It is well-known that in structural models with occasionally binding constraints, equilibria may not exist (incoherency) or there may be multiple equilibria (incompleteness). Gourieroux et al. (1980) (henceforth GLM) studied this problem in the context of simultaneous equations models with endogenous regime switching, and derived conditions for existence and uniqueness of solutions, which are known as 'coherency and completeness' (CC) conditions. Mavroeidis (2019) applied their methodology to structural vector autoregressions with occasionally binding constraints. However, to the best of our knowledge, there are no general results about the conditions for existence and uniqueness of equilibria in forward-looking dynamic equilibrium models with rational expectations. 1 This is despite the fact that there is a large and expanding literature on solution algorithms for such models (see Fernández-Villaverde et al., 2016) applied for example to models with a zero lower bound (ZLB) to the interest rate (see e.g., Fernández-Villaverde et al., 2015; Aruoba et al., 2018; Gust et al., 2017; Aruoba et al., 2020) . In this paper, we attempt to fill that gap in the literature.
To make progress, we focus mostly on linear models with discretely-distributed shocks in order to apply the methodology of GLM. The discrete support assumption on the shocks is not uncommon in the literature. It has been used extensively in theoretical papers on the implications of the ZLB for the effectiveness of macroeconomic policy. 2 It is also indirectly used by various solution methods that use discrete approximations to continuously-distributed state variables, see Fernández-Villaverde et al. (2016) . The (piecewise) linearity assumption is more restrictive.
We use it because it does not seem possible to make much progress on the problem 1 For monetary models with a ZLB constraint, existence and uniqueness results have been obtained so far under perfect foresight (Holden, 2019) or in the case of no (endogenous or exogenous) dynamics (Mendes, 2011) . A very recent contribution by Eggertsson and Giannoni (2020) shows that an equilibrium might not exists in models exhibiting long-term money neutrality.
2 See, e.g., Eggertsson and Woodford (2003) , Eggertsson (2011) , Mertens and Ravn (2014) , Bilbiie (2018) . without it.
To provide straightforward intuition about the source and the implications of the problem, we start with a graphical analysis of a simple equilibrium model that has been used, amongst others, by Eggertsson and Woodford (2003) and Aruoba et al. (2018) . The model consists of a Fisher equation and a Taylor rule subject to a ZLB constraint, and is driven by a single exogenous shock to the discount factor that can take only two possible values. We show that the model is generally incoherent, meaning there can be no equilibrium at all if the shock is too large, or it is incomplete if the shock is not too large, meaning there are multiple equilibria.
We then propose an algorithm, derived from GLM, that can be used to characterize the CC condition and solve linear forward-looking rational expectations models subject to a ZLB constraint when the distribution of the shocks has discrete support consisting of k states. The algorithm involves calculating the determinants of 2 k matrices corresponding to all possible configurations of regime allocations to the k states. If all determinants have the same sign, then the model is coherent and complete. Otherwise, the model cannot have a solution for all possible values of the shocks. In other words, a nonzero probability of sufficiently large (or 'unfavorable') shocks will destroy the existence of an equilibrium. We can restore existence of equilibria only if we place restrictions on the support of the distribution of the shocks.
Applying the GLM analysis to the simple equilibrium model above with a k-state Markov Chain, we find that the CC condition depends on the reaction coefficient on inflation in the Taylor rule, and is violated for all values of that parameter when the number of states k is large. Moreover, we show that the model is incoherent when the discount factor shock can take large values. When the support of the distribution of this shock is restricted sufficiently, the model generates up to 2 k minimum state variable (MSV, henceforth) solutions, which are many more than typically considered in the literature. 3 Our proposed methodology can also find sunspot equilibria, and, with sufficient restrictions, sunspots can act as equilibrium selection devices.
The above findings qualitatively hold also for a prototypical three-equation NK model that has been used by Eggertsson (2011) , Mertens and Ravn (2014) , Bilbiie (2018) and Eggertsson and Singh (2019) , amongst others. The CC condition for this model is more involved, because, in addition to the reaction coefficient on inflation in the Taylor rule, it also depends on the relative slopes of the aggregate demand and aggregate supply curves. However, in all cases, the CC condition violates the Taylor principle. This means that with the typical calibrations in the aforementioned literature, the model is generically incoherent and existence of a solution can only be obtained by restricting the support of the distribution of the shock. Under appropriate support restrictions, again we can find several MSV solutions to the model, i.e., there are multiple equilibria even in the absence of sunspot shocks.
From the analysis of the case of discrete support therefore we draw the following main lesson. With an active Taylor rule, if we do not impose support restrictions, the model is generally incoherent, and if we do, the model has more solutions than typically reported in the literature. This raises two questions: (i) what happens when the shock support is continuous? (ii) what are the implications of support restrictions?
Regarding the first question, the big complication with extending the above analysis to the case of continuously-distributed shocks is that the expectations operator becomes infinite-dimensional, and the problem cannot be approximated by the discrete GLM algorithm above to arbitrary precision, because the number of necessary computations makes it NP hard. It is possible to reduce the dimensionality of the problem and make it manageable for some particular model specifications. An example of this is given by the simple two-equation model of the Fisher equation and 3 For example, Aruoba et al. (2018) only look at two equilibria.
the Taylor rule if we make the Taylor rule purely forward-looking, i.e., interest rates react to expected future inflation instead of current inflation. With such modifications, the model then becomes a system of piecewise linear dynamic simultaneous equations studied by Mavroeidis (2019), and the computation of the CC condition is straightforward. In the present example, it is simply the opposite of the Taylor principle. Moreover, it is possible to characterize analytically the requisite support restrictions on the distribution of the shocks for existence of equilibria when the CC condition does not hold. It is also possible to characterize the multiple equilibria analytically: there are only two MSV solutions. 4
On the second question, this analysis highlights that the support of the shock distributions must not only be bounded, but that these bounds will generally be time-varying. Hence, structural shocks cannot be independent and identically distributed over time. In models with multiple shocks, the time-variation of these bounds is also cross-correlated, because the bounds for each shock depend on the realizations of the other shocks. Hence, the support of their distribution cannot be rectangular in the incomplete case, so the assumption of orthogonality of structural shocks is incompatible with multiple equilibria. The intuition is that with rectangular support we cannot rule out incoherency, i.e., the possibility of shocks that are sufficiently large to destroy the equilibrium completely.
