Regression is a main technique in scientific research, which is widely used in exploring the linear relationship between observable quantities, and in analyzing the structure of variability.
The connection between regression and finance originated from the capital asset pricing model (CAPM, Markowitz (1952Markowitz ( , 1959)), Sharpe (1964), Lintner (1965), Black (1972)). Over time, the connection has expanded to multiple factors, such as in Fama and MacBeth (1973), and Ross (1976). The literature has gradually split into regression (observed factors) and principal component analysis (PCA, unobserved factors). We are here concerned with the former. For literature ✩ Financial support from the United States National Science Foundation under grants DMS-2015530 (Zhang), and DMS-2015544 (Mykland), is gratefully acknowledged. reviews, see, e.g., Campbell et al. (1997) and Cochrane (2005). Recent developments in high frequency PCA are reviewed in Chen et al. (2020).
The importance of time-varying betas (regression coefficients) has received increasing attention in the finance and econometrics literature. Such betas reflect time-varying conditional information. Research in this direction includes Hansen and Richard (1987), Bollerslev et al. (1988), Jagannathan and Wang (1996), Boguth et al. (2011), Ang and Kristensen (2012), Engle (2016), and Gagliardini et al. (2016).
WIth the advent of high-frequency data, a literature has started to develop where time-varying betas are estimated from intraday data. Important empirical contributions are Andersen et al. (2006), who investigated the persistence and predictability of time-varying beta estimates, and Patton and Verardo (2012), who explored the effect of information flows on stock returns.
The purpose of this paper is to develop the theory for how to estimate betas in fixed and increasing dimension, for high frequency data. If we let c X,X t and c X,Y t be the (unobserved) spot (instantaneous) covariance matrices of (latent) efficient prices (or other semi-martingales) X and Y, the spot and integrated beta are given by, 1
where [0, T ] is the fixed interval under observation. By considering data with microstructure noise, as well as letting observations (such as transactions and quotes) happen asyncronously across dimensions, we bring the theory to the point where it can accommodate real data.
In finite dimension (Section 3), our theory focuses on the integrated beta. The integrated beta ∫ T 0 β t dt is consistently estimated by aggregating estimates of spot beta. The aggregation is similar to the constructions in the papers cited at the beginning of Section 1.1 We show asymptotic normality in finite dimension (Theorem 2), preceded by a bias correction which is needed for this normality to hold.
In increasing dimension (Section 4), our theory estimates the spot (instantaneous) β t , 2 and from there estimates the spot precision matrix, which has a role in determining asset allocation, cf. (Fan et al., 2016a). We derive the rate of convergence as the dimensions of X and Y tend to infinity. Both these developments take as their points of departure spot covariance matrices that are calculated by the S-TSRV procedure (pre-averaging followed by two-scales, Section 2 in this paper, and Mykland et al. (2019)). The basic preaveraging is done over time blocks of length ∆τ n , and spot covariance matrices are calculated over time blocks of length ∆T n . To get a sense of the magnitudes we have in mind, in the simulation we have used ∆T n = 2340 seconds, and ∆τ n is 5, 15 or 60 seconds. In the empirical application, ∆τ n = 5 seconds, and ∆T n is (in most cases) hourly.
On the theoretical side, the rate of convergence in the CLT (Theorem 2) is a -1 n , which is allowed to be arbitrarily close to n -1/4 . The latter is previously known as the standard efficient rate for covariances in estimation problems with microstructure. 3 A precise explanation of the rate a n is given in Eq. (2.10) and Remark 1 in Section 2.2. As described there, a n is closely related to ∆τ n .
In the increasing dimension setting, the rates of convergence also depend crucially on a n , but we defer discussion of this to Section 4.
Closely related literature. The theory of estimation the two betas in (1.1) has previously been studied in the case of no microstructure noise and synchronous observations, in Mykland and Zhang (2006, 2008), and Zhang (2012), with a jump-robust version in Aït-Sahalia et al. (2020) and Aït-Sahalia et al. (2021). In this setting, the estimator of integrated beta is simply a sum of ordinary least squares regression estimators (Mykland and Zhang, 2009, Section 4.2, pp. 1424-1426). More generally, all proposed estimators of (1.1) are local in time, so that covariance at time t is only compared with variance around t. This is also the case for the estimators developed in the current paper.
In the presence of asynchronous and noisy observations, the development of a feasible spot beta estimator has become increasingly necessary. As shown by Monte Carlo simulation (Table 5.1 in Section 5.2), integrated beta estimates become biased when the data is noisy. By applying the spot-version of the smoothed TSRV (S-TSRV), this paper proposes feasible estimators for spot beta under both fixed and increasing dimension.
