# Statistical Estimation for CAPM with Long-Memory Dependence

Authors: Tomoyuki Amano; Tsuyoshi Kato; Masanobu Taniguchi
Journal: Advances in Decision Sciences
 (2012)

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## Abstract

We investigate the Capital Asser Pricing Model (CAPM) with time dimension. By using time series analysis, we discuss the estimation of CAPM when market portfolio and the error process are long-memory process and correlated with each other. We give a sufficient condition for the return of assets in the CAPM to be short memory. In this setting, we propose a two-stage least squares estimator for the regression coefficient and derive the asymptotic distribution. Some numerical studies are given. They show an interesting feature of this model.


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## Body

## 1. Introduction

The CAPM is one of the typical models of risk asset’s price on equilibrium market and has been used for pricing individual stocks and portfolios. At first, Markowitz [1] did the groundwork of this model. In his research, he cast the investor’s portfolio selection problem in terms of expected return and variance. Sharpe [2] and Lintner [3] developed Markowitz’s idea for economical implication. Black [4] derived a more general version of the CAPM. In their version, the CAPM is constructed based on the excess of the return of the asset over zero-beta return E[Ri]=E[R0m]+βim(E[Rm]-E[R0m]), where Ri and Rm are the return of the ith asset and the market portfolio and R0m is the return of zero-beta portfolio of the market portfolio. Campbell et al. [5] discussed the estimation of CAPM, but in their work they did not discuss the time dimension. However, in the econometric analysis, it is necessary to investigate this model with the time dimension; that is, the model is represented as Ri,t=αim+βimRm,t+ϵi,t. Recently from the empirical analysis, it is known that the return of asset follows a short-memory process. But Granger [6] showed that the aggregation of short-memory processes yields long-memory dependence, and it is known that the return of the market portfolio follows a long-memory process. From this point of view, first, we show that the return of the market portfolio and the error process ϵt are long-memory dependent and correlated with each other.For the regression model, the most fundamental estimator is the ordinary least squares estimator. However, the dependence of the error process with the explanatory process makes this estimator to be inconsistent. To overcome this difficulty, the instrumental variable method is proposed by use of the instrumental variables which are uncorrelated with the error process and correlated with the explanatory variable. This method was first used by Wright [7], and many researchers developed this method (see Reiersøl [8], Geary [9], etc.). Comprehensive reviews are seen in White [10]. However, the instrumental variable method has been discussed in the case where the error process does not follow long-memory process, and this makes the estimation difficult.For the analysis of long-memory process, Robinson and Hidalgo [11] considered a stochastic regression model defined by yt=α+β′xt+ut, where α,β=(β1,…,βK)′ are unknown parameters and the K-vector processes {xt} and {ut} are long-memory dependent with E(xt)=0,E(ut)=0. Furthermore, in Choy and Taniguchi [12], they consider the stochastic regression model yt=βxt+ut, where {xt} and {ut} are stationary process with E(xt)=μ≠0, and Choy and Taniguchi [12] introduced a ratio estimator, the least squares estimator, and the best linear unbiased estimator for β. However, Robinson and Hidalgo [11] and Choy and Taniguchi [12] assume that the explanatory process {xt} and the error process {ut} are independent.In this paper, by the using of instrumental variable method we propose the two-stage least squares (2SLS) estimator for the CAPM in which the returns of the individual asset and error process are long-memory dependent and mutually correlated with each other. Then we prove its consistency and CLT under some conditions. Also, some numerical studies are provided.This paper is organized as follows. Section2 gives our definition of the CAPM, and we give a sufficient condition that return of assets as short dependence is generated by the returns of market portfolio and error process which are long-memory dependent and mutually correlated each other. In Section 3 we propose 2SLS estimator for this model and show its consistency and asymptotic normality. Section 4 provides some numerical studies which show interesting features of our estimator. The proof of theorem is relegated to Section 5.


