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The asymptotic distribution of canonical correlations and vectors in higherorder cointegrated models

Contributed by T. W. Anderson
Abstract
The study of the largesample distribution of the canonical correlations and variates in cointegrated models is extended from the firstorder autoregression model to autoregression of any (finite) order. The cointegrated process considered here is nonstationary in some dimensions and stationary in some other directions, but the first difference (the “errorcorrection form”) is stationary. The asymptotic distribution of the canonical correlations between the first differences and the predictor variables as well as the corresponding canonical variables is obtained under the assumption that the process is Gaussian. The method of analysis is similar to that used for the firstorder process.
Cointegrated stochastic processes are used in econometrics for modeling macroeconomic time series that have both stationary and nonstationary properties. The term “cointegrated” means that in a multivariate process that appears nonstationary some linear functions are stationary. Many economic time series may show inflationary tendencies or increasing volatility, but certain relationships are not affected by these tendencies. Statistical inference is involved in identifying these relationships and estimating their importance.
The family of stochastic processes studied in this paper consists of vector autoregressive processes of finite order. A vector of contemporary measures is considered to depend linearly on earlier values of these measures plus random disturbances or errors. The dependence may be evaluated by the canonical correlations between the contemporary values and the earlier values.
The nonstationarity of a process may be eliminated by treating differences or higherorder differences (over time) of the vectors. This paper treats processes in which firstorder differencing accomplishes stationarity. The firstorder difference is represented as a linear combination of the first lagged variable and lags of the difference variable. The stationary linear combinations are the canonical variables corresponding to the nonzero process canonical correlations between the difference variable and the first lagged variable not accounted for by the lagged differences. The number of these is defined as the degree of cointegration.
Statistical inference of the model is based on a sample of observations; that is, a vector time series over some period of time. The estimator of the parameters of the original autoregressive model is a transformation of the estimator of the (stationary) errorcorrection form. In the latter, one coefficient matrix is of lower rank (the degree of cointegration). It is estimated efficiently by the reduced rank regression estimator introduced by me (1). It depends on the larger canonical correlations and corresponding canonical vectors. The smaller correlations are used to determine the rank of this matrix. Inference is based on the largesample distribution of these correlations and variables.
The asymptotic distribution of the canonical correlations and coefficients of the variates for the firstorder autoregressive process was derived by me (2). The distribution for the higherorder process (that is, several lags) is obtained in this paper, using similar algebra. Hansen and Johansen (3) have independently obtained the asymptotic distribution of the canonical correlations, but by a different method and expressed in a different form.
The likelihood ratio test for the degree of cointegration that I found (1) is given in Asymptotic Distribution of the Smaller Roots; its asymptotic distribution under the null hypothesis was found by Johansen (4). To evaluate the power of such a test, one needs to know the distribution or asymptotic distribution of the sample canonical correlations corresponding to process canonical correlations different from 0. See ref. 5, for example.
For further background, the reader is referred to Johansen (6) and Reinsel and Velu (7).
The Model
The general cointegrated model is an autoregressive process {Y_{t}} of order m defined by 1 where Z_{t} is unobserved with ℰZ_{t} = 0, ℰZ_{t}Z′_{t} = Σ_{ZZ}, and ℰY_{t−i}Z′_{t} = 0, i = 1, … . Let B(λ) = λ^{m}I − λ^{m−1}B_{1} − … − B_{m}. If the roots λ_{1}, … , λ_{pn} of B(λ) = 0 satisfy λ_{i} < 1, a stationary process {Y_{t}} can be defined by 1. If some of the roots are 1, the process will be nonstationary. In this paper, we assume that n (0 < n < p) roots of B(λ) = 0 are 1(λ_{1} = … = λ_{p} = 1), and the other pm − n roots satisfy λ_{i} < 1, i = n + 1, … , pm. The first difference of the process, the “errorcorrection” form, is 2 Here Π = B_{1} + … + B_{m} − I = −B(1), Π_{j} = −(B_{j+1} + … + B_{m}), j = 1, … , m − 1, Π̄ = (Π_{1}, … , Π_{m−1}), and Δ̄Y_{t−1} = (ΔY′_{t−1}, … , ΔY′_{t−m+1})′.
