Statistical Estimation for CAPM with Long-Memory Dependence

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.

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[𝑅_𝑖]=𝐸[𝑅_0𝑚]+𝛽_𝑖𝑚(𝐸[𝑅_𝑚]−𝐸[𝑅_0𝑚]), where 𝑅_𝑖 and 𝑅_𝑚 are the return of the 𝑖th asset and the market portfolio and 𝑅_0𝑚 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 𝑅_𝑖,𝑡=𝛼_𝑖𝑚+𝛽_𝑖𝑚𝑅_𝑚,𝑡+𝜖_𝑖,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 𝜖_𝑡 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 𝑦_𝑡=𝛼+𝜷^′𝐱_𝑡+𝑢_𝑡, where 𝛼,𝛽=(𝛽₁,…,𝛽_𝐾)′ are unknown parameters and the 𝐾-vector processes {𝐱_𝑡} and {𝑢_𝑡} are long-memory dependent with 𝐸(𝐱_𝑡)=𝟎,𝐸(𝑢_𝑡)=0. Furthermore, in Choy and Taniguchi [12], they consider the stochastic regression model 𝑦_𝑡=𝛽𝑥_𝑡+𝑢_𝑡, where {𝑥_𝑡} and {𝑢_𝑡} are stationary process with 𝐸(𝑥_𝑡)=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 {𝑥_𝑡} and the error process {𝑢_𝑡} 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. Section 2 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 𝐸𝑅_𝑖=𝑅_𝑓+𝛽_𝑖𝑚𝐸𝑅_𝑚−𝑅_𝑓, (2.1) where 𝛽_𝑖𝑚=Cov𝑅_𝑖,𝑅_𝑚𝑉𝑅_𝑚, (2.2)𝑅_𝑚 is the return of the market portfolio, and 𝑅_𝑓 is the return of the risk-free asset. Another Sharpe-Lintner’s CAPM (see Sharpe [2] and Lintner [3]) is defined for 𝑍_𝑖≡𝑅_𝑖−𝑅_𝑓, 𝐸𝑍_𝑖=𝛽_𝑖𝑚𝐸𝑍_𝑚, (2.3) where 𝛽_𝑖𝑚=Cov𝑍_𝑖,𝑍_𝑚𝑉𝑍_𝑚 (2.4) and 𝑍_𝑚=𝑅_𝑚−𝑅_𝑓.

Black [4] derived a more general version of CAPM, which is written as 𝐸𝑅_𝑖=𝛼_𝑖𝑚+𝛽_𝑖𝑚𝐸𝑅_𝑚, (2.5) where 𝛼_𝑖𝑚=𝐸[𝑅_0𝑚](1−𝛽_𝑖𝑚) and 𝑅_0𝑚 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: 𝑌_𝑖,𝑡=𝛼_𝑖+𝛽_𝑖𝑍_𝑡+𝜖_𝑖,𝑡, (2.6) where 𝑖 denotes the asset, 𝑡 denotes the period, and 𝑌_𝑖,𝑡 and 𝑍_𝑡, 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 Figures 1 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 an 𝑛-dimensional process {𝐘_𝑡=(𝑌_1,𝑡,…,𝑌_𝑛,𝑡)^′} is generated by 𝐘_𝑡=𝜶+𝐁^′𝐙_𝑡+𝝐_𝑡(𝑡=1,2,…,𝑇), (2.7) where 𝜶=(𝛼₁,…,𝛼_𝑛)^′ and 𝐁={𝛽_𝑖𝑗;𝑖=1,…,𝑝,𝑗=1,…,𝑛} are unknown vector and matrix; respectively, {𝐙_𝑡=(𝑍_1,𝑡,…,𝑍_𝑝,𝑡)^′} is an explanatory stochastic regressor process, and {𝝐_𝑡=(𝜖_1,𝑡,…,𝜖_𝑛,𝑡)′} is a sequence of disturbance process. The 𝑖th component is written as𝑌_𝑖,𝑡=𝛼_𝑖+𝜷^′_𝑖𝐙_𝑡+𝜖_𝑖,𝑡, (2.8) where 𝜷^′_𝑖=(𝛽_𝑖,1,…,𝛽_𝑖,𝑝).