## 2. CAPM (Capital Asset Pricing Model)

For Sharpe and Lintner version of the CAPM (see Sharpe [2] and Lintner [3]), the expected return of asset i is given by
(2.1)E[Ri]=Rf+βim(E[Rm-Rf]),
where
(2.2)βim=Cov[Ri,Rm]V[Rm],Rm is the return of the market portfolio, and Rf is the return of the risk-free asset. Another Sharpe-Lintner’s CAPM (see Sharpe [2] and Lintner [3]) is defined for Zi≡Ri-Rf,
(2.3)E[Zi]=βimE[Zm],
where
(2.4)βim=Cov[Zi,Zm]V[Zm]
and Zm=Rm-Rf.Black [4] derived a more general version of CAPM, which is written as
(2.5)E[Ri]=αim+βimE[Rm],
where αim=E[R0m](1-βim) and R0m is the return on the zero-beta portfolio.Since CAPM is single-period model, (2.1) and (2.5) do not have a time dimension. However, for econometric analysis of the model, it is necessary to add assumptions concerning the time dimension. Hence, it is natural to consider the model:
(2.6)Yi,t=αi+βiZt+ϵi,t,
where i denotes the asset, t denotes the period, and Yi,t and Zt, i=1,…,n and t=1,…,T, are, respectively, the returns of the asset i and the market portfolio at t.Empirical features of the realized returns for assets and market portfolios are well known.We plot the autocorrelation function (ACF(l) (l: time lag)) of returns of IBM stock and S&P500 (squared transformed) in Figures 1 and 2, respectively.Figure 1
ACF of return of the IBM stock.Figure 2
ACF of return of S&P500 (square transformed).From Figures1 and 2, we observe that the return of stock (i.e., IBM) shows the short-memory dependence and that a market index (i.e., S&P500) shows the long-memory dependence.Suppose that ann-dimensional process {Yt=(Y1,t,…,Yn,t)′} is generated by
(2.7)Yt=α+B′Zt+ϵt(t=1,2,…,T),
where α=(α1,…,αn)′ and B={βij;i=1,…,p,j=1,…,n} are unknown vector and matrix; respectively, {Zt=(Z1,t,…,Zp,t)′} is an explanatory stochastic regressor process, and {ϵt=(ϵ1,t,…,ϵn,t)′} is a sequence of disturbance process. The ith component is written as(2.8)Yi,t=αi+βi′Zt+ϵi,t,
where βi′=(βi,1,…,βi,p).In the CAPM,Yt is the return of assets and Zt is the return of the market portfolios. As we saw, empirical studies suggest that {Yt} is short-memory dependent and that {Zt} is long-memory dependent. On this ground, we investigate the conditions that the CAPM (2.7) is well defined. It is seen that, if the model (2.7) is valid, we have to assume that {ϵt} is also long-memory dependent and is correlated with {Zt}.Hence, we suppose that{Zt} and {εt} are defined by
(2.9)Zt=∑j=0∞γjat-j+∑j=0∞ρjbt-j,ϵt=∑j=0∞ηjet-j+∑j=0∞ξjbt-j,
where {at}, {bt}, and {et} are p-dimensional zero-mean uncorrelated processes, and they are mutually independent. Here the coefficients {γj} and {ρj} are p×p-matrices, and all the components of γj are ℓ1-summable, (for short, γj∈ℓ1) and those of ρj are ℓ2-summable (for short, ρj∈ℓ2). The coefficients {ηj} and {ξj} are n×p-matrices, ηj∈ℓ1, and ξj∈ℓ2. From (2.9) it follows that(2.10)Yt=α+∑j=0∞(B′γjat-j+ηjet-j)+∑j=0∞(B′ρj+ξi)bt-j.Although(B′ρj+ξj)∈ℓ2 generally, if B′ρj+ξj=O(1/jα), α>1, then (B′ρj+ξj)∈ℓ1, which leads to the following.Proposition 2.1.
IfB′ρj+ξj=O(j-α),α>1, then the process  {Yt} is short-memory dependent.Proposition2.1 provides an important view for the CAPM; that is, if we assume natural conditions on (2.7) based on the empirical studies, then they impose a sort of “curved structure”: B′ρj+ξj=O(j-α) on the regressor and disturbance. More important view is the statement implying that the process {βi′Zt+ϵi,t} is fractionally cointegrated. Here βi and ϵi,t are called the cointegrating vector and error, respectively, (see Robinson and Yajima [13]).


