Conclusion

We have proposed a spatial generalized autoregressive conditional heteroskedas-ticity (S-GARCH) model as extension of a spatial autoregressive conditional het-eroskedasticity (S-ARCH) model by Sato and Matsuda (2017). By re-expressing S-GARCH as spatial autoregressive moving average (SARMA) models, we em-ploy spatial econometrics methodology to estimate the parameters by the two step procedure, and establish rigorous asymptotic results. Applications to land price data in Tokyo demonstrate that S-GARCH models detect several interest-ing features of spatial volatilities caused by the Great East Japan Earthquake in 2011.

Finally let us introduce possible extensions S-GARCH models. We employed the first-order contiguity relations to construct a spatial weight matrix, which is the simplest choice. It is desired to check what kind of spatial weight matrix can improve the fitting of S-GARCH models. Spatio-temporal extension of the S-GARCH models are surely our next target that can provide much better ways for land price data analysis than the year by year fitting of S-GARCH models in this paper.

Table3.1:Theempiricalmeansandrootmeansquarederrors(RMSE)oftheestimators. normalchi(3)lognormal n=100n=400n=100n=400n=100n=400 ϕBiasRMSEBiasRMSEBiasRMSEBiasRMSEBiasRMSEBiasRMSE 0.90.0290.0820.0070.0290.0320.0780.0090.0300.0310.0800.0090.029 0.05-0.0390.069-0.0090.026-0.0400.066-0.0100.027-0.0380.068-0.0110.026 0.50.0390.3780.0060.1050.0150.310-0.0030.100-0.0370.310-0.0180.101 1.00.0210.1880.0090.0890.0230.1760.0040.0820.0200.1730.0040.077 0.45-0.0600.238-0.0150.098-0.0650.243-0.0150.103-0.0530.224-0.0160.097 0.45-0.0010.1550.0020.0720.0020.1590.0020.0750.0020.1510.0030.073 0.5-0.0140.292-0.0020.092-0.0540.313-0.0070.113-0.2770.595-0.0860.255 1.00.0340.2320.0120.1130.0440.2290.0180.1090.0410.2160.0070.103 0.05-0.0270.139-0.0110.080-0.0230.141-0.0110.079-0.0290.132-0.0130.079 0.9-0.0110.1080.0020.069-0.0170.1150.0020.068-0.0060.1080.0030.068 0.5-0.4310.829-0.1000.240-0.6271.089-0.1140.295-1.0091.660-0.3130.630 1.00.0130.2360.0070.1140.0220.2280.0060.1090.0070.2150.0040.105 Note:ϕ=(λ,ρ,α,β)′

Table 3.2: Estimated values and standard errors ofλ,ρ, αand β in S-ARCH and S-GARCH models, which are applied year by year to the residuals by fitting SAR models to land priced data.

S-ARCH S-GARCH

2010 2011 2012 2013 2014 2010 2011 2012 2013 2014

ˆλ 0.772 0.845 0.874 0.893 0.601

se(λ) 0.206 0.139 0.128 0.100 0.415

ˆ

ρ 0.240 0.244 0.274 0.279 0.184 0.110 0.076 0.059 0.060 0.104

se(ρ) 0.083 0.081 0.082 0.083 0.084 0.077 0.055 0.048 0.045 0.086 ˆ

α 0.569 -0.518 -0.606 -0.193 -0.804 0.162 -0.121 -0.130 -0.021 -0.412 βˆ -0.022 0.212 0.232 0.109 0.225 -0.001 0.052 0.049 0.025 0.120 AIC 1538.7 1481.7 1549.8 1573.8 1537.7 1536.6 1475.3 1547.9 1570.4 1537.9

Figure 3.1: The identified volatilities in 2010 and 2011. The great earth quake occurred in 2011.

Figure 3.2: A comparison between identified volatilities by ARCH and S-GARCH models.

A. Hessian, average Hessian and symmetric ma-trix Ω

_ψ,n

The Hessian matrixH_n(ψ)≡ _∂ψ∂ψ^∂2 ′ logL_n(ψ) has the elements:

Hββ^′ = −1

σ²X_n^′R_n^′−1(λ)R⁻_n¹(λ)Xn, Hβσ² = −1

σ⁴X_n^′R_n^′−1(λ)V(θ), Hβρ = −1

σ²X_n^′R_n^′−1(λ)R⁻_n¹(λ)WnYn, Hβλ = 1

σ²X_n^′R^′−_n ¹(λ)(W_n^′R^′−_n ¹(λ)Vn(θ) +R⁻_n¹(λ)WnVn(θ)−R⁻_n¹(λ)WnYn), H_σ2σ² = n

2σ⁴ −V_n^′(θ)Vn(θ) σ⁶ , H_σ2ρ = −1

σ⁴Y_n^′W_n^′R^′−_n ¹(λ)V(θ), H_σ2λ = 1

σ⁴(V_n^′(θ)−Y_n^′)W_n^′R^′−_n ¹(λ)Vn(θ), Hρρ = −1

σ²Y_n^′W_n^′R^′−_n ¹(λ)R⁻_n¹(λ)WnYn−tr(S⁻_n¹(θ)WnS_n⁻¹(θ)Wn), Hρλ = 1

σ²Y_n^′W_n^′R^′−_n ¹(λ)(W_n^′R^′−_n ¹(λ)V(θ) +R⁻_n¹(λ)WnVn(θ)−R⁻_n¹(λ)WnYn)

−tr(S_n⁻¹(θ)W_nS_n⁻¹(θ)W_n), Hλλ = 1

σ²(Y_n^′−V_n^′(θ))W_n^′R^′−_n ¹(λ)(2W_n^′R^′−_n ¹(λ)Vn(θ) +R_n⁻¹(λ)WnVn(θ)−R⁻_n¹(λ)WnYn) +tr(R⁻_n¹(λ)WnR⁻_n¹(λ)Wn)−tr(S_n⁻¹(θ)WnS_n⁻¹(θ)Wn).