The structure of the paper is as follows. After a review of the related literature, Section 2 presents the main findings of the paper in the context of the simple forwardlooking model consisting of a Fisher equation and a Taylor rule subject to a ZLB constraint. Section 3 extends the results to a prototypical three equation New Keynesian model. Section 4 provides further analytical results on the requisite support restrictions in incomplete models using a modified version of the toy model 4 There are, however, infinitely many other sunspot equilibria that can be easily characterized by a sunspot process that acts as an equilibrium selection device. This is similar to Lubik and Schorfheide's (2004) use of sunspots to determine the equilibrium path under indeterminacy in linear rational expectations models.
in Section 2. Section 5 concludes. Proofs and additional results are in the Appendix.
A number of studies deal with the issue of existence and multiplicity of solutions mostly in NK models with a ZLB constraint, and find that no solution may exist if the variance of the shocks is too high. Mendes (2011) considers a three-equation NK model without any dynamics (endogenous or exogenous) and a single independently and identically distributed (i.i.d.) Gaussian demand shock, and shows that there is no rational expectations solution if the variance of the demand shock is too large. Basu and Bundick (2015) draw a similar conclusion using a simulated NK model, where they cannot find a solution when the variance of the shock gets large. The problem arises because the expected policy rate is increasing in the volatility of the exogenous shocks when monetary policy follow a ZLB-constrained standard Taylor rule. 5
Our analysis provides a theoretical underpinning of the above finding in the literature that NK models with a ZLB might have no solution at all if the variance of the shocks is too high. However, it highlights that the necessary restrictions are on the support of the distribution of the shocks, rather than their variance.
For example, very large negative real interest rate shocks can cause non-existence of equilibria, but very large positive shocks will not. In that sense, the link of existence of equilibria to the variance of shocks is quite special: it requires both symmetry of the shock distribution and i.i.d. shocks. Thus, the intuition from the above studies does not carry over to more general settings, and it can, therefore, be misleading. 6
Once the correct support restrictions are imposed, the variance of the shock does not 5 Reifschneider and Williams (2000) first discussed this implication, and Nakov (2008) labelled it 'deflationary bias' when monetary policy does not reach its inflation target.
6 For example, Aruoba et al. (2018, footnote 5) refer to the need to restrict the magnitude of exogenous shocks for existence of a solution. Yet, in their calibrated example, they use Guassian shocks with a low variance, which is not enough for existence as our analysis shows.
matter. Hence, restrictions on variance are generally neither necessary nor sufficient for a solution to exists. 7
The same issue is related to another common finding in the literature that studies models with a ZLB using nonlinear solution methods: policy function iterations do not converge for high values of the variance of the shocks. The simulations in Richter and Throckmorton (2015) show a trade-off between the persistence and the standard deviation of the stochastic shock processes. As the persistence of a shock increases, its standard deviation must decline for the numerical algorithm to converge to a minimum state variable solution. The failure to converge occurs because the economy either remains at the ZLB too long when the shocks are very persistent or falls to the ZLB too frequently when the processes are highly volatile. The boundary of the convergence region hence shows a trade-off between the expected frequency and average duration of episodes at the ZLB. 8 While non-convergence of policy function iteration does not prove non-existence, it might suggest so (see also Gavin et al., 2015 , for a similar finding). 9
Furthermore, we stress in this paper that the issue of multiplicity of solutions is pervasive. The literature recognized from the outset the issue of the multiplicity of solutions, starting from the seminal papers by Benhabib et al. (2001a,b) . It is worth stressing again, however, that the issue we highlight is about the multiplicity of MSV solutions, while we do not tackle the multiplicity of steady states or the 7 Some earlier studies have characterized the conditions for the existence of Taylor-rule equilibria in models with the ZLB with two states, assuming that the economy eventually reverts back to an absorbing state and the ZLB does not bind in the absorbing state (e.g., Eggertsson, 2011; Christiano et al., 2018; Boneva et al., 2016) . By setting the shock in the absorbing state to zero, these papers implicitly assumed an 'asymmetric' shock distribution and imposed a restriction on the support that guarantees existence.
8 Nakata and Schmidt (2019) show analytically the same result in an NK model with a two-state Markov process for the shock under discretionary optimal monetary policy. 9 Gavin et al. (2015) investigate why technology shocks may have unconventional effects at the ZLB numerically solving a non-linear NK model with capital. They show that the presence of capital changes the qualitative effects of demand shocks and alters the impact of monetary policy rules. Many other works used non-linear solution methods to solve NK models with a ZLB constraint (e.g., Wolman, 2005; Mertens and Ravn, 2014; Fernández-Villaverde et al., 2015; Boneva et al., 2016; Nakata, 2016 Nakata, , 2017 Gust et al., 2017; Aruoba et al., 2018) . multiplicity due to indeterminacy. In this respect, a natural question that our paper raises is to which solution and why the non-linear solution algorithms converge. Richter and Throckmorton (2015) show that their numerical algorithm converges to the inflationary equilibrium -as long as there is a sufficient expectation of returning to an active monetary policy rule -while it never converges to the deflationary equilibrium. However, we show that there can be many MSV solutions when the (unconstrained) monetary policy is active.
Last but not least, a recent important work by Holden (2019) provides a host of results on existence and uniqueness or multiplicity of solutions in a standard linear NK model with a ZLB constraint. He also provides both a handy numerical solution method and a novel analytical approach to the problem using results from the mathematics of the "Linear Complementarity Problem" (LCP). While this approach seems similar to the GLM approach we use in this paper, it is actually rather different. The fundamental difference lies in the fact that Holden (2019) looks at the perfect foresight problem, while we tackle the stochastic problem, which cannot be cast in LCP form. Thus, ours and Holden's (2019) work are highly complementary.
We illustrate the main results of the paper using the simplest possible model that is analytically tractable and suffices for our purposes. Section 3 shows that similar results apply to the basic three equations New Keynesian model. The model is taken from Section 2 in ACS. It consists of two equations: a consumption Euler equation
(1) and a simple Taylor rule
where r is the steady-state gross real interest rate, R t is the gross nominal interest rate, π t is the gross inflation rate, π * is the target of the central bank for the gross inflation rate, and M t+1 is the stochastic discount factor. The monetary policy rule (2) features a ZLB constraint given by the max operator that prevents the net interest rate to fall below zero. The only difference from ACS is that they consider only active Taylor rules ψ > 1, while we also allow also for passive rules, ψ ≤ 1.
