A new generalized volatility proxy via the

Applied Economics

ISSN: 0003-6846 (Print) 1466-4283 (Online) Journal homepage: http://www.tandfonline.com/loi/raec20

A new generalized volatility proxy via the stochastic volatility model Jong-Min Kim, Hojin Jung & Li Qin To cite this article: Jong-Min Kim, Hojin Jung & Li Qin (2017) A new generalized volatility proxy via the stochastic volatility model, Applied Economics, 49:23, 2259-2268, DOI: 10.1080/00036846.2016.1237751 To link to this article: http://dx.doi.org/10.1080/00036846.2016.1237751

View supplementary material

Published online: 18 Nov 2016.

Submit your article to this journal

Article views: 44

View related articles

View Crossmark data

Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=raec20 Download by: [University of Minnesota - Morris]

Date: 25 February 2017, At: 07:59

APPLIED ECONOMICS, 2017 VOL. 49, NO. 23, 2259–2268 http://dx.doi.org/10.1080/00036846.2016.1237751

A new generalized volatility proxy via the stochastic volatility model Jong-Min Kima, Hojin Jungb and Li Qina a Statistics Discipline, Division of Science and Mathematics, University of Minnesota at Morris, Morris, MN, USA; bSchool of Economics, Henan University, Kaifeng, China

ABSTRACT

KEYWORDS

This article proposes power transformation of absolute returns as a new proxy of latent volatility in the stochastic model. We generalize absolute returns as a proxy for volatility in that we place no restriction on the power of absolute returns. An empirical investigation on the bias, mean square error and relative bias is carried out for the proposed proxy. Simulation results show that the new estimator exhibiting negligible bias appears to be more efficient than the unbiased estimator with high variance.

Volatility; stochastic volatility; relative bias; mean square error

I. Introduction The volatility process is essential in order to measure financial asset returns. Two main streams of volatility models are the generalized autoregressive conditional heteroscedasticity (GARCH) model and the stochastic volatility (hereafter SV) model in the literature. However, these two models are very different because of their assumptions on the volatility states and on the error term. The former assumes deterministic volatility states and a single error term while the latter assumes stochastic volatility states and two error processes which can make the model more flexible than the GARCH model. Since volatility is latent, proxies are required when estimating volatility models. The SV model proposed by Taylor (1982) has received substantial attention for modelling volatility of financial time series in the last three decades. The most common proxy variables for volatility in financial markets are squared returns and absolute returns. The previous literature (see Taylor 1986; Triacca 2007; Giles 2008) uses the basic SV model to examine properties of the squared returns and the absolute returns as an implicit estimator of the true unobserved volatility in financial markets.1 In this article, we propose power transformation of absolute returns as a proxy of volatility. While we

JEL CLASSIFICATION

C32; C52

undertake a similar analysis of Giles (2008), we generalize his results to a case in which the power of absolute returns is allowed to vary freely. This study evaluates the accuracy and precision by using the standard metrics considered in the literature. Our empirical results show that our proposed estimator is the most efficient estimator compared to the unbiased estimator in terms of mean square error (hereafter MSE). This also infers that the new estimator improves proxies. Commonly used volatility proxies are the absolute returns and the squared returns in the financial economics literature. Previous empirical studies found that the proxy of absolute returns has superior forecasting power than that of squared returns, see, for example, Ederington and Guan (2005) and Ghysels, Santa-Clara, and Valkanov (2006). Given this stylized empirical fact, we show in this study that power transformations of absolute returns are generally superior to the traditional use of absolute returns. The layout of the article is as follows. In Section II we shed light on some properties of the proposed power transformation of absolute returns. Section III is devoted to empirical analysis. Concluding remarks are presented in Section IV.

CONTACT Hojin Jung [email protected] School of Economics, Henan University, Kaifeng, Henan 475001, China 1 Kalman filter proposed by Kalman (1960) through a state space representation is a common tool for the SV model. Moreover, empirical studies of Andersen, Benzoni, and Lund (2002) and Eraker, Johannes, and Polson (2003) improve model fit of the SV model by incorporating price jumps. Recently, Harvey (2013) proposes the dynamic conditional score model which is close to SV specification, where the volatility is treated as an unobserved stochastic process. Supplemental data for this article can be accessed here. © 2016 Informa UK Limited, trading as Taylor & Francis Group

2260

J.-M. KIM ET AL.

