Dynamic conditional angular correlation

Journal of Econometrics xxx (xxxx) xxx

Contents lists available at ScienceDirect

Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom

Dynamic conditional angular correlation Riad Jarjour, Kung-Sik Chan

∗

The University of Iowa, Iowa City, IA, USA

article

info

Article history: Available online xxxx JEL classification: C590 C510 Keywords: Instantaneous correlation matrix Portfolio construction Positive definiteness Robustness Volatility

a b s t r a c t We introduce the concept of angular correlation for estimating the instantaneous correlation matrix with a single multivariate realization. The proposed estimator is generally a positive definite correlation matrix and robust in that for bivariate normal data, the sample angular correlation is equally likely to be above or below the population correlation coefficient. We then generalize the dynamic conditional correlation (DCC) model to the dynamic conditional angular correlation (DCAC) model. We demonstrate the efficacy and robustness of the proposed methods against leptokurticity, with some numerical experiments. In particular, a real application illustrates the better performance of the DCAC model than the DCC model in portfolio construction. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Since the introduction of univariate volatility models by Engle (1982) and Bollerslev (1986), a number of attempts have been made to generalize these models to jointly study the volatility and correlation dynamics in multivariate time series data. Engle (1982) introduced the vector autoregressive conditional heteroskedasticity (ARCH) model (sometimes denoted as VEC) and diagonal VEC, which have some technical problems, particularly the issue of ensuring the positive definiteness of the conditional covariance matrices. To remedy this, the BEKK model (named after Baba, Engle, Kraft and Kroner) was proposed by Engle and Kroner (1995), though this model suffers from the curse of dimensionality. The generalized orthogonal generalized ARCH (GO-GARCH model) by Van der Weide (2002) is more parsimonious and generalizes the Orthogonal GARCH model (Alexander, 2001), but its assumptions are questionable for some applications (Tsay, 2013). For a survey of multivariate GARCH models see Bauwens et al. (2006) and for multivariate stochastic volatility models see Asai et al. (2006). Engle’s seminal paper on the dynamic conditional correlation (DCC) model (Engle, 2002) solves a major hurdle in multivariate volatility modeling. It uses only two parameters to model the correlation dynamics, yet seems to perform similarly to models that use a lot more parameters. However, owing to the lack of a ‘‘natural’’ yet informative instantaneous correlation matrix estimator from a single realization of a multivariate distribution, the DCC model and its variants generally require a normalization step in the specification of the correlation dynamics or use some window-based local correlation matrix as a proxy for the instantaneous correlation matrix; such procedures may likely compromise the model fit and/or complicate the theoretical analysis. Here, we introduce the concept of angular correlation coefficient for measuring the linear association between two random variables, with a single pair of data from a bivariate population with known zero mean and identical variance. The angular correlation enjoys two interesting properties: (i) the sample angular correlation matrix is generally a positive definite correlation matrix even though it is computed from a single multivariate realization, and (ii) the sample angular correlation coefficient from a bivariate normal population has equal ∗ Corresponding author. E-mail address: [email protected] (K.-S. Chan). https://doi.org/10.1016/j.jeconom.2020.01.010 0304-4076/© 2020 Elsevier B.V. All rights reserved.

Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

2

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

probability of being above or below the population correlation coefficient. Thus, the sample angular correlation matrix may serve as a useful estimator of the instantaneous population correlation matrix in a multivariate time series setting. In practice, financial time series are generally highly leptokurtic. In Bai et al. (2003), Professor George C. Tiao and his collaborators derived a beautiful formula encapsulating how the GARCH leptokurticity and the innovation leptokurticity interact symmetrically to produce the process leptokurticity. They demonstrated that while a GARCH model generally introduces leptokurticity in the stationary distribution of the process even with normal innovations (referred to as the GARCH leptokurticity), the GARCH leptokurticity is far lower than the typical observed kurtosis values, so they argued for the need of non-normal innovations for financial time series analysis. In particular, the angular correlation is expected to be a more robust estimator of the conditional correlation for leptokurtic data, as will be illustrated in the simulation studies reported below. We now outline the content of the paper. We review the DCC model and set up some notations in Section 2. The concept of angular correlation and its main properties are elaborated in Section 3, with results of simulation studies demonstrating the robustness of the angular correlation in comparison with other estimators summarized in Section 4. The new DCAC model with or without correlation targeting (CT) is formulated in Section 5 which also contains a proposed estimation procedure. The efficacy of the proposed DCAC models in terms of tracking the correlation dynamics is studied via simulation in Section 6. Section 7 reports results of an empirical study showcasing the better performance of DCAC model based portfolio than other approaches including DCC model based portfolio construction. We briefly conclude in Section 8. All proofs and mathematical derivations are relegated to Section 9. 2. Review of the DCC model Let yt be a K -dimensional time series of log returns and consider the model yt = E [yt |Ft −1 ] + εt ,

(1)

where Ft −1 = σ (yt −1 , yt −2 , . . .) denotes the sigma field generated by the past y’s and conditional on Ft −1 , εt is of zero mean and variance–covariance matrix Σt > 0; the decorrelated standardized errors −1/2

zt = Σ t

εt ,

(2)

are i.i.d. random vectors independent of Ft −1 , with:

E [zt ] = 0, Var[zt ] = IK ,

(3)

the K × K identity matrix and Σt is Ft −1 measurable. Thus, Var (εt |Ft −1 ) = Σt .

(4)

The constant conditional correlation (CCC) model (Bollerslev, 1990) factorizes Σt as Σt = Dt RDt ,

(5)

where R is the constant conditional correlation matrix and Dt is a diagonal matrix containing the conditional standard deviations σ1t , . . . , σKt that can tentatively be modeled via GARCH(p, q),

σit2 = ωi +

p ∑

αi,h εi2,t −h +

h=1

q ∑

βi,h σi2,t −h ,

(6)

h=1

where the ω’s, α ’s and β ’s are generally non-negative parameters. Engle (2002) and Tse and Tsui (2002) extended the CCC model by allowing R to be time-varying, Σ t = Dt Rt Dt .

