Appendices to Chapter 2

2.A. Finding optimal policies and welfare gains

This section describes the algorithm that we use to derive the optimal federal RI scheme.

As discussed in Section 2.4.2, we wish to have closed-form expressions for the objective function values (for given member-state policies and a given federal RI scheme). To obtain optimal federal RI in Sections 2.5.1 and 2.5.2, we find federal and member-state policies that maximize the unconditional mean of the objective function. When accounting for the transi-tion, in Sections 2.5.3 and 2.5.4, we find policies that maximize the conditional expectation of the objective. In these cases, we condition on the initial state being the non-stochastic steady state implied by our calibrated model (Table 2.4).

The next section describes how we find the optimal federal RI scheme. We describe this for the case when the member state can choose policies once and we account for the transi-tions. The other cases are handled analogously.

2.A.1. Finding the optimal federal RI scheme. The algorithm proceeds as follows.

(1) Fix Chebyshev nodes for the federal RI scheme as described in Section 2.4.2. Keep nodes fixed.

(2) The goal is to find valuesφ=[φ1,φ2,φ3,φ4]⁰∈RandτFthat solve the federal govern-ment’s problem (2.21) anticipating the member-states’ policy choices. The values of φinduce payout functionBF(·;·).

(3) Findφby numerical optimization. For this, for each tryφ, evaluate the federal gov-ernment’s objective function either using unconditional expectations (Sections 2.5.1 to 2.5.2) or conditional expectations for a given initial state (Sections 2.5.3 and 2.5.4).

(4) In order to evaluate the federal planner’s objective function for givenφ, make sure that the scheme is feasible in light of the member states’ optimal response. In partic-ular, a federal RI policy has to be self-financing in light of member-states’ responses, recall (2.19). For fixedφwe iterateas follows.

(a) mark the iteration by⁽ⁿ⁾. Setn=0. Start from an initial value ofτ⁽⁻¹⁾_F .

(b) setτF =τ⁽ⁿ_F⁻¹⁾. For given federal RI policyφandτF =τ⁽ⁿ_F⁻¹⁾, let the member state solve (2.23).

(c) label the maximizing member-state policies {τ^i,(n)v ,τ^i,(n)_ξ ,b^i,(n)}. These induce a law of motionµ⁽ⁿ⁾₀ and a value for the objective function ofR

W₀⁽ⁿ⁾dµ⁽ⁿ⁾₀ . (d) for given member-state policies {τ^i,(n)v ,τ^i,(n)_ξ ,b^i,(n)}, and given federal policyφ,

find a valueτ⁽ⁿ⁾_F that solves the federal RI scheme’s financing constraint (2.19) for these given policies and given the induced dynamics for the member-state economies.

(e) ifτ⁽ⁿ⁾_F is not sufficiently close toτ⁽ⁿ_F⁻¹⁾, setn=n+1 and go to step 4b. Else, set τF=τ⁽ⁿ⁾_F and go to step 5.

(5) The federal policy implied byφandτFis feasible. SetR

W0dµ0=R

W₀⁽ⁿ⁾dµ⁽ⁿ⁾₀ . (6) Continue numerical optimization started in step 3 until the maximum for the federal

government’s objective is found.

2.A.2. Welfare gains from federal UI. The consumption-equivalent welfare gains reported in Table 2.5 are computed as follows. For each policy setting, compute welfare under optimal federal RI as detailed in Section 2.A.1. This givesR

W₀dµ0. Follow the program as in Sec-tion 2.A.1, but settingφ=0 (BF(·;·)=0) andτF=0. That is, solve for optimal member-state policy in autarky. Denote the induced welfare byR

W₀^autdµ^aut₀ . The consumption equivalent welfare gains are computed as the value of the direct transfer received every period which would make households indifferent between living in autarky and living under the federal RI.

The welfare gain is expressed in terms of percent of the steady state consumption level.

2.B. Calculating fourth-order-accurate unconditional first moments

In this section we consider a pruned perturbation solution to a dynamic stochastic gen-eral equilibrium (DSGE) model. We derive closed-form solutions for fourth-order-accurate unconditional first moments of the model’s endogenous variables. The exposition here heav-ily builds on Andreasen et al. (2018) who already provide the formulae up to third-order of accuracy. Our sole contribution is to provide formulae for fourth-order moments.

2.B.1. Preliminaries. We consider the following class of DSGE models. Letyt∈R^nybe a vector of control variables,xt∈R^nx+1a vector of state variables which includes a perturbation parameterσ≥0. Consider a perturbation solution to a DSGE model around the steady state xSS=0. The exact solution to the model is given by

y_t=g(x_t),

x_t+1=h(xt)+ση²t+1, (2.41)

where²t+1followsn_²dimensional multivariate normal distribution and is independently and identically distributed in each period. Solving a DSGE model amounts to finding unknown functionsgandh.

For most DSGE models, the full solution to system (2.41) cannot be found explicitly. The perturbation solution approximates the true solution using a Taylor series expansion around the steady state,x_t=x_t+1=0. Up to fourth order, we have

(2.42) xt+1=hxxt+1

2hxxx^⊗2_t +1

6hxxxx^⊗3_t + 1

24hxxxxx^⊗4_t +σηεt+1,

wherehx,hxx, . . . denote first, second, etc, order derivatives of functionhwith respect to vec-torx. Superscript^⊗nrepresents then-th Kronecker power, i.e.x^⊗n=

ntimes

z }| { x⊗x⊗. . ..