II. A new measure of volatility in the SV model Giles (2008) uses absolute returns as the proxy for unobserved volatility. We propose a power transformation method of absolute returns based on his analysis: jrt jk as an estimator of σ kt , where k > 0, including two special values of k of 1 and 2.2 Let pt be the spot price of a financial asset at time t and its
 one-period return is defined as rt ¼ loge ppt1t , then the latent volatility is defined as σ 2t ;varðrt jIt1 Þ, where It1 is the information set observed at time t  1. In this study, we adapt a basic SV model (Taylor 2005, 278–83) and use Giles (2008) notation for comparability. rt ¼ μ þ σ t zt ;

zt , iid

where zt and ut are independent processes. The unconditional distribution of loge ðσ 2t Þ is norγ0 1γ1

and variance

σ 2u , 1γ21

  k 22 kþ1 ; EðjZj Þ ¼ pffiffiffi Γ 2 π k

(5)

0

jγ1 j 0. This specification is the main novelty of this article. Using the properties of the integral representation of the gamma function, Giles (2008) shows that the absolute moments of a standard normal variate, Z, are

where Gamma function ΓðαÞ is as follows: ð1 ● ΓðαÞ ¼ tα1 et dt if the real part of the com-

Nð0; 1Þ

loge ðσ 2t Þ ¼ γ0 þ γ1 loge ðσ 2t1 Þ þ ut ; ut , iid Nð0; σ 2u Þ;

By substituting the value of 2k, instead of k, in Equation (3), we obtain the following general form of the SV model:   km k2 ν k þ ; (4) Eðσ t Þ ¼ exp 2 8

given the

stationarity condition in Equation (1). If Y is a Nðm; νÞ random variable, then X ¼ expðYÞ has a log-normal distribution with central moments given by
ν (2) EðXk Þ ¼ exp km þ k2 : 2 The model given by Equations (1) and (2) can be adapted for our setting. Let Y be equal to loge ðσ 2t Þ, then we have X ¼ σ 2t . Thus, Equation (2) can be easily rewritten as follows:
 2ν : (3) Eðσ 2k t Þ ¼ exp km þ k 2

plex number (α) is positive, i.e. (ReðαÞ>0Þ π ● ΓðαÞ ¼ Γðαþ1Þ ; ΓðαÞΓð1 α  pffiffiffi αÞ ¼ sinðπαÞ 1 ● Γðn þ 1Þ ¼ n!; Γ 2 ¼ π and the corresponding expression in the (nonstandardized) Student-t case with degrees of freedom, p, is 
pk k  p2 Γ kþ1 2 Γ 2 EðjTjk Þ ¼ ; p > k: (6) pffiffiffi p πΓ 2 The results allow us to obtain the bias and MSE of jrt jk . Note that here consistent with the previous literature (see Giles 2008; Triacca 2007), we set μ ¼ 0. Case 1. jrt jk as an estimator of σ kt We start by considering that jrt jk is an estimator of σ kt . When zt is normally distributed, with Equation (5), the MSE is equal to

h i h i h i 2 2 2 MSEðjrt jk Þ ¼ E ðjrt jk  σ kt Þ ¼ E ðjZt jk σ kt  σ kt Þ ¼ E ðjZt jk  1Þ σ 2k t h i

h i

2 ¼ E jZt j2k  2jZt jk þ 1 E σ 2k ¼ E ðjZt jk  1Þ E σ 2k t t h i

: ¼ EðjZt j2k Þ  2EðjZt jk Þ þ 1 E σ 2k t " #      kþ2  kγ0 2k 1 22 kþ1 k2 σ 2u ¼ pffiffiffi Γ k þ  pffiffiffi Γ þ 1 exp þ 1  γ1 2ð1  γ21 Þ 2 2 π π

(7)

APPLIED ECONOMICS

We define the relative bias (RB) as the bias divided by its true value. The empirical RB is then given by Biasðjrt jk Þ Eðjrt jk  σ kt Þ ¼ Eðσ kt Þ Eðσ kt Þ ¼ EðjZt jk  1Þ " k  #  22 kþ1 1 : ¼ pffiffiffi Γ 2 π

Biasðjrt jk Þ Eðjrt jk  σ kt Þ ¼ Eðσ kt Þ Eðσ kt Þ ¼ EðjTt jk  1Þ 2 k  
pk 3 p2 Γ kþ1 2 Γ 2  15: ¼4 pffiffiffi p πΓð2Þ

2261

RBðjrt jk Þ ¼

RBðjrt jk Þ ¼

(8)

(10)