Without any approximation and simply by taking logs, (2) can be re-written aŝ
where hatted variables are log-deviations from steady state values π t = π * , M t = 1/r, R t = rπ * , i.e.,X t = log (X t /x) andX t+1|t := E tXt+1 . (1) can be approximated by a standard Fisher equationR
which relates the nominal interest rate to the ex-ante real interest rate, i.e., (−M t+1|t ), and expected inflation. 10 Substituting forR t in (4) using (3) yields the following univariate dynamic rational expectations equation forπ t (where µ := log (rπ * ))
This equation is simple to analyze as it is completely forward-looking and it features no endogenous persistence. However, as we will see, even such a simple model can be incoherent and/or incomplete.
Here we consider a very simple case that can be illustrated by a graphical analysis,
to gain an intuition about which type of solutions could characterize the dynamic expected first difference equation (5). Following Eggertsson and Woodford (2003) , many papers in the literature on DSGE models with a ZLB constraint assume that an exogenous shock pushes the economy to the ZLB. The shock persists with a given probability and when the shock disappears the economy exits from the ZLB forever, going back to steady state. Accordingly, let us assume that the stochastic discount factor can assume only two possible values:M t = −r L > 0, which is a transitory state which persists with probability p; andM t = 0, which is an absorbing state.
r L < 0 can be interpreted as negative real interest rate shock, which captures the possibility of a temporary liquidity trap. This state is transitory and its duration is a random variable T with expected value E(T ) = (1 − p) −1 . After date T, the economy will be forever at the absorbing state. We want to characterize the solutions to (5). We define a solution, or an equilibrium, a mapping from each of the possible values of the state variable,M t , to the values of the endogenous variables, (π,R), that satisfies (5).
The absorbing state. As it is well known, in the absorbing state, two steady state outcomes are possible. The first one features a positive interest rate (PIR, in what follows), the economy is at the intended steady state inflation target, and However, Panel B shows that when ψ < 1, the only possible equilibrium outcome in the absorbing state is a PIR.
The temporary state. Assume to be in a period t, where the negative shock hits the economy, i.e.,M t = −r L > 0. To solve for the possible equilibria of (5), one needs to solve for the expectations terms, that takes into account the possibility of ending up in the absorbing steady state. When ψ > 1, however, there are two possible steady state outcomes, because agents might expect to end up in the PIR or in the ZIR. Panel C and D in Figure 1 show these two possibilities by plotting the LHS (blue line) and the RHS (kinked red line) of (5). The negative real interest rate shock r L (i.e., positiveM ) shifts the kinked red curve upwards in both panels.
The two panels differ from each other because of the blue line, given the different expectation about the absorbing state. In both cases, however, there are two possible temporary equilibrium outcomes in period t: a PIR (the yellow dots) and a ZIR (green dots). Therefore, combining all the possibilities, the simple equation (5) admits four possible equilibria described in Table 1 (see Appendix A.1 for the derivation).
It is immediate to grasp what happens when ψ < 1, by looking at Figure 1 or Table 1 . In this case, PIR is the only possible absorbing steady state, and we need to distinguish two cases, shown in Panel E and F of Figure 1 . If p < ψ < 1, the red line is steeper than the blue line, and there are two possible temporary outcomes in t, similar to the previous case. Hence, both the equilibria (PIR, PIR) and (ZIR, PIR) survive. If ψ < p < 1, the red line is flatter than the blue line and there will 
be a unique equilibrium (PIR, PIR). Moreover, note that this unique equilibrium implies inflation above target in the temporary state t, while in all the other cases inflation was below equilibrium in the temporary state t.
We have characterized all the possible equilibria in this very simple example.
However simple, this example provides us with the intuition of the two main takeaways of this paper. In what follows, we will show that these two main points hold generally in a ZLB environment.
First, the model is generally incomplete, meaning there are multiple equilibria.
The kink in (5) defines two possible outcomes for each realization of the exogenous stateM t . Since we assume two possible states, then the model delivers 2 2 equilibria.
When ψ > 1, there are up to four distinct solutions: the first and the second have the property that in the good (r L = 0) absorbing state the economy is at the intended steady state, orπ t = 0. The other two have the property that in the good (r L = 0) absorbing state the economy is in a liquidity trap andπ t = −µ. In all these equilibria, inflation is below target in the transitory state. We will show below that in case of k possible values of the state the model would deliver up to 2 k possible equilibria.
Second, the model is generally incoherent, meaning there can be no equilibrium at all. All the four equilibria in the case ψ > 1, need a support restriction for the exogenous stateM t . They are defined provided that the discount factor (i.e., real interest rate) shock is not too large (small). As evident from Panels A to D in Figure 1 , a large enough shock would move the red line above the blue one, so that there is no equilibrium. Table 1 shows that the first and the second equilibria In what follows, we generalize these results and we show that the incoherency and incompleteness problem is pervasive in ZLB (kinked) models of rational expectations. Hence, either we impose support restrictions on the structural exogenous shocks, and we live with multiple equilibria, or we give away rational expectations to make these type of models coherent.
The previous subsection considered a special case when the distribution ofM t has a two-point support with an absorbing state. We now generalize this to the case whenM t follows a k-state Markov chain. Let m = (m 1 , ..., m k ) denote the support of the distribution, and the k × k matrix K denote transition probability kernel with
As in the previous subsection, we consider minimum state variable solutions (MSV), that is, solutions of the formπ t = f M t , i.e.,π t only depends on the current value of the state variableM t . Under this assumption,π t will have k points of support and it will also follow a first-order Markov chain with the same transition kernel K. 12 Let π = (π 1 , ..., π k ) denote the support ofπ t , Then, equation (5) can be written as the system of k equations
where ι q is the q × 1 vector of ones. 13 We can write this generically in the form
where B = K, y = π + ι k µ/ψ, b = Km + 1−ψ ψ µι k and D = ψI k . This is a system of simultaneous piecewise linear equations with endogenous regime-switching. Conditions for existence (coherency) and uniqueness (completeness) of solutions were derived by Gourieroux et al. (1980) (henceforth GLM). We will show in Section 3 below that eq. (7) can also be used to analyze the generic three-equation New Keynesian model.
