(1*− α*2*) p*_{11}+ (1*− α*1*) p*_{22}*+ α*_{1}*α*_{2} *> 1.* (57)
*For given p*_{11} *and p*_{22}*, the LRTP deﬁnes a hyperbola in (α*_{1}*, α*_{2})*−space with the*
*vertical asymptote where α*_{1}*= p*_{11} *and the horizontal asymptote where α*_{2} *= p*_{22}. As
in ﬁgure 1, our analysis focuses on the hyperbola in the region of the parameter space
*where α*_{i}*> p*_{ii}*, for i = 1, 2. This region captures the economically interesting set*
*of monetary policy processes and has the intuitive implication that α*_{i}*> 1 for some*
*i = 1, 2 is a necessary condition for the LRTP to imply determinacy. For example,*
*α*_{i}*< 0, for i = 1, 2, can satisfy the LRTP, but is at odds with the way central banks*
*set policy and does not result in all the eigenvalues of M being inside the unit circle,*
so fails to deliver a unique bounded equilibrium.

**Lemma 1. If α**_{i}*> p*_{ii}*for all i = 1, 2 and LRTP, then α*_{i}*> 1 for some i = 1, 2.*

*Proof. For p*_{ii}*= 1 for some i, then α*_{i}*> 1 for some i = 1, 2 by the condition α*_{i}*> p*_{ii}*for all i = 1, 2. For p*_{ii}*< 1 for both i, take α*_{2} *> p*_{22}*≥ 0 and rewrite the LRTP as*

*α*_{1} *>* 1*− p*11*− p*22*+ α*_{2}*p*_{11}

*α*_{2}*− p*22 *.* (58)

*Note that the right side of (58), expressed as a function of α*_{2} and the transition
*probabilities, is monotonically decreasing in α*_{2}*. We now show that over the range*
*p*_{22}*< α*_{2} *< 1, α*_{1} *> 1 for all p*_{11}*∈ [0, 1) and p*22 *∈ [0, 1). Letting α*2 *→ 1, (58) implies*

*α*_{1} *>* 1*− p*22

1*− p*22 *= 1.* (59)

*Letting α*_{2} *→ p*22 implies the right side of (58) approaches*∞ for any p*11 *∈ [0, 1) and*

*p*_{22}*∈ [0, 1).*

**Proposition 2. Given α**_{i}*> p*_{ii}*for i = 1, 2, the following statements are equivalent:*

*(A) All the eigenvalues of M lie inside the unit circle.*

*(B) α*_{i}*> 1, for some i = 1, 2, and the long-run Taylor principle (LRTP) is *
*satis-ﬁed.*

**Proof. (Statement A implies statement B)**

*We know λ*_{1} *> 0, so the restriction on that root is 0 < λ*_{1} *< 1. Hence we seek the*
*implications for (α*_{1}*, α*_{2}*, p*_{11}*, p*_{22}*) of the conditions that λ*_{1} *> 0 and λ*_{1} *< 1. Considering*
each case,

*λ*_{1} *> 0 : This is true by inspection and imposes no additional restrictions on the*
policy process.

*λ*_{1} *< 1 : This condition implies*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2}*+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21} *< 2α*_{1}*α*_{2} *− (α*2*p*_{11}*+ α*_{1}*p*_{22}*).* (60)
The restriction that *−1 < λ*2 *< 1, written as λ*_{2} *>−1 and λ*2 *< 1, carries further *
im-plications for the policy process that delivers a determinate equilibrium. Considering
each case,

*λ*_{2} *< 1 : This condition implies*

*−*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2} *+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21} *< 2α*_{1}*α*_{2}*− (α*2*p*_{11}*+ α*_{1}*p*_{22}*) .* (61)
*λ*_{2} *>−1 : This condition implies*

*2α*_{1}*α*_{2}*+ (α*_{2}*p*_{11}*+ α*_{1}*p*_{22}*) >*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2}*+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21}*.* (62)
Squaring both sides and simplifying yields

*α*_{1}*α*_{2}*+ p*_{11}*(1 + α*_{2}*) + p*_{22}*(1 + α*_{1}*) > 1* (63)
Note that (60) and (61) together imply that

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2}*+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21}

