Backus, Chernov & Zin

Notes on Recursive Models, part 3January 16, 2012; last revision July 17, 2012

Value function approximation:

log ut b0 + b1 logt(gt+1ut+1) (1)

The pricing kernel (marginal rate of substitution) is

mt+1 = (ct+1/ct)1 [Ut+1/t(Ut+1)]

= g1t+1 [gt+1ut+1/t(gt+1ut+1)] . (2)

1 Recursive models with persistent jumps

1.1 The pricing kernel with recursive preferences

We derive the pricing kernel for a representative agent model with recursive preferences,consumption growth dynamics, stochastic volatility, and jumps with time-varying intensity.The models in Sections ?? and ?? are special cases.

We posit a consumption growth process with

log gt = log g + (B)v1/2t1wgt + (B)zgt

vt = v + (B)wvt

ht = h+ (B)wht.

Here {wgt, wvt, wht} are N (0, 1) and independent of each other and over time. zgt is a jumpcomponent. Conditional on the number of jumps j, zgt is normal: zgt|j N (j, j2). Theprobability of j 0 jumps at date t is eht1hjt1/j!. This covers most of the models inthe literature, but we can generalize further by (for example) adding dynamics to the jumpcomponent, jumps to volatility and intensity, or interdependence among the three processes.

Given a value of b1, we solve the model by using the Bellman equation (1) to character-ize the value function and substituting the result into the marginal rate of substitution cumpricing kernel (2). Our use of value functions mirrors Hansen, Heaton, and Li (2008) andHansen and Scheinkman (2009). Our use of lag polynomials mirrors Hansen and Sargent(1980) and Sargent (1987).

We use two results about certainty equivalents, evaluations of the power certainty equiv-alent function (??). The first one applies to normal risks. If wt+1|t N (1t, 2t), then thelog of the certainty equivalent of eawt+1 is logt(e

awt+1) = 1t + (/2)a22t. The second

applies to Poisson-normal mixtures like the jump component zgt+1. The analogous certainty

equivalent result is logt(eazt+1) = ht(e

a+(a)2/2 1)/.We find the value function by guess and verify:

1

Guess. We guess a value function of the form

log ut = log u+ pg(B)v1/2t1wgt + pv(B)wvt + ph(B)wht + pz(B)zgt

with parameters (u, pg, pv, ph, pz) to be determined.

Compute. We need the certainty equivalent t(gt+1ut+1) for (1). Given our guess,log(gt+1ut+1) is

log(gt+1ut+1) = log g + log u+ [(B) + pg(B)]v1/2t wgt+1 + [(B) + pz(B)]zgt+1

+ pv(B)wvt+1 + ph(B)wht+1

= log(gu) + [(B) + pg(B) (0 + pg0)]v1/2t wgt+1+ [pv(B) pv0]wvt+1 + [ph(B) ph0]wht+1 + [(B) + pz(B) (0 + pz0)]zgt+1+ (0 + pg0)v

1/2t wgt+1 + pv0wvt+1 + ph0wht+1 + (0 + pz0)zgt+1.

We use a clever trick here from Sargent (1987, Section XI.19): we rewrite (for example)pv(B)wvt+1 as the sum of (pv(B)pv0)wvt+1 and pv0wvt+1. As of date t, the former isconstant (despite appearances, it doesnt depend on wvt+1) and the latter is not. Theother terms are treated the same way. As a result, the last line consists of innovations,the others of (conditional) constants. The certainty equivalent treats them differently:

logt(gt+1ut+1) = log(gu) + [(B) + pg(B) (0 + pg0)]v1/2t wgt+1+ [(B) + pz(B) (0 + pz0)]zgt+1+ [pv(B) pv0]wvt+1 + [ph(B) ph0]wht+1+ (/2)(0 + pg0)

2vt + (/2)(p2v0 + p

2h0)

+ [(e(0+pz0)+((0+pz0))2/2 1)/]ht

= log(gu) + [(B) + pg(B) (0 + pg0)]v1/2t wgt+1+ [(B) + pz(B) (0 + pz0)]zgt+1+ [pv(B) pv0]wvt+1 + [ph(B) ph0]wht+1+ (/2)(0 + pg0)

2[v + (B)wvt] + (/2)(p2v0 + p

2h0)

+ [(e(0+pz0)+((0+pz0))2/2 1)/][h+ (B)wht].