(7)

hence allowing for richer correlation dynamics. The commonly used version due to Engle (2002) uses an auxiliary process Qt to obtain the correlation matrix Rt as follows: Qt = (1 − θ1 − θ2 )S + θ1 ηt −1 ηt −1 + θ2 Qt −1 , ⊺

Rt = [diag(Qt )]

−1/2

Qt [diag(Qt )]

(8)

−1/2

(9)

where S is positive definite, ηt := Dt εt and diag(Qt ) is the diagonal matrix comprising the diagonal elements of Qt . Thus ηt are the residuals standardized to have unit variance. Qt has to be normalized at every step to obtain Rt . S can be set to be the unconditional covariance matrix of ηt for covariance targeting. Furthermore, we must have θ1 > 0, θ2 > 0 and θ1 + θ1 < 1 for positive definiteness and stationarity. The varying conditional correlation (VCC) model introduced by Tse and Tsui (2002) avoids the normalization and obtains Rt in one step: −1

Rt = (1 − θ1 − θ2 )R + θ1 Ψt −1 + θ2 Rt −1 . Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

(10) Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

3

Here R is generally the unconditional sample correlation matrix of ηt , and Ψt −1 is a local sample correlation matrix based on lagged values of ηt , i.e., Ψt −1

⎫ ⎬ η η i , t − h j , t − h = √∑ h=1 ∑M 2 ⎭ , 1 ≤ i < j ≤ K , ⎩ M 2 h=1 ηj,t −h h=1 ηi,t −h ⎧ ⎨

∑M

(11)

ij

where ηi,t is the ith element of ηt , and M must be not smaller than K to obtain positive definiteness. Larger M values widen the window used for the local correlation matrix and hence lead to smoother but less spontaneous correlations. It is interesting to note that letting M = 1 in (11), Ψt −1 becomes the sign correlation matrix, i.e., its (i, j)th element is the sign of ηi,t −1 ηj,t −1 . Thus, Ψt −1 is always non-negative definite, even though it need not be positive definite. Consequently, Rt defined by (10) is always positive definite whenever R is positive definite, assuming the aforementioned conditions on θi , i = 1, 2. Nonetheless, the sign correlation contains little information on the instantaneous correlation since it is either 1 or −1. Preliminary data analysis (unreported) also indicates that the model of Tse and Tsui (2002) with M = 1 does not fit data well. From the previous discussion we see that Engle’s approach specifies the correlation dynamics via an auxiliary process with a normalization step, but only uses new information at time t − 1 in the auxiliary equation. The normalization ⊺ step makes the updating implicit but it is necessary because ηt −1 ηt −1 is generally not a correlation matrix. On the other hand, Tse and Tsui’s approach does not require normalization, but uses a window of several lagged values for the local correlation matrix in the updating equation. The general recipe of requiring M to be not less than the timeseries dimension to ensure the local correlation matrix positive definite may then be deemed as a criterion for a ‘‘sufficiently’’ informative estimator of the instantaneous correlation matrix. This raises the question as to the existence of an instantaneous yet ‘‘informative’’ estimator of the instantaneous correlation matrix for updating the correlation dynamics in the DCC model. Below, we introduce and argue for the use of the angular correlation matrix for addressing this question. Henceforth, the DCC model defined by (8) and (9) will be simply referred to as the DCC model. 3. Angular correlation as a measure of instantaneous correlation Throughout this section, let X and Y be two random variables of zero mean and identical variance. The angular correlation is defined as the following ratio: G=

2XY (X 2

+ Y 2)

.

(12)

Without loss of generality, the common variance of X and Y will be assumed to be 1. Assuming X and Y are bivariate normal with population correlation ρ , the expected value of G can be shown to equal (details in Section 9.1)

φ := E Hence by

2XY X 2 +Y 2

[

]

2XY X2 + Y 2

[ =ρ×

]

1 1+

√

.

1 − ρ2

(13)

is actually a biased estimator of ρ . However, we note that the sample Pearson correlation coefficient given

∑n r X ,Y = √ ∑ n

i=1 (xi

− x¯ )(yi − y¯ ) √∑ n i=1 (yi

¯2 i=1 (xi − x)

.

(14)

− y¯ )2

is also biased (Hotelling, 1953), and in fact has expectation

[ E(ˆ rX ,Y ) = ρ × 1 −

1 − ρ2

(

2n

)

( +O

1 n2

)]

.

(15)

From (15) we can see that the bias will be largest for small values of the sample size n, and the simulations in the next section will show that the angular correlation actually performs better in such cases. Note that we can invert the relationship in (13) to get

ρ=

2φ 1 + φ2

.

(16)

In general, it is readily seen that G = sin(2θ ) where θ is the angle between the vector (X , Y )⊺ and the abscissa measured counterclockwise from the latter, see Fig. 1. We can interpret θ as the angle measuring the strength of correlation between the two variables. θ = π4 points in the direction of perfect positive correlation, and the (sample) angular correlation defined by (12) evaluates to 1. Similarly an angle of − π4 would make the angular correlation (12) evaluate to −1. Angles that point in directions of weak correlation give values of (12) that are close to zero. θ = π2 for example yields an angular correlation of 0. A similar concept has also been used in the physics of gamma ray emissions, where the angular correlation between gamma rays is studied (Brady and Deutsch, 1950). Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

4

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

Fig. 1. Visualizing angular correlation.

Fig. 2. Contour plot of f as a function of g and ρ .

We now establish a relationship between the population median of the angular correlation and the familiar Pearson correlation coefficient ρ . Assume a bivariate standard normal distribution for X and Y . The probability density function is then given by (pdf) of G = X 22XY +Y 2 f (g) =

√

1

π (1 − g ρ )

1 − ρ2 1 − g2

− 1 < g < 1;

(17)

see Section 9.1 for a derivation. Fig. 2 shows the contour plot of f as a function of g and ρ . Using this density, we can calculate the corresponding cumulative distribution function (cdf) for −1 ≤ x ≤ 1 as

∫ F (x) =

x

1

√

1 − ρ2

dg

(18)

1 − g2 −1 π (1 − g ρ ) ⎧ (√ ) ⎪ (1−ρ 2 )(1−x2 ) 1 −1 ⎪ , x≤ρ ⎨ π tan ρ−x (√ ) = ⎪ (1−ρ 2 )(1−x2 ) ⎪ , x > ρ. ⎩1 − π1 tan−1 x−ρ

(19)

Hence we can see that F (ρ ) = 0.5, thus establishing that the median of the sample angular correlation G is the true population correlation ρ . This lends credibility to the usage of the angular correlation as an alternative, robust estimator of ρ . 4. Comparing measures of correlation We report some simulation results for gauging the performance of the angular correlation G as an estimator of ρ . We focus on small sample sizes since that is the context under which we will ultimately use G in our models. Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

5

Table 1 Comparing performance the quasi-likelihood estimator (QLE) and angular correlation (AC) for normal and t-distributed bivariate data (n = 1), in terms of mean squared error (MSE) on Fisher scale and original scale, as well as mean absolute error on original scale, based on 10,000 replications. MSE (Fisher Scale)

