@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Mathematics, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Chou, Steve Che-Ming"@en ; dcterms:issued "2009-12-11T18:21:15Z"@en, "2005"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """We consider a multi-stock market, where daily return process of each stock together with its mean rate of daily return are assumed to follow a continuous diffusion process, which is similar to a state-space system with linear Gaussian dynamics. Our major objective is to estimate the model parameters based on historical data. Our estimation method is an iterative method bases on the expectation maximization (or EM) algorithm."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/16495?expand=metadata"@en ; skos:note "Modeling the Return Rates of Stock via E M Algorith by Steve Che-Ming Chou B . A . S c , University of British Columbia, 2001 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in T H E F A C U L T Y O F G R A D U A T E STUDIES (Mathemat ics) The University of British Columbia August 2005 © Steve Che-Ming Chou, 2005 Abstract We consider a multi-stock market, where daily return process of each stock together with its mean rate of daily return are assumed to follow a continuous diffusion process, which is similar to a state-space system with linear Gaussian dynamics. Our major objective is to estimate the model parameters based on historical data. Our estimation method is an iterative method bases on the expectation maximization (or E M ) algorithm. Contents Abstract ii Contents iii List of Tables v List of Figures vi Notations and Definitions vii 1 Introduction 1 2 Basic Theory 5 2.1 State-space Model with Linear Gaussian Dynamics 5 2.2 Random Behaviour of Jit 7 2.3 Interpolation of conditionally Gaussian processes 12 3 Estimations of Parameters 16 3.1 Estimation of a 16 3.2 Estimation of (5 and mo or /IQ 19 3.3 Estimation of a 21 3.4 Estimation of (3 25 3.5 Initial Value of a 26 3.6 Iterative Procedure for Parameter Estimation 27 ii i 4 Estimation Results 30 4.1 Description of \"Experiment\" or Sampling Procedure 30 4.2 Estimated Values 33 4.3 Convergence of Iteration 40 5 Future Works 48 5.1 . Estimation Methods 48 5.2 Model Validation and Performance 50 5.3 Statistical Analysis of Parameters 51 Bibliography 53 Appendix A Solving I V P (2.23) for 7, 55 Appendix B Historical Data 58 Appendix C Computational Details 59 C . l Differential Equations (2.22) and (2.23) for fht and 7t in Theorem 2.2 59 C.2 Backward Equations (2.37) and (2.38) for m(s,t) and 7(s,t) and dif-ferential Equation (2.39) for in Theorem 2.3 • • • • 60 C.3 Definition and Computation of Rt 61 C.4 Calculation of Yh in the Estimation of a2 63 iv List of Tables 4.1 Estimates of Parameters with Rank 34 4.2 Summaries of Estimates for Parameters 35 4.3 Stocks in Ascending Order of Estimates 37 4.4 95% Confidence Interval of a 41 4.5 95% Confidence Interval of exp(—a) in 10-thousandth 42 4.6 95% Confidence Interval of S 43 4.7 95% Confidence Interval of (3 44 4.8 95% Confidence Interval of a 45 v List of Figures Pairwise Scatter plots: Estimates for Parameters from 20 Stocks . . 39 95% Confidence Intervals for Parameters 46 vi Notation and Definitions {Ct '• to < t < ti} denotes a continuous time process £ in from time to to time t\\. So, & is value of the continuous time process at time t in years. {£o>£i) • • • >£ni • • • j£ /v- i} denotes a set of TV discrete observations from the continuous time process : 0 < t < T} , taken at uniform frequency A t in years. Thus, while T denotes the length of duration in which the continuous time process is observed, N represents the number of discrete observations taken,starting from time 0. Thus, T = (N — l ) A t . Here, we somewhat abuse the notations in the sense that the similar notations (i.e., subscripts) are used with slightly different meaning: while £t denotes the value of the continuous