In the CAPM, 𝐘_𝑡 is the return of assets and 𝐙_𝑡 is the return of the market portfolios. As we saw, empirical studies suggest that {𝐘_𝑡} is short-memory dependent and that {𝐙_𝑡} 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 {𝝐_𝑡} is also long-memory dependent and is correlated with {𝐙_𝑡}.

Hence, we suppose that {𝐙_𝑡} and {𝜺_𝑡} are defined by 𝐙_𝑡=^∞_𝑗=0𝜸_𝑗𝐚_𝑡−𝑗+^∞_𝑗=0𝝆_𝑗𝐛_𝑡−𝑗,𝝐_𝑡=^∞_𝑗=0𝜼_𝑗𝐞_𝑡−𝑗+^∞_𝑗=0𝝃_𝑗𝐛_𝑡−𝑗, (2.9) where {𝐚_𝑡}, {𝐛_𝑡}, and {𝐞_𝑡} are 𝑝-dimensional zero-mean uncorrelated processes, and they are mutually independent. Here the coefficients {𝜸_𝑗} and {𝝆_𝑗} are 𝑝×𝑝-matrices, and all the components of 𝜸_𝑗 are ℓ¹-summable, (for short, 𝜸_𝑗∈ℓ¹) and those of 𝝆_𝑗 are ℓ²-summable (for short, 𝝆_𝑗∈ℓ²). The coefficients {𝜼_𝑗} and {𝝃_𝑗} are 𝑛×𝑝-matrices, 𝜼_𝑗∈ℓ¹, and 𝝃_𝑗∈ℓ². From (2.9) it follows that𝐘_𝑡=𝜶+^∞_𝑗=0𝐁^′𝜸_𝑗𝐚_𝑡−𝑗+𝜼_𝑗𝐞_𝑡−𝑗+^∞_𝑗=0𝐁^′𝝆_𝑗+𝝃_𝑖𝐛_𝑡−𝑗. (2.10)

Although (𝐁^′𝝆_𝑗+𝝃_𝑗)∈ℓ² generally, if 𝐁^′𝝆_𝑗+𝝃_𝑗=𝑂(1/𝑗^𝛼), 𝛼>1, then (𝐁^′𝝆_𝑗+𝝃_𝑗)∈ℓ¹, which leads to the following.

Proposition 2.1. If 𝐁^′𝝆_𝑗+𝝃_𝑗=𝑂(𝑗^−𝛼),𝛼>1, then the process {𝐘_𝑡} is short-memory dependent.

Proposition 2.1 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^′𝝆_𝑗+𝝃_𝑗=𝑂(𝑗^−𝛼) on the regressor and disturbance. More important view is the statement implying that the process {𝜷^′_𝑖𝐙_𝑡+𝜖_𝑖,𝑡} is fractionally cointegrated. Here 𝜷_𝑖 and 𝜖_𝑖,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(𝐙_𝑡𝝐^′_𝑡)≠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 𝝐_𝑡, which we call instrumental variables, and to overcome this difficulty. Without instrumental variables, correlations between the observables {𝐙_𝑡} and unobservables {𝝐_𝑡} persistently contaminate our estimator for 𝐁. Hence, instrumental variables are useful in allowing us to estimate 𝐁.

Let {𝐗_𝑡} be 𝑟×1-dimensional vector (𝑝≤𝑟) instrumental variables with 𝐸[𝐗_𝑡]=𝟎, Cov(𝐗_𝑡,𝐙_𝑡)≠𝟎, and Cov(𝐗_𝑡,𝝐_𝑡)=𝟎. Consider the OLS regression of 𝐙_𝑡 on 𝐗_𝑡. If 𝐙_𝑡 can be represented as 𝐙_𝑡=𝜹^′𝐗_𝑡+𝐮_𝑡, (3.1) where 𝜹 is a 𝑟×𝑝 matrix and {𝐮_𝑡} is a 𝑝-dimensional vector process which is independent of {𝐗_𝑡}, 𝜹 can be estimated by the OLS estimator 𝜹=^𝑇_𝑡=1𝐗_𝑡𝐗^′_𝑡^−1𝑇_𝑡=1𝐗_𝑡𝐙^′_𝑡. (3.2) From (2.7) with 𝜶=𝟎 and (3.1), 𝐘_𝑡 has the form: 𝐘_𝑡=𝐁^′𝜹^′𝐗_𝑡+𝐁^′𝐮_𝑡+𝝐_𝑡, (3.3) and 𝜹^′𝐗_𝑡 is uncorrelated with 𝐁^′𝐮_𝑡+𝝐_𝑡; hence, 𝐁 can be estimated by the OLS estimator:𝐁_OLS=^𝑇_𝑡=1𝜹^′𝐗_𝑡𝜹^′𝐗_𝑡^′−1𝑇_𝑡=1𝜹^′𝐗_𝑡𝐘^′_𝑡. (3.4)

