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Licensed Unlicensed Requires Authentication Published by De Gruyter February 27, 2014

Efficient bond price approximations in non-linear equilibrium-based term structure models

  • Martin M. Andreasen EMAIL logo and Pawel Zabczyk

Abstract

This paper develops an efficient method to compute higher-order perturbation approximations of bond prices. At third order, our approach can significantly shorten the approximation process and its precision exceeds the log-normal method and a procedure using consol bonds. The efficiency gains greatly facilitate any estimation which is illustrated by considering a long-run risk model for the US. Allowing for an unconstrained intertemporal elasticity of substitution enhances the model’s fit, and we see further improvements when incorporating stochastic volatility and external habits.

JEL: C68; E0

Corresponding author: Martin M. Andreasen, Aarhus University, Fuglesangs Allé 4, 8210 Aarhus V, Denmark, e-mail:

Acknowledgment

For helpful comments and suggestions, we thank Michel Juillard, Paul Klein, Simon Price, Juan Rubio-Ramirez, Bruce Mizrach (the Editor), and an anonymous referee. Andreasen gratefully acknowledges access to computer facilities provided by the Danish Center for Scientific Computing (DCSC) and financial support to CREATES – Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation.

Appendix A

A general transformation of bond prices

This appendix considers the general case of an invertible transformation function R(‧)∈CN, implying that R(pt, k)≡Pt,k. Here

Fk(x,σ)=Et[R(pt,k(x,σ))(g(h(x,σ)+σηεt+1,σ),g(x,σ),h(x,σ)+σηεt+1,x)×R(pt+1,k1(h(x,σ)+σηεt+1,σ))].

First order terms

Derivative of pk with respect to x

Consider [Fxk(xt,σ)]α1=0 which implies

Rp(pk)[pxt,k]α1[x]α1R(pt+1,k1)Rp(pt+1,k1)[pxt+1,k1]γ1[hxt]α1γ1=0

For k=1 we have Rp(p1)[px1]α1=[x]α1 and M=R(p1). Hence

Rp(pk)[pxk]α1=[px1]α1Rp(p1)R(pk1)+R(p1)Rp(pk1)[pxk1]γ1[hx]α1γ1

Derivative of pk with respect to σ

Computing Fσk(xss,0)=0 implies

Et[Rp(pk)[pσk][σ]R(pk1)Rp(pk1)([pxk1]γ1([hσ]γ1+ηt+1)+[pσk1])]=0

For k=1 we have Et[Rp(p1)[pσ1]]=Et[σ]. Similar arguments as in the text then implies pσk=0 for all values of k.

Second order terms

Derivative of pk with respect to (x, x)

We have that [Fxxk(xss,0)]α1,α2=0 implies

Rpp(pk)[pxk]α2[pxk]α1+Rp(pk)[pxxk]α1α2[xx]α1α2R(pk1)[x]α1Rp(pk1)[pxk1]γ2[hx]α2γ2[x]α2Rp(pk1)[pxk1]γ1[hx]α1γ1Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1Rp(pk1)[pxk1]γ1[hxx]α1α2γ1=0

The value of M equals R(p1) and Mx is computed above. Moreover, for k=1 we have

[xx]α1α2=Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1

Thus

Rp(pk)[pxxk]α1,α2=Rpp(pk)[pxk]α2[pxk]α1+(Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1)R(pk1)+[px1]α1Rp(p1)Rp(pk1)[pxk1]γ2[hx]α2γ2+[px1]α2Rp(p1)Rp(pk1)[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk1)[pxk1]γ1[hxx]α1α2γ1

Derivative of pk with respect to (σ, σ)

Next, [Fσσ(xss,0)]=0 implies

Et[Rp(pk)[pσσk]+[σσ]R(pk1)+[σ]Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2+[σ]Rp(pk1)[pxk1]γ1[η]ϕ1γ1[εt+1]ϕ1+Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxk1]γ1[η]ϕ1γ1[εt+1]ϕ1+Rp(pk1)[pxxk1]γ1γ2[η]ϕ2γ2[εt+1]ϕ2[η]ϕ1γ1[εt+1]ϕ1+Rp(pk1)[pxk1]γ1[hσσ]γ1+Rp(pk1)[pσσk1]]=0

For k=1 we have Et[σσ]=[pσσ1]Rp(p1). Using this and previous results, we then obtain