Case 2. jrt jk as an estimator of σ 2t Considering jrt jk as an estimator of σ 2t , our purpose now is to investigate the statistical properties of jrt jk . In the same way as in Case 1, using Equation

We also derive the Student-t MSE of jrt jk by using Equation (6) when zt is a standardized Student-t variate, that is,

h i h i h i 2 2 2 MSEðjrt jk Þ ¼ E ðjrt jk  σ kt Þ ¼ E ðjTt jk σ kt  σ kt Þ ¼ E ðjTt jk  1Þ σ 2k t h i h i



2 ¼ E jTt j2k  2jTt jk þ 1 E σ 2k ¼ E ðjTt jk  1Þ E σ 2k t t h i

¼ EðjTt j2k Þ  2EðjTt jk Þ þ 1 E σ 2k t 2  3  
pk k  p    2 Γ kþ1 Γ k 1 2 2 2p Γ p Γ k þ  k 2 kγ 2 k σ u 0 2 2 : ¼4 þ  þ 15 exp pffiffiffi p pffiffiffi p 1  γ1 2ð1  γ21 Þ πΓð2Þ πΓð2Þ

(9)

(5) and the normality assumption, we can derive the estimator errors as follows:

Under the standard Student-t case, the RB can be written as

Biasðjrt jk Þ Eðjrt jk  σ 2t Þ Eðjrt jk Þ ¼ ¼ 1 Eðσ 2t Þ Eðσ 2t Þ Eðσ 2t Þ h k  i
 kγ0 k2 σ 2u 22ffiffi kþ1 p exp Γ þ 2ð1γ1 Þ 2 8ð1γ21 Þ π
  1: ¼ γ0 σ 2u exp 1γ þ 2ð1γ2 Þ

RBðjrt jk Þ ¼

1

(12)

1

We also write the RB of jrt jk as

k

h

k

2 σ 2t Þ

i

h

ðjZt jk σ kt

2 σ 2t Þ

i

¼E MSEðjrt j Þ ¼ E ðjrt j   h i

h i



 2E jZt jk E σ kþ2 þ E σ 4t ¼ E jZt j2k E σ 2k t t    k     kγ0 2γ0 2 1 k2 σ 2u 2σ 2u þ exp ¼ pffiffiffi Γ k þ exp þ þ 1  γ1 2ð1  γ21 Þ 1  γ1 1  γ21 2 π " kþ2      # kγ0 γ0 22 kþ1 k2 σ 2u σ 2u  pffiffiffi Γ þ þ exp ; exp 2ð1  γ1 Þ 8ð1  γ21 Þ 1  γ1 2ð1  γ21 Þ 2 π

(11)

2262

J.-M. KIM ET AL.

The following equations are derived for the two metrics of the statistical estimator’s accuracy and precision under the Student-t distribution, using Equation (6).

previous cases, we obtain the MSE and RB for the different distributions of zt . Using Equation (5), the normal MSE and RB are also directly obtained:

h i h i 2 2 MSEðjrt jk Þ ¼ E ðjrt jk  σ 2t Þ ¼ E ðjTt jk σ kt  σ 2t Þ h i

h i



¼ E jTt j2k E σ 2k  2E jTt jk E σ kþ2 þ E σ 4t t t "    #     pk Γ k þ 12 Γ p2  k kγ0 2γ0 k2 σ 2u 2σ 2u þ exp ¼ exp þ þ pffiffiffi p 1  γ1 2ð1  γ21 Þ 1  γ1 1  γ21 πΓ 2 2 k  
pk3     2p2 Γ kþ1 kγ0 γ0 2 Γ 2 k2 σ 2u σ 2u 4 5 exp ; exp  þ þ pffiffiffi p 2ð1  γ1 Þ 8ð1  γ21 Þ 1  γ1 2ð1  γ21 Þ πΓ 2

Biasðjrt jk Þ Eðjrt jk  σ 2t Þ Eðjrt jk Þ ¼ ¼ 1 Eðσ 2t Þ Eðσ 2t Þ Eðσ 2t Þ k
 pk p2 Γðkþ1 kγ0 k2 σ 2u 2 Þ Γð 2 Þ pffiffi p exp þ 2ð1γ1 Þ 8ð1γ21 Þ π Γð 2 Þ
  1: ¼ γ σ 2u exp 1γ0 þ 2ð1γ 2Þ

(13)

RBðjrt jk Þ ¼

1

Case 3. jrt jk as an estimator of σ t We turn to the last case, in which jrt jk is considered as an estimator of σ t . As we stated in the two