Verify. We substitute the certainty equivalent into (1) and solve for the parameters.Matching like terms, we have

constant : log u = b0 + b1[log(gu) + (/2)(p2v0 + p

2h0)]

v1/2t1wgt : pg(B) = b1

[(B) + pg(B) (0 + pg0)

B

]wvt : pv(B) = b1

[pv(B) pv0

B+ (/2)(0 + pg0)

2(B)

]wht : ph(B) = b1

[ph(B) ph0

B+ [(e(0+pz0)+((0+pz0))

2/2 1)/](B)]

zt : pz(B) = b1

[(B) + pz(B) (0 + pz0)

B

].

2

We skipped the constants v and h in the first equation, because were going to includethem in vt and ht below.

We solve the equations one at a time. The second equation leads to forward-lookinggeometric sums like those in Hansen and Sargent (1980, Section 2) and Sargent(1987, Section XI.19). Following their lead, we set L = b1 to get 0 + pg0 = (b1).The other coefficients of pg(B) are of no concern to us: they dont show up inthe pricing kernel. The third equation has a similar structure. Setting B = b1gives us pv0 = (/2)(b1)

2b1(b1). In the fourth equation, setting B = b1 gives usph0 = [(e

(0+pz0)+((0+pz0))2/2 1)/]b1(b1). The last equation is similar to thesecond and yields: 0 + pz0 = (b1). Finally, the first equation implies

log u = (1 b1)1{b0 + b1

[log g + (/2)(p2v0 + p

2h0)]}.

We can substitute for pv0 and ph0 with some loss of simplicity.

Now that we know the value function, we construct the pricing kernel from (2). Onecomponent is

log(gt+1ut+1) logt(gt+1ut+1) = (b1)v1/2t wgt+1 + (b1)zgt+1 + pv0wvt+1 + ph0wht+1 (/2)[(b1)2vt + (p2v0 + p2h0)] [(e(b1)+((b1))

2/2 1)/]ht,

a combination of innovations to future utility (the first line) and adjustments for risk (thesecond). The pricing kernel then becomes

logmt+1 = log + ( 1) log g ( )(/2)(p2v0 + p2h0)+ [( 1)0 + ( )(b1)]v1/2t wgt+1 + [( 1)0 + ( )(b1)]zgt+1+ ( 1)[(B)/B]+v1/2t1wgt + ( 1)[(B)/B]+zgt+ ( )[pv0wvt+1 (/2)(b1)2vt]+ ( ){ph0wht+1 [(e(b1)+((b1))

2/2 1)/]ht}.

The examples in Sections ?? and ?? follow from setting various combinations of (B), (B),and (B) equal to zero and rearranging terms.

Therefore, the conditional entropy is

Lt(mt+1) = logEtmt+1 Et logmt+1= [( )(b1) + ( 1)0]2vt/2+ (e[()(b1)+(1)0]+[()(b1)+(1)0]

22/2 [( )(b1) + ( 1)0] 1)ht+ ( )2([b1(b1)(b1)2/2]2 + [b1(b1)(e(b1)+

2(b1)22/2 1)/]2)/2

1.2 Horizon dependence with recursive models

We derive horizon dependence for the model described in Appendix 1.1. The results areused on Sections ?? and ??. The pricing kernel has the form

logmt+1 = logm+ ag(B)(vt/v)1/2wgt+1 + av(B)wvt+1 + az(B)zgt+1 + ah(B)wht+1

3

vt = v + (B)wvt

ht = h+ (B)wht

with {wgt, wvt, zgt, wht} defined above. This differs from the Vasicek model in the roles ofvt in scaling wgt and of the intensity ht in the jump component zgt. For future reference,we define the partial sums Axn =

nj=0 axj for x = g, v, h, z.