time process at time t (in years), £,i refers to the n t h discrete observation. However, the meaning of the notation should be clear from the context and from the different subscripts (\"n\" and \"t\"). Thus, ambiguity usually should not occur. In case unusual ambiguity does arise, we still use subscript n for the discrete observation but use the time in parentheses for the value of the process: £ n as £(nAt) . (9, £) = (9t, with 0 < t < T, represents a continuous time random process with unobservable first component 9 and observable second component £, cf. p . l of Liptster and Shiryayev [11] {St : 0 < t < T} denotes the observable price process of a stock on the finite time interval [0, T]. vi i • Rt denotes the observable cumulative return of a stock starting from time 0 up to time t, relative to the price St itself; i.e., dRt = 4^ , cf. Equation (2.12) on p.81 of Lakner [10] or p.4 of Sass and Haussmann [14]. For briefness, it is simply referred as the \"return\" of a stock and the cumulative, relative or proportional sense are assumed by default. Note that, by definition, i?y = 0. • Following the definitions above, {Ro, Ri, • • • , Rn, • • • i R-N-i} denotes a set of N discrete observations of the continuous time process {Rt : 0 < t < T} at uniform frequency At.1 That is, Rn = R(nAt). By definition, RQ — 0. • fit == E (^§L^j, the unobservable mean rate of change of a stock's return over time. For briefness, it is simply referred as the \"mean rate of return\". Here, the mean or expectation is taken over a hypothetical population of all the possible replicates or \"sample paths\" of a stock which could have been taken over time. A — A • fit = ^ — S and Rt = Rt — St are two transformed processes as defined in (2.3) and (2.4) on page 6, respectively. Note that Ro = Ro — 0 x t = 0. • (fit, Rt) takes the role of (8t, £t) as the working process. • J-f = a{^s : s < t}, the sigma field generated by {£, : s < t}\\ here, we mostly work with = o~{Rs : s < t}. Occasionally, also use 34 to denote a{ys : s s, r(t , s) denotes the counterpart of 7(s,i) defined above but with s fixed and t varying with t > s, cf. p.22 and p.36 of [11]. 9 denotes the maximum likelihood estimator or estimate for the unknown parameter 9. . ' C2(T>) denotes a set of functions that are twice continuously differentia ble in V. CT represents a space of continuous functions x — {xs, s < T}, cf. p.2 of [11] [z]+ = z, if z > 0; e, otherwise. Here, e denotes a small positive random number. A ' denotes transpose of matrix A . A® or Oj j denotes the element on the i t h row and j t h column of matrix A . ix Chapter 1 Introduction We consider the unobservable mean rate of return {fit • 0 < t < T} of a stock, when the return process {R,t : 0 < t < T} is observable.1 This leads to a random pro-cess (nt,Rt) with unobservable first component and observable second component.2 Particularly, we consider (fit, Rt) as a continuous time process of the diffusion type specified by a pair of Ito stochastic differential equations: dut = a(5- n^dt + PdW^, ( l . l ) dRt = ntdt + adwf\\ (1.2) where a, <5, f3 and a are assumed to be some unknown \"constants\", wf 1 ^ and w[\"^ denote two mutually independent Wiener processes, and \\i§ is assumed to be deterministic. Moreover, RQ = 0 by definition, so we take it as given. 3 See next xSome clarifications about the terminologies used here are required: 1) the \"rate of re-turn\" refers to the rate of change of return over time (per year). 2) The mean or expectation could be taken over a hypothetical population of all the stocks which could have possibly been considered or one of all the possible replicates or \"sample paths\" of a single stock that could have been taken over time. 3) The \"return\" refers to the cumulative change of price from time 0, relative to the price itself. So, if St denotes the price at time t. N,t can be defined as dRt = dSt/St in equation (2.12) on p.81 of Lakner [10] or p.4 of Sass and Ha.ussmann [14]. 