of America Merrill Lynch, New York, USA School of Economics, London, UK c CEPR, London, UK

b London

Abstract The Appendix provides details of the rational expectations solution, interpretation of risk premia, approximate bond valuation, computation of the likelihood function and long-run means of the state variables, local identification of structural parameters, importance of using bonds for model estimation, and fit of the estimated model.

Appendix A. Rational Expectations Solution This appendix suggests a method for finding a rational expectations solution to the problem (2.1)-(2.3). First, the problem can be rewritten in the following compact matrix form m m m e m d B0 (sm t )xt = m0 (st ) + B−1 (st )xt−1 + B1 (st )E t xt+1 + Γ(st , st , st )εt

(A.1)

where

B0

=

B1

=

m0

=

1 0 −δ 1 −(1 − ρ(sm ))β(sm ) 0 t t µg φ 0 µπ 0 (1 − ρ(sm ))α(sm ) t

t

φ 0 , 1 0 0 , 0

1 − µg 0 0 B−1 = 0 1 − µπ 0 0 0 ρ(sm t ) σg (se ) 0 0 t e Γ = , 0 σπ (st ) 0 0 0 σr (sdt )

,

(mg , mπ , mr )0 and εt = (εgt , επt , εrt )0

Since matrix B0 (sm t ) is generally non-singular, (A.1) can be recast in a more convenient form as xt = m1 (S t ) + F1 (S t )xt−1 + Et [A1 (S t )xt+1 ] + Σ1 (S t )εt

(A.2)

m m −1 m m −1 m m −1 m e m d with m1 (S t ) = B−1 0 (st )m0 (st ), F 1 (S t ) = B0 (st )B−1 (st ), A1 (S t ) = B0 (st )B1 (st ), Σ1 (S t ) = B0 (st )Γ(st , st , st ) and where d S t = (set , sm t , st ) denotes the compound regime variable.

We first demonstrate that the rational expectations solution (A.2) has the form of a regime-switching vector autoregressive process (2.4). Substituting (2.4) in (A.2) and keeping in mind that S t is independent of the state variable xt we can see that ∗ Corresponding

author: London School of Economics, Department of Finance, Houghton Street, London WC2A 2AE. Email addresses: [email protected] (Ruslan Bikbov), [email protected] (Mikhail Chernov)

Preprint submitted to Journal of Econometrics

February 9, 2017

(2.4) is the solution of (A.2) if there exist reduced-form parameters µ, Φ and Σ that satisfy the following set of equations: [I − A1 (S t )Et Φ(S t+1 )] Φ(S t ) = F1 (S t ) [I − A1 (S t )Et Φ(S t+1 )] µ(S t ) = m1 (S t ) + Et µ(S t+1 ) Σ(S t ) = [I − A1 (S t )Et Φ(S t+1 )]−1 Σ1 (S t ), which can be rewritten in terms of probabilities that regimes will change πi j as [I − A1 (i)

X πi j Φ( j)]Φ(i) = F1 (i)

(A.3)

j

[I − A1 (i)

X

πi j Φ( j)]µ(i) = m1 (i) +

X

j

πi j µ( j)

(A.4)

j

Σ(i) = [I − A1 (i)

X

πi j Φ( j)]−1 Σ1 (i).

(A.5)

j

To find the solution directly from the above equations, one would have to solve the system of coupled quadratic matrix equations (A.3) for Φ(i) and then use Φ(i) to find µ(i) and Σ(i) from (A.4) and (A.5). However, because the direct solution of (A.3) is complicated, we do not follow this approach. Instead, we generalize the forward method proposed in Cho and Moreno (2011) to the case of changing regimes. This method solves for a particular fundamental equilibrium, referred to as forward solution. The idea of the solution method is to substitute the left-hand side of (A.2) into the expectation in the right-hand side recursively. At step n of this recursive procedure, we obtain xt = mn (S t ) + Fn (S t )xt−1 + Et [An (S t+1 , . . . , S t+n−1 )xt+n ] + Σn (S t )εt

(A.6)

By substituting (A.6) into the right-hand side of (A.2) we obtain the following recursive formulae for coefficients mn , Fn , An and Σn : Cn (S t ) ≡