The coherency and completeness condition (CC) for this system of equations (7) is given by GLM Theorem 1, see Appendix A.2. In Appendix A.4 we present a simple
algorithm for checking whether the CC condition holds. This algorithm involves 2 k calculations, i.e., it is NP hard, and it is therefore not suitable for very large k.
When k = 2, the CC condition can be derived analytically. Letting K 11 = p and
The example of the previous subsection is a special case with an absorbing state q = 1, so the CC condition (8) reduces to ψ < p, as was shown graphically in the previous subsection. It is also interesting to consider another special case when p = q = 1+ρ 2 for ρ ∈ (−1, 1), in which case the Markov Chain has autocorrelation ρ. Then, the CC condition becomes ψ < ρ if ρ > 0 and ρ < ψ < 1 if ρ < 0.
The last example is a special case of a k-point Rouwenhorst (1995) approximation of an AR(1) process with continuous support. 14 For k > 2, we can evaluate the CC condition numerically. We find that the CC condition is given by ψ <ψ k , wherē ψ k is a function of ρ and k. Figure 2 reports the cutoff valueψ k as a function of k for ρ = 0.9 and ρ = −0.9. We see that as k increases,ψ k → 0, irrespective of the sign of ρ. Therefore, we conjecture that the coherency condition is ψ = 0 in the case of continuous support, though we have not managed to prove this formally, see Subsection 2.3. Figure 2: Value ofψ k for which the coherency and completeness condition is satisfied for ψ <ψ k , where k is the number of states in the Rouwernhorst discrete approximation to a continuous AR(1) process for the state variableM t , with AR coefficient ρ.
distinguish between multiplicity of steady states and multiplicity of MSV solutions.
In the case ψ > 1 there are two steady states, but up to four MSV equilibria, while in the case p + q − 1 < ψ < 1, there is a unique steady state but up to two MSV equilibria.
In general, when the CC condition is not satisfied, there can be up to 2 k MSV solutions. Using the algorithm in Appendix A.4, we can compute the various equilibria of the model (5) for any given value of the parameters. We consider the discrete k-state Rouwenhorst approximation to the AR(1) process in ACS, with parameter values ρ = 0.9 and σ = 0.0007, and we set ψ = 1.5 and µ = 2 log(1.005) following the calibration in ACS, so that the CC condition (8) For any k, it is possible to impose restrictions on the support of the distribution of the shocks such that we are always at ZIR or always at PIR.
SupposeM t is first-order Markovian and continuously distributed with conditional to (5) must satisfy
for all m ∈ . This is the continuous, infinite-dimensional, analogue of the piecewise linear equation (7) above. It is difficult to make general statements about the CC condition for this model or about the existence of such a solution, even specifying a particular distribution of the shocks, as the typical example (cf. ACS) ofM t following a Gaussian AR(1). The analysis of the previous subsection showed that existence of equilibria requires restrictions on the discrete support of the shocks,
and that under such support restrictions, the model is generically incomplete. The same might (or might not) hold true in the case of continuous support. It is a very hard problem, on which so far we have not been able to make much progress, unless we avoid the non-linearity caused by the max operator by confining the analysis to an always PIR or always ZIR solution. This is illustrated in the following example.
Suppose thatM t follows the AR(1) process:M t = ρM t−1 + σε t with E t−1 ε t = 0, but ε t is not iid Normal, but instead, a truncated non-iid Normal:
, where φ(·) and Φ(·) are the density and distribution functions of a standard Normal, respectively. 15 Then, if ψ > ρ, two solutions of (9) are
See Appendix A.6 for a derivation. Solution (10) is always on the PIR, while (11) is always on the ZIR (and they correspond to the ones in Table 1 ). ACS refer to them as targeted inflation equilibrium and liquidity trap equilibrium, respectively. 15 m t is unique but not analytically available.
The analysis of the previous subsection suggests there are many other solutions that move endogenously between the two regimes, though we do not have analytical expressions for them in the case of continuous support. Note that the upper bound on the support of the distribution of the shocks, a t−1 = −µ ρ−ψ ρψ − ρM t−1 , is timevarying and depends on past shocks, so the innovations ε t cannot be independent over time. We return to this issue in Section 4. 
ACS include a sunspot shock s t that determines which of the two equilibria is selected at any point in time. It can be thought of as an equilibrium selection device.
Obviously, when one considers the full set of solutions, with multiplicity n, say, the sunspot shock should have n states. We can apply this generalization to the k-state example of the subsection 2.2, whereM t follows the Markov chain characterized by (m, K) . Let s follow the Markov Chain ({0, 1, ...n − 1} , K s ), and assume, for simplicity as in ACS, that s t is independent ofM t .
Conjecture a solution of the formπ t = f M t , s t . Thus,π t follows a nk-state Markov chain with support π ∈ nk and transition matrix K π = K s ⊗ K. Hence, E t (π t+1 ) = K π π, and the model equation (6) becomes
and can be written as (7) K π y = b + max (0, ψy) .
where y = π + ι nk µ/ψ and b = ι n ⊗ Km + 1−ψ ψ µι k . Hence, the redefinition of K and b is the only difference with respect to equation (7) without the sunspots.
We can therefore apply the same methodology to solve (13) as we did to solve (7). Without further restrictions, therefore, the introduction of sunspots will not necessarily reduce the complexity of the problem, and in fact, we need to perform 2 nk instead of 2 k computations.
However, sunspots can reduce the cardinality of the set of solutions if we impose further constraints. For example, suppose that we further consider only two solutions: one always PIR and one always ZIR. Hence, n = 2 and the solution f M t , s t is such that f M t , 1 ≥ −µ/ψ (PIR, above ZLB) and f M t , 0 < −µ/ψ (ZIR, at ZLB). Then defining the selection matrices
(where e 1 = (1, 0) and e 2 = (0, 1) ), (13) becomes as K π y = b + ψS 1 y, (e 1 ⊗ I k ) y < 0, and (e 2 ⊗ I k ) y ≥ 0. Therefore, the unique solution, if it exists, would simply be
In their general approach, ACS do not actually impose this restriction, but they instead allow the economy to be at or above the ZLB irrespective of the value of s t . However, they do impose that when s t = 1, there must be some states above the ZLB and when s = 0 there are some states below the ZLB, de facto reducing the set of possible solutions.
