6.3.1 Results and comments
In this section we will specialise the notation to the univariate case, delaying the discussion of the multivariate case for a couple of sections. In particular we will work in terms of volatility, variance and integrated variance processes. We will review results on this topic, giving an intuitive understanding of them and illustrate them on Monte Carlo and real data. The next section will then give a formal proof the result. This is a starred section and so can be skipped at first reading without the loss of the thread of the book for those readers put off by its higher mathematical level.
In the special case ofy∗ being univariate y∗(t) =α∗(t) +
Z t
0 σ(u)dw(u) (6.16)
the following three results hold under assumptions (A)-(C). The first result is that as M → ∞ so, recallingτ(t) =σ2(t),
qM
¯ h
nPM
j=1y2j,i−R¯h(i¯hi−1)τ(u)duo q2R(ii¯h−1)¯hτ2(u)du
→L N(0,1). (6.17)
The second result is that
PM
j=1yj,i2 −R¯h(i¯hi−1)τ(u)du q2
3
PM j=1yj,i4
→L N(0,1). (6.18)
These two limit theorems are linked together by the third result which is that M
3¯h XM j=1
y4j,i→p Z i¯h
(i−1)¯hτ2(u)du. (6.19)
The result (6.18) is statistically feasible, while (6.17) is perhaps more informative from a theo- retical viewpoint. In particular the two results imply:
• PMj=1yj,i2 converges to R¯h(i¯hi−1)τ(u)du at rate√
M. This considerably strengthens the QV result, for now we know the rate of convergence, not just that it converges.
• The limit theorem is unaffected by the form of the drift processα∗, smoothness condition (6.14) is sufficient that its effect becomes negligible. Again this considerably strengthens the QV result which says the p-lim is unaffected by the drift. Now we know this result extends to the next order term as well.
• Knowledge of the form of the volatility dynamics is not required in order to use this theory.
• The fourth moment of returns need not exist for the asymptotic normality to hold. In such heavy tailed situations, the stochastic denominator R(ii¯h−1)¯hτ2(u)du loses its unconditional mean. However, this property is irrelevant to the workings of the theory.
• The volatility processτ(t) can be non-stationary, exhibit long-memory or include intra-day effects.
• PMj=1yj,i2 −R¯h(i¯hi−1)τ(u)du has a mixed Gaussian limit implying that marginally it will have heavier tails than a normal.
• The magnitude of the error PMj=1yj,i2 −R¯h(i¯hi−1)τ(u)duis likely to be large in times of high volatility.
• Conditionally on R¯h(i¯hi−1)τ2(u)du and R¯h(k¯hk−1)τ2(u)du, the errors XM
j=1
yj,i2 − Z ¯hi
¯
h(i−1)τ(u)du and
XM j=1
yj,k2 − Z ¯hk
¯
h(k−1)τ(u)du are asymptotically independent and jointly normal fori6=k.
• Some of the features of (6.17) appear in the usual cross-section asymptotic theory of the estimation of σ2 when zi ∼N ID(0, σ2). Then
√MnM1 PMj=1zi2−σ2o
√2σ4
→L N(0,1), whose natural feasible version is
√MnM1 PMj=1zi2−σ2o q 2
3M
PM j=1zi4
→L N(0,1).
This has quite a few differences from (6.18). In particular the denominator divides byM rather than multiplies byM, while in the numeratorPMj=1zi2 is divided byM where as in the theory for realised variance it is left unscaled.
• These results are also quite closely related to some work on the asymptotic distribution theory for an estimator of Σ(t), the spot (not integrated) variance. The idea there is to compute a local variance from the lagged data, e.g.,
Σ(t) = ¯b h−1 XM j=1
ny∗³t−¯hjM−1´−y∗³t−¯h(j−1)M−1´o2. (6.20)
They the behaviour of this estimator can then studied as M → ∞and ¯h ↓0 under some assumptions. This “double asymptotics” yields a Gaussian limit theory so long as ¯h ↓ 0 andM → ∞ at the right, related rates. The double asymptotics makes it harder to use in practice than our own simpler analysis, which just needsM → ∞. This is made possible because our goal is to estimate the easier integrated covariation rather than the harder spot covariation.
6.3.2 Intuition about the result
The proof of the result spells out the details of why (6.18) and (6.17) hold. Here we build some intuition for the result in the case where the drift processα∗ is set to zero. This maybe helpful to readers before delving into the proof or for readers who do not want to read the proof. As we can see from the result, it holds for all values of i. In order to simplify the notation in the exposition it is helpful to set i= 1 and drop reference to that subscript.
We start with
y∗(t) = Z t
0 σ(u)dw(u), whereσ andw are independent processes. This implies
yj|τj ∼N(αj, τj), where
τj =τ∗³¯hjM−1´−τ∗³¯h(j−1)M−1´, (6.21) the high-frequency increment to integrated variance. Conditional on the path of the variance processτ
u= XM j=1
yj2− Z ¯h
0
τ(u)du=L XM j=1
τj³ε2j−1´,
whereεj i.i.d.∼ N(0,1). Thus u, the realised variance error, is a mixture of weighted centred chi- squared variables. The terms in the sumτj³ε2j −1´are zero means and independent, conditional on the weights. If the weights do not trend upwards or collapse to zero then we might expect
√M uto be roughly Gaussian with a mean of zero and variance of
2M XM j=1
τ2j.
For largeM the locally bounded variation assumption on τ implies, writing τj = M
¯ h
Z ¯hjM−1
¯
h(j−1)M−1τ(u)du,
that M
¯ h
PM
j=1τ2j = M¯h PMj=1τj2
→p ¯hR0¯hτ2(u)du.