I − A1 (S t )Et Fn (S t+1 )

Fn+1 (S t ) = Cn−1 (S t )F1 (S t ) mn+1 (S t ) = Cn−1 (S t ) [m(S t ) + A1 (S t )Et mn (S t+1 )] An+1 (S t , . . . , S t+n ) = Cn−1 A1 (S t )An (S t , . . . , S t+n−1 ) Σn+1 (S t ) = Cn−1 (S t )Σ1 (S t ) Suppose the above iteration procedure converges, i.e., forward convergence condition (FCC) is satisfied: there exist Φ(S t ), µ(S t ) and Σ(S t ) such that 1. For S t = 1, . . . , N lim Fn (S t ) = Φ(S t ) n→∞

2. For S t = 1, . . . , N lim µn (S t ) = µ(S t ) n→∞

3. For S t = 1, . . . , N lim Σn (S t ) = Σ(S t ) n→∞

In addition, consider the following no-bubble condition (NBC): lim Et [An (S t+1 , . . . , S t+n−1 )xt+n ] = 0.

n→∞

2

(A.7)

Proposition 3 in Cho (2011) implies that the forwards solution (2.4) to the model (A.2) and, therefore, of the original model (2.1)-(2.3), exists if and only if the model satisfies the FCC. It is the unique minimum state variable (MSV) solution that satisfies the NBC. Further, Cho establishes conditions under which (i) the forward solution is mean-square stable (MSS); (ii) there are no other MSS solutions with non-fundamental components. These results imply that if all these conditions hold, then the unique MSV solution is determinate in MSS sense and, therefore, the forward solution is the determinate equilibrium. We have implemented Cho’s method (using the code available from his website) at the estimated parameter values and established that the forward solution exists and it is the the determinate equilibrium.

Appendix B. Risk premia This appendix presents the motivation for our formulation of risk premia Λt,t+1 in (2.11). From the perspective of structural modeling, (2.11) can be seen as a local approximation of a general nonlinear form of the stochastic discount factor log Mt,t+1 = g(xt+1 , xt ) that does not explicitly depend on regime S t : log Mt,t+1

≈ = = −

∂g (xt+1 − xt ) ∂x ∂g ∂g ∂g g(xt , xt ) + µ(S t+1 ) xt (Φ(S t+1 ) − I) + Σ(S t+1 )εt+1 ∂x ∂x ∂x !0 ∂g ∂g 1 ∂g ∂g g(xt , xt ) + µ(S t+1 ) + xt (Φ(S t+1 ) − I) + Σ(S t+1 ) Σ(S t+1 ) ∂x ∂x 2 ∂x ∂x !0 1 ∂g ∂g ∂g Σ(S t+1 ) Σ(S t+1 ) + Σ(S t+1 )εt+1 2 ∂x ∂x ∂x g(xt , xt ) +

It must be the case that Et [Mt,t+1 ] = exp(−rt ). Using log-linearization, this property implies that " !0 # X ∂g ∂g 1 ∂g ∂g πS t , j −rt ≈ g(xt , xt ) + µ( j) + (Φ( j) − I)xt + Σ( j) Σ( j) ∂x ∂x 2 ∂x ∂x j

(B.1)

(B.2)

On the basis of these expressions, we conclude that the continuous-time formulation of (B.1) would imply the reduced-form formulation of the stochastic discount factor in the expressions (2.10)-(2.11), where Π(xt ) =

∂g ∂x (xt , xt ).

In discrete time, the

drift of the log stochastic discount factor, that is, the next-to-last line in (B.1), must be close to −rt , because its expectation is equal to −rt .

Appendix C. Bond Pricing In order to facilitate the bond pricing we introduce risk-neutral notation. Following Dai et al. (2007), we introduce an equivalent risk-neutral measure Q such that the Radon-Nikodym derivative is given by ! ! dQ 1 = exp − Λ0t,t+1 Λt,t+1 − Λ0t,t+1 εt+1 dP t,t+1 2

(C.1)

where P denotes the physical measure. It can be shown that a random variable εQ t+1 = εt+1 − Λt,t+1 has a standard Gaussian distribution under Q.1 Thus, the risk-neutral dynamics of the state variable xt has the following form: xt = µQ (S t ) + ΦQ (S t )xt−1 + Σ(S t )εQ t 1 See,

(C.2)

for example, Dai et al. (2007) for details. Dai et al. use a different timing for regime changes but the measure can be changed in the same way for

the timing used in our paper.