A sample consists of T observations: Y_{1}, … , Y_{T}. Because the rank of Π is k, it is to be estimated by the reduced rank regression estimator introduced by me (1) as the maximum likelihood estimator when Z_{1}, … , Z_{T} are normally distributed and Y_{0}, Y_{−1}, … , Y_{−m+1} are nonstochastic and known. The matrices Π_{1}, … , Π_{m−1} are unrestricted except for the condition λ_{i} < 1, i = n + 1, … , pm. The estimator depends on the canonical correlations and vectors of ΔY_{t} and Y_{t−1} conditioned on ΔY_{t−1}, … , ΔY_{t−m+1}.
Define where S_{Δ̄Y,Δ̄Y} = T^{−1} ∑Δ̄Y_{t−1}Δ̄Y′_{t−1}, S_{ΔY,Δ̄Y} = T^{−1} ∑ ΔY_{t}Δ̄Y′_{t−1}, and S_{Ȳ,Δ̄Y} = T^{−1} ∑ Y_{t−1}Δ̄Y′_{t−1}. The vectors ΔŶ and Ŷ are the sample residuals of ΔY_{t−1} and Y_{t−1} regressed on Δ̄Y_{t−1}. Define Ŝ = T^{−1} ∑ ΔŶΔŶ = S_{ΔY,ΔY} − S_{ΔY,Δ̄Y}SS_{Δ̄Y,ΔY}, Ŝ = T^{−1} ∑ ΔŶŶ = S_{ΔY,Ȳ} − S_{ΔY,Δ̄Y}SS_{Δ̄Y}, and Ŝ = T^{−1} ∑ ŶŶ = S_{ȲȲ} − S_{Ȳ,Δ̄Y}SS_{Δ̄Y,Ȳ}, where S_{ΔY,ΔY} = T^{−1} ∑ ΔY_{t}ΔY′_{t}, S_{ΔY,Ȳ} = T^{−1} ∑ ΔY_{t}Y′_{t−1}, and S_{ȲȲ} = T^{−1} ∑ Y_{t−1}Y′_{t−1}. The sample canonical correlations between ΔŶ and Ŷ and variates are defined by 3 4 More information on canonical analysis is covered in chapter 12 of ref. 8. One form of the reduced rank regression estimator is Π̂_{(k)} = ŜΓ̂_{2}Γ̂′_{2}, where Γ̂_{2} = (γ̂_{n+1}, … , γ̂_{p}) and r < … < r.
We shall assume that there are exactly n linearly independent solutions to ω′B(1) = 0; that is, ω′Π = 0. Then the rank of Π is p − n = k and there exists a p × n matrix Ω_{1} of rank n such that Ω′_{1}Π = 0. See Anderson (9). There is also a p × k matrix Ω_{2} of rank k such that Ω′_{2}Π = Υ_{2}Ω′_{2}, where Υ_{2} (k × k) is nonsingular, and Ω = (Ω_{1}, Ω_{2}) is nonsingular.
To distinguish between the stationary and nonstationary coordinates, we make a transformation of coordinates. Define Ψ_{j} = Ω′B_{j}(Ω′)^{−1}, j = 1, … , m. Then the process 1 is transformed to 5 If we define Υ = Ψ_{1} + … + Ψ_{m} − I = Ω′Π(Ω′)^{−1}, Υ_{j} = −∑ Ψ_{j} = Ω′Π_{j}(Ω′)^{−1}, Ῡ = (Υ_{1}, … , Υ_{m−1}), and Δ̄X_{t−1} = (ΔX′_{t−1}, … , ΔX′_{t−m+1})′, the form 2 is transformed to 6 Note that Υ = diag(0, Υ^{22}).
