Abstract:
JEL classification: C19, C51, C62, C63.
Keywords: Indeterminacy, General Equilibrium, Solution method, Bayesian methods.
Sunspot shocks and multiple equilibria have been at the center of economic thinking at least since the seminal work of Cass and Shell, 1983, Farmer and Guo, 1994 and Farmer and Guo, 1995. The zero lowerbound has brought renovated interest to the problem of indeterminacy (Aruoba et al., 2018). Furthermore, in many of the Linear Rational Expectation (LRE) models used to study the properties of the macroeconomy the possibility of multiple equilibria arises for some parameter values, but not for others. This paper proposes a novel approach to solve LRE models that easily accommodates both the case of determinacy and indeterminacy. As a result, the proposed methodology can be used to easily estimate a LRE model that could potentially be characterized by multiplicity of equilibria. Our approach is implementable even when the analytic conditions for determinacy or the degrees of indeterminacy are unknown. Importantly, the proposed method can be easily implemented to study indeterminacy in standard software packages, such as Dynare and Sims' (2001) code Gensys.
To understand how our approach works, it is useful to recall the conditions for determinacy as stated by Blanchard and Kahn, 1980. Indeterminacy arises when the parameter values are such that the number of explosive roots is smaller than the number of non-predetermined variables. The key idea behind our methodology consists of augmenting the original model by appending additional autoregressive processes that can be used to provide the missing explosive roots. The innovations of these exogenous processes are assumed to be linear combinations of a subset of the forecast errors associated with the expectational variables of the model and a newly defined vector of sunspot shocks. When the Blanchard-Kahn condition for determinacy is satisfied, all the roots of the auxiliary autoregressive processes are assumed to be within the unit circle and the auxiliary process is irrelevant for the dynamics of the model. The law of motion for the endogenous variables is in this case equivalent to the solution obtained using standard solution algorithms (King and Watson, 1998, Klein, 2000, Sims, 2001). When the model is indeterminate, the appropriate number of appended autoregressive processes is assumed to be explosive. For example, if there are two degrees of indeterminacy, two of the auxiliary processes are assumed to be explosive. The solution that we obtain for the endogenous variables is equivalent to the one obtained with the methodology of Lubik and Schorfheide, 2003 or, equivalently, Farmer et al., 2015.
Our methodology can be used with standard estimation packages such as Dynare. The solution or estimation under indeterminacy is not generally implementable in standard packages. Our method solves this problem by expanding the state space and making sure that in this expanded state space the conditions for determinacy always hold. Thus, our approach allows the researcher to solve and estimate a model under indeterminacy using standard software packages. Our methodology also simplifies the common approach used to deal with indeterminacy. The common procedure requires the researcher to solve the model differently depending on the area of the parameter space that is being studied. Under indeterminacy, existing methods require to construct the solution ex-post following the seminal contribution of Lubik and Schorfheide, 2003 or to rewritethe model based on the existing degree of indeterminacy (Farmer et al., 2015). In itself, this is not an insurmountable task, but it implies that the researcher interested in a structural estimation of the model would need to write the estimation codes and not just the solution codes. Our proposed method only requires the researcher to augment the original system of equations to reflect the maximum degree of indeterminacy and can therefore be used with no modification of the solution approach. Finally, we show that our approach can facilitate the transition between the determinacy and indeterminacy regions of the parameter space. This method works because our auxiliary processes can be used to make more likely a draw that crosses the threshold of determinacy and to keep track of the distance from such threshold. This idea is particularly easy to implement when the threshold of the determinacy region is known.
Our work is related to the vast literature that studies the role of indeterminacy in explaining the evolution of the macroeconomy. Prominent examples in the monetary policy literature include the work of Clarida et al., 2000 and Kerr and King,1996, that study the possibility of multiple equilibria as a result of violations of the Taylor Principle in New-Keynesian (NK) models. Applying the methods developed in Lubik and Schorfheide, 2003 to the canonical NK model, Lubik and Schorfheide, 2004 test for indeterminacy in U.S. monetary policy. Using a calibrated small-scale model, Coibon and Gorodnichenko, 2011 find that the reduction of the target inflation rate in the United States also played a key role in explaining the Great Moderation, and Arias et al., 2017 support this finding in the context of a medium-scale model a la Christiano et al., 2005. In a similar spirit, Arias, 2013 studies the dynamic properties of medium-sized NK models with trend inflation. More recently, Aruoba et al., 2018 study inflation dynamics at the Zero Lower Bound (ZLB) and during an exit from the ZLB.
The paper closest to our is Farmer et al., 2015. As explained above, the main difference between the two approaches is that our method accommodates the case of both determinacy and indeterminacy while considering the same augmented system of equations.Instead, the method proposed by Farmer et al., 2015 requires us to rewrite the model based on the existing degree of indeterminacy. With respect to Lubik and Schorfheide, 2003, the main novelty of our approach is to provide a unified approach to study determinacy and indeterminacy of different degrees.1 Finally, we deliberately use Dynare in all the examples presented inthis paper to show that our method can be combined with standard packages. However, our solution method can be combined with more sophisticated estimation techniques such as the ones developed in Herbst and Schorfheide, 2015.
To show how to use our methodology in practice, we estimate the small-scale NK model of Galí, 2017 using Bayesian techniques using U.S. data over the period 1982:Q4 until 2007:Q3. Gali's model extends a conventional NK model to allow for the existence of rational bubbles. An interesting aspect of the model is that it displays up to two degrees of indeterminacy for realistic parameter values. We find that the data support the version of the model with two degrees of indeterminacy, implying that the central bank was not reacting strongly enough to the bubble component.
The remainder of the paper is organized as follows. Section 2 builds the intuition by using a univariate example in the spirit of Lubik and Schorfheide, 2004. Section 3 describes the methodology and shows that the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, and under indeterminacy to those obtained using the methodology provided by Lubik and Schorfheide (2003, 2004) and Farmer et al., 2015. In Section 4, we provide guidance on how to properly implement our methodology, and suggestions on how it could be used to improve the efficiency of existing estimation algorithms. In Section 5, we apply our theoretical results to estimate NK model with rational bubbles of Galí, 2017 using Bayesian techniques. We present our conclusions in Section 6.
Before presenting the theoretical results of the paper, this section builds the intuition behind our approach by considering a univariate example similar to the one proposed in Lubik and Schorfheide, 2004. While Section 2.1explains our approach from an analytical perspective, Section 2.2 addresses questions which could arise at the time of its practical implementation.
Consider a classical monetary model characterized by the Fisher equation
$$\displaystyle i_{t}=E_{t}(\pi _{t+1})+r_{t},$$ | (1) |
$$\displaystyle i_{t}=\phi _{\pi }\pi _{t},$$ | (2) |
$$\displaystyle \eta _{t}\equiv \pi _{t}-E_{t-1}(\pi _{t}).$$ | (3) |
$$\displaystyle E_{t}(\pi _{t+1})=\phi _{\pi }\pi _{t}-r_{t}.$$ | (4) |
$$\displaystyle \pi _{t}$$ | $$\displaystyle =$$ | $$\displaystyle \frac{1}{\phi _{\pi }}E_{t}(\pi _{t+1})+\frac{1}{\phi _{\pi }}r_{t}$$ | |
$$\displaystyle =$$ | $$\displaystyle \frac{1}{\phi _{\pi }}r_{t}.$$ | (5) |
where the last equality is obtained by solving the equation forward and recalling the assumptions on $$r_{t}$$. The strong response of the monetary authority to changes in inflation ( $$\phi _{\pi }>1$$) guarantees that inflation is pinned down as a function of the exogenous real interest $$r_{t}$$. From a technical perspective, when $$\phi _{\pi }>1$$ the Blanchard-Kahn condition for uniqueness of a solution is satisfied: The number of explosive roots matches the number of expectational variables, that in this univariate case is one.
The second case corresponds to $$\phi _{\pi }\leq 1$$. The solution is obtained by combining (4) with (3), and it corresponds to any process that takes the following form
$$\displaystyle \pi _{t}=\phi _{\pi }\pi _{t-1}-r_{t-1}+\eta _{t}.$$ | (6) |
The previous equation also holds under determinacy, but in that case the central bank's behavior induces restrictions on the expectation error $$\eta _{t}$$. Instead, when the monetary authority does not respond aggressively enough to changes in inflation ( $$\phi _{\pi }\leq 1$$), there are multiple solutions for the inflation rate, $$\pi _{t}$$, each indexed by the expectations that the representative agent holds about future inflation, $$\eta _{t}$$. Equivalently, the solution to the univariate model is indeterminate: The Blanchard-Kahn solution is not satisfied as there is no explosive root to match the number of expectational variables.
The simple model considered here can be solved pencil and paper. However, when considering richer models with multiple endogenous variables, indeterminacy represents a challenge from a methodological and computational perspective. Standard software packages such as Dynare do not allow for indeterminacy. Of course, a researcher could in principle code an estimation algorithm herself, following the methods outlined in Lubik and Schorfheide, 2004. However, this approach requires a substantial amount of time and technical skills. The researcher would need to write a code that not only finds the solution, but also implements the estimation algorithm. Hence, the result is that in practice most of the papers simply rule out the possibility of indeterminacy, even if the model at hand could in principle allow for such a feature.
The problem that a researcher faces when solving a LRE model under indeterminacy using standard solution algorithms can be easily understood based on the example provided above. Under determinacy, the model already has a sufficient number of unstable roots to match the number of expectational variables. However, under indeterminacy, the model is missing one explosive root. Thus, we propose to augment the original state space of the model by appending an independent process which could be either stable or unstable.
The key insight consists of choosing this auxiliary processes in a way to deliver the correct solution. When the original model is determinate, the auxiliary process must be stationary so that also the augmented representation satisfies the Blanchard-Kahn condition. In this case, the auxiliary process represents a separate block that does not affect the law of motion of the model variables. When the model is indeterminate, the additional process should however be explosive so that the Blanchard-Kahn condition is satisfied for the augmented system, even if not for the original model. By choosing the auxiliary process in the appropriate way, the solution under determinacy in this expanded state space corresponds to the solution under indeterminacy under the original state space. In what follows, we apply this intuition to the example considered above. In Section 3, we show that the approach can be easily extended to richer models to accommodate any degree of indeterminacy.
Our methodology proposes to solve an augmented system of equations which can be dealt with by using standard solution algorithms such as Sims, 2001 under both determinacy and indeterminacy. Consider the following augmented system
$$\displaystyle \left\{ \begin{array}{c} E_{t}(\pi _{t+1})=\phi _{\pi }\pi _{t}-r_{t}, \\ \omega _{t}=\left( \frac{1}{\alpha }\right) \omega _{t-1}-\nu _{t}+\eta _{t},\end{array}\right.$$ | (7) |
Table 1: Blanchard-Kahn condition in the augmented representation
The table reports the regions of the parameter space for which the Blanchard-Kahn condition in the augmented representation is satisfied, even when the original model is indeterminate.
Determinacy $$\phi _{\pi }>1$$ in original model (4)
Unstable Roots | B-K condition in augmented model (7) | Solution | |
---|---|---|---|
$$\frac{1}{\alpha }{ <1}$$ | 1 | Satisfied | $$\left\{ \pi _{t}=\frac{1}{\phi _{\pi }}r_{t},{ \ }\omega_{t}=\alpha \omega _{t-1}-\nu _{t}+\varepsilon _{t}\right\} $$ |
$$\frac{1}{\alpha }{ >1}$$ | 2 | Not satisfied | - |
Indeterminacy $$\phi _{\pi }\leq 1$$ in original model (4)
Unstable Roots | B-K condition in augmented model (7) | Solution | |
---|---|---|---|
$$\frac{1}{\alpha }{ <1}$$ | 0 | Not satisfied | - |
$$\frac{1}{\alpha }{ >1}$$ | 1 | Satisfied | $$\{\pi _{t}=\phi _{\pi }\pi _{t-1}-r_{t-1}+\eta _{t}$$, $$\omega_{t}=0\}$$ |
Table 1 summarizes the intuition behind our approach. When the original LRE model in (4) is determinate,
$$\phi _{\pi }>1$$, the Blanchard-Kahn condition for the augmented representation in (7) is satisfied when
$$1/\alpha <1$$. Indeed, for
$$\phi _{\pi }>1$$ the original model has the same number of unstable roots as the number of expectational variables. Our methodology thus suggests to append a stable autoregressive process such that
$$1/\alpha <1$$. In this case, the method of Sims, 2001 delivers the same solution for the endogenous variable $$\pi _{t}$$ as in equation (5) and for the autoregressive process
$$\omega _{t}$$. Importantly,
$$\omega _{t}$$ is an independent autoregressive process, and its dynamics do not impact the endogenous variable $$\pi _{t}$$.
Considering the case of indeterminacy (i.e. $$\phi _{\pi }\leq 1$$), the original model has one expectational variable, but no unstable root, thus violating the Blanchard-Kahn condition. By appending an explosive autoregressive process, the augmented representation that we propose satisfies the Blanchard-Kahn condition and delivers the same solution as the one resulting from the methodology of Lubik and Schorfheide, 2003 or Farmer et al., 2015 described by equation(6). Moreover, stability imposes conditions such that $$\omega _{t}$$ is always equal to zero at any time $$t$$, and even in this case, the solution for the endogenous variable does not depend on the appended autoregressive process.
Summarizing, the choice of the coefficient $$\frac{1}{\alpha }$$ should be made as follows. For values of $$\phi _{\pi }$$ greater than 1, the Blanchard-Kahn condition for the augmented representation is satisfied for values of $$\alpha $$ greater than 1. Conversely, under indeterminacy (i.e. $$\phi _{\pi }\leq 1$$) the condition is satisfied when $$\alpha $$ is smaller than 1. The choice of parametrizing the auxiliary process with $$1/\alpha $$ instead of $$\alpha $$ induces a positive correlation between $$\phi _{\pi }$$ and $$\alpha $$ that facilitates the implementation of our method when estimating a model.
Finally, note that under both determinacy and indeterminacy, the exact value of $$1/\alpha $$ is irrelevant for the law of motion of $$\pi _{t}$$. Under determinacy, the auxiliary process $$\omega _{t}$$ is stationary, but its evolution does not affect the law of motion of the model variables. Under indeterminacy, $$\omega _{t}$$ is always equal to zero. Thus, the introduction of the auxiliary process does not affect the properties of the solution in the two cases. However, this process serves two important purposes: It provides the correct number of explosive roots under indeterminacy and creates a mapping between the sunspot shock and the expectation errors. As we will see in Section 3, this result can be generalized and applies to more complicated models with potentially multiple degrees of indeterminacy.
Before presenting detailed suggestions for the practical implementation of our method in Section 4, it is useful to provide the intuition for the choice of the parameter $$\alpha $$ in the context of the simple model presented above. First of all, from the discussion above, it should be clear that what matters is only if this parameter is smaller or larger than 1. Its exact value does not affect the solution for $$\pi _{t}$$. Thus, if a researcher wants to solve the model only under indeterminacy (determinacy), it can simply fix the parameter to a value smaller (larger) than 1. In this way, standard solution algorithms proceed to solve the model in the augmented state space only when the model under the original state space is characterized by indeterminacy (determinacy).