Rp(pk)[pσσk]=[pσσ1]Rp(p1)R(pk1)+2[yt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[xt+1]γ1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+R(p1)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk1)[pxk1]γ1[hσσ]γ1+R(p1)Rp(pk1)[pσσk1]

Third order terms

Derivative of pk with respect to (x, x, x)

Applying the chain rule to the definition of Fk one can show that [Fxxx(xss,0)]α1α2α3=0 equals

Rp(pk)[pxxxk]α1α2α3=Rppp(pk)[pxk]α3[pxk]α2[pxk]α1Rpp(pk)[pxxk]α2α3[pxk]α1Rpp(pk)[pxk]α3[pxxk]α1α2Rpp(pk)[pxk]α3[pxxk]α1α2+[xxx]α1α2α3R(pk1)+(Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1)Rp(pk1)[pxk1]γ3[hx]α3γ3+(Rp(p1)[pxx1]α1α3+Rpp(p1)[px1]α3[px1]α1)Rp(pk1)[pxk1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rp(pk1)[pxk1]γ2[hxx]α2α3γ2+(Rp(p1)[pxx1]α2,α3+Rpp(p1)[px1]α3[px1]α2)Rp(pk1)[pxk1]γ1[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rppp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxxk1]γ2γ3[hx]α3γ3[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hxx]α2α3γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxxk1]γ1γ3[hx]α3γ3[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[hx]α3γ3[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[hxx]α2α3γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hxx]α1α2γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hxx]α1α2γ1+R(p1)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hxx]α1α2γ1+R(p1)Rp(pk1)[pxk1]γ1[hxxx]α1α2α3γ1

Note that we can also eliminate [xxx]α1α2α3 from this expression. Again, the trick is to observe that for k=1 we have P0=1 for all values of (xt, σ) and so all derivatives have to equal zero. Thus

Rp(p1)[pxxx1]α1α2α3=Rppp(p1)[px1]α3[px1]α2[px1]α1Rpp(p1)[pxx1]α2α3[px1]α1Rpp(p1)[px1]α2[pxx1]α1α3Rpp(p1)[px1]α3[pxx1]α1α2+[xxx]α1α2α3R(p0)

(Rp(p1)[pxxx1]α1α2α3+Rppp(p1)[px1]α3[px1]α2[px1]α1+Rpp(p1)[pxx1]α2α3[px1]α1Rpp(p1)[px1]α2[pxx1]α1α3+Rpp(p1)[px1]α3[pxx1]α1α2)=[xxx]α1α2α3

because R(p0)=1.

Thus we get for k>1

Rp(pk)[pxxxk]α1α2α3=Rppp(pk)[pxk]α3[pxk]α2[pxk]α1Rpp(pk)[pxk]α2α3[pxk]α1Rpp(pk)[pxk]α2[pxk]α1α3Rpp(pk)[pk]α3[pxxk]α1α2+(Rp(p1)[pxxx1]α1α2α3+Rppp(p1)[px1]α3[px1]α2[px1]α1+Rpp(p1)[pxx1]α2α3[px1]α1+Rpp(p1)[px1]α2[pxx1]α1α3+Rpp(p1)[px1]α3[pxx1]α1α2)R(pk1)+(Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1)Rp(pk1)[pxk1]γ3[hx]α3γ3+(Rp(p1)[pxx1]α1α3+Rpp(p1)[px1]α3[px1]α1)Rp(pk1)[pxk1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rp(pk1)[pxk1]γ2[hxx]α2α3γ2+(Rp(p1)[pxx1]α2,α3+Rpp(p1)[px1]α3[px1]α2)Rp(pk1)[pxk1]γ1[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hx]α1γ1+[px1]α2Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rppp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxxk1]γ2γ3[hx]α3γ3[hx]α2γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hxx]α2α3γ2[pxk1]γ1[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxxk1]γ1γ3[hx]α3γ3[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ2[hx]α2γ2[pxk1]γ1[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxxk1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[hx]α3γ3[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[hxx]α2α3γ2[hx]α1γ1+R(p1)Rp(pk1)[pxxk1]γ1γ2[hx]α2γ2[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hxx]α1α2γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hxx]α1α2γ1+R(p1)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hxx]α1α2γ1+R(p1)Rp(pk1)[pxk1]γ1[hxxx]α1α2α3γ1

With a log-transformation R(pt,k)=Mk, Rp(pt,k)=Mk, Rpp(pt,k)=Mk, and Rppp(pt,k)=Mk in the deterministic steady state. Using the expressions for first and second order derivatives of bond prices derived above, we get, after simplifying, the expression stated in the body of the text.