Using (3.2) and (3.4), we can propose the 2SLS estimator:𝐁_2SLS=^𝑇_𝑡=1𝜹^′𝐗_𝑡𝜹^′𝐗_𝑡^′−1𝑇_𝑡=1𝜹^′𝐗_𝑡𝐘^′_𝑡. (3.5)

Now, we aim at proving the consistency and asymptotic normality of the 2SLS estimator 𝐁_2SLS. For this we assume that {𝝐_𝑡} and {𝐗_𝑡} jointly constitute the following linear process: 𝝐_𝑡𝐗_𝑡=^∞_𝑗=0𝐆𝑗𝚪𝑡−𝑗=𝐀_𝑡say, (3.6) where {Γ(𝑡)} is uncorrelated (𝑛+𝑟)-dimensional vector process with𝐸[𝚪(𝑡𝐸𝚪(𝑡)𝚪(𝑠))]=𝟎,_∗()𝛿()==𝛿𝑡,𝑠𝐊,𝑡,𝑠1,𝑡=𝑠,0,𝑡≠𝑠, (3.7) and 𝐆(𝑗)^,s are (𝑛+𝑟)×(𝑛+𝑟) matrices which satisfy ∑^∞_𝑗=0𝑡𝑟{𝐆(𝑗)𝐊𝐆(𝑗)^∗}<∞. Then {𝐀_𝑡} has the spectral density matrix: 𝐟(𝜔)=1𝐤(𝜔)(𝜔)2𝜋𝐊𝐤^∗=𝑓_𝑎𝑏(𝜔);1≤𝑎,𝑏≤(𝑛+𝑟)(−𝜋<𝜔≤𝜋), (3.8) where 𝐤(𝜔)=^∞_𝑗=0𝐆𝑗𝑒^𝑖𝜔𝑗=𝑘_𝑎𝑏(𝜔);1≤𝑎,𝑏≤(𝑛+𝑟)(−𝜋<𝜔≤𝜋). (3.9) Further, we assume that ∫^𝜋_−𝜋logdet𝐟(𝜔)𝑑𝜔>−∞, so that the process {𝐀_𝑡} is nondeterministic. For the asymptotics of 𝐁_2SLS, from page 108, line↑1–page 109, line 7 of Hosoya [14], we impose the following assumption.

Assumption 3.1. (i) There exists 𝜖>0 such that, for any 𝑡<𝑡₁≤𝑡₂≤𝑡₃≤𝑡₄ and for each 𝛽₁,𝛽₂,Var𝐸Γ_𝛽1𝑡₁Γ_𝛽2𝑡₂∣𝐁(𝑡)−𝛿𝑡₁−𝑡₂,0𝐾_𝛽1𝛽2=𝑂𝑡₁−𝑡^−2−𝜖, (3.10) and also 𝐸𝐸Γ_𝛽1𝑡₁Γ_𝛽2𝑡₂Γ_𝛽3𝑡₃Γ_𝛽4𝑡₄∣𝐁(𝑡)−𝐸Γ_𝛽1𝑡₁Γ_𝛽2𝑡₂Γ_𝛽3𝑡₃Γ_𝛽4𝑡₄=𝑂𝑡₁−𝑡^−1−𝜖, (3.11) uniformly in 𝑡, where 𝐁(𝑡) is the 𝜎-field generated by {Γ(𝑠);𝑠≤𝑡}.
(ii) For any 𝜖>0 and for any integer 𝑀≥0, there exists 𝐵_𝜖>0 such that 𝐸𝑇(𝑛,𝑠)²𝑇(𝑛,𝑠)>𝐵_𝜖<𝜖, (3.12) uniformly in 𝑛, 𝑠, where 𝑇(𝑛,𝑠)=^𝑝𝑀_{𝛼,𝛽=1𝑟=0}1√𝑇^𝑇_𝑡=1Γ_𝛼(𝑡+𝑠)Γ_𝛽(𝑡+𝑠+𝑟)−𝐾_𝛼𝛽𝛿(0,𝑟)^21/2, (3.13) and {𝑇(𝑛,𝑠)>𝐵_𝜖} is the indicator, which is equal to 1 if 𝑇(𝑛,𝑠)>𝐵_𝜖 and equal to 0 otherwise.
(iii) Each 𝑓_𝑎𝑏(𝜔) is square-integrable.

