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Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns

Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns

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Realized Variance: Four Stylized Facts



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• The top panel of Figure 5.3 shows a histogram of the RVs

from Figure 5.1

• The bottom panel of Figure 5.3 shows the histogram of

the natural logarithm of RV

• Figure 5.3 shows that the logarithm of RV is very close to

normally distributed whereas the level of RV is strongly

positively skewed with a long right tail

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Figure 5.3. Histogram of Realized Variance and log

Realized Variance



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Realized Variance: Four Stylized Facts

• The approximate log normal property of RV is the third

stylized fact. We can write



• The fourth stylized fact of RV is that daily returns

divided by the square root of RV is very close to

following an i.i.d. (independently and identically

distributed) standard normal distribution:



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



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Realized Variance: Four Stylized Facts

RVmt+1 can only be computed at the end of day t+1, So this

result is not immediately useful for forecasting purposes

• If a good forecast RVmt+1/t can be made using information

available at time t then a normal distribution assumption

of

will be a decent first modeling strategy.

Approximately:



• where we have now standardized the return with the RV

forecast

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Realized Variance: Four Stylized Facts



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• When constructing a good forecast for RVmt+1 , we need to

keep in mind the four stylized facts of RV:

– RV is a more precise indicator of daily variance than is

the daily squared return.

– RV has large positive autocorrelations for many lags.

– The log of RV is approximately normally distributed.

– The daily return divided by the square root of RV is

close to i.i.d. standard normal.



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Forecasting Realized Variance



• Realized variances are very persistent

• So we need to consider forecasting models that

allow for current RV to matter for future RV



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Simple ARMA Models of Realized Variance



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• AR(1) model allows for persistence in a time series

• If we treat the estimated RVmt as an observed time series,

then we can assume the AR(1) forecasting model



• where t+1 is assumed to be uncorrelated over time and

have zero mean

• The parameters 0 and 1 can easily be estimated using

OLS.

• The one-day-ahead forecast of RV is then

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Simple ARMA Models of Realized Variance

• Since the log of RV is close to normally distributed we

may be better off modeling the RV in logs rather than

levels. We can therefore assume



• The normal property of ln (RVmt+1) will make the OLS

estimates of 0 and 1 better than those in the AR(1)

model for RVmt+1

• The AR(1) errors, t+1, are likely to have fat tails, which

in turn yield noisy parameter estimates

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Simple ARMA Models of Realized Variance



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• As we have estimated it from intraday squared returns, the

RVmt+1 is not truly an observed time series but it can be

viewed as the true RV observed with a measurement error.

• If the true RV is AR(1) but we observed true RV plus an

i.i.d. measurement error then an ARMA(1,1) model is

likely to provide a good fit to the observed RV. We can

write



• which due to the MA term must be estimated using

maximum likelihood techniques

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Simple ARMA Models of Realized Variance

• As the exponential function is not linear, we have in the

log RV model that



• Assuming normality of the error term we can use:

• In the AR(1) model the forecast for tomorrow is



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Simple ARMA Models of Realized Variance



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• And for the ARMA(1,1) model we get,



• More sophisticated models such as long-memory ARMA

models can be used to model realized variance

• These models may yield better longer horizon variance

forecasts than the short-memory ARMA models

considered here

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns

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