Derivative of pk with respect to (σ, σ, x)

It is possible to show that [Fσσx(xss,0)]α3=0 implies

Et{Rpp(pk)[pxk]α3[pσσk]Rp(pk)[pσσxk]α3+[σσx]R(pk1)+[σσ]Rp(pk1)[pxk1]γ3[hx]α3γ3+2[σx]α3Rp(pk1)[pxk1]γ2[η]ϕ2γ2[t+1]ϕ2+2[σ]Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[t+1]ϕ2+2[σ]Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2+[x]α3Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[t+1]ϕ2[pxk1]γ1[η]ϕ1γ1[t+1]ϕ1+Rppp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[t+1]ϕ2[pxk1]γ1[η]ϕ1γ1[t+1]ϕ1+Rpp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2[pxk1]γ1[η]ϕ1γ1[t+1]ϕ1+Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[t+1]ϕ2[pxxk1]γ1γ3[hx]α3γ3[η]ϕ1γ1[t+1]ϕ1+[x]α3Rp(pk1)[pxxk1]γ1γ2[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1+Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxxk1]γ1γ2[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1+Rp(pk1)[pxxxk1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1+[x]α3Rp(pk1)[pxk1]γ1[hσσ]γ1+Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hσσ]γ1+Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hσσ]γ1+Rp(pk1)[pxk1]γ1[hσσx]α3γ1+[x]α3Rp(pk1)[pσσk1]+Rpp(pk1)[pxk1]γ3[hx]α3γ3[pσσk1]+Rp(pk1)[pσσxk1]γ3[hx]α3γ3}=0

Rp(pk)[pσσxk]α3=Rpp(pk)[pxk]α3[pσσk]+Et[σσx]R(pk1)+[pσσ1]Rp(p1)/R(p0)Rp(pk1)[pxk1]γ3[hx]α3γ3+2Et([σx]α3Rp(pk1)[pxk1]γ2[η]ϕ2γ2[et+1]ϕ2)+2Et([σ]Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[et+1]ϕ2)+2Et([σ]Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[et+1]ϕ2)+[px1]α3Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rppp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxxk1]γ1γ3[hx]α3γ3[η]ϕ1γ1[I]ϕ2ϕ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxxk1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hσσ]γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hσσ]γ1+R(p1)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hσσ]γ1+R(p1)Rp(pk1)[pxk1]γ1[hσσx]α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pσσk1]+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pσσk1]+R(p1)Rp(pk1)[pσσxk1]γ3[hx]α3γ3

where we have used

=R(p1)

[x]α3=[px1]α3Rp(p1)/R(p0)

Et[σσ]=[pσσ1]Rp(p1)/R(p0)

We now compute the terms with derivatives of σ. Here we recall that

[σ]([yt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1+[xt+1]γ1[η]ϕ1γ1[εt+1]ϕ1).

So

2Et([σ]Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2Et([yt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2)+2Et([xt+1]γ1[η]ϕ1γ1[εt+1]ϕ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2[yt+1]β1[gx]γ1β1[η]ϕ1γ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[xt+1]γ1[η]ϕ1γ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2

and

2Et([σ]Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)=2Et([yt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)+2Et([xt+1]γ1[η]ϕ1γ1[εt+1]ϕ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)=2[yt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2+2Et[xt+1]γ1[η]ϕ1γ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2

To compute the [σx]α3 term, we need to find an expression for [σx]α3. Hence

[σx]α3=([yt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[yt+1yt]β1β3[gx]α3β3+[yt+1xt+1]β1γ3[hx]α3γ3+[yt+1xt]β1α3)×[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1+[yt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1[εt+1]ϕ1+([xt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[xt+1yt]γ1β3[gx]α3β3+[xt+1xt+1]γ1γ3[hx]α3γ3+[xt+1xt]γ1α3)×[η]ϕ1γ1[εt+1]ϕ1