3

where the risk-neutral parameters are µQ (S t ) =

µ(S t ) − Σ(S t )Σ0 (S t )Π0

(C.3)

ΦQ (S t )

Φ(S t ) − Σ(S t )Σ0 (S t )Π x

(C.4)

=

Standard arguments then imply that the price of a zero-coupon bond with maturity n is given by Bnt (xt , S t ) = E Q [e−

Pn−1 i=0

rt+i

|xt , S t ]

(C.5)

with rt = δ0 xt and δ = (0, 0, 1)0 . We can reinterpret the price (C.5) as being equal to Bnt (x, i) = E Q [exp(−ξt,n )|xt = x, S t = i] where x and i are time-t values of the state and regime variables respectively and ξt,n = δ0 (xt + . . . + xt+n−1 ). The yield of the bond can be represented in term of cumulants of random variable ξt,n . ∞ 1 1 X (−1)k (k) Ytn (x, i) = − log E Q [exp(−ξt,n )|x, i] = − µ (x, i) n n k=1 k! n

(C.6)

(k) Q k Cumulants µ(k) n (x, i) are related to the moments mn (x, i) = E [ξt,n |xt = x, S t = i] of ξt,n in the following recursive way:

µ(k) n (x, i)

=

m(k) n (x, i)

! k−1 X k − 1 ( j) j) − µn (x, i)m(k− (x, i) n j − 1 j=1

(C.7)

Note that although the price of a bond Bnt (x, i) does not have a closed-form solution, the moments of ξt,n do. It can be shown Q k that the n-th moment m(k) n (x, i) = E [ξt,n |xt = x, S t = i] can be represented as 0 m(k) n (x, i) = An (i) +

X

i

An1 (i)x(i1 ) +

X

i ,i

An1 2 (i)x(i1 ) x(i2 ) + · · · +

i1 ,i2

i1

X

i ,i2 ,...,ik

An1

(i)x(i1 ) x(i2 ) · · · x(ik )

(C.8)

i1 ,i2 ,...,ik

By substituting (C.8) into the definition of moment m(k) n+1 (x, i) for maturity n + 1, one can find the recursive formulas for coefficients Ani1 ,i2 ,...,ik . The idea of our approximation method is to compute a few first moments to approximate bond prices. In what follows, we demonstrate that taking just the two first cumulants in (C.6) produces prices that are sufficiently accurate. The first two moments m(k) n (x, i) can be represented using the following matrix notation: m(1) n (x, i) =

An (i) + B0n (i)x

0 0 m(2) n (x, i) = C n (i) + Dn (i)x + x F n (i)x

where coefficients An (i), Bn (i), Cn (i), Dn (i) and Fn (i) satisfy the following recursive equations:2 An+1 (i) =

X

h i πi j An ( j) + µQ ( j)0 Bn ( j)

j

Bn+1 (i) =

δ+

Cn+1 (i) =

X

X

πi j ΦQ ( j)0 Bn ( j)

j

h i πi j Cn ( j) + µQ ( j)0 Dn ( j) + µQ ( j)0 Fn ( j)µQ ( j) + tr(Fn ( j)Σ( j)Σ( j)0 )

j

Dn+1 (i)

= 2An+1 (i)δ +

Fn+1 (i)

= −δδ +

X

h i πi j ΦQ ( j)0 Dn ( j) + 2Fn0 ( j)µQ ( j)

j 0

δB0n+1 (i)

+ Bn+1 (i)δ0 +

X

πi j ΦQ ( j)0 Fn ( j)ΦQ ( j)

j

2 There

was a typo in the expression for F that was corrected on February 9, 2017. We thank David Leather for pointing this out.

