Intraday high frequency data have shown important values in econometric modeling and have been extensively studied. Following this point, in this paper, we study the linear regression model for variables which have intraday high frequency data. In order to overcome the nonstationarity of the intraday data, intraday sequences are aggregated to the daily series by weighted mean. A lower bound for the trace of the asymptotic variance of model estimator is given, and a data-driven method for choosing the weight is also proposed, with the aim to obtain a smaller sum of asymptotic variance for parameter estimators. The simulation results show that the estimation accuracy of the regression coefficient can be significantly improved by using the intraday high frequency data. Empirical studies show that introducing intraday high frequency data to estimate CAPM can have a better model fitting effect.
Citation: Wenhui Feng, Xingfa Zhang, Yanshan Chen, Zefang Song. Linear regression estimation using intraday high frequency data[J]. AIMS Mathematics, 2023, 8(6): 13123-13133. doi: 10.3934/math.2023662
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Intraday high frequency data have shown important values in econometric modeling and have been extensively studied. Following this point, in this paper, we study the linear regression model for variables which have intraday high frequency data. In order to overcome the nonstationarity of the intraday data, intraday sequences are aggregated to the daily series by weighted mean. A lower bound for the trace of the asymptotic variance of model estimator is given, and a data-driven method for choosing the weight is also proposed, with the aim to obtain a smaller sum of asymptotic variance for parameter estimators. The simulation results show that the estimation accuracy of the regression coefficient can be significantly improved by using the intraday high frequency data. Empirical studies show that introducing intraday high frequency data to estimate CAPM can have a better model fitting effect.
With the development of electronic technology, intraday high frequency data become easily available. Such data are valuable in statistical modeling and financial risk assement [1,2,3,4]. By utilizing high frequency data, potential risks can be identified more efficiently and accurately due to an increased level of detail in the analysis of the markets. However, due to the nonstationarity and periodicity of the intraday high frequency data, generally it is not appropriate to directly introduce these data into a stationary model [5]. A possible way is aggregating the intraday high frequency data to a daily stationary quantity or constructing a low frequency stationary proxy [6,7]. To illustrate this idea, we plot three return series related to CSI (China Shanghai-Shenzhen) 300 index in Figure 1. We collected the returns of the CSI 300 index from 01 Sep 2017 to 12 July 2019, including 466 daily observations. There are 240 observations each day based on the intrady sampling frequency of 1 min. Subplot (a) in the Figure 1 is the time series plot of the intraday sequence on the seventh day of the data set, which shows an obvious time trend and nonstationarity; subplot (b) is the time series plot of the seventh intraday observation for the first 240 days of the data set; subplot (c) is the time series plot of the mean of intraday sequences for the first 240 days of the data set. It is seen that the series in subplots (b) and (c) tend to be stationary sequences, which implies the fact that although the intraday sequence can be nonstationary while its aggregation (weighted mean) might be stationary.
Linear regression model has been extensively applied in the area of daily financial time series analysis, such as ARMA model, linear pricing model, factor model and other linear forecasting models, see [8,9,10,11,12,13,14,15,16,17,18]. Let yt be the observation of dependent variable and Xt=(1,xt1,⋯,xtp)τ be the the observation of independent variable vector at day t. Then the classic linear regression model has the form
yt=Xτtβ+εt, | (1.1) |
where β=(β0,β1,⋯,βp)τ is the regression coefficient vector and εt is the error term. To avoid the spurious regression problem, {yt,Xt} are normally assumed to be stationary [19].
The main goal of this paper is to study the following linear regression model with intraday high frequency data
yt(ui)=Xτt(ui)β+εt(ui), | (1.2) |
where yt(ui),Xτt(ui),εt(ui) are the observations at time ui on the t-th day, 1≤i≤k, ui is the scaled time with 0≤ui≤1. When ui=1, the series become the daily sequence, namely, (yt(1),Xτt(1),εt(1))=(yt,Xτt,εt). εt(u) is assumed to be an independent and identically distributed errors process with zero mean and finite variance for each fixed u. For model (1.2), one can directly use the daily data (yt,Xτt) to estimate the coefficient without introducing the intraday data information. However the information is not efficiently used in this occasion. Alternatively, one can aggregate the intraday high frequency data to a daily quantity and then obtain a more precise parameter estimator for the model.