The average Hessian matrix Σψ,n ≡ −E(₁

n

∂²

∂ψψ′logLn(ψ0))

has the

ele-ments:

Σββ^′ = 1

nσ₀²X_n^′R^′−_n ¹R⁻_n¹Xn, Σ_βσ2 = 0,

Σβρ = 1

nσ₀²X_n^′R^′−_n ¹R⁻_n¹WnS_n⁻¹Xnβ0, Σβλ = 1

nσ₀²X_n^′R^′−_n ¹R⁻_n¹WnS_n⁻¹Xnβ0, Σσ²σ² = 1

2σ⁴₀, Σσ²ρ = 1

nσ₀²tr(WnS_n⁻¹), Σσ²λ = 1

nσ₀²tr(WnS_n⁻¹−WnR⁻_n¹), Σρρ = 1

nσ₀²β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹R⁻_n¹WnS⁻_n¹Xnβ0+ 1

ntr(R^′_nS^′−_n ¹W_n^′R_n^′−1R_n⁻¹WnS_n⁻¹Rn+S_n⁻¹WnS_n⁻¹Wn), Σ_ρλ = 1

nσ₀²β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS⁻_n¹X_nβ₀+ 1

ntr(R^′_nS^′−_n ¹W_n^′R_n^′−1R_n⁻¹W_nS_n⁻¹R_n+S_n⁻¹W_nS_n⁻¹W_n)

−1

ntr(R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_n+S_n⁻¹W_nR_n⁻¹W_n), Σ_λλ = 1

nσ₀²β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS⁻_n¹X_nβ₀+ 1

ntr(R^′_nS^′−_n ¹W_n^′R_n^′−1R_n⁻¹W_nS_n⁻¹R_n+S_n⁻¹W_nS_n⁻¹W_n)

−2

ntr(R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_n+S_n⁻¹W_nR_n⁻¹W_n) +1

ntr(R⁻_n¹W_nR⁻_n¹W_n+W_n^′R^′−_n ¹R⁻_n¹W_n).

The symmetric matrix Ωψ,n has the elemetns:

Ωββ^′ = 0, Ω_βσ2 = µ₃

2nσ⁶₀X_n^′R^′−_n ¹1_n, Ω_βρ = µ₃

nσ₀⁴

∑n i

{(R⁻_n¹X_n)_i}^′(R⁻_n¹W_nS_n⁻¹R_n)_ii,

Ωβλ = µ3

nσ₀⁴

∑n i

{(R⁻_n¹Xn)i}^′(R⁻_n¹WnS_n⁻¹Rn−R⁻_n¹Wn)ii,

Ωσ²σ² = µ₄−3σ⁴₀ 4σ⁸₀ , Ω_σ2ρ = µ3

2nσ⁶₀β₀^′X_n^′S^′−_n ¹W_n^′R^′−_n¹1n+µ4−3σ⁴₀

2nσ₀⁶ tr(S_n⁻¹Wn), Ω_σ2λ = µ₃

2nσ⁶₀β₀^′X_n^′S^′−_n ¹W_n^′R^′−_n¹1_n+µ₄−3σ⁴₀

2nσ₀⁶ tr(S_n⁻¹W_n−R⁻_n¹W_n), Ω_ρρ = 2µ₃

nσ₀⁴

∑n i=1

(R⁻_n¹W_nS_n⁻¹X_nβ₀)_i(R⁻_n¹W_nS_n⁻¹R_n)_ii+µ₄−3σ⁴₀ nσ₀⁴

∑n i=1

{(R_n⁻¹W_nS_n⁻¹R_n)_ii}²,

Ω_ρλ = µ3

nσ₀⁴

∑n i=1

(R⁻_n¹W_nS_n⁻¹X_nβ₀)_i(2R⁻_n¹W_nS_n⁻¹R_n−R⁻_n¹W_n)_ii

+µ4−3σ₀⁴ nσ₀⁴

∑n i=1

(R⁻_n¹WnS_n⁻¹Rn)ii(R⁻_n¹WnS_n⁻¹Rn−R⁻_n¹Wn)ii,

Ωλλ = 2µ3

nσ₀⁴

∑n i=1

(R⁻_n¹WnS_n⁻¹Xnβ0)i(R⁻_n¹WnS_n⁻¹Rn−R⁻_n¹Wn)ii

+µ₄−3σ₀⁴ nσ₀⁴

∑n i=1

{(R⁻_n¹W_nS_n⁻¹R_n−R⁻_n¹W_n)_ii}²,

whereµ₃andµ₄are the third and fourth moments ofv_is, respectively, (R⁻_n¹X_n)_i is thei-th row of (R⁻_n¹X_n), (R⁻_n¹W_nS_n⁻¹X_nβ₀)_iis thei-th element of (R⁻_n¹W_nS⁻_n¹X_nβ₀) and (R⁻_n¹WnS⁻_n¹Rn)ii, (R⁻_n¹WnS_n⁻¹Rn −R⁻_n¹Wn)ii and (2R⁻_n¹WnS_n⁻¹Rn − R⁻_n¹Wn)iiare the (i, j)th element of (R⁻_n¹WnS_n⁻¹Rn), (R⁻_n¹WnS⁻_n¹Rn−R⁻_n¹Wn) and (2R⁻_n¹WnS_n⁻¹Rn−R⁻_n¹Wn), respectively.

B. Some useful Lemmas

Lemma 3.5.1(Proposition 8.4.13, Bernstein (2009)). Let A and B be matrices.

We useγmaxandγminto denote the largest and smallest eigenvalues of a matrix.

If A is symmetric and B is positive semi definite, then γmin(A)tr(B)≤tr(AB)≤γmax(A)tr(B).

Lemma 3.5.2 (Lee, 2002, p.256; Lee, 2004, p1918). Let {An} and {Bn} be two two sequences of n×n matrices that are uniformly bounded in both row and column sums and the elements of ann×nmatrix{Cn}beO(1)uniformly.

Then

1. the sequence{A_nB_n}are uniformly bounded in both row and column sums, 2. the elements of C_nB_n have the uniform orderO(1), and

3. the elements of An are uniformly bounded andtr(An) =O(n).

Lemma 3.5.3(Lee, 2004, p1918). The elements, thev^′_isofVn are assumed to be i.i.d. with zero mean and a finite variance and the fourth moment of thev^′s is assumed to exist. Suppose thatAn is a square matrix with tis column sums being uniformly bounded and elements of the n×K matrix Zn are uniformly bounded. Let {Bn} be uniformly bounded either in row or column sums and their elementsbn,ij haveO(1) uniformly in i and j. Then

1. √¹nZ_n^′A_nV_n=O_p(1)and

2. _n¹E(V_n^′B_nV_n) =O(1) and ¹_n[V_n^′B_nV_n−E(V_n^′B_nV_n)] =o_p(1).

C. Proofs of Theorems 3-5

Proof of Theorem3

The consistency of ˆθ will follow from the uniform convergence of _n¹(logLn(θ)− Qn(θ)) to zero on Θ and the uniqueness identification condition that, for any ϵ > 0,lim sup_n→∞maxθ∈N_ϵ^c(θ₀) 1

n(Qn(θ)−Qn(θ0)) < 0, where N_ϵ^c(θ0) is the complement of an open neighborhood ofθ0in Θ of diameterϵ(Theorem 3.4 of white (1994)).

Proof of the uniform convergence of _n¹(logLn(θ)−Qn(θ))

First, we shall prove the uniform convergence of _n¹(logL_n(θ)−Q_n(θ)) to zero on Θ. The proof follows from:

(a) inf_θ∈Θσ_n^∗2(θ) is bounded away from zero, (b) sup_θ∈Θ|σˆ²_n(θ)−σ^∗_n²(θ)|=op(1),

(c) sup_θ∈Θ|_n¹(logLn(θ)−Qn(θ))|=op(1).