4

with initial conditions A1 (i) = 0, B1 (i) = δ, C1 (i) = 0, D1 (i) = 0 and F1 (i) = δδ0 . We performed a simulation study to check how accurate our quadratic approximation is. To find a bond price by simulation, we note that the bond price has a closed-form solution conditional on a regime path. Indeed, the price of a bond (C.5) can be represented as h i Bnt (x, i) = E Q E Q [exp(−ξt,n )|xt = x, St+n−1 ]|S t = i t

(C.9)

where the outer expectation is taken with respect to regime paths St+n−1 t = {S t , . . . , S t+n−1 }. The inner expectation in (C.9) can be represented as Bnt (x, St+n−1 ) ≡ E Q [exp(−ξt , n)|xt = x, St+n−1 ] = exp −An (St+n−1 ) − Bn (St+n−1 )0 x t t t t

(C.10)

where coefficients An (St+n−1 ) and Bn (St+n−1 ) satisfy the following recursive equations: t t An (St+n−1 ) t

= −

Bn (St+n−1 ) t

=

t+n−1 0 An−1 (St+1 ) + µQ (S t+1 )Bn−1 (St+n−1 t+1 ) 1 0 t+n−1 0 Bn−1 (St+n−1 t+1 )Σ(S t+1 )Σ(S t+1 ) Bn−1 (St+1 ) 2 t+n−1 δ + ΦQ (S t+1 )0 Bn−1 (St+1 )

t+n−1 with boundary conditions A1 (St+n−1 t+n−1 ) = 0 and B1 (St+n−1 ) = δ. The simulation strategy will be clear from the above formulae:

we simulate regime paths St+n−1 = {S t , . . . , S t+n−1 }, compute the bond prices (C.10) conditional on the paths, and take t their average over a number of simulations. This “conditional Monte-Carlo” strategy improves the convergence properties drastically compared with the direct sampling of the state variables. Equipped with the pricing method just presented, we compared the accuracy of the quadratic approximation suggested above. The design of the experiment was as follows. We constructed a range of parameter values and state variables within three standard deviations from the estimated parameters and unconditional means of state variables. We generated 5,000 Sobol points inside this range. For each generated set of parameters, we computed bond prices based on simulation, quadratic approximation, and a log-linear approximation proposed in Bansal and Zhou (2002). Then, we calculated mean, median, and maximal absolute deviations of yields obtained with both approximate methods from the exact (simulated) yields. To make sure that the simulated prices were sufficiently accurate, we repeated the exercise several times, increasing the number of simulated paths each time. We found that 100,000 simulations are sufficient: the results did not change when the number of simulations was increased further. Table 1 reports the results. The maximum error produced by our method is 2.8 basis points, which is less than that yielded by the Fama-Bliss zero-yield bootstrapping procedure and much less than the pricing errors generated by our model (see Table 2).

Appendix D. Likelihood Function Let Ωt = {w1 , . . . , wt } be the econometrician’s dataset at time t, t = 1, . . . , T , where T is the sample size. The econometrician observes the state variable xt that consists of detrended real GDP, inflation, and the short rate (equal to the 3-month bill yield) as well as the set of yields Yt = (Yt1 , . . . , Ytn ) with maturities longer than 3 months: τ1 , . . . , τn . Thus, wt = (xt , Yt ). The state vector xt satisfies a regime-changing VAR (2.4) with parameters µ, Φ and Σ being the non-linear functions of structural parameters from model (2.1)-(2.3). Yields Yt are observed with errors and have the following form: Yt(i) = f (xt , S t , τi ) + ti , 5

(D.1)

where f (xt , S t , τi ) are the yields generated by the model. Measurement errors are assumed to be mutually independent i.i.d. normally distributed random shocks with standard errors σy . Let Qit = Pr(S t = i|Ωt ) be the filtered probability that policy i is implemented. Following Ang et al. (2008) and Dai et al. (2007) we use a recursive algorithm to compute the likelihood function. At time t = 1 we initialize Qit at the stationary probabilities π∗i of the Markov chain S t . At time t + 1 we compute the following conditional density: p(wt+1 , S t+1 = j|Ωt ) =

X

πi j Qit p(xt+1 |xt , S t = i, S t+1 = j)p(Yt+1 |xt+1 , S t = i, S t+1 = j),

(D.2)

i

where πi, j are the transition probabilities of S t and the conditional densities of xt+1 and Yt+1 are equal to p(xt+1 |xt , S t = i, S t+1 = j)