Different from the well known mixed data sampling (MIDAS) regression model of Ghysels et al. [20], the dependent and independent variables in model (1.2) have the same sampling frequency which makes the regression coefficients keep unchanged after the high frequency data are aggregated by a weighted mean form. Such a property enables us to estimate low frequency regression
yt=Xτtβ+εt,namely,yt(1)=Xτt(1)β+εt(1) |
by taking the intraday high frequency data into account.
The contributions of this paper are as follows. First, this paper proposes a linear regression model which aggregates the intraday high-frequency data to a daily quantity. Second, a lower bound for the trace of the asymptotic variance of model estimator is given. Third, we propose a simple data-driven method for choosing the weight for aggregation of the high frequency data, with the aim to obtain a smaller sum of asymptotic variance for parameter estimators. Different from the existent methods, the weight is not restricted to certain parametric form and can be obtained by simple restricted quadratic programming.
The rest of the paper is organized as follows. Section 2 introduces the model and estimation. Section 3 investigates the estimation performance based on simulation studies. An empirical study is provided in Section 4. We conclude the paper in Section 5.
For model (1.1), define
Y=(y1y2⋮yn),X=(1x11⋯x1p1x21⋯x2p⋯⋯⋯⋯1xn1⋯xnp)=(Xτ1Xτ2⋮Xτn),ˆβ=(ˆβ0ˆβ1⋮ˆβp). |
The least squared estimator for β is given by
ˆβ=(XτX)−1XτY | (2.1) |
and under regularity conditions, the following asymptotic normality holds:
ˆβ∼N(β,(XτX)−1σ2), | (2.2) |
where σ2 is the variance of εt in (1.1) and it is also equivalent to var(yt|Xt). In practice, we always hope the asymptotic variance of each parameter will not be too large. Equivalently, we hope the trace of the asymptotic variance matrix tr[(XτX)−1σ2] is small. The following proposition gives an approximate lower bound for tr[(XτX)−1σ2] based on the samples {yt,Xt}nt=1.
Proposition 1. Suppose {yt,Xt} are stationary processes and all the eigenvalues of XτX are positive, for fixed sample size n, then a lower bound for tr[var(ˆβ)] is given by
var(yt|Xt)E(XτtXt)(p+1)2n(1+op(1)). |
Proof. Before the statement of detailed proof, we first list two properties of matrix trace.
P1 Suppose A is a symmetric m×m matrix and all the the eigenvalues of A are positive, then tr(A−1)≥m2tr(A).
P2 Suppose A is a s×m matrix and B is a m×s matrix, then tr(AB)=tr(BA).
According to P1,
tr[(XτX)−1]≥(p+1)2tr(XτX). | (2.3) |
By ergodicity theorem for stationary time series,
XτX=n1nn∑t=1XtXτt=n[E(XtXτt)+op(1)] |
and
(p+1)2tr(XτX)=(p+1)2n1+op(1)tr[E(XtXτt)]. | (2.4) |
Further, according to P2,
tr[E(XtXτt)]=E[tr(XtXτt)]=E[tr(XτtXt)]=E[XτtXt]. | (2.5) |
Recall tr[var(ˆβ)]=tr[(XτX)−1σ2]=tr[(XτX)−1var(yt|Xt)]. Then the result of Proposition 1 is proved based on (2.3)–(2.5).
Denote {yt(ui),xt1(ui),⋯,xtp(ui),εt(ui)} to be observations at time ui on the t-th day, 1≤i≤k, namely there are k intraday observations for each variable. For demonstration, we rewrite (1.2) as followed:
yt(ui)=β0+β1xt1(ui)+β2xt2(ui)+⋯+βpxtp(ui)+εt(ui). | (2.6) |
Let y∗t=k∑i=1yt(ui)wi, x∗tq=k∑i=1xtq(ui)wi(q=1,2,⋯,p), ε∗t=k∑i=1εt(ui)wi, k∑i=1wi=1,wi≥0. From (2.6),
y∗t=β0+β1x∗t1+β2x∗t2+⋯+βpx∗tp+ε∗t. | (2.7) |
It is easy to see that ε∗t is still i.i.d sequence with zero mean and finite variance based on the assumption on εt(ui) and the daily sequences y∗t,x∗tq(q=1,2,⋯,p) are supposed to be stationary after aggregation. Consequently, β0,β1,⋯,βp can be estimated based on y∗t,x∗tq(q=1,2,⋯,p) and we denote the corresponding estimator as ˜β. It is hoped that ˜β would be more precise than ˆβ which only uses low frequency information. To construct proper y∗t,x∗tq(q=1,2,⋯,p) is equivalent to find a proper weight vector w=(w1,⋯,wp+1)τ. Next, we give a method to choose the weight w.