* < 2α*^{1}*α*_{2}*− (α*2*p*_{11}*+ α*_{1}*p*_{22}*),* (64)
*so it must be the case that 2α*_{1}*α*_{2}*− (α*2*p*_{11}*+ α*_{1}*p*_{22}*) > 0 and squaring both sides of*
(60) preserves the inequality. Doing this and rearranging yields the LRTP

*α*_{1}*α*_{2}*+ p*_{11}(1*− α*2*) + p*_{22}(1*− α*1*) > 1.* (65)
*Since, as shown above, when λ*_{1} *< 1,*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2}*+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21} *< 2α*_{1}*α*_{2} *− (α*2*p*_{11}*+ α*_{1}*p*_{22}*),* (66)
and

*−*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2}*+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21}*<*

*(α*_{2}*p*_{11}*− α*1*p*_{22})^{2} *+ 4α*_{1}*α*_{2}*p*_{12}*p*_{21}*,* (67)
(61) also yields the LRTP

*α*_{1}*α*_{2}*+ p*_{11}(1*− α*2*) + p*_{22}(1*− α*1*) > 1.* (68)
It is straightforward to show that condition (68) implies condition (63). To see this,
rewrite the conditions as

*α*_{1}*α*_{2}*+ p*_{11}*+ p*_{22} *> 1 + α*_{2}*p*_{11}*+ α*_{1}*p*_{22} (69)
*α*_{1}*α*_{2}*+ p*_{11}*+ p*_{22}*> 1− α*_{2}*p*_{11}*− α*_{1}*p*_{22} (70)

*By the non-negativity of (α*_{1}*, α*_{2}*, p*_{11}*, p*_{22}*), it is clear that if (69) holds, then (70) also*
*holds. Given α*_{i}*> p*_{ii}*for i = 1, 2 and LRTP, Lemma 1 implies that α*_{i}*> 1, for some*
*i = 1, 2.*

**(Statement B implies statement A)**
Write the LRTP as

1*−*

*p*_{11}
*α*_{1} +*p*_{22}

*α*_{2}

*>* 1*− p*11*− p*22

*α*_{1}*α*_{2} *,* (71)

and note the eigenvalues satisfy

*λ*_{1}*+ λ*_{2} = *p*_{11}
*α*_{1} + *p*_{22}

*α*_{2} *≥ 0,* (72)

*since α*_{i}*> p*_{ii}*for i = 1, 2 and*

*λ*_{1}*λ*_{2} = *p*_{11}*+ p*_{22}*− 1*

*α*_{1}*α*_{2} *.* (73)

Substituting (72) and (73) into (71), yields

*λ*_{1}*+ λ*_{2}*− 1 < λ*1*λ*_{2}*.* (74)

*Now (74) implies that λ*_{1} *= 1 and λ*2 *= 1 because under the maintained assumption*
of the LRTP, equality of the roots to unity implies the obvious contradiction that
*λ*_{i}*< λ*_{i}*.*

*Since λ*_{1} *> 0, (74) is*

1*− 1/λ*_{1} *< λ*_{2}(1*− 1/λ*_{1}*),* (75)
*and (75) implies if 0 < λ*_{1} *< 1 then λ*_{2} *< 1 or if λ*_{1} *> 1 then λ*_{2} *> 1. Thus, (75) states*
that both eigenvalues are either outside or inside the unit circle.

*Now using the assumption that α*_{i}*> p*_{ii}*for i = 1, 2, expression (72) becomes*
0*≤ λ*1*+ λ*_{2} = *p*_{11}

*α*_{1} +*p*_{22}

*α*_{2} *< 2.* (76)

*imposing a restriction on λ*_{2}. We now have that

*0 < λ*_{1} *< 1 and* *|λ*2*| < 1* (77)
*because the alternative of λ*_{1} *> 1 and λ*_{2} *> 1 violates (76). Clearly, if 0 < λ*_{1} *< 1 and*
*λ*_{2} *≤ −1, then (76) is violated, leading to |λ*2*| < 1.*
Uniqueness of the bounded solution, which is established by proposition 1, need
*not imply there is a unique stationary solution, as Farmer, Waggoner, and Zha (2006)*
show. Those authors require a solution to be mean square stable, ensuring the
exis-tence of ﬁnite ﬁrst and second moments, and argue that the long-run Taylor principle
admits a continuum of solutions, including sunspots. Their solution, which allows
lagged states to enter, is not bounded, as it can exceed any ﬁnite bound with positive

probability.^{23} This implication of stationarity makes their solution at odds with the
standard deﬁnition of determinacy applied to linear rational expectations models.