However, a researcher might want to allow for both determinacy and indeterminacy when solving the model. We consider the following two cases: (1) The analytic condition defining the region of determinacy are known; (2) The analytic condition defining the region of determinacy are unknown. We consider the two cases separately.
We first consider the case in which the researcher is able to analytically derive the condition which defines when the model is determinate or indeterminate. For the example considered in this section, this case corresponds to knowing that when $$\phi _{\pi }\leq 1$$ the model in (4) is indeterminate. We thus suggest to write the parameter $$\alpha $$ as a function of the parameter $$\phi _{\pi }$$ so that the augmented representation in (7) always satisfies the Blanchard-Kahn condition. In this example, we set $$\alpha \equiv \phi _{\pi } $$. When the original model is determinate ( $$\phi _{\pi }>1$$), the appended autoregressive process is stationary because $$1/\alpha <1$$. If the original model is indeterminate ( $$\phi _{\pi }\leq 1$$), the coefficient $$1/\alpha $$ is greater than 1 and the appended process is therefore explosive. Hence, when the region of determinacy is known, the researcher can easily choose $$\alpha $$ such that the augmented representation always delivers a solution under both determinacy and indeterminacy. Note that in this case $$\alpha $$ is a transformation of $$\phi _{\pi }$$ and effectively no auxiliary extra parameters are introduced.
There are however instances in which the researcher does not know the exact properties of the determinacy region. In this case, the researcher can start with an arbitrary value of $$\alpha $$ for a given sets of parameters $$\theta $$. Suppose that the researcher starts with a value less than 1 and finds that the model is indeterminate for the given set of parameters $$\theta $$. Then, the researcher can just change $$\alpha $$ to a value larger than 1, for example $$\alpha ^{\prime }=1/\alpha $$. A similar logic applies to the case with multiple degrees of indeterminacy that we discuss below: If the solution algorithm returns a solution with $$m$$ degrees of indeterminacy, $$m$$ explosive auxiliary processes are necessary.
We now present the main contribution of the paper generalizing the intuition provided above to a multivariate model with potentially multiple degrees of indeterminacy. Given the general class of LRE models described in Sims, 2001, this paper proposes an augmented representation which embeds the solution for the model under both determinacy and indeterminacy. In particular, the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, and under indeterminacy to those obtained using the methodology provided by Lubik and Schorfheide (2003, 2004) or equivalently Farmer et al., 2015. In the following, we generalize the intuition built in the previous section. Consider the following LRE model
$$\displaystyle \Gamma _{0}(\theta )X_{t}=\Gamma _{1}(\theta )X_{t-1}+\Psi (\theta )\varepsilon _{t}+\Pi (\theta )\eta _{t},$$ | (8) |
$$\displaystyle E_{t-1}(\varepsilon _{t})=0,$$ and $$\displaystyle E_{t-1}(\eta _{t})=0.$$ |
We also define the $$\ell \times \ell $$ matrix $$\Omega _{\varepsilon \varepsilon }$$,
$$\displaystyle \Omega _{\varepsilon \varepsilon }\equiv E_{t-1}(\varepsilon _{t}\varepsilon _{t}^{\prime }),$$ |
Consider a model whose maximum degree of indeterminacy is denoted by $$m$$.3The proposed methodology appends to the original LRE model in (8) the following system of $$m$$ equations
$$\displaystyle \omega _{t}=\Phi \omega _{t-1}+\nu _{t}-\eta _{f,t},\Phi \equiv \begin{bmatrix}\frac{1}{\alpha _{1}} & & \mathbf{0} \\ & \ddots & \\ \mathbf{0} & & \frac{1}{\alpha _{m}}\end{bmatrix}$$ | (9) |
The intuition behind the proposed methodology works as in the example considered in the previous section. Let $$m^{\ast }\left( \theta \right) $$ denote the actual degree of indeterminacy associated with the parameter vector $$\theta $$. Under indeterminacy the Blanchard-Kahn condition for the original LRE model in (8) is not satisfied. Given that the system is characterized by $$m^{\ast }\left( \theta \right) $$ degrees of indeterminacy, it is necessary to introduce $$m^{\ast }\left( \theta \right) $$ explosive roots to solve the model using standard solution algorithms. In this case, $$m^{\ast }\left( \theta \right) $$ of the diagonal elements of the matrix $$\Phi $$ are assumed to be outside the unit circle (in absolute value), and the augmented representation is therefore determinate because the Blanchard-Kahn condition is now satisfied. On the other hand, under determinacy the (absolute value of the) diagonal elements of the matrix $$\Phi $$ are assumed to be all inside the unit circle, as the number of explosive roots of the original LRE model in (8) already equals the number of expectational variables in the model ( $$m^{\ast }\left( \theta \right) =0$$). Also, in this case the augmented representation is determinate due to the stability of the appended auxiliary processes. Importantly, as shown for the univariate example in Section 2, the block structure of the proposed methodology guarantees that the autoregressive process, $$\omega _{t}$$, never affects the solution for the endogenous variables, $$X_{t}$$.
Denoting the newly defined vector of endogenous variables $$\hat{X}_{t}\equiv (X_{t},\omega _{t})^{\prime }$$ and the newly defined vector of exogenous shocks $$\hat{\varepsilon}_{t}\equiv (\varepsilon _{t},\nu _{t})^{\prime }$$, the system in (8) and (9) can be written as
$$\displaystyle \hat{\Gamma}_{0}\hat{X}_{t}=\hat{\Gamma}_{1}\hat{X}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t},$$ | (10) |
$$\displaystyle \hat{\Gamma}_{0}\equiv \begin{bmatrix}\Gamma _{0}(\theta ) & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Gamma}_{1}\equiv \begin{bmatrix}\Gamma _{1}(\theta ) & \mathbf{0} \\ \mathbf{0} & \Phi\end{bmatrix},\hat{\Psi}\equiv \begin{bmatrix}\Psi (\theta ) & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Pi}\equiv \begin{bmatrix}\Pi _{n}(\theta ) & \Pi _{f}(\theta ) \\ 0 & -\mathbf{I}\end{bmatrix},$$ |
and without loss of generality the matrix $$\Pi $$ in (8) is partitioned as $$\Pi =\left[ \Pi _{n}\Pi _{f}\right] $$, where the matrices $$\Pi _{n}$$ and $$\Pi _{f}$$ are respectively of dimension $$k\times (p-m)$$ and $$k\times m$$.4
Section 3.1 and 3.2 show that the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, and under indeterminacy to those obtained using themethodology provided by Lubik and Schorfheide (2003, 2004) and Farmer et al., 2015. In order to simplify the exposition, when analyzing the case of indeterminacy we assume, without loss of generality, $$m^{\ast }(\theta )=m$$. As it will become clear, the case of $$m^{\ast }(\theta )<m$$ is a special case of what we present below.
This section considers the case in which the original LRE is determinate, and shows the equivalence of the solution obtained using the proposed augmented representation with the one from the standard solution method described in Sims, 2001.
Consider the LRE model in (8) and reported in the following equation
$$\displaystyle \underset{k\times k}{\Gamma _{0}}\underset{k\times 1}{X_{t}}=\underset{k\times k}{\Gamma _{1}}\underset{k\times 1}{X_{t-1}}+\underset{k\times l}{\Psi }\underset{l\times 1}{\varepsilon _{t}}+\underset{k\times p}{\Pi }\underset{p\times 1}{\eta _{t}}.$$ | (11) |
The method described in Sims, 2001 delivers a solution, if it exists, by following four steps. First, Sims, 2001 shows how to write the model in the form
$$\displaystyle SZ^{\prime }X_{t}=TZ^{\prime }X_{t-1}+Q\Psi \varepsilon _{t}+Q\Pi \eta _{t},$$ | (12) |
where $$\Gamma _{0}=Q^{\prime }SZ^{\prime }$$ and $$\Gamma _{1}=Q^{\prime }TZ^{\prime }$$ result from the QZ decomposition of $$\{\Gamma _{0},\Gamma _{1}\}$$, and the $$k\times k$$ matrices $$Q$$ and $$Z$$ are orthonormal, upper triangular and possibly complex. Also, the diagonal elements of $$S$$ and $$T$$ contain the generalized eigenvalues of $$\{\Gamma _{0},\Gamma _{1}\}$$.
Second, given that the QZ decomposition is not unique, Sims' algorithm chooses a decomposition that orders the equations so that the absolute values of the ratios of the generalized eigenvalues are placed in an increasing order, that is
$$\displaystyle \left\vert t_{jj}\right\vert /\left\vert s_{jj}\right\vert \geq \left\vert t_{ii}\right\vert /\left\vert s_{ii}\right\vert$$ for $$\displaystyle j>i$$. |
The algorithm then partitions the matrices $$S$$, $$T$$, $$Q$$ and $$Z$$ as
$$\displaystyle S=\begin{bmatrix}S_{11} & S_{12} \\ 0 & S_{22}\end{bmatrix},T=\begin{bmatrix}T_{11} & T_{12} \\ 0 & T_{22}\end{bmatrix},Z^{\prime }=\begin{bmatrix}Z_{1} \\ Z_{2}\end{bmatrix},Q=\begin{bmatrix}Q_{1} \\ Q_{2}\end{bmatrix},$$ |
where the first block corresponds to the system of equations for which $$\left\vert t_{jj}\right\vert /\left\vert s_{jj}\right\vert \leq 1$$ and the second block groups the equations which are characterized by explosive roots, $$\left\vert t_{jj}\right\vert /\left\vert s_{jj}\right\vert >1$$.
The third step imposes conditions on the second, explosive block to guarantee the existence of at least one bounded solution. Defining the transformed variables
$$\displaystyle \xi _{t}\equiv Z^{\prime }X_{t}=\begin{bmatrix}\underset{\left( k-n\right) \times 1}{\xi _{1,t}} \\ \underset{n\times 1}{\xi _{2,t}}\end{bmatrix},$$ |
$$\displaystyle \widetilde{\Psi }\equiv Q^{\prime }\Psi ,$$ and $$\displaystyle \widetilde{\Pi }\equiv Q^{\prime }\Pi ,$$ |
$$\displaystyle \xi _{2,t}=S_{22}^{-1}T_{22}\xi _{2,t-1}+S_{22}^{-1}(\widetilde{\Psi }_{2}\varepsilon _{t}+\widetilde{\Pi }_{2}\eta _{t}).$$ |
$$\displaystyle \underset{n\times 1}{\xi _{2,0}}$$ | $$\displaystyle =$$ | 0 | (13) |
$$\displaystyle \underset{n\times \ell }{\widetilde{\Psi }_{2}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{n\times p}{\widetilde{\Pi }_{2}}\underset{p\times 1}{\eta _{t}}$$ | $$\displaystyle =$$ | $$\displaystyle 0,$$ | (14) |
$$\displaystyle \eta _{t}=-\widetilde{\Pi }_{2}^{-1}\widetilde{\Psi }_{2}\varepsilon _{t}.$$ | (15) |
$$\displaystyle \xi _{1,t}$$ | $$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}(\widetilde{\Psi }_{1}\varepsilon _{t}+\widetilde{\Pi }_{1}\eta _{t})$$ | |
$$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}\left( \widetilde{\Psi }_{1}-\widetilde{\Pi }_{1}\widetilde{\Pi }_{2}^{-1}\widetilde{\Psi }_{2}\right) \varepsilon _{t}$$ | (16) |
Using the algorithm by Sims, 2001, we can describe the solution under determinacy of the LRE model in (11) with equations (13), (15), and (16).
We now consider the methodology proposed in this paper, and we augment the LRE model in (11) with the following system of $$m$$ equations
$$\displaystyle \omega _{t}=\Phi \omega _{t-1}+\nu _{t}-\eta _{f,t},\Phi \equiv \begin{bmatrix}\frac{1}{\alpha _{1}} & & \mathbf{0} \\ & \ddots & \\ \mathbf{0} & & \frac{1}{\alpha _{m}}\end{bmatrix}$$ |
$$\displaystyle \hat{\Gamma}_{0}\hat{X}_{t}=\hat{\Gamma}_{1}\hat{X}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t},$$ | (17) |
where
$$\displaystyle \hat{\Gamma}_{0}\equiv \begin{bmatrix}\Gamma _{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Gamma}_{1}\equiv \begin{bmatrix}\Gamma _{1} & \mathbf{0} \\ \mathbf{0} & \Phi\end{bmatrix},\hat{\Psi}\equiv \begin{bmatrix}\Psi & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Pi}\equiv \begin{bmatrix}\Pi _{n} & \Pi _{f} \\ 0 & -\mathbf{I}\end{bmatrix},$$ |
and without loss of generality the matrix $$\Pi $$ is partitioned as $$\Pi =\left[ \Pi _{n}\Pi _{f}\right] $$, where the matrices $$\Pi _{n}$$ and $$\Pi _{f}$$ are respectively of dimension $$k\times (p-m)$$ and $$k\times m$$.
We can find a solution to the augmented representation in (17) by using Sims' algorithm. Similarly to the previous section, we follow the four steps which describe the algorithm. First, the solution algorithm performs the QZ decomposition of the matrices $$\{\hat{\Gamma}_{0},\hat{\Gamma}_{1}\}$$ and the augmented representation takes the form
$$\displaystyle \hat{S}\hat{Z}^{\prime }\hat{X}_{t}=\hat{T}\hat{Z}^{\prime }\hat{X}_{t-1}+\hat{Q}\hat{\Psi}\hat{\varepsilon}_{t}+\hat{Q}\hat{\Pi}\eta _{t},$$ | (18) |
where $$\hat{\Gamma}_{0}=\hat{Q}^{\prime }\hat{S}\hat{Z}^{\prime }$$ and $$\hat{\Gamma}_{1}=\hat{Q}^{\prime }\hat{T}\hat{Z}^{\prime }$$ result from the QZ decomposition of $$\{\hat{\Gamma}_{0},\hat{\Gamma}_{1}\}$$, and
$$\displaystyle \hat{S}=\begin{bmatrix}S_{11} & \mathbf{0} & S_{12} \\ \mathbf{0} & \mathbf{I} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & S_{22}\end{bmatrix},\hat{T}=\begin{bmatrix}T_{11} & \mathbf{0} & T_{12} \\ \mathbf{0} & \Phi & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & T_{22}\end{bmatrix},\hat{Z}^{T}=\begin{bmatrix}Z_{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \\ Z_{2} & \mathbf{0}\end{bmatrix},\hat{Q}=\begin{bmatrix}Q_{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \\ Q_{2} & \mathbf{0}\end{bmatrix}.$$ |
Second, the algorithm chooses a QZ decomposition which groups the equations in a stable and an explosive block. Because this section assumes that the original model is determinate and that the diagonal elements of the matrix $$\Phi $$ are within the unit circle, the explosive block corresponds to the third system of equations in (18) which is characterized by explosive roots. Recalling the definition of the matrices $$\hat{\Psi}$$ and $$\hat{\Pi}$$, the system of equations in the third block is
$$\displaystyle \xi _{2,t}=S_{22}^{-1}T_{22}\xi _{2,t-1}+S_{22}^{-1}(\widetilde{\Psi }_{2}\varepsilon _{t}+\widetilde{\Pi }_{2}\eta _{t}).$$ | (19) |
The third step imposes conditions to guarantee the existence of a bounded solution. However, the explosive block in (19) is identical to the system of equations found in the previous section. Therefore, the algorithm imposes the same restrictions to guarantee the existence of a bounded solution, that is
$$\displaystyle \xi _{2,0}=0$$ | (20) |
$$\displaystyle \eta _{t}=-\widetilde{\Pi }_{2}^{-1}\widetilde{\Psi }_{2}\varepsilon _{t}.$$ | (21) |
$$\displaystyle \xi _{1,t}$$ | $$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}\left( \widetilde{\Psi }_{1}-\widetilde{\Pi }_{1}\widetilde{\Pi }_{2}^{-1}\widetilde{\Psi }_{2}\right) \varepsilon _{t},$$ | (22) |
$$\displaystyle \omega _{t}$$ | $$\displaystyle =$$ | $$\displaystyle \Phi \omega _{t-1}+\nu _{t}-\eta _{f,t}.$$ | (23) |
Two remarks should be made when comparing the two solutions. First, as shown in (21), the forecast errors are only a function of the exogenous shocks $$\varepsilon _{t}$$, and $$\mathit{not}$$ of the newly defined sunspot shocks, $$\nu _{t}$$. It is therefore clear that the endogenous variables, $$X_{t}$$, of the original LRE model do not respond to sunspot shocks either, as expected under determinacy. Second, (22) and (23) indicate that under determinacy the appended system of equations constitutes a separate block, which does not affect the dynamics of the endogenous variables, $$X_{t}$$. Thus, the likelihood associated with a vector of observables $$Z_{t}$$ that represents a linear transformation of the variables in $$X_{t}$$ is invariant with respect to the method used to compute the solution. This statement holds because the latent processes do not affect $$Z_{t}$$.