We derive multiperiod entropy from bond prices as described in (??). Suppose bondprices have the form

log qnt+1 = n0 +

ng (B)(vt/v)

1/2wgt+1 + nv (B)wvt+1 +

nh (B)wht+1 +

nz (B)zt+1. (3)

Then the pricing relation (??) tells us

log qn+1t = logEt(mt+1qnt+1).

The log of the term in brackets is

logmt+1 + log qnt+1 = logm+

n0 +

[ag(B) +

ng (B)

](vt/v)

1/2wgt+1 + [av(B) + nv (B)]wvt+1

+ [az(B) + nz (B)] zgt+1 + [ah(B) +

nh (B)]wht+1.

Evaluating the expectation and lining up terms gives us

n+10 = logm+ n0 +

[(ag0 +

ng0)

2 + (av0 + nv0)

2 + (ah0 + nh0)

2]/2 + h(e(az0+

nz0)+((az0+

nz0))

2/2 1)n+1gj =

ngj+1 + agj+1

n+1vj = nvj+1 + avj+1 + (ag0 +

ng0)

2j/(2v)

n+1hj = nhj+1 + ahj+1 + (e

(az0+nz0)+((az0+nz0))

2/2 1)jn+1zj =

nzj+1 + azj+1.

The second and forth equations mirror the Vasicek model:

ngj =ni=1

agj+i = Agn+j Agj

nzj =ni=1

azj+i = Azn+j Azj .

The third equation implies

nvj = Avn+j Avj + (2v)1n1i=0

j+n1iA2gi.

The fourth equation implies

nhj = Ahn+j Ahj +n1i=0

j+n1i(eAzi+(Azi)

2/2 1).

4

And at long last, the first equation implies

n0 = n logm+1

2

nj=1

A2gj1 +1

2

nj=1

A2zj1 + hnj=1

(eAzj1+(Azj1)2/2 1)

+1

2

nj=1

[Avj1 + (2v)

1j2i=0

j2iA2gi

]2+

1

2

nj=1

[Ahj1 +

j2i=0

j2i(eAzi+(Azi)

2/2 1)

]2.

If subscripts are beyond their bounds, the expression is zero.Horizon dependence is determined by unconditional expectations of yields. The last

component in the log - price (3) is predictable, so we have to take this into account:

E(nz (B)zt+1) = hnz (1) = h

j=0

(Azn+j Azj).

Horizon dependence is therefore

H(n) = (2n)1nj=1

(A2gj1 A2g0) + (2n)1nj=1

(A2zj1 A2z0)

+ hn1nj=1

(eAzj1+(Azj1)

2/2 eAz0+(Az0)2/2)

+ (2n)1nj=1

(Avj1 + (2v)1 j2i=0

j2iA2gi

)2A2v0

+ (2n)1

nj=1

(Ahj1 + j2i=0

j2i(eAzi+(Azi)

2/2 1)

)2A2h0

+ n1hnz (1) h1z (1).

The first term is a Vasicek-like expression for wgt. The second reflects the interactionbetween wgt and volatility. The last term is the interaction between the jump componentzgt and the intensity process.

2 Recursive models based on ARG(1) processes

2.1 The pricing kernel

We like the simplicity and transparency of linear processes; expressions like (b1) summarizeclearly and cleanly the impact of volatility dynamics. A less appealing feature is that theyallow the conditional variance vt and intensity ht to be negative. Here we describe andsolve an analogous model based on ARG(1) processes, discrete-time analogs of well-knowncontinuous-time processes. The analysis parallels Appendix 1.1.