Boundeness precludes lagged states from entering the solution.

Appendix B. Determinacy and Solution for the New Keynesian Model The proof of determinacy and the solution method described in appendix A can be applied directly to any purely forward-looking linear model to show that the minimum state variable solution is the unique bounded solution. The equations of the model are

*π*_{t}*= βE*_{t}*π*_{t+1}*+ κx*_{t}*+ u*^{S}_{t}*,* (78)

*x*_{t}*= E*_{t}*x*_{t+1}*− σ*^{−1}*(α(s*_{t}*)π*_{t}*+ γ(s*_{t}*)x*_{t}*− E**t**π*_{t+1}*) + u*^{D}_{t}*.* (79)
The state-contingent expectations are

*E*_{t}*π*_{t+1}*= E[π*_{t+1}*s*_{t}*= i, Ω*^{−s}_{t}*] = p*_{i1}*E[π** _{1t+1}*Ω

^{−s}

_{t}*] + p*

_{i2}*E[π*

*Ω*

_{2t+1}

^{−s}

_{t}*],*(80)

*E*

_{t}*x*

_{t+1}*= E[x*

_{t+1}*s*

_{t}*= i, Ω*

^{−s}

_{t}*] = p*

_{i1}*E[x*

*Ω*

_{1t+1}

^{−s}

_{t}*] + p*

_{i2}*E[x*

*Ω*

_{2t+1}

^{−s}

_{t}*].*(81) The model can be rewritten as

*π*_{it}*= β (p*_{i1}*E*_{t}*π*_{1t+1}*+ p*_{i2}*E*_{t}*π*_{2t+1}*) + κx*_{it}*+ u*^{S}_{t}*,* (82)
*x*_{it}*= p*_{i1}*E*_{t}*x*_{1t+1}*+ p*_{i2}*E*_{t}*x*_{2t+1}*−* (83)
*σ*^{−1}*((α*_{i}*π*_{it}*+ γ*_{i}*x** _{it}*)

*− (p*

_{i1}*E*

_{t}*π*

_{1t+1}*+ p*

_{i2}*E*

_{t}*π*

_{2t+1}*)) + u*

^{D}

_{t}*,*(84)

*where π*

_{it}*and x*

_{it}*are inﬂation and output at t for s*

_{t}*= i, where i = 1, 2. The*information set Ω

^{−s}*=*

_{t}*{u*

^{S}

_{t}*, u*

^{S}

_{t−1}*, . . . , u*

^{D}

_{t}*, u*

^{D}

_{t−1}*, . . . s*

_{t−1}*, s*

_{t−2}*, . . .} denotes the agents’*

*information set at t, not including the current regime, and Ω** _{t}* = Ω

^{−s}

_{t}*∪ {s*

*t*

*}. All*ex-pectations in (78) and (79) are formed conditional on Ω

*. Deﬁne the state-contingent forecast errors*

_{t}*η*_{1t+1}^{π}*= π*_{1t+1}*− E**t**π*_{1t+1}*,* *η*_{2t+1}^{π}*= π*_{2t+1}*− E**t**π*_{2t+1}*,* (85)
*η*_{1t+1}^{x}*= x*_{1t+1}*− E**t**x*_{1t+1}*,* *η*_{2t+1}^{x}*= x*_{2t+1}*− E**t**x*_{2t+1}*.* (86)
and use them to eliminate the conditional expectations in (82)-(83), yielding the
system

*AY*_{t}*= BY*_{t−1}*+ Aη*_{t}*+ Cu*_{t}*,* (87)

23One aspect of Farmer, Waggoner, and Zha’s indeterminate solution includes*x** _{t+1}* =

_{p}

^{α}^{2}

22*x** _{t}*+

*shocks. |α*

_{2}

*/p*

_{22}

*| > 1 is ruled out by boundedness, while |α*

_{2}

*/p*

_{22}

*| < 1 produces a multiplicity of*bounded solutions.

where

*The roots of the system are the generalized eigenvalues of (B, A), where a unique,*
bounded equilibrium requires all four eigenvalues to lie inside the unit circle.
Obtain-ing analytical restrictions on the roots that deliver determinacy are more complicated
in the new Keynesian model because, as was highlighted in the text, the determinacy
regions vary with private sector parameters. To establish determinacy, note that the
model can be written so current state-contingent variables depend only on the
ex-pectations of future state-contingent variables, as in (11). Writing the model in this
form highlights that the model has the same structure as the simple Fisherian model.