= ×

p(Yt+1 |xt+1 , S t = i, S t+1 = j)

=

1 (2π)3/2 |Σ j Σ j 0 |1/2 " # 1 exp − (xt+1 − µ j − Φ j xt )0 (Σ j Σ j 0 )−1 (xt+1 − µ j − Φ j xt ) 2 n 1 X 1 (k) 2 − exp (Y − f (x , j, τ )) t+1 k t+1 2 2 n/2 (2πσ ) 2σ y

y k=1

The filtered probabilities of regimes are then updated using the Bayes rule: p(wt+1 , S t+1 = j|Ωt ) p(wt+1 |Ωt )

j Qt+1 = Pr(S t+1 = j|Ωt+1 ) =

(D.3)

The density of wt+1 conditional on the information set at time t is given by p(wt+1 |Ωt ) =

X

p(wt+1 , S t+1 = j|Ωt ).

(D.4)

j

Finally, the log-likelihood function is equal to T −1

L=

1 X log p(wt+1 |Ωt ). T − 1 t=1

(D.5)

Appendix E. Long-run means of the state variables We now demonstrate how to compute the unconditional mean of the state variable xt based on the reduced-form representation (2.4) of the model dynamics. We essentially follow Ang et al. (2008). First, let us compute conditional means mi = E[xt |S t = i], i = 1, . . . , N. From (2.4) it follows that E[xt+1 |S t+1 = i] = µi + Φi E[xt |S t+1 = i].

(E.1)

The conditional expectation in the right-hand side can be rewritten as E[xt |S t+1 = i] =

X

Pr(S t = j|S t+1 = i)E[xt |S t = j].

(E.2)

j

Probabilities b ji = Pr(S t = j|S t+1 = i) define the “backward” transition matrix. These probabilities are related to the “forward” transition probabilities πi j = Pr(S t+1 = i|S t = j): b ji = πi j

π∗j π∗i

,

(E.3)

where π∗i are the stationary probabilities of πi j . Combining (E.1) and (E.2) we obtain the following equations for conditional means mi : mi = µ i + Φ i

X j

6

b ji m j

(E.4)

By introducing matrix notation m ¯ = [m01 , . . . , m0N ]0 , µ¯ = [µ01 , . . . , µ0N ]0 and b11 Φ1 bΦ = b Φ 1N N

··· ··· ···

bN1 Φ1 bNN ΦN

we can represent the solution to (E.4) as m ¯ = (I − bΦ)−1 µ. ¯ Finally, the unconditional mean of xt is given by Ext =

P i

π∗i mi .

Appendix F. Local identification of structural parameters We start with the discussion of a single-regime model. We use a form and the rational expectations solution of the model as in equations (2.5) - (2.9). Structural parameters S = (B0 , B−1 , B1 , m0 , Γ) are identified if these parameters can be deduced uniquely from the estimated reduced-form parameters R = (µ, Φ, ΣΣ0 ). It is easy to check that it is sufficient to focus on the identification of (B0 , B−1 , B1 ) given Φ. Indeed, if (B0 , B−1 , B1 ) can be identified from (2.7), then m0 can be uniquely identified from (2.8) and Γ can be identified from (2.9). We rewrite (2.7) in the following generic form: F(S, R) = 0, where F : Rn+m → Rm , S ∈ Rn , R ∈ Rm , n ≤ m. Given the above discussion, we specialize structural and reduced-form parameters to S = (B0 , B−1 , B1 ) and R = Φ. In the case of a single-regime version of our model, m = 9, and n = 7. Let R0 be a solution of (2.7) for given a set of structural parameters S0 . In what follows, we focus on the local identification of structural parameters in the neighborhood of point S0 . Specifically, S is locally identified if there is a neighborhood of S0 such that for any point S , S0 from this neighborhood F(S, R0 ) , 0. Note that the absence of local identification covers such cases as 1. A structural parameter does not appear in the reduced-from model (the first example discussed in Cochrane, 2011) 2. A subset of structural parameters appear in the reduced form model only in the form of parameter convolution By establishing local identification we can rule out such cases. Proposition Appendix F.1. If rank (FS (S0 , R0 )) = n than S is locally indentified. Proof: Suppose S is not locally identified. Then there is a sequence {Sk } → S0 , Sk , S0 , such that F(Sk , R0 ) = 0. Thus, for any k 0 = F(Sk , R0 ) = FS (S0 , R0 )(Sk − S0 ) + o(Sk − S0 ). A sequence ek = (Sk − S0 )/||Sk − S0 || is bounded (||ek || = 1) and, therefore, has a converging subsequence lim ek j = λ.