However, the system (2.42) may display explosive dynamics and may not have any finite unconditional moments (Andreasen et al. 2018) . The solution to this problem suggested by Kim et al. (2008) is to prune the state space of the approximated solution so as to remove explosive paths. As shown by Andreasen et al. (2018) the pruned 4th-order approximation to the perturbation solution reads

xt+1=x_t^f₊₁+x^s_t+1+x^{r d}_t+1+x_t^{4t h}₊₁, where

x_t^f₊₁=hxx_t^f+σηεt+1, x_t^s₊₁=hxx^s_t+1

2hxx

³x_t^f´⊗2

, x^{r d}_t+1=hxx^{r d}_t +1

2hxx

³2³

(x_t^f ⊗x_t^s)´´

+1 6hxxx

³x_t^f´⊗3

, and

x^{4t h}_t+1=h_xx^{4t h}_t +1 2h_xx³

2³

x_t^f ⊗x^{r d}_t ´ +¡

x^s_t¢_⊗2´ +1

6h_xxx µ

3³ x_t^f´_⊗2

⊗x_t^s

¶ + 1

24h_xxxx³ x^f´_⊗4

.

Note that if the shock is drawn from the standard normal distribution, as is the case in the model developed in the current paper, then

E(εt)^⊗2=vec(In_e), E(εt)^⊗3=0, and E(εt)^⊗5=0.

LetM⁴≡E(εt)^⊗4be the kurtosis of the standard multivariate normal distribution.

In the course of the proofs we will use extensively the following (well-known) properties of the Kronecker product (for example, Magnus and Neudecker 1999). These are:

A⊗(B+C)=A⊗B+A⊗C, (A+B)⊗C=A⊗C+B⊗C,

(kA)⊗B=A⊗(kB)=k(A⊗B), (A⊗B)⊗C=A⊗(B⊗C),

(AC)⊗(BD)=(A⊗B)(C⊗D), vec(ABC)=(C⁰⊗A)vec(B).

We say that matrixKm,nof sizemn×mnis acommutation matrixif it has the following prop-erty: LetAbe (m×n) matrix andBis (p×q) matrix. Then

Km,p(A⊗B)Kq,n=B⊗A.

That is, the commutation matrix reverses the order of Kronecker product. The commutation matrixKm,ncan be defined explicitly as

Kn,m=

m

X

i=1 n

X

j=1

³

(e^m_i (eⁿ_j)⁰)⊗(eⁿ_j(e^m_i )⁰)´ ,

wheree_i^mis theith unit column vector of orderm. For any commutation matrixKp,q=K_q⁻¹_,p.

2.B.2. Analytical expressions for the first moments. We are ready to derive formulas for unconditional first moments of the endogenous variables. Our goal is to characterize the following expression

E0x_t=E0x_t^f+E0x_t^s+E0x_t^{r d}+E0x_t^{4t h},

Andreasen et al. (2018) showed thatE0x_t^f =E0x^{r d}_t =0. The first equality is the certainty equiv-alence of the linear approximation. The second equality,E0x^{r d}_t =0, results from the symme-try of the normal distribution (i.e. skewness is zero).

We write perturbation parameterσas a separate variable, not included in statext. Note thath_σ=hxσ=hxxσ =hxxxσ=hxxxxσ=0 (see, for example, Theorem 7 in Jin and Judd (2002)).

For completeness, we derive expressions for the unconditional first moments for the so-lution approximations of all orders from one up to four. Derivations for orders of approxima-tion up to three are based on Andreasen et al. (2018). Formulas for the fourth order are our contribution.

2.B.2.1. First and second-order of approximation. We start with the formulae accurate up to the second order. We have

x_t^f₊₁=hxx_t^f+σηεt+1

x_t+1^s =hxx^s_t+1 2hxx

³x_t^f´_⊗2

+1 2h_σσσ² Ex_t^f =0

E³ x_t^f´_⊗2

=(I_n²

x−hx⊗hx)⁻¹(σ²η⊗η)vec(I)) Ex_t^s=(Inx−hx)⁻¹

·1 2hxx(I_n²

x−hx⊗hx)⁻¹(σ²η⊗η)vec(I))+1 2h_σσσ²

¸ PROOF.

x_t+1^f =hxx_t^f+ση²t+1

Ex_t+1^f =Ehxx_t^f (stationarity) Ex_t^f(I−hx)=0 ⇒ Ex_t^f=0.

³x_t^f´_⊗2

=³

hxx_t^f+ση²t+1

´

⊗³

hxx_t^f +ση²t+1

´

=hxx_t^f⊗hxx_t^f +(hxx_t^f)⊗(ση²t+1)+(ση²t+1)⊗(hxx_t^f)+(ση²t+1)⊗(ση²t+1) E³

x_t^f´_⊗2

=hx⊗hxE³ x_t^f´_⊗2

+σ²η⊗ηE(²t+1)^⊗2 E³

x_t^f´⊗2

=(I−hx⊗hx)⁻¹(σ²η⊗η)vec(I))

x^s_t+1=hxx^s_t+1 2hxx

³ x_t^f´_⊗2

+1 2h_σσσ² Ex^s_t=hxEx_t^s+1

2hxxE³ x_t^f´_⊗2

+1 2h_σσσ² Ex^s_t=(I−hx)⁻¹

·1

2hxx(I−hx⊗hx)⁻¹(σ²η⊗η)vec(I))+1 2h_σσσ²

¸ .