(14)

1

In the same manner, using Equation (6) we obtain the Student-t MSE and RB as follows:

h i h i 2 2 MSEðjrt jk Þ ¼ E ðjrt jk  σ t Þ ¼ E ðjZt jk σ kt  σ t Þ h i

h i



k ¼ E jZt j2k E σ 2k  2E jZ þ E σ 2t E σ kþ1 j t t t      k   kγ0 γ0 2 1 k2 σ 2u σ 2u þ exp exp ¼ pffiffiffi Γ k þ þ þ 1  γ1 2ð1  γ21 Þ 1  γ1 2ð1  γ21 Þ 2 π " kþ2      # kγ0 γ0 22 kþ1 k2 σ 2u σ 2u  pffiffiffi Γ exp ; exp þ þ 2ð1  γ1 Þ 8ð1  γ21 Þ 2ð1  γ1 Þ 8ð1  γ21 Þ 2 π

Biasðjrt jk Þ Eðjrt jk  σ t Þ Eðjrt jk Þ ¼ ¼ 1 Eðσ t Þ Eðσ t Þ Eðσ t Þ h k  i
 kγ0 k2 σ 2u 22ffiffi kþ1 p exp Γ þ 2 2ð1γ1 Þ 8ð1γ2 Þ π
 1  1: ¼ 2 γ0 σu exp 2ð1γ Þ þ 8ð1γ2 Þ

(15)

RBðjrt jk Þ ¼

1

1

(16)

APPLIED ECONOMICS

h i h i 2 2 MSEðjrt jk Þ ¼ E ðjrt jk  σ t Þ ¼ E ðjTt jk σ kt  σ t Þ h i

h i



k  2E jT þ E σ 2t ¼ E jTt j2k E σ 2k E σ kþ1 j t t t "    #     pk Γ k þ 12 Γ p2  k kγ0 γ0 k2 σ 2u σ 2u þ exp ¼ exp þ þ pffiffiffi p 1  γ1 2ð1  γ21 Þ 1  γ1 2ð1  γ21 Þ πΓ 2 2 k  
pk3     2 2 2 2p2 Γ kþ1 2 Γ 2 kγ γ k σ σ u u 0 0 5 exp exp ; 4 þ þ pffiffiffi p 2ð1  γ1 Þ 8ð1  γ21 Þ 2ð1  γ1 Þ 8ð1  γ21 Þ πΓ 2

(17)

generated from the parameter ranges deduced from the evidence compiled by Taylor (2005, 287–8), such as γ0 2 ð0:95; 0:99Þ; γ1 2 ð0:54; 0:1Þ and σ 2u 2 ð0:0018; 0:0478Þ.3 Given the typical values of the parameters, there are many possible combinations to construct datasets. In this study, we report empirical results from only the case in which we use the most conservative set of values for the parameters such as k 2 f0:3; 0:8; 1; 1:3; 1:8; 2g; γ0 2 f0:95; 0:97; 0:99g; γ1 2 f0:54; 0:25; 0:1g 2 and σ u 2 f0:0018; 0:025; 0:0478g.4,5 We report the normal RB and MSE of Case 1 by using Equations (7) and (8). Table 1 shows that the RB is as low as −0.065 when k ¼ 1:8, and in

Biasðjrt jk Þ Eðjrt jk  σ t Þ Eðjrt jk Þ ¼ ¼ 1 Eðσ t Þ Eðσ t Þ Eðσ t Þ k
 pk p2 Γðkþ1 kγ0 k2 σ 2u 2 Þ Γð 2 Þ pffiffi p exp þ 2 2ð1γ1 Þ 8ð1γ1 Þ π Γð 2 Þ
  1: ¼ γ0 σ 2u exp 2ð1γ Þ þ 8ð1γ2 Þ

RBðjrt jk Þ ¼

1

2263

1

(18)

III. Empirical analysis of a new proxy measure In order to assess the efficiency of our proposed proxy along several dimensions, we utilize simulated data. In the simulation design, the datasets are Table 1. RB, MSE and RE of jrt jk in Case 1 (normal distribution).