Proof of (a) By the definition ofV_n^∗(θ), V_n^∗(θ) = R_n⁻¹(λ)(Sn(θ)Yn−Xnβ_n^∗(θ)),

= R_n⁻¹(λ)Sn(θ)Yn−R⁻_n¹Xn(X_n^′R^′−_n ¹(λ)R_n⁻¹(λ)Xn)⁻¹X_n^′R^′−_n ¹(λ)R_n⁻¹(λ)Sn(θ)E(Yn),

= R_n⁻¹(λ)S_n(θ)Y_n−P_nR_n⁻¹(λ)S_n(θ)E(Y_n),

= MnR⁻_n¹(λ)Sn(θ)Yn+PnR⁻_n¹(λ)Sn(θ)(Yn−E(Yn)),

where,Pn=R⁻_n¹Xn(X_n^′R^′−_n ¹(λ)R⁻_n¹(λ)Xn)⁻¹X_n^′R_n^′−1 andMn=In−Pn. From the orthogonality between the two symmetric idempotent matricesMn

andPn, we have, σ_n^∗2(θ) = 1

nE(V_n^′∗(θ)V_n^∗(θ)),

= 1

nE[Y_n^′S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)Yn+

(Yn−E(Yn))^′S^′_n(θ)R^′−_n ¹(λ)PnR⁻_n¹(λ)Sn(θ)(Yn−E(Yn))],

= 1

nE(Y_n^′)S_n^′(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)E(Y_n) + 1

ntr(R^′−_n ¹(λ)R⁻_n¹(λ)V ar(S_n(θ)Y_n)).

The matrix M_n is positive semi definite because M_n is a symmetric idem-potent matrix (Lemma 14.2.14 of Harville (1997)). Thus, the first term is non-negative uniformly inθ∈Θ.

Because the matrixV ar(S_n(θ)Y_n) is symmetric andγ_minV ar(S_n(θ)Y_n)>0 from the assumption, the matrix is positive semi definite (Theorem 3.25 of Schott (2005)). By Lemma 3.5.1, the second term is

1

ntr(R^′−_n ¹(λ)R⁻_n¹(λ)V ar(Sn(θ)Yn)) ≥ 1

nγmin(R^′−_n ¹(λ)R⁻_n¹(λ))tr(V ar(Sn(θ)Yn)),

≥ 1 nc_rc_y,

> 0,uniformly inθ∈Θ.

It follow that inf_θ∈Θσ_n^∗2(θ) is bounded away from zero.

Proof of (b) Noting that

Vˆn(θ) = R_n⁻¹(λ)(Sn(θ)Yn−Xnβˆn(θ)),

= R_n⁻¹(λ)Sn(θ)Yn−R⁻_n¹Xn(X_n^′R^′−_n ¹(λ)R⁻_n¹(λ)Xn)⁻¹X_n^′R^′−_n ¹(λ)R⁻_n¹(λ)Sn(θ)Yn,

= MnR⁻_n¹(λ)Sn(θ)Yn. Hence,

ˆ

σ²_n(θ) = 1 n

Vˆ^′n(θ) ˆVn(θ),

= 1

nY_n^′S_n^′(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)Y_n. It follows that

ˆ

σ_n²(θ)−σ_n^∗2(θ) = 1

nY_n^′S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)Yn− 1 nE(

Y_n^′S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)Yn

)

−1 nE(

(Yn−E(Yn))^′S_n^′(θ)R^′−_n ¹(λ)PnR⁻_n¹(λ)Sn(θ)(Yn−E(Yn))) ,

= (Q1−EQ1)−EQ2,

where,Q1= _n¹Y_n^′S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)YnandEQ2= _n¹E(

(Yn−E(Yn))^′S_n^′(θ)R^′−_n ¹(λ) P_nR⁻_n¹(λ)S_n(θ)(Y_n−E(Y_n)))

.

To show the result, it sufficient to show Q1−EQ1

−→p 0 and EQ2 −→ 0, uniformly inθ∈Θ.

First, we show thatQ1−EQ1

−→p 0 uniformly in θ∈Θ. By Theorem 1 of Andrews (1992), the uniform convergence ofQ1−EQ1 to zero in probability follows from the pointwise convergence for eachθ∈Θ and stochastic equicon-tinuity of Q₁, i.e., for any ϵ > 0, there exists a positive number δ such that lim sup_n→∞P(sup_θ∈Θsup_θ′∈B(θ,δ)> ϵ)< ϵ, whereB(θ, δ) denote a closed ball in Θ of radiusδ≥0 centered atθ.

First of all, the pointwise convergence ofQ₁−EQ₁will be shown. We have, by the identity: Yn=S_n⁻¹Xnβ0+S_n⁻¹RnVn,

Q₁ = 1

n(S_n⁻¹X_nβ₀+S_n⁻¹R_nV_n)^′S^′_n(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)(S_n⁻¹X_nβ₀+S_n⁻¹R_nV_n),

= 1

n(β^′₀X_n^′S_n^′−1S_n^′(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)S_n⁻¹X_nβ₀+ 2β^′₀X_n^′S_n^′−1S^′_n(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹RnVn

+V_n^′R^′_nS_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹RnVn),

= Q_1,1(θ) + 2Q_1,2(θ) +Q_1,3(θ),

whereQ1,1(θ) = ¹_n(β₀^′X_n^′S_n^′−1S_n^′(θ)R^′−_n¹(λ)MnR_n⁻¹(λ)Sn(θ)S_n⁻¹Xnβ0), Q1,2(θ) =_n¹(β₀^′X_n^′S_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹RnVn) and

Q1,3(θ) =_n¹(V_n^′R^′_nS_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹RnVn). The two terms Q_1,2(θ) andQ_1,3(θ) are stochastic.

For the second term, the column sums ofS^′−_n ¹S_n^′(θ)R^′−_n¹(λ)M_nR⁻_n¹(λ)S_n(θ)S⁻_n¹R_n are uniformly bounded from assumption 8 and Lemma 3.5.2 andE(Q_1,2(θ)) = 0.

Thus, the pointwise convergence of Q_1,2(θ)−E(Q_1,2)(θ) follow from Lemma 3.5.3. Similarly, the column sums ofR^′_nS_n^′−1S_n^′(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)S_n⁻¹R_n are uniformly bounded and the pointwise convergence of Q1,3(θ)−E(Q1,3)(θ) follows from Lemma 3.5.3. Therefore,Q1−EQ1

−→p 0, for eachθ∈Θ.