Define
Yt(u)=(yt(u1)yt(u2)⋯yt(uk)),Xt(u)=(xt0(u)xt1(u)⋯xtp(u))=(11⋯1xt1(u1)xt1(u2)⋯xt1(uk)⋯xtp(u1)xtp(u2)⋯xtp(uk)). |
Then
X∗t=Xt(u)w=(11⋯1xt1(u1)xt1(u2)⋯xt1(uk)⋯xtp(u1)xtp(u2)⋯xtp(uk))(w1w2⋯wk)=(k∑i=1xt0(ui)wik∑i=1xt1(ui)wi⋯k∑i=1xtp(ui)wi), y∗t=wτYt(u)=k∑i=1yt(ui)wi.
According to Proposition 1, ˜β based on y∗t,x∗tq(q=1,2,⋯,p) has the property that
tr[var(˜β)]≥var(y∗t|X∗t)E(X∗τtX∗t)(p+1)2n(1+op(1)). |
Intuitively, if the above right bound is smaller, then we can expect to obtain a less tr[var(˜β)]. Such an intuition gives a way to choose the weight: finding a w which can get the smallest value for var(y∗t|X∗t)/E(X∗τtX∗t). Note that when X∗t is deterministic, we have var(y∗t|X∗t)=var(y∗t). On the other hand, from
var(y∗t)=E[var(y∗t|X∗t)]+var[E(y∗t|X∗t)], |
smaller var(y∗t) will bring smaller var(y∗t|X∗t). Consequently, the rule to choose the weight can be transformed to that: finding a w which can get the smallest value for var(y∗t)/E(X∗τtX∗t). Further,
var(y∗t)E(X∗τtX∗t)=wτvar(Yt(u))wwτE(Xτt(u)Xt(u))w. | (2.8) |
Let Σyu=var(Yt(u)),Σxu=E(Xτt(u)Xt(u)). From the above, if the quantity
wτΣyuwwτΣxuw |
is small, then ˜β is supposed to be more precise. In practice, Σyu and Σxu can be respectively estimated by their corresponding sample variance or sample mean, namely,
ˆΣyu=1nn∑t=1(Yt(u)−ˉY)(Yt(u)−ˉY)τ,ˆΣxu=1nn∑t=1Xτt(u)Xt(u), |
where ˉY is the sample mean vector of Yt(u). Hence the weight vector w is chosen as the minimizer of the following objective function
argminw∈Rp+1wτˆΣyuwwτˆΣxuw | (2.9) |
such that k∑i=1wi=1,wi≥0.
Theoretically, without restriction wi≥0, the solution of w in (2.9) is the eigenvector of the smallest eigenvalue for matrix ˆΣ−1xuˆΣyu, denoted as ˆw. Hence ˆw is also the solution of
argminw∈Rp+1wτˆΣ−1xuˆΣyuw. | (2.10) |
From the above, the solution in (2.9) can be approximated by the solution of the quadratic programming in (2.10) with restrictions: k∑i=1wi=1,wi≥0. More detail about the quadratic programming with restrictions can be referred to Huyer and Neumaier [21]. The simulation studies in the following section show that such an approximation works well.