2.B.2.2. Third order. Next, we tackle the third-order approximation. We replicate the re-sults in Andreasen et al. (2018) in the following

x^{r d}_t+1=hxx_t^{r d}+1 2hxx

³ 2³

(x_t^f⊗x^s_t)´´

+1 6hxxx

³ x_t^f´_⊗3

+3·1

6hxσσx_t^fσ²+1

6h_σσσσ³

Ex_t^f₊₁⊗x_t^s₊₁=¡

I−h^⊗_x²¢−1µ hx⊗1

2hxx

¶ E³

x_t^f´⊗3

E³ x_t^f₊₁´⊗3

=0 Ex_t^{r d}₊₁=0.

PROOF. x_t+1^f ⊗x^s_t+1=³

hxx_t^f+ση²t+1

´

⊗ µ

hxx^s_t+1 2hxx

³ x_t^f´_⊗2

+1 2h_σσσ²

¶

=(hx⊗hx)³

x_t^f⊗x_t^s´ +

µ hx⊗1

2hxx

¶³ x_t^f´⊗3

+(hxx_f)⊗1 2h_σσσ² +¡

ση⊗hx¢ ¡

²t+1⊗x^s_t¢ +¡

ση⊗.5hxx¢ µ

²t+1⊗³ x_t^f´_⊗2¶

+(η²t+1)⊗1 2h_σσσ³. E(x_t^f ⊗x_t^s)=¡

I−h_x^⊗2¢−1µ hx⊗1

2hxx

¶ E³

x_t^f´_⊗3 .

³ x_t+1^f ´_⊗3

=³ x_t^f₊₁´_⊗2

⊗³

hxx_t^f +ση²t+1

´

=³

hxx_t^f+ση²t+1

´

⊗h

(hx⊗hx) µ³

x_t^f´_⊗2¶ +¡

ση⊗ση¢ ¡

²^⊗2t+1

¢ +¡

ση⊗h_x¢³

²t+1⊗x_t^f´ +¡

h_x⊗ση¢³

x_t^f⊗²t+1

´ i

=(hx⊗hx⊗hx)³ x_t^f´_⊗3

+(ση²t+1)⊗(hx⊗hx) µ³

x_t^f´_⊗2¶ +(hxx_t^f)⊗¡

ση⊗ση¢ ¡

²^⊗t+²1

¢+(ση²t+1)⊗¡

ση⊗ση¢ ¡

²^⊗t+²1

¢ +(hxx_t^f)⊗¡

ση⊗hx¢³

²t+1⊗x_t^f´

+(ση²t+1)⊗¡

ση⊗hx¢³

²t+1⊗x_t^f´ +(hxx_t^f)⊗¡

hx⊗ση¢³

x_t^f⊗²t+1

´

+(ση²t+1)⊗¡

hx⊗ση¢³

x_t^f⊗²t+1

´

=(hx⊗hx⊗hx)³ x_t^f´_⊗3

+terms zero in expectation.

The last equality follows from the fact thatx_t^f and²t+1are independent, thereforeE³

x_t^f⊗²t+1

´

= Ex_t^f ⊗E²t+1=0,E(hxx_t^f)⊗¡

ση⊗ση¢ ¡

²^⊗2_t+1¢

=0 sinceE(hxx_t^f)=0.

Therefore,

E µ³

x_t^f₊₁´_⊗3¶

=E(h_x⊗h_x⊗h_x)³ x_t^f´_⊗3

and by stationarityE(h_x⊗h_x⊗h_x)³ x_t^f´_⊗3

=0, x^{r d}_t+1=hxx^{r d}_t +1

2hxx

³ 2³

(x_t^f⊗x_t^s)´´

+1 6hxxx

³ x_t^f´_⊗3

+3·1

6h_xσσx_t^fσ²+1

6h_σσσσ³ Ex^{r d}_t+1=h_xEx_t^{r d}+1

6h_σσσσ³ Ex^{r d}_t (I−hx)=0,

the last line following sinceh_σσσ=0 for symmetric distribution (see Andreasen, 2012) 2.B.2.3. Fourth order. Finally, we derive the solutions accurate up to fourth order.

We have that the fourth-order accurate law of motion of the states is given by x_t^{4t h}₊₁=hxx^{4t h}_t +1

2hxx

³ 2³

x_t^f⊗x^{r d}_t ´ +¡

x_t^s¢_⊗2´ +1

6hxxx

µ 3³

x_t^f´_⊗2

⊗x^s_t

¶ + 1

24hxxxx

³ x^f´_⊗4 +3

6h_σσxσ²x_t^s+6· 1

24h_σσxxσ²³ x_t^f´⊗2

+4· 1

24h_σσσxσ³x_t^f+ 1

24h_σσσσσ⁴. The fourth raw moment of the state variables is given by.

Ex^{4t h}_t =¡

Inx−hx¢₋1h1 2hxxE³

2³

x_t^f⊗x^{r d}_t ´ +¡

x_t^s¢_⊗2´ +1

6hxxxE µ

3³ x_t^f´_⊗2

⊗x^s_t

¶ + 1

24hxxxxE³ x^f´_⊗4 +3

6h_σσxσ²Ex^s_t+6· 1

24h_σσxxσ²E³ x_t^f´⊗2

+ 1

24h_σσσσσ⁴i .