Next, we show thatQ1 is stochastic equicontinuous. We have by the mean value theorem:

Q_1,ℓ(θ₁)−Q_1,ℓ(θ₂) = ∂

∂θ^′Q_1,ℓ(¯θ)(θ₂−θ₁),

≤ sup

θ∈Θ

∂

∂θ^′Q_1,ℓ(θ)

(θ₂−θ₁),

where ℓ = 1,2,3 and ¯θ lies between θ₁ and θ₂. For stochastic equicontinu-ous, it suffices to show that sup_θ∈Θ ^∂

∂θ^′Q_1,ℓ(θ) = O_p(1) by Theorem 21.10 of Davidson (1994). Let Π₁ be S_n^′−1S^′_n(θ)R^′−_n ¹(λ)M_nR⁻_n¹(λ)S_n(θ)S_n⁻¹, Π₂ be β₀^′X_n^′S_n^′−1S_n^′(θ)R^′−_n¹(λ)M_nR_n⁻¹(λ)S_n(θ)S_n⁻¹R_nand Π₃beR^′_nS_n^′−1S_n^′(θ)R^′−_n ¹(λ) M_nR⁻_n¹(λ)S_n(θ)S_n⁻¹R_n. The partial derivatives _∂θ^∂_′Π_1,ℓ take simple form and consequently_∂θ^∂_′Π1,ℓare also uniformly bounded in both row and column sums.

ForQ1,1, for anyθ, the elements ofβ₀^′X_n^′ _∂θ^∂_′S_n^′−1S^′_n(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹

andXnβ0 are uniformly bounded. Thus, there exists constantsc1 andc2such that|{β₀^′X_n^′(_∂θ^∂_′S_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S⁻_n¹)}i|≤c1and|(Xnβ0)i| ≤ c2 where {β₀^′X_n^′(_∂θ^∂_′S_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹)}i and (Xnβ0)i are thei-th elements of each vector. It follows that ^∂

∂θ^′Q_1,1≤c₁c₂ =O(1). For Q_1,2, for any θ, _∂θ^∂_′Π_1,2,i≤c₃ where _∂θ^∂_′Π_1,2,i is the i-th element of _∂θ^∂_′Π_1,2. Therefore, from Lemma 3.5.3, P(_∂θ^∂_′Q1,2 > M)

≤ P(_n¹∑n

i=1c3vi > M)

= O(

n⁻¹²)

. For Q1,3, for any θ, ^∂

∂θ^′Π1,3,ij ≤ c4 where where _∂θ^∂_′Π1,3,ij is the (i, j)th element of _∂θ^∂_′Π1,3. Thus, from Lemma 3.5.3, P(_∂θ^∂_′Q1,3 > M) P(_n¹∑n ≤

i=1

∑

j=1c4vivj > M)

= O(1). Thus, sup_θ∈Θ ^∂

∂θ^′Q1,ℓ(θ) = Op(1) It follow thatQ1 is stochastic equicontinuous. Hence, by Theorem 1 of Andrews (1992),Q₁−EQ₁−→^p 0 uniformly inθ∈Θ.

Secondly, we show thatEQ₂−→0, uniformly inθ∈Θ. There existc_xsuch that

0 < c_x ≤ infλ∈Λγmin

(₁

nX_n^′R_n^′−1R_n⁻¹X)

from assumption. By Assumption, Lemma 3.5.1 and 3.5.2 and theorem 3.4 of Schott (2005), We have,

EQ2 = 1 nE(

(Yn−E(Yn))^′S_n^′(θ)R^′−_n ¹(λ)PnR⁻_n¹(λ)Sn(θ)(Yn−E(Yn))) ,

= 1

ntr(R^′−_n ¹(λ)PnR⁻_n¹(λ)V ar(Sn(θ)Yn)),

= 1

ntr(R^′−_n ¹(λ)R⁻_n¹Xn(X_n^′R_n^′−1(λ)R⁻_n¹(λ)Xn)⁻¹X_n^′R^′−_n¹R_n⁻¹(λ)V ar(Sn(θ)Yn)),

≤ 1

nγ_min⁻¹ (X_n^′R^′−_n¹R_n⁻¹X)γ_max² (R_n^′−1(λ)R⁻_n¹(λ))γmax(V ar(Sn(θ)Y))tr(X_n^′Xn)),

= 1

nγ_min⁻¹

(X_n^′R_n^′−1R_n⁻¹X n

)

γ_max² (R_n^′−1(λ)R⁻_n¹(λ))γmax(V ar(Sn(θ)Y))1

ntr(X_n^′Xn)),

≤ 1

nc⁻_x¹c²_rcy

1

ntr(X_n^′Xn)),

= O( n⁻¹)

Hence,EQ2−→0, uniformly inθ∈Θ.

Therefore, sup_θ∈Θ|σˆ²_n(θ)−σ^∗_n²(θ)|=op(1), completing the proof of (b).

Proof of (C) We show that sup_θ∈Θ¹

n(logLn(θ)−Qn(θ))=op(1). Note that 1

n(logLn(θ)−Qn(θ)) =−1

2(log ˆσ_n²(θ)−logσ^∗_n²(θ)).

By the Taylor expansion,

log ˆσ_n²(θ)−logσ_n^∗2(θ)= 1

˜

σ²_n(θ)σˆ_n²(θ)−σ_n^∗2(θ),

where ˜σ_n²(θ) lies between ˆσ²_n(θ) andσ^∗_n²(θ). From the proof (a) and (b), it follow that ˆσ²_n(θ) is uniformly bounded away from zero on Θ. Moreover, ˜σ_n²(θ) is also uniformly bounded away from zero on Θ because ˜σ²_n(θ) exists between ˆσ_n²(θ)

andσ^∗_n²(θ) and thereby _˜σ2¹

n(θ) is uniformly bounded. As ˆσ²_n(θ)−σ^∗_n²(θ) coverges in probability to zero uniformly on Θ,|log ˆσ_n²(θ)−logσ_n^∗2(θ)|=op(1) uniformly on Θ.

Consequently, sup_θ∈Θ¹

n(logL_n(θ)−Q_n(θ))=o_p(1).

Proof of the identification uniqueness condition

Secondly, we shall prove the identification uniqueness condition. The proof follow from:

(i) _n¹Qn(θ) is uniformly equicontinuous on Θ.

(ii) Show some properties of an auxiliary model.

(iii) Show that the identification uniqueness condition holds.

Proof of (i) We show that_n¹Q_n(θ) = ¹₂(log 2π+1)−¹₂logσ^∗_n²(θ)−_n¹log|R_n(λ)|+

1

nlog|S_n(θ)|is uniformly equicontinuous on Θ. It is sufficient to show that par-tial derivatives of each term are uniformly bounded. The uniform continuity of logσ^∗_n²(θ) on Θ follows because _σ∗2¹

n(θ) is uniformly bounded since σ^∗_n²(θ) is uniformly bounded away form zero on Θ. For ¹_nlog|Rn(λ)|, _dλ^d _n¹log|Rn(λ)|=

1

ntr(R⁻_n¹(λ)Wn). From assumption and Lemma 3.5.2, the elements ofR⁻_n¹(λ)Wn

are uniformly bounded. Thus, ¹_ntr(R⁻_n¹(λ)Wn) =O(1) from Lemma 3.5.2. Sim-ilarly, _∂θ^{∂ 1}_nlog|Sn(θ)| =O(1). Hence, ¹_nQn(θ) is uniformly equicontinuous on Θ.