In this section, we assess the finite-sample performance of the proposed estimator ˜β. The sample was simulated from the model below
yt(ui)=0.1+0.3xt1(ui)+0.4xt2(ui)+εt(ui), | (3.1) |
1≤i≤20,ui=i/20. Following Visser [7], xt1(ui) and xt2(ui) were independently simulated from the following process ξt(u) with different parameter setting:
dγt(u)=−δ(γt(u)−μ)du+σγdB(2)t(u),dξt(u)=eγt(u)dB(1)t(u),u∈[0,1]. | (3.2) |
The Brownian motions B(1)t and B(2)t were uncorrelated, ξt(0)=0, and γ(0) was sampled from N(μ,σ2r). We divided the unit time interval [0,1] into 20 small intervals, set δ=1/2, σγ=1/4, μ=−1/16 for xt1(u) and δ=1/3, σγ=1/5, μ=−1/14 for xt2(u). εt(ui)∼i.i.dN(0,0.64), and then yt(ui) can be obtained based on (3.1). When ui=1, we also get the daily sample {yt(1),xt1(1),xt2(1),εt(1)}, namely {yt,xt1,xt2,εt}, such that
yt=0.1+0.3xt1+0.4xt2+εt. | (3.3) |
Let ˜β and ˆβ be the estimator from (3.1) and (3.3) respectively. Here ˜β introduces the intraday high frequency information, as discussed in Section 2, while ˆβ only uses the daily sequence. Hence, ˜β is expected to be more precise than ˆβ. The sample sizes of n=50, 100 and 150 are considered, and the replication time is 1000. Table 1 reports the sample bias and the sample standard deviation of ˜β and ˆβ, denoted as BS1, BS2, SD1 and SD2, respectively. From the table, we can receive several observations as follows. The biases of ˜β are smaller than those of ˆβ, and both become smaller when the sample size n increases. This implies that both estimators are asymptotically unbiased. The sample standard deviations of ˜β are also significantly smaller than those of ˆβ, and both become smaller when the sample size n increases. This implies that ˜β performs better than ˆβ does in our simulations.
sample size | BS1 | BS2 | SD1 | SD2 |
Result for β0 | ||||
n=50 | −0.0027 | −0.0037 | 0.0791 | 0.1239 |
n=100 | −0.0026 | 0.0003 | 0.0488 | 0.0789 |
n=150 | 0.0001 | 0.0035 | 0.0365 | 0.0700 |
Result for β1 | ||||
n=50 | −0.0073 | −0.0034 | 0.0066 | 0.0132 |
n=100 | −0.0048 | −0.0038 | 0.0039 | 0.0086 |
n=150 | −0.0038 | −0.0034 | 0.0030 | 0.0071 |
Result for β2 | ||||
n=50 | −0.0091 | −0.0074 | 0.0052 | 0.0124 |
n=100 | −0.0064 | −0.0070 | 0.0034 | 0.0087 |
n=150 | −0.0051 | −0.0072 | 0.0025 | 0.0071 |
†Number of replications=1000. |
Let Sσ≡var(y∗t)/E(X∗τtX∗t). From (2.8), for smaller Sσ, a less tr[var(˜β)] is expected. To justify this expectation, for each replication, Sσ for ˜β and ˆβ are respectively estimated by sample variance and sample mean for each sample size. Figure 2 shows the box plots of the Sσ series. It can be seen that the median Sσ values of ˜β are smaller than those of ˆβ, and both become smaller when the sample size n increases. Such a result is consistent with Table 1, justifying the intuition: the smaller Sσ is, the less tr[var(˜β)] would be. According to the simulation results, it is shown that introducing the intraday high frequency data can significantly improve the estimation of the regression coefficient.
In this section, the proposed method is applied to study the relationship between single stock and stock index based on the classic Capital Asset Pricing Model (CAPM), see Latunde et al. [22]. Denote Pt(u) as the t-th intraday price sequence. We calculate the intraday log-return as
Yt(u)=100[logPt(u)−logPt−1(1)],u∈[0,1]. | (4.1) |
According to the CAPM, we choose the Shanghai Composite Index as the market portfolio and randomly choose JCHX Mining Management (stock code: 603979) as the single asset, from China Shanghai Stock Exchange. Let rmt(u) and rt(u) be the intraday high frequency return series for Shanghai Composite Index and JCHX Mining stock respectively, which can be computed based on (4.1). And rmt and rt, namely rmt(1) and rt(1) are the daily return series. Classic CAPM implies the following relationship between rmt and rt:
rt=β0+β1rmt+εt, | (4.2) |
and β1 is the famous beta coefficient used in the CAPM, and it describes the relationship between systematic risk and expected return for assets (usually stocks). The beta coefficient can be used to help investors understand whether a stock moves in the same direction as the rest of the market. It also provides insights into how risky a stock is relative to the rest of the market. Consequently, it makes sense to get a more precise estimation for the beta coefficient by using extra information. Following this motivation, we introduce the intraday high frequency data in model (4.2):
rt(u)=β0+β1rmt(u)+εt(u), | (4.3) |
Note that models (4.2) and (4.3) share the same regression coefficient while (4.3) takes intraday high frequency information into account and can have a more precise estimation for the parameters, as discussed in the Sections 2 and 3.