Under the above assumptions, we can establish the following theorem.

Theorem 3.2. Under Assumption 3.1, it holds that (i)𝐁⟶_2SLS^𝑃𝐁, (3.14)(ii)√𝑇𝐁_2SLS−𝐁⟶^𝑑𝐐⁻¹𝐸𝐙_𝑡𝐗^′_𝑡𝐸𝐗_𝑡𝐗^′_𝑡⁻¹𝐔, (3.15) where 𝐐=𝐸𝐙_𝑡𝐗^′_𝑡𝐸𝐗_𝑡𝐗^′_𝑡⁻¹𝐸𝐗_𝐭𝐙^′_𝑡, (3.16) and 𝐔={𝑈_𝑖,𝑗;1≤𝑖≤𝑟,1≤𝑗≤𝑛} is a random matrix whose elements follow normal distributions with mean 0 and 𝐶𝑜𝑣𝑈_𝑖,𝑗,𝑈_𝑘,𝑙=2𝜋^𝜋_−𝜋𝑓_{𝑛+𝑖,𝑛+𝑘}(𝜔)𝑓_𝑗,𝑙(𝜔)+𝑓_{𝑛+𝑖,𝑙}(𝜔)𝑓_{𝑗,𝑛+𝑘}(𝜔)𝑑𝜔+2𝜋^𝑝_{𝛽1,…,𝛽4=1}^𝜋_−𝜋^𝜋_−𝜋𝜅_{𝑛+𝑖,𝛽1}𝜔₁𝜅_𝑗,𝛽2−𝜔₁𝜅_{𝑛+𝑘,𝛽3}𝜔₂𝜅_𝑙,𝛽4−𝜔₂𝑄^Γ_{𝛽1,…,𝛽4}×𝜔₁,−𝜔₂,𝜔₂𝑑𝜔₁𝑑𝜔₂. (3.17)

The next example prepares the asymptotic variance formula of 𝐁_2SLS to investigate its features in simulation study.

Example 3.3. Let {𝑍_𝑡} and {𝑋_𝑡} be scalar long-memory processes, with spectral densities {2𝜋|1−𝑒^𝑖𝜆|^2𝑑𝑍}⁻¹ and {2𝜋|1−𝑒^𝑖𝜆|^2𝑑𝑋}⁻¹, respectively, and cross spectral density (1/2𝜋)(1−𝑒^𝑖𝜆)^−𝑑𝑋(1−𝑒^−𝑖𝜆)^−𝑑𝑍, where 0<𝑑_𝑍<1/2 and 0<𝑑_𝑋<1/2. Then 𝐸𝑋_𝑡𝑍_𝑡=12𝜋^𝜋_−𝜋11−𝑒^{𝑖𝜆𝑑𝑋}11−𝑒^{−𝑖𝜆𝑑𝑍}𝑑𝜆. (3.18)
Suppose that {𝜖_𝑡} is a scalar uncorrelated process with 𝜎²_𝜖≡𝐸{𝜖²_𝑡}. Assuming Gaussianity of {𝐴_𝑡}, it is seen that the right hand of (3.17) is 2𝜋^𝜋_−𝜋12𝜋1−𝑒^{𝑖𝜆2𝑑𝑋}𝜎²_𝜖2𝜋𝑑𝜆, (3.19) which entails lim_𝑇→∞Var√𝑇𝐵_2SLS−𝐵=2𝜋∫^𝜋_−𝜋1/2𝜋1−𝑒^{𝑖𝜆2𝑑𝑋}𝜎²_𝜖/2𝜋𝑑𝜆1/2𝜋∫^𝜋_−𝜋1/1−𝑒^{𝑖𝜆𝑑𝑋}1/1−𝑒^{−𝑖𝜆𝑑𝑍}𝑑𝜆²=𝜎²_𝜖∫2𝜋^𝜋−𝜋1/1−𝑒^{𝑖𝜆2𝑑𝑋}𝑑𝜆∫^𝜋_−𝜋1/1−𝑒^{𝑖𝜆𝑑𝑋}1/1−𝑒^{−𝑖𝜆𝑑𝑍}𝑑𝜆²=𝜎²_𝜖×𝑉_∗𝑑_𝑋,𝑑_𝑍. (3.20)