So

2Et([σx]α3Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2Et{([yt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[yt+1yt]β1β3[gx]α3β3+[yt+1xt+1]β1γ3[hx]α3γ3+[yt+1xt]β1α3)×[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2+[yt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1[εt+1]ϕ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2+([xt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[xt+1yt]γ1β3[gx]α3β3+[xt+1xt+1]γ1γ3[hx]α3γ3+[xt+1xt]γ1α3)×[η]ϕ1γ1[εt+1]ϕ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2}=2([yt+1yt+1]β1β3[gx]α3β3[hx]α3γ3+[yt+1yt]β1β3[gx]α3β3+[yt+1xt+1]β1γ3[hx]α3γ3+[yt+1xt]β1α3)×[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[yt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2([xt+1yt+1]γ1β3[gx]α3β3[hx]α3γ3+[xt+1yt]γ1β3[gx]α3β3+[xt+1xt+1]γ1γ3[hx]α3γ3+[xt+1xt]γ1α3)×[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2

We finally note that Et([σσx]α3) can be solved for and then substituted out by exploiting the fact that for k=1 we have P0=1 for all values of (xt, σ) and so all derivatives have to equal zero. Thus

Rp(p1)[pσσx1]α3=Rpp(p1)[px1]α3[pσσ1]+Et([σσx]α3)R(p0)

(Rp(p1)[pσσx1]α3+Rpp(p1)[px1]α3[pσσ1])/R(p0)=Et([σσx]α3)

So for k>1 we get

Rp(pk)[pσσxk]α3=Rpp(pk)[pxk]α3[pσσk]+(Rp(p1)[pσσx1]α3+Rpp(p1)[px1]α3[pσσ1])R(pk1)/R(p0)+[pσσ1]Rp(p1)/R(p0)Rp(pk1)[pxk1]γ3[hx]α3γ3+2([yt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[yt+1yt]β1β3[gx]α3β3+[yt+1xt+1]β1γ3[hx]α3γ3+[yt+1xt]β1α3)×[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]φ1ϕ2+2[yt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2([xt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[xt+1yt]γ1β3[gx]α3β3+[xt+1xt+1]γ1γ3[hx]α3γ3+[xt+1xt]γ1α3)×[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[yt+1]β1[gx]γ1β1[η]ϕ1γ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[xt+1]γ1[η]ϕ1γ1Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[yt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2+2Et[xt+1]γ1[η]ϕ1γ1Rp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2+[px1]α3Rp(p1)/R(p0)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rppp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxxk1]γ2γ3[hx]α3γ3[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxxk1]γ1γ3[hx]α3γ3[η]ϕ1γ1[I]ϕ2ϕ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxxk1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxxk1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pxk1]γ1[hσσ]γ1+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pxk1]γ1[hσσ]γ1+R(p1)Rp(pk1)[pxxk1]γ1γ3[hx]α3γ3[hσσ]γ1+R(p1)Rp(pk1)[pxk1]γ1[hσσx]α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk1)[pσσk1]+R(p1)Rpp(pk1)[pxk1]γ3[hx]α3γ3[pσσk1]+R(p1)Rp(pk1)[pσσxk1]γ3[hx]α3γ3

For a logarithm transformation R(pt,k)=Mk, Rp(pt,k)=Mk, Rpp(pt,k)=Mk, and Rppp(pt,k)=Mk. Using the expressions for first and second order derivatives of bond prices derived above, we get, after simplifying,

pσσxk(1,α3)=21yt+1gxηη(pxk1)px1(1,α3)21xt+1ηη(pxk1)px:1(1,α3)+pσσx1(1,α3)+21pxk1ηη(gx)×(yt+1yt+1gxhx(:,α3)+yt+1ytgx(:,α3)+yt+1xt+1hx(:,α3)+yt+1xt(:,α3))+β1=1ny21yt+1(1,β1)pxk1ηηgxx(β1,:,:)hx(:,α3)+21pxk1ηη×(xt+1yt+1gxhx(:,α3)+xt+1ytgx(:,α3)+xt+1xt+1hx(:,α3)+xt+1xt(:,α3))+21yt+1gxηηpxxk1hx(:,α3)+21xt+1ηηpxxk1hx(:,α3)+pxk1ηηpxxk1hx(:,α3)+pxk1ηηpxxk1hx(:,α3)+γ1=1nxη(γ1,:)ηpxxxk1(γ1,:,:)hx(:,α3)+(hσσ)'pxxk1hx(:,αk)+pxk1hσσx(:,α3)+pσσxk1hx(:,α3)