5

Consider the consumption process

log gt = log g + (B)v1/2t1wgt + zgt

vt ARG(cv, v, v)ht ARG(ch, h, h)

The autoregressive Gamma of order assumption about vt and ht implies that

vt = vcv + vvt1 + vt

ht = hch + hht1 + ht,

where vt and ht are m.d.s. with conditional variances equal to vc2v + 2vcvvt1 and

hc2h + 2hchht1. The cgfs for vt and ht are:

kt(s; vt+1) = vs(1 scv)1vt v log(1 scv)kt(s;ht+1) = hs(1 sch)1ht h log(1 sch)

If one selects the ARG inputs as follows:

vt ARG(2v/2, v, (1 v)v/(2v/2))ht ARG(2h/2, h, (1 h)h/(2h/2))

then

vt = (1 v)v + vvt1 + vtht = (1 h)h+ hht1 + ht

with variances of shocks equal to 2v [(1 v)v/2 + vvt1] and 2h[(1 h)h/2 + hht1]and cgfs:

kt(s; vt+1) = vs(1 s2v/2)1vt (1 v)v log(1 s2v/2)/(2v/2)kt(s;ht+1) = hs(1 s2h/2)1ht (1 h)h log(1 s2h/2)/(2h/2)

We start with the value function:

Guess. We guess a value function of the form

log ut = log u+ pg(B)v1/2t1wgt + pvvt + phht

with parameters to be determined.

Compute. Since log(gt+1ut+1) is

log(gt+1ut+1) = log(gu) + [(B) + pg(B)]v1/2t wgt+1 + zgt+1 + pvvt+1 + phht+1

= log(gu) + [(B) + pg(B) (0 + pg0)]v1/2t wgt+1+ (0 + pg0)v

1/2t wgt+1 + zgt+1 + pvvt+1 + phht+1,

6

its certainty equivalent is

logt(gt+1ut+1) = log(gu) + [(B) + pg(B) (0 + pg0)]v1/2t wgt+1+ (/2)(0 + pg0)

2vt + [(e+()2/2 1)/]ht

v/ log(1 pvcv) + vpv(1 pvcv)1vt h/ log(1 phch) + hph(1 phch)1ht

Verify. We substitute the certainty equivalent into (1) and collect similar terms:

constant : log u = b0 + b1[log(gu) v/ log(1 pvcv) h/ log(1 phch)]

v1/2t1wgt : pg(B) = b1

[(B) + pg(B) (0 + pg0)

B

]vt : pv = b1[(/2)(0 + pg0)

2 + vpv(1 pvcv)1]

ht : ph = b1[(e+()2/2 1)/+ hph(1 phch)1].

The second equation is the same one we saw in Appendix 1.1 and has the samesolution: 0 + pg0 = (b1).

The third and fourth equations are new. Their quadratic structure is different fromanything weve seen so far, but familiar to anyone who has worked with square-rootprocesses. The quadratic terms arise because risk to future utility depends on ht andvt through their innovations. We solve them using value function iterations: startingwith zero, we substitute a value into the right side and generate a new value on theleft. If this converges, we have the solution as the limit of a finite-horizon problem.

A more formal approach is to solve the quadratic equations directly and select theappropriate root. The third equation implies

0 = cvp2v + bpvpv + b1(0 + pg0)

2/2,

bpv = b1v b1cv2(0 + pg0)2/2 1.

It has two real roots :

pv =bpv

[b2pv 2b1cv2(0 + pg0)2

]1/22cv

.

If the variance of log gt is equal to zero, pv = 0 only if we select the smaller root.

Similar logic applies to ph. The fourth equation implies:

0 = chp2h + bphph + b1(e

+()2/2 1)/,bph = b1h b1ch(e+()

2/2 1) 1.

The two roots are

ph =bph

[b2ph 4b1ch(e+()

2/2 1)]1/2

2ch.

7

Again, the discriminant must be positive. If it is, stability leads us to choose thesmaller root.

Given these value function coefficients, the pricing kernel is

logmt+1 = log + ( 1) log g + ( )(v log(1 pvcv)/+ h log(1 phch)/)+ ( 1)zgt+1 + [( 1)0 + ( )(b1)]v1/2t wgt+1 + ( 1)[(B)/B]+v

1/2t1wgt

+ ( ){pvvt+1 [(0 + pg0)2/2 + vpv(1 cvpv)1]vt

}+ ( )

{phht+1 [(e+()

2/2 1)/+ hph(1 chph)1]ht}.

8

Recursive models with persistent jumpsThe pricing kernel with recursive preferencesHorizon dependence with recursive models

Recursive models based on ARG(1) processesThe pricing kernel