Where the respective terms for each basis vector of the 4th-order pruned state space are listend in the following.

E³ x_t+1^f ´_⊗4

=σ²³

I_n⁴_x−h^⊗4_x ´₋₁h

σ²η^⊗4M⁴+³

(h^⊗2_x ⊗η^⊗2)K_ne²_,n²_x +(hx⊗η⊗hx⊗η)¡

In_x⊗Kn_x,n_e⊗In_e¢ K_n²

e,n²_x

+(hx⊗η⊗η⊗hx)³

Inx⊗K_nx,n²_e´ K_n²_e,n²_x +¡

η⊗hx⊗hx⊗η¢³

In_e⊗K_ne,n²

x

´ +¡

η⊗hx⊗η⊗hx¢ ¡

In_e⊗Kn_e,n_x⊗In_x¢ +¡

η^⊗2⊗h^⊗2_x ¢´ ³

vec(I_ne)⊗E(x_t^f)^⊗2´ i . This we can calculate givenE³

x_t^f´_⊗2

=σ²(I−h^⊗2_x )⁻¹(η^⊗2)vec(Ine)) from further above.

Regarding the remaining vectors that span the 4th-order pruned state space, we have Eh

¡x^s_t+1¢_⊗2i

=³ I_n2

x−h^⊗2_x ´₋1³ +.5³

Kn_x,n_x+I_n2 x

´

(hxx⊗hx)E µ³

x_t^f´_⊗2

⊗x_t^s

¶ +1

4h^⊗2_xxE(x_t^f)^⊗4 +³

Kn_x,n_x+I_n2 x

´·1

2[(hx⊗h_σσ)(σ²x^s_t)]+1

4(hxx⊗h_σσ)(σ²(x_t^f)^⊗2)

¸ +1

4h_σσ⊗h_σσσ⁴´ ,

E

·³ x_t^f´_⊗2

⊗x_t^s

¸

=³ I_n3

x−h^⊗3_x ´₋1³

(σ²η^⊗2⊗hx)(E£

²^⊗2t+1

¤⊗Ex_t^s)+1

2(h^⊗2_x ⊗hxx)E(x_t^f)^⊗4 +1

2(σ²η^⊗2⊗h_xx)(vec(I_ne)⊗E(x_t^f)^⊗2)+1

2(h^⊗_x²⊗h_σσ)σ²Eh (x_t^f)^⊗2i +1

2(η^⊗2⊗h_σσ)σ⁴E²^⊗2t+1

´ ,

Eh

x_t^f₊₁⊗x^{r d}_t+1i

=

³ I_n2

x−h^⊗_x²´₋₁³

(h_x⊗h_xx)E((x_t^f)^⊗2⊗x^s_t)+1

6(h_x⊗h_xxx)E³ x_t^f´_⊗4 +3

6(h_x⊗h_σσ)σ²Eh

(x_t^f)^⊗2i ´ . PROOF. We derive each of the terms.

³ x_t^f₊₁´_⊗4

=³

h_xx_t^f+ση²t+1

´

⊗³ x_t^f₊₁´_⊗3

=(h_xx_t^f)⊗³ x_t^f₊₁´_⊗3

+(ση²t+1)⊗³ x_t^f₊₁´_⊗3

. (2.43)

We tackle separately each of the summands in (2.43).

(h_xx_t^f)⊗³ x_t^f₊₁´_⊗3

=(h_xx_t^f)⊗³h

h_xx_t^f +ση²t+1

i

⊗h

h_xx_t^f+ση²t+1

i

⊗h

h_xx_t^f +ση²t+1

i´

=(hxx_t^f)⊗³ h

hxx_t^f+ση²t+1

i

⊗ h

h_xx_t^f ⊗h_xx_t^f+h_xx_t^f ⊗ση²t+1+ση²t+1⊗h_xx_t^f+ση²t+1⊗ση²t+1

i ´

=(h_xx_t^f)⊗

³ h

h_xx_t^f+ση²t+1

i

⊗h

h^⊗2_x (x_t^f)^⊗2+σ(hx⊗η)(x_t^f⊗²t+1)+σ(η⊗hx)(²t+1⊗x_t^f)+σ²η^⊗2²^⊗2t+1

i ´ . Further,

(h_xx_t^f)⊗³ x_t^f₊₁´_⊗3

=(h_xx_t^f)⊗³

h^⊗_x³(x_t^f)^⊗3+σ(h^⊗_x²⊗η)((x_t^f)²⊗²t+1)

+σ(hx⊗η⊗hx)(x_t^f ⊗²t+1⊗x_t^f)+σ²(hx⊗η^⊗2)(x_t^f ⊗²^⊗2t+1) +σ(η⊗h^⊗2_x )(²t+1⊗(x_t^f)^⊗2)+σ²(η⊗h_x⊗η)(²t+1⊗x_t^f⊗²t+1) +σ²(η^⊗2⊗h_x)(²^⊗t+²1⊗x_t^f)+σ³η^⊗3²^⊗t+³1

´ .