Bias in the integrated beta. Expanding the Riemann sum of spot beta estimates to higher order, a bias term naturally arises, which is analogous to the aggregated second order expansion term of the non-linear functional of stochastic volatilities in Jacod and Rosenbaum (2013) and Aït-Sahalia and Xiu (2017). This bias term becomes the main barrier to the central limit theorem. By properly selecting the range of the smoothing window ∆T n over which the spot β is calculated, and then applying the extended bias-correction technique based on Chen et al. (2020), the central limit theorem (CLT) for the bias-corrected estimator (Theorem 2) follows.
An earlier approach to the assessment of integrated beta is to estimate
The theory for the estimation (1.2) would seem to go back to Barndorff-Nielsen and Shephard (2004), and natural estimators were considered empirically by Andersen et al. (2006) and Patton and Verardo (2012). The advantage of this formulation is that it permits results for covariance (co-volatility) matrices to be directly extended to the estimation of integrated β. This reduces the problem to one that has been given substantial consideration in the literature, and for which there are now already results that cover noise and asynchronicity. (See Footnote 3.) A main disadvantage of estimating (1.2) is that natural estimators are not local in time: if the time interval is a day, then, for example, covariance at 10:45 am is compared with variance at 3:20 pm.
Notwithstanding the distinction between (1.1) and (1.2), the two quantities are similar if the time span T is comparatively short. They are also the same if β t is constant in t. Constancy tests for betas have been proposed by Todorov and Bollerslev (2010), Kalnina (2012), Reiß et al. (2015) and Kong and Liu (2018).
We also point out that the estimators in the current paper are based on the assumption that the latent semi-martingales are continuous. This is substantially more complex for the case where there is microstructure noise and asynchronous observation, and we hope to approach this topic in a later paper.
When estimating a high dimensional spot (cross-sectional) covariance matrix, the rank of the estimated matrix is bounded by 2∆T n /∆τ n +b,foot_0 by construction. This is a severe constraint, even more so than when estimating an integrated matrix. It is thus possible that the rank of the true spot covariance matrix may grow much faster than the given bound.
To resolve such a contradiction, the main approach in the literature is to rely on sparsity. Our high dimensional realized regression makes use of a time-varying (observed) factor model, where we threshold the residual based on sparsity. This goes back to Bickel and Levina (2008). Our development of the large spot precision matrix estimator may be regarded as the ''realized'' and spot (high-frequency) version of Fan et al. (2011).
An estimation theory for high dimensional high frequency integrated covariance matrices has been derived with blockwise-diagonal residual covariance structure in Fan et al. (2016a), which was further improved by considering the asynchronous and noisy observations in Dai et al. (2019). In both these papers, the factor loadings are assumed to be time-invariant, which is unlike in the current paper.
As an application in Section 6, we use high-frequency beta estimation to study the variation of stock betas on earnings announcement days. It is well known in the literature that stock betas tend to be higher around the event days. For example, Ball and Kothari (1991) documented an increase in daily average beta during the three-day earnings announcement period. Vijh (1994) found that after being added to S&P 500 index between 1975 and 1989, those stocks displayed higher market beta at daily and weekly frequency. More recently, Patton and Verardo (2012) estimated daily variations in betas around earnings announcements for all the S&P 500 constituent stocks over the period 1996-2006. They found that the beta increase on announcement day was short-lived and it reverted to average levels two to five days later.
We investigate hourly beta variation within 5 days of the earnings announcement. Our study follows the spirit of Patton and Verardo (2012). While the earlier paper uses daily betas, our current technology permits us to find hourly betas, and thus to understand intra-day variation as well as overnight change in beta. Also, the construction of the beta estimate differs. Patton and Verardo (2012) used 25-minute intra-day returns (plus the overnight return from the previous day) to construct daily beta estimates. As the authors mentioned, they used the 25-minute sampling interval to reduce the impact of microstructure noise but at the cost of the accuracy of the estimate.
In the current paper, we construct beta estimates from 5-second pre-averaged returns of S&P 100 constituent stocks from 2007 to 2017, while taking account of the microstructure noise and the cross-sectional asynchronicity. Our hourly betas are unbiased and consistent, thus can more precisely capture the beta dynamics in a shorter time window around the announcements. With the definition of ''Day 0'' as the calendar day of each earnings announcement, we are able to separate the before-and post-market announcement impact on beta change. When the earnings are released in the morning prior to market open on ''Day 0'', we observe substantial beta jump in the first hour (i.e. 10am). On the other hand, when the earnings are announced after market close (4pm), we notice a significant beta jump the following day, again at the first hour. Within the 5-day window (from ''Day -2'' to ''Day +2''), most hourly beta stays at the non-earnings level.