Derivative of pk with respect to (σ, σ, σ)

It is possible to show that Fσσσ(xss, 0)=0 implies

[Fσσσ(xss,0)]=Et{Rp(pk)[pσσσk]+[σσσ]R(pk1)+3[σσ]Rp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2+3[σ]Rpp(pk1)[pxk1]γ3[η]ϕ3γ3[εt+1]ϕ3[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2+3[σ]Rp(pk1)[pxxk1]γ2γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ2γ2[εt+1]ϕ2+R(p1)Rppp(pk1)[pxk1]γ3[η]ϕ3γ3[εt+1]ϕ3[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxk1]γ1[η]ϕ1γ1[εt+1]ϕ1+3R(p1)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxxk1]γ1γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ1γ1[εt+1]ϕ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ2γ2[εt+1]ϕ2[η]ϕ1γ1[εt+1]ϕ1+R(p1)Rp(pk1)[pxk1]γ1[hσσσ]γ1+R(p1)Rp(pk1)[pσσσk1]}=0

We next use the expression for [Mσ] found previously. We also have from differentiation of that

[σσ]=([yt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[εt+1]ϕ3+[yt+1xt+1]β1γ3[η]ϕ3γ3[εt+1]ϕ3)[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1+[yt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[εt+1]ϕ3[η]ϕ1γ1[εt+1]ϕ1+[yt+1]β1[gx]γ1β1[hσσ]γ1+[yt+1]β1[gσσ]β1+[yt]β1[gσσ]β1+([xt+1yt+1]γ1β3[gx]γ3β3[η]ϕ3γ3[εt+1]ϕ3+[xt+1xt+1]γ1γ3[η]ϕ3γ3[εt+1]ϕ3)[η]ϕ1γ1[εt+1]ϕ1+[xt+1]γ1[hσσ]γ1

For [Mσσσ], we exploit the fact P0=1 for all values of (xt, σ) and so all derivatives have to equal zero. Thus Rp(p1)[pσσσ1]=Et{[σσσ]}.

To evaluate the expectations in the term for [Fσσσ(xss, 0)], we define

[m3(εt+1)]ϕ2ϕ3ϕ1={m3(εt+1(ϕ1))ifϕ1=ϕ2=ϕ30otherwise

where m3(ϵt+1) denotes the third moment of ϵt+1(φ1) for φ1=1, 2, …, nε. Notice that m3(ϵt+1) is a nε×nε×nε matrix. Following some simplifications we finally get

Rp(pk)[pσσσk]=+Rp(p1)[pσσσ1]R(pk1)+3[yt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+6[yt+1xt+1]β1γ3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+3[yt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+3[xt+1xt+1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1Rp(pk1)[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+3([yt+1]β1[gx]γ1β1[η]ϕ1γ1+[xt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1Rpp(pk1)[pxk1]γ3[η]ϕ3γ3[pxk1]γ2[η]ϕ2γ2+3([yt+1]β1[gx]γ1β1[η]ϕ1γ1+[xt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1Rp(pk1)[pxxk1]γ2γ3[η]ϕ3γ3[η]ϕ2γ2+R(p1)Rppp(pk1)[pxk1]γ3[η]ϕ3γ3[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[m3(et+1)]ϕ2ϕ3ϕ1+3R(p1)Rpp(pk1)[pxk1]γ2[η]ϕ2γ2[pxxk1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1+R(p1)Rp(pk1)[pxxxk1]γ1γ2γ3[η]ϕ3γ3[η]ϕ2γ2[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1+R(p1)Rp(pk1)[pxk1]γ1[hσσσ]γ1+R(p1)Rp(pk1)pσσσk1

For a logarithm transformation, it is straightforward to show that

pσσσk=pσσσ1+31[yt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+61[yt+1xt+1]β1γ3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+31[yt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[η]ϕ1γ1[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+31[xt+1xt+1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[pxk1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1+31([yt+1]β1[gx]γ1β1[η]ϕ1γ1+[xt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1[pxk1]γ3[η]ϕ3γ3[pxk1]γ2[η]ϕ2γ2+31([yt+1]β1[gx]γ1β1[η]ϕ1γ1+[xt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1[pxxk1]γ2γ3[η]ϕ3γ3[η]ϕ2γ2+[pxk1]γ3[η]ϕ3γ3[pxk1]γ2[η]ϕ2γ2[pxk1]γ1[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1+3[pxk1]γ2[η]ϕ2γ2[pxxk1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1+[pxxxk1]γ1γ2γ3[η]ϕ3γ3[η]ϕ2γ2[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1+[pxk1]γ1[hσσσ]γ1+pσσσk1