This section shows the equivalence of the solutions obtained for a LRE model under indeterminacy using the proposed augmented representation and the methodology of Lubik and Schorfheide (2003, 2004).
As in Section 3.1, we consider the LRE model in (11) and reported below as (24)
$$\displaystyle \Gamma _{0}X_{t}=\Gamma _{1}X_{t-1}+\Psi \varepsilon _{t}+\Pi \eta _{t}.$$ | (24) |
In this section we assume that the model is indeterminate, and we present the method used by Lubik and Schorfheide, 2003. The authors implement the first two steps of the algorithm by Sims, 2001 and described in Section3.1.1.5 They proceed by first applying the QZ decomposition to the LRE model in (24) and then ordering the
resulting system of equations in a stable and an explosive block as defined in equation (12). However, their approach differs in the third step when the algorithm imposes restrictions to guarantee the existence of a bounded solution. In particular, the restrictions in (13) and (14) reported below as (25) and (26) require that
$$\displaystyle \underset{n\times 1}{\xi _{2,0}}$$ | $$\displaystyle =$$ | $$\displaystyle 0,$$ | (25) |
$$\displaystyle \underset{n\times \ell }{\widetilde{\Psi }_{2}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{n\times p}{\widetilde{\Pi }_{2}}\underset{p\times 1}{\eta _{t}}$$ | $$\displaystyle =$$ | $$\displaystyle 0.$$ | (26) |
Nevertheless, it is clear that the system of equation in (26) is indeterminate as the number of forecast errors exceeds the number of explosive roots ($$p>n$$). Equivalently, there are less equations ($$n$$) than the number of variables to solve for ($$p$$). To characterize the full set of solutions to equation (26), Lubik and Schorfheide, 2003 decompose the matrix $$\tilde{\Pi}_{2}$$ using the following singular value decomposition
$$\displaystyle \underset{n\times p}{\widetilde{\Pi }_{2}}~\equiv ~\underset{n\times n}{U}\left[ \begin{array}{cc} \underset{n\times n}{D_{11}} & \underset{n\times m}{\mathbf{0}}\end{array}\right] \underset{p\times p}{V^{\prime }},$$ |
where $$m$$ represents the degrees of indeterminacy. Given the partition $$$\underset{p\times p}{V}\equiv \left[ \begin{array}{cc} \underset{p\times n}{V_{1}} & \underset{p\times m}{V_{2}}\end{array}\right] $$$, equation (26) can be written as
$$\displaystyle \underset{n\times n}{D_{11}^{-1}}\underset{n\times n}{U^{\prime }}\underset{n\times \ell }{\widetilde{\Psi }_{2}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{n\times p}{V_{1}^{\prime }}\underset{p\times 1}{\eta _{t}}=0.$$ | (27) |
$$\displaystyle \underset{m\times \ell \;}{\widetilde{M}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{m\times m}{M_{\zeta }}\underset{m\times 1}{\zeta _{t}}=\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times 1}{\eta _{t}}.$$ | (28) |
$$\displaystyle E\left[ \zeta _{t}\right] =0,\;\ \ E\left[ \zeta _{t}\varepsilon _{t}^{\prime }\right] =0,\;\;E\left[ \zeta _{t}\zeta _{t}^{\prime }\right] =\Omega _{\zeta \zeta }.$$ |
$$\displaystyle \underset{p\times 1}{\eta _{t}}=\left( \underset{p\times n}{-V_{1}}\underset{n\times n}{D_{11}^{-1}}\underset{n\times n}{U_{1}^{\prime }}\underset{n\times \ell }{\widetilde{\Psi }_{2}}+\underset{p\times m\;}{V_{2}}\underset{m\times \ell \;}{\widetilde{M}}\right) \underset{\ell \times 1}{\varepsilon _{t}}+\underset{p\times m\;}{V_{2}}\underset{m\times 1}{\zeta _{t}}.$$ |
$$\displaystyle \underset{p\times 1}{\eta _{t}}\;=~\left( \underset{p\times n}{V_{1}}\underset{n\times \ell }{N}\;+\underset{p\times m\;}{V_{2}}\underset{m\times \ell \;}{\widetilde{M}}\right) \underset{\ell \times 1}{\varepsilon _{t}}+\underset{p\times m\;}{V_{2}}\underset{m\times 1}{\zeta _{t}},$$ | (29) |
$$\displaystyle \underset{n\times \ell }{N}\;\equiv \;\underset{n\times n}{-D_{11}^{-1}}\underset{n\times n}{U_{1}^{\prime }}\underset{n\times \ell }{\widetilde{\Psi }_{2}.}$$ |
$$\displaystyle \xi _{1,t}$$ | $$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}(\widetilde{\Psi }_{1}\varepsilon _{t}+\widetilde{\Pi }_{1}\eta _{t})$$ | |
$$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}\left( \widetilde{\Psi }_{1}+\widetilde{\Pi }_{1}V_{1}N+\widetilde{\Pi }_{1}V_{2}\widetilde{M}\right) \varepsilon _{t}+S_{11}^{-1}\left( \widetilde{\Pi }_{1}V_{2}\right) \zeta _{t}.$$ | (30) |
Using the method in Lubik and Schorfheide, 2003, we can describe the solution for the original LRE model under indeterminacy with equations (25), (29) and (30).
We now consider the augmented representation as in (17) and reported below as
$$\displaystyle \hat{\Gamma}_{0}\hat{X}_{t}=\hat{\Gamma}_{1}\hat{X}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t},$$ | (31) |
where $$\hat{X}_{t}\equiv (X_{t},\omega _{t})^{\prime }$$, $$\hat{\varepsilon}_{t}\equiv (\varepsilon _{t},\nu _{t})^{\prime }$$ and
$$\displaystyle \hat{\Gamma}_{0}\equiv \begin{bmatrix}\Gamma _{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Gamma}_{1}\equiv \begin{bmatrix}\Gamma _{1} & \mathbf{0} \\ \mathbf{0} & \Phi\end{bmatrix},\hat{\Psi}\equiv \begin{bmatrix}\Psi & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{\Pi}\equiv \begin{bmatrix}\Pi _{n} & \Pi _{f} \\ 0 & -\mathbf{I}\end{bmatrix}.$$ |
where the matrix $$\Pi $$ is partitioned as $$\Pi =\left[ \Pi _{n}\Pi _{f}\right] $$ without loss of generality.
The novelty of our approach is that, given our representation, we can easily obtain the solution by using Sims' algorithm even when the original LRE is assumed to be indeterminate. It is enough to assume that the auxiliary processes $$\omega _{t}$$ are characterized by explosive roots, or equivalently that the diagonal elements of the matrix $$\Phi $$ are outside the unit circle. This approach guarantees that the Blanchard-Kahn condition for the augmented representation is satisfied and, given the analytic form that we propose for the auxiliary processes, we show that the solution for the endogenous variables of interest, $$X_{t}$$, is equivalent to the method of Lubik and Schorfheide, 2003.
To show this result, we simply apply the four steps of the algorithm described in Sims, 2001 to the proposed augmented representation. First, the QZ decomposition of (31) takes the form
$$\displaystyle \hat{S}\hat{Z}^{\prime }\hat{X}_{t}=\hat{T}\hat{Z}^{\prime }\hat{X}_{t-1}+\hat{Q}\hat{\Psi}\hat{\varepsilon}_{t}+\hat{Q}\hat{\Pi}\eta _{t},$$ | (32) |
$$\displaystyle \hat{S}=\begin{bmatrix}S_{11} & S_{12} & \mathbf{0} \\ \mathbf{0} & S_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{T}=\begin{bmatrix}T_{11} & T_{12} & \mathbf{0} \\ \mathbf{0} & T_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \Phi\end{bmatrix},\hat{Z}^{T}=\begin{bmatrix}Z_{1} & \mathbf{0} \\ Z_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix},\hat{Q}=\begin{bmatrix}Q_{1} & \mathbf{0} \\ Q_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix}.$$ | (33) |
Second, the QZ decomposition chosen by the algorithm groups the explosive dynamics of the model in the second and third system of equations in (32), which are reported below as (34)
$$\displaystyle \begin{bmatrix}S_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix}\begin{bmatrix}\xi _{2} \\ \omega _{t}\end{bmatrix}=\begin{bmatrix}T_{22} & \mathbf{0} \\ \mathbf{0} & \Phi\end{bmatrix}\begin{bmatrix}\xi _{2,t-1} \\ \omega _{t-1}\end{bmatrix}+\begin{bmatrix}Q_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix}\left( \hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t}\right) .$$ | (34) |
$$\displaystyle \underset{n\times 1}{\xi _{2,0}}$$ | $$\displaystyle =$$ | $$\displaystyle 0,$$ | (35) |
$$\displaystyle \underset{m\times 1}{\omega _{0}}$$ | $$\displaystyle =$$ | $$\displaystyle 0,$$ | (36) |
$$\displaystyle \begin{bmatrix} Q_{2} & \mathbf{0} \ \mathbf{0} & \mathbf{I}\end{bmatrix}\left( \hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t}\right)$$ | $$\displaystyle =$$ | $$\displaystyle 0.$$ | (37) |
$$\displaystyle \underset{p\times (\ell +m)}{\underbrace{\begin{bmatrix}\widetilde{\Psi }_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}\end{bmatrix}}}\underset{(\ell +m)\times 1}{\hat{\varepsilon}_{t}}+\underset{p\times p}{\underbrace{\begin{bmatrix}\widetilde{\Pi }_{n,2} & & \widetilde{\Pi }_{f,2} \\ \mathbf{0} & & -\mathbf{I}\end{bmatrix}}}\underset{p\times 1}{\eta _{t}}=0,$$ | (38) |
$$\displaystyle \eta _{t}=-\begin{bmatrix}\widetilde{\Pi }_{n,2}^{-1}\widetilde{\Psi }_{2} & & \widetilde{\Pi }_{n,2}^{-1}\widetilde{\Pi }_{f,2} \\ \mathbf{0} & & -\mathbf{I}\end{bmatrix}\hat{\varepsilon}_{t}.$$ |
$$\displaystyle \eta _{t}=C_{1}\varepsilon _{t}+C_{2}\nu _{t},$$ | (39) |
The last step of Sims' algorithm combines the restrictions in (35), (36) and (39) with the stationary block derived from the QZ decomposition in (32),
$$\displaystyle \xi _{1,t}$$ | $$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}(\widetilde{\Psi }_{1}\varepsilon _{t}+\widetilde{\Pi }_{1}\eta _{t})$$ | |
$$\displaystyle =$$ | $$\displaystyle S_{11}^{-1}T_{11}\xi _{1,t-1}+S_{11}^{-1}\left( \widetilde{\Psi }_{1}+\widetilde{\Pi }_{1}C_{1}\right) \varepsilon _{t}+S_{11}^{-1}\left( \widetilde{\Pi }_{1}C_{2}\right) \nu _{t}.$$ | (40) |
The indeterminate equilibria found using the methodology of Lubik and Schorfheide, 2003 are parametrized by two sets of parameters. The first set is defined by $$\theta _{1}\in \Theta _{1}$$, where $$\theta _{1}\equiv vec(\Gamma _{0},\Gamma _{1},\Psi ,\Omega _{\varepsilon \varepsilon })^{\prime }$$ is a vector of structural parameters of the model as well as the covariance matrix of the exogenous shocks. The second set corresponds to $$\theta _{2}\in \Theta _{2}$$, where $$\theta _{2}\equiv vec\left( \Omega _{\zeta \zeta },\widetilde{M}\right) ^{\prime }$$ is a parameter vector related to the additional equations introduced in (28) and reported below as (41),
$$\displaystyle \underset{m\times \ell \;}{\widetilde{M}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{m\times m}{M_{\zeta }}\underset{m\times 1}{\zeta _{t}}=\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times 1}{\eta _{t}}.$$ | (41) |
The characterization of a Lubik-Schorfheide equilibrium is a vector $$\theta ^{LS}\in \Theta ^{LS}$$, where $$\Theta ^{LS}$$ is defined as
$$\displaystyle \Theta ^{LS}\equiv \left\{ \Theta _{1},\Theta _{2}\right\} .$$ |
$$\displaystyle \Theta ^{BN}\equiv \left\{ \Theta _{1},\Theta _{3}\right\} .$$ |
The following theorem establishes the equivalence between the characterizations of indeterminate equilibria obtained by using the methodology in Lubik and Schorfheide, 2003 and the proposed augmented representation.
$$\displaystyle \Gamma_0 X_t = \Gamma_1 X_{t-1} + \Psi\varepsilon_t + \Pi \eta_t.$$ |
In the paper Farmer et al., 2015, the authors also show that their characterization of indeterminate equilibria is equivalent to Lubik and Schorfheide, 2003. Therefore, the following corollary holds.