This paper is organized as follows: we first set up the general data structure and define the spot-version of the Smoothed TSRV (S-TSRV, Mykland et al. (2019)) estimator in Section 2. For fixed dimension, consistency and asymptotic normality are shown theoretically in Section 3, and for high dimension, consistency is shown in Section 4. The results are corroborated by Monte Carlo simulation in Section 5.2. Section 6 conducts an empirical study that applies our methodology to the cross-sectional intraday returns of the components of S&P 100 Index.
For a matrix A p×q , (A) k,• denotes its kth row, (A) •,r denotes its rth column, A (r,k) denotes its (r, k)th element,
and A ⊺ denotes its transpose. We denote the largest and smallest eigenvalue of matrix A by λ max (A) and λ min (A), respectively. We denote by ∥A∥ , ∥A∥ 1 , ∥A∥ F , ∥A∥ max the spectral norm, L 1 -norm, Frobenius norm and elementwise max norm of matrix A, defined as ∥A∥ = λ
If A is a vector, then ∥A∥ and ∥A∥ F are equal to its Euclidean norm. For two sequences, we write
A number of processes, such as the martingale M, is fully indexed as M (r,s) n,t , where the superscript (r, s) refers to matrix element, and the subscript t refers to time, t ∈ [0, T ], and n is an index referring to the number of observations. In order to not overburden the paper with super-and subscripts, we do on occasion omit one or several of these. (i) M n,t is a matrix martingale. Further notation in this direction is introduced in Section 3. (ii) Meanwhile, we introduce dependence on n when we gradually get close to asymptotics in Eqs. (2.11)-(2.12), and therefore also in (2.8). However, one should bear in mind that every ingredient in (2.8) depends on sample size n, with the single exception of the latent process (2.1)-(2.2). (iii) In certain equations, such as in Remark 3, the time variable t is omitted in the subscript of the martingale M (r,s) n,t , because the quadratic variation [•, •] t is an operation on the entire path of the martingale, and t is conventionally moved to become a subscript of the quadratic variation instead. Note in particular that M ∞ (with possibly further indices) always refers to a limit when n has gone to infinity. This is because time t is always finite (≤ T ). -Similar considerations apply to other stochastic variables and processes in the following.
We here provide a description of the data generating process, as well as assumptions that we make on these processes.
The latent process. For two positive integers q, d ≥ 1, we work with data discretely sampled from the continuous process
(2.1)
The separation of Ξ t into an X t and a Y t process is irrelevant in this section, which is concerned with the estimation of the covariance (volatility) matrix process for Ξ t , but it plays a rôle when studying regression in subsequent sections. We assume that the (Ξ t ) process is a (q + d)-dimensional continuous Itô process, i.e., of the following form
where W is a (q + d)-dimensional standard (F t ) 0≤t≤T -Brownian motion, and X 0 is F 0 -measurable. The coefficients µ u and σ u are predictable and
where we use
(2.4)
Thus, the integrated covariance matrix of Ξ t may be expressed as:
(2.5)
The volatility matrix. We also suppose that c (r,s) t is itself an Itô process for any 1 ≤ r, s ≤ q + d. In other words, it has the same structure as described above, but is a matrix and not a vector.
The Observed Process. For 1 ≤ r ≤ q+d, the process
. This yields an observed process Ξ * = ( Ξ * ,(1) , . . . , Ξ * ,(q) , Ξ * ,(q+1) , . . . , Ξ * ,(q+d) )
, as follows:
Our assumptions on the data are summarized as follows:
Condition 1 (Structure of the Data). The data generating process and the observations are as laid out in Section 2.1. The processes Ξ t , µ t and σ t are adapted to a filtration (F t ). The observation times t n,j are (F t )-stopping times. For each (n, j), the noise ϵ n,t n,j is F t n,j -measurable, and sup n,j Eϵ 2 n,t n,j < ∞, and Eϵ n,t n,j = 0. The signal Ξ t may not depend on n.
In order to estimate the integrated covariance matrix ⟨Ξ, Ξ⟩ t , we construct the smoothed TSRV (S-TSRV) estimator ⟨Ξ, Ξ⟩ t on a synchronous grid, as follows.
where
and where, 1 ≤ i ≤ N and 1 ≤ r ≤ q + d, the pre-averaged price is defined as:
We similarly define J [
by switching J and K . The Smoothed-TSRV is defined as:
(2.8)
We assume the following about the block structure (imposed by the econometrician) and its interface with the data.
Condition 2 (Structure of Blocks). We assume that the block separation times τ n,i are (F t )-stopping times that are ''exogenous'' (independent of the Ξ -process), and that for each n, there are nonrandom ∆τ
, and that
(2.9)
See Remark 1 below for some clarification of Condition 2.