Proof of (ii) It is useful to establish an auxiliary process:

Yn=λWnYn+ρWnYn+Rn(λ)Vn,

where Vn ∼ N(0, σ²₀In). The log-likelihood function of the above auxiliary process is given by

logLp,n(θ, σ²) = −n

2log(2π)−n

2log(σ²(θ))−log|Rn(λ)|+ log|Sn(θ)|

− 1

2σ²Y_n^′S_n^′(θ)R^′−_n ¹(λ)R⁻_n¹(λ)Sn(θ)Yn.

Let E_p be the expectation under this auxiliary process. Define Q_p,n(θ) = max_σ2E_p(logL_p,n(θ)). The optimal solutions of this maximization problem is

σ_n²(θ) = 1

nEp(Y_n^′S_n^′(θ)R^′−_n¹(λ)R⁻_n¹(λ)Sn(θ)Yn),

= σ²

ntr(R_nS_n⁻¹S_n^′(θ)R^′−_n ¹(λ)R⁻_n¹(λ)S_n(θ)S_n⁻¹R_n).

Hence,

Qp,n(θ) =−n

2 log(2π+ 1) +n

2logσ_n²(θ)−log|Rn(λ)|+ log|Sn(θ)|.

By Shannon-Kolmogorov Information Inequality (Ferguson (1996), p113), Qp,n(θ)≤Qp,n(θ0) for all θ∈Θ. This implies that _n¹(Qp,n(θ)−Qp,n(θ0)≤0 for allθ∈Θ.

Proof of (iii) We show that the identification uniqueness condition holds by contradiction.

1

n(Qn(θ)−Qn(θ0)) = −1

2logσ_n^∗2(θ)−log|Rn(λ)|+ log|Sn(θ)| − (

−1

2logσ₀²−log|Rn|+ log|Sn| )

= (

−1

2(logσ²_n(θ)−logσ²₀)−1

n(log|Rn(λ)| −log|Rn|) + 1

n(log|S_n(θ)| −log|S_n|) )

−1

n(logσ_n^∗2(θ)−logσ_n²(θ)),

= 1

n

(Q_p,n(θ)−Q_p,n(θ₀))

−1

2(logσ^∗_n²(θ)−logσ²_n(θ)).

Moreover,

σ^∗_n²(θ)−σ²_n(θ) = 1

nβ₀^′X_n^′S_n^′−1S_n^′(θ)R^′−_n ¹(λ)MnR⁻_n¹(λ)Sn(θ)S_n⁻¹Xnβ0. Mn is positive semi definite and thereby σ^∗_n²(θ)−σ²_n(θ) ≥ 0. This implies

−¹₂(logσ^∗_n²(θ)−logσ_n²(θ))≤0.

Now, suppose that the identification uniqueness condition does not hold.

Then, there exists anϵ >0 and a sequence{θn}inN_ϵ^c(θ0) such that limn→∞ 1 n

(Qn(θ)− Q_n(θ₀))

= 0. By the compactness of N_ϵ^c(θ₀), there exists a convergent subse-quence {θn_m} of {θn} with the limit θ+ of θn_m being in N_ϵ^c(θ0). This implies thatθ+̸=θ0. As _n¹Qn(θ) is uniformly equicontinuous, limn_m→∞ 1

n_m

(Qn_m(θ+)− Qn_m(θ0))

= 0. Because _n¹(

Qp,n(θ)−Qp,n(θ0))

≤ 0 and −¹₂(

logσ^∗_n²(θ) − logσ_n²(θ))

≤0, this is possible only if lim_nm→∞n¹

m

(Q_nm(θ₊)−Q_nm(θ₀))

= 0 and−¹₂(

logσ_n^∗2(θ)−logσ²_n(θ))

≤0. However, lim_n→∞¹_nβ₀^′X_n^′S^′−_n ¹S_n^′(θ)R^′−_n¹(λ)M_nR_n⁻¹(λ)S_n(θ)S⁻_n¹X_nβ₀̸= 0 from the assumption in Theorem 3 . Thus, −¹₂(

logσ_n^∗2(θ)−logσ_n²(θ))

< 0 and consequently

lim_nm→∞n¹

m

(Q_nm(θ₊)−Q_nm(θ₀))

̸

= 0. This is a contradiction. Therefore, the identification uniqueness condition must hold.

The consistency of ˆθfollow form uniform convergence and the identification uniqueness condition. This completes the proof of the theorem.

Proof of Theorem 4

We have by the Taylor expansion,

0 = 1

√n

∂logLn( ˆψn)

∂ψ ,

= 1

√n

∂logLn(ψ0)

∂ψ +

(1 n

∂²logLn( ¯ψn)

∂ψ∂ψ^′ )√

n( ˆψn−ψ0),

where ¯ψnlies between ˆψnandψ0. Thus, the asymptotic normality of ˆψnfollows if

(a) √¹n

∂logL_n(ψ₀)

∂ψ

−→D N(

0,limn→∞Γ(ψ0)) , (b) _n^1∂2log_∂ψ∂ψ^Ln(ψ_′ ⁰⁾−E(₁

n

∂²logLn(ψ0)

∂ψ∂ψ^′

) p

−→0, and (c) _n^1∂2log_∂ψ∂ψ^{Ln( ¯}_′^ψn)−_n^1∂2log_∂ψ∂ψ^Ln(ψ′ ⁰⁾

−→p 0.

Proof of (a) The asymptotic normality of √¹n

∂logLn(ψ0)

∂ψ follows from the central limit theorems for linear-quadratic forms in Kelejian and Prucha (2001).

We need to check that the score vector holds Assumptions in Kelejian and Prucha (2001). To check assumptions for asymptotic normality, it is sufficient to show some matrices hold desired boundaly conditions. From assumptions of this paper and Lemma 3.5.2, (R^′_nS_n^′−1W_n^′R^′−_n ¹−W_n^′R_n^′−1) andR^′_nS_n^′−1W_n^′R^′−_n ¹ are uniformly bounded in column sums, and the elements of X_n^′S_n^′−1W_n^′R^′−_n ¹ are uniformly bounded. Thus, each score function holds the assumptions and the asymptotic normality of each score function follows. Finally, the Cram´ er-Wold devise (Proposition 6.3.1 of Brockwell and Davis (1991)) leads to the joint asymptotic normality.

