allow for long-run mean
reversion in implied volatility as well as leverage effects. Model (4) allows for the presence
of nonlinearities in the relationship between expected volatility and market returns across
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