MSE

QLE

AC

QLE

AC

QLE

MAE AC

2.43 2.57 2.44 2.46

0.62 0.64 0.67 0.49

0.50 0.51 0.52 0.38

0.73 0.71 0.61 0.34

0.63 0.62 0.55 0.35

0.73 0.74 0.72 0.52

0.50 0.50 0.51 0.39

0.82 0.78 0.65 0.35

0.64 0.61 0.54 0.35

0.65 0.67 0.67 0.49

0.50 0.51 0.51 0.37

0.76 0.73 0.61 0.34

0.64 0.61 0.54 0.34

0.63 0.65 0.66 0.48

0.49 0.51 0.51 0.37

0.74 0.72 0.61 0.33

0.63 0.62 0.55 0.34

Bivariate normal distribution

ρ ρ ρ ρ

=0 = 0.3 = 0.6 = 0.9

3.69 3.87 3.89 3.95

Bivariate t-distribution with 3 degrees of freedom

ρ ρ ρ ρ

=0 = 0.3 = 0.6 = 0.9

5.23 5.15 5.30 5.31

2.49 2.38 2.47 2.48

Bivariate t-distribution with 7 degrees of freedom

ρ ρ ρ ρ

=0 = 0.3 = 0.6 = 0.9

4.06 4.16 4.36 4.22

2.43 2.45 2.51 2.43

Bivariate t-distribution with 15 degrees of freedom

ρ ρ ρ ρ

=0 = 0.3 = 0.6 = 0.9

3.74 3.95 4.06 4.12

2.37 2.46 2.49 2.44

4.1. Comparison to the MLE when sample size is 1 We compare the sample angular correlation given by (12) for measuring correlation to the MLE for the case of sample size n = 1. The log-likelihood of a bivariate standard normal with correlation ρ is

ℓ(ρ|x, y) = − ln(2π ) −

1 2

ln(1 − ρ 2 ) −

1 2(1 − ρ 2 )

(x2 + y2 − 2ρ xy).

Setting the derivative w.r.t. ρ to 0 yields the following equation for the maximum likelihood estimator (MLE):

ρˆ 3 − xyρˆ 2 − (1 − x2 − y2 )ρˆ − xy = 0. Fosdick and Raftery (2012) gave closed form solutions for the three roots of this equation, while Kendall et al. (1977) noted that at least one of those roots will be real and lie in the interval [−1, 1]. If more than one root lies in this interval then we pick the root that gives the largest value for the likelihood. We replicated the experiment 10,000 times with observations drawn from the bivariate standard normal distribution using different correlation coefficients, for comparing the two methods using the mean squared error (MSE), the mean absolute error (MAE), and the MSE on the Fisher scale as induced by the Fisher transformation given by f (ρ ) =

1 2

( ln

1+ρ 1−ρ

)

= arctanh(ρ ).

(20)

The upper part of Table 1 shows the sample angular correlation performs uniformly better for all tested values of ρ , except when ρ = 0.9 (in which case the two methods have similar MAE). To gauge the robustness of the methods to leptokurtosis, we then repeated the experiment but with data from a bivariate t distribution with different degrees of freedom and correlation coefficients. The lower 3 sub-tables in Table 1 summarize the results. The angular correlation performs surprisingly well, with little difference from the case of normality. The quasi-MLE now performs considerably worse, with the errors more pronounced when the degree of freedom is low. 4.2. Comparison with the sample Pearson correlation coefficient For n > 1 the sample Pearson correlation coefficient ˆ rX ,Y coincides with the MLE, and we compare its performance ¯ the mean angular correlation, and that of G, ˜ the median angular correlation, for n ∈ {2, 3, 4, 5} and to that of G, ρ ∈ {0.0, 0.3, 0.6, 0.9}. Note, by symmetry, we need only consider non-negative values of ρ . Each experiment was replicated 1000 times. Table 2 lists the averages of the estimates and their mean squared errors (MSE) which are enclosed ¯ = G, ˜ and rˆX ,Y is either −1 or 1. Table 2 confirms that G¯ is a biased estimator of ρ . in parentheses. Note that for n = 2, G Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

6

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx Table 2 Comparing the Pearson correlation, the mean angular correlation and the median angular correlation in terms of their averages and their mean squared error (MSE) which are enclosed in parentheses, based on 1000 replications. n=2

rˆX ,Y ¯ G ˜ G

ρ=0

ρ = 0.3

ρ = 0.6

ρ = 0.9

−0.052 (1) −0.018 (0.238) −0.018 (0.238)

0.16 (0.994) 0.14 (0.281) 0.14 (0.281)

0.392 (0.89) 0.319 (0.311) 0.319 (0.311)

0.702 (0.546) 0.616 (0.237) 0.616 (0.237)

ρ=0

ρ = 0.3

ρ = 0.6

ρ = 0.9

−0.008 (0.501) −0.007 (0.18) 0.002 (0.349)

0.268 (0.449) 0.157 (0.186) 0.198 (0.353)

0.521 (0.343) 0.336 (0.222) 0.421 (0.32)

0.841 (0.111) 0.629 (0.178) 0.748 (0.158)

n=3

rˆX ,Y ¯ G ˜ G n=4

rˆX ,Y ¯ G ˜ G

ρ=0

ρ = 0.3

ρ = 0.6

ρ = 0.9

−0.015 (0.328) −0.017 (0.127) −0.013 (0.226)

0.267 (0.297) 0.158 (0.14) 0.208 (0.235)

0.539 (0.21) 0.336 (0.178) 0.433 (0.213)

0.865 (0.042) 0.631 (0.146) 0.762 (0.1)

ρ=0

ρ = 0.3

ρ = 0.6

ρ = 0.9

0.016 (0.252) 0 (0.1) 0 (0.27)

0.283 (0.212) 0.162 (0.111) 0.255 (0.239)

0.567 (0.139) 0.342 (0.149) 0.5 (0.191)

0.874 (0.032) 0.633 (0.129) 0.819 (0.073)

n=5

rˆX ,Y ¯ G ˜ G

¯ has the largest bias. On the other hand, G¯ appears to generally Among the three estimators, rˆX ,Y has the least bias, while G have the smallest variance so much so that its MSE are often the smallest, especially for the case of small n and small ρ . ¯ and G˜ perform uniformly better than rˆX ,Y for n = 2. For n = 3, G¯ outperforms the other two estimators over the G range of ρ examined, though with some deterioration in performance at ρ close to 1 or −1 (not shown). This deterioration ¯ appears to outperform the other two estimators for −.5 ≤ ρ ≤ .5 and becomes larger as n increases to 5, where G underperform relative to rˆX ,Y otherwise. The selection of which estimator to use will thus depend on the sample size, as ¯ is clearly larger when the magnitude well as any prior belief about where the true value of ρ lies (the benefit of using G of ρ is small). 5. The dynamic conditional angular correlation model We propose two variations of the dynamic conditional angular correlation model (DCAC). The first does not employ correlation targeting, and uses the following as the updating equation for the correlation matrix: Rt = (1 − θ1 − θ2 )R + θ1 Γt −1 + θ2 Rt −1

