6.6 Distribution theory for realised covariation
6.6.3 Distribution theory for derived quantities
Regression plays a central role both in theoretical and empirical financial economics. For exam- ple, the regression of the returns of an individual asset on a wide market index is often called a
“beta.” In this Section we use our distribution theory for realised covariation to derive a theory for univariate regression. Again this will be based on fixed intervals of time and allowing the number of high frequency observations to go to infinity within that interval. We regress variable l on variablek, then again surpressing subscriptsi,
βb(lk)= PM
j=1yj(k)yj(l) PM
j=1y2j(k) . (6.60)
This involves just elements of the realised covariation and so we can use the asymptotic theory of the previous section to derive its asymptotic distribution. The probability limit of regression is known by the theory of QV. In particular
βb(lk)→p
hy(k)∗ , y∗(l)i
hy(k)∗ i =β(lk). (6.61)
Here we extend the theoretical results to derive the asymptotic distribution, under our additional assumptions given above. In this caseβ(lk) has the simpler form of
β(lk)= R¯hi
¯
h(i−1)Σkl(u)du R¯hi
¯
h(i−1)Σkk(u)du. (6.62)
The asymptotic distribution can be derived using the delta method6which yields, asM → ∞, the infeasible limit theory
qM
¯ h
³βb(lk)−β(lk)´ r³R¯hi
¯
h(i−1)Σkk(u)du´−2g(lk)
→L N(0,1),
where
g(lk)=d0(lk)Ψ(lk)d(lk), d(lk)=³ 1 −β(lk) ´ (6.63) and
Ψ(lk)= Z ¯hi
¯ h(i−1)
( Σkk(u)Σll(u) + Σ2kl(u) 2Σkk(u)Σkl(u) 2Σkk(u)Σkl(u) 2Σ2kk(u)
)
du. (6.64)
In practice we have to replace Ψ(lk) and d(lk) by estimators to make the above regression theory feasible. However, the previous section implies this is straightforward. In particular as M → ∞
βb(lk)−β(lk) r³PM
j=1yj(k)2 ´−2bg(lk)
→L N(0,1). (6.65)
6This is based on approximatingx/yby àx
ày +x−àx ày −
Ăy−ày àx à2y = àx
ày + x ày −yàx
à2y , whereàx andàyare the p-lims of xandyrespectively.
where
xj =yj(k)yj(l)−βb(lk)yj(k)2 and bg(lk) = XM j=1
x2j −
MX−1 j=1
xjxj+1. (6.66) An attractive feature of this theory is that all of the required terms are straightforward to compute. It is interesting to note thatPMj=1xj = 0 exactly in this context.
Realised correlation
The same strategy can be used to derive the asymptotic distribution of the realised correlation coefficient. We define
b ρ(lk)=
PM
j=1yj(k)yj(l) qPM
j=1y2j(k)PMj=1yj(l)2
→p
hy(k)∗ , y∗(l)i rhy∗(k)i hy(l)∗ i
=ρ(lk)=
R¯hi
¯
h(i−1)Σkl(u)du qR¯hi
¯
h(i−1)Σkk(u)duR¯h(i¯hi−1)Σll(u)du. (6.67) The infeasible asymptotic distribution can be derived using standard linearisation methods7. In particular as M → ∞so
qM
¯ h
³bρ(lk)−ρ(lk)´ r³R¯hi
¯
h(i−1)Σkk(u)duR¯h(i¯hi−1)Σll(u)du´−1gi(l,k)
→L N(0,1), (6.68)
where
g(lk)=d0(lk)Π(lk)d(lk), d(lk)=³ −12β(lk) 1 −12β(kl) ´0 (6.69) and
Π(lk)= Z ¯hi
¯ h(i−1)
2Σ2kk(u) 2Σkk(u)Σkl(u) 2Σ2kl(u) 2Σkk(u)Σkl(u) Σkk(u)Σll(u) + Σ2kl(u) 2Σll(u)Σkl(u) 2Σ2k2(u) 2Σll(u)Σkl(u) 2Σ2ll(u)
du. (6.70) The feasible limit theory is that as M → ∞ so
b
ρ(lk)−ρ(lk) r³PM
j=1yj(k)2 PMj=1y2j(l)´−1bg(lk)
→L N(0,1). (6.71)
where
xj =yj(k)yj(l)−1
2βb(lk)y2j(k)−1
2βb(kl)yj(l)2 and bg(lk)= XM j=1
x2j −
MX−1 j=1
xjxj+1. (6.72) Example 20 We continue with Example 18. Figures 6.7(a), (b) and (c) show the sample path of ρ(lk) for this model, together with its realised estimator bρ(lk) based on a variety of values of M. We see that for small values of M the estimator is poor, but by the time M = 288 it is reasonably precise except when the correlation is low. The bottom row of pictures in Figure 6.7 shows the corresponding errors bρ(lk)−ρ(lk) together with their standard errors computed using (6.71). We see the intervals increasing in size whenρ(lk) is close to zero and reducing otherwise.