4. Numerical Studies

In this section, we evaluate the behaviour of 𝐁_2SLS in the case 𝑝=1 in (2.7) numerically.

Example 4.1. Under the condition of Example 3.3, we investigate the asymptotic variance behaviour of 𝐁_2SLS by simulation. Figure 3 plots 𝑉_∗(𝑑_𝑋,𝑑_𝑍) for 0<𝑑_𝑋<1/2 and 0<𝑑_𝑍<1/2.

Figure 3 

𝑉∗(𝑑𝑥,𝑑𝑧) in Section 4.

From Figure 3, we observe that, if 𝑑_𝑍↘0 and if 𝑑_𝑋↗1/2, then 𝑉_∗ becomes large, and otherwise 𝑉_∗ is small. This result implies only in the case that the long-memory behavior of 𝑍_𝑡 is weak and the long-memory behavior of 𝑋_𝑡 is strong, 𝑉_∗ is large. Note that long-memory behaviour of 𝑍_𝑡 makes the asymptotic variance of the 2SLS estimator small, but one of 𝑋_𝑡 makes it large.

Example 4.2. In this example, we consider the following model: 𝑌_𝑡=𝑍_𝑡+𝜖_𝑡,𝑍_𝑡=𝑋_𝑡+𝑢_𝑡,𝜖_𝑡=𝑤_𝑡+𝑢_𝑡, (4.1) where 𝑋_𝑡, 𝑤_𝑡, and 𝑢_𝑡 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 𝑍_𝑡 and 𝜖_𝑡 are correlated, 𝑋_𝑡 and 𝑍_𝑡 are correlated, but 𝑋_𝑡 and 𝜖_𝑡 are independent. Under this model we compare 𝐵_2SLS with the ordinary least squares estimator 𝐵_OLS for 𝐵, which is defined as 𝐵_OLS=^𝑇_𝑡=1𝑍²_𝑡^−1𝑇_𝑡=1𝑍_𝑡𝑌_𝑡. (4.2) The lengths of 𝑋_𝑡, 𝑌_𝑡, and 𝑍_𝑡 are set by 100, and based on 5000 times simulation we report the mean square errors (MSE) of 𝐵_2SLS and 𝐵_OLS. We set 𝑑₁,𝑑₂=0.1,0.2,0.3 in Table 1.
In most cases of 𝑑₁ and 𝑑₂ in Table 1, MSE of 𝐵_2SLS is smaller than that of 𝐵_OLS. Hence, from this Example we can see that our estimator 𝐵_2SLS is better than 𝐵_OLS in the sense of MSE. Furthermore, from Table 1, we can see that MSE of 𝐵_2SLS and 𝐵_OLS increases as 𝑑₂ becomes large; that is, long-memory behavior of 𝑤_𝑡 makes the asymptotic variances of 𝐵_2SLS and 𝐵_OLS large.

Table 1 

MSE of 𝐵_2SLS and 𝐵_OLS.

Example 4.3. In this example, we calculate 𝐁_2SLS based on the actual financial data. We choose S&P500 (square transformed) as 𝐙_𝑡 and the Nikkei stock average as an instrumental variable 𝐗_𝑡. Assuming that 𝐘_𝑡(5×1) consists of the return of IBM, Nike, Amazon, American Expresses and Ford; the 2SLS estimates for 𝐵_𝑖,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 assumption Cov(𝐗_𝑡,𝜺_𝑡)=𝟎.