Solutions for the model are derived using the method of undetermined coeﬃcients.

However, given that the expectational errors in (87) are conditionally mean zero, standard methods for solving linear rational expectations models can be used to compute the solution. For example, Sims’s (2001) method and corresponding gensys code produces a solution matching the method of undetermined coeﬃcients solution.

McCallum (2004) notes that in purely forward looking models, such as (87), the method of undetermined coeﬃcients using the minimum set of state variables yields the unique, bounded solution.

In obtaining the solution, we assume supply and demand shocks are uncorrelated, so the coeﬃcients on the demand shocks and those on the supply shocks can be solved separately. Coeﬃcients on the supply shock come from solving for the undetermined

coeﬃcients in⎡

and those on demand shocks from solving for the coeﬃcients in

⎡

Analytical expressions for the coeﬃcients are not easy to interpret, but are straight-forward to compute. These coeﬃcients are the impact elasticities of the various shocks on output and inﬂation.

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**Demand** **Supply**
Inﬂation Output Inﬂation Output
Active Regime 1.152 0.936 1.090 1.022
Passive Regime 2.650 1.980 2.866 0.359

Table 1. **Standard deviation relative to ﬁxed active regime.**

*Uses Lubik and Schorfheide’s estimated policy parameters—α*_{1} =
*2.19, γ*_{1} *= .30, α*_{2} *= .89, γ*_{2} *= .15—and baseline parameters—β =*
*.99, σ = 1, κ = .17. Transition probabilities are p*_{11} *= p*_{22} *= .95. Fixed*
*active regime is α = 2.19, γ = .30.*

*p*_{11}*= .95* *p*_{11} *= .975*

**Demand** **Supply** **Demand** **Supply**

Inﬂation Output Inﬂation Output Inﬂation Output Inﬂation Output
*p*_{22} = 0

*α*_{2} *= .5* 1.044 1.008 1.075 .995 1.022 1.004 1.037 .998
*α*_{2} *= .25* 1.060 1.011 1.092 .994 1.030 1.005 1.045 .997
*α*_{2} = 0 1.073 1.014 1.110 .992 1.037 1.007 1.054 .997
*p*_{22} *= .5*

*α*_{2} *= .5* 1.084 .988 1.143 1.008 1.042 .993 1.071 1.004
*α*_{2} *= .25* 1.120 .983 1.185 1.010 1.059 .990 1.091 1.006
*α*_{2} = 0 1.165 .977 1.238 1.013 1.080 .987 1.115 1.007
*p*_{22} *= 2/3*

*α*_{2} *= .5* 1.123 .961 1.209 1.025 1.061 .979 1.104 1.014
*α*_{2} *= .25* 1.188 .940 1.290 1.034 1.092 .968 1.142 1.018
*α*_{2} = 0 1.283 .910 1.408 1.048 1.135 .953 1.194 1.025
*p*_{22} *= .75*

*α*_{2} *= .5* 1.162 .931 1.275 1.044 1.080 .963 1.137 1.024
*α*_{2} *= .25* 1.268 .886 1.412 1.066 1.129 .940 1.199 1.034
*α*_{2} = 0 1.454 .807 1.653 1.104 1.210 .903 1.302 1.052

Table 2. **Standard deviation in active regime 1 relative to**
**ﬁxed regime. Active and ﬁxed regimes set α**_{1} *= α = 1.5 and γ*_{1} =
*γ = .25. Passive regime sets γ*_{2} *= .5. Ergodic probability of active*
*regime ranges from .83 (p*_{11} *= .95, p*_{22} *= .75) to .98 (p*_{11} *= .975, p*_{22} = 0).