(21)

where Γt is an n × n matrix whose (i, j)th entry is 2ηi,t ηj,t

(22)

ηi2,t + ηj2,t

where ηi,t is the ith element of ηt , the standardized residual defined just below (9). During estimation, R is typically set to be the empirical correlation matrix of ηt and is therefore positive definite; also, the recursion in (21) starts from t = 2 with R1 = R. Below, we prove the positive definiteness of Γt under very general conditions. Hence, assuming the same conditions on θ1 and θ2 as required by the DCC models (i.e., θi ≥ 0, i = 1, 2; θ1 + θ2 < 1) we can ensure that Rt is positive definite. Theorem 1. Let x = (x1 , x2 , . . . , xK )⊺ . The K × K matrix Γ, whose (i, j)th entry is equal to 2xi xj /(x2i + x2j ) for all 1 ≤ i, j ≤ K , is always positive semi-definite. Moreover, it is positive definite if and only if xi , i = 1, . . . , K are distinct, non-zero real numbers. The other version of the DCAC uses correlation targeting (meaning the updating equation is set up such that E [Rt ] is the unconditional correlation matrix of ηt ), and the updating equation of Rt is given by: Rt = (1 − θ2 )R − θ1 Γ + θ1 Γt −1 + θ2 Rt −1

(23)

where during estimation Γ is set to the empirical average of Γt . The aforementioned DCC conditions on θ1 and θ2 are no longer sufficient to ensure positive definiteness of Rt . Instead, we have the following result. Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

7

Theorem 2. For the DCAC model with correlation targeting, assuming R is positive definite, {Rt } defined by (23) is always positive definite if θi ≥ 0, i = 1, 2 and λm θ1 + θ2 < 1 where λm is the maximum eigenvalue of R −1/2 ΓR −1/2 . Note that the equation E ηt η′t = ERt holds for the DCC model, the DCAC model and its correlation targeting variant which is denoted as the DCAC (CT) model; indeed, the preceding equality holds under very mild conditions: (i) Dt is Ft −1 -measurable, and Eqs. (4) and (7) hold, because ′ −1 1 E ηt η′t = ED− t εt εt Dt −1 ′ 1 = E {D− t E(εt εt |Ft −1 )Dt } −1 −1 = E {Dt Dt Rt Dt Dt } = ERt .

Similar to the DCC model, one of the convenient aspects of the DCAC model is the fact that its log-likelihood can be decomposed into a correlation term and a volatility term, with the former only involving the GARCH parameters. This is done as follows:

ℓ=−

T ) 1 ∑( ⊺ 1 n log (2π) + log |Σt | + εt Σ− t εt 2 t =1

=−

T 1 ∑(

2

1 −1 n log (2π) + log |Dt |2 + εt D− t Dt εt ⊺

) (24)

t =1

−

T ) 1 ∑( ⊺ ⊺ log |Rt | + ηt Rt−1 ηt − ηt ηt 2 t =1

:= ℓv (θ v ) + ℓc (θ v , θ c ) where |Σt | is the determinant of Σt , θ v is the vector of parameters involved in the volatility term of the likelihood (namely, ω1 , ω2 , . . . , ωn , α1 , α2 , . . . , αn , β1 , β2 , . . . , βn in (6)), θ c is the vector of parameters involved in the correlation term of the likelihood (namely, θ1 and θ2 in (21)). The volatility term is given by

ℓv (θ v ) = −

T 1 ∑(

2

1 −1 n log (2π ) + log |Dt |2 + εt D− t Dt ε t ⊺

)

(25)

t =1

and the correlation term by

ℓ c (θ v , θ c ) = −

T ) 1 ∑( ⊺ log |Rt | + ηt Rt−1 ηt − η⊺ t ηt . 2

(26)

t =1

Thus the maximization can be done in two steps. We first obtain the MLE θˆ v of the volatility term, which is the same as the MLE of fitting individual GARCH models to the variables in εt . We then use this information to find the value of θ c that maximizes ℓc (θˆ v , θ c ). Both R and R1 are set to be the sample unconditional correlation matrix of D−1 εt . This enables us to begin the recursion in (21). Since ηt do not depend on the θ ’s, we can use the linearity of the derivative operator to get

∂ ⊺ −1 ∂ −1 η R η = η⊺t R η ∂θi t t t ∂θi t t ( ) ⊺ −1 ∂ Rt −1 = η t − Rt R ηt . ∂θi t

(27)

Thus, we can now differentiate ℓc (θˆ v , θ c ) with respect to θ c as follows:

( ) ) T ( ∂ℓc (θˆ v , θ c ) 1∑ ∂ Rt ∂ Rt −1 ⊺ =− tr(Rt−1 ) − ηt Rt−1 Rt ηt ∂θi 2 ∂θi ∂θi t =1

∂ Rt ∂ Rt − 1 = −R + Γt −1 + θ2 ∂θ1 ∂θ1 ∂ Rt ∂ Rt − 1 = −R + Rt −1 + θ2 . ∂θ2 ∂θ2 Using standard optimization techniques we can use the aforementioned likelihood and gradient to estimate the correlation parameters as well as obtain an estimate of the Hessian that can be used to obtain standard errors. However, Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

8

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

due to the two-step method of estimation, an adjustment is needed to the Hessian obtained from maximizing ℓc (θˆ v , θ c ) with respect to θ c . The adjustment can be achieved by following the outline in Engle and Sheppard (2001), with the proof and regularity conditions given by Newey and McFadden (1994). The adjusted covariance matrix of θˆ c becomes