2In filtering terminology, fit is the signal, whereas Rt is the observation. 3Herc, the roles of fig and RQ differ from those usually assumed in literature, where they are usually assumed to be two random variables independent of and W^2\\ Sec the beginning of Chapter 3 on page 16. 1 / paragraph that clarifies the constancy of the parameters. In fact, this model had been used by Lakner for the return process R and drift process n in the context of multiple stocks, c.f. equations (2.13) and (4.1) of [10]. However, Lakner basically worked with this model directly. Instead, we simplify our work by first re-parameterizing the model into a simpler one; see Chapter 2 on page 6 below. The interpretations of (3 and a are relatively obvious. Because they are the multiples or scalars of the two random increments of the Wiener processes, ,8 and a denote the volatilities of fit and Rt, respectively. For the interpretations of 6 and a, let us first take expectation on both sides of (1.2) to get E{dRt)= (Ent)dt, o r E ( i § L ) = (1-3) As deprived later on page 19 in Section 3.2, (1.1) implies Ein = 8 + {Ena-S)e-at. ' (3.6) Because we expect a > 0, (3.6) assures that E/it remains bounded and Efj,t. —> S as t —> oo. That is, S is the long-term or asymptotic value of Em or E (^§L^J-Moreover, as observed in Section 3.5 on page 26, a somehow controls the rate of such convergence: (Em — 5) —+ 0 monotonically as a geometric sequence with a retarding ratio e~a per year. For instance, if a = 1, then Em approaches to its limit 5; otherwise, it is a negative excess. we interpret the mean rate of return is taken over a hypothetical population of all replicates or \"sample paths\" of a single stock that could have been taken over time. Our major objective of this work is to understand the mechanism that drives the return of stocks. One application of such understanding is to help forecasting the future return and hence the prices of the stocks. This leads to the optimal investment strategy in the stock market: determine the proportion of funds to invest on various stocks that constitute a portfolio and the stopping time for investing on the portfolio. To this end, we would use historical data to estimate the unknown parameters in (1.1) and (1.2). To simplify the current model specified by (1.1) and (1.2), we first re-parameterize the random process, as we will discuss in Section 2.1 on page 5, into one that follows a state-space model with linear Gaussian dynamics for the state fit and observation Rt: dfit = -afitdt + fidW^, (1.4) dRt = fitdt + adWP. (1.5) More specifically, based on the state-space.model specified by Equations (1.4) and (1.5), we estimate • er using result of Sass and Haussmann in [14]; • a using filter-based maximum likelihood estimators via the expectation max-imization algorithm derived by Elliott and Krishnamurthy in [5]; • 5 and E/M) using the method of least-squares; • f32 using results on interpolation of conditionally Gaussian processes, well doc-umented by Liptster arid'Shiryayev in [11]. Except for a, the other four parameters are estimated iteratively. Chapter 2 provides the foundations for the parameter estimations, based on the process (fit,Rt)- Section- 2.1 first starts with a model specified by (1.1) and 3 (1.2). It then introduces a re-parametrization to simplify the model into a state-space model. Section 2.2 provides Theorem 2.1, Corollary 2.1 and Theorem 2.2 to examine the conditional behaviour of fit given the sigma field of {Rs : 0 < s < t] (denoted as J-^). Although these results generally impose certain assumptions on the data, in our simple setting, those assumptions all hold automatically, as justi-fied in Section 2.2. Section 2.3 discusses some results on interpolation of conditional Gaussian processes, as the foundation of estimation of (3. Particularly, Lemma 