So that

(h_xx_t^f)⊗

³ x_t+1^f ´_⊗3

=h^⊗4_x (x_t^f)^⊗4+σ²(h^⊗2_x ⊗η^⊗2)((x_t^f)^⊗2⊗²^⊗2_t+1) +σ²¡

h_x⊗η⊗h_x⊗η¢³

x_t^f⊗²t+1⊗x_t^f⊗²t+1

´

+σ²¡

h_x⊗η^⊗2⊗h_x¢³

x_t^f⊗²^⊗2_t+1⊗x_t^f´

+terms zero in expectation.

(2.44)

The last equality follows since²t+1⊥x_t^f andEx_t^f =E³ (x_t^f)^⊗3´

E²t+1=0.

Using the commutation matrixK_nx,neto change the order of⊗. For instance x_t^f ⊗²t+1⊗²t+1⊗x_t^f =x_t^f ⊗I_ne²t+1⊗K_nx,ne(x_t^f ⊗²t+1)

=x_t^f ⊗(I_ne⊗K_nx,ne)(²t+1⊗x_t^f⊗²t+1)

=x_t^f ⊗(In_e⊗Kn_x,n_e)(Kn_x,n_e(x_t^f⊗²t+1)⊗²t+1)

=x_t^f ⊗(I_ne⊗K_nx,ne)((K_nx,ne⊗I_ne)(x_t^f ⊗²t+1⊗²t+1))

=£

I_nx⊗(I_ne⊗K_nx,ne)(K_nx,ne⊗I_ne)¤h

(x_t^f)^⊗2⊗²^⊗t+1²

i . This can be simplified further using (I_r⊗K_m,s)(K_m,s⊗I_s)=K_{m,r s}. Thus

Eh

x_t^f⊗²t+1⊗²t+1⊗x_t^fi

=

³ I_nx⊗K_n

x,n²_e

´E³

(x_t^f)^⊗2⊗²^⊗_t+1²´ . Similarly,

E³

x_t^f ⊗²t+1⊗x_t^f ⊗²t+1

´

=¡

I_nx⊗K_nx,ne⊗I_ne¢ E³

(x_t^f)^⊗2⊗²^⊗t+1²

´ . Going back to (2.44) we have

E

·

(h_xx_t^f)⊗³

x_t^f₊₁´_⊗3¸

=h^⊗_x⁴E(x_t^f)^⊗4+σ²h

(h_x^⊗2⊗η^⊗2) +(h_x⊗η⊗h_x⊗η)¡

I_nx⊗K_nx,ne⊗I_ne¢ +(h_x⊗η⊗η⊗h_x)³

I_nx⊗K_n

x,n²_e

´ i ³ Eh

(x_t^f)^⊗2i

⊗vec(I_ne)´ , sinceE³

(x_t^f)^⊗2⊗²^⊗2_t+1´

=Eh (x_t^f)^⊗2i

⊗vec(In_e).

Regarding the other summand in (2.43), (ση²t+1)⊗

³ x_t+1^f ´_⊗3

=(ση²t+1)⊗

³

h^⊗3_x (x_t^f)^⊗3+σ(h^⊗2_x ⊗η)((x_t^f)²⊗²t+1)

+σ(h_x⊗η⊗h_x)(x_t^f⊗²t+1⊗x_t^f)+σ²(h_x⊗η^⊗2)(x_t^f⊗²^⊗t+1²) +σ(η⊗h^⊗2_x )(²t+1⊗(x_t^f)^⊗2)+ση²t+1⊗σ(hx⊗η)(x_t^f ⊗²t+1) +ση²t+1⊗σ(η⊗h_x)(²t+1⊗x_t^f)+σ³η^⊗3²^⊗3_t+1´

=σ²¡

η⊗h_x⊗h_x⊗η¢³

²t+1⊗x_t^f⊗x_t^f⊗²t+1

´

+σ²¡

η⊗hx⊗η⊗hx¢³

²t+1⊗x_t^f⊗²t+1⊗x_t^f´ +σ²¡

η^⊗2⊗h^⊗2_x ¢³

²^⊗2_t+1⊗(x_t^f)^⊗2´

+σ⁴η^⊗4²^⊗4_t+1 +terms that are zero in expectation.

(2.45)

Using the commutation matrices E³

²^⊗t+1² ⊗(x_t^f)^⊗2´

=vec(I_ne)⊗E(x_t^f)^⊗2 E³

²t+1⊗x_t^f ⊗²t+1⊗x_t^f´

=¡

I_ne⊗K_ne,nx⊗I_nx¢³

vec(I_ne)⊗E(x_t^f)^⊗2

´

E³

²t+1⊗x_t^f ⊗x_t^f⊗²t+1

´

=(I_ne⊗K_n

e,n²_x)(vec(I_ne)⊗E(x_t^f)^⊗2).

Plugging into (2.45) delivers E

·

(ση²t+1)⊗

³

x_t+1^f ´_⊗3¸

=σ²h

¡η⊗h_x⊗h_x⊗η¢³ I_ne⊗K_n

e,n²_x

´

+¡

η⊗hx⊗η⊗hx¢ ¡

In_e⊗Kn_e,n_x⊗In_x¢ +¡

η^⊗2⊗h^⊗2_x ¢i ³

vec(I_ne)⊗E(x_t^f)^⊗2´

+σ⁴η^⊗4M⁴, whereM⁴≡E£

²^⊗_t+1⁴¤ .