7This is based on approximatingx/√yzby àx
√àyàz +(x−àx)
√àyàz −1 2
Ăy−ày àx ày√àyàz −1
2
(z−àz)àx àz√àyàz
= àx
√àyàz +ρ àx
àx −1 2
y ày −1
2 z àz
ả ,
whereàx,ày,àz are the p-lims ofx,yandzrespectively.
0 100 200
−0.5 0.0 0.5
1.0 Figure a: M=12
Realised correlation Actual correlation
0 100 200
−0.25 0.00 0.25 0.50 0.75
Figure b: M=48
Realised correlation Actual correlation
0 100 200
0.00 0.25 0.50 0.75
Figure c: M=288
Realised correlation Actual correlation
0 100 200
−1.0
−0.5 0.0 0.5
1.0 Error
0 100 200
−0.25 0.00 0.25
0.50 Error
0 100 200
−0.1 0.0 0.1
Error
Figure 6.7: Results from a simulation from a bivariate factor SV model. Plotted on the top graphs are the actual correlation ρ(lk) each day. Also drawn are the estimated values bρ(lk) based on M=12, 48 and 288 in graphs (a), (b) and (c) respectively. On the bottom row of graphs is ρ(lk)−bρ(lk) together with their asymptotic standard errors. Code is available at: simple.ox
One possible way of improving the finite sample behaviour of the asymptotic distribution of b
ρ(lk) is by using the Fisher-z transformation z(lk) = 1
2log1 +bρ(lk)
1−bρ(lk) and ζ(lk) = 1
2log1 +ρ(lk) 1−ρ(lk).
Recall Fisher’s analysis was based on M multivariate, independent and identically distributed Gaussian data, in which case his transformation has the important feature that√
M³z(lk)−ζ(lk)´ has a standard normal limit distribution and it is well known its asymptotic distribution provides an excellent approximation to the exact distribution. In the present case
z(lk)−ζ(lk) sẵ
1−³bρ(lk)´2
ắ−2³PM
j=1yj(l)2 PMj=1y2j(k)´−1³PMj=1x2j −PMj=1−1xjxj+1´
→L N(0,1). (6.73)
Example 21 We continue with Example 20 but now we focus on the performance of the Fisher based asymptotics (6.73). Using the same simulations as reported in the previous example we now plot z(lk)−ζ(lk) for each day, with their corresponding 95% confidence intervals. This is given in Figure 6.8. The results show that the confidence intervals are now much more stable.
In particular they do not seem to move in and out with the value of ρ(lk). The finite sample behaviour of the asymptotic theory is studied in the bottom row of Figure 6.8. It shows the QQ
0 100 200
−1.0
−0.5 0.0 0.5 1.0
Figure a: M=12
0 100 200
−0.50
−0.25 0.00 0.25
Figure b: M=48
0 100 200
−0.1 0.0 0.1
Figure c: M=288
−2.5 0.0 2.5
−5 0 5
Observed 45 degree line
−2.5 0.0 2.5
−2.5 0.0 2.5
Observed 45 degree line
−2.5 0.0 2.5
−2.5 0.0 2.5
Observed 45 degree line
Figure 6.8: Results from a simulation from a bivariate factor SV model. Plotted on the top graphs are the Fisher transformed z(lk)−ζ(lk) for each of the first 200 days based on M=12, 48 and 288 in graphs (a), (b) and (c) respectively, together with the associate standard errors.
On the bottom row of graphs are the corresponding QQ plots to assess normality. The y-axis is the sorted simulations, on the x-axis are the corresponding expected values. Code is available at:
simple.ox
plot for (6.73) computed based on 2,400 days. The QQ plot draws the sorted standardised errors (on the y-axis) against that expected under a Gaussian assumption (on the x-axis). Hence if the asymptotics is a perfect description of the finite sample behaviour the QQ plot should be on a 45 degree line. This line is given in the Figure for comparison. We see that for very small values of M the approximation is poor, but by the time M = 48 the approximation is quite good and is very accurate by the time M = 288.
An important aspect of the improved finite sample behaviour of the Fisher-based asymptotics for the realised correlation is that it can be used in combination with the theory of the log- realised variances to produce an improved asymptotic theory for the realised covariation matrix.
This could be useful in making inference off any function of the covariance matrix. Given the centrality of this measure in financial economics this seems to be of some importance.
Efficiency frontiers NEIL: TO BE ADDED
Portfolio weights NEIL: TO BE ADDED Partial correlation NEIL: TO BE ADDED