Relative bias k 0.3 0.8 1 1.3 1.8 2 Mean square error 0.3 0.8 1 1.3 1.8 2 2 % of RE 1:8

ðγ0 ¼ 0:95; γ1 ¼ 0:54; σ2u ¼ 0:0018Þ; –0.1331 –0.2045 –0.2021 –0.1740 –0.0659 0.0000

ðγ0 ¼ 0:97; γ1 ¼ 0:25; σ2u ¼ 0:025Þ; –0.1331 –0.2045 –0.2021 –0.1740 –0.0659 0.0000

ðγ0 ¼ 0:99; γ1 ¼ 0:1; σ2u ¼ 0:0478Þ; –0.1331 –0.2045 –0.2021 –0.1740 –0.0659 0.0000

Monte Carlo DGP –0.1331 –0.2045 –0.2021 –0.1740 –0.0659 0.0000

0.0901 0.4760 0.7493 1.4319 4.3268 7.1040 164.19%

0.0945 0.5416 0.8812 1.7700 5.8182 9.6856 166.47%

0.0982 0.5991 1.0003 2.0891 7.3372 12.3949 168.93%

0.0936 0.5287 0.8555 1.7053 5.5414 9.0753 163.77%

A power term (k) is raised to construct the series of jrt jk for an estimator of σkt . In particular, the underlying data is generated by a combination of three parameters (γ0 ; γ1 and σ2u ), which designate to increase their values across the specifications. The calculated RB and MSE are obtained by using Equations (7) and (8) under a normal distribution assumption. A value is highlighted in the upper panel if RB is less than the critical value of 10% and for its corresponding MSE in the lower panel. Note that Giles (2008) analysis is a special case in our model when we assume that k = 1. The use of these ranges of the parameters is valid for the following reasons: (1) As discussed in Taylor (2005), the estimates of these parameters given by previous empirical studies are usually between the ranges. (2) This study chooses the same parameter ranges used in Giles (2008) for comparison with his study. 4 For simplicity, k takes such values, although in principle k can be extended to accommodate any values. 5 We find that the results shown in this study are not sensitive to other datasets constructed by different parameter values. The complete datasets used in this study and all of our empirical results are available in the Supplemental data. 2 3

2264

J.-M. KIM ET AL.

particular, they are zero across the columns when k ¼ 2. If RB is less than 10%, then it is negligible in Cochran’s (1963) sense. Now we have one unbiased estimator and another biased, but acceptable, estimator. Then, it is reasonable to compare variance or relative efficiency (hereafter RE) of two estimators. The normal MSE and RE of jrt jk are presented at the bottom of Table 1. The empirical RE is defined as the ratio of two different MSEs. When k changes from 2 to 1.8, the empirical RE in percentage terms is much greater than 164% across specifications. Our proposed estimator with an acceptable level of bias could lead to a systematically significant increase in efficiency. This empirical result strongly supports that under normality, the negligible biased estimator of jrt jk when k ¼ 1:8 is more efficient than the unbiased estimator of jrt jk when k ¼ 2. We also conduct a simulation exercise. The RB and MSE statistics in the last column are computed via stochastic simulations in a Monte Carlo setting with 1000 replications, where artificial series for returns are simulated according to the proposed data-generating process (DGP). The figures in the column are the average values of the statistics obtained by implementing the simulation. We find that the empirical results are consistent with the theoretical values delivered by Equations (7) and (8). A closer inspection of how our proposed estimator performs in a fat tailed returns distribution is investigated in a further simulation. For that purpose, different degrees of freedom are induced, such as p 2 f5; 7; 10g. Table 2 provides the RB and MSE of the new estimator jrt jk in the Student-t case by using Equations (9) and (10). Keeping degree of freedom p as constant when power k increases, bias shows a tendency of becoming negative from k ¼ 0:3 to k ¼ 1:0 and then positive from k ¼ 1:3 to k ¼ 2 across all specifications.6 This finding implies that the bias is close to zero around k ¼ 1:3. The RB of jrt jk is less than 10% when k ¼ 1:3. It is interesting to note here that the MSE is much smaller when k ¼ 0:3 than when k takes other values. When we choose jrt j0:3 as an estimator instead of jrt j1:3 , it provides significant improvement in precision, which is desirable, even at the expense of losing only a little accuracy. In accor-