Condition 3 (Assumption on the Interface Between Noise and Blocks, and on Averaged Noise). We suppose that E
). Also let εn,i = εi be the averaged noise across the block from τ n,i-1 to τ n,i . Assume that the ϵ n,t n,j process is stationary, exponentially α mixing, and that there is a constant κ > 0 so that Eϵ 4+κ n,t n,j < ∞. 5 Define the sequence {a n } n≥1 by
(2.10)
5 Condition 3 is one of several ways to assure Cov For the relationship to the latter, observe that since
.
and note that a n → 0 as the number of observations n → ∞ by Condition 2. Under Conditions 1-3, it follows from Mykland et al. (2019, Section 5, pp. 110-111) 6 that
where c (r,s) is the (r, s)th element of c from (2.4), and there the M n,t /a n converges stably in law to a continuous martingale limit.
Remark 1 (The Meaning and Size of K n , J n , and a n ). We here explain that the order of convergence a n can be up to n -1/4 , but that this rate cannot be attained within the development of this paper. To see this, return to Condition 2, and consider the simplified case where M n,i only depends on n, i.e., M n,i = M n . In this case, K n -
) is desirable since it assures an optimal tradeoff between statistical error due to signal and to noise (Mykland et al., 2019, end of Section 5, p. 111). The same discussion shows that if Eq. (2.9) were removed from Condition 2, one might choose N n and M n to be of exact order O(n 1/2 ), and K n and J n would be finite. In this case, a n is of exact order n -1/4 . However, (2.9) is necessary for the representations (2.13)-(2.15), cf. Chen et al. (2020, Appendix A). We believe that it is possible to create an asymptotic development that does not require (2.9), since the finite sample calculations in Mykland et al. (2019) remain valid in this case, but this is beyond the scope of this paper. Meanwhile, the current paper should be read with the understanding that a n is almost n -1/4 , and that K n and J n are approximately finite (they grow arbitrarily slowly).
Remark 2. The selection of the tuning parameters (''scales'') K n and J n is an area which remains more art than science.
For low dimensional problems, one can proceed through signature plots on estimated volatilities, introduced by Andersen et al. (2000) and their co-authors. Signature plot was used to determine K n and J n in multiple dimensions in Zhang (2011, Fig. 2, p. 42). For moderate dimension regression problems, one option is the signature plot of integrated beta, as in Fig. 5.2 in Section 5. For truly high dimensional problems, an attractive approach is to use signature plots on eigenvalues (Chen et al., 2020, Fig. 2, p. 13). We have not gone into this detail in this paper, but Fig. 5.3 (also Section 5) plots the spectral norm (also an eigenvalue) of the error of the final precision matrix estimator (the red curve in the plot). Finally, note that for the S-TSRV, the scales are expected to be approximately finite (Remark 1 above), while for the original TSRV ( Zhang et al. (2005)), especially K n will grow with sample size n.
We shall need a slightly sharper representation under the same assumptions. For 1 ≤ r ≤ q+d, define ∆Ξ (r)
τ i-1 , the estimation error can be written as follows:
(2.12)
where the subscript n has been omitted on the right hand side, and below until Eq. (2.15), for notational convenience, and where the martingale term may be expressed as:
and the edge effect terms e
(r,s) 0 and ẽ(r,s) t has the order of O p ( a 2 n )
, which may be further expressed as:
(2.14)
(2.15)
The representation and rates in (2.13)-(2.15) follow from Chen et al. (2020, Appendix A).
Remark 3 (Assumption on Asymptotic Covariance). Let M (r,s) n,t be as defined in (2.12) and (2.13). Since the above development guarantees that a -1 n M (r,s) n , 1 ≤ r, s ≤ q + d, converge jointly in law (as continuous martingales) to a limit M (r,s)
∞ . Following Jacod and Shiryaev (2003, Corollary 6.30, p. 385), it is then also the case that for the optional (''observed'') quadratic variations,
For the simplicity of discussion, we define the spot volatility estimator ĉ(r,s) ∆Tn,t for some ∆T n > 0 as follows:
ĉ(r,s)
(2.16)
where {∆T n } n≥1 is a sequence of positive numbers satisfying
(2.17)
Moreover, to facilitate the theory development, we define c(r,s) ∆Tn,t = 1 ∆T n ∫ t+∆Tn t c (r,s) u du, π(r,s) ∆Tn,t = c(r,s) ∆Tn,t -c (r,s) t and π (r,s) ∆Tn,t = ĉ(r,s) ∆Tn,t -c(r,s) ∆Tn,t , (2.18) and π(r,s) ∆Tn,t = 1 ∆T n ( M (r,s) t+∆Tn -M (r,s) t ) and η (r 1 ,s 1 ,r 2 ,s 2 ) ∆Tn,t = π (r 1 ,s 1 ) ∆Tn,t π (r 2 ,s 2 ) ∆Tn,t -π(r 1 ,s 1 ) ∆Tn,t π(r 2 ,s 2 ) ∆Tn,t .