Appendix B

Matlab implementation of the POP method

The approximation method presented in the body of the text is implemented in Matlab. For the first perturbation step, we apply the codes accompanying Schmitt-Grohé and Uribe (2004) to compute first and second order derivatives, while the routines underlying Andreasen (2012b) are used for all third-order terms. For the second perturbation step, the user only needs to specify the stochastic discount factor in Anal_PricingKernel_derivatives.m and the position of the one-period bond price in yt. Analytical derivatives of the pricing kernel are then computed based on symbolic differentiation, and these derivatives are evaluated in the steady state by num_eval_PricingKernel.m. Bond prices are then computed in Get_Bond_Prices_3rd.m up to third order, either for the level of bond prices or for a log-transformation.

Appendix C

Closed-form solution to the endowment model with habits

For the considered habit model, we have

t+1:=β(Ct+1hZt+1)γ(CthZt)γ=β(1hexp{zt+1})γ(1hexp{zt})γexp{γxt+1}.

A closed-form solution for zero-coupon bond prices is given by (see Zabczyk 2014)

Ptk=(1hexp{zt})γβkexpγkμ+xtμρ(1ρk)(1ρ)×n=0+γn(h)exp(1φ)kztμ1(1φ)kφ(xtμ)ρρk(1φ)kρ(1φ)n×j=1k𝕃ξγ(1ρj)(1ρ)+nρj(1φ)jρ(1φ).

Here, Lξ is the Laplace transform of ξ, and (αn) denotes a generalized binomial coefficient, i.e.,

(αn):=k=1n(αk+1)/k,forn>0and(α0):=1,

where αℝ and n∈𝒩. The condition for convergence of this solution is βexp{Var(ξ)2(γ1ρ)2}<1.

References

Amisano, G., and O. Tristani. 2009. “A DSGE Model of the Term Structure with Regime Shifts.” Working paper.Search in Google Scholar

Andreasen, M. M. 2010. “Stochastic Volatility and DSGE Models.” Economics Letters 108: 7–9.10.1016/j.econlet.2010.03.007Search in Google Scholar

Andreasen, M. M. 2012a. “An Estimated DSGE Model: Explaining Variation in Nominal Term Premia, Real Term Premia, and Ination Risk Premia.” European Economic Review 56: 1656–1674.Search in Google Scholar

Andreasen, M. M. 2012b. “On the Effects of Rare Disasters and Uncertainty Shocks for Risk Premia In Non-Linear DSGE Models.” Review of Economic Dynamics 15: 295–316.10.1016/j.red.2011.08.001Search in Google Scholar

Anh, L., Q. Dai, and K. Singleton. 2010. “Discrete-Time Dynamic Term Structure Models with Generalized Market Prices of Risk.” Review of Financial Studies 23: 2184–2227.10.1093/rfs/hhq007Search in Google Scholar

Arouba, S. B., J. Fernández-Villaverde, and J. F. Rubio-Ramírez. 2005. “Comparing Solution methods for Dynamic Equilibrium Economies.” Journal of Economic Dynamics and Control 20 (2): 891–910.Search in Google Scholar

Backus, D. K., A. W. Gregory, and S. E. Zin. 1989. “Risk Premiums in the Term Structure: Evidence from Artificial Economics.” Journal of Monetary Economics 24: 371–399.10.1016/0304-3932(89)90027-5Search in Google Scholar

Bansal, R., and A. Yaron. 2004. “Risks for The Long Run: A Potential Resolution of Asset Pricing Puzzles.” Journal of Finance 59 (4): 1481–1509.10.1111/j.1540-6261.2004.00670.xSearch in Google Scholar

Barillas, F., L. P. Hansen, and T. J. Sargent. 2009. “Doubts or Variability.” Journal of Economic Theory 144: 2388–2418.10.1016/j.jet.2008.11.014Search in Google Scholar