Going back to the original formula (2.43), E³

x_t+1^f ´_⊗4

=E

·

(h_xx_t^f)⊗

³

x_t+1^f ´_⊗3¸ +E

·

(ση²t+1)⊗

³

x_t+1^f ´_⊗3¸

=h^⊗4_x E(x_t^f)^⊗4+σ²h

(h^⊗2_x ⊗η^⊗2) +(h_x⊗η⊗h_x⊗η)¡

I_nx⊗K_nx,ne⊗I_ne¢ +(h_x⊗η⊗η⊗h_x)³

I_nx⊗K_n

x,n²_e

´ i ³ Eh

(x_t^f)^⊗2i

⊗vec(I_ne)´ +σ²h

¡η⊗h_x⊗h_x⊗η¢³ I_ne⊗K_n

e,n²_x

´

+¡

η⊗hx⊗η⊗hx¢ ¡

In_e⊗Kn_e,n_x⊗In_x¢ +¡

η^⊗2⊗h^⊗2_x ¢i ³

vec(I_ne)⊗E(x_t^f)^⊗2´

+σ⁴η^⊗4M⁴.

Hence, using stationarity and the fact thatEh (x_t^f)^⊗2i

⊗vec(I_ne)=K_n2 e,n²_x

³

vec(I_ne)⊗E(x_t^f)^⊗2´

, we have that

E³ x_t^f₊₁´_⊗4

=σ²³ I_n4

x−h^⊗_x⁴´₋₁h

σ²η^⊗4M⁴+³

(h^⊗_x²⊗η^⊗2)K_n2 e,n²_x

+(hx⊗η⊗hx⊗η)¡

In_x⊗Kn_x,n_e⊗In_e¢ K_n2

e,n²_x

+(h_x⊗η⊗η⊗h_x)³ I_nx⊗K_n

x,n²_e

´ K_n2

e,n²_x

+¡

η⊗h_x⊗h_x⊗η¢³ I_ne⊗K_n

e,n²_x

´

+¡

η⊗h_x⊗η⊗h_x¢ ¡

I_ne⊗K_ne,nx⊗I_nx¢ +¡

η^⊗2⊗h^⊗2_x ¢´ ³

vec(I_ne)⊗E(x_t^f)^⊗2´ i .

Continued on next page.

Next, we calculateE³

¡x_t+1^s ¢_⊗2´ .

¡x^s_t+1¢_⊗2

= µ

h_xx^s_t+1 2h_xx³

x_t^f´_⊗2 +1

2h_σσσ²

¶

⊗ µ

h_xx_t^s+1 2h_xx³

x_t^f´_⊗2 +1

2h_σσσ²

¶

=h^⊗2_x (x_t^s)^⊗2+(h_xx_t^s)⊗(1 2h_xx³

x_t^f´_⊗2 )+(1

2h_xx³ x_t^f´_⊗2

)⊗(h_xx_t^s) +

µ1 2h_xx³

x_t^f´_⊗2¶

⊗ µ1

2h_xx³ x_t^f´_⊗2¶ +1

2(h_σσ⊗h_x)(σ²x_t^s)+1

4(h_σσ⊗h_xx)(σ²(x_t^f)^⊗2)+1

4h_σσ⊗h_σσσ⁴ +1

2(h_x⊗h_σσ)(σ²x_t^s)+1

4(h_xx⊗h_σσ)(σ²(x_t^f)^⊗2)

=h^⊗_x²(x_t^s)^⊗2+(1

2h_x⊗h_xx) µ

x_t^s⊗³ x_t^f´_⊗2¶

+ µ1

2h_xx⊗h_x

¶ µ³ x_t^f´_⊗2

⊗x_t^s

¶

+1

4h^⊗_xx²(x_t^f)^⊗4+1

2(h_x⊗h_σσ+h_σσ⊗h_x) (σ²x^s_t) +1

4(h_σσ⊗hxx+hxx⊗h_σσ) (σ²(x_t^f)^⊗2)+1

4h_σσ⊗h_σσσ⁴

=h^⊗2_x (x_t^s)^⊗2+(1

2hx⊗hxx)K_n2 x,n_x

µ³ x_t^f´_⊗2

⊗x_t^s

¶ +

µ1

2hxx⊗hx

¶ µ³ x_t^f´_⊗2

⊗x_t^s

¶

+1

4h^⊗_xx²(x_t^f)^⊗4+1 2

³

Kn_x,n_x+I_n2 x

´

[(hx⊗h_σσ)(σ²x^s_t)]

+1 4

³

Kn_x,n_x+I_n2 x

´

([hxx⊗h_σσ)(σ²(x_t^f)^⊗2)]+1

4h_σσ⊗h_σσσ⁴

=h^⊗2_x (x_t^s)^⊗2+1

2(h_xx⊗h_x)

³ K_n2

x,nx+I_n2 x

´µ

³ x_t^f´_⊗2

⊗x_t^s

¶ +1

4h^⊗2_xx(x_t^f)^⊗4 +³

Kn_x,n_x+I_n2 x

´·1

2[(hx⊗h_σσ)(σ²x^s_t)]+1

4(hxx⊗h_σσ)(σ²(x_t^f)^⊗2)

¸ +1

4h_σσ⊗h_σσσ⁴. where the last line follows since (hx⊗hxx)K_n2

x,n_x =Kn_x,n_x(hxx⊗hx) (Note: K_p,q⁻¹ =Kq,p, soKn_x,n_x = K_n⁻¹_x,nx andK1,n=Kn,1=In).