(2.19)
We now list several useful results of spot volatility estimator.
Lemma 1. Assume Conditions 1-3, as well as Condition (2.17). Then we have: (i)
(2.20)
(2.21)
(2.22)
(2.23)
Proof. The proof of this lemma is collected in Appendix A. □
In multiple regression, it is possible that q, d > 1 in the definition (2.1) of (Ξ t ) 0≤t≤T . Without loss of generality, we denote X = ( X (1) , . . . , X (q) ) = ( Ξ (1) , . . . , Ξ (q) ) and we let Y be a scalar process, so that Y = Ξ (q+l) for some 1 ≤ l ≤ d. It is natural to use the following notations: ⟨X,
, and ⟨X,
For the convenience of notation, we define
c X,X t = { c (r,s) t } 1≤r,s≤q q×q matrix process and c X,Y t = { c (r,q+1) t } 1≤r≤q q×1 column vector process . (3.1)
We analogously define the related matrix and vector quantities for M, c, ĉ, π , π , π , φ, ẽ an e.
Suppose that the processes are related by
for all t and k.
(3.2)
If we assume that β = ( β (1) , . . . , β (q) ) is a q × 1 column vector process, then the quadratic variation of the residual process may be expressed as:
To find min β ⟨Z, Z ⟩ T , and assuming c X,X s is positive definite almost surely for all 0 ≤ t ≤ T , we solve the identity -2c X,Y s + 2c X,X s β s = 0, and finally obtain the unique solution as follows:
The spot beta estimator is naturally constructed as:
) -1 ĉX,Y ∆Tn,T i-1 .
(3.5)
The quantity in which we are interested is:
and its estimator is given by,foot_3 :
We first show the consistency of θn . For the simplicity of discussion, we define an intermediate process:
With this smoothed beta, the estimation error of β∆Tn,T i-1 can also be decomposed into two parts, Moreover, the estimation error may be decomposed as follows:
Discretization error, R Discrete .
(3.7)
Then we can show the representations of these two types of estimation error. There representations matter both in the proofs, and also in Section 3.1.
We presently state the consistency of spot beta estimator β∆Tn,T i-1 . For this, we need an additional assumption about spot covariance matrix.
Condition 4. There are constants ϑ 1 , ϑ 2 > 0 such that inf 0≤t≤T λ min ( c X,X t ) > ϑ 1 and sup 0≤t≤T ∥c t ∥ max < ϑ 2 almost surely.
Condition 4 can, obviously, be localized just as in (2.3), cf. Jacod and Protter (2012, Chapter 4.4.1, pp. 114-121) and Mykland and Zhang (2012, Chapter 2.4.5, pp. 160-161).
Lemma 2 (Consistency of θn ). Assume Conditions 1-4. Assume that the number of regressors q is finite, and ∆T n satisfies (2.17). Then, for any ϵ, 0 < ϵ < 1/2, we have:
and θnθ = o p (1) .
Proof. The proof of this lemma is collected in the Appendix B. □
When ∆T n → 0 and inf n a -1 n ∆T n > 0, the discretization error R Discrete (Eq. (3.7)) becomes the dominating term in the estimation error of θn . However, in this scenario, it cannot achieve the optimal convergence rate. Consequently, we consider the setting of a -1 n ∆T n → 0 and a -2 n ∆T n → ∞. In this scenario, the aggregated error of β∆Tn,T i-1 , R Spot becomes the dominating term. By further analyzing the aggregated error R Spot , it is easy to show that there is a bias term arises in R Spot , which has bigger size than the martingale term. In the following theorem, we provide the representation of the bias term so that we can design the bias-corrected estimator in the subsequent subsection. For ease of exposition, this result is stated in the simple regression case only.
Theorem 1 (Second Order Behavior of θn in the Univariate Case).
Assume that q = d = 1, as well as Conditions 1-4. and also that a -1 n ∆T n → 0 and a -2 n ∆T n → ∞. Then we have:
where
) .
Proof. The proof of this theorem is collected in the Appendix C. □
Similar to the single regressor case, the size of bias term is bigger than the martingale term when a -1 n ∆T n → 0. Thus, in order to develop the CLT, we need to construct the bias corrected estimator. For 1 ≤ r, s ≤ q + d, we define
) .
(3.8)
The bias corrected estimator is defined as:
and φX,X,X,X ∆Tn,T i-1 = φX,X ∆Tn,T i-1
where φX,Y ∆Tn,T i-1
and φX,X ∆Tn,T i-1 is defined in (3.8) and with the notations "X,Y " and "X, X" that follow from the conventions of (3.1), respectively.