## 3. Two-Stage Least Squares Estimation

This section discusses estimation of (2.7) satisfying Proposition 2.1. SincE(Ztϵt′)≠0, the least squares estimator for B, is known to be inconsistent. In what follows we assume that α=0 in (2.7), because it can be estimated consistently by the sample mean. However, by use of the econometric theory, it is often possible to find other variables that are uncorrelated with the errors ϵt, which we call instrumental variables, and to overcome this difficulty. Without instrumental variables, correlations between the observables {Zt} and unobservables {ϵt} persistently contaminate our estimator for B. Hence, instrumental variables are useful in allowing us to estimate B.Let{Xt} be r×1-dimensional vector (p≤r) instrumental variables with E[Xt]=0, Cov(Xt,Zt)≠0, and Cov(Xt,ϵt)=0. Consider the OLS regression of Zt on Xt. If Zt can be represented as
(3.1)Zt=δ′Xt+ut,
where δ is a r×p matrix and {ut} is a p-dimensional vector process which is independent of {Xt}, δ can be estimated by the OLS estimator
(3.2)δ̂=[∑t=1TXtXt′]-1[∑t=1TXtZt′].
From (2.7) with α=0 and (3.1), Yt has the form:
(3.3)Yt=B′δ′Xt+B′ut+ϵt,
and δ′Xt is uncorrelated with B′ut+ϵt; hence, B can be estimated by the OLS estimator:(3.4)B̂OLS=[∑t=1T(δ′Xt)(δ′Xt)′]-1[∑t=1T(δ′Xt)Yt'].Using (3.2) and (3.4), we can propose the 2SLS estimator:(3.5)B̂2SLS=[∑t=1T(δ̂′Xt)(δ̂′Xt)′]-1[∑t=1T(δ̂′Xt)Yt'].Now, we aim at proving the consistency and asymptotic normality of the 2SLS estimatorB̂2SLS. For this we assume that {ϵt} and {Xt} jointly constitute the following linear process:
(3.6)(ϵtXt)=∑j=0∞G(j)Γ(t-j)=At(say),
where {Γ(t)} is uncorrelated (n+r)-dimensional vector process with(3.7)E[Γ(t)]=0,E[Γ(t)Γ(s)*]=δ(t,s)K,δ(t,s)={1,t=s,0,t≠s,
and G(j),s are (n+r)×(n+r) matrices which satisfy ∑j=0∞tr{G(j)KG(j)*}<∞. Then {At} has the spectral density matrix:
(3.8)f(ω)=12πk(ω)Kk(ω)*={fab(ω);1≤a,b≤(n+r)}(-π<ω≤π),
where
(3.9)k(ω)=∑j=0∞G(j)eiωj={kab(ω);1≤a,b≤(n+r)}(-π<ω≤π).
Further, we assume that ∫-ππlogdetf(ω)dω>-∞, so that the process {At} is nondeterministic. For the asymptotics of B̂2SLS, from page 108, line↑1–page 109, line 7 of Hosoya [14], we impose the following assumption.Assumption 3.1.
(i) There existsϵ>0 such that, for any t<t1≤t2≤t3≤t4 and for each β1,β2,(3.10)Var⁡[E{Γβ1(t1)Γβ2(t2)∣B(t)}-δ(t1-t2,0)Kβ1β2]=O{(t1-t)-2-ϵ},
and also
(3.11)E|E{Γβ1(t1)Γβ2(t2)Γβ3(t3)Γβ4(t4)∣B(t)}-E{Γβ1(t1)Γβ2(t2)Γβ3(t3)Γβ4(t4)}|=O{(t1-t)-1-ϵ},
uniformly in t, where B(t) is the σ-field generated by {Γ(s);s≤t}.
(ii) For anyϵ>0 and for any integer M≥0, there exists Bϵ>0 such that
(3.12)E[T(n,s)2{T(n,s)>Bϵ}]<ϵ,
uniformly in n, s, where
(3.13)T(n,s)=[∑α,β=1p∑r=0M{1T∑t=1T(Γα(t+s)Γβ(t+s+r)-Kαβδ(0,r))}2]1/2,
and {T(n,s)>Bϵ} is the indicator, which is equal to 1 if T(n,s)>Bϵ and equal to 0 otherwise.
(iii) Eachfab(ω) is square-integrable.Under the above assumptions, we can establish the following theorem.Theorem 3.2.
Under Assumption3.1, it holds that (i)
(3.14)B̂2SLS⟶PB,(ii)
(3.15)T(B̂2SLS-B)⟶dQ-1E[ZtXt′]E[XtXt′]-1U,
where
(3.16)Q=[E(ZtXt′)][E(XtXt′)]-1[E(XtZt′)],
and U={Ui,j;1≤i≤r,1≤j≤n} is a random matrix whose elements follow normal distributions with mean 0 and
(3.17)Cov[Ui,j,Uk,l]=2π∫-ππ[fn+i,n+k(ω)f̅j,l(ω)+fn+i,l(ω)f̅j,n+k(ω)]dω+2π∑β1,…,β4=1p∫-ππ∫-ππκn+i,β1(ω1)κj,β2(-ω1)κn+k,β3(ω2)κl,β4(-ω2)Qβ1,…,β4Γ×(ω1,-ω2,ω2)dω1dω2.The next example prepares the asymptotic variance formula ofB̂2SLS to investigate its features in simulation study.Example 3.3.
Let{Zt} and {Xt} be scalar long-memory processes, with spectral densities {2π|1-eiλ|2dZ}-1 and {2π|1-eiλ|2dX}-1, respectively, and cross spectral density (1/2π)(1-eiλ)-dX(1-e-iλ)-dZ, where 0<dZ<1/2 and 0<dX<1/2. Then
(3.18)E(XtZt)=12π∫-ππ1(1-eiλ)dX1(1-e-iλ)dZdλ.
Suppose that{ϵt} is a scalar uncorrelated process with σϵ2≡E{ϵt2}. Assuming Gaussianity of {At}, it is seen that the right hand of (3.17) is
(3.19)2π∫-ππ12π|1-eiλ|2dXσϵ22πdλ,
which entails
(3.20)lim⁡T→∞Var⁡[T(B̂2SLS-B)]=2π∫-ππ(1/2π|1-eiλ|2dX)(σϵ2/2π)dλ(1/2π∫-ππ(1/(1-eiλ)dX)(1/(1-e-iλ)dZ)dλ)2=σϵ2(2π∫-ππ(1/|1-eiλ|2dX)dλ(∫-ππ(1/(1-eiλ)dX)(1/(1-e-iλ)dZ)dλ)2)=σϵ2×V*(dX,dZ).