Define ΔX̂, X̂, S_{Δ̄X,Δ̄X}, S_{ΔX,Δ̄X}, S_{X̄,Δ̄X}, Ŝ, Ŝ, and Ŝ in a manner analogous to the definitions in the Ycoordinates. The reduced rank regression estimator of Υ is based on the canonical correlations and canonical variates between ΔX̂ and X̂ defined by 7 8 The estimator of Υ of rank k is Υ̂_{(k)} = ŜG_{2}G′_{2}, where G_{2} = (g_{n+1}, … , g_{p}) and g_{i} is the solution for g in 8 when r = r_{i}, the solution to 7 and r_{1} < … < r_{p}. The rest of this paper is devoted to finding the asymptotic distribution of {g_{i}, r_{i}}. Note that Υ̂_{(k)} = Ω′Π̂_{(k)}(Ω′)^{−1}.
The vectors ΔX̂ = ΔX_{t} − S_{ΔX,Δ̄X}SΔ̄X_{t−1} and X̂ = X_{t−1} − S_{X̄,Δ̄X}SΔ̄X_{t−1} are the residuals of ΔX_{t} and X_{t−1} regressed on Δ̄X_{t−1}, and r_{1} is the maximum correlation between ΔX̂ and X̂, which is the correlation between ΔX_{t} and X_{t−1} after taking account of the dependence “explained” by Δ̄X_{t−1}. The canonical correlations are the canonical correlations between (ΔX′_{t}, Δ̄X_{t−1}) and (X_{t−1}, Δ̄X_{t−1}) other than ±1.
The Process
The process {X_{t}} defined by 5 can be put in the form of the Markov model 9 (section 5.4, ref. 10). Multiplication of 9 on the left by yields a form that includes the errorcorrection form 6 10 The first n components of 10 constitute 11 Here Υ_{j} has been partitioned into n and k rows and columns. Assume X_{10} = X_{1,−1} = … = 0 and W_{10} = W_{1,−1} = … = 0. The sum of 11 for t = −∞ to t = s is X_{1s} = ∑ [ΥX_{1,s−j} + ΥX_{2,s−j}] + ∑ W_{1t}, or 12 Write 12 as 13 where Γ = (I − ∑ Υ)^{−1}, Γ^{−1}H is a linear combination of Υ_{1}, … , Υ_{m−1}, and X̃_{s} = (X′_{2s}, Δ̄X′_{s})′. [The matrix on the lefthand side of 12 is nonsingular because otherwise there would be a linear combination of the righthand side identically 0.] The righthand side of 13 is the sum of a stationary process and a random walk (∑ W_{1t}).
The last pm − n = k + p(m − 1) components of 10 constitute a stationary process satisfying 14 where X̃′_{t} = (X′_{2t}, Δ̄X′_{t}), W̃′_{t} = (W′_{2t}, W′_{t}, 0), and Υ̃ consists of the last pm − n rows and columns of the coefficient matrix in 10. Note that the first n columns and last pm − n rows of that matrix consist of 0s. Because the eigenvalues of Υ̃ are less than 1 in absolute value (9), X̃_{t} = ∑ Υ̃^{s}W̃_{t−s}, ℰX̃_{t}X̃′_{t} = Σ̃ = ∑ Υ̃^{s}Σ̃_{WW}Υ̃′^{s}, ℰX̃_{t}X̃′_{t−h} = Υ̃^{h}Σ̃. The covariance Σ̃ satisfies 15 Given Υ̃ and Σ̃_{WW}, 15 can be solved for Σ̃ [Anderson (10), section 5.5]. Further we write 13 as X_{1t} = Γ ∑ W_{1,t−s} + H ∑ Υ̃^{s}W̃_{t−s}. Then since I − Υ̃^{t} → I. Here ℰW_{1t}W̃′_{t} = Σ is the second set of rows in Σ̃_{WW}. Then T^{−1}ℰS = T^{−2} ∑ ℰX_{1t}X′_{1t} → 2^{−1}ΓΣΓ′ because ∑ t = T(T + 1)/2. Further Define 16 where Σ_{ΔX,Δ̄X} = ℰΔX_{t}Δ̄X′_{t−1}, Σ_{X̄,Δ̄X} = ℰX_{t−1}Δ̄X′_{t−1} depends on t, and Σ_{Δ̄X,Δ̄X} = ℰΔ̄X_{t−1}Δ̄X′_{t−1} does not depend on t. Note that ΔX and X correspond to ΔX̂ and X̂ with S_{ΔX,Δ̄X}, S_{X̄,Δ̄X} and S_{Δ̄X,Δ̄X} replaced by Σ_{ΔX,Δ̄X}, Σ_{X̄,Δ̄X} and Σ_{Δ̄X,Δ̄X}, respectively. Then 6 can be written as a regression model 17 with ℰXW′_{t} = 0. Note that this model has the form of 2.10 in Anderson (2).