Beaudry, P., and E. V. Wincoop. 1996. “The Intertemporal Elasticity of Substitution: An Exploration Using A US Panel of State Data.” Economica 63: 495–512.10.2307/2555019Search in Google Scholar

Bekaert, G., S. Cho, and A. Moreno. 2010. “New Keynesian Macroeconomics and The Term Structure.” Journal of Money, Credit and Banking 42 (1): 33–62.10.1111/j.1538-4616.2009.00277.xSearch in Google Scholar

Benigno, P. (2007). “Discussion of Equilibrium Yield Curves.” In NBER Macroeconomics Annual 2006, edited by Monika Piazzesi and Martin Schneider, 443–457. Cambridge (MA), London: MIT Press.Search in Google Scholar

Binsbergen, J. H. V., J. Fernandez-Villaverde, R. S. Koijen, and J. Rubio-Ramirez. 2012. “The Term Structure of Interest Rates in a DSGE Model with Recursive Preferences.” Journal of Monetary Economics 59: 634–648.10.1016/j.jmoneco.2012.09.002Search in Google Scholar

Caldara, D., J. Fernandez-Villaverde, J. F. Rubio-Ramirez, and W. Yao. 2012. “Computing DSGE Models with Recursive Preferences and Stochastic Volatility.” Review of Economic Dynamics 15: 188–206.10.1016/j.red.2011.10.001Search in Google Scholar

Campbell, J. Y., and R. J. Shiller. 1991. “Yield Spread and Interest Rate Movements: A Bird′s Eye View.” The Review of Economic Studies 58 (3): 495–514.10.2307/2298008Search in Google Scholar

Chernov, M., A. R. Gallant, E. Ghysels, and G.Tauchen. 2003. “Alternative Models for Stock Price Dynamics.” Journal of Econometrics 116: 225–257.10.1016/S0304-4076(03)00108-8Search in Google Scholar

Cochrane, J. H. 2001. Asset Pricing. Princeton, NJ: Princeton University Press.Search in Google Scholar

Cochrane, J. H., and M. Piazzesi. 2005. “Bond Risk Premia.” American Economic Review 95 (1): 138–160.10.1257/0002828053828581Search in Google Scholar

De Paoli, B., A. Scott, and O. Weeken. 2010. “Asset Pricing Implications of A New Keynesian Model.” Journal of Economic Dynamic and Control 34 (10): 2056–2073.10.1016/j.jedc.2010.05.012Search in Google Scholar

Doh, T. 2011. “Yield Curve in An Estimated Nonlinear Macro Model.” Journal of Economic Dynamic and Control 35 (8): 1229–1244.10.1016/j.jedc.2011.03.003Search in Google Scholar

Duffie, D., and K. J. Singleton. 1993. “Simulated Moments Estimation of Markov Models of Asset Prices.” Econometrica 61 (4): 929–952.10.2307/2951768Search in Google Scholar

Epstein, L. G., and S. E. Zin. 1989. “Substitution, Risk Aversion, and The Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework.” Econometrica 57 (4): 937–969.10.2307/1913778Search in Google Scholar

Gürkaynak, R., B. Sack, and J. Wright. 2007. “The U.S. Treasury Yield Curve: 1961 to The Present.” Journal of Monetary Economics 54: 2291–2304.10.1016/j.jmoneco.2007.06.029Search in Google Scholar

Hordahl, P., O. Tristani, and D. Vestin. 2008. “The Yield Curve and Macroeconomic Dynamics.” The Economic Journal 118: 1937–1970.10.1111/j.1468-0297.2008.02197.xSearch in Google Scholar

Jermann, U. J. 1998. “Asset Pricing in Production Economics.” Journal of Monetary Economics 41: 257–275.10.1016/S0304-3932(97)00078-0Search in Google Scholar

Juillard, M., and S. Villemot. 2011. “Multi-Country Real Business Cycle Models: Accuracy Tests and Test Bench.” Journal of Economic Dynamic and Control 35: 178–185.10.1016/j.jedc.2010.09.011Search in Google Scholar

Kamenik, O. 2005. “Solving SDGE Models: A New Algorithm for the Sylvester Equation.” Computational Economics 25: 167–187.10.1007/s10614-005-6280-ySearch in Google Scholar