## 4. Numerical Studies

In this section, we evaluate the behaviour ofB̂2SLS in the case p=1 in (2.7) numerically.Example 4.1.
Under the condition of Example3.3, we investigate the asymptotic variance behaviour of B̂2SLS by simulation. Figure 3 plots V*(dX,dZ) for 0<dX<1/2 and 0<dZ<1/2.Figure 3
V*(dx,dz) in Section 4.From Figure3, we observe that, if dZ↘0 and if dX↗1/2, then V* becomes large, and otherwise V* is small. This result implies only in the case that the long-memory behavior of Zt is weak and the long-memory behavior of Xt is strong, V* is large. Note that long-memory behaviour of Zt makes the asymptotic variance of the 2SLS estimator small, but one of Xt makes it large.Example 4.2.
In this example, we consider the following model:(4.1)Yt=Zt+ϵt,Zt=Xt+ut,ϵt=wt+ut,
where Xt, wt, and ut are the scalar long-memory processes which follow FARIMA(0,d1,0), FARIMA(0,d2,0), and FARIMA(0,0.1,0), respectively. Note that Zt and ϵt are correlated, Xt and Zt are correlated, but Xt and ϵt are independent. Under this model we compare B̂2SLS with the ordinary least squares estimator B̃OLS for B, which is defined as
(4.2)B̃OLS=[∑t=1TZt2]-1[∑t=1TZtYt].
The lengths of Xt, Yt, and Zt are set by 100, and based on 5000 times simulation we report the mean square errors (MSE) of B̂2SLS and B̃OLS. We set d1,d2=0.1,0.2,0.3 in Table 1.
In most cases ofd1 and d2 in Table 1, MSE of B̂2SLS is smaller than that of B̃OLS. Hence, from this Example we can see that our estimator B̂2SLS is better than B̃OLS in the sense of MSE. Furthermore, from Table 1, we can see that MSE of B̂2SLS and B̃OLS increases as d2 becomes large; that is, long-memory behavior of wt makes the asymptotic variances of B̂2SLS and B̃OLS large.Table 1
MSE ofB̂2SLS and B̃OLS.
d20.10.20.3B̂2SLS (d1=0.1)0.030.0520.189B̃OLS (d1=0.1)0.2590.2710.34B̂2SLS (d1=0.2)0.030.0750.342B̃OLS (d1=0.2)0.1780.1930.307B̂2SLS (d1=0.3)0.0190.0520.267B̃OLS (d1=0.3)0.0690.0890.23Example 4.3.
In this example, we calculateB̂2SLS based on the actual financial data. We choose S&P500 (square transformed) as Zt and the Nikkei stock average as an instrumental variable Xt. Assuming that Yt(5×1) consists of the return of IBM, Nike, Amazon, American Expresses and Ford; the 2SLS estimates for Bi,i=1,…,5 are recorded in Table 2. We chose the Nikkei stock average as the instrumental variable, because we got the following correlation analysis between the residual processes of returns and Nikkei.Correlation of IBM’s residual and Nikkei’s return: −0.000311Correlation of Nike’s residual and Nikkei’s return: −0.00015Correlation of Amazon’s residual and Nikkei’s return: −0.000622Correlation of American Express’s residual and Nikkei’s return: 0.000147Correlation of Ford’s residual and Nikkei’s return: −0.000536,which supports the assumptionCov(Xt,εt)=0.Table 2
B̂2SLS based on the actual financial data.
StockIBMNikeAmazonAmerican ExpressFordB̂2SLS0.751.391.712.61−1.89From Table2, we observe that the return of the finance stock (American Express) is strongly correlated with that of S&P500 and the return of the auto industry stock (Ford) is negatively correlated with that of S&P500.