2.1 and Theorem 2.3 present the forward and backward equations, respectively, for the posterior variance of p,s given Tf-. Chapter 3 continues with estimation of param-eters. They include estimation of a in Section 3:1, estimation of 5 in Section 3.2, estimation of a in Section 3.3, the estimation of P in Section 3.4. Section 3.5 justifies the choice of initial estimates for the iterative procedure that estimates the unknown parameters. Chapter 3 ends in Section 3.6 with a summary of the iterative procedure for the parameter estimation. The estimation algorithm introduced in Chapter 3 is then employed to analyze the historical data; see Appendix B for the description of such a data set. Chapter 4 documents the estimation results of the historical data, as well as some aspects on the convergence of iteration. Finally, Chapter 5 gives some directions of future works. These include some possible improvements, as well as some future applications or developments, from the current work. A l l computations are done using Matlab 6.5 Release 13. The two graphical summaries are produced using R version 2.0.1, a free-ware version of S-Plus, a statistical language and computing environment. 4 Chapter 2 Basic Theory 2.1 State-space Model with Linear Gaussian Dynamics We consider the unobservable mean rate of return {/it : 0 < t < T} of a stock, when the return process {Rt : 0 < t < T} is observable. This leads to a random pro-cess -(fit, Rt)1 with unobservable first component and observable second component. Particularly, we consider (fj,t,Rt), or (fi,R) in short unless clarity is needed, as a continuous time process of the diffusion type with a pair of Ito stochastic equations dut. = 'a(S - m) dt + P dW^], (2.1) dRt = ntdt + odW^, (2.2) where a, j3 and a are assumed to be constant both in time and stocks, 5 is time-independent but may differ among the stocks, and denote two mutually independent Wiener processes, and /io and RQ usually are assumed to be two random variables 2 that are independent of and W^2\\ We can apply Kalman filter to (2.2) (2.3), cf. [11], but to use the result of [5] we want to have 5 = 0. In fact, Lakner had used this model in the context of multiple stocks, cf. equations (2.13) and (4.1) in [10]. However, instead of working with the model directly as Lakner 1or (Ot,£t) in the notation of Liptster and Shiryayev in [11]. 2Sce the beginning of Chapter 3: here, fio is assumed as deterministic and Ru is observed. 5 did, we first simplify the model by re-parameterizing the process into (fit, Rt) with flt = fit- 6 and (2.3) . Rt = Rt-St. (2.4) Since 5, being the limit of Efi,t as t —> oo (as discussed in Chapter 1)', is a constant in time, the diffusion model specified by Equation (2.1) and Equation (2.2) then reduces to dp = d/j, = -apdt + i3dW(1\\ dR = dR-6dt = (fidt + adW'Wy-Sdt = (n - 5) dt + G dW{2) = fidt + o-dWW-that is, dfit = ~afltdt + /3dW^\\ (2.5) dRt = futdt + adWJ;2). (2.6) This model now has Gaussian dynamics as used in [5], having the state p, and observation process R. We will see later, cf. Section 3.2 on page 19, how to estimate S as the long run mean of fit, i.e. of Putting this model in Liptster and Shiryayev's notations as on p.2 of [11], with 9 and £ replaced by p and R, respectively, we have dpi = [a0{t,R)+ai(t,R)pt}dt + bi(t,R)d,W^(t)+b2(t,R)d,W^(t), dRt = [AQ(t, R) + Ai(t, R)pt] dt + D(t, R) dW^{t), with ao{t, R) = 0, oj {t, R) = -a, bx{t, R) = [5, b2{t, 0 = 0; A0(t, R) = 0, Ax(t, R\\) = 1 and B(t, R) = a. 6 The next few sections examine the conditional behaviour of fit given o~{Rs : 0 < s < i). The results are summarized as Theorems-2.1, 2.2 & 2.3 and Corol-lary 2.1. They provide the theoretical bases for the parameter estimations in Chap-ter 3. 