Lecture 12 Heteroscedasticity 6

RV Models: Log Rules • The log approximations rules for the variance and SD are used to change frequencies for the RV and RVOL. For example, suppose we are calculating RV based on frequency 𝑗, 𝑅𝑉 . Using the 𝑗-period 𝑅𝑉 , we can compute the annualized variance as: 𝐽 ∗ 𝑅𝑉 𝑅𝑉 The 𝑅𝑉𝑂𝐿 is the square root of 𝑅𝑉 . RV Models: Log Rules Example: We calculated using 10’ data the daily realized variance, 𝑅𝑉 . Then, the annual variance can be calculated as 260 ∗ 𝑅𝑉 𝑅𝑉 where 260 is the number of trading days in the year. The annualized RVOL is the squared root of 𝑅𝑉 : 𝑅𝑉𝑂𝐿 𝑠𝑞𝑟𝑡 260 ∗ 𝑅𝑉𝑂𝐿 We can use time series models –say, an ARIMA model- for 𝑅𝑉 to forecast daily volatility. 61 RV Models: Quarterly RV From Daily Data Example: Using daily data we calculate 3-mo Realized Volatility (𝑘 = 66 days) for log returns for the MSCI (1970: March – 2020: Oct). > mean(rvol) [1] 0.07725361 > sd(rvol) [1] 0.02592653 # average monthly Rvol in the sample  log approximation: sqrt(3) * 0.04326 = 0.07493 (close!) # standard deviation of monthly Rvol in the sample RV Models: Properties • Under some conditions (bounded kurtosis and autocorrelation of squared returns less than 1), 𝑅𝑉 is consistent and m.s. convergent. • Realized volatility is a measure. It has a distribution. • For returns, the distribution of RV is non-normal (as expected). It tends to be skewed right and leptokurtic. For log returns, the distribution is approximately normal. • Daily returns standardized by RV measures are nearly Gaussian. • RV is highly persistent. • The key problem is the choice of sampling frequency (or number of observations per day). 62 Realized Volatility (RV) Models - Properties • The key problem is the choice of sampling frequency (or number of observations per day). — Bandi and Russell (2003) propose a data-based method for choosing frequency that minimizes the MSE of the measurement error. — Simulations and empirical examples suggest optimal sampling is around 1-3 minutes for equity returns. RV Models - Variation • Another method: AR model for volatility: |  t |    |  t 1 |  t The εt are estimated from a first step procedure -i.e., a regression. Asymmetric/Leverage effects can also be introduced. OLS estimation possible. Make sure that the variance estimates are positive. 63 Other Models - Parkinson’s (1980) estimator • The Parkinson’s (1980) estimator: 𝑠 = { ∑ ln 𝐻 – ln 𝐿 /(4ln(2)T) }, where 𝐻 is the highest price and 𝐿 is the lowest price. • There is an RV counterpart, using HF data: Realized Range (RR): / 4ln 2 }, RRt = { ∑ 100 ∗ ln 𝐻 , – ln 𝐿 , where 𝐻 , and 𝐿 , are the highest and lowest price in the 𝑗th interval. • These “range” estimators are very good and very efficient. Stochastic volatility (SV/SVOL) models • Now, instead of a known volatility at time t, like ARCH models, we allow for a stochastic shock to 𝜎𝒕 , η or υ : 𝜎𝒕 𝜔 𝛽 𝜎 +η , η ~ 𝑁 0, 𝜎η Or using logs: log 𝜎𝒕 𝜔 𝛽 log 𝜎 +υ , υ ~ 𝑁 0, 𝜎 • The difference with ARCH models: The shocks that govern the volatility are not necessarily the shocks to the mean process, ε ’s. • Usually, the standard model centers log volatility around ω: log 𝜎𝒕 𝜔 𝛽 log 𝜎 – ω +υ , Then, E[log(𝜎𝒕 )] = 𝜔 Var[log(𝜎𝒕 )] = κ2 𝜎 /(1 – 𝛽 2).  Unconditional distribution: log(𝜎𝒕 ) ~ N(ω, κ2) 12 8 64 Stochastic volatility (SV/SVOL) models • We have 3 SVOL parameters to estimate: φ = (𝜔, 𝛽 , 𝜎 ). • Like ARCH models, SV models produce returns with kurtosis > 3 (and, also, positive autocorrelations between squared excess returns): Var[𝑟 = E[(𝑟 – E[𝑟 ])2] = E[σ 𝑧 ] = E[σ ] E[𝑧 ] = E[σ ] = exp(2𝜔 + 2 κ2 ) (property of log normal) kurt[𝑟 = E[(𝑟 - E[𝑟 ])4] / {(E[(𝑟 - E[𝑟 ])2] )2 } = E[σ ] E[𝑟 4] / {(E[σ ])2 (E[𝑧 ])2 } = 3 exp(4𝜔 + 8κ2 ) / exp(4𝜔 + 4κ2 ) = 3 exp(4κ2 ) > 3! • Estimation: - GMM: Using moments, like the sample variance and kurtosis of returns. Complicated -see Anderson and Sorensen (1996). - Bayesian: Using MCMC methods (mainly, Gibbs sampling). Modern approach. 12 9 Stochastic volatility (SV/SVOL) models • The Bayesain approach takes advantage of the idea of hierarchical structure: - f(𝒚 |ℎ ) (distribution of the data given the volatilities) - f(ℎ |φ) (distribution of the volatilities given the parameters) - f(φ) (distribution of the parameters) 65