)]−1 [ ( × E ∇θ c θ c ℓc ( E {∇θc ℓc − E ∇θc θv ℓc

(

)[ ( )]−1 E ∇θ v θ v ℓv ∇θv ℓv }×

) )]−1 )[ ( ( ⊺ ∇θv ℓv } × {∇θc ℓc − E ∇θc θv ℓc E ∇θv θv ℓv E ∇θc θc ℓc

[ (

)]−1

where for a function f (x1 , x2 ) of vectors x1 , x2 , we define

∇xi f (x1 , x2 ) =

∂ f (x1 , x2 ) , ∂ xi

∇xi xj f (x1 , x2 ) =

∂ 2 f (x1 , x2 ) , 1 ≤ i, j ≤ 2. ∂ xi ∂ xTj

Notice that if[ ℓc( did not)]involve parameters in θ v , the expression above will reduce to just the inverse of ℓc ’s expected −1 . Hessian, i.e., E ∇θc θ c ℓc 6. Simulations comparing the DCC and the DCAC models We first study the empirical performance of the proposed estimation method for estimating a DCAC model with correlation targeting. Bivariate time series of length n = 1, 000 were simulated from the DCAC model with correlation targeting, i.e., Rt updated according to (23) where θ1 = 0.1, θ2 = 0.8 and the conditional variances following the following GARCH(1,1) specification:

σ12,t = 0.01 + 0.05ε12,t −1 + 0.94σ12,t −1 ,

(28)

σ22,t = 0.5 + 0.2ε22,t −1 + 0.5σ22,t −1 ,

(29)

with i.i.d. standard normal innovations. We set the stationary population correlation coefficient ρ = 0.6, with the corresponding population angular correlation φ = ρ/(1 +

( R=

1 3/5

3/5 , 1

)

( Γ=

1 1/3

√

1 − ρ 2 ) = 1/3, i.e.,

1/3 . 1

)

Moreover, we initialized R0 = R and Γ0 = Γ. In practice, the return process may have an asymmetric impact on the volatility clustering, which is often modeled by the GJR-GARCH model (Glosten et al., 1993); the GJR-GARCH(1,1) model differs from the GARCH(1,1) model in replacing the term αεt2 by {α + γ I(εt < 0)}εt2 , where I(·) equals 1 if the enclosed expression holds and 0 otherwise. For each simulated bivariate time series, we fitted a GJR-GARCH(1,1) model componentwise to get θˆ v , and then estimated θ c by numerically maximizing ℓc (θˆ v , ·) whose contour plot for a simulated series is shown in Fig. 3. From the contour plot, the profile likelihood appears to be a unimodal function. It follows from Theorem 2 that the feasible parametric region is the interior of a triangle defined by the constraints: θi > 0, i = 1, 2 and λm θ1 + θ2 < 1 where λm = 5/3. We repeated the experiment 500 times. Table 3 summarizes the simulation results. The GARCH estimates are more concentrated around the true values, with a more symmetric sampling distribution under a more persistent GARCH specification, i.e., more so for the first-component GARCH estimates than their second-component counterparts. In fact, the intercept term of the GARCH model for the second component has substantial variation; see Fig. S.1 in Supplementary Material. In comparison, the estimates of the θ ’s approximately center around their true values although there is a lone outlier. Fig. 4 shows the scatter plot of θˆ2 versus θˆ1 , with the outlier excluded. The scatter diagram shows that the distribution of the estimates resembles bivariate normality to some extent. Financial time series data are often heavy-tailed. For instance, among the 8 daily log return series examined in Tsay (2010, Table 1.2), the excess kurtosis ranges between about 7 to 33. Because the (sample) angular correlation is equally likely to be below or above the population correlation coefficient for standard normal data, it may be expected that the DCAC and DCAC (CT) models provide more robust tracking of the conditional correlation dynamics. We examine the tracking performance of the proposed models by performing simulations along the lines of Engle (2002), specifically,

(

z1,t z2,t

) ∼N

[( ) 0 , 0

√ ε1,t = σ1,t κt z1,t ,

(

1

ρt

ρt 1

)]

,

(30)

√ ε2,t = σ2,t κt z2,t ,

Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

(31) Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

9

Fig. 3. Contour plot of ℓc (θ˜v , ·), with the true coefficient vector superimposed as a cross.

Fig. 4. Scatter plot of θˆ2 vs. θˆ1 , with the outlier (0.144, 0.261) removed. The true parametric vector is superimposed as a red cross.

Table 3 Summary of the DCAC (CT) parametric estimates, with 500 replications. True value

Average

Median

SD

0.0141 0.0480 0.9360 0.0014

0.028 0.021 0.046 0.025

0.4529 0.1792 0.5500 0.0007

0.4724 0.1954 0.5110 −0.0029

0.218 0.085 0.196 0.069

0.1000 0.7895

0.0992 0.7962

0.019 0.055

GJR-GARCH parameters of the first series

ω1 α1,1 β1,1 γ1

0 0.05 0.94 0

0.0179 0.0486 0.9315 0.0012

GJR-GARCH parameters of the second series

ω2 α2,1 β2,1 γ2

0 0.2 0.5 0.0 Correlation parameters

θ1 θ2

0.1 0.8

Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

10

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

where the σ ’s follow (28) and (29), κt ≡ 1 for normality, otherwise {κt } are independent and identically Gamma distributed with identical shape parameter and rate parameter equal to κ > 0; κt is independent of the past εt ’s, as well as the current and past z’s; ρt follows a variety of different processes: 1. 2. 3. 4. 5.

Constant: ρt = 0.9. Sine: ρt = 0.5 + 0.4cos(2π t /200). Fast sine: ρt = 0.5 + 0.4cos(2π t /20). Ramp: ρt = (t mod 200)/200. Step: ρt = 0.9 − 0.5I(t > 500).

It is readily checked that conditional on the past εt ’s, ρt is the correlation between ε1,t and ε2,t and the excess kurtosis of the marginal distribution of the standardized ε ’s is equal to 3/κ . We tried five κ values: κ = +∞, 1, 1/2, 1/3 and 1/5 corresponding to the excess kurtosis being 0, 3, 6, 9, 15. We fitted 4 models to each simulated bivariate series of length n = 1000: the VCC model with M = 2 in (11), the DCC as in (8) and (9), DCAC defined by (21) and with correlation targeting in (23). We compared these 4 models using the MAE of the within-sample estimates of ρt ’s. The simulation was repeated 500 times per each model setting and the averaged results are reported in Table 4. We can draw the following conclusions from Table 4: The MAE increases with the kurtosis. For the case of normal innovations, the DCC model outperforms all other models except for the case of constant ρt , with the DCAC (CT) or DCAC model being the second best. In the case of high excess kurtosis (six or above), the DCAC (CT) model outperforms all other models, when ρt takes the sine, fast sine or ramp specification, although the greatest reduction in the MAE is between 5% to 16% even for the case of the excess kurtosis equal to 15. Thus, we conclude that for strongly heavy-tailed data, the DCAC (CT) model may render more accurate tracking of the underlying conditional correlation dynamics that displays large, smooth variation. On the other hand, even for the constant or step specification of ρt , the DCAC (CT) model is very competitive and often achieved the second best MAE. We note that similar conclusions are borne out using RME instead of MAE for assessing the tracking performance, see Table S.1 in Supplementary Material. Since financial time series data are often strongly heavy-tailed, the DCAC (CT) model may achieve consistent though small to moderate gain over the DCC model in tracking the conditional correlation dynamics with financial time series. 7. Empirical studies In this section we compare the out-of-sample performance of the various models discussed so far in constructing minimum-variance portfolios with (or without) pre-specified target return rate, using a Markowitz approach. Our study was conducted along the lines of the empirical application in Billio et al. (2006). Consider a portfolio w ⊺ yt where yt is an K -dimensional vector of asset returns with the conditional moments E [yt ] = rt , Var[yt ] = Σt , given the past data. Then, the optimal weights w of the minimum-variance portfolio with target rate of return µr comprise the first K elements of