2.2 Random Behaviour of p,t This section examines the random behaviour of (fit, Ri) or more precisely, fit given the observation on {Rs : 0 < s < t}, i.e., given the sigma field T^'. In the later discussions, the following conditions will be assumed (first expressed in terms of Liptster and Shiryayev's notations as in [11]). See equations (11.4)-(11.11) on p.2, 3 of [11]. Suppose CT denotes a space of continuous functions x = {xs, s < T). (1) Vx G C T , rT I Y, i\\^(t,x)\\ + \\Ai(t,x)\\}+ Y tf(t,x) + B2(t,x) ] ,dt < oo; ' / 0 \\ f c O , l j = l , 2 J rT / [A20(t,x)+A2(t,x)}dt< oo; • - • Jo |2 Vt\\ , inf D2(t,x) > C > 0, 0 < t < T; xeC (2)Vx,y£CT, \\B(t,x)-B(t,y)\\2 C > 0, 0 < t < T, or a2 > 0; (2.9) (2) Vx,y€CT, 0 = \\a - • d \\eat = Peai dWfl\\ because a is a constant eatfit / x 0 + [' PeasdWPds Jo =• fit = e~at fi0 + e~at P f eas dW™ ds E \\TH I = E e~at fi0 + e~at fi f eas dW™ ds Jo Since a. > 0 and t > 0 so that e a t < 1 and since /l*O — (MO - <5) is deterministic, we have By the virtue of inequality (2.15), assumptions (2.12) and (2.13) follow. Finally, assumption (2.13) holds because Therefore, in conclusion, all the assumptions (2.7)-(2.14) are valid in our setting. W i t h the assumptions specified in Equations (2.7)-(2.14), the following theo-rem, as stated on p.3 of [11], assures that the random process (p,t, Rt) is conditionally Gaussian. Theorem 2.1 (Joint Conditional Gaussian of /z's) Suppose conditions (2.7)-(2.14) are fulfilled and let (with probability 1) the initial conditional distribution EFi0(a) ~ P{Po < a\\Ro) be Gaussian, iV(mo,7o), with 0 < 70 < 00. Then the future Thus, with, a > 0 and for 0 < t < T < 00 E\\tn\\ < 00. (2.15) 9 random process (p,t,Rt), 0 < t < T, satisfying (2.5) and (2.6) is conditional Gaus-sian in the sense that for any t, and 0 < to < t\\ < • • • < tn < t, the conditionally « distributions FRi0(XQ, xn) = P(pt0 < x0, • • •, fHn < XnlF?) are (P-a.s.) Gaussian. Corollary 2.1 (Conditional Gaussian of jit) 3 In accordance with Theorem 2.1, the conditional distribution F}^(a) — P(fit < a\\jc[1') will also be Gaussian. That is (fii\\!Ft') ~ N(mt)7t), for some parameters mt and \"ft-As a consequence of Corollary 2.1, we conclude that the posterior mean, m<, will be an optimal estimate of jit from {Rs : s < t], where optimality is in L2 norm or in the mean-square sense. Thus, a procedure to obtain m/ is called for. The next theorem, as stated on p.21 of [11], provides a pair of differential equations for the posterior mean and variance of the conditional distribution of jXt given {fis : s < t} (i.e., nit and jt)- However, this theorem requires the following assumptions, cf. equations (12.12)-(12.15) on p.17 of [11]. sup E(9f) < oo; 0<1; frE{A0(s,t:) +A1(s,00s}2ds < oo; Jo In our case, these assumptions become: sup £(/J. 4) < oo;' (2.16) 0\\ i 1 b2(t,R)B(t,R)+~ftAl{t.R) dmt = [a,0(t,R) + ai{t,R)mt] dt + ' v ' ; ' u ' ' B*(t,R) dRt - (A0(t, R) + Ai{t, R)mt) dt] (2.20) h(t,R)B(t,R)+'ytAl{t,R)'\" B(t, R) jt = 2aj(t,R)Jt + b2x(t,R) + b22(t,R) (2.21) 11 Note that Kt = (a + ^ is a function of t. Using the integrating factor e-h K'zdz, we have d[mteSZK'*]=eftK'*(j£)dRt m t e / o . ^ = fh0 + a-2 J* ( 7 s e /o cLR, m0 Jo 7s& dR, where It Kt= la+-r (2.24) (2.25) Also, as shown in Appendix A , solving Equation (2.23) gives jt. in closed form: It — c l + £t p- (TQ . 1 - £t where p = ^(aa)2+p2, Vo = — + era, and - ( V ° - P \\ f-W (2.26) (2.27) (2.28) (2.29) 2.3 Interpolation of conditionally Gaussian processes This section develops the basic theory for the estimation of (3 introduced in Sec-tion 3.4. Suppose (p,R), specified in the stochastic differential equations (2.5) and (2.6), satisfies conditions given'by (2.7)-(2.14). Then, by the virtue of Theorem 2.1, the conditional distribution P(ps\\pf-), s < t, is (P-a.s.) Gaussian. Let's denote the parameters as m{s,t) = E{ps\\F?), ^s,t)^E{{p,s-m{s,t)]2\\^}. Theorem 2.3 below gives m(s,t) and \"y(s,t), for s < t;in close forms. 