j→∞

Equation (F.1) implies that as j → ∞, we have FS (S0 , R0 )λ = 0, λ , 0. Thus, the column rank of FS (S0 , R0 ) is less than n. G.E.D. We check numerically the local identification of the following models: 7

(F.1)

Benchmark Model. gt

=

mg + (1 − µg )gt−1 + µg Et gt+1 − φ(rt − Et πt+1 ) + σg εgt

πt

=

rt

=

mπ + (1 − µπ )πt−1 + µπ Et πt+1 + δgt + σπ επt mr + (1 − ρ) αEt πt+1 + βgt + ρrt−1 + σr εrt

This is a single-regime version of our model. In this model n = 7. As an example, we use estimated parameter values. The matrix FS has seven positive singular values, so the model is locally identified. Model 1. gt

=

mg + (1 − µg γ)gt−1 + µg γEt gt+1 − φ(rt − Et πt+1 ) + σg εgt

πt

=

rt

=

mπ + (1 − µπ )πt−1 + µπ Et πt+1 + δgt + σπ επt mr + (1 − ρ) αEt πt+1 + βgt + ρrt−1 + σr εrt

In this model n = 8. Parameters µg and γ enter the model only as a product, so we know for sure that this model is not fully identified. As an example, we use estimated parameter values and set γ = 1. The matrix FS has one zero and seven positive singular values, so the model is not locally identified. Model 2. gt

= mg + (1 − µg )gt−1 + γg Et gt+1 − φ(rt − Et πt+1 ) + σg εgt

πt

= mπ + (1 − µπ )πt−1 + γπ Et πt+1 + δgt + σπ επt = mr + (1 − ρ) αEt πt+1 + βgt + ρrt−1 + σr εrt

rt

In this model n = 9. As an example, we use estimated parameter values and set γg = µg , and γπ = µπ . The matrix FS has nine positive singular values, so the model is locally identified. For the regime-switching model, we start with a generic representation in (A.1). Reduced-form parameters in (2.4) solve the following system of equations: B−1 (i) =

[B0 (i) − B1 (i)

X

πi j Φ( j)]Φ(i)

(F.2)

j

m0 (i) =

[B0 (i) − B1 (i)

X

πi j Φ( j)]µ(i) − B1 (i)

X

j

Γ(i) =

[B0 (i) − B1 (i)

πi j µ j

(F.3)

j

X

πi j Φ( j)]Σ(i).

(F.4)

j

These equations are the same as equations (A.3)-(A.5). Here, we use a different notation, e.g., m0 and B0 show up separately instead of jointly as in m1 = B−1 0 m0 , to highlight parameters that need to be identified. In this case, the parameters that can be estimated are µ(i), Φ(i), Σ(i)Σ0 (i), and πi j . Again, it is sufficient to check the identification of the structural parameters contained in (B0 (i), B−1 (i), B1 (i)), because m0 (i) and Γ(i) are uniquely identified from (F.3) and (F.4). Our regime model has 10 structural parameters pertaining to the B matrices. We evaluate numerically the rank of matrix FS at the estimated parameter values and find it to be equal to 10. 8

Appendix G. The effect of incorporating bonds into a model We now motivate, using a simple example, how adding bonds affects the estimation results. Suppose that inflation follows and AR(1) process under the objective probability measure P : πt

=

φπt−1 + σεt .

(G.1)

We assume that inflation is observable and, therefore, that all the parameters controlling the inflation dynamics are known (first-stage estimates). Next, assume that under the risk-neutral probability measure Q inflation follows: πt = φQ πt−1 + σεQ t .