Before stating the Central Limit Theorem (CLT), we introduce the following notation. Recall the definition of
Finally, the CLT for θn can be stated as follows.
Theorem 2 (Central Limit Theorem for θn ). Assume all conditions in Lemma 2 and further assume that a -1 n ∆T n → 0 and a -3/2 n ∆T n → ∞. Then we know that there is a q × q matrix process (Σ t ) 0≤t≤T defined in (3.11), such that
where the convergence is stable in law, N q (0, Σ T ) is a q-dimensional normal distribution with mean 0 and covariance matrix as Σ T .
Proof. The proof of this theorem is collected in the Appendix E. □ Moreover, following the idea of Mykland and Zhang (2017), it is straightforward to see that the asymptotic variance estimator could be constructed as follows:
) -1 ,
where
and φX,X ∆Tn,T i-1
being defined in (3.8) and the notations "X,Y " and "X, X" following from the conventions of (3.1), respectively.
We again start by adjusting the notation. In the case of high dimensional factor model, we assume that q, d > 1, with d typically much larger than q. Specifically, q is asymptotically ''almost'' finite (see Condition 5 below), while d → ∞ as n → ∞. As foreshadowed in (2.1), denote
) .
It is then also natural to use the following notations:
⟨X, X⟩ t = {⟨ Ξ (r) , Ξ (s) ⟩ t } 1≤r,s≤q , ⟨X, Y⟩ t = {⟨ Ξ (r) , Ξ (q+l) ⟩ t } 1≤r≤q,1≤l≤d , ⟨X, X⟩ t = { ⟨Ξ (r) , Ξ (s) ⟩ t } 1≤r,s≤q , and ⟨X, Y⟩ t = { ⟨Ξ (r) , Ξ (q+l) ⟩ t } 1≤r≤q,1≤l≤d . For the spot quantities, we define c X,X t as in (3.1), and define c X,Y t = { c (r,q+l) t } 1≤r≤q,1≤l≤d
, which is a q × d matrix process, and
, which is a d × d matrix process.
(4.2)
Following the similar convention, we define the related matrix and vector quantities for M, c, ĉ, π , π , π , φ, ẽ an e. Then it is easy to see that in matrix form,
which is defined in (2.16).
The log-price process
of d stocks is generated from a multiple regression, also known as a ''supervised'' factor model:
where
is a q × 1 vector process, denoting a set of time-varying common regressors or factors, B t is a d × q matrix process of time-varying factor loadings and
⟨X, Z⟩ t = 0 for all t , (4.5) cf. Mykland and Zhang (2006). The difference from an unsupervised factor model, is that in our current case, the factors X t are observed, though with noise, and at asynchronous discrete times, as in Section 2.1. It is straightforward to see that
where s t = ⟨Z, Z⟩ ′ t , in view of (3.1), and (4.2). In this paper, we adopt the sparsity structure for s t , which is measured by
for some ν ∈ (0, 1) ,
)
. This measure is widely used in existing literature, i.e., Bickel and Levina (2008) and Cai and Liu (2011) and as pointed out by Fan et al. (2013).
In this case, the factors are observable. Thus, in order to get the estimates of factor loadings B t , we use the least quadratic variation (LQV) method:
tr ⟨Z, Z⟩ t .
Based on the similar derivation of (3.4), the LQV solution of factor loading can be expressed as:
). Therefore, by the formula (4.7), the LQV solution for the spot idiosyncratic covariance matrix is:
where
(4.9)
We define the related estimators as follows:
where ĉX,X t , ĉX,Y t and ĉY,Y t are defined in (4.3). In the case of high dimensional factor models, we allow the number of common factors to diverge slowly, as the cross-sectional dimension d goes to infinity. The detailed technical assumption is stated as follows.
Condition 5. For the number of common factors q, we assume that q = o (d) and q 4 ∆T n log d = o (1).
We now show the result for convergence rate of ĉB•X t under elementwise max norm. Define:
with ∆T n ≍ a n . Assume Conditions 1-5. The following is then valid:
where ω n is defined in (4.11). Consequently by the triangular inequality and formulas (4.8) and (4.10), we obtain:
Proof. The proof of this theorem is collected in the Appendix G. Specifically, (4.12) is a consequence of Lemma 3, while (4.13) follows from (G.6). □ Now we apply the adaptive thresholding on ŝt . Denote the thresholding estimator by ŝ * t , defined as follows:
where φ ij is the adaptive thresholding rule, for z ∈ R,
, soft thresholding, SCAD and the adaptive lasso, i.e., see Rothman et al. (2009) and Fan et al. (2016b). Because of the absence of residuals, the standard error estimator of ŝ(i,j) t cannot be easily obtained. Thus, in contrast to the settings of χ ij in Fan et al. (2013), the thresholding parameter are set to be elementwise constant, i.e., defined as:
with a sufficiently large C > 0. Before stating the results of the thresholding estimator, we first make one assumption about the spot residual covariance matrix. Condition 6. For the spot residual covariance matrix s t , there exist constants ϑ
Based on the result in Theorem 3, we obtain the following proposition.