(

2Σt ⊺ rt 1⊺

rt 0 0

)−1 (

1 0 0

0

)

µr , 1

whereas those of the global minimum-variance portfolio are the first K elements of

(

2Σt 1⊺

)−1 ( )

1 0

0 , 1

see Jarjour (2018) for a derivation. We consider five sectors in the American stock exchange separately to reflect the DCC limitation of having only two parameters modeling the correlation evolution, as we expect those to be different across sectors. 1000 observations (approximately 4 years of data since there are about 252 days of trading per year) were obtained from Google Finance, stretching from the beginning of 2014 till the end of 2017. An out-of-sample rolling estimation was used, initially using the first 750 observations for estimation and construction of portfolios. Over the next 25 days the portfolios are adjusted daily using the models’ 1-step ahead forecasts (assuming no transaction costs), after which all the models are re-estimated using the latest 750 observations. The cycle is repeated 10 times for the out-of-sample data which spans the year 2017. The portfolio weights w can be negative, thus shorting is allowed. Similar to Billio et al. (2006), we set the expected returns to the average return of the last 2 months of data, and fitted GJR-GARCH(1,1) model componentwise to account for the volatility clustering in the data. For the required rate of return µr , we consider many cases: the average annual return of the S&P; a 20% annual return; 40% annual return and the last year return of an equally weighted portfolio. The global minimum variance portfolio is also included. The models considered are the CCC, DCC, DCAC, DCAC (CT), as well as a portfolio constructed using equal weights for all stocks. Five sectors were considered: technology, airline, retail, consumer goods and automotive. We looked at 10 stocks from the technology sector, including majors like Google, Apple and Microsoft. In the airline sector all major US airlines were included along with smaller domestic carriers for which data is available. The automotive dataset includes the major US auto companies as well as some foreign companies that trade on American stock exchanges. Similarly, the other sectors Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

11

Table 4 The mean absolute error (MAE) of the within-sample estimates of the ρt ’s. The smallest MAE in each row is bold-faced whereas the second best is in italics. Constant Excess kurtosis

VC

DCC

DCAC

DCAC (CT)

0 3 6 9 15

0.00595 0.01102 0.01313 0.01580 0.02170

0.00633 0.01102 0.01255 0.01552 0.02039

0.00607 0.01087 0.01305 0.01663 0.02224

0.00637 0.01128 0.01278 0.01608 0.02059

Sine Excess kurtosis

VCC

DCC

DCAC

DCAC (CT)

0 3 6 9 15

0.19073 0.19350 0.19660 0.19844 0.20319

0.13565 0.15869 0.17374 0.18631 0.20679

0.19963 0.20500 0.20849 0.21349 0.21582

0.16201 0.16586 0.16648 0.17159 0.17437

Fast sine Excess kurtosis

VCC

DCC

DCAC

DCAC (CT)

0 3 6 9 15

0.24625 0.24748 0.24879 0.24998 0.25148

0.22667 0.23962 0.24530 0.24950 0.25397

0.23587 0.23843 0.24065 0.24309 0.24580

0.23218 0.23429 0.23587 0.23799 0.24087

Ramp Excess kurtosis

VCC

DCC

DCAC

DCAC (CT)

0 3 6 9 15

0.21305 0.21523 0.22024 0.22386 0.22702

0.15584 0.18066 0.19615 0.20899 0.22230

0.21852 0.22449 0.22946 0.23301 0.23496

0.18741 0.19040 0.19474 0.19588 0.19998

Step Excess kurtosis

VCC

DCC

DCAC

DCAC (CT)

0 3 6 9 15

0.16685 0.16895 0.17170 0.17240 0.17400

0.07140 0.08650 0.09644 0.10759 0.12182

0.17260 0.18115 0.18828 0.19359 0.19619

0.13240 0.13417 0.13260 0.13754 0.13927

included all the major companies as well as some of the smaller ones. See Tables S.2 and S.3 in Supplementary Material for the entire list of companies included in each of the 5 sectors, as well as Fig. S.2–S.5 for the time plots of the data. We computed the excess kurtosis of the standardized data, i.e., ηt , series by series and obtain the following median excess kurtosis by sector: Technology, 8.08, Airline, 3.75, Retail, 14.46, Consumer Goods, 7.43 and Automotive, 3.22; see Table S.4 in Supplementary Material for the kurtosis of individual series. Table 5 summarizes the sector by sector results of the rolling portfolios with various target return rates, constructed using each model. The results across the 5 sectors show the empirical rates of return and their standard deviations. The best number in each column (i.e. the highest return or the lowest standard deviation) is made bold, while the second best is made italic. Results for the global optimal portfolio are only included for completeness, and do not serve as a good measure for comparison since it only focuses on minimizing the portfolio variance. Overall, the DCAC or its correlation targeting (CT) variant appears to almost always perform better than the DCC; for instance, the empirical mean return of the DCAC-based portfolios with a 20% target return rate exceeds those of their DCC counterparts by anywhere from about 0.4% to about 1.5%. Its standard deviation is also often very competitive, being either the best or second best. A few interesting observations can be noted from the results. In a booming sector like technology, one can do well by simply using an equal weight allocation (though using a Markowitz approach and aiming for a high required rate of return can yield even better results). However, in a sector like the airline, where half the stocks closed lower than they opened during the out-of-sample period, the Markowitz approach clearly outperforms using equal weights. In the retail and consumer goods sectors, the Markowitz approach quickly outperforms using equal weights once we target a rate of return of 20% or higher. When one considers that the S&P return for the year was about 10%, we find that the targeted returns are in many cases relatively close to the realized portfolio returns. This was surprising since the portfolios were constructed on an out-of-sample basis. Also surprising was the performance of the CCC, which often did very well and achieved returns only marginally below the DCAC. Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