Kim, J., S. Kim, E. Schaumburg, and C. A. Sims. 2008. “Calculating and Using Second-Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models.” Journal of Economic Dynamics and Control 32: 3397–3414.10.1016/j.jedc.2008.02.003Search in Google Scholar

Klein, P. 2000. “Using the Generalized Schur Form to Solve A Multivariate Linear Rational Expectations Model.” Journal of Economic Dynamic and Control 24: 1405–1423.10.1016/S0165-1889(99)00045-7Search in Google Scholar

Malloy, C. J., T. J. Moskowitz, and A. Vissing-Jørgensen. 2009. “Long-Run Stockholder Consumption Risk and Asset Returns.” The Journal of Finance LXIV (6): 2427–2479.10.1111/j.1540-6261.2009.01507.xSearch in Google Scholar

Martin, I. W. R. 2008. “Disasters and The Welfare Cost of Uncertainty.” American Economic Review: Papers and Proceedings 98 (2): 74–78.10.1257/aer.98.2.74Search in Google Scholar

Piazzesi, M., and M. Schneider. 2007. “Equilibrium Yield Curves.” NBER Macro Annual 21: 389–442.10.1086/ma.21.25554958Search in Google Scholar

Rudebusch, G. D., and E. T. Swanson. 2008. “Examining the Bond Premium Puzzle with a DSGE Model.” Journal of Monetary Economics 55: 111–126.10.1016/j.jmoneco.2008.07.007Search in Google Scholar

Rudebusch, G. D., and E. T. Swanson. 2012. “The Bond Premium in a DSGE Model With Long-Run Real and Nominal Risks.” American Economic Journal: Macroeconomics 4 (1): 1–43.10.1257/mac.4.1.105Search in Google Scholar

Schmitt-Grohé, S., and M. Uribe. 2004. “Solving Dynamic General Equilibrium Models Using a Secondorder Approximation to the Policy Function.” Journal of Economic Dynamics and Control 28: 755–775.10.1016/S0165-1889(03)00043-5Search in Google Scholar

Swanson, E. 2012. “Risk Aversion and the Labor Margin in Dynamic Equilibrium Models.” American Economic Review 102 (4): 1663–1691.10.1257/aer.102.4.1663Search in Google Scholar

Swanson, E., G. Anderson, and A. Levin. 2005. “Higher-Order Perturbation Solutions to Dynamic, Discrete-Time Rational Expectations Models.” Working Paper .10.2139/ssrn.892369Search in Google Scholar

Tsionas, E. G. 2003. “Exact Solution of Asset Pricing Models with Arbitrary Shock Distributions.” Journal of Economic Dynamic and Control 27: 843–851.10.1016/S0165-1889(02)00017-9Search in Google Scholar

Uhlig, H. 2007. “Explaining Asset Prices with External Habits and Wage Rigidities.” American Economic Review: Papers and Proceedings 97 (2): 239–243.10.1257/aer.97.2.239Search in Google Scholar

Vissing-Jorgensen, A. 2002. “Limited Asset Market Participation and the Elasticity of Intertemporal Substitution.” Journal of Political Economy 110 (4): 825–853.10.1086/340782Search in Google Scholar

Wachter, J. A. 2006. “A Consumption-Based Model of the Term Structure of Interest Rates.” Journal of Financial Economics 79: 365–399.10.1016/j.jfineco.2005.02.004Search in Google Scholar

Weil, P. 1990. “Nonexpected Utility in Macroeconomics.” Quarterly Journal of Economics 105 (1): 29–42.10.2307/2937817Search in Google Scholar

Wu, T. 2006. “Macro factors and the Affine Term Structure of Interest Rates.” Journal of Money, Credit, and Banking 30: 1847–1875.10.1353/mcb.2006.0097Search in Google Scholar

Zabczyk, P. 2014. “Asset Prices Under Persistent Habits and Arbitrary Shocks to Consumption Growth: Closed-form Solutions and Their Implications.” Working Paper.Search in Google Scholar


Article note

A previous version of the paper was entitled: “An Efficient Method of Computing Higher Order Bond Price Perturbation Approximations.”



Supplemental Material

The online version of this article (DOI: 10.1515/snde-2012-0005) offers supplementary material, available to authorized users.


Published Online: 2014-2-27
Published in Print: 2015-2-1

©2015 by De Gruyter

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