Hence Eh

¡x^s_t+1¢_⊗2i

=³ I_n2

x−h^⊗2_x ´₋1³ +.5³

K_nx,nx+I_n2 x

´(h_xx⊗h_x)E µ³

x_t^f´_⊗2

⊗x_t^s

¶ +1

4h^⊗2_xxE(x_t^f)^⊗4 +³

Kn_x,n_x+I_n2 x

´·1

2[(hx⊗h_σσ)(σ²x_t^s)]+1

4(hxx⊗h_σσ)(σ²(x_t^f)^⊗2)

¸ +1

4h_σσ⊗h_σσσ⁴´ .

Continued on next page.

Next we calculateE µ³

x_t+1^f ´_⊗2

⊗x^s_t+1

¶ .

³ x_t^f₊₁´_⊗2

⊗x^s_t+1=³

hxx_t^f ⊗hxx_t^f+(hxx_t^f)⊗(ση²t+1)+(ση²t+1)⊗(hxx_t^f)+(ση²t+1)⊗(ση²t+1)´

⊗ µ

h_xx_t^s+1 2h_xx³

x_t^f´_⊗2 +1

2h_σσσ²

¶

=h^⊗3_x ((x_t^f)^⊗2⊗x^s_t)+(σ²η^⊗2⊗h_x)(²^⊗2_t+1⊗x_t^s)+1

2(h_x^⊗2⊗h_σσ)σ²(x_t^f)^⊗2 +1

2(η^⊗2⊗h_σσ)σ⁴²^⊗2_t+1+1

2(h^⊗2_x ⊗h_xx)(x_t^f)^⊗4+1

2(σ²η^⊗2⊗h_xx)(²^⊗2_t+1⊗(x_t^f)^⊗2) +terms zero in expectation.

Hence E

·³ x_t^f´_⊗2

⊗x_t^s

¸

=

³ I_n3

x−h^⊗3_x ´₋₁³

(σ²η^⊗2⊗h_x)(E£

(²t+1)^⊗2¤

⊗Ex_t^s)+1

2(h_x^⊗2⊗h_xx)E(x_t^f)^⊗4 +1

2(σ²η^⊗2⊗h_xx)(vec(I_ne)⊗E(x_t^f)^⊗2)+1

2(h^⊗2_x ⊗h_σσ)σ²Eh (x_t^f)^⊗2

i

+1

2(η^⊗2⊗h_σσ)σ⁴E²^⊗2_t+1´ , where the equality follows since²t+1andx_t^sare orthogonal.

For the last missing terms.

x_t^f₊₁⊗x^{r d}_t+1=³

h_xx_t^f +σηεt+1

´

⊗³

h_xx^{r d}_t +1 2h_xx³

2³

(x_t^f⊗x_t^s)´´

+1 6h_xxx³

x_t^f´_⊗3 +3

6h_σσxσ²x_t^f+1

6h_σσσσ³´

=h^⊗2_x (x_t+1^f ⊗x_t^{r d}₊₁)+(hx⊗hxx)((x_t^f)^⊗2⊗x_t^s)+1

6(hx⊗hxxx)³ x_t^f´_⊗4

+3

6(hx⊗h_σσ)σ²(x_t^f)^⊗2+terms zero in expectation so that

Eh

x_t+1^f ⊗x^{r d}_t+1i

=³ I_n2

x−h^⊗2_x ´₋1³

(hx⊗hxx)E((x_t^f)^⊗2⊗x^s_t)+1

6(hx⊗hxxx)E³ x_t^f´_⊗4

+3

6(h_x⊗h_σσ)σ²Eh (x_t^f)^⊗2

i ´

This completes the proof.

2.B.2.4. Simplifying the 4th-order expressions. For the model used in the current paper, the 4th-order Kronecker products can be simplified further. Namely,

¡h^⊗2_x ⊗η^⊗2¢ K_n2

e,n²_x=K_n2 x,n²_x

¡η^⊗2⊗h^⊗2_x ¢

For the current model we haven_e=1 so thatK_ne,nx=K_nx,ne=I_nx,I_ne=vec(I_ne)=1.

E³ x_t^f₊₁´_⊗4

=σ²³ I_n4

x−h^⊗_x⁴´₋₁h

σ²η^⊗4M⁴+K_n2 x,n²_x

¡η^⊗2⊗h^⊗_x²¢

+(hx⊗η⊗hx⊗η)¡

In_x⊗In_x¢ +(h_x⊗η⊗η⊗h_x)¡

I_nx⊗I_nx¢ +¡

η⊗h_x⊗h_x⊗η¢ +¡

η⊗hx⊗η⊗hx¢ ¡

In_x⊗In_x¢ +¡

η^⊗2⊗h^⊗2_x ¢´ ³

E(x_t^f)^⊗2´ i

=σ²³ I_n4

x−h^⊗4_x ´₋₁h

σ²η^⊗4M⁴ +(h_x⊗η⊗h_x⊗η)¡

I_nx⊗I_nx¢ +(hx⊗η⊗η⊗hx)¡

In_x⊗In_x¢ +¡

η⊗h_x⊗h_x⊗η¢ +¡

η⊗h_x⊗η⊗h_x¢ ¡

I_nx⊗I_nx¢ +³

K_n2 x,n²_x+I_n2

x

´¡

η^⊗2⊗h^⊗2_x ¢´ ³

E(x_t^f)^⊗2´ i . Note thatIn_x⊗In_x=I_n2

x so that E³

x_t^f₊₁´_⊗4

=σ²³ I_n4

x−h^⊗_x⁴´₋₁h

σ²η^⊗4M⁴

+(hx⊗η⊗hx⊗η) +(h_x⊗η⊗η⊗h_x) +¡

η⊗h_x⊗h_x⊗η¢ +¡

η⊗h_x⊗η⊗h_x¢ +

³ K_n2

x,n²_x+I_n2 x

´¡

η^⊗2⊗h^⊗2_x ¢´ ³

E(x_t^f)^⊗2´ i .