From 16 and 17 we calculate The process analogs of 7 and 8 are 18 19 These define the process canonical correlations and variates in the Xcoordinates.
Sample Statistics
The canonical correlations and vectors depend on Ŝ, Ŝ, and Ŝ, which in turn depend on the submatrices of S_{X̄X̄}, S_{X̄,Δ̄X}, and S_{Δ̄X,Δ̄X} (equivalently S, S̃, S̃). The vector X̃_{t} satisfies the firstorder stationary autoregressive model 14. The sample covariance matrices S̃_{XX}, S̃_{WX}, and S_{WW} are consistent estimators of Σ̃, 0, and Σ_{WW}, and S̃ = (S̃_{XX} − Σ̃), S̃ = S̃_{WX}, S = (S_{WW} − Σ_{WW}) have a limiting normal distribution with means 0 and covariances that have been given in refs. 2 and 11.
Let W(u) be the Brownian motion process defined by T^{−}^{} ∑ W_{t} →^{w}W(u). Define I_{11} by See Anderson (2) and theorem B.12 of Johansen (6). Define J_{j1} by Then T^{−1}S →^{d}ΓI_{11}Γ′ by 13, T^{−1}S̃_{XX} →^{p}0, and the Cauchy–Schwarz inequality.
We shall find the limit in distribution of S from the limit of S by using equation B.20 of theorem B.13 of Johansen (6). A specialization to the model here is where W̃(u) = [W′_{2}(u), W′(u), 0]′. [In theorem B.13, let θ_{i} = (I, 0), ψ_{i} = (0, Υ̃^{i}), ɛ′_{t} = (W′_{1t}, W̃′_{t}), and Ω = ℰɛ_{t}ɛ′_{t}, V_{t} = X̃_{t}.] Then Because {X̃_{t}} is stationary, T^{−1} ∑ W_{t}X̃′_{t−1} →^{p}0 and 20 Now we wish to show that ΔX and X lead to the same asymptotic results as ΔX̂ and X̂. First note that T^{−1}S →^{d}ΓI_{11}Γ′ and T^{−1} times any other sample covariance converges in probability to 0. Hence T^{−1}S →^{d}ΓI_{11}Γ′ and T^{−1}Ŝ →^{d}ΓI_{11}Γ′. Because {X̃_{t}} is stationary, {X} is stationary, and S →^{p}Σ, Ŝ →^{p}Σ. Moreover S̃ = (S̃_{X̄X̄} − Σ̃) has a limiting normal distribution. Expansion of S and Ŝ in terms of the submatrices of S̃ shows that S and Ŝ have the same limiting normal distribution. (See Asymptotic Distribution of the Larger Roots.) Finally, T^{−}^{}S →^{p}0 and T^{−}^{}Ŝ →^{p}0 because S, S, S_{Δ̄X,Δ̄X}, and S and hence S have finite limits in distribution.
From 17 we find that plim_{T→∞}S = plim_{T→∞}Ŝ = Σ and where S = S_{WX̄} − S_{W,Δ̄X}ΣΣ_{Δ̄X,X̄}, which converges in distribution to the righthand side of 20.