12 As stated below, representation of 7(-,-) depends on (i) whether the first argument is larger or less than the second argument and (ii) which of the two arguments is fixed and which is varying. In [11], t is always used to denote the larger argument, whereas s is always used to denote smaller one. Also, the second argument of 7(-, •) is always fixed and the first one is varying (or running) 5. When the first argument is larger with the second argument fixed, [11] refers the representation as forward equation (of interpolation) for 7(4, s). In the other case when the second argument is larger with the first augment fixed, [11] refers the representation as the backward equation for 7(s, i) . When the notations for the forward equations were first introduced on p.22 of [11], the subscript p,s was used in m and 7 to distinguish the difference as: mii.(t,s) = E[jit\\J?\"fi], 7A„(*,s) = E{\\fia - (t,s)}2 IJf \"*} . However, these subscripts are suppressed later in [11]. For clarity, we use -/{s,t) in the case s is varying in the backward equation and r ( i , s ) when t is varying in the forward equation; in both cases, t > s. Note that in the discussion of interpolation and extrapolation of compo-nents, Liptster and Shiryayev extend the l td stochastic differential equation for the observable process from dRt = [A0(t, R) + Ai (t, R)pt] dt + B(t, R) dW® (t) of equation (11.3) on p.2 of [11] to 2 dR\\ = [Mt, R) + M(t, R)ih} dt + Y, Bi(t, R) dW®(t) of equation (12.60) on p.30 of [11]. Thus, in the later discussion, we extend from B(t, R) = a to B-i (t, R) = 0 and B2(t, R) = a. 5cf. p.35 of [11] 13 Theorem 12.9 on p.37 of [11] provides representations of m(s,t) and 7(.s,£), for s < t: m(s,t) = ms + f\\(s,u)(tf(R))'A[(u,R){B o Befall) Js x\\dRu - {A0{u,R) + A^fyfhujdu}, (2.30) 7 ( M ) = + % fy^m'A'^R^BoBr'iu^R) x AY{u,R)^{R)du}~X 7.,, (2.31) where / is the identity matrix, B o B denotes B\\B[ -f B2B'2 6 and for t > s, the triangular matrix s, r(t, s) is the unique solution to the differential equation: ^ = (2.33) at az with F(s, s) = 0. Note that the differential equation (2.33) for T(t,s) is the same as (2.23), just subjected to different initial or boundary condition. Modifying the solution for lip.32 of [11]. 7Lemma 12.2 on p.36 of [11]. 8See p.36 and Note 3 on p.22 of [11]. 14 7t given in (2.26)-(2.29) for boundary condition F(s, s) = 0 gives F(t,s) = a where 1 + £t,s p- aa 1 — £t,s P £t, s. aa — p N exp -2p (t - s) (2.34) (2.35) (2.36) , aa + p because, with F(s, s) = 0, (2.28) gives Vb = r ^ ' 6 ^ + aa = aa. Note that even though T(t, s) = 7 A „(t , s) = £ {[£., - % , ( t , s)] 2 | j f s , R | de-pends on p,,* by definition, (2.34)-(2.35) show that r(r.,.s) does not depend on ps. Theorem 2.3 (Backward Equations for m(s,t) and 7(s,t)) Suppose (p,R), spec-ified in the stochastic differential equations (2.5) and (2.6), satisfies (2.7)-(2.14). Then, for s < t, the conditional mean and variance of ps given observations from process {Rs : s < t} are: rt m{s,t) = ms + a 2 f j(s,u)ip'^[dRn J s 7(M) = 4 + mudu], du = 1 7 7 ' + W W J s where ip™, with u > s, is the solution to the IVP d^ du du = -{a + a-2r(u,s)}s = l. Solving the IVP gives V?\" = e x p | - a(u-s)+a~2 j T(z,s)dz j . (2.37)-(2.38) (2.39) (2.40) because ipss = l. 15 Chapter 3 Estimations of Parameters Because R is the cumulative return starting from time 0, Ro = 0 by definition. Moreover, because Rt = Rt — St, this leads to Ro = Ro = 0. Consequently, E(.\\Ro) or E(.\\J-Q) is simply E(.) — it is so, because RQ assumes its value by definiton and provides no information on the process. Thus, mo = E(no\\^:o') ~ -^(MO)- Further-more, we also assume the initial process is fixed; that is, fio = rn,Q is deterministic. Because fit is defined as Ht — S for some constant 5, p,Q = fio — S is also deterministic. Therefore, 70, the conditional variance of flo, is zero. 3.1 Estimation of a Observe from model equation (1.2) on page 1 dRi = ntdt + adWt2), (1.2) the parameter a is the volatility of the return process Rt- Thus, one can estimate