Proof of (b) LetD_ψψ be ¹_n^∂2log_∂ψ∂ψ^Ln(ψ_′ ⁰⁾−E(₁

n

∂²logL_n(ψ₀)

∂ψ∂ψ^′

). Then, D_ψψ has the elements:

Dββ^′ = 0, D_βσ2 = − 1

nσ₀⁴X_n^′R^′−_n ¹Vn, Dβρ = − 1

nσ₀²X_n^′R^′−_n ¹R⁻_n¹WnS_n⁻¹RnVn, Dβλ = 1

nσ²₀X_n^′(R^′−_n¹W_n^′R^′−_n ¹+R^′−_n ¹R⁻_n¹W −R^′−_n ¹R⁻_n¹WnS_n⁻¹Rn)Vn, Dσ²σ² = 1

σ⁴₀ − 1 nσ₀⁶V_n^′Vn, Dσ²ρ = − 1

nσ₀⁴β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹Vn− 1

nσ₀⁴(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹Vn−σ²₀tr(S^′−_n ¹W_n^′)), Dσ²λ = − 1

nσ₀⁴β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹Vn+ 1

nσ₀⁴(V_n^′W_n^′R^′−_n ¹Vn−σ²₀tr(W_n^′R^′−_n ¹))

− 1

nσ₀⁴(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹V_n−σ₀²tr(S_n^′−1W_n^′)), D_ρρ = − 2

nσ0

β₀^′X_n^′S_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS_n⁻¹R_nV_n

− 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹WnS_n⁻¹RnVn−σ₀²tr(R^′_nS^′−_n ¹W_n^′R^′−_n¹R_n⁻¹WnS_n⁻¹Rn)), Dρλ = 1

nσ²₀β₀^′X_n^′(S^′−_n ¹W_n^′R_n^′−1W_n^′R^′−_n ¹+S_n^′−1W_n^′R^′−_n ¹R⁻_n¹Wn−2S_n^′−1W_n^′R^′−_n ¹R⁻_n¹WnS_n⁻¹Rn)Vn

+ 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹W_n^′R^′−_n ¹Vn−σ₀²tr(S_n^′−1W_n^′R^′−_n ¹W_n^′)) + 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nV_n−σ²₀tr(R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_n))

− 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS_n⁻¹R_nV_n−σ₀²tr(R^′_nS^′−_n ¹W_n^′R^′−_n¹R_n⁻¹W_nS_n⁻¹R_n)), D_λλ = 1

nσ²₀β₀^′X_n^′(2S_n^′−1W_n^′R^′−_n ¹W_n^′R^′−_n ¹+S_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_n−2S_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS_n^′−1R_n

−2R^′−_n ¹W_n^′R^′−_n ¹W_n^′R_n^′−1−R^′−_n ¹W_n^′R^′−_n ¹R⁻_n¹Wn+ 2R^′−_n ¹W_n^′R_n^′−1R_n⁻¹WnS_n⁻¹Rn

+2R^′−_n ¹W_n^′R^′−_n ¹W_n^′R_n⁻¹+R^′−_n ¹W_n^′R^′−_n ¹R⁻_n¹Wn−R^′−_n ¹W_n^′R^′−_n¹R_n⁻¹WnS_n⁻¹Rn)Vn

+ 2

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹W_n^′R^′−_n ¹V_n−σ₀²tr(S_n^′−1W_n^′R^′−_n ¹W_n^′)) + 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nV_n−σ²₀tr(R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_n))

− 1

nσ₀²(V_n^′R^′_nS_n^′−1W_n^′R^′−_n ¹R⁻_n¹W_nS_n⁻¹R_nV_n−σ₀²tr(R^′_nS^′−_n ¹W_n^′R^′−_n¹R_n⁻¹W_nS_n⁻¹R_n))

− 2

nσ₀²(V_n^′W_n^′R^′−_n ¹W_n^′R^′−_n ¹V_n−σ₀²tr(W_n^′R_n^′−1W_n^′R^′−_n ¹)) + 1

nσ₀²(V_n^′W_n^′R^′−_n ¹R⁻_n¹W_nV_n−σ²₀tr(W_n^′R^′−_n ¹R⁻_n¹W_n)) 1 _{′ ′ ′−1 −1 −1}

− ^{2 ′ ′−1 −1 −1} 54

Thus, the elements ofDψψare decomposed into sums of the forms: ¹_nX_n^′An(θ)Vn,_n¹β₀^′X_n^′An(θ)Vn,

1

n(V_n^′An(θ)Vn−E(V_n^′An(θ)Vn)) and_σ¹4 0−_nσ¹6

0

V_n^′Vn, where a matrixAn(θ) is

uni-formly bounded in both row and column sums. From Lemma 3.5.3, ¹_nX_n^′An(θ)Vn,_n¹β^′₀X_n^′An(θ)Vn

and _n¹(V_n^′An(θ)Vn −E(V_n^′An(θ)Vn)) are convergence to zero in probability.

Moreover, _σ¹₄

0 − _nσ¹6 0

V_n^′V_n −→^p 0 because _n¹V_nV_n −→^p σ²₀ by the law of large numbers. Therefore, it follow that _n^1∂2log_∂ψ∂ψ^Ln(ψ_′ ⁰⁾−E(₁

n

∂²logL_n(ψ₀)

∂ψ∂ψ^′

) p

−→0.

Proof of (c) From Lemma 3.5.2 and 3.5.3, it is easy to show that_n^1∂2log_∂ψ∂ψ^{Ln( ¯}_′^ψn) = Op(1) and ¹_n^∂2log_∂ψ∂ψ^Ln(ψ_′ ⁰⁾ = Op(1). Here, ¯σ^−r = σ⁻₀^r+op(1), r = 2,4,6 be-cause ¯σ² −→^p σ₀² and σ^r appears inHn(ψ) ≡ _∂ψ∂ψ^∂2 ′logLn(ψ) multiplicatively, thus it results in an asymptotically negligible error to replace ¯σ² by σ₀². The elements of the Hessian matrix, Hn(ψ) ≡ _∂ψ∂ψ^∂2 ′logLn(ψ), are decomposed into sums of terms of the forms: X_n^′An(θ)Xn, X_n^′An(θ)Yn, X_n^′An(θ)V(θ), Y_n^′An(θ)Yn,_2σⁿ4−_σ¹6V_n^′(θ)Vn(θ), Y_n^′An(θ)Vn(θ), V_n^′(θ)An(θ)Vn(θ) andtr(An(θ)), where a matrix An(θ) is uniformly bounded in both row and column sums.

Therefore, it is sufficient to show that the difference between each term at ¯ψ andψ0 converges to zero in probability and moreover this can be easily shown.

We show some examples corresponding each term of the Hessian matrix.