As noted above, S →^{d}S, which consists of the first k rows of the weak limit of S. Then
Asymptotic Distribution of the Smaller Roots
Let Q^{+} = S S S. The n smaller roots of Q^{+} − r^{2}S = 0 converge in probability to 0 and the k larger roots converge to the roots of by the analysis of ref. 2. Let d = Tr^{2}. The n smaller roots of Q − T^{−1} dS = 0 converge in distribution to the roots of by the algebra used in ref. 2, section 5, resulting in the same distribution as in ref. 2. The likelihood ratio criterion for testing that the rank of Υ is k (2) is −2 log λ = −T ∑(1 − r) = ∑ d_{i} + o_{p}(1), the limiting distribution of which was found by Johansen (4, 6) and was given in equation 5.4 in ref. 2.
Asymptotic Distribution of the Larger Roots
We now turn to deriving the asymptotic distribution of the k larger roots of Q^{+} − r^{2}S = 0 and the associated vectors solving Q^{+}g = r^{2}Sg. First we show that the asymptotic distribution of r, … , r is the same as the asymptotic distribution of the zeros of Q − r^{2}S. Then we transform from the Xcoordinates to the coordinates of the process canonical correlations and vectors.
Let R̂ = diag(r, … , r) and G′_{2} = (G′_{12}, G′_{22})′ consist of the corresponding solutions to Q^{+}G_{2} = SG_{2}R̂. The normalization of the columns of G_{2} is G′_{2}SG_{2} = I, that is, 21 The probability limit of 21 shows that G_{12} = O_{p}(1) and G_{22} = O_{p}(1). The submatrix equations in Q^{+}G_{2} = SG_{2}R̂ can be written as 22 23 Because T^{−1}Q →^{p}0, T^{−}^{}Q →^{p}0, T^{−}^{}S →^{p}0, T^{−1}S →^{d}ΓI_{11}Γ′ and R̂ →^{p}R = diag(ρ, … , ρ), the probability limit of the lefthand side of 22 is 0; this shows that G_{12} →^{p}0. Then the asymptotic distribution of G_{22} is the asymptotic distribution of G_{22} defined by 24 where the elements of R̂ are defined by Q − r^{2}S = 0. Note that when G_{12} →^{p}0 is combined with 23, we obtain QG_{22} = SG_{22}R̂ + o_{p} (T^{−}^{}).
We proceed to find the asymptotic distribution of G_{22} and R̂ defined by 24 in the manner of ref. 2. Let where W_{2⋅1,t} = W_{2t} − Σ(Σ)^{−1}W_{1t} and ℰW_{2⋅1,t}W′_{2⋅1,t} = Σ = Σ − Σ (Σ)^{−1}Σ. We expand {Q − [Σ(Σ)^{−1}Σ]_{22}} to obtain 25 where Λ = Υ^{22}ΣΥ^{22′} + Σ. See equation 6.5 of ref. 2.
To express the covariances of the sample matrices, we use the “vec” notation. For A = (a_{1}, … a_{n}), we define vec A = (a′_{1}, … , a′_{n})′. The Kronecker product of two matrices A = (a_{ij}) and B is A ⊗ B = (a_{ij}B). A basic relation is vec ABC = (C′ ⊗ A) vec B, which implies vec xy′ = vec x1y′ = (y ⊗ x) vec 1 = y ⊗ x. Define the commutator matrix K as the (square) permutation matrix such that vec A′ = K vec A for every square matrix of the same order as K.
Define C = (I, −Σ_{2,Δ̄X}Σ) and D = [I, −Σ(Σ)^{−1}, 0]. Then X = CX̃_{t}, W_{2⋅1,t} = DW̃_{t}, DΣ̃_{WW} = ΣJ′_{(k)}, CΣ̃ = ΣI′_{(k)}, J′_{(k)} = (I, 0, I, 0), I′_{(k)} = (I, 0), Σ = CΣC′, and Σ = DΣ̃_{WW}D′.