For the considered series rmt(u) and rt(u), the data span the period from 19 Nov 2019 to 17 Jan 2020, which consist of 43 daily observations. For each day, the intraday sampling frequency: 1min, 5min, 15min, 30min and 60min are considered. We can get the estimations for models (4.2) and (4.3) by applying the method given in Section 2. Table 2 lists the results for (4.2) (in the last column) and (4.3) with different sampling frequency, where CIL and CIU denote the 95% confidence lower bound and upper bound respectively, R2 is the R-squared coefficient of linear regression, Tr is the estimated values for var(ˆβ0)+var(ˆβ1) in Table 2.
results | 1min | 5min | 15min | 30min | 60min | daily |
ˆβ0 | −0.1038 | −0.1319 | −0.1418 | −0.1185 | −0.1035 | 0.1553 |
CIL for ˆβ0 | −0.1858 | −0.2240 | −0.2692 | −0.2921 | −0.2822 | −0.1026 |
CIU for ˆβ0 | −0.0218 | −0.0398 | −0.0143 | 0.0551 | 0.0752 | 0.4132 |
ˆβ1 | 0.5314 | 0.5355 | 0.4429 | 0.6260 | 0.7149 | 0.6379 |
CIL for ˆβ1 | 0.2429 | 0.2273 | 0.0464 | 0.2070 | 0.3093 | 0.2519 |
CIU for ˆβ1 | 0.8199 | 0.8436 | 0.8394 | 1.0451 | 1.1206 | 1.0239 |
R2 | 0.2524 | 0.2310 | 0.1104 | 0.1817 | 0.2360 | 0.2136 |
Tr | 0.0288 | 0.0322 | 0.0467 | 0.0602 | 0.0616 | 0.0656 |
From Table 2, different sampling frequencies get different beta values from 0.4429 to 0.7149. According to the computed R2 and Tr, the fitting effect is the best under the sampling frequency of 1min and the estimated beta value is 0.5341, which is smaller than the value 0.6379 estimated only by the daily information.
The above results imply that using the intraday high frequency data to estimate the CAPM can have a better model fitting effect and this is helpful for the investor to make rational decisions. And such empirical studies can be easily extended to other pricing models such as ICAPM and factor model, see [23,24,25,26].
It is valuable to introduce high frequency data into low frequency standard models. These data can provide insights into trends, patterns, and correlations that may not be visible with lower frequency data. Additionally, they can help identify anomalies or outliers that may indicate risk. Analyzing high frequency data makes it possible to detect subtle changes or shifts.
The linear regression model for variables which have intraday high frequency data is studied in this paper. A method is given to estimate the model based on the idea of time series aggregation. Simulation results show the proposed approach performs well. Empirical studies imply that our model can have many potential applications in linear forecasting models.
Our research findings will provide insights for studying other linear or nonlinear time series models, such as threshold autoregression models. The method can be applied to different pricing and factor models in our future study, and it is expected to perform better.
This work is partially supported by Guangdong Basic and Applied Basic Research Foundation 2022A1515010046 (Xingfa Zhang) and Funding by Science and Technology Projects in Guangzhou SL2022A03J00654 (Xingfa Zhang), 202201010276 (Zefang Song).
The authors declare no conflict of interest.