Corollary #.:
Moreover, the following two considerations support Corollary 3 below, which describes a relevant result on the likelihood function of the augmented representation. First, as emphasized in this section, the solution of the model in the augmented state space has a block structure which ensures that the evolution of the endogenous variables in $$X_{t}$$ is not a function of the autoregressive processes, $$\omega _{t}$$. Second, note that the appended autoregressive processes in $$\omega _{t}$$ only serves the purpose of providing the necessary explosive roots under indeterminacy and creating a mapping from the sunspot shocks to the expectational errors. These auxiliary processes are not mapped into the observable variables through the measurement equation. These two considerations imply that the parameters of the matrix $$\Phi $$ introduced with the augmented representation are not identified within certain parameter region. The algorithm only requires them to be inside or outside the unit circle. Corollary 3 then follows.7
Corollary #.:
While Section 3.1 shows that the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, Theorem 1 proves that under indeterminacy the solutions of ourmethodology are equivalent to those obtained using Lubik and Schorfheide (2003, 2004) and Farmer et al., 2015. This theoretical result is crucial for the application of our methodology to the New-Keynesian (NK) model with rational bubbles of Galí, 2017 in Section 5.
In this section, we present some suggestions for the practical implementation of our method with an emphasis on the use of standard software packages such as Dynare. We consider both the model and the data that Lubik and Schorfheide, 2004 use totest for indeterminacy in U.S. monetary policy, as it is possible to derive analytically the boundary of the determinacy region and to estimate the model under determinacy and indeterminacy. The model is described by equations (42)$$\sim $$(47) below and consists of a dynamic IS curve,
$$\displaystyle y_{t}=E_{t}\left( y_{t+1}\right) -\tau \left( i_{t}-E_{t}\left( \pi _{t+1}\right) \right) +g_{t},$$ | (42) |
$$\displaystyle \pi _{t}=\beta E_{t}\left( \pi _{t+1}\right) +\kappa \left( y_{t}-z_{t}\right) ,$$ | (43) |
$$\displaystyle i_{t}=\rho _{i}i_{t-1}+\left( 1-\rho _{i}\right) \left( \psi _{1}\pi _{t}+\psi _{2}\left( y_{t}-z_{t}\right) \right) +\varepsilon _{i,t}.$$ | (44) |
The demand shock, $$g_{t}$$, and the supply shock, $$z_{t}$$, follow univariate AR(1) processes
$$\displaystyle g_{t}$$ | $$\displaystyle =$$ | $$\displaystyle \rho _{g}g_{t-1}+\varepsilon _{g,t},$$ | (45) |
$$\displaystyle z_{t}$$ | $$\displaystyle =$$ | $$\displaystyle \rho _{z}z_{t-1}+\varepsilon _{z,t},$$ | (46) |
$$\displaystyle \eta _{y,t}\equiv y_{t}-E_{t-1}\left[ y_{t}\right] ,\;\ \ \ \eta _{\pi ,t}\equiv \pi _{t}-E_{t-1}\left[ \pi _{t}\right] .$$ | (47) |
We define the vector of endogenous variables as $$X_{t}\equiv \left( y_{t},\pi _{t},i_{t},E_{t}\left( y_{t+1}\right) ,E_{t}\left( \pi _{t+1}\right) ,g_{t},z_{t}\right) ^{\prime }$$, the vectors of fundamental shocks and expectation errors,
$$\displaystyle \varepsilon _{t}=\left( \varepsilon _{i,t},\varepsilon _{g,t},\varepsilon _{z,t}\right) ^{\prime },\ \;\;\;\ \eta _{t}=\left( \eta _{y,t},\eta _{\pi ,t}\right) ^{\prime }$$ |
$$\displaystyle \Gamma _{0}(\theta )X_{t}=\Gamma _{1}(\theta )X_{t-1}+\Psi (\theta )\varepsilon _{t}+\Pi (\theta )\eta _{t}.$$ | (48) |
The LRE model in (48) is determinate when the following analytic condition is satisfied,
$$\displaystyle \left\vert \psi ^{\ast }\right\vert \equiv \left\vert \psi _{1}+\frac{(1-\beta )}{\kappa }\psi _{2}\right\vert >1.$$ | (49) |
$$\displaystyle \omega _{t}=\left( \frac{1}{\alpha }\right) \omega _{t-1}+\nu _{t}-\eta _{\pi ,t}.$$ | (50) |
$$\displaystyle \hat{\Gamma}_{0}\hat{X}_{t}=\hat{\Gamma}_{1}\hat{X}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t}.$$ | (51) |
As in Lubik and Schorfheide, 2004, we estimate the model using Bayesian methods using three series as observables: The percentage deviations of (log) real GDP per capita from an HP-trend ($$y_{obs,t}$$), the annualized percentage change in the Consumer Price Index for all Urban Consumers ( $$\pi _{obs,t}$$), and the annualized Federal Funds Rate ($$i_{obs,t} $$). We focus on the data for the pre-Volcker period (1960Q1 - 1979Q2) as Lubik and Schorfheide, 2004 show that during this period the monetary authority did not respond aggressively enough to changes in inflation, thus not suppressing self-fulfilling inflation expectations. We repeat the estimation of the modelof Lubik and Schorfheide, 2004 by adopting the same prior distributions for the structural parameters. The Bayesian estimation is conducted using conventional Metropolis-Hastings algorithm in Dynare.
Priors for the auxiliary parameters. As a first step, we discuss how we choose the prior distribution for the additional parameters introduced under our methodology. Our augmented representation introduces the vector of parameters $$\alpha $$ and parametrizes the continuum of equilibria under indeterminacy by introducing the standard deviation of a sunspot shock and its correlations with the exogenous shocks.
Regarding the vector of parameters $$\alpha $$, we can distinguish three cases. Case 1: When the determinacy threshold is known, then $$\alpha $$ can be expressed as a function of the other parameters. In this case, there is no need to specify a prior on $$\alpha $$ and the prior probability of determinacy is given by the prior on the parameter vector $$\theta $$. Case 2: When the threshold is unknown and the researcher writes her own code, she can start with all the roots inside the unit circle for $$\alpha $$ at each draw of $$\theta $$ and then flip the appropriate number of elements in the vector $$\alpha $$. Thus, even in this case, there is no need to specify a prior on $$\alpha $$ and the prior probability of indeterminacy depends on the prior on the parameter vector $$\theta $$. Case 3: The researcher is using Dynare and the region of the parameter state is unknown. In this case, we suggest to choose priors that are symmetric between the two regions, i.e. that attach 50% probability to determinacy, and orthogonal with respect to the priors on the other parameters.
In what follows, we focus on Case 3 and discuss how to proceed under the assumption that the researcher does not know the region of determinacy and might be interested in using an estimation package such as Dynare. For a given draw of the structural parameters, the researcher would like to make draws of $$\alpha $$ smaller or greater than 1 with equal probabilities. In this case, the researcher could use a uniform distribution over the interval $$[0,2]$$ or any symmetric interval around 1 as a prior distribution.9 Note that when the determinacy region is not known, the efficiency of the algorithm can be improved when the researcher writes her own estimation/solution algorithm. We describe how to improve the efficiency of traditional MCMC algorithms in the subsection "Efficiency" below.
The correlations of the sunspot shocks with the exogenous disturbances are crucial parameters that affect the fit of the model. In line with the theoretical results, a given set of correlations under the representation that includes the forecast error for the inflation rate has a unique mapping to (different values of) the correlations in the representation with the forecast errors for the output gap, and vice versa. Therefore, in order for the two representations to deliver the same fit to the data, a researcher has to leave the correlations unrestricted. One simple option is to set a uniform prior distributions over the interval (-1,1) for the correlations of the sunspot shocks. This approach guarantees that the fit of the model does not depend on which forecast error is included in the auxiliary process.10This is the approach that we follow in the estimation of the model of Galí, 2017 in Section 5.
Lubik and Schorfheide, 2004 center the prior distributions for the additional parameters introduced in their representation on values that minimize the distance between the impulse responses of the model under indeterminacy and determinacy evaluatedat the boundary of the region of determinacy. Thus, they estimate parameters controlling deviations from this centering point. We showed the mapping between the two representations in Section 3 and we illustrate this equivalence with an analytical example in Appendix B. Given this equivalence, the priors for the correlations between sunspot shocks and structural shocks could also be specified in a way to replicate the approach of Lubik and Schorfheide, 2004. Specifically, we could specify a prior on the auxiliarymatrices used in Lubik and Schorfheide, 2004 and then map the matrices into the correlations used in our approach. However, to easily implement our approach using standard packages, we suggest choosing a flat prior. Thus, we do not center our priors on the impulse response at the boundary of the region of determinacy, but still cover this case as equally likely with respect to the others.
Convergence. We are interested in showing that the methodology allows a standard estimation algorithm such as the one implemented in Dynare to travel to the correct region of the parameter space. At the same time, we also want to emphasize the importance of conducting standard convergence diagnostics. To achieve these goals, we set the initial parametrization in the "wrong" region of the parameter space and consider 1,000,000 draws to show that the methodology accommodates the case of determinacy and indeterminacy, as well as to highlight the importance of checking convergence before interpreting the estimation results. Figure 1 reports the posterior distribution for the parameter $$\psi _{1}$$ and $$\alpha $$ obtained for an initial parametrization such that the Taylor Principle holds (i.e. we set $$\psi _{1}=2$$). At first glance, the posterior distribution of the parameter $$\psi _{1}$$ would appear to be bimodal. This finding is consistent with the fact that the proposed augmented representation allows the Metropolis-Hastings algorithm to visit both regions of the parameter space. At the same time, the posterior distribution for the parameter $$\alpha $$ is very similar to the prior distribution, which is specified as a uniform distribution over the interval $$[0,2]$$. Such result conveys the same evidence derived from the posterior for $$\psi _{1}$$ because the algorithm explores both regions by considering draws of $$\alpha $$ which are within as well as outside the unit circle.
Figure 1: Posterior distribution of parameter $$\psi _{1}$$ and $$\alpha $$
Initial
parametrization $$\psi _1=2$$. The grey line represents the prior distribution and the black line is the posterior distribution.
A researcher should then verify the occurrence of either of the following two circumstances. This bimodal distribution could arise because the log-likelihood is highly discontinuous between the two regions. In this case, the algorithm could have jumped towards the region where the peak of the posterior lies, without having spent a significant time there. In other words, convergence has not occurred yet. Alternatively, if the log-likelihood function varies smoothly between the two regions of the parameter space, the posterior distribution plotted in Figure 1 could be the result of the algorithm traveling across the two regions multiple times.
We therefore recommend the researcher to analyze the draws of the parameter $$\alpha $$ which have been accepted during the MCMC algorithm. By inspecting the behavior of the auxiliary parameter $$\alpha $$, a researcher can detect if the algorithm reached convergence or not. We report the draws that we obtained during our exercise in Figure 2. After approximately 400,000 draws of $$\alpha $$ in the region of determinacy (i.e. outside the unit circle), the algorithm jumps to the indeterminate region and never visits the determinacy region again.
Figure 2: Draws of the parameter $$\alpha $$
Sequence of draws for $$\alpha $$ given an initial
parametrization $$\psi _1=2$$.
Thus, Figure 1 and 2 suggest that the algorithm is in fact able to jump toward the correct region of the parameter space, but also that convergence has not occurred yet. Therefore, the researcher should repeat the estimation exercise, increase the number of draws, and make sure that the parameter $$\alpha $$ stabilizes on one region of the parameter space. Under different circumstances, the researcher could face the second scenario, for which the log-likelihood function transitions smoothly between the two regions. In this case, the parameter $$\alpha $$ would repeatedly transition between the two areas of the parameter space and could be used to infer the probability attached to determinacy. Below we discuss how our methodology can be used to efficiently facilitate such transition.
Only (in)determinacy. In some cases, a researcher might want to estimate the model exclusively under determinacy or exclusively under indeterminacy. Our approach easily accommodates this need. If the researcher is only interested in the solution under determinacy, the parameter vector of $$\alpha $$ should be chosen in a way to guarantee stationarity of the auxiliary process (for example, fixing all values of the alphas to 2). Furthermore, all parameters that are relevant only under indeterminacy should be fixed to zero or any other constant, given that they do not affect the fit of the model under determinacy. If instead the researcher is only interested in estimating the model under indeterminacy, the parameters of the auxiliary process can be chosen in a way to guarantee that the correct number of explosive roots are provided. In this case, the parameters describing the properties of the sunspot disturbances should also be estimated.
Model comparison. A researcher might also be interested in comparing the fit of the model under determinacy and under indeterminacy. Model comparison can be conducted by using standard techniques, such as the harmonic mean estimator proposed by Geweke, 1999. If the researcher is interested in comparing the same model under determinacy and under indeterminacy, we recommend the following procedure that adapts the approach used by Lubik and Schorfheide, 2004:
Efficiency. As mentioned above, our method can also be used to make traditional MCMC algorithms more efficient. We have left the discussion of this point last because, unlike the procedures described above, it generally requires the researcher to write its own code. The key idea is that in many cases the auxiliary parameter $$\alpha $$ can be used to summarize the distance of the current parameter vector from the threshold separating determinacy and indeterminacy regions. This approach is substantially easier when the partition of the parameter space is known, as in Lubik and Schorfheide, 2004. However, the idea can be used even when the region of the parameter space is not known. For the sake of presenting the general idea, we focus on the former case.
As illustrated above, our method prescribes to set the parameter $$\alpha $$ to a value smaller than 1 when the original model presents indeterminacy. Therefore, the value of this auxiliary parameter can be considered an indicator variable for the presence of indeterminacy in the original model. When determinacy or indeterminacy depends on a large number of parameters, access to this indicator variable can facilitate transition between the two regions of the parameter space. To see why, suppose that indeterminacy depends on $$k$$ parameters and that the threshold for indeterminacy is known. Then, we can easily obtain a draw for $$\alpha $$ and $$k-1$$ of the parameters that control determinacy, check whether the drawn $$\alpha $$ implies determinacy or indeterminacy, and, finally, solve for the $$k-th$$ parameter. Therefore, the probability of jumping between the two regions is controlled by a single parameter. The proposal distribution for this parameter can then be chosen to ensure that once it approaches the threshold, the proposal distribution is such that a jump is more likely. Instead, in the standard approach the $$k$$ parameters are drawn without consideration of how far the current parameter vector is from the threshold separating the two areas of the parameter space.
There are of course many possible ways to choose the proposal distribution for $$\alpha $$ in a way that jumps between the two regions is more likely. One simple way consists of choosing a mixture of normals and then using a standard Metropolis-Hastings algorithm that corrects for the asymmetry in the proposal distribution. In what follows, we present this approach in the context of the model of Lubik and Schorfheide, 2004.