The interesting question in ACS was about the possibility of distinguishing between fundamental-driven and sunspot-driven equilibria. In principle, we can apply the previous framework to the same question in order to derive a test of the null hypothesis of a fundamental-driven against the alternative of a sunspot-driven equilibrium, say. With discrete support, such a test will be effectively comparing the supports of the data under the two hypotheses, and will therefore have zero size and power one except when the parameters are such that the two equilibria become observationally equivalent. However, given that we don't have a solution to (9), we cannot apply this methodology to data with continuous support.
The results in the previous section apply also to the classic three-equation New Keynesian model loglinearized around the usual (targeted inflation) steady state:
wherex t is output gap. For simplicity, as in the previous section, the Taylor rule features no inertia and no output, and there is only one shock, t , to the Euler equation, that can be interpreted as a discount factor shock. This model has been extensively used in the literature on ZLB. Appendix A.7 analytically characterizes the solutions for this model when t follows a k-state Markov Chain. We start with k = 2, and (following, e.g., Eggertsson and Woodford, 2003; Christiano et al., 2011; Eggertsson, 2011) we assume that the state t = 0 is absorbing for simplicity, while Eggertsson and Singh (2019) , GD: Great Depression, GR: Great Recession. These papers assume an absorbing state and ρ corresponds to the persistence probability of the transitory state.
the state t = −a, with a > 0, is persistent with probability p. 16
It is shown in Appendix A.7.1 that MSV solutions must solve equation (7) for an appropriate choice of parameters B, D and b that depend on the model parameters β, λ, σ, ψ and the distribution of t . Using GLM Theorem 1, we then derive the CC condition for this model as:
see Appendix A.7.1 for the details. Table 2 reports the CC condition for various calibrations of the model found in Mertens and Ravn (2014) , Eggertsson and Singh (2019) and Bilbiie (2018) . Note that p is replaced by ρ in the table in anticipation of the more general results below. 17
As in the previous simple model analysed in Section 2, the CC condition requires that the Taylor principle should not be satisfied, that is, a value of ψ < 1. This is not the case usually analyzed in the literature because it implies indeterminacy of the REE equilibria. Unlike the model of Section 2, the CC condition now also crucially depends on the slopes of the AS (i.e., NKPC) and AD (i.e., EE) curves, λ and σ. Appendix A.7.2 presents a graphical analysis similar to Section 2.1 and Figure 1 . It provides a visual and intuitive interpretation of the CC condition (17) in the two sub-cases, related to the ones in Eggertsson (2011) and Bilbiie (2018) . 18
Next, we study the CC condition when the assumption of an absorbing state is relaxed, and when there are more than two possible states of the shock.
Consider first the case of k = 2, with transition probabilities p and q, and define
In Appendix A.7.1, we show that the CC condition becomes
In the special case when p = q = (1 + ρ)/2, where ρ ∈ [−1, 1] is the autocorrelation coefficient of the shock t , the CC condition has exactly the same form as (17) when p is replaced by ρ.
Finally, we study numerically the CC condition for k > 2 when t follows a k-point Rouwenhorst (1995) approximation of an AR(1) process with continuous support, as in Subsection 2.2. Unlike the results in subsection 2.2 for the twoequation model, we find that the cutoff valueψ k say, does not go to zero, and it 18 The condition pσλ < (1 − p) (1 − pβ) in (17b) exactly corresponds to condition C2 in Proposition 1 of Eggertsson (2011) . It implies that the slope of the aggregate supply curve is steeper than the one of the aggregate demand curve under ZLB in the temporary state, as shown in Figure 10 (and vice versa for condition (17a)). See the discussion in Appendix A.7.2.
generally depends on all the parameters of the model in a complicated manner. For example, when ρσλ < (1 − ρ) (1 − ρβ), the CC condition remains ψ < 1, that is, (19b) holds for all k. 19 In the opposite case, the CC condition remains (19a) when ρ and σλ are small, but can get considerably smaller for larger values of ρ and σλ.
However, for any given values of ρ and σλ,ψ k seems to converge to some value that is bounded away from zero (see the last column of Table 2 ). This is different from the model of Section 2, where we found thatψ k → 0 as k increased, for any value of ρ.
In the case generally considered in the literature, i.e., ψ > 1, the model is generally neither complete nor coherent. The condition in (17) relates to the relative slopes of the aggregate demand curve under ZLB and the aggregate supply curve in the temporary state, when the shock is on. The literature stressed the importance of these relative slopes in the case of ψ > 1. Eggertsson (2011) assumes that the slope of the AD curve is flatter than the one of the AS curve (see Figure 10 and the discussion in Appendix A.7.2), hence generating a unique temporary equilibrium in the temporary state (sometimes called a "fundamental" equilibrium). If the shock is large enough, this temporary equilibrium implies a ZIR. On the contrary, Mertens and Ravn (2014) show that if the slope of the AD curve is steeper than the one of the AS curve, the temporary state admits two possible temporary equilibria, with two very different implications for fiscal policy (sometimes called "confidence driven" equilibria). 20 Above, we just showed that the relative slopes of these two curves is also important for the CC condition (compare (17a) with (17b)), which in any case never holds if ψ > 1. Moreover, under Eggertsson's (2011) and Ravn's (2014) assumption the model is incomplete, because it admits up to four MSV equilibria, depending on the magnitude of the temporary shock. If the shock is sufficiently negative, there will be no equilibria.
We can extend the above findings to the case when the distribution of the shock has k states. In particular, consider again the Rouwenhorst approximation to an AR(1) process for the AD shock t . Figure 4 plots the decision rules associated with various MSV equilibria of the model using the parameter values from Mertens and Ravn (2014) . The support of the distribution of the shock t has been carefully chosen to avoid incoherency. In this case, this is achieved through the standard deviation of the shock, denoted by σ . Larger values yield more dispersion, so when σ gets sufficiently large, there are no MSV equilibria. This exercise also highlights that the restrictions on the support that are required for existence of equilibria depend in a complicated way on the model parameters.