(G.2)

Suppose that the nominal interest rate is determined by the Taylor rule: rt = αEt (πt+1 ) = αφπt .

(G.3)

Therefore, yields are equal to Q

yt (τ) = αφ

1 − e−(1−φ )τ πt (1 − φQ )τ

(G.4)

We want to estimate the unknown parameters φQ and α using the information in yields. It is customary to place i.i.d. measurement errors on yields. As a result, the term of the log-likelihood that corresponds to time t is 2 Q 1−e−(1−φ )τ y (τ) − αφ π 2 2 X t t Q (1−φ )τ (πt − φπt−1 ) (rt − αφπt ) − − log Lt = − 2 2σ2 2σ2e 2σ e τ

(G.5)

Then the term of the score that corresponds to the Taylor rule parameter α is −(1−φQ )τ

X yt (τ) − αφ 1−e πt 1 − e−(1−φQ )τ ∂ log Lt it − αφπt (1−φQ )τ φπ + · = φπt t ∂α σ2e σ2e (1 − φQ )τ τ

(G.6)

The terms showing up in the summation appear because yields are added to the estimation. If one computes the information matrix as a product of scores, all the cross-product terms will vanish under the assumption that measurement errors are i.i.d. One is left with squared individual terms in the expression above. Therefore, all the extra terms that arise from adding yields increase the information matrix. This leads us to the conclusion that adding yields reduces the standard error of α. Modifying the likelihood with yields leads also to different point estimates of the parameters, in general. Thus, under the null hypothesis of the model, precision should come together with less finite sample bias in point estimates.

Appendix H. Goodness of Fit To evaluate the model’s ability to fit the data, we first review the pricing errors of the yields of different maturities. Table 2 indicates that the absolute pricing errors do not exceed 30 basis points. This is a reasonable degree of accuracy, given the magnitude of the noise in the approximated zero yields (e.g., Dai et al., 2004). Table 2 also reports the correlation of model-implied slope and curvature with their data counterparts.3 The correlations are 97.5% and 64.1%, respectively. Thus, 3 In

this paper, the slope of the yield curve is defined as y(40) − y(1), i.e. the 10-year yield minus the 3-month yield. The curvature is defined as

y(40) + y(1) − 2y(8), where y(8) is the 2-year yield.

9

the model captures the slope almost perfectly and does a decent job of fitting the curvature. This performance is on a par with single-regime term structure models that use several latent variables as well as observed macro variables. We also evaluate the model’s ability to fit various unconditional moments, such as the mean, standard deviation, skewness, kurtosis, and autocorrelations of the state variables, yields, slope, and curvature. We use the parametric bootstrap strategy to compute the finite sample confidence bounds of the moments implied by the model.4 Table 3 reports the results. The model is able to match all aspects of the data except for the kurtosis of the curvature. Although the state dynamics (2.4) involves a lag in the state variables of only one quarter, the 1-year autocorrelations are matched perfectly by our model. We conclude that, overall, model T S M fits the data well.

References Ang, A., Bekaert, G., Wei, M., 2008. The term structure of real rates and expected inflation. Journal of Finance 63, 797–849. Bansal, R., Zhou, H., 2002. Term structure of interest rates with regime shifts. Journal of Finance 57, 1997–2043. Cho, S., October 2011. Characterizing markov-switching rational expectations models, working paper, Yonsei University. Cho, S., Moreno, A., 2011. The forward method as a solution refinement in rational expectations models. Journal of Economic Dynamics and Control 35, 257–272. Cochrane, J., 2011. Determinacy and identification with Taylor rules. Journal of Political Economy 119, 565–615. Dai, Q., Singleton, K., Yang, W., June 2004. Predictability of bond risk premia and affine term structure models, working Paper, Stanford University. Dai, Q., Singleton, K., Yang, W., 2007. Regime shifts in a dynamic term structure model of U.S. Treasury bond yields. Review of Financial Studies 20, 1669–1706.

4 We

do not take into account parameter uncertainty. With parameter uncertainty the confidence bounds would be even wider.