Table 2 

𝐵_2SLS based on the actual financial data.

From Table 2, 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 Theorem 3.2. First for convenience we define 𝐙_𝑡=(𝑍_1,𝑡,…,𝑍_𝑝,𝑡)^′≡𝜹^′𝐗_𝑡. Let 𝐮_𝑡=(𝑢_1,𝑡,…,𝑢_𝑝,𝑡)^′ be the residual from the OLS estimation of (3.1); that is, 𝑢_𝑖,𝑡=𝑍_𝑖,𝑡−𝑍_𝑖,𝑡. (5.1)

The OLS makes this residual orthogonal to 𝐗_𝑡: ^𝑇_𝑡=1𝐗^′_𝑡𝑢_𝑖,𝑡=𝟎, (5.2) which implies the residual is orthogonal to 𝑍_𝑗,𝑡, ^𝑇_𝑡=1𝑍_𝑗,𝑡𝑢_𝑖,𝑡=^𝑇_𝑡=1𝐗^′_𝑡𝑢_𝑖,𝑡𝜹_𝑗=0, (5.3) where 𝜹_𝑗 is 𝑗th column vector of 𝜹. Hence, we can obtain ^𝑇_𝑡=1𝑍_𝑗,𝑡𝑍_𝑖,𝑡=^𝑇_𝑡=1𝑍_𝑗,𝑡𝑍_𝑖,𝑡+𝑢_𝑖,𝑡=^𝑇_𝑡=1𝑍_𝑗,𝑡𝑍_𝑖,𝑡, (5.4) for all 𝑖 and 𝑗. This means ^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡=^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡. (5.5) So, the 𝑖th column vector of the 2SLS estimator (3.5) 𝜷_2SLS,𝑖 (say) can be represented as 𝜷_2SLS,𝑖=^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡^−1𝑇_𝑡=1𝐙_𝑡𝑌_𝑖,𝑡, (5.6) which leads to 𝜷_2SLS,𝑖−𝜷_𝑖=1𝑇^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡⁻¹1𝑇^𝑇_𝑡=1𝐙_𝑡𝜖_𝑖,𝑡. (5.7) Hence, we can see that √𝑇𝐁_2SLS−𝐁=1𝑇^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡⁻¹1√𝑇^𝑇_𝑡=1𝐙_𝑡𝝐^′_𝑡. (5.8)

Note that, by the ergodic theorem (e.g., Stout [15] p179–181), 1𝑇^𝑇_𝑡=1𝐙_𝑡𝐙^′_𝑡=1𝑇𝜹^′𝑇_𝑡=1𝐗_𝑡𝐙^′_𝑡=1𝑇^𝑇_𝑡=1𝐙_𝑡𝐗^′_𝑡1𝑇^𝑇_𝑡=1𝐗_𝑡𝐗^′_𝑡⁻¹1𝑇^𝑇_𝑡=1𝐗_𝑡𝐙^′_𝑡𝑃⟶𝐐. (5.9)

Furthermore, the second term of the right side of (5.8) can be represented as 1√𝑇^𝑇_𝑡=1𝐙_𝑡𝝐^′_𝑡=𝜹^′1√𝑇^𝑇_𝑡=1𝐗_𝑡𝝐^′_𝑡, (5.10) and by the ergodic theorem (e.g., Stout [15] p179–181), we can see 𝜹^′=^𝑇_𝑡=1𝐙_𝑡𝐗^′_𝑡^𝑇_𝑡=1𝐗_𝑡𝐗^′_𝑡^𝑃−1⟶𝐸𝐙_𝑡𝐗^′_𝑡𝐸𝐗_𝑡𝐗^′_𝑡⁻¹. (5.11)

Proof of (i). From the above, 𝐁_2SLS−𝐁=𝑂_𝑃1𝑇^𝑇_𝑡=1𝐗_𝑡𝝐^′_𝑡. (5.12)
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 Theorem 3.2 of Hosoya [14], if Assumption 3.1 holds, it follows that (1/√𝑇)∑^𝑇_𝑡=1𝐗_𝑡𝝐−→^′_𝑡^𝑑𝐔. Hence, Theorem 3.2 is proved.

Acknowledgments

The author would like to thank the Editor and the referees for their comments, which improved the original version of this paper.