Noting that

R⁻_n¹(λ)−R⁻_n¹ = R⁻_n¹(λ)(R_n−R_n(λ))R⁻_n¹,

= (λ0−λ)R⁻_n¹(λ)WnR⁻_n¹. ForX_n^′An(θ)Xn,

1

nX_n^′R^′−_n ¹(¯λ)R⁻_n¹(¯λ)X_n−1

nX_n^′R^′−_n ¹R⁻_n¹X_n = 1

nX_n^′(R_n^′−1(¯λ)−R^′−_n ¹+R^′−_n ¹)R⁻_n¹(¯λ)X_n− 1

nX_n^′R^′−_n ¹R⁻_n¹X_n,

= 1

nX_n^′(R_n^′−1(¯λ)−R^′−_n ¹)R⁻_n¹(¯λ)X_n+ 1

nX_n^′R^′−_n ¹R⁻_n¹(¯λ)X_n−1

nX_n^′R^′−_n ¹R⁻_n¹X_n,

= (λ₀−¯λ)1

nX_nR^′−_n ¹(λ)W_n^′R⁻_n¹R⁻_n¹(¯λ)X_n +(λ₀−λ)¯ 1

nX_n^′R^′−_n ¹R⁻_n¹(λ)W_nR⁻_n¹X_n,

= op(1)O(1) +op(1)O(1),

= o_p(1).

Moreover, the convergence ofX_n^′An(θ)Yn is shown similarly.

Noting that

Vn(θ) = R⁻_n¹(λ)Rn(λ)Vn(θ),

= R⁻_n¹(λ)(S(θ)Yn−Xnβ),

= R⁻_n¹(λ)((λ₀−λ)W_nY_n+ (ρ₀−ρ)W_nY_n+X_n(β₀−β) +R_nV_n).

Thus, forX_n^′A_n(θ)V(θ), 1

nX_n^′R^′−_n ¹(¯λ)V_n(¯θ)− 1

nX_n^′R^′−_n ¹V_n = (

(λ₀−λ) + (ρ¯ ₀−ρ)¯)1

nX_n^′R^′−_n ¹(¯λ)W_nY_n+ 1

nX_n^′R^′−_n ¹(¯λ)X_n(β₀−β) +1

nX_n^′R^′−_n ¹(¯λ)R_nV_n− 1

nX_n^′R^′−_n¹V_n,

= op(1)Op(1) +Op(1)op(1) +op(1) +op(1),

= op(1),

where the convergence of last two terms follow from Lemma 3.5.3.

Here, 1

nV_n^′(¯θ)Vn(¯θ) = (

(λ0−λ) + (ρ¯ 0−ρ)¯)21

nY_n^′W_n^′R^′−_n ¹(¯λ))R⁻_n¹(¯λ)WnYn

+(β0−β)^′1

nX_n^′R^′−_n ¹(¯λ)R⁻_n¹(¯λ)Xn(β0−β) +1

nV_n^′R^′_nR^′−_n ¹(¯λ)R_n⁻¹(¯λ)RnVn

+2 n

((λ0−λ) + (ρ¯ 0−ρ)¯)

Y_n^′W_n^′R^′−_n ¹(¯λ)R_n⁻¹(¯λ)Xn(β0−β) +2

n

((λ0−λ) + (ρ¯ 0−ρ)¯)

Y_n^′W_n^′R^′−_n ¹(¯λ)R_n⁻¹(¯λ)RnVn+ (β0−β)^′2

nX_n^′R_n^′−1(¯λ)R⁻_n¹(¯λ)RnVn,

= op(1)Op(1) +op(1)O(1)op(1) +σ²₀+op(1)Op(1)op(1) +op(1)Op(1) +op(1)op(1),

= σ₀²+o_p(1).

It follows that _2σ¹4 0 −_nσ¹6

0

V_n^′(θ)V_n(θ) =o_p(1).

Before next proof, we show an example. Y_n^′Sn(θ)Vn = β^′X_n^′S_n⁻¹S(θ)Vn + V_n^′R^′_nS_n⁻¹Sn(θ)Vn and

1

nV_n^′R_n^′S_n⁻¹Sn(θ)Vn− 1

nV_n^′R^′_nS_n⁻¹SnVn = (

(λ0−λ) + (ρ0−ρ))1

nV_n^′R_n^′S_n⁻¹Vn,

= op(1)Op(1),

= o_p(1).

It follows that _n¹Y_n^′S_n(θ)V_n−_n¹Y_n^′S_nV_n =o_p(1) and similarly _n¹Y_n^′A_n(θ)V_n−

1

nY_n^′A_nV_n =o_p(1) and _n¹Y_n^′A_n(θ)Y_n−¹_nY_n^′A_nY_n=o_p(1) where A_n isA_n(θ) at true valueθ₀.

Now, forY_n^′An(θ)Vn(θ), 1

nY_n^′W_n^′R^′−_n ¹(λ)V_n(θ)−1

nY_n^′W_n^′R^′−_n ¹V_n = (

(λ₀−λ) + (ρ¯ ₀−ρ)¯)1

nY_n^′W_n^′R^′−_n ¹(λ)R⁻_n¹(λ)W_nY_n +1

nY_n^′W_n^′R^′−_n ¹(λ)R⁻_n¹(λ)X_n(β₀−β)¯ +1

nY_n^′W_n^′R^′−_n ¹(¯λ)R⁻_n¹(¯λ)R_nV −1

nY_n^′W_n^′R^′−_n ¹V_n

= op(1)Op(1) +Op(1)op(1) +op(1)

= op(1).

Moreover, the convergence ofVn(θ)^′An(θ)Vn(θ) is also shown similary.

Finally, fortr(A_n(θ)), by the Taylor expansion, 1

ntr(R⁻_n¹(λ)WnR⁻_n¹(λ)Wn)−1

ntr(R⁻_n¹WnR⁻_n¹Wn) = d

dλtr(R_n⁻¹(˜λ)WnR_n⁻¹(˜λ)Wn)(¯λ−λ0),

= O(1)o_p(1),

= op(1), where ˜λlies between ¯λandλ0.

The convergence of the other elements of the Hessian matrix are shown similarly, hence _n^1∂2log_∂ψ∂ψ^{Ln( ¯}_′^ψn)−_n^1∂2log_∂ψ∂ψ^Ln(ψ′ ⁰⁾

−→p 0.

This completes the proof of the theorem.

Proof of Theorem 5

The estimator forαis ˆ

α_n= (1−λ) logˆ (1

n

∑n i=1

exp{(R⁻_n¹(ˆλ)[S(ˆθ)Y_n−Z_nδ])ˆ _i} )

, Here,

S(ˆθ)Y_n−Z_nδˆ = Y_n−λWˆ _nY_n−ρWˆ _nY_n−Z_nδ,ˆ

= (λ₀−λ)Wˆ _nY_n+ (ρ₀−ρ)Wˆ _nY_n+Z_n(δ₀−δ) +ˆ α₀1_n+R_nV_n,

= D+α01n+RnVn,

whereD= (λ₀−λ)Wˆ _nY_n+ (ρ₀−ρ)Wˆ _nY_n+Z_n(δ₀−δ).ˆ BecauseR⁻_n¹(ˆλ)(S(ˆθ)Yn−Znδ) =ˆ ^α0

1−λˆ1n+R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn, 1

n

∑n i=1

exp{(R⁻_n¹(ˆλ)[S(ˆθ)Yn−Znδ])ˆ i}= exp ( α

1−λ )1

n

∑n i=1

exp{(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i}. Thus,

ˆ

α−α0= (1−ˆλ) log (1

n

∑n i=1

exp{(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i} )

. (3.10)

To prove consistency, it is sufficient that the right side of (3.10) converges to zero in probability.