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1. | Guglielmo Maria Caporale, Alex Plastun, Persistence in high frequency financial data: the case of the EuroStoxx 50 futures prices, 2024, 12, 2332-2039, 10.1080/23322039.2024.2302639 | |
2. | Lu Zhang, Lei Hua, Major Issues in High-frequency Financial Data Analysis: A Survey of Solutions, 2024, 1556-5068, 10.2139/ssrn.4834362 |
sample size | BS1 | BS2 | SD1 | SD2 |
Result for β0 | ||||
n=50 | −0.0027 | −0.0037 | 0.0791 | 0.1239 |
n=100 | −0.0026 | 0.0003 | 0.0488 | 0.0789 |
n=150 | 0.0001 | 0.0035 | 0.0365 | 0.0700 |
Result for β1 | ||||
n=50 | −0.0073 | −0.0034 | 0.0066 | 0.0132 |
n=100 | −0.0048 | −0.0038 | 0.0039 | 0.0086 |
n=150 | −0.0038 | −0.0034 | 0.0030 | 0.0071 |
Result for β2 | ||||
n=50 | −0.0091 | −0.0074 | 0.0052 | 0.0124 |
n=100 | −0.0064 | −0.0070 | 0.0034 | 0.0087 |
n=150 | −0.0051 | −0.0072 | 0.0025 | 0.0071 |
†Number of replications=1000. |
results | 1min | 5min | 15min | 30min | 60min | daily |
ˆβ0 | −0.1038 | −0.1319 | −0.1418 | −0.1185 | −0.1035 | 0.1553 |
CIL for ˆβ0 | −0.1858 | −0.2240 | −0.2692 | −0.2921 | −0.2822 | −0.1026 |
CIU for ˆβ0 | −0.0218 | −0.0398 | −0.0143 | 0.0551 | 0.0752 | 0.4132 |
ˆβ1 | 0.5314 | 0.5355 | 0.4429 | 0.6260 | 0.7149 | 0.6379 |
CIL for ˆβ1 | 0.2429 | 0.2273 | 0.0464 | 0.2070 | 0.3093 | 0.2519 |
CIU for ˆβ1 | 0.8199 | 0.8436 | 0.8394 | 1.0451 | 1.1206 | 1.0239 |
R2 | 0.2524 | 0.2310 | 0.1104 | 0.1817 | 0.2360 | 0.2136 |
Tr | 0.0288 | 0.0322 | 0.0467 | 0.0602 | 0.0616 | 0.0656 |
sample size | BS1 | BS2 | SD1 | SD2 |
Result for β0 | ||||
n=50 | −0.0027 | −0.0037 | 0.0791 | 0.1239 |
n=100 | −0.0026 | 0.0003 | 0.0488 | 0.0789 |
n=150 | 0.0001 | 0.0035 | 0.0365 | 0.0700 |
Result for β1 | ||||
n=50 | −0.0073 | −0.0034 | 0.0066 | 0.0132 |
n=100 | −0.0048 | −0.0038 | 0.0039 | 0.0086 |
n=150 | −0.0038 | −0.0034 | 0.0030 | 0.0071 |
Result for β2 | ||||
n=50 | −0.0091 | −0.0074 | 0.0052 | 0.0124 |
n=100 | −0.0064 | −0.0070 | 0.0034 | 0.0087 |
n=150 | −0.0051 | −0.0072 | 0.0025 | 0.0071 |
†Number of replications=1000. |
results | 1min | 5min | 15min | 30min | 60min | daily |
ˆβ0 | −0.1038 | −0.1319 | −0.1418 | −0.1185 | −0.1035 | 0.1553 |
CIL for ˆβ0 | −0.1858 | −0.2240 | −0.2692 | −0.2921 | −0.2822 | −0.1026 |
CIU for ˆβ0 | −0.0218 | −0.0398 | −0.0143 | 0.0551 | 0.0752 | 0.4132 |
ˆβ1 | 0.5314 | 0.5355 | 0.4429 | 0.6260 | 0.7149 | 0.6379 |
CIL for ˆβ1 | 0.2429 | 0.2273 | 0.0464 | 0.2070 | 0.3093 | 0.2519 |
CIU for ˆβ1 | 0.8199 | 0.8436 | 0.8394 | 1.0451 | 1.1206 | 1.0239 |
R2 | 0.2524 | 0.2310 | 0.1104 | 0.1817 | 0.2360 | 0.2136 |
Tr | 0.0288 | 0.0322 | 0.0467 | 0.0602 | 0.0616 | 0.0656 |