12

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

Table 5 Sector by sector empirical out-of-sample rates of return and SD, for 5 portfolio construction schemes. Target return:

S&P return

20% annual

40% annual

Last year

Global optimal

Technology sector

CCC DCC DCAC DCAC (CT) Equal Wts

Return

SD

Return

SD

Return

SD

Return

SD

Return

SD

7.34 5.99 7.53 7.3 27.58

10.72 10.55 10.5 10.56 12.64

17.27 16.12 17.48 17.28 27.58

10.84 10.67 10.63 10.69 12.64

37.14 36.38 37.36 37.25 27.58

11.89 11.74 11.72 11.77 12.64

24.8 23.8 25.01 24.85 27.58

11.12 10.95 10.92 10.98 12.64

28.74 27.56 28.28 27.97 27.58

12.18 12.01 11.94 11.99 12.64

12.06 12.09 12.63 12.48 6.22

22.63 22.64 22.57 22.6 22.08

22.58 22.71 23.17 23.05 6.22

22.45 22.43 22.38 22.42 22.08

43.6 43.95 44.25 44.17 6.22

22.75 22.69 22.66 22.7 22.08

8.09 8.07 8.64 8.49 6.22

22.76 22.77 22.7 22.73 22.08

4.69 3.9 4.87 4.78 6.22

22.16 22.11 22.08 22.12 22.08

10.86 10.59 10.98 11.04 12.67

16.65 16.59 16.62 16.55 17.94

20.64 20.47 20.84 20.9 12.67

15.67 15.6 15.63 15.58 17.94

40.2 40.24 40.58 40.62 12.67

14.67 14.59 14.63 14.62 17.94

13.47 13.22 13.61 13.67 12.67

16.36 16.3 16.33 16.26 17.94

23.09 22.8 23.13 23.02 12.67

15.83 15.8 15.82 15.89 17.94

10.61 9.61 10.93 10.85 20.64

10.43 10.39 10.37 10.4 11.11

25.55 24.74 25.85 25.76 20.64

10.03 9.96 9.92 9.96 11.11

55.42 54.99 55.69 55.56 20.64

11.98 11.89 11.81 11.86 11.11

26.51 25.71 26.81 26.71 20.64

10.03 9.97 9.92 9.96 11.11

17.29 16.73 17.79 17.69 20.64

10.15 9.96 10.01 10.03 11.11

4.89 4.28 4.49 4.75 16.44

11.64 11.57 11.56 11.62 11.15

15.54 15.17 15.24 15.63 16.44

11.65 11.57 11.58 11.63 11.15

36.83 36.95 36.75 37.4 16.44

12.72 12.65 12.69 12.73 11.15

11.74 11.29 11.41 11.76 16.44

11.6 11.53 11.53 11.59 11.15

14.9 15.52 15.2 15.14 16.44

10.87 10.8 10.72 10.73 11.15

Airline sector CCC DCC DCAC DCAC (CT) Equal Wts Retail sector CCC DCC DCAC DCAC (CT) Equal Wts

Consumer goods sector CCC DCC DCAC DCAC (CT) Equal Wts Automotive sector CCC DCC DCAC DCAC (CT) Equal Wts

8. Conclusion We have shown that the sample angular correlation coefficient is an efficient and robust estimator of the population Pearson correlation coefficient for very small samples. In particular, our empirical study suggests that the DCAC model yields better performance in portfolio construction than other approaches including the DCC model and the CCC model. An interesting future research problem is to explore non-parametric or semi-parametric approaches for using the angular correlation matrix in portfolio construction with high-dimensional time series data. 9. Proofs 9.1. Derivation of the probability density function and the expectation of the angular correlation We begin with the density of the bivariate standard normal distribution: 1

f (x, y) =

√

2π

1 − ρ2

( exp −

x2 + y2 − 2ρ xy 2(1 − ρ 2 )

)

,

and change to polar coordinates to obtain r

f (r , θ ) =

2π

√

1 − ρ2

( exp −

r 2 (1 − 2ρ sin(2θ ))

)

2(1 − ρ 2 )

r > 0, 0 < θ < 2π.

We integrate out r to get the density of θ , f (θ ) =

∫

√

∞

f (r , θ )dr = −

0

1 − ρ2

2π (1 − ρ sin(2θ ))

√ =

(

1 − ρ2

2π (1 − ρ sin(2θ ))

Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

exp −

,

K.-S. Chan,

r 2 (1 − 2ρ sin(2θ )) 2(1 − ρ 2 )

) ⏐⏐∞ ⏐ ⏐ ⏐

(32)

0

0 < θ < 2π. Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

13

We can now perform a change in variable, g = sin(2θ ), 0 < θ < 2π , which is generally a 4 to 1 function. Note

θ=

sin−1 (g)

H⇒

2

∂θ 1 = √ , ∂g 2 1 − g2

hence the density of g is given by

√

4 1 − ρ2

(

2π

1

)

1

=

√

1 − pg 2 1 − g 2

√

1

π (1 − g ρ )

1 − ρ2 1 − g2

,

for −1 < g < 1. We can use the density of θ from (32) to calculate the expectation of g = sin(2θ ) as follows:

√ E[sin(2θ )] =

1 − ρ2 2π

√

2π

∫

sin(2t) 1 − ρ sin(2t)

0

dt

2π

ρ sin(2t) − 1 1 + dt 1 − ρ sin(2t) 1 − ρ sin(2t) 0 √ ∫ ∫ 2π 2π 1 1 − ρ2 =− dt + f (θ )dθ 2ρπ ρ 0 0 √ 1 − ρ2 1 =− + ρ ρ √ 1 − 1 − ρ2 = ρ

=

1 − ρ2

∫

2ρπ

=ρ×

1 1+

√

1 − ρ2

.