*α* **¯***γ* *δ* *θ*
Regime 1 2.182 0.30 -1.690 0.409
Regime 2 0.885 0.15 -0.750 1.675
Full Sample 1.375 0.225 -1.476 0.657

Table 3. **Estimated parameters from an identiﬁed VAR in**
**(42) using simulated data from regime-switching new **
**Keyne-sian model. Regime 1 is conditional on remaining in regime with**
*α*_{1} *= 2.19; Regime 2 is conditional on remaining in regime with*
*α*_{2} *= 0.89; Full sample is recurring changes from regime 1 to regime*
* 2. α is the estimated response of monetary policy to inﬂation; ¯γ is the*
policy response to output, held ﬁxed in estimation.

0.8 1 1.2 1.4 0.8

1 1.2 1.4

α_{1}

α 2

p11 = 0.95 ; p

22 = 0.95

0.8 1 1.2 1.4

0.8 1 1.2 1.4

α_{1}

α 2

p11 = 0.8 ; p

22 = 0.95

0.8 1 1.2 1.4

0 0.5 1 1.5

α_{1}

α 2

p11 = 0.95 ; p

22 = 0

0 1 2 3 4 5

0 1 2 3 4 5

α_{1}

α 2

p11 = 0 ; p

22 = 0

Figure 1. **Determinacy regions: Fisherian model. Parameter**
combinations in the light-shaded regions imply a unique equilibrium
in ﬁxed-regime model; combinations in dark-shaded plus light-shaded
regions imply a unique equilibrium in regime-switching model.

0.8 1 1.2 1.4 0.8

1 1.2 1.4

α_{1}

α 2

p11 = 0.95 , p

22 = 0.95

0.5 1 1.5

0.8 1 1.2 1.4

α_{1}

α 2

p11 = 0.8 , p

22 = 0.95

0 1 2 3 4 5

−6

−4

−2 0 2 4

α_{1}

α 2

p11 = 0.95 , p

22 = 0

−2 0 2 4

−2 0 2 4

α_{1}

α 2

p11 = 0 , p

22 = 0

Figure 2. **Determinacy regions: New Keynesian model. **
Pa-rameter combinations in the light-shaded regions imply a unique
equi-librium in ﬁxed-regime model; combinations in dark-shaded plus
light-shaded regions imply a unique equilibrium in regime-switching model.

1 1.5 2

Figure 3. **Determinacy regions and private parameters: New**
**Keynesian model. Parameter combinations in the light-shaded **
re-gions imply a unique equilibrium in ﬁxed-regime model; combinations
in dark-shaded plus light-shaded regions imply a unique equilibrium in
*regime-switching model for various settings of ω and σ.*

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.5

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

p_{11}
p 22

Figure 4. **Determinacy regions for Lubik and Schorfheide’s**
**estimates. Shaded regions give (p**_{11}*, p*_{22}) combinations that yield a
determinate equilibrium. Dark region is for parameters implying high
*degree of ﬂexibility and substitution (σ = 1.04, κ = 1.07); light region*
*plus dark region for a low degree of ﬂexibility and substitution (σ =*
*2.84, κ = 0.27); β = .99 in both regions. Using Lubik and Schorfheide’s*
*(2004) estimates: α*_{1} *= 2.19, γ*_{1} *= .30, α*_{2} *= .89, γ*_{2} *= .15.*

0 10 20 30 40 50 0

0.1 0.2 0.3 0.4 0.5

Inflation

Demand

0 10 20 30 40 50

0 0.5 1 1.5 2 2.5 3 3.5

Inflation

Supply

0 10 20 30 40 50

0 0.05 0.1 0.15 0.2

Output

0 10 20 30 40 50

−10

−8

−6

−4

−2 0

Output

Figure 5. **Demand and supply shocks under Lubik and**
**Schorfheide’s estimates of policy parameters. Solid line is **
*con-ditional on active regime initially (α*_{1} *= 2.19, γ*_{1} *= .30) when other*
*regime is passive (α*_{2} *= .89, γ*_{2} *= .15). Transition probabilities are*
*p*_{11} *= .95, p*_{22} *= .93. Dashed line is ﬁxed regime with α = α*_{1}*, γ = γ*_{1}*.*
Figures plot the mean responses from 50,000 draws of regime, beginning
in the second period.