Specifically, the largest value of σ that can support an equilibrium for the calibration on the left with ρ = 0.7 is 0.0032 up to four decimals, while for the calibration on the right, with ρ = 0.4, it is 0.0083. That is, the support restriction becomes more stringent as the persistence of the shock increases. 21
One interesting question is which of the solutions in Figure 4 survives as we 21 Moreover, changing the parameters of the structural model or of the shocks yields a different number of solutions. For example, with a low variance of the shock and σ = 4, the "MR2014 CD" case in Table 2 delivers 8 solutions. lower the value of ψ to satisfy the CC condition, i.e., ψ <ψ in Table 2 
We have seen above that it is difficult to derive all the equilibria and characterize the requisite support restrictions in incomplete models when the distribution of the shocks is continuous. To shed further light on this, we will now consider a modification to the basic example of Section 2 that enables us to get analytical results. Specifically, we replace the contemporaneous Taylor rule (2) with a purely forward-looking one. We will also append a monetary policy shock to the Taylor rule to discuss the relationship between the shocks in the case when the CC condition is not satisfied. The full log-linearized model is given by the equationŝ
This model fits in the framework of Mavroeidis (2019), and can be seen as a kinked simultaneous equations model in the two endogenous variablesπ t+1|t andR t , subject to a lower boundR t ≥ −µ. 
These decision rules are notably piecewise linear and continuous at the kink, as in Aruoba et al. (2020) .
Incompleteness We now look at what happens when the ψ > 1, so the coherency and completeness condition is violated. 23 When ψ > 1, the model may be incoherent (nonexistence of a solution) or incomplete (two solutions) a.s.. 24 First, we need to derive restrictions on the support of the distribution of the exogenous variable to rule out incoherency. The problem is illustrated in Figure 5 . Substituting forπ t+1|t in (20) using (4), the model is defined by the intersection of curves corresponding 23 We do not consider the case ψ = 1 because it is non-generic and can be ruled out a.s. when there are any unconstrained observations. This is because at ψ = 1, the two structural equations (4) and (20) have exactly the same slope above the constraint, so there is either a unique solution at the boundaryR t = −µ, infinite solutionsR t ≥ −µ when the two equations fall on top of each other (an event that occurs with probability zero) or no solution if the two equations do not intersect. The last case is incoherency, which we can rule out with restrictions on the support of the distribution of the errors. These are the same as in the generic case ψ > 1, so we do not need to discuss them separately here. 24 There is a non-generic case where the two structural equations intersect exactly at the kink, where, provided ψ > 1, the two distinct solutions will coincide, so in that sense the model becomes complete. But this is an event of measure zero when the distribution of the shocks is continuous.
to each side of the following equation
When ψ > 1, the above equation (i) may have no solution (i.e., the model is incoherent), an example of which is shown on the left graph in Figure 5 ; or (ii) it may have two solutions (i.e., the model is incomplete), which is shown in the graph on the right in Figure 5 . The last graph shows the range of values of the shocks corresponding to incoherency -when the positively sloped part of the red curve lies in the grey area -and the ones for which two solution exists -when the positively sloped part of the red curve lies on the right of the grey area. The support restrictions required for existence of a solution are:
which, using (21), can be equivalently rewritten as:
When condition (26) holds, the two distinct solutions of the model given by equations (4) and (21) areπ t+1|t =M t+1|t + µ, andπ t+1|t = ψM t+1|t +νt 1−ψ , or a 'liquidity trap equilibrium' and a 'targeted-inflation equilibrium', respectively, in the language of ACS. These solutions can still be represented by equations (22) and (23), except now the indicator variable D t that selects the regime is completely arbitrary, unlike the CC case, when it was a deterministic function of the structural shocks. In other words, under incompleteness, D t acts as an equilibrium selection device, and any distribution for it will yield a valid solution to the model. In particular, D t can be driven by the fundamental shocks of the model, or it can be driven by sunspots. Figure 5 : Illustration of the restriction on the support ofM t+1|t , ν t in the model given by the intersection between the LHS of (25), blue line, and the RHS of (25), red line.
Condition (26) has important implications for the structural model that are sometimes overlooked. It says that the shocks t and ν t cannot be independent of each other, nor can they be independently distributed over time. To clarify the last point, suppose ν t = 0, so the only shock driving the model is t . Condition (26) says that t cannot be independently and identically distributed over time, since the support of its distribution depends on pastM t , and hence, past t . In the presence of a monetary policy shock, the incompleteness condition says that the monetary policy shock cannot be independent of the real shock t , since the support of their distribution cannot be rectangular. Specifically, the monetary policy shock cannot be too big relative to current and past shocks to the discount factor if we are to rule out incoherency.
If these shocks are structural shocks in a DSGE model, such needed support restrictions are very difficult to justify. Structural shocks are generally assumed to be orthogonal. In our opinion, it is very hard to make sense of structural shocks whose supports depend on the value of the other shocks in a time-dependent way.
On the one hand, this is a very serious problem for any DSGE model with a ZLB constraint, and, actually, more generally, with almost any kinked constraint. 
This paper highlights a possibly overlooked problem in rational expectation models with a ZLB (or any other kind of kinked) constraint. This constraint might make the model incoherent or incomplete.
We propose a method for checking the coherency and completeness (CC) condition, that is, the existence and uniqueness of equilibria in piecewise linear DGSE models with a ZLB constraint based on GLM, and an algorithm for finding all the solutions. When applied to the typical NK model, this method shows that the CC condition generally violates the Taylor principle. Hence, the case typically analysed in the literature is either incoherent or incomplete. This raises two main issues, regarding the estimation and the solution of these models, that should be the focus of future research. 