Proposition 1. Assume Conditions 1-6. Then, for a sufficiently large C > 0 in the thresholding parameter (4.14), the estimator for the sparse residual covariance matrix satisfies:
) .
If ω 1-ν n m d = o p (1) is assured, then the eigenvalues of ŝ * qt ,t are all bounded away from 0 with probability approaching 1, and
) .
Proof. The proof of this proposition directly follows from the similar discussions in the proof of Theorem 5 of Fan et al. (2013). □
Next, let us define the spot covariance matrix estimator based on the thresholding estimator as follows:
ĉY,Y, *
Then we consider the estimation performance of precision matrix based on
. The theoretical development is based on the Sherman-Morrison-Woodbury formula, i.e., recall the formulas (4.8) and (4.9), we obtain:
We first assume the pervasiveness of the common factors by the following technical assumption, which is parallel to the Assumption 3.5 in Fan et al. (2011).
(1) for some q × q symmetric positive definite matrix Ω t and some constants ϑ ′ 3 , ϑ ′ 4 > 0 such that (i) inf 0≤t≤T λ min (Ω t ) > ϑ ′ 3 almost surely;
(ii) if q → ∞ as n, d → ∞, we further assume sup 0≤t≤T λ max (Ω t ) < ϑ ′ 4 almost surely.
Then we show the convergence rate for the estimator of the precision matrix as follows.
Theorem 4. Assume Conditions 1-7. Also suppose that ω 1-ν n m d = o p (1).
Then, for a sufficiently large C > 0 in thresholding parameter (4.14), ( ĉY,Y, * t ) -1 is non-singular with probability approaching 1, and
) .
∆Tn,t with simple CV: ĉt = 1 kn∆τn ∑ N * (t)+kn j=N * (t)+1 ∆Ξ τ j ∆Ξ ⊺ τ j (without noise) and ĉt = 1 kn∆τn ∑ N * (t)+kn j=N * (t)+1 ∆Ξ * τ j ( ∆Ξ * τ j ) ⊺
(with noise). The number of simulation trials is 10000. The examination is conducted under different settings of sampling frequency. The sampling frequency is set in two scenarios:
1. ∆τ n = 5 seconds and ∆T n = 468∆τ n , with K = 20, J = 3. 2. ∆τ n = 15 seconds and ∆T n = 156∆τ n , with K = 10, J = 3.
Table 5.1 shows that in the presence of microstructure noise, the estimator based on Simple CV becomes inconsistent: it tends to under-estimate the market beta
t dt, and over-estimate the other non-market betas
t dt and ∫ T 0 β (3) t dt. When ∆τ n = 5 seconds, the magnitude of the estimation bias for ∫ T 0 β (1) t dt is 26.8% of the averaged true value and the bias magnitude for ∫ T 0 β (2) t dt and ∫ T 0 β (3) t dt are 81.6% and 230.2% comparing to their averaged true values. It also appears that the estimation bias (under the market microstructure noise) becomes more severe as the length of the sampling interval ∆τ n decreases from 15 to 5 seconds. On the other hand, our proposed estimator (based Smoothed TSRV) is well behaved, regardless of the sampling interval. To validate the asymptotic behavior of the bias corrected estimator, the finite sample distribution of the standardized statistics are reported in Fig. 5.1, where ∆τ n = 5 seconds. Note that the standardized statistics are calculated by the This figure presents the signature plot for the estimates of integrated market beta
t dt in the presence of market microstructure noise. ''Estimator 1'' denotes the integrated beta estimate based on the Simple CV estimator with subsampled data. ''Estimator 2'' denotes integrated beta estimate proposed in this paper which is based on Smoothed TSRV. following formulas
where θ(k) n is defined in (3.9) and Σ(k,k)
T is defined in (3.13). In Fig. 5.1, the finite sample distributions are approximately normal, with slight fat-tailed. It is worth to emphasize that the edge effects can greatly affect the validity of the asymptotic normality in this scenario (i.e., in practice, T /∆T n should be a positive integer exactly).
As sampling interval increases, say, to 5 minutes or 10 minutes, one could expect the bias of the integrated beta estimate using simple CV goes down. However, its variance increases at the same time. This phenomenon is demonstrated in the signature plot Fig. 5.2. So, even when one samples very 10 minutes, we still recommend our proposed estimator because of its precision.