6

dance with our previous results, the proposed estimator jrt jk is a better volatility proxy even when zt is a standardized Student-t variate. Note that the upper panel in Table 1 reports identical figures because the RB depends on the power (k) of absolute returns as shown in Equation (8). For a similar reason, we observe the same RB across the degrees of freedom (p) at (k) in Table 2. k and p are critical factors that determine the RBs in a fat tailed returns distribution, as shown in Equation (10). We also find from our simulation exercise that the empirical results obtained for the standardized Student-t converge towards the results obtained for the standardized normal processes as the degrees of freedom (p) increase. For example, when p ¼ 100, the estimated RBs are –0.1316, –0.2000, –0.1961, –0.1650, –0.0496 and 0.0204, respectively. The complete simulation results in this study can be available upon request. Figure 1 illustrates that holding γ0 ; γ1 and σ 2u constant, as k increases, MSE also increases when zt is a standardized Student-t variate. An interesting feature of the figure is that the RB and MSE are significantly larger when p ¼ 5, which means that fatter tails in the returns distribution cause the estimator to become more biased and imprecise. Since we have already discussed in detail the application of the proposed estimator to empirical analysis in Case 1, we now focus directly on the empirical results in Cases 2 and 3. Tables 3 and 5 report the RB and MSE obtained from Case 2 under different types of returns distributions. Under the normal specification, all RBs for any given value of k are greater than 10%, except when k ¼ 2. Note that Triacca (2007) considers jrt j2 as an unbiased estimator of σ 2t and our empirical findings support Triacca’s result. On the other hand, under the Student-t specification, the sign and magnitude of our proposed estimator’s RB depend on the parameter values. Our empirical results also indicate that jrt j2 becomes a biased estimator. A comparison of the MSEs of jrt j2 and the others shows that it performs worse as a proxy of volatility in this more general Student-t case. However, the RBs of jrt jk are lower than 10% when k ¼ 1:8 and p ¼ 10, suggesting that jrt jk is a better estimator than the unbiased estimator considered in Triacca (2007).

We also find that for the typical parameter value ranges, as degrees of freedom increase the MSE of jrt jk monotonically decreases.

0.0931 0.5636 0.9660 2.1590 8.9531 16.3953

0.0951 0.6225 1.1196 2.7331 13.6978 27.6131

Relative bias

–0.1101 –0.1317 –0.1017 –0.0184 0.2408 0.4000 0.0999 0.7125 1.3286 3.4304 18.9675 39.8362

0.1037 0.8391 1.6925 5.0743 41.2865 112.8691

p=7

–0.0999 –0.0965 –0.0510 0.0659 0.4322 0.6667

p=5

0.0978 0.6451 1.1464 2.7098 12.3975 23.6527

–0.1173 –0.1556 –0.1353 –0.0721 0.1287 0.2500

p = 10

(γ0 ¼ 0:97; γ1 ¼ 0:25; σ2u ¼ 0:025)

0.1077 0.9330 1.9367 6.0718 53.4504 151.0274

–0.0999 –0.0965 –0.0510 0.0659 0.4322 0.6667 0.1038 0.7923 1.5203 4.1047 24.5557 53.3038

–0.1101 –0.1317 –0.1017 –0.0184 0.2408 0.4000

p=7

0.1016 0.7173 1.3118 3.2425 16.0501 31.6491

–0.1173 –0.1556 –0.1353 –0.0721 0.1287 0.2500

p = 10

(γ0 ¼ 0:99; γ1 ¼ 0:1; σ2u ¼ 0:0478) p=5

0.1027 0.8194 1.6443 4.8947 39.4143 107.3658

–0.0999 –0.0965 –0.0510 0.0659 0.4322 0.6667

p=5

0.0990 0.6958 1.2908 3.3089 18.1074 37.8938

–0.1101 –0.1317 –0.1017 –0.0184 0.2408 0.4000

p=7

(Monte Carlo DGP) p = 10

0.0968 0.6299 1.1137 2.6139 11.8353 22.4995

−0.1173 –0.1556 –0.1353 –0.0721 0.1287 0.2500

A power term (k) is raised to construct the series of jrt jk for an estimator of σkt . In particular, the underlying data is generated by a combination of four parameters (γ0 ; γ1 ; σ2u and p), which designate to increase their values across the specifications. The calculated RB and MSE are obtained by using Equations (9) and (10) under a Student-t distribution assumption. A value is highlighted in the upper panel if RB is less than the critical value of 10% and for its corresponding MSE in the lower panel.

–0.1173 –0.1556 –0.1353 –0.0721 0.1287 0.2500

–0.1101 –0.1317 –0.1017 –0.0184 0.2408 0.4000

0.3 –0.0999 0.8 –0.0965 1 –0.0510 0.0659 1.3 1.8 0.4322 2 0.6667 Mean square error 0.3 0.0987 0.8 0.7331 1 1.4262 1.3 4.0429 1.8 29.8159 2 78.2371

p = 10

p=7

p=5

k

(γ0 ¼ 0:95; γ1 ¼ 0:54; σ2u ¼ 0:0018)

Table 2. RB and MSE of jrt jk in Case 1 (t-distribution).