10

Table 1 : Accuracy of the Approximate Pricing Method. This table presents the results of a simulation exercise designed to evaluate the accuracy of the bond pricing method proposed in this paper. We generated 5,000 Sobol points of parameters and state variables inside a range of three standard deviations around estimated parameter values and long-run means of the state variables. For each generated point, we computed an exact yield based on simulations as well as the yields obtained with the quadratic approximation method used in this paper and the log-linear method proposed in Bansal and Zhou (2002). The table reports the mean, median, and maximal absolute deviations of the approximate yields from the exact one.

mean, bp

median, bp

maximum, bp

0.04

0.01

0.53

0.13e-4

3.78e-4

0.02

Log-Linear Approximation

0.19

0.04

2.91

Quadratic Approximation

0.01

0.17e-4

0.19

Log-Linear Approximation

0.49

0.05

16.20

Quadratic Approximation

0.09

0.32e-4

2.77

8 quarters maturity Log-Linear Approximation Quadratic Approximation 20 quarters maturity

40 quarters maturity

Table 2 : Pricing Errors. The table reports the pricing errors measured as mean absolute errors (MAE) for yields of maturities at 2, 5, and 10 years for model T S M. We also report the correlations of slope and curvature implied by the model with their data counterparts.

Yield maturity, qtrs

MAE, bp

1

0.00∗

8

22.71

20

19.20

40

23.40 Correlations, %

∗

slope

97.56

curvature

66.66

one quarter yield is assumed to be observed exactly.

Table 3 : Unconditional Moments Tests. This table reports various unconditional moments of the observables computed from the dataset (quarterly observations from 1970 to 2004) and implied by the estimated model T S M. Bootstrapped 95% confidence bounds are shown in parentheses. Means, % output

inflation

y(1)

y(8)

y(40)

slope

curvature

data

0.00

3.97

5.88

6.56

7.29

1.41

0.05

model

0.01

4.10

6.18

7.26

8.25

2.07

-0.09

(-0.6,0.6)

(2.1,6.2)

(3.1,9.2)

(4.3,10.2)

(5.8,10.6)

(1.0,3.2)

(-0.6,0.4)

y(1)

y(8)

y(40)

slope

curvature

Standard Deviation, % output

inflation

data

1.54

2.31

2.99

2.88

2.41

1.41

0.87

model

1.48

2.02

3.74

3.66

3.12

1.55

1.01

(1.0,2.0)

(1.0,3.5)

(2.1,6.0)

(2.1,5.9)

(1.9,4.9)

(1.0,2.2)

(0.8,1.3)

output

inflation

y(1)

y(8)

y(40)

slope

curvature

Skewness

data

-0.38

1.07

0.85

0.63

0.86

-0.57

0.12

model

-0.01

0.01

0.04

0.12

0.18

0.01

-0.12

(-0.8,0.8)

(-1.0,1.0)

(-0.9,1.0)

(-0.8,1.1)

(-0.7,1.1)

(-0.8,0.8)

(-0.6,0.3)

output

inflation

y(1)

y(8)

y(40)

slope

curvature

3.82

3.15

4.14

3.56

3.49

3.35

4.03

Kurtosis

data model

3.01

2.46

2.64

2.68

2.65

2.84

2.90

(2.1,4.6)

(1.6,4.2)

(1.8,4.2)

(1.8,4.3)

(1.8,4.3)

(2.0,4.6)

(2.3,3.8)

One Quarter Autocorrelation

data model

output

inflation

y(1)

y(8)

y(40)

slope

curvature

0.87

0.98

0.92

0.94

0.96

0.78

0.69

0.83

0.97

0.93

0.93

0.92

0.81

0.42

(0.7,0.9)

(0.9,1.0)

(0.8,1.00)

(0.9,1.00)

(0.9,1.0)

(0.6,0.9)

(0.2,0.7)

Four Quarter Autocorrelation

data model

output

inflation

y(1)

y(8)

y(40)

slope

curvature

0.29

0.86

0.78

0.82

0.85

0.41

0.42

0.46

0.86

0.77

0.76

0.74

0.56

0.28

(0.2,0.7)

(0.7,1.0)

(0.5,0.9)

(0.5,0.9)

(0.5,0.9)

(0.2,0.8)

(0.1,0.6)