By the Taylor expansion, 1

n

∑n i=1

exp{(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i} = 1 + 1 n

∑n i=1

exp(bi){

(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i

}

= 1 + 1

nb^′(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn), wherebi lies between 0 and (R_n⁻¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i, andb= (b1, . . . , bn)^′.

From Assumptions, Theorem 3 and Lemma 3.5.2 and 3.5.3, 1

nb^′(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn) = (λ0−λ)ˆ 1

nb^′R⁻_n¹(ˆλ)WnYn+ (ρ0−ρ)ˆ 1

nb^′R⁻_n¹(ˆλ)WnYn

+1

nb^′R⁻_n¹(ˆλ)Zn(δ0−δ) +ˆ 1

nb^′R⁻_n¹(ˆλ)RnVn,

= op(1)Op(1) +op(1)Op(1) +O(1)op(1) +op(1),

= o_p(1).

Thus, _n¹∑n

i=1exp{(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)R_nV_n)_i}−→^p 1 and (1−λ) logˆ (₁

n

∑n

i=1exp{(R⁻_n¹(ˆλ)D+R⁻_n¹(ˆλ)RnVn)i}) p

−→0.

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Chapter 4

SARAR-GARCH models

Abstract

This study proposes spatio-temporal extensions of time series generalized au-toregressive conditional heteroskedasticity (GARCH) models. We call spatio-temporally extended GARCH models as spatial autoregressive models with spa-tial autoregressive error and generalized autoregressive conditional heteroskedas-ticity processes, namely SARAR-GARCH models. One important problem which multivariate volatility models contain is the curse of dimensionality. To overcome the problem, we adopt a spatial weight matrix which expresses the dependence relation between observations. A spatial weight matrix is usually determined by geographical information of spatial data. However, financial data doesn’t include geographical information. Therefore, we propose a method to make spatial weight matrix from financial data by stepwise backward regres-sions. Parameters are estimated by a two step procedure. First step is the estimation of spatial parameters and second step is that of GARCH param-eters. In real data analysis, We apply the SARAR-GARCH model to daily returns of the Nikkei 225 stock price data and S&P 500 stock price data. We compare the in-sample and out-sample performances of SARAR-GARCH mod-els with those of CCC modmod-els which is a benchmark. The results show the in-sample performance of the CCC model is better because the CCC model contains many more parameters. However, the out-sample performance of the SARAR-GARCH model are better than that of the CCC model in both markets analysis.

4.1 Introduction

Volatility which is a conditional variance in a model is one of the most important concepts in financial econometrics because it is used in widely areas such as risk management, option pricing and portfolio selection. Financial market data often exhibits volatility clustering (i.e., volatility may be high for certain time periods

and low for other periods) This means time-varying volatility is more common than constant volatility. Therefore, accurate modeling of time-varying volatility is important in financial econometrics.

The seminal work of Engle (1982) proposes autoregressive conditional het-eroscedasticity (ARCH) models and the most important extension of the model is generalized ARCH (GARCH) models proposed by Bollerslev (1986). The models have been widely used to identify volatilities. After that, many ex-tended GARCH models have been proposed. For example, integrated GARCH models ( Engle and Bollerslev (1986)), exponential GARCH models (Nelson (1991)), threshold GARCH models (Glosten, etal (1993)), GARCH in the mean models, and GJR-GARCH models are proposed.

Univariate volatility models are generalized to multivariate cases in many ways. One important problem which multivariate volatility models contain is the curse of dimensionality. We estimate a conditional covariance matrix which has ⁿ⁽ⁿ⁺¹⁾₂ quantities for a n-dimensional time series, therefor it is difficult to estimate all quantities. Thus, we attempt to give a conditional covariance ma-trix some simple structures to reduce the number of parameters. For example, exponentially weighted moving average models, constant conditional correlation models (Bollerslev (1990)), BEKK models (Engle and Kroner (1995)), orthogo-nal GARCH models (Alexander (2001) ), dynamic conditioorthogo-nal correlation mod-els (Tse and Tsui (2002)), dynamic orthogonal component modmod-els, and factor GARCH models are proposed.

The ideas of spatial econometrics have been applied to volatility models to reduce number of parameters in a covariance matrix in recent years. Caporin and Paruolo (2008) and Borovkova and Lopuhaa (2012) have applied the ideas of spatial econometrics to time series multivariate GARCH models. Yan (2007) and Robinson (2009) have done spatial extensions of stochastic volatility models which are another kind of volatility models. Sato and Matsuda (2017, 2018) have extend time series GARCH models to spatial models.

This paper contributes to extend GARCH models to spatiotemporal mod-els which we call spatial autoregressive modmod-els with spatial autoregressive error and generalized autoregressive conditional heteroskedasticity processes, namely SARAR-GARCH models by using spatial econometrics ideas. The model is characterized by a spatial weight matrix which express cross-section correla-tions between assets and used to reduce the number of parameters. A spatial weight matrix is usually determined by geographical information of spatial data.

However, financial data doesn’t include geographical information. Therefore, we propose a method to make spatial weight matrix from financial data. we ap-ply the multiple linear regression model and stepwise backward regression to calculate spatial weights in spatial weight matrices. Parameters are estimated by a two step procedure. First step is the estimation of spatial parameters and second step is that of GARCH parameters. Spatial parameters are estimated in first step. We regard volatilities in the model as constant variance and we apply quasi-maximum likelihood method with the model. After that we ap-ply GARCH models with residuals derived from first step in second step. In

real data analysis, We apply the SARAR-GARCH model to daily returns of the Nikkei 225 stock price data and S&P 500 stock price data. We compare the in-sample and out-sample performances of SARAR-GARCH models with those of CCC models. First, we check the in-sample performances based on log-likelihood. The results show the log-likelihood of the CCC model is grater than that of SARAR-GARCH. This means model fitting of the CCC model is better. One reason is that the number of parameters in CCC models is more than five times of those of SARAR-GARCH models. Secondly, we compare out-sample performances by using quasi-likelihood loss function. The result shows the quasi-likelihood loss function of SARAR-GARCH models are smaller than that of CCC models. Then, the out-sample performance of SARAR-GARCH models is better. One reason is the CCC model may be over-fitting and it cause lower forecasting performance. Moreover, SARAR-GARCH models have bet-ter prediction performance in U.S. market analysis because stock price in U.S.

market are more volatile and proposed models can capture sharp fluctuations.

The rest of paper proceeds as follows. Section 4.2 introduces SARAR-GARCH models. The estimation procedures are described in section 4.3. Sec-tion 4.4 examines empirical properties of SARAR-GARCH models by applying the models to real data such as stock price in the Japanese and the U.S. market.

Section 4.5 discusses some concluding remarks.