APPLIED ECONOMICS 2265

2266

J.-M. KIM ET AL.

Figure 1. RB (left panel) and MSE (right panel) of jrt jk in Case 1 (t-Distribution). The panels plot when we restrict γ0 ¼ 0:95; γ1 ¼ 0:54 and σ2u ¼ 0:0018.

Table 3. RB and MSE of jrt jk in Case 2 (normal distribution). Relative bias k ðγ0 ¼ 0:95; γ1 0.3 0.8 1 1.3 1.8 2 Mean square error 0.3 0.8 1 1.3 1.8 2

¼ 0:54; σ2u ¼ 0:0018Þ –0.4875 –0.4511 –0.4144 –0.3349 –0.1220 0.0000 0.8958 1.1177 1.2751 1.7631 4.3924 6.9208

ðγ0 ¼ 0:97; γ1 ¼ 0:25; σ2u ¼ 0:025Þ –0.5576 –0.5062 –0.4641 –0.3753 –0.1378 0.0000

ðγ0 ¼ 0:99; γ1 ¼ 0:1; σ2u ¼ 0:0478Þ –0.6060 –0.5457 –0.5004 –0.4055 –0.1502 0.0000

Monte Carlo DGP –0.5446 –0.4963 –0.4553 –0.3682 –0.1351 0.0000

1.7112 1.8449 1.9848 2.5505 6.2933 10.2211

2.7218 2.7338 2.8385 3.4568 8.3969 13.9541

1.5974 1.7468 1.8901 2.4446 6.0199 9.7413

A power term (k) is raised to construct the series of jrt jk for an estimator of σ2t . In particular, the underlying data is generated by a combination of three parameters (γ0 ; γ1 and σ2u ), which designate to increase their values across the specifications. The calculated RB and MSE are obtained by using Equations (11) and (12) under a normal distribution assumption. A value is highlighted in the upper panel if RB is less than the critical value of 10% and for its corresponding MSE in the lower panel.

Lastly, we also present empirical evidence regarding the performance of jrt jk as a predictor of σ t . Table 4 shows that the RB in absolute value is less than 10% when k ¼ 1:3 with the normal distribution in Case 3. Note that jrt j is a biased estimator of σ t when zt is normally distributed. This is consistent with the findings in Giles (2008). With this particular parameter, k ¼ 1:3, the proposed proxy of volatility exhibits good performance for a standardized Student-t distribution with 10 degrees of freedom. Its RB becomes less than 10% when considering the Student-t distribution with large degrees of freedom. This provides additional support that jrt jk is a good proxy for measuring volatility. Overall, the accuracy and precision of jrt jk as an estimator of volatility 7

are remarkable when we compare RB and MSE across cases. When estimating volatility, jrt jk has smaller MSE than the absolute return or the squared return often used in literature, at least for the typical parameter settings. It is worth mentioning that our findings concur with McKenzie’s (1999) empirical findings of forecasting performance relative to other power terms. McKenzie (1999) found that a power term of 1.25 is optimal based on the standard root MSE when modelling the exchange rate volatility. Further insights into the differences across cases can be gained by comparing figures. Figures 2 and 3 display the RB and MSE obtained from Cases 2 and 3, respectively.7 They are consistent with the pattern we

We also obtain figures by employing different parameter values in Cases 1, 2 and 3. These figures are omitted to save space but the Supplementary data provides these figures.

APPLIED ECONOMICS

2267

Table 4. RB and MSE of jrt jk in Case 3 (normal distribution). Relative bias k ðγ0 ¼ 0:95; γ1 0.3 0.8 1 1.3 1.8 2 Mean square error 0.3 0.8 1 1.3 1.8 2

0:54; σ2u

¼ –0.3017 –0.2521 –0.2021 –0.0937 0.1964 0.3626 0.2388 0.5272 0.7519 1.3857 4.4051 7.1570

¼ 0:0018Þ

ðγ0 ¼ 0:97; γ1 ¼ 0:25; σ2u ¼ 0:025Þ –0.3413 –0.2647 –0.2021 –0.0699 0.2837 0.4888

ðγ0 ¼ 0:99; γ1 ¼ 0:1; σ2u ¼ 0:0478Þ –0.3708 –0.2745 –0.2021 –0.0506 0.3571 0.5970