Let's choose $$\alpha $$ in a way that every draw implies existence and uniqueness of a solution in the augmented parameter space:
$$\displaystyle \alpha \equiv \psi _{1}+\frac{1-\beta }{\kappa }\psi _{2}$$ |
$$\displaystyle \psi _{1}=\alpha -\frac{1-\beta }{\kappa }\psi _{2}.$$ |
Let $$d\equiv \alpha -\overline{\alpha }=\alpha -1$$ be the distance between the current value for $$\alpha $$ and the boundary of the determinacy region for this auxiliary parameter, $$\overline{\alpha }\equiv 1$$. Note that when the distance is negative, $$\alpha $$ is below the threshold and the model is under indeterminacy. Suppose that we specify the proposal distribution to be a mixture of normals: One centered on the current parameter value and one just beyond the threshold of the determinacy region. Specifically, we assume the following proposal distribution for the proposed draw $$\widetilde{\alpha }$$, given the current value, $$\alpha _{n}$$:
$$\displaystyle \widetilde{\alpha }$$ | $$\displaystyle \sim$$ | $$$\left\{ \begin{array}{ccc} N\left( 1-sign\left( d\right) \mu _{c},\sigma _{c}^{2}\right) & & \text{with probability }w \ N\left( \alpha _{n},\sigma _{\alpha }^{2}\right) & & \text{with probability }1-w\end{array}\right.$$$ | |
$$\displaystyle w$$ | $$\displaystyle =$$ | $$\displaystyle K_{0}\exp \left( -\left\vert d\right\vert K_{1}\right) ,$$$$\displaystyle w_{m}=\exp \left( -\left\vert d\right\vert K_{2}\right)$$ | |
$$\displaystyle \mu _{c}$$ | $$\displaystyle =$$ | $$\displaystyle (1-w_{m})\mu _{b}+w_{m}\ast .5$$ | |
$$\displaystyle \sigma ^{2}$$ | $$\displaystyle =$$ | $$\displaystyle (1-w_{m})\sigma _{b}+w_{m}\ast .1$$ |
Figure 3: Proposal distribution for different values of $$\alpha $$.
The proposal distribution is chosen to facilitate crossing the determinacy threshold and is obtained with a
mixture of normals. The upper (lower) panel assumes that $$\alpha $$ is currently above (below) the threshold of the determinacy region.
Figure 3 presents the proposal distribution for different values of the current $$\alpha _{n}$$. To facilitate the interpretation of the graphs, the top panel plots the proposal distribution for a series of values implying determinacy, while the lower panel considers a series of values implying indeterminacy.11 When $$\alpha _{n}$$ is far from the threshold separating the two regions (dotted black line), the proposal distribution is symmetric. As the current $$\alpha _{n}$$ becomes closer to 1, the weight on the auxiliary normal increases and more and more mass is assigned to drawing a value of $$\alpha $$ that implies a jump between the two regions. Furthermore, as the the current $$\alpha _{n}$$ gets closer to 1, the mean of the auxiliary normal distribution moves further from the determinacy threshold.
Figure 4:
The figure reports the distribution for the number of draws necessary to cross the determinacy threshold for the first time when using a Metropolis-Hastings algorithm to estimate the model of Lubik and Schorfheide (2004). Two cases are
considered. In the first case (blue/dark colored bars), the algorithm is implemented by drawing values for the auxiliary parameter $$\alpha $$. The value of $$\alpha $$ is then used to obtain the corresponding value of
$$\psi _{\pi }$$. In the second case (yellow/light colored bars), the algorithm is implemented by drawing directly the parameters of the model. The distribution is truncated at 100,000 draws.
To understand how the algorithm helps in crossing the determinacy threshold, we estimate the model of Lubik and Schorfheide, 2004 for the post-1982 period using the modified Metropolis algorithm involving the parameter $$\alpha $$ and the traditional algorithm that only involves the model parameters and a symmetric proposal distribution. We start the two algorithms 1,000 times by making a draw from the posterior mode. For each iteration, we count the number of draws necessary for the parameters to cross the determinacy threshold for the first time. We stop when the algorithm has reached 100,000 iterations.
Figure 4 reports the distribution for the number of draws necessary to cross the determinacy threshold for the first time in the two cases. The blue/dark colored bars correspond to the algorithm implemented by drawing values for the auxiliary parameter $$\alpha $$ and then using the value of $$\alpha $$ to obtain the corresponding value of $$\psi _{1}$$. Instead, the yellow/light colored bars correspond to the traditional algorithm that makes draws for the original parameter space. The distribution is truncated at 100,000 draws. From the graph, it is clear that the modified algorithm greatly facilitates crossing the determinacy region. The median value for the number of draws necessary to cross the determinacy region is only 16,555 for the modified algorithm. Instead, for the traditional algorithm in 74,2% of the cases the parameters have not crossed the determinacy threshold after 100,000 iterations.
Finally, we also verify that the modified MCMC algorithm is able to repeatedly jump back and forth between the two regions of the parameter space. We then make $$2,100,000$$ draws from the posterior using the modified algorithm. We find that the algorithm transitions a total of 34 times between the two regions. The posterior probability of being under determinacy, computed as the fraction of draws for which $$\alpha >1$$, is 98.9%. Therefore, the algorithm is able to explore the entire area of the parameter space despite the fact that the determinacy region is overwhelmingly favored by the data.
When conducting the same exercise for the pre-1979 period, we found that the algorithm was able to quickly move to the indeterminacy region independently of the starting point. However, once the algorithm had reached such region, it was not able to leave it because of a large discontinuity in the likelihood. This result is important to highlight that while our approach can facilitate the transition across regions, it cannot overcome the fact that for some models and some data samples the boundary of the determinacy regions might imply a large discontinuity in the posterior. In this case, jumping between the two regions becomes extremely unlikely, even when a clever proposal distribution is used. In these cases, more recent methods, such as the ones described in Herbst and Schorfheide, 2015, can be used to make sure that the entire parameter space is explored. However, for the example considered in this paper, it is worth emphasizing that the conclusions of the analysis are unlikely to change because the lack of jumps between the two regions reflects the fact that the data strongly favor indeterminacy.
In this section, we implement the proposed methodology to estimate the small-scale NK model of Galí, 2017 using Bayesian techniques. The model extends a conventional NK model to allow for the existence of rational expectations equilibria with asset price bubbles. Interestingly, the model displays up to two degrees of indeterminacy for realistic parameter values.
We estimate the model using U.S. data over the period 1982:Q4 until 2007:Q3, and we consider the case that the U.S. monetary policy aimed at stabilizing the inflation rate and leaning against the bubble. We find that the strength of such responses was not enough to guarantee a stabilization of the U.S. economy and to avoid that unexpected changes in expectations could drive U.S. business cycles. In particular, we show that the model specification that provides the best fit to the data is characterized by two degrees of indeterminacy.12
The model of Galí, 2017 is described by the following equations. First, equation (52) represents a dynamic IS curve
$$\displaystyle y_{t}=\Phi E_{t}\left( y_{t+1}\right) -\Psi \left( i_{t}-E_{t}\left( \pi _{t+1}\right) \right) +\Theta q_{t},$$ | (52) |
The aggregate bubble plays the role of demand shifter and is defined as
$$\displaystyle q_{t}=b_{t}+u_{t}^{q},$$ |
$$\displaystyle u_{t}^{q}=\rho _{q}u_{t-1}^{q}+\varepsilon _{t}^{q},$$ $$\displaystyle \varepsilon _{t}^{q}\overset{iid}{\sim }N(0,\sigma _{q}^{2}).$$ |
Equation (53) defines the evolution of the value of the asset bubble $$q_{t}$$ as
$$\displaystyle q_{t}=\Lambda \Gamma E_{t}\left( b_{t+1}\right) -q\left( i_{t}-E_{t}\left( \pi _{t+1}\right) \right) ,$$ | (53) |
where $$q\equiv \frac{\gamma (\beta -\Lambda \Gamma v)}{(1-\beta \gamma )(1-\Lambda \Gamma v\gamma )}$$ represents the steady state bubble-to-output ratio, $$\Lambda \equiv 1/(1+r)$$ is the steady state stochastic discount factor for one-period ahead payoffs derived from a portfolio of securities and $$\Gamma \equiv (1+g)$$ is the gross rate of productivity growth. To guarantee that newly created bubbles along the BGP are non-negative, the model requires that $$\Lambda \Gamma =\frac{1+g}{1+r}\geq 1$$. Equivalently, it must hold that $$r\leq g$$ on a BGP characterized by the creation of (non-negative) new asset bubbles. Equation (53) shows how "optimistic" expectations about the future value of the bubble lead to a higher price for those assets today.
The model is then closed by the following NK Phillips curve
$$\displaystyle \pi _{t}=\Lambda \Gamma v\gamma E_{t}\left( \pi _{t+1}\right) +\kappa y_{t}+u_{t}^{s},$$ | (54) |
$$\displaystyle i_{t}=\rho _{i}i_{t-1}+(1-\rho _{i})\left( \phi _{\pi }\pi _{t}+\phi _{q}q_{t}\right) +\varepsilon _{t}^{i},$$ | (55) |
$$\displaystyle \eta _{y,t}\equiv y_{t}-E_{t-1}\left[ y_{t}\right] ,\;\ \ \ \eta _{\pi ,t}\equiv \pi _{t}-E_{t-1}\left[ \pi _{t}\right] ,\;\ \ \ \eta _{b,t}\equiv b_{t}-E_{t-1}\left[ b_{t}\right] .$$ | (56) |
Equations (52)$$\sim $$(56) describe the equilibrium dynamics of the model economy around a given BGP. We define the vector of endogenous variables as $$X_{t}\equiv (y_{t},\pi _{t},b_{t},i_{t},q_{t},E_{t}\left( y_{t+1}\right) ,E_{t}\left( \pi _{t+1}\right) $$, $$E_{t}\left( b_{t+1}\right) ,u_{t}^{q},u_{t}^{s})^{\prime }$$, and the vectors of fundamental shocks, $$\varepsilon _{t}$$, and non-fundamental errors, $$\eta _{t}$$, as
$$\displaystyle \varepsilon _{t}\equiv \left( \varepsilon _{t}^{q},\varepsilon _{t}^{s},\varepsilon _{t}^{i}\right) ^{\prime },\ \;\;\;\ \eta _{t}\equiv \left( \eta _{y,t},\eta _{\pi ,t},\eta _{b,t}\right) ^{\prime }.$$ |
The model can therefore be represented as
$$\displaystyle \Gamma _{0}(\theta )X_{t}=\Gamma _{1}(\theta )X_{t-1}+\Psi (\theta )\varepsilon _{t}+\Pi (\theta )\eta _{t},$$ | (57) |
$$\displaystyle \omega _{1,t}$$ | $$\displaystyle =$$ | $$\displaystyle \left( \frac{1}{\alpha _{1}}\right) \omega _{1,t-1}+\nu _{1,t}-\eta _{1,t},$$ | (58) |
$$\displaystyle \omega _{2,t}$$ | $$\displaystyle =$$ | $$\displaystyle \left( \frac{1}{\alpha _{2}}\right) \omega _{2,t-1}+\nu _{2,t}-\eta _{2,t},$$ | (59) |
$$\displaystyle \hat{\Gamma}_{0}\hat{X}_{t}=\hat{\Gamma}_{1}\hat{X}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}+\hat{\Pi}\eta _{t}.$$ |
We estimate the model to match U.S. data over the period 1982:Q4 until 2007:Q3. We consider a subset of three macroeconomic quarterly time series used in Smets and Wouters, 2007 to match the number of exogenous shocks in the model. In particular, we use the growth rate in real GDP, the inflation rate measured by the GDP deflator and the Federal Funds rate. We implement Bayesian techniques, and the measurement equations that relate the macroeconomic data to the endogenous variables of the model are defined as
$$\displaystyle \begin{bmatrix}dlGDP_{t} \\ dlP_{t} \\ FFR_{t}\end{bmatrix}=\begin{bmatrix}g \\ \pi ^{\ast } \\ i^{\ast }\end{bmatrix}+\begin{bmatrix}y_{t}-y_{t-1} \\ \pi _{t} \\ i_{t}\end{bmatrix},$$ |
We follow Galí, 2017 and set the discount factor of each individual, $$\beta $$, to 0.998. We estimate the remaining structural parameters of the model using Bayesian techniques. We reportthe prior distributions for the parameters in Table 2. As mentioned when studying equation (53) for the evolution of the value of the asset bubble $$q_{t}$$, the model requires that the real interest rate, $$r$$, and the growth rate of output, $$g$$, satisfy $$r\leq g$$ to ensure that newly created bubbles along the BGP are non-negative. To guarantee that this inequality holds for each draw of the Metropolis-Hastings algorithm, we express the real interest rate, $$r$$, as $$r=\lambda g$$, where $$\lambda \in \left( 0,1\right) $$. We then set the prior for the quarterly growth rate of output, $$g$$, as a gamma distribution centered at 0.45, and the prior for $$\lambda $$ as a beta distribution with mean 0.8. These priors imply that the annualized growth rate of output is 1.6% and the annualized real interest rate is approximately 1.3% over the considered period.
We center the prior for the employment ratio, $$\alpha $$, to 0.6. Following the calibration in Galí, 2017, the prior distribution for the probability that an individual survives to the next period, $$\gamma $$, is centered at 0.996 . The prior for the slope of the New Keynesian Phillips Curve, $$\kappa $$,, is set at 0.04, a value chosen for the calibration in Galí, 2017 and consistent with an average duration of individual prices of 4 quarters. The parameter describing the response of the monetary authority to changes in inflation, $$\phi _{\pi }$$, follows a gamma distribution with mean 1 and standard error 0.4. The response to deviations of the bubble relative to its value along the BGP follows a gamma distribution with mean 0.3 and standard error 0.15. The parameter which governs the degree of interest rate inertia, $$\rho _{i}$$, follows a beta distribution centered at 0.7.
The priors on the stochastic processes that define the fundamental shocks are inverse gamma distributions centered at 0.3 with a standard deviation of 0.15. Finally, when we estimate the model under indeterminacy, we specify uniform prior distributions for both the standard deviations of the non-fundamental shocks $$\{{\small\sigma }_{\nu _{l}}\}$$ where $$l=\{\pi ,y,b\}$$, and their correlations with the exogenous shocks of the model $$\{\varphi _{\nu _{l,j}}\}$$ where $$j=\{i,q,s\}$$.
Table 2: Prior distribution for model parameters
The table reports the prior distribution for the model parameters.