Theorem 1. If the W_{t}are independently normally distributed, S, S, and Shave a limiting normal distribution with means 0, 0, and 0 and covariances262728293031Lemma 1. If X is normally distributed with ℰX = 0 and ℰXX′ = Σ, then ℰ vec XX′(vec XX′)′ = (I + K)(Σ ⊗ Σ) + vec Σ(vec Σ)′. If X and Y are independent, ℰ vec XX′(vec YY′)′ = vec ℰXX′ ⊗ (vec ℰ′YY′)′, ℰ vec XY′(vec XY′)′ = ℰYY′ ⊗ ℰXX′, and ℰ vec XY′(vec(YX′)′ = KℰXX′ ⊗ ℰYY′.
Proof of Theorem 1: First 26 is equivalent to the first expression in Lemma 1. Next vec S = T^{−}^{} ∑(X ⊗ W_{2⋅1,t}) implies 27 because X and W_{2⋅1,s} are independent for t − 1 ≤ s. Similarly 28 follows. To prove 29, 30, and 31, we use the following lemma.
Lemma 2. Proof of Lemma 2: We have from X̃_{t} = Υ̃X̃_{t−1} + W̃_{t}32 Because S̃_{XX} − S̃_{X̄X̄} = (1/T)(X̃_{T}X̃′_{T} − X̃_{0}X̃′_{0}) and {X̃_{t}} is a stationary process, S̃_{X̄X̄} in 32 can be replaced by S̃_{XX} + o_{p}(1). Then Lemma 2 results from 32 and vec Υ̃S̃_{X̄W} = K vec S̃_{WX̄}Υ̃′.▪
Lemma 3. Proof of Lemma 3: Write W_{2t} = W_{2⋅1,t} + Σ(Σ)^{−1}W_{1t}. Then from which the lemma follows.▪
Proof of Theorem 1 Continued: Then 29 follows from Lemma 2, 26, and 28, and 30 follows from Lemma 2, 27, and 28 for X̃_{t}. To prove 31, use Lemma 2, 26, 27, and 10 to obtain 33 Then substitution of Σ̃_{WW} = Σ̃ − Υ̃Σ̃Υ̃′ in 33 yields 31.▪
Let Ξ be a k × k matrix such that Ξ′(Υ^{22}ΣΥ^{22′})Ξ = Θ and Ξ′ΣΞ = I, where Θ = diag(θ_{n+1}, … , θ_{p}) = R(I − R)^{−1}, R = diag(ρ, … , ρ), and ρ is a root of 18 with 0 < ρ < … < ρ. Let U = Ξ′X, V_{2t} = Ξ′W_{2t}, V_{1t} = W_{1t}, Δ_{2} = Ξ′(Υ^{22} + I)(Ξ′)^{−1}, M_{2} = Ξ′Υ^{22}(Ξ′)^{−1}, Ξ̃ = diag[Ξ, I_{m−1} ⊗ diag(I_{n}, Ξ)], Δ̃ = Ξ̃′Υ̃(Ξ̃′)^{−1}, Ũ_{t} = Ξ̃′X̃_{t}, C_{U} = Ξ′C(Ξ̃′)^{−1}. Then {Ũ_{t}} is generated by Ũ_{t} = Δ̃Ũ_{t−1} + Ṽ_{t}, where Ṽ_{t} = Ξ̃′W̃_{t} and U satisfies U = Δ_{2}U + V_{2t}, ΔU = M_{2}U + V_{2t}. Multiplication of 25 on the left by Ξ′ and right by Ξ yields 34 Theorem 2. If the V_{t} are independently normally distributed, S, S, and S have a limiting normal distribution with means 0, 0, and 0 and covariances 35 Let L_{2,t−1} = M_{2}U_{2,t−1}(= Ξ′Υ^{22}X_{2,t−1}). Then 34 becomes The covariances of the limiting normal distribution of vec S, vec S = (M_{2} ⊗ I) vec S, and vec S = (M_{2} ⊗ M_{2}) vec S are found from Theorem 2. We write the transform of 35 as 36 where 37 Let H_{22} = (M′_{2})^{−1}Ξ^{−1}G_{22} [= Ξ^{−1}(Υ^{22}′)^{−1}G_{22}]. Then QG_{22} = SG_{22}R̂ and G′_{22}SG_{22} = I transform to 38 Because (SSS)_{22} →^{p}ΘR and S →^{p}Θ, the probability limits of 38 and h_{ii} > 0 imply H_{22} →^{p}Θ^{−}^{}.