Monte Carlo DGP –0.3336 –0.2624 –0.2021 –0.0742 0.2682 0.4668

0.3422 0.6396 0.9134 1.7620 6.2609 10.6274

0.4487 0.7469 1.0668 2.1353 8.2874 14.5574

0.3264 0.6230 0.8891 1.7036 5.9751 10.1005

A power term (k) is raised to construct the series of jrt jk for an estimator of σt . In particular, the underlying data is generated by a combination of three parameters (γ0 ; γ1 and σ2u ), which designate to increase their values across the specifications. The calculated RB and MSE are obtained by using Equations (15) and (16) under a normal distribution assumption. A value is highlighted in the upper panel if RB is less than the critical value of 10% and for its corresponding MSE in the lower panel.

Figure 2. RB (left panel) and MSE (right panel) of jrt jk in Case 2 (t-Distribution). The panels plot when we restrict γ0 ¼ 0:95; γ1 ¼ 0:54 and σ2u ¼ 0:0018.

Figure 3. RB (left panel) and MSE (right panel) of jrt jk in Case 3 (t-Distribution). The panels plot when we restrict γ0 ¼ 0:95; γ1 ¼ 0:54 and σ2u ¼ 0:0018.

2268

J.-M. KIM ET AL.

observed in Figure 1. Given Figures 1–3, it is clear that the bias and MSE in Case 1 are relatively smaller than those of the other cases. This provides strong evidence that Case 1, jrt jk as an estimator of σ kt , is more efficient than Case 2, jrt jk as an estimator of σ 2t , and Case 3, jrt jk as an estimator of σ t . IV. Conclusion This article proposes power transformation of absolute returns as a proxy for the unobservable volatility. The RB and MSE serve as yardsticks for measuring the performance of such a proxy in the various parameter conditions. We provide empirical evidence that the new proposed estimator outperforms two popular proxies such as the absolute and squared returns in terms of efficiency. The power transformation of absolute returns is a biased estimator with negligible bias and small variance. Moreover, it exhibits better fits in a heavy tailed returns distribution which is typical for data from financial markets.8

Disclosure statement No potential conflict of interest was reported by the authors.

References Andersen, T. G., L. Benzoni, and J. Lund. 2002. “An Empirical Investigation of Continuous-Time Equity Return Models.” The Journal of Finance 57 (3): 1239– 1284. doi:10.1111/1540-6261.00460.

8

Cochran, W. 1963. Sampling Techniques. New York: John Wiley & Sons. Ederington, L. H., and W. Guan. 2005. “Forecasting Volatility.” Journal of Futures Markets 25 (5): 465–490. doi:10.1002/(ISSN)1096-9934. Eraker, B., M. Johannes, and N. Polson. 2003. “The Impact of Jumps in Volatility and Returns.” The Journal of Finance 58 (3): 1269–1300. doi:10.1111/1540-6261.00566. Ghysels, E., P. Santa-Clara, and R. Valkanov. 2006. “Predicting Volatility: Getting the Most Out of Return Data Sampled at Different Frequencies.” Journal of Econometrics 131 (1–2): 59–95. doi:10.1016/j. jeconom.2005.01.004. Giles, D. E. 2008. “Some Properties of Absolute Returns as a Proxy for Volatility.” Applied Financial Economics Letters 4 (5): 347–350. doi:10.1080/17446540701720709. Harvey, A. C. 2013. Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series, 52. Cambridge: Cambridge University Press. Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering 82 (1): 35–45. doi:10.1115/1.3662552. McKenzie, M. D. 1999. “Power Transformation and Forecasting the Magnitude of Exchange Rate Changes.” International Journal of Forecasting 15 (1): 49–55. doi:10.1016/S0169-2070(98)00066-1. Taylor, S. J. 1982. “Financial Returns Modelled by the Product of Two Stochastic Processes-A Study of the Daily Sugar Prices 1961–75.” Time Series Analysis: Theory and Practice 1: 203–226. Taylor, S. J. 1986. Modelling Financial Time Series. New York, NY: Wiley. Taylor, S. J. 2005. Asset Price Dynamics, Volatility, and Prediction. Princeton, NJ: Princeton university press. Triacca, U. 2007. “On the Variance of the Error Associated to the Squared Return as Proxy of Volatility.” Applied Financial Economics Letters 3 (4): 255–257. doi:10.1080/ 17446540601118343.

Note: Additional information for this article can be found in the Supplementary data at http://facultypages.morris.umn.edu/ jongmink/research/ OnlineAppendix.pdf