Name | Density | Mean | Std. Dev. |
---|---|---|---|
$$g$$ | $${ Gamma}$$ | 0.45 | 0.04 |
$$\lambda $$ | $${ Beta}$$ | 0.80 | 0.10 |
$$\alpha $$ | $${ Beta}$$ | 0.60 | 0.10 |
$${ 100(\gamma }^{-1}-1{ )}$$ | $${ Gamma}$$ | 0.4 | 0.10 |
$${ \kappa }$$ | $${ Gamma}$$ | 0.04 | 0.005 |
$$\pi ^{\ast }$$ | $${ Gamma}$$ | 0.9 | 0.30 |
$${ i}^{\ast }$$ | $${ Gamma}$$ | 1.2 | 0.30 |
$$\phi _{\pi }$$ | $${ Gamma}$$ | 1 | 0.40 |
$$\phi _{q}$$ | $${ Gamma}$$ | 0.3 | 0.10 |
$${ \rho }_{i}$$ | $${ Beta}$$ | 0.70 | 0.10 |
$${ \sigma }_{q}$$ | $${ Inv.Gamma}$$ | 0.30 | 0.15 |
$${ \sigma }_{s}$$ | $${ Inv.Gamma}$$ | 0.30 | 0.15 |
$${ \sigma }_{i}$$ | $${ Inv.Gamma}$$ | 0.30 | 0.15 |
$${ \rho }_{q}$$ | $${ Beta}$$ | 0.70 | 0.10 |
$${ \rho }_{s}$$ | $${ Beta}$$ | 0.70 | 0.10 |
$${ \sigma }_{\nu _{i}}$$ | $${ Uniform[0,10]}$$ | 5 | 2.89 |
$$\varphi _{\nu _{i},j}$$ | $${ Uniform[-1,1]}$$ | 0 | 0.57 |
We estimate the model in each region of the parameter space: Determinacy, one degree of indeterminacy and two degrees of indeterminacy. When the model is indeterminate, we run the estimation for the different combinations consisting of one or two of the forecast errors defined by the vector
$$\eta _{t}\equiv \left( \eta _{y,t},\eta _{\pi ,t},\eta _{b,t}\right) ^{\prime }$$ depending on the degree of indeterminacy. In line with our theoretical results, we show that, given the degree of indeterminacy, the
estimation delivers the same marginal data densities regardless of which forecast error(s) we include in our representation. In Appendix C, we also show that the posterior distributions for the model parameters are equivalent up to a transformation of the correlations between the exogenous shocks
and the sunspot disturbances considered in each specification.16
Table 3 reports the (log) marginal data density for each of the model specification. We find that the data favor the specification of the model with two degrees of indeterminacy. We attribute this result to the stylized nature of the model, and the observation that indeterminate models are consistent with a richer dynamic and stochastic structure. In future work, it would be valuable to study whether the findings would carry over in the context of a more realistic, medium-scale model that could explain the persistence and volatility in the data without recurring to indeterminate dynamics.
Table 3: Model comparison
The table reports the (log) marginal data densities for each model specification.
Specification | Marginal data densities |
---|---|
Indet-2 | -72.3 |
Indet-1 | -83.0 |
Determinacy | -158.3 |
Table 4 reports the mean and 90% probability interval of the posterior distribution of the estimated structural parameters. The probability of surviving to the next period, $$\gamma $$, is estimated to be approximately 99%. The posterior of the slope of the NK Phillips curve is 0.039, which in this model is consistent with a probability of 24.1% that a firm keeps its price unchanged in any given period. The steady-state inflation rate and nominal interest
rate are about 0.7% and 1.4% on an quarterly basis. We also find that the strength of the responses of U.S. monetary policy to stabilize the inflation rate and lean against the bubble was not enough to guarantee a stabilization of the U.S. economy and to avoid that unexpected changes in
expectations could drive U.S. business cycles.
The mean of the standard error of the bubble component is 0.28, and larger than the standard deviation of the supply and monetary policy shocks that are estimated to be 0.11 and 0.12, respectively. The data also provide evidence that the bubble shock is less persistent than the supply shock.
Finally, we report the standard deviation of the sunspot shocks for the representation that includes the forecast error for the output gap and the inflation rate. The posterior estimates show that the standard error related to forecast errors for the output gap is approximately twice as large as the standard deviation of the sunspot shock associated with the inflation rate. The data also appear to be informative on the correlations of both sunspot shocks with the exogenous shocks of the model. A monetary policy shock is negatively correlated with both sunspot shocks, implying a contemporaneous impact on both inflation and output. A shock due to a new bubble can be interpreted as a demand shifter, and it has no significant correlation with unexpected changes in expectations about future inflation and economic activity. A supply shock has a positive correlation with the sunspot shock associated with inflation, as well as a negative correlation with the sunspot shock for output. These correlations are crucial to interpret the impact that each shock has on the model economy as described next.
Table 4: Posterior distribution for model parameters
The table reports the posterior distribution of the model parameters under two degrees of indeterminacy
$$\{\nu_{\pi},\nu_{y}\}$$.
Mean | 90% prob. int. | |
---|---|---|
$$g$$ | 0.46 | [ 0.41,0.51] |
$$\lambda $$ | 0.79 | [0.64,0.94] |
$$\alpha $$ | 0.59 | [0.51,0.68] |
$${ 100(\gamma }^{-1}-1{ )}$$ | 0.44 | [ 0.30,0.58] |
$${ \kappa }$$ | 0.039 | [0.032,0.047] |
$$\pi ^{\ast }$$ | 0.69 | [0.38,1.01] |
$${ i}^{\ast }$$ | 1.41 | [1.07,1.71] |
$$\phi _{\pi }$$ | 0.35 | [0.16,0.53] |
$$\phi _{q}$$ | 0.13 | [0.06,0.19] |
$${ \rho }_{i}$$ | 0.68 | [0.54,0.84] |
$${ \sigma }_{q}$$ | 0.28 | [0.13,0.43] |
$${ \sigma }_{s}$$ | 0.11 | [0.09,0.13] |
$${ \sigma }_{i}$$ | 0.12 | [0.09,0.14] |
$${ \rho }_{q}$$ | 0.70 | [0.54,0.86] |
$${ \rho }_{s}$$ | 0.89 | [0.84,0.95] |
$${ \sigma }_{\nu _{\pi }}$$ | 0.28 | [0.24,0.32] |
$${ \sigma }_{\nu _{y}}$$ | 0.69 | [0.60,0.78] |
$$\varphi _{\nu _{\pi },i}$$ | - 0.42 | [-0.67,-0.16] |
$$\varphi _{\nu _{\pi },q}$$ | 0.07 | [-0.43,0.59] |
$$\varphi _{\nu _{\pi },s}$$ | 0.61 | [0.48,0.73] |
$$\varphi _{\nu _{y},i}$$ | - 0.14 | [-0.40,0.13] |
$$\varphi _{\nu _{y},q}$$ | - 0.01 | [-0.52,0.55] |
$$\varphi _{\nu _{y},s}$$ | - 0.68 | [-0.77,-0.59] |
Figure 5 plots the impulse response of output, inflation and nominal interest rate. We orthogonalize the fundamental shocks using a Cholesky decomposition with the same order as in the plots
$$\{\varepsilon _{q},\varepsilon _{s},\varepsilon _{i}\}$$. The last two panels report the impulse response functions in which each sunspot shock is the most exogenous shock in the Cholesky decomposition. We plot the
impulse responses to a one-standard-deviation shock. The solid lines represent the posterior means, while the dashed line correspond to the 90% probability intervals.
Considering the estimated correlations reported in Table 4, we observe that a shock due to the creation of a new bubble generates no significant effect on the economy in line with the estimated correlations. A positive supply shock has both inflationary and contractionary effects on impact. The persistence of the shock on output is then associated to deflationary effects to which the monetary authority responds by decreasing the nominal interest rate. A monetary policy tightening generates contractionary and deflationary pressures. The persistence of these effects on the inflation rate then requires the monetary authority to adopt an accommodative stance to stabilize the economy.
The last two panels show the impulse response to the sunspot shocks that we assume to be uncorrelated. In this economy, a positive shock to inflation expectations generates self-fulfilling effects on inflation. Given that in this panel the sunspot shock, $$\varepsilon _{\nu _{\pi }}$$, is assumed to be the most exogenous, economic activity does not respond on impact, while the increase in the nominal interest rate triggers a contractionary effect in the medium term. Finally, a positive sunspot shock to the expectation about future deviations of output from its trend leads to a rise in economic activity due to its self-fulfilling nature. Given that in the last panel we assume that the sunspot shock, $$\varepsilon _{\nu _{y}}$$, is the most exogenous, the inflation rate does not respond on impact, while it is characterized by a mild deflationary effect in the medium term that leads to a decrease in the nominal interest rate.
Table 5 reports the variance decompositions for output, inflation and interest rate. The means and the 90% probability intervals are calculated from the output of the Metropolis-Hastings algorithm. Because the estimated correlations of the two sunspotshocks with the exogenous shocks are nonzero, the reported variance decomposition results from the orthogonalization of the shocks using a Cholesky factorization in which the order of the shocks follows the list in Table 5. The results are in line with those in the literature and in particular with Lubik and Schorfheide, 2004. The deviations of output from its trend are mostly explained by supply shocks. In addition to supply-side shocks, fluctuations in inflations are also accounted for by unexpected changes in monetary policy. Similar conclusions can be drawn for the decomposition of the nominal interest rate. Interestingly, both sunspot shocks play only a marginal role in explaining business cycle fluctuations for each of the three endogenous variables.
Table 5: Variance Decomposition
The table reports the means and 90-percent probability intervals for the unconditional variance decomposition. Since the estimated correlations with the two sunspot shocks are nonzero, the decomposition of the orthogonalized shocks via Cholesky decomposition follows the order of
the shocks as listed in the Table.
Output dev. from trend Mean |
Output dev. from trend 90% prob. int. |
Inflation Mean |
Inflation 90% prob. int. |
Interest rate Mean |
Interest rate 90% prob. int. |
|
---|---|---|---|---|---|---|
$$\varepsilon _{s}$$ | 0.68 | [0.37,0.93] | 0.27 | [0.01,0.57] | 0.27 | [0.01,0.56] |
$$\varepsilon _{i}$$ | 0.22 | [0.02,0.45] | 0.61 | [0.37,0.85] | 0.60 | [0.36,0.85] |
$$\varepsilon _{q}$$ | 0.02 | [0.01,0.03] | 0.02 | [0.01,0.04] | 0.02 | [0.01,0.04] |
$$\varepsilon _{\nu _{y}}$$ | 0.07 | [0.01,0.14] | 0.08 | [0.01,0.21] | 0.09 | [0.01,0.21] |
$$\varepsilon _{\nu _{\pi }}$$ | 0.01 | [0.001,0.02] | 0.02 | [0.01,0.03] | 0.02 | [0.01,0.03] |
Figure 5:
The figures plot the posterior means (solid lines) and 90-percent probability intervals (dashed lines) for the impulse responses of output, inflation and nominal interest rate to a shock of one standard deviation for each orthogonalized
disturbance using a Cholesky decomposition with the same order as in the plots.
In this paper, we propose a generalized approach to solve and estimate LRE models over the entire parameter space. Our approach accommodates both cases of determinacy and indeterminacy and it does not require the researcher to know the analytic conditions describing the region of determinacy or the degrees of indeterminacy.
When a LRE model is characterized by $$m$$ degrees of indeterminacy, our approach augments it by appending $$m$$ autoregressive processes whose innovations are linear combinations of a subset of endogenous shocks and a vector of newly defined sunspot shocks. We show that the solution for the resulting augmented representation embeds both the solution which is obtained under determinacy using standard solution methods and that delivered by solving the model under indeterminacy using the approach of Lubik and Schorfheide, 2003 and equivalently Farmer et al., 2015.
We apply our methodology to estimate the small-scale NK model of Galí, 2017 using Bayesian techniques. Gali's model extends a conventional NK model to allow for the existence of rational bubbles. An interesting aspect of the model is that it displays up to two degrees of indeterminacy for realistic parameter values. We estimate the model using U.S. data over the period 1982:Q4 until 2007:Q3. Using Bayesian model comparison we find that the data support the version of the model with two degrees of indeterminacy, implying that the central bank was not reacting strongly enough to the bubble component. One caveat to note, however, is that the model of Galí, 2017 is quite stylized, but the results are intriguing and merit further exploration in future research.
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We prove the equivalence between the parametrization of the Lubik-Schorfheide indeterminate equilibrium $$\theta ^{LS}\in \Theta ^{LS}$$ and the Bianchi-Nicolo equilibrium parametrized by $$\theta ^{BN}\in \Theta ^{BN}$$. In particular, we show that there is a unique mapping between the linear restrictions imposed in each of the two methodologies on the forecast errors to guarantee the existence of at least a bounded solution. As shown in Section 3.2.1, the method by Lubik and Schorfheide, 2003 imposes the following restrictions on the non-fundamental shocks, $$\eta _{t}$$, as a function of the exogenous shocks, $$\varepsilon _{t}$$, and the sunspot shocks introduced in their specification, $$\zeta _{t}$$,
$$\displaystyle \underset{p\times 1}{\eta _{t}}\;=~\left( \underset{p\times n}{V_{1}}\underset{n\times \ell }{N}\;+\underset{p\times m\;}{V_{2}}\underset{m\times \ell \;}{\underset{m\times \ell \;}{\widetilde{M}}}\right) \underset{\ell \times 1}{\varepsilon _{t}}+\underset{p\times m\;}{V_{2}}\underset{m\times 1}{\zeta _{t}}.$$ | (60) |
Using the methodology proposed in this paper, Section 3.2.2 shows that the restrictions on the non-fundamental shocks, $$\eta _{t}$$, as a function of the exogenous shocks, $$\varepsilon _{t}$$, and the sunspot shocks, $$v_{t}$$, are
$$\displaystyle \underset{p\times 1}{\eta _{t}}=\underset{p\times \ell }{C_{1}}\underset{\ell \times 1}{\varepsilon _{t}}+\underset{p\times m}{C_{2}}\underset{m\times 1}{\nu _{t}},$$ | (61) |
$$\displaystyle C_{1}\equiv -\begin{bmatrix}\widetilde{\Pi }_{n,2}^{-1}\widetilde{\Psi }_{2} \\ \mathbf{0}\end{bmatrix}$$ and $$\displaystyle C_{2}\equiv -\begin{bmatrix}\widetilde{\Pi }_{n,2}^{-1}\widetilde{\Pi }_{f,2} \\ -\mathbf{I}\end{bmatrix}.$$ |
Post-multiplying equation (60) and (61) by
$$\varepsilon _{t}^{\prime }$$ and taking expectations on both sides,
$$\displaystyle \underset{p\times l}{\Omega _{\eta \varepsilon }}\;$$ | $$\displaystyle =$$ | $$\displaystyle ~\underset{p\times n}{V_{1}}\underset{n\times \ell }{N}\underset{\ell \times l}{\Omega _{\varepsilon \varepsilon }}\;+\underset{p\times m\;}{V_{2}}\underset{m\times \ell \;}{\widetilde{M}}\underset{\ell \times l}{\Omega _{\varepsilon \varepsilon }},$$ | |
$$\displaystyle \underset{p\times l}{\Omega _{\eta \varepsilon }}\;$$ | $$\displaystyle =$$ | $$\displaystyle \underset{p\times \ell }{C_{1}}\underset{\ell \times l}{\Omega _{\varepsilon \varepsilon }}\;+\underset{p\times m}{C_{2}}\underset{m\times l}{\Omega _{\nu \varepsilon }}\;$$ |
Pre-multiplying by $$V_{2}^{\prime }$$ and equating the equations,
$$\displaystyle \underset{m\times \ell \;}{\widetilde{M}}\underset{\ell \times l}{\Omega _{\varepsilon \varepsilon }}\;=\left( \underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times \ell }{C_{1}}-\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times n}{V_{1}}\underset{n\times \ell }{N}\right) \underset{\ell \times l}{\Omega _{\varepsilon \varepsilon }}+\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times m}{C_{2}}\underset{m\times l}{\Omega _{\nu \varepsilon }}.$$ | (62) |
Using the properties of the vec operator, the following result holds
$$ \begin{equation}\underset{\left( m\times \ell \right) \times 1\;}{vec(\widetilde{M})}=\underset{\left( m\times \ell \right) \times \left( m\times \ell \right) }{\left( \Omega _{\varepsilon \varepsilon }\otimes I_{m}\right) ^{-1}}\left[\underset{\left( m\times \ell \right) \times \ell ^{2}}{\left[ I_{l}\otimes \left( V_{2}^{\prime }C_{1}-V_{2}^{\prime }V_{1}N\right) \right] }\underset{\ell ^{2}\times 1}{vec\left( \Omega _{\varepsilon \varepsilon }\right) }+\underset{\left( m\times \ell \right) \times \left( m\times \ell \right) }{\left( I_{l}\otimes V_{2}^{\prime }C_{2}\right) }\underset{\left( m\times \ell \right) \times 1}{vec\left( \Omega _{\nu \varepsilon }\right) }\right] .\end{equation}$$ | (63) |
Equation (63) is the first relevant equation to show the mapping between the representation in Lubik and Schorfheide, 2003 and our representation. For a given variance-covariance matrix of the exogenous shocks, $$\Omega _{\varepsilon \varepsilon }$$, that is common between the two representations, equation (63) tells us that the covariance structure, $$\Omega _{\nu \varepsilon }$$, of the sunspot shock in our representation with the exogenous shocks has a unique mapping to the matrix, $$\widetilde{M}$$, in Lubik and Schorfheide, 2003. Clearly, equation (62) can also be used to derive the mapping from their representation to our method.