A.1 Derivation of the equilibria in the simple example
Here we derive the analytical expressions for the equilibria in Table 1 . Assume to be in a period t, where the negative shock hits the economy, i.e.,M t = −r L > 0. To solve for the possible equilibria of (5), one needs to solve for the expectations terms, that takes into account the possibility of ending up in the absorbing steady state. As we saw in the main text, when ψ > 1, there are two possible steady state outcomes in the absorbing state: PIR where the economy is at the intended steady state inflation target, i.e., (M ,π,R) = (0, 0, 0); ZIR where the economy steady state hits the ZLB constraint, i.e., (M ,π,R) = (0,π ZIR = −µ, −µ). Hence, in the temporary state in t, agents might expect to end up in PIR or in ZIR. If the agents expect to end up in PIR in the absorbing state, then the expectations terms will be:
and thus (5) becomes
Panel C in Figure 1 displays this equation in a graph. There are two changes with respect to Panel A, that shows the absorbing state, that is the equationπ = max {−µ, ψπ} . First the blue line is flatter, because the slope is p rather than 1. Second, the negativer t (i.e., positiveM ) shifts the red curve upwards. The two equilibria in Panel C survive only if the real interest rate is not too low, in which case the red line shifts above the blue line and there is no possible equilibrium (incoherence). It is easy to show that the two equilibria in Panel C are given bŷ
These are the (PIR, PIR) and (ZIR, PIR) equilibria in Table 1 . The second one implies a liquidity trap equilibrium in the temporary state. If r L < − ψ−p ψp µ, there is no equilibrium.
If the agents expect to end up in ZIR in the absorbing state, instead, then the expectations terms will be:
and thus (5) becomes
Panel D shows this case. With respect to Panel C, the blue line (LHS) now shifts down, because of the expectation of the possibility of a (permanent) liquidity trap equilibrium in the future (i.e., (1 − p)(−µ)). The two possible equilibria arê
These are the (PIR, ZIR) and (ZIR, ZIR) equilibria in Table 1 . Again, the second one implies a liquidity trap in the temporary state, and if r L < −µ ψ−1 ψ there is no equilibrium.
When ψ < 1, there is only one possible equilibrium in the absorbing state which is a PIR. Hence, expectations are given by (A1) and (A2), so that (5) is (A3). So the two possible equilibria are again given by (PIR, PIR) and (ZIR, PIR), that is, equations (A4) and (A5) respectively. However, we need to distinguish two cases, as shown in Panel E and F of Figure 1 . If p < ψ < 1,, then the red line is steeper than the blue line and the two equilibria are possible, again provided that the shock is sufficiently small. On the contrary, if ψ < p < 1, the red line is flatter than the blue line and there will be a unique equilibrium: (PIR, PIR) given by (A4). Note that know the equilibrium exists whatever the value of the shock, because r L p ψ−p > 0, hence ψπ > −µ. Therefore if ψ < p < 1, the model is coherent and complete. In this unique equilibrium implies inflation is above target in the temporary state, as evident from Panel F and (A4). Hence more precisely, if ψ < p < 1, the unique solution isπ
We briefly explain the application of GLM's Theorem 1 for obtaining the CC condition for the system of piecewise linear simultaneous equations (7). In GLM notation, this is a function of the form f = n i=1 A i 1 C i , where C i is a 'cone' in k , e.g., an orthant, and n = 2 k . In the bivariate example of Subsection 2.2 given in eq. (6) with k = 2, the four orthants C i and coefficient matrices A i are
C 3 = {(y 1 , y 2 ) : y 1 < 0, y 2 < 0} , A 3 = B = K, and (A14)
where we used the notation D j· to denote the jth row of a matrix D. Let S i be a k × k selection matrix that consists of unit and zero vectors arranged such that certain elements of S i y are zero while the others are unchanged. In the above example, S 1 = I 2 for (A12), S 2 = 1 0 0 0 for (A13), so that S 2 D = D 1· , S 3 = 0 for (A14) and S 4 = I 2 − S 2 for (A15). Thus, A i = B − S i D, GLM's theorem 1 states that the system of equations (7) is coherent and complete if and only if det A i have the same sign for all i, i.e.,
(A16)
Consider the generic two-state Markov chain with transition kernel K = p 1 − p 1 − q q .
Then,
There are two cases to consider.
Case p+q −1 > 0 This happens, for example when p, q > 1/2, so that the process is 'positively autocorrelated' (the process is more likely to stay in each current state than switch). Then, p + q − 1 > 0, so the CC requires that all determinants must be positive, i.e., q (1 − ψ) > 1 − p, p (1 − ψ) > 1 − q, which both require ψ < 1, and hence, det A 1 > 0 then implies
Case p + q − 1 < 0 Now, CC requires all determinants to be negative. From det A 1 < 0 we get p + q − 1 < ψ < 1.
det A 2 < 0 and det A 4 < 0 require ψ > p+q−1 q and ψ > p+q−1 p , respectively, which are both satisfied by (A18).
Next, consider the case when K is set according to the Rouwenhorst approximation of an AR(1) process. 26 With k = 2, this requires setting p = q = 1+ρ 2 , so that (8) becomes ψ < ρ if ρ > 0 and ρ < ψ < 1 if ρ < 0.
Under CC, the unique solution to the ACS example of Section 2 with k = 2 is given by π = y − ι 2 µ/ψ, where
and (x) j denotes the jth element of a vector x.
A.5 Further numerical results on multiple equilibria A.6 Derivation of multiple equilibria of the ACS model with continuous support
We wish to prove that (10) and (11) f
are solutions to (9) whenM t = ρM t−1 + σε t with ε t = v t 1 {vt<a t−1 } , where v t ∼ N (m t−1 , 1), a t−1 := −µ 1−ψ σψ − ρ σM t−1 , and ψ > ρ. m t−1 is found by solving m t−1 = φ(m t−1 −a t−1 ) Φ(m t−1 −a t−1 ) , where φ(·) and Φ(·) are the density and distribution functions of a standard Normal, respectively, so that E t−1 t = 0, whence we obtainM t+1|t = ρM t , or g(m) = ρm. The support restriction also impliesM t ≤ ρM t
Consider first the PIR-only solution (10), which must satisfy (9) with max(−µ, ψf (m)) = ψf (m). Substituting this candidate solution and g(m) = ρm into (9), we need to verify
which holds. We also need to verify that (10) is always above the ZLB, i.e., ψf (m) ≥ −µ, or m < ψ−ρ ψρ µ for ψ > ρ. This follows from (A20) because ∂ ∂ρ ψ−ρ ψρ = − 1 ρ 2 , so ψ−ρ ψρ is decreasing in ρ, and therefore, ψ−1 ψ < ψ−ρ ρψ for ρ < 1. Finally, turn to the ZIR-only solution (11). It is immediate to see that (9) indeed holds at (11) solution is always at the ZLB, i.e., ψf (M ) = ψm − ψµ ≤ −µ, which follows from (A20).