Moreover

h_x⊗η⊗h_x⊗η=h_x⊗£ K_nx,nx¡

h_x⊗η¢ K_ne,nx¤

⊗η

=¡

I_nx⊗K_nx,nx⊗I_nx¢ ¡

h^⊗_x²⊗η^⊗2¢

=¡

I_nx⊗K_nx,nx⊗I_nx¢³ K_n2

x,n²_x

¡η^⊗2⊗h_x^⊗2¢´ ,

hx⊗η⊗η⊗hx=hx⊗η⊗£ Kn_x,n_x¡

hx⊗η¢ Kn_e,n_x¤

=

³ I_n2

x⊗K_nx,nx´

£h_x⊗η⊗h_x⊗η¤

=³ I_n2

x⊗K_nx,nx´

£¡I_nx⊗K_nx,nx⊗I_nx¢ ¡

h^⊗_x²⊗η^⊗2¢¤

=³ I_n2

x⊗K_nx,nx´ h

¡I_nx⊗K_nx,nx⊗I_nx¢³ K_n2

x,n²_x

¡η^⊗2⊗h^⊗2_x ¢ ,´i

η⊗hx⊗η⊗hx=η⊗£ Kn_x,n_x¡

η⊗hx¢ Kn_x,n_e¤

⊗hx

=(I_nx⊗K_nx,nx⊗I_nx)(η^⊗2⊗h^⊗2_x ),

η⊗h_x⊗h_x⊗η=η⊗h_x⊗[K_nx,nx(η⊗h_x)K_nx,ne]

=(I_n2

x⊗K_nx,nx)(η⊗h_x⊗η⊗h_x)

=(I_n2

x⊗Kn_x,n_x)£

(In_x⊗Kn_x,n_x⊗In_x)(η^⊗2⊗h_x^⊗2)¤ . Hence

E³ x_t+1^f ´_⊗4

=σ²³ I_n4

x−h^⊗4_x ´₋1h

σ²η^⊗4M⁴ +¡

I_nx⊗K_nx,nx⊗I_nx¢³ K_n2

x,n²_x

¡η^⊗2⊗h^⊗2_x ¢´

+³ I_n2

x⊗K_nx,nx´ h

¡I_nx⊗K_nx,nx⊗I_nx¢³ K_n2

x,n²_x

¡η^⊗2⊗h^⊗_x²¢´i

+(I_n2

x⊗Kn_x,n_x)£

(In_x⊗Kn_x,n_x⊗In_x)(η^⊗2⊗h^⊗2_x )¤ +(I_nx⊗K_nx,nx⊗I_nx)(η^⊗2⊗h^⊗2_x )

+³ K_n2

x,n²_x+I_n2 x

´¡

η^⊗2⊗h^⊗_x²¢´ ³

E(x_t^f)^⊗2´ i ,

or

E³ x_t+1^f ´_⊗4

=σ²³ I_n4

x−h^⊗4_x ´₋₁h

σ²η^⊗4M⁴ +

n¡

I_nx⊗K_nx,nx⊗I_nx¢ K_n2

x,n²_x

+³ I_n2

x⊗Kn_x,n_x

´ h¡

In_x⊗Kn_x,n_x⊗In_x¢³ K_n2

x,n²_x

´i

+(I_n2

x⊗K_nx,nx)£

(I_nx⊗K_nx,nx⊗I_nx)¤ +(I_nx⊗K_nx,nx⊗I_nx)

+³ K_n2

x,n²_x+I_n4 x

´ o¡

η^⊗2⊗h^⊗2_x ¢´ ³

E(x_t^f)^⊗2´ i ,

E³ x_t+1^f ´_⊗4

=σ²³ I_n4

x−h_x^⊗4´₋1h

σ²η^⊗4M⁴ +

n ³ I_n2

x⊗K_nx,nx´

¡I_nx⊗K_nx,nx⊗I_nx¢ +I_n4

x

+I_nx⊗K_nx,nx⊗I_nxo ³ K_n2

x,n²_x+I_n4 x

´¡

η^⊗2⊗h^⊗2_x ¢³

E(x_t^f)^⊗2´ i , hence

E³ x_t+1^f ´_⊗4

=σ²³ I_n4

x−h^⊗4_x ´₋₁h

σ²η^⊗4M⁴ +n ³

I_n2

x⊗K_nx,nx+I_n4 x

´¡

I_nx⊗K_nx,nx⊗I_nx¢ +I_n4

x

o ³ K_n2

x,n²_x+I_n4 x

´¡

η^⊗2⊗h^⊗_x²¢³

E(x_t^f)^⊗2´ i .

CHAPTER 3