We now show how to derive the mapping between the variance-covariance matrix,
$$\Omega _{\nu \nu }$$, of the sunspot shocks in our representation to the variance-covariance matrix,
$$\Omega _{\zeta \zeta }$$, of the sunspot shocks in Lubik and Schorfheide, 2003. Considering again equation (60) and (61), we
post-multiply by
$$\zeta _{t}^{\prime }$$ and take expectations on both sides,
$$\displaystyle \underset{p\times m}{\Omega _{\eta \zeta }}\;$$ | $$\displaystyle =$$ | $$\displaystyle \underset{p\times m\;}{V_{2}}\underset{m\times m}{\Omega _{\zeta \zeta }},$$ | |
$$\displaystyle \underset{p\times m}{\Omega _{\eta \zeta }}\;$$ | $$\displaystyle =$$ | $$\displaystyle \underset{p\times m}{C_{2}}\underset{m\times m}{\Omega _{\nu \zeta }}\;$$ |
Pre-multiplying both equations by $$V_{2}^{\prime }$$ and equating them,
$$\displaystyle \underset{m\times m}{\Omega _{\zeta \zeta }}\;=\underset{m\times m}{\Omega _{\zeta \nu }}\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) ^{\prime }}.$$ | (64) |
Finally, to obtain an expression for
$$\Omega _{\zeta \nu }$$, we post-multiply equation (60) and (61) by
$$\nu _{t}^{\prime }$$ and taking expectations
$$\displaystyle \underset{p\times m}{\Omega _{\eta \nu }}\;$$ | $$\displaystyle =$$ | $$\displaystyle ~\left( \underset{p\times n}{V_{1}}\underset{n\times \ell }{N}+\underset{p\times m\;}{V_{2}}\underset{m\times \ell \;}{\widetilde{M}}\right) \underset{\ell \times m}{\Omega _{\varepsilon \nu }}+\underset{p\times m}{V_{2}}\underset{m\times m}{\Omega _{\zeta \nu }},$$ | |
$$\displaystyle \underset{p\times m}{\Omega _{\eta \nu }}\;$$ | $$\displaystyle =$$ | $$\displaystyle \underset{p\times \ell }{C_{1}}\underset{\ell \times m}{\Omega _{\varepsilon \nu }}\;+\underset{p\times m}{C_{2}}\underset{m\times m}{\Omega _{\nu \nu }}\;$$ |
Pre-multiplying both equations by $$V_{2}^{\prime }$$ and solving for $$\Omega _{\zeta \nu }$$,
$$\displaystyle \underset{m\times m}{\Omega _{\zeta \nu }}=\left( \underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times \ell }{C_{1}}-\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times n}{V_{1}}\underset{n\times \ell }{N}-\underset{m\times \ell \;}{\widetilde{M}}\right) \underset{\ell \times m}{\Omega _{\varepsilon \nu }}+\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) }\underset{m\times m}{\Omega _{\nu \nu }}.$$ | (65) |
Post-multiplying (65) by $$\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) ^{\prime }}$$ and using (64), then
$$\displaystyle \underset{m\times m}{\Omega _{\zeta \zeta }}\;=\left( \underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times \ell }{C_{1}}-\underset{m\times p\;}{V_{2}^{\prime }}\underset{p\times n}{V_{1}}\underset{n\times \ell }{N}-\underset{m\times \ell \;}{\widetilde{M}}\right) \underset{\ell \times m}{\Omega _{\varepsilon \nu }}\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) ^{\prime }}+\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) }\underset{m\times m}{\Omega _{\nu \nu }}\underset{m\times m}{\left( V_{2}^{\prime }C_{2}\right) ^{\prime }}.$$ | (66) |
Therefore, equation (66) defines the mapping between the variance-covariance matrix, $$\Omega _{\nu \nu }$$, of the sunspot shocks in our representation to the variance-covariance matrix, $$\Omega _{\zeta \zeta }$$, of the sunspot shocks in Lubik and Schorfheide, 2003. Together with equation (63), we show that this equation defines the one-to-one mapping between the parametrization in Lubik and Schorfheide $$\{\Theta ,\Theta ^{LS}\}$$ and the parametrization in Bianchi-Nicolo $$\{\Theta ,\Theta ^{BN}\}$$.
In this Appendix, we provide an analytical example to show the equivalence between the solutions for an indeterminate LRE model using two alternative methodologies: Lubik and Schorfheide, 2003 and our proposed method. In particular, we consider the
following simple model
$$\displaystyle y_{t}$$ | $$\displaystyle =$$ | $$\displaystyle \frac{1}{\theta _{y}}E_{t}(y_{t+1})+\frac{1}{\theta _{y}}E_{t}(x_{t+1})+\varepsilon _{t}$$ | (67) |
$$\displaystyle x_{t}$$ | $$\displaystyle =$$ | $$\displaystyle \frac{1}{\theta _{x}}E_{t}(x_{t+1})$$ | (68) |
$$\displaystyle \eta _{y,t}$$ | $$\displaystyle \equiv$$ | $$\displaystyle y_{t}-E_{t-1}(y_{t})$$ | (69) |
$$\displaystyle \eta _{x,t}$$ | $$\displaystyle \equiv$$ | $$\displaystyle x_{t}-E_{t-1}(x_{t})$$ | (70) |
The LRE model in (67)$$\sim $$ (70) can be written in the following matrix form
$$\displaystyle \Gamma _{0}S_{t}=\Gamma _{1}S_{t-1}+\Psi \varepsilon _{t}+\Pi \eta _{t},$$ | (71) |
where $$S_{t}\equiv (y_{t},x_{t},E_{t}(y_{t+1}),E_{t}(x_{t+1}))^{\prime }$$ and $$\eta _{t}\equiv (\eta _{y,t},\eta _{x,t})^{\prime }$$.
As the matrix $$\Gamma _{0}$$ is non-singular, the LRE model in (71) can be written as
$$\displaystyle S_{t}=\Gamma _{1}^{\ast }S_{t-1}+\Psi ^{\ast }\varepsilon _{t}+\Pi ^{\ast }\eta _{t},$$ | (72) |
where
$$\displaystyle \Gamma _{1}^{\ast }\equiv \Gamma _{0}^{-1}\Gamma _{1}=\begin{bmatrix}\mathbf{0}_{\mathbf{_{4\times 2}}} & \mathbf{A}_{\mathbf{_{4\times 2}}}\end{bmatrix},\Pi ^{\ast }\equiv \Gamma _{0}^{-1}\Pi =\mathbf{A}_{\mathbf{_{4\times 2}}}$$ |
$$\displaystyle \Psi ^{\ast }\equiv \Gamma _{0}^{-1}\Psi =\begin{bmatrix}0 \\ 0 \\ -\theta _{x} \\ 0\end{bmatrix},\mathbf{A}_{\mathbf{_{4\times 2}}}=\begin{bmatrix}1 & 0 \\ 0 & 1 \\ \theta _{y} & -\theta _{x} \\ 0 & \theta _{x}\end{bmatrix}$$ |
Note that equation (72) corresponds to equation (20) in Lubik and Schorfheide, 2004. We now show how to solve the model and obtain equation (26) in Lubik and Schorfheide, 2004.
Applying the Jordan decomposition, the matrix $$\Gamma _{1}^{\ast }$$ can be decomposed as $$\Gamma _{1}^{\ast }\equiv J\Lambda J^{-1}$$, where the elements of the diagonal matrix $$\Lambda $$ denote the roots of the system
$$\displaystyle \Lambda \equiv \begin{bmatrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \theta _{x} & 0 \\ 0 & 0 & 0 & \theta _{y}\end{bmatrix}=\begin{bmatrix}\Lambda _{11} & \mathbf{0} \\ \mathbf{0} & \theta _{y}\end{bmatrix}.$$ |
Assuming without loss of generality that $$\left\vert \theta _{x}\right\vert \leq 1$$ and $$\left\vert \theta _{y}\right\vert >1$$, the system in (72) is indeterminate because the number of expectational variables, $$\{E_{t}(y_{t+1}),E_{t}(x_{t+1})\}$$, exceeds the number of explosive roots, $$\theta _{y}$$. Defining the vector $$w_{t}\equiv J^{-1}S_{t}$$, the model can be represented as
$$\displaystyle w_{t}\equiv \begin{bmatrix}w_{1,t} \\ w_{2,t}\end{bmatrix}=\begin{bmatrix}\Lambda _{11} & \mathbf{0} \\ \mathbf{0} & \theta _{y}\end{bmatrix}\begin{bmatrix}w_{1,t-1} \\ w_{2,t-1}\end{bmatrix}+\begin{bmatrix}\tilde{\Psi}_{1} \\ \tilde{\Psi}_{2}\end{bmatrix}\varepsilon _{t}+\begin{bmatrix}\tilde{\Pi}_{1} \\ \tilde{\Pi}_{2}\end{bmatrix}\eta _{t},$$ | (73) |
where the first block denotes the stationary block of the system and the second block is unstable.
The adoption of Sims' (2002) code, Gensys, to solve this model is not appropriate as it deals with determinate models. After having obtained the representation in (73), Gensys would construct a matrix $$\Phi $$ such that premultiplying the system by a matrix $$[I$$ $$\ -\Phi ]$$ would eliminate the effect of non-fundamental shocks. Equivalently, the matrix has to satisfy the condition
$$\displaystyle \lbrack I\ -\Phi ]\begin{bmatrix}\tilde{\Pi}_{1} \\ \tilde{\Pi}_{2}\end{bmatrix}=\tilde{\Pi}_{1}-\Phi \tilde{\Pi}_{2}=0.$$ | (74) |
Under determinacy, the matrix $$\tilde{\Pi}_{2}$$ is square and, assuming that it is also non-singular17, it is possible to solve for $$\Phi =\tilde{\Pi}_{1}\left( \tilde{\Pi}_{2}\right) ^{-1}$$.
The approach in Lubik and Schorfheide, 2003 modifies this intuition to account for the indeterminacy that characterizes the model in (73). Under indeterminacy, the matrix $$\tilde{\Pi}_{2}$$ is a vector with more columns than rows, implying that it is not possible to obtain a matrix $$\Phi $$ that satisfies the above condition in (74). Nevertheless, Lubik and Schorfheide, 2003 apply a singular value decomposition (SVD) to the matrix $$\tilde{\Pi}_{2}$$ to obtain
$$\displaystyle \tilde{\Pi}_{2}\equiv UDV^{\prime }=\begin{bmatrix}U_{.1} & U_{.2}\end{bmatrix}\begin{bmatrix}D_{11} & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}V_{.1}^{\prime } \\ V_{.2}^{\prime }\end{bmatrix}=U_{.1}D_{11}V_{.1}^{\prime },$$ | (75) |
$$\displaystyle \tilde{\Pi}_{2}\equiv UDV^{\prime }=1\begin{bmatrix}d & 0\end{bmatrix}\begin{bmatrix}\frac{a}{d} & \frac{b}{d} \\ \frac{b}{d} & -\frac{a}{d}\end{bmatrix},$$ | (76) |
where $$d\equiv \sqrt{a^{2}+b^{2}}$$. Lubik and Schorfheide, 2003 then proceed by defining the matrix $$\Phi $$ as
$$\displaystyle \Phi =\tilde{\Pi}_{1}\left( V_{.1}d^{-1}U_{.1}^{\prime }\right) =\begin{bmatrix}0 & 0 \\ 0 & 0 \\ 0 & \theta _{x}\end{bmatrix}\begin{bmatrix}\frac{a}{d} \\ \frac{b}{d}\end{bmatrix}\frac{1}{d}=\begin{bmatrix}0 & 0 \\ 0 & 0 \\ 0 & \theta _{x}\frac{b}{d^{2}}\end{bmatrix},$$ |
and premultiply the system in (73) by the following matrices
$$\displaystyle \begin{bmatrix} I & -\Phi \ 0 & 1\end{bmatrix}\begin{bmatrix} w_{1,t} \ w_{2,t}\end{bmatrix}$$ | $$\displaystyle =$$ | $$\displaystyle \begin{bmatrix} I & -\Phi \ 0 & 0\end{bmatrix}\begin{bmatrix} \Lambda _{11} & 0 \ 0 & \theta _{y}\end{bmatrix}\begin{bmatrix} w_{1,t-1} \ w_{2,t-1}\end{bmatrix}+$$ | |
$$\displaystyle +\begin{bmatrix} I & -\Phi \ 0 & 0\end{bmatrix}\begin{bmatrix} \tilde{\Psi}_{1} \ \tilde{\Psi}_{2}\end{bmatrix}\varepsilon _{t}+\underset{\neq 0}{\underbrace{\begin{bmatrix} I & -\Phi \ 0 & 0\end{bmatrix}\begin{bmatrix} \tilde{\Pi}_{1} \ \tilde{\Pi}_{2}\end{bmatrix}}}\eta _{t},$$ | (77) |
where the second block represents the constraint that guarantees the boundedness of the solution,
$$\displaystyle w_{2,t}=0\Longleftrightarrow E_{t}(y_{t+1})=-\frac{b}{a}E_{t}(x_{t+1}).$$ | (78) |
$$\displaystyle S_{t}=\tilde{\Gamma}_{1}^{\ast }S_{t-1}+\tilde{\Psi}^{\ast }\varepsilon _{t}+\tilde{\Pi}^{\ast }\eta _{t},$$ | (79) |
where
$$\displaystyle \tilde{\Gamma}_{1}^{\ast }\equiv \begin{bmatrix}\mathbf{0}_{\mathbf{_{4\times 2}}} & \mathbf{B}_{\mathbf{_{4\times 2}}}\end{bmatrix},\tilde{\Psi}^{\ast }\equiv \left( \frac{a}{d}\right) ^{2}\begin{bmatrix}1 \\ b/a \\ -\theta _{x}(b/a)^{2} \\ \theta _{x}b/a\end{bmatrix},$$ |
$$\displaystyle \tilde{\Pi}^{\ast }\equiv \mathbf{B}_{\mathbf{_{4\times 2}}}=\begin{bmatrix}\left( b^{2}/d^{2}\right) & -\frac{b}{a}(1-b^{2}/d^{2}) \\ -ab/d^{2} & (1-b^{2}/d^{2}) \\ \theta _{x}\left( b^{2}/d^{2}\right) & -\theta _{x}\frac{b}{a}(1-b^{2}/d^{2}) \\ -\theta _{x}ab/d^{2} & \theta _{x}(1-b^{2}/d^{2})\end{bmatrix}.$$ |
The last step that Lubik and Schorfheide, 2003 implement is to express the forecast errors as a function of the fundamental shock, $$\varepsilon _{t}$$, and a sunspot shock, $$\zeta _{t}$$, as
$$\displaystyle \eta _{t}=-V_{.1}D_{11}^{-1}U_{.1}^{\prime }\tilde{\Psi}_{2}\varepsilon _{t}+V_{.2}\left( \overset{\backsim }{M}\varepsilon _{t}+M_{\zeta }\zeta _{t}\right) ,$$ | (80) |
where $$V_{.2}^{\prime }=\begin{bmatrix} \frac{b}{d} & -\frac{a}{d}\end{bmatrix}$$. Combining (79) with (80) and normalizing $$M_{\zeta }=1$$, the solution to the LRE model is18
$$\displaystyle S_{t}=\tilde{\Gamma}_{1}^{\ast }S_{t-1}+\tilde{\Psi}^{\ast }\varepsilon _{t}+\tilde{\Pi}^{\ast }V_{.2}\left( \overset{\backsim }{M}\varepsilon _{t}+\zeta _{t}\right) .$$ | (81) |
This solution can be equivalently written in a form that explicitly includes the boundedness condition in (78) for which $$w_{2,t}=0$$ and therefore $$E_{t}(y_{t+1})=-\frac{b}{a}E_{t}(x_{t+1})$$. Recalling that $$S_{t}=(y_{t},x_{t},E_{t}(y_{t+1}),E_{t}(x_{t+1}))^{\prime }$$, the dynamics of the solution in (81) are now expressed as a function of only one state variable,
$$\begin{eqnarray} S_{t} &=&\begin{bmatrix} -b/a \\\ 1 \\ -\theta _{x}b/a \\\ \theta _{x}\end{bmatrix}E_{t-1}(x_{t})+\tilde{\Psi}^{\ast }\varepsilon _{t}+\tilde{\Pi}^{\ast}V_{.2}\left( \overset{\backsim }{M}\varepsilon _{t}+\zeta _{t}\right)\nonumber \\\ &=&\begin{bmatrix}\frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\\ 1 \\\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ \theta _{x}\end{bmatrix}E_{t-1}(x_{t})+\frac{\theta _{y}^{2}}{d^{2}}\begin{bmatrix} 1 \\\ \frac{\theta _{x}}{(\theta _{x}-\theta _{y})} \\\ -\frac{\theta _{x}^{3}}{(\theta _{x}-\theta _{y})^{2}} \\\ \frac{\theta _{x}^{2}}{(\theta _{x}-\theta _{y})}\end{bmatrix}\varepsilon _{t}+\frac{\theta _{y}}{d}\begin{bmatrix} \frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\\ 1 \\\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ \theta _{x}\end{bmatrix}\left( \overset{\backsim }{M}\varepsilon _{t}+\zeta _{t}\right) ,\end{eqnarray}$$ | (82) |
where $$d=\sqrt{\theta _{y}^{2}+\left( \theta _{x}\theta _{y}\right) ^{2}/(\theta _{x}-\theta _{y})^{2}}$$.