A.7.1 Coherency and completeness condition
We look for a solution of the formπ t = f π ( t ) andx t = f x ( t ) . Let denote the vector of k states of the shock and similarly for the solutions π and x. As before, denote by K the transition kernel of the Markov chain for t . Then, with some abuse of notation, π t+1|1 = Kπ and x t+1|t = Kx. Substituting into (14)- (15) we have
The last equation can be written as
and multiplying the first by (I − K)
and using the equation above, we get
Re-arranging, we get
We can write this in the form (7) with
The CC condition is obtained by solving the system of inequalities (A16). We can derive the CC condition analytically in the case k = 2.
As in the bivariate ACS example, we have a piecewise linear system in four orthants that can be written as
For CC, we need all determinants below to be of the same sign:
Hence, the CC condition depends on the sign of det B, which can be written as
So, we have the following two cases to consider.
Case (p + q − 1) σλ > (1 + β − pβ − qβ) (2 − p − q) Note that the RHS of this inequality is always positive, so given σλ > 0, this means p + q − 1 > 0.
In that case, we need det A i > 0 for all i. For det A 1 , this means
For det A 2 , we need
An entirely symmetric argument applies for det A 4 . Hence, going back to (A21), we
The CC now requires det A i < 0 for all i. For det A 1 < 0, because (det B − σ 2 λ 2 ψ) < 0 for all ψ ≥ 0, we need ψ < 1. Next, we turn to det A 2 < 0. This satisfied for all ψ ≥ 0 if (1 − q) (1 + β − pβ − qβ) − qσλ < 0, so the only interesting case is the opposite. In that case, we have
where the last inequality follows from det B < 0 and (A22). An entirely symmetric argument applies for det A 4 < 0. Hence, we have established that the CC condition in this case is ψ < 1. Putting these cases together yields (19).
Here we sketch the graphical analysis of the NK model (14)- (16). Let's first look at the absorbing state (or the steady state), where t = 0. Then x = 1 − β λπ AS ;π = max {−µ, ψπ} AD There are two possible steady states: 1) PIR: (π,x,R) = (0, 0, 0); 2) ZIR: (π,x,R) = (−µ, −µ (1−β) λ , −µ). A ZIR, however, must satisfy:π =R = max {−µ, ψπ} = −µ, which can be true only if ψ ≥ 1. Hence, a ZIR steady state does not exists when ψ < 1. Figure 8 shows these two possibilities given by the intersection between the AS and the AD curves, where the AD is defined by bothR =π andR = max {−µ, ψπ} , as in the previous simple model in Section 2. Again, it is evident that a support restrictions is needed for the absorbing state to have a solution (actually two solutions) in the case ψ > 1.
As for the simple model in Section 2, the temporary state lasts for a random time T , after which the economy jumps to the steady state, because the model is completely forward-looking with no endogenous persistence. In the temporary equilibrium t = −a < 0, the equilibrium will be constant (π L ,x L ) and with probability (1 − p) we are back in the absorbing state. The latter can be a PIR one or a ZIR one (if ψ > 1).
Let's start with the case ψ < 1. In this case, the only possible absorbing state is PIR, so the system becomeŝ 
The AD curve is piecewise linear and exhibits a kink at π = −µ/ψ. Graphically, it is easy to distinguish the type of temporary equilibria that can arise depending on the relative slopes of the three curves involved: AS, AD T R and AD ZLB . The AD ZLB is always steeper than the AD T R , which becomes negative if ψ > p.
Case ψ < p. Figure 9 show the possible cases when ψ < p, that is when AD T R has a positive slope.
Panel (a) arises when the AS is steeper than the AD ZLB (so the AS is steeper than both the AD s); that is: σλ − 1 + σµ > 0. Panel (b) displays the case where slope AD ZLB > slope AS > slope AD T R . This happens when: 1 > Θ(p) λσ > 1 − ψ p > 0. As evident from the graph, in this case, the CC is not satisfied and there can be two temporary equilibria, but one needs to < 1, which, given that ψ < p, implies: ψ < p 1 − Θ(p) λσ , which is condition (A23). As evident from the graph, in this case, the CC is satisfied and there is only one temporary equilibrium (a PIR since a is positive, because a ZIR requires a negative a), and the solution is given by (A28) and (A29).
Case p < ψ < 1. In this case the slope of AD T R is negative so it is always below the slope of the AS. It follows that the case in panel (c) in Figure 9 is now impossible. We are left thus with the other two MSV solutions, depending if AS is steeper than AD ZLB (i.e., the unique solution (A26)-(A27)) or vice versa (i.e., the multiple solutions (A28)-(A29) and (A30)-(A31)). The relevant graphs are in panel (a) and (b) in Figure 10 .
Case p < 1 < ψ. This is the case the literature usually analyzes. There are two steady states in the absorbing case.
When the absorbing state is PIR, this is similar to before, as shown in Figure  10 : (i) when the AS is steeper than AD ZLB (panel (a)), there is one MSV solution given by (A26)-(A27); (ii) when the AS is flatter than AD ZLB (panel (b)), there two MSV solutions given by (A28)-(A29) and (A30)-(A31).
When the absorbing state is ZIR, instead, expectations in the temporary equilibrium are different, so the system to solve for iŝ
The system is very similar to the previous case (A24)-(A25). Expectation of a ZIR in the absorbing state, rather than a PIR, just modifies the intercepts of the two curves, but it does not modify the slopes. Hence, with respect to the panels (a) and (b) in Figure 10 , in panels (c) and (d): (i) the AS is shifted upward; (ii) the AD is shifted downward or upward depending whether 1−β λ ≷ σ. These shifts do not affect the CC condition, because the relative slope of the AD ZLB and the AS does not change. From the graphs it is then easy to see that the case is qualitatively the same as the previous one. The only things that differ are the threshold value of a, i.e.,ã, at which the kink in the AD is on the AS, which is now given by:
λψ − βµ λ , as well as the equilibrium values ofπ L andx L . Hence: (i) when the AS is steeper than AD ZLB , there are two MSV solutions (panel (a) and panel (c)); (ii) when the AS is flatter than AD ZLB (panel (b) and panel (d)), there four MSV solutions. 