For d large, we next show the performance of the estimator of the precision matrix
, as d gets large. The simulation setting remains the same as in Section 5.1. For ease of demonstration, we fix ∆τ n = 5 seconds.
Consider the spectral norm of the estimation error of the precision matrix, as defined in Theorem 4, as the error measure, i.e.
(5.3) As in Fig. 5.3, we see that in the presence of microstructure noise, the precision matrix using Smoothed TSRV performs satisfactorily, with contained error (red line) even at high dimensionality situation. However, the precision matrix using simple CV gets worse as d increases from 5 to 200, with error increasing in logarithmic shape.
In this section, we apply high frequency beta estimation to study the variation of stock betas on earnings announcement days. We implement the high frequency regression of the intraday returns of the S&P 100 constituent stocks on the returns of exchange-traded fund OEF. The latter serves as a proxy for the large-cap market returns. The trade prices are extracted from the Trade and Quote (TAQ) database of the New York Stock Exchange (NYSE). In particular, we collect the intraday trade prices of OEF as well as those of the S&P 100 Index constituents, between 9:35 a.m. EST and 3:55 p.m. EST of each This figure compares the estimation performance of the large precision matrix in the presence of microstructure noise. The precision matrix using Smooth TSRV is indicated by red solid line, while the one using simple CV is in blue dashed line. The error measure on y-xis is as defined in (5.3). trading day, from January 2007 to December 2017 (2769 trading days in total). Our spot beta estimate is then applied to explore the change in betas around the earning announcements.
For the earning data, the dates and times of quarterly earnings announcements are downloaded from the Thomson Reuters I/B/E/S database for the components of S&P 100 Index ranging from January 2007 to December 2017. The earnings announced at non-trading days are deleted. At the end, 3845 earnings announcements are collected during this sampling period. We can see from Fig. 6.1 that for the stocks in our sample, most earnings announcements arrived right before the market open (6-8 AM) or right after market close (4-5 PM).
To investigate the beta changes on earnings announcement days, we extended the model in Patton and Verardo (2012) 9 by adding the hourly effects. Specifically, we regress the market beta estimates β OEF i,t on event time dummies using the following model:
2 16 ∑ k=10 δ j,k I i,j,k,t + γ i,1 D 1,t + γ i,2 D 2,t + • • • γ i,10 D 10,t + ε i,t , (6.1) where β OEF i,t is the spot beta estimates of stock i on time t by using the following formula, with ĉOEF, stock i and ĉOEF,OEF being the Smoothed TSCV and Smoothed TSRV estimates, β OEF i,t = ĉOEF, stock i ∆Tn,t ĉOEF,OEF ∆Tn,t , 9
We should note that we deviate from Patton and Verado (2012) in how to define event day. The former paper relabeled the announcement at or after 4:00 p.m. on a given date to have the following trading day's date. In contrast, we follow the exact calendar day when labeling the announcement day. In other words, ''Day 0'' in our paper is the day when the earnings are announced, no matter the announcement time is pre-market, during market open, or post-4 pm. of market open time. The news in the final hour of trading period may also contribute to the beta increase in the next morning.
The central contribution of this paper is a feasible estimator of spot beta, which is robust to noise and asynchronicity. With the help of the spot-version of the Smoothed TSRV estimator, spot beta can be consistently estimated. There are two direct applications of the spot beta estimates in the current paper. In the first application, the integrated beta can be consistently estimated by aggregating the spot beta estimates. After a bias-correction procedure, a fixed dimension central limit theorem is established for the bias-corrected estimator, with convergence rate which may be arbitrarily close to O p (n -1/4 ). In the second application we assume time-varying factor structure and conditional sparsity. The spot beta matrix estimator enables the estimation of high dimensional spot covariance and precision matrices. Simulation results show that our proposed estimators perform well.
As an empirical application, this paper explores the hourly change in beta around earnings announcements of the S&P 100 constituents. The hourly beta was constructed with the help of Smooth TSRV using 5-second pre-averaged returns from 2007 to 2017. We separate the impact of pre-and post-market announcement on beta change, and find that significant beta change takes place in the first hour of market open.
Here ∆T n and ∆τ n are as described above, and b is a very slowly growing number, cf. Eq. (2.7) and Remark 1 in Section 2.2.
Journal of Econometrics 239 (2024) 105446
Here and below, the effect of the drift term is negligible, cf. Mykland and Zhang (2009, Section 2.2, pp. 1407-1409).
Cf.Mykland and Zhang (2009, Section 4.2, pp. 1424-1428) Zhang (2012, Section 4).
this is similar to Aït-Sahalia and Xiu (2019) with θ ν = 0.06 and η ν = 0.3