9.2. Proof of Theorem 1 To demonstrate the positive semi-definiteness of Γ, consider the quadratic form α⊺ Γα where α = (α1 , α2 , . . . , αK )⊺ is an arbitrary Below we derive an integral representation for the quadratic form: ∫ 1 but fixed K -dimensional ∑ real vector. 2 α⊺ Γα = 2 0 h2 (t)dt where h(t) = Ki=1 αi xi t xi −1/2 , which can be seen as follows:

{ α Γα = α ⊺

=

}

2xi xj

⊺

x2i + x2j

α ij

K K ∑ ∑ 2αi αj xi xj

x2i + x2j

i=1 j=1

=2

K K ∑ ∑

αi αj xi xj

1

∫ 0

=2 0

αi xi t xi −1/2 2

i=1 1

∫

K ∑

(

t

x2i +x2j −1

dt

0

i=1 j=1

=2

1

∫

K ∑

K ∑

α j xj t

x2j −1/2

dt

j=1

)2 αi xi t

x2i −1/2

dt ≥ 0.

i=1

Hence, Γ is positive semi-definite. Next, we prove that the positive definiteness of Γ holds if and if the x’s are distinct non-zero real numbers. That the latter condition is necessary follows from the preceding integral representation of the quadratic form and the fact that if ∑K x2i −1/2 some of the x’s are zero or the x’s are non-distinct, then there exists an α ̸ = 0 such that h(t) = ≡ 0. For i=1 αi xi t instance, if x1 = x2 , then we can take α = (1, −1, 0, . . . , 0)⊺ , whereas if x1 = 0, we can set α = (1, 0, . . . , 0)⊺ . The proof on the sufficiency part relies on the fact that any real analytic function w (t) with an open, connected domain ⊆ R and whose set of zeros, i.e., {t : w(t) = 0} has an accumulation point must be identically zero (Krantz and Parks, 2002, Section 4.1); a real analytic function is a function which is locally equal to its Taylor series expansion. Suppose the quadratic form α⊺ Γα equals zero. We proceed to show that α = 0 under the condition that the x’s are distinct non-zero real numbers. The integral representation of the quadratic form entails that the corresponding h(t) = 0 for almost all t ∈ [0, 1], w.r.t. the Lebesgue measure. Since h is an analytic function, this implies that h(t) ≡ 0 for all real t. With no loss Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),

14

R. Jarjour and K.-S. Chan / Journal of Econometrics xxx (xxxx) xxx

of generality, assume the distinct, non-zero x’s are in ascending order in magnitude: |x1 | < |x2 | < · · · < |xK |. Suppose α1 ̸= 0. Then, h(t) = t x1 −1/2 (α1 x1 + 2

K ∑

2

2

αi xi t xi −x1 ).

i=2

∑ 2 2 ̸= 0 for all t ̸= 0, the analytic function h1 (t) = α1 x1 + Ki=2 αi xi t xi −x1 must be equal to 0 for all t ̸= 0 and ∑K x2i −1/2 . hence identically a zero function. Therefore α1 x1 = h1 (0) = 0, hence α1 = 0 because x1 ̸ = 0. Thus, h(t) = i=2 αi xi t The same argument can be repeated to show that all α ’s are zero. Since t

x21 −1/2

9.3. Proof of Theorem 2 Note that the constant term on the RHS of (23) is equal to R − θ2 R − θ1 Γ = R 1/2 (I − θ2 I − θ1 R −1/2 ΓR −1/2 )R 1/2 .

(33) 1/2

1/2

−1/2

−1/2

and A BA are positive definite. The First, we note that if two matrices A and B are positive definite, then A BA smallest eigenvalue of the matrix in parenthesis in (33) is 1 − θ2 − λm θ1 > 0, hence the matrix is positive definite. This ensures positive definiteness of the constant term in (23) and hence the same for Rt . Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jeconom.2020.01.010. References Alexander, C., 2001. Orthogonal GARCH. Master. Risk 2, 21–38. Asai, M., McAleer, M., Yu, J., 2006. Multivariate stochastic volatility: a review. Econometric Rev. 25 (2–3), 145–175. Bai, X., Russell, J.R., Tiao, G.C., 2003. Kurtosis of GARCH and stochastic volatility models with non-normal innovations. J. Econometrics 114 (2), 349–360. Bauwens, L., Laurent, S., Rombouts, J.V., 2006. Multivariate GARCH models: a survey. J. Appl. Econometrics 21 (1), 79–109. Billio, M., Caporin, M., Gobbo, M., 2006. Flexible dynamic conditional correlation multivariate GARCH models for asset allocation. Appl. Financ. Econ. Lett. 2 (2), 123–130. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 (3), 307–327. Bollerslev, T., 1990. Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Rev. Econ. Stat. 498–505. Brady, E., Deutsch, M., 1950. Angular correlation of successive gamma-rays. Phys. Rev. 78 (5), 558. Engle, R., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 987–1007. Engle, R., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J. Bus. Econom. Statist. 20 (3), 339–350. Engle, R., Kroner, K.F., 1995. Multivariate simultaneous generalized ARCH. Econometric Theory 11 (01), 122–150. Engle, R., Sheppard, K., 2001. Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. Technical Report, National Bureau of Economic Research. Fosdick, B.K., Raftery, A.E., 2012. Estimating the correlation in bivariate normal data with known variances and small sample sizes. Amer. Statist. 66 (1), 34–41. Glosten, L.R., Jagannathan, R., Runkle, D.E., 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. J. Finance 48 (5), 1779–1801. Hotelling, H., 1953. New light on the correlation coefficient and its transforms. J. R. Stat. Soc. Ser. B Stat. Methodol. 15 (2), 193–232. Jarjour, R., 2018. Clustering Financial Time Series for Volatility Modeling (Ph.D. thesis). The University of Iowa. Kendall, M., Stuart, A., Ord, J., 1977. The Advanced Theory of Statistics: Inference and Relationship. In: Kendall’s Advanced Theory of Statistics: Classical Inference and the Linear Model, C. Griffin, URL https://books.google.com/books?id=kBbvAAAAMAAJ. Krantz, S.G., Parks, H.R., 2002. A primer of Real Analytic Functions. Springer Science & Business Media. Newey, W.K., McFadden, D., 1994. Large sample estimation and hypothesis testing. Handb. Econom. 4, 2111–2245. Tsay, R., 2010. Analysis of financial time series: Third edition. In: Analysis of Financial Time Series: Third Edition. http://dx.doi.org/10.1002/ 9780470644560. Tsay, R., 2013. Multivariate Time Series Analysis: With R and Financial Applications. In: Wiley Series in Probability and Statistics, Wiley, URL https://books.google.com/books?id=A4QVAgAAQBAJ. Tse, Y.K., Tsui, A.K.C., 2002. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. J. Bus. Econom. Statist. 20 (3), 351–362. Van der Weide, R., 2002. GO-GARCH: a multivariate generalized orthogonal GARCH model. J. Appl. Econometrics 17 (5), 549–564.

Please cite this article as: R. Jarjour and https://doi.org/10.1016/j.jeconom.2020.01.010.

K.-S. Chan,

Dynamic

conditional

angular

correlation.

Journal

of

Econometrics

(2020),