Define H = (H_{22} − Θ^{−}^{}) and R̂ = (R̂ − R). Then we can write 38 as 39 40 where Lemma 4. 41 Lemma 5. 42 Proof of Lemma 5: We use the facts that M_{2} = Δ_{2} − I, J_{(k)}M_{2} = Δ̃I_{(k)} − I_{(k)} = (Δ̃ − I)I_{(k)}, and (I + K)K = I + K. Then the lefthand side of 42 is which is the righthand side of 42.▪
Theorem 3. If Z_{t}are normally distributed and the roots of 18 are distinct,43Proof: Theorem 4 follows from Theorem 2, 37, 41, 42, and the transpose of 42 and the fact that K(R ⊗ R) = [(I − R) ⊗ R]K(Θ ⊗ Θ)[(I − R) ⊗ R].▪
Note that 43 is equation 6.14 of ref. 2 with Φ^{+} replacing Φ.
Let Ẽ = ∑ ɛ_{i}(ɛ′_{i} ⊗ ɛ′_{i}), where ɛ_{i} is the kvector with 1 in the ith position and 0s elsewhere. The matrix Ẽ has 1 in the ith row and i,ith column and 0s elsewhere. Define r^{2∗} = (r, … , r)′. Then The matrix Ẽ has the effect of selecting the i,ith element of Θ^{−}^{}PΘ^{−}^{} and placing it in the ith position of r^{2∗}.
Theorem 4. If the Z_{t}vectors are independently normally distributed and the roots of 18 are distinct, the limiting distribution of r^{2∗}is normal with mean 0 and covariance matrix44 In terms of the components of r^{2∗} the asymptotic covariance of r and r is 2[(1 − ρ)^{2}φρ + ρ(φ(1 − ρ)^{2}]. Here φ denotes the element in the ith row of the ith block of rows and the jth column of the jth block of columns in Φ^{+}.
We now derive the limiting distribution of H = H + H, where H = diag(h, … , h). From vec HR = (R ⊗ I) vec H, and vec RH = (I ⊗ R) vec H we obtain vec(HR − RH) = NH = NH, where The Moore–Penrose generalized inverse of N (denoted N^{+}) has a 0 where N has a 0 and has (ρ − ρ)^{−1} where N has (ρ − ρ), i ≠ j. Note that NN^{+} = (I ⊗ I) − E, where E = ∑ (ɛ_{i} ⊗ ɛ_{i})(ɛ′_{i} ⊗ ɛ′_{i}). The k^{2} × k^{2} matrix E is idempotent of rank k; the k^{2} × k^{2} matrix NN^{+} is idempotent of rank k^{2} − k; and E is orthogonal to N and N^{+}.
From 39 we obtain vec H = N^{+}(Θ^{−}^{} ⊗ Θ^{−1})vec P. From 40 we find H = −½Θ^{−}^{} diagS and vec H = −½ EΘ^{−}^{} vec S.
Theorem 5. If the Z_{t}vectors are independently normally distributed and the roots of 18 are distinct, vec Hand vec Hhave a limiting normal distribution with means 0 and 0 and covariancesandrespectively.
From G_{22} = Υ^{22}′ΞH_{22} we can transform Theorem 5 into the asymptotic covariances of vec G_{22} = (I ⊗ Υ^{22}′Ξ) vec H_{22}.
 Accepted February 6, 2001.
 Copyright © 2001, The National Academy of Sciences
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