## 5. Proof of Theorem

This section provides the proof of Theorem3.2. First for convenience we define Ẑt=(Ẑ1,t,…,Ẑp,t)′≡δ̂′Xt. Let ût=(û1,t,…,ûp,t)′ be the residual from the OLS estimation of (3.1); that is, (5.1)ûi,t=Zi,t-Ẑi,t.TheOLS makes this residual orthogonal to Xt:
(5.2)∑t=1TXt'ûi,t=0,
which implies the residual is orthogonal to Ẑj,t,
(5.3)∑t=1TẐj,tûi,t=(∑t=1TXt'ûi,t)δ̂j=0,
where δ̂j is jth column vector of δ̂. Hence, we can obtain
(5.4)∑t=1TẐj,tZi,t=∑t=1TẐj,t(Ẑi,t+ûi,t)=∑t=1TẐj,tẐi,t,
for all i and j. This means
(5.5)∑t=1TẐtZt′=∑t=1TẐtẐt′.
So, the ith column vector of the 2SLS estimator (3.5) β̂2SLS,i (say) can be represented as
(5.6)β̂2SLS,i=[∑t=1TẐtZt′]-1[∑t=1TẐtYi,t],
which leads to
(5.7)β̂2SLS,i-βi=[1T∑t=1TẐtZt′]-1[1T∑t=1TẐtϵi,t].
Hence, we can see that
(5.8)T(B̂2SLS-B)=[1T∑t=1TẐtZt′]-1[1T∑t=1TẐtϵt′].Note that, by the ergodic theorem (e.g., Stout [15] p179–181), (5.9)1T∑t=1TẐtZt′=1Tδ̂′∑t=1TXtZt′=[1T∑t=1TZtXt′][1T∑t=1TXtXt′]-1[1T∑t=1TXtZt′]⟶PQ.Furthermore, the second term of the right side of (5.8) can be represented as
(5.10)[1T∑t=1TẐtϵt′]=δ̂′1T∑t=1TXtϵt′,
and by the ergodic theorem (e.g., Stout [15] p179–181), we can see (5.11)δ̂′=[∑t=1TZtXt′][∑t=1TXtXt′]-1⟶P[E(ZtXt′)][E(XtXt′)]-1.Proof of (i).
From the above,(5.12)B̂2SLS-B=OP[1T∑t=1TXtϵt′].
In view of Theorem 1.2 (i) of Hosoya [14], the right-hand side of (5.12) converges to 0 in probability.Proof of (ii).
From Theorem3.2 of Hosoya [14], if Assumption 3.1 holds, it follows that (1/T)∑t=1TXtϵt′→dU. Hence, Theorem 3.2 is proved.



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