The theory and proof firms up these approximations, yielding the infeasible limit result given in (6.17). As this result has a limiting distribution which does not depend upon Σ, it holds unconditionally as well as conditionally onτ.
The step to making the result feasible is to prove (6.19). However,
M 3¯h
PM
j=1yj4 =L 3M¯h PMj=1τj2ε4j
→p ¯hR0¯hτ2(u)du.
We call
M 3¯h
XM j=1
y4j (6.22)
thequarticityof the high frequency data. The application of Slutsky’s theorem yields the desired feasible limit theory (6.18).
6.3.3 Asymptotically equivalent results
The quarticity is not the only consistent estimator of R0¯hτ2(u)du, although in practice we have found it to be the most accurate of all the different options we have considered. In particular
M
¯ h
PM
j=2yj2yj+12 =L M¯h PMj=1−1(τj) (τj+1)ε2jε2j+1
→p ¯hR0¯hτ2(u)du.
Lagging by a single time unit is not particularly important here, some other small lag could have been used. This implies the non-negative estimator
M
¯ h
XM j=1
y4j −M
¯h
MX−1 j=1
yj2y2j+1→p 2¯h Z ¯h
0 τ2(u)du,
and so provides an alternative denominator in the infeasible limit theory (6.17). In particular this delivers the feasible theories
PM
j=1yj2−R0¯hτ(u)du qPM
j=1yj4−PMj=1−1y2jyj+12
→L N(0,1),
and P
Mj=1yj2−R0¯hτ(u)du q2PMj=1−1y2jyj+12
→L N(0,1).
The latter is interesting for it avoids the use of fourth moments of the data in the denominator.
6.3.4 Log transforms and realised volatilities
The basic limit theory results (6.17) and (6.18) can be embellished in a number of ways. One ap- proach, which Monte Carlo experiments suggest improves the finite sample behaviour of asymp- totic approximation, is to take a logarithmic transform. A straightforward application of the delta-method1 yields the infeasible limit theory
qM
¯ h
nlogPMj=1y2j −logR0¯hτ(u)duo r
2R0¯hτ2(u)du/³R0¯hτ(u)du´2
→L N(0,1).
The denominator is invariant to scaling the returns so we might expect the denominator not to vary so much through time even when there is volatility clustering. In practice we have to replace the unobserved denominator by an estimator, yielding the feasible approximation
logPMj=1yj2−logR0¯hτ(u)du r
2PMj=1y4j/³PMj=1y2j´2
→L N(0,1). (6.23)
Confidence limits forPMj=1y2j based on this theory will be non-symmetric due to the curvature of the log-function.
Example 17 We continue with the simulation from Example 16 which was based on an SV model with a Γ(4,8)-OU process for τ. Based on a sample of 12,000 days, we study the perfor- mance of the asymptotic theory based on the original feasible version (6.18) and the log-version
1This is based on approximating logxby
logàx+x−àx àx , whereàx is the p-lims ofx.
0 100 200
−0.25 0.00 0.25
0.50 Figures a: M=48
0 100 200
−0.5 0.0 0.5
Log based approximations
−2.5 0.0 2.5
−5.0
−2.5 0.0 2.5
QQ plots: y axis is observed, x axis expected
standard theory log−based theory 45 degree line
0 100 200
−0.2 0.0 0.2
Figures b: M=96
0 100 200
−0.5 0.0 0.5
−2.5 0.0 2.5
−2.5 0.0
2.5 standard theory log−based theory 45 degree line
0 100 200
−0.1 0.0 0.1
Figures c: M=288
0 100 200
−0.2 0.0 0.2
−2.5 0.0 2.5
−2.5 0.0
2.5 standard theory log−based theory 45 degree line
Figure 6.2: Left graphs: Actual PMj=1yj,i2 −R¯h(i¯hi−1)τ(u)du plotted again i and twice asymptotic S.E.s. Middle graphs: logPMj=1yj,i2 −logR¯h(i¯hi−1)τ(u)du plotted against i and twice asymptotic S.E.s. Right graphs: QQ plot of the standardised realised volatility error (X-axis has the expected quantiles, Y-axis the observed). Code: simple.ox.
(6.23). Column one of Figure 6.2 the time series plot ofPMj=1yj,i2 −R¯h(i¯hi−1)τ(u)duagainsti, given by crosses, together with their 95% confidence intervals based on using two asymptotic standard deviations. These are computed using M = 48,96 and288. The pictures show a number of fea- tures. First, asM increases so the confidence intervals shrink. More interestingly, the size of the intervals vary dramatically through time. This was predicted by the theory, but the practical im- plication is clear that they vary considerably not just in theory. The second column repeats these experiments but based on the log-theory. Now the crosses depictlogPMj=1yj,i2 −logR¯h(i¯hi−1)τ(u)du.
The confidence limits are now almost constant through time. This holds for any value ofM. This implies that on the log-scale the realised variance error is approximately Gaussian. Finally, the third column displays the QQ plots of standardised errors (that is it plots the sorted standardised residuals against the expected quantile from the normal distribution). By standardised errors we mean the left hand side of (6.18) and (6.23). This assesses the Gaussianity of the finite sample distribution and so the performance of the asymptotic distribution. The graphs suggest both the original and log-version have a long left hand tail, but the log-version is much more precise.
The above theory can be used to provide confidence limits for the realised volatilities vu
utXM
j=1
yj2,
by just square rooting the confidence limits for the realised variance . It is of some interest to have a limiting theory directly in terms of the realised volatility however. The resulting theory, again based on the delta method2 has
qM
¯ h
ẵqPM
j=1y2j −qR0¯hτ(u)du
ắ r
2R0¯hτ2(u)du/³R0¯hτ(u)du´
→L N(0,1).
Here the denominator is not scale invariant.