We now provide the derivation of the solution for the LRE model in (71) and reported below in equation (83) using the methodology proposed in this paper
$$\displaystyle \Gamma _{0}S_{t}=\Gamma _{1}S_{t-1}+\Psi \varepsilon _{t}+\Pi \eta _{t}.$$ | (83) |
The methodology consists of appending the following equation to the original LRE model
$$\displaystyle \omega _{t}=\frac{1}{\alpha }\omega _{t-1}+\nu _{x,t}-\eta _{x,t},$$ |
where $$v_{t}$$ denotes a newly defined sunspot shock and without loss of generality $$\alpha \equiv \left\vert \theta _{x}\right\vert $$. Denoting the newly defined vector of endogenous variables $$\hat{S}_{t}\equiv (S_{t},\omega _{t})^{\prime }=(y_{t},x_{t},E_{t}(y_{t+1}),E_{t}(x_{t+1}),\omega _{t})^{\prime }$$, and the newly defined vector of exogenous shocks $$\hat{\varepsilon}_{t}^{x}\equiv (\varepsilon _{t},\nu _{x,t})^{\prime }$$, the augmented representation of the LRE model is
$$\displaystyle \hat{\Gamma}_{0}\hat{S}_{t}=\hat{\Gamma}_{1}\hat{S}_{t-1}+\hat{\Psi}\hat{\varepsilon}_{t}^{x}+\hat{\Pi}\eta _{t}.$$ | (84) |
Pre-multiplying the system in (84) by $$\hat{\Gamma}_{0}^{-1}$$, we obtain
$$\displaystyle \hat{S}_{t}=\hat{\Gamma}_{1}^{\ast }\hat{S}_{t-1}+\hat{\Psi}^{\ast }\hat{\varepsilon}_{t}^{x}+\hat{\Pi}^{\ast }\eta _{t},$$ | (85) |
where
$$\displaystyle \hat{\Gamma}_{1}^{\ast }\equiv \begin{bmatrix}\Gamma _{1}^{\ast } & \mathbf{0_{4\times 1}} \\ & \\ \mathbf{0_{1\times 4}} & \frac{1}{\alpha }\end{bmatrix},\hat{\Psi}^{\ast }\equiv \begin{bmatrix}\Psi ^{\ast } & \mathbf{0_{4\times 1}} \\ & \\ 0 & -1\end{bmatrix},\hat{\Pi}^{\ast }\equiv \begin{bmatrix}& \Pi _{4\times 2}^{\ast } \\ & \\ 0 & 1\end{bmatrix}.$$ |
and the matrices $$\{\Gamma _{1}^{\ast },\Psi ^{\ast },\Pi ^{\ast }\}$$ are the same as those found in (72). Applying the Jordan decomposition, the matrix $$\hat{\Gamma}_{1}^{\ast }$$ can be decomposed as $$\hat{\Gamma}_{1}^{\ast }\equiv \hat{J}\hat{\Lambda}\hat{J}^{-1}$$, where the elements of the diagonal matrix $$\hat{\Lambda}$$ denote the roots of the system
$$\displaystyle \hat{\Lambda}\equiv \begin{bmatrix}\Lambda & 0 \\ 0 & \frac{1}{\alpha }\end{bmatrix}=\begin{bmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta _{x} & 0 & 0 \\ 0 & 0 & 0 & \theta _{y} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{\alpha }\end{bmatrix}=\begin{bmatrix}\Lambda _{11} & 0 \\ 0 & \Lambda _{22}\end{bmatrix}.$$ |
$$\displaystyle \hat{w}_{t}\equiv \begin{bmatrix}\hat{w}_{1,t} \\ \hat{w}_{2,t}\end{bmatrix}=\begin{bmatrix}\Lambda _{11} & 0 \\ 0 & \Lambda _{22}\end{bmatrix}\begin{bmatrix}\hat{w}_{1,t-1} \\ \hat{w}_{2,t-1}\end{bmatrix}+\begin{bmatrix}\hat{\Psi}_{1}^{\ast \ast } \\ \hat{\Psi}_{2}^{\ast \ast }\end{bmatrix}\hat{\varepsilon}_{t}^{x}+\begin{bmatrix}\hat{\Pi}_{1}^{\ast \ast } \\ \hat{\Pi}_{2,x}^{\ast \ast }\end{bmatrix}\eta _{t},$$ | (86) |
where the first block is stationary. Given that the second block is unstable, the following two conditions have to be imposed to guarantee the boundedness of the solution. First, the linear combination of the endogenous variables, $$\hat{w}_{2,t}$$, is set to zero,
$$\displaystyle \hat{w}_{2,t}=0$$ $$\displaystyle \Longleftrightarrow$$ $$\displaystyle \left\{ \begin{array}{l} E_{t}(y_{t+1})=-\frac{b}{a}E_{t}(x_{t+1}) \\ \omega _{t}=0\end{array}\right.$$ | (87) |
Second, the linear combination of fundamental and non-fundamental shocks also has to equal zero. Therefore, the non-fundamental shocks, $$\eta _{t}$$, become a function of the augmented vector of exogenous shocks, $$\hat{\varepsilon}_{t}^{x}$$,
$$\displaystyle \eta _{t}=-\left( \hat{\Pi}_{2,x}^{\ast \ast }\right) ^{-1}\hat{\Psi}_{2}^{\ast \ast }\hat{\varepsilon}_{t}^{x}$$ $$\displaystyle \Longleftrightarrow$$ $$\displaystyle \eta _{t}=\begin{bmatrix}1 & -\frac{\theta _{x}}{\theta _{x}-\theta _{y}} \\ 0 & 1\end{bmatrix}\begin{bmatrix}\varepsilon _{t} \\ \nu _{x,t}\end{bmatrix}$$ | (88) |
Considering equation (86), it is relevant to point out that the matrix $$\hat{\Pi}_{2,x}^{\ast \ast }$$ differs from the corresponding matrix for the representation in which we incorporate the forecast error, $$\eta _{y,t}$$, defined as $$\hat{\Pi}_{2,y}^{\ast \ast }$$,
$$\displaystyle \hat{\Pi}_{2,x}^{\ast \ast }\equiv \begin{bmatrix}\theta _{y} & \frac{\theta _{x}\theta _{y}}{\theta _{x}-\theta _{y}} \\ 0 & -1\end{bmatrix}$$ $$\displaystyle \hat{\Pi}_{2,y}^{\ast \ast }\equiv \begin{bmatrix}\theta _{y} & \frac{\theta _{x}\theta _{y}}{\theta _{x}-\theta _{y}} \\ -1 & 0\end{bmatrix}.$$ |
Therefore, when the auxiliary process is written as a function of the non-fundamental shock, $$\eta _{y,t}$$, the restriction imposed on $$\eta _{t}$$ to guarantee the boundedness of the solution also differs from the one found in (88)
$$\displaystyle \eta _{t}=-\left( \hat{\Pi}_{2,y}^{\ast \ast }\right) ^{-1}\hat{\Psi}_{2}^{\ast \ast }\hat{\varepsilon}_{t}^{y}$$ $$\displaystyle \Longleftrightarrow$$ $$\displaystyle \eta _{t}=\begin{bmatrix}0 & 1 \\ \frac{\theta _{x}-\theta _{y}}{\theta _{x}} & -\frac{\theta _{x}-\theta _{y}}{\theta _{x}}\end{bmatrix}\begin{bmatrix}\varepsilon _{t} \\ \nu _{y,t}\end{bmatrix}$$ | (89) |
Importantly, from equations (88) and (89) it is possible to establish a relationship that links the two non-fundamental disturbances $$\{\nu _{x,t},\nu _{y,t}\}$$ and the exogenous shock $$\varepsilon _{t}$$,
$$\displaystyle \nu _{x,t}=\frac{\theta _{x}-\theta _{y}}{\theta _{x}}\varepsilon _{t}-\frac{\theta _{x}-\theta _{y}}{\theta _{x}}\nu _{y,t}.$$ | (90) |
We show below that equations (87) and (90) are crucial for the equivalence between the augmented representations that include different non-fundamental shocks in the auxiliary processes that our methodology proposes.
The augmented model in (86) is determinate as the second block has two explosive roots to match the two expectational variables of the model. It is therefore possible to apply the approach in Sims'(2002) to construct a matrix $$\hat{\Phi}_{x}$$ such that premultiplying the system by a matrix $$[I$$ $$\ -\hat{\Phi}_{x}]$$ would eliminate the effect of non-fundamental shocks. Equivalently, the matrix has to satisfy the condition
$$\displaystyle \lbrack I\ -\hat{\Phi}_{x}]\begin{bmatrix}\hat{\Pi}_{1}^{\ast \ast } \\ \hat{\Pi}_{2,x}^{\ast \ast }\end{bmatrix}=\hat{\Pi}_{1}^{\ast \ast }-\hat{\Phi}_{x}\hat{\Pi}_{2,x}^{\ast \ast }=0.$$ | (91) |
Importantly, the matrix $$\hat{\Pi}_{2,x}^{\ast \ast }$$ is square under determinacy and, assuming that it is also non-singular19, it is possible to solve for $$\hat{\Phi}_{x}=\hat{\Pi}_{1}^{\ast \ast }\left( \hat{\Pi}_{2,x}^{\ast \ast }\right) ^{-1}$$.
To solve the model, the system in (86) is then premultiplied by the following matrices
$$\displaystyle \begin{bmatrix} I & -\hat{\Phi}_{x} \ 0 & I\end{bmatrix}\begin{bmatrix} \hat{w}_{1,t} \ \hat{w}_{2,t}\end{bmatrix}$$ | $$\displaystyle =$$ | $$\displaystyle \begin{bmatrix} I & -\hat{\Phi}_{x} \ 0 & 0\end{bmatrix}\begin{bmatrix} \Lambda _{11} & 0 \ 0 & \Lambda _{22}\end{bmatrix}\begin{bmatrix} \hat{w}_{1,t-1} \ \hat{w}_{2,t-1}\end{bmatrix}+$$ | |
$$\displaystyle +\begin{bmatrix} I & -\hat{\Phi}_{x} \ 0 & 0\end{bmatrix}\begin{bmatrix} \hat{\Psi}_{1}^{\ast \ast } \ \hat{\Psi}_{2}^{\ast \ast }\end{bmatrix}\hat{\varepsilon}_{t}^{x}+\underset{=0}{\underbrace{\begin{bmatrix} I & -\hat{\Phi}_{x} \ 0 & 0\end{bmatrix}\begin{bmatrix} \hat{\Pi}_{1}^{\ast \ast } \ \hat{\Pi}_{2,x}^{\ast \ast }\end{bmatrix}}}\eta _{t},$$ | (92) |
where the second block represents the constraint that guarantees the boundedness of the solution,
$$\hat{w}_{2,t}=0$$. Importantly, the augmented representation is determinate, and the last term of the system in (92) equals zero. Nevertheless, the non-fundamental disturbance,
$$\nu _{x,t}$$, affects the dynamics of the original model through vector of exogenous shocks,
$$\hat{\varepsilon}_{t}^{x}\equiv (\varepsilon _{t},\nu _{x,t})^{\prime }$$. Solving (91) for the endogenous variables,
$$\hat{S}_{t}\equiv (S_{t},\omega _{t})^{\prime }=(y_{t},x_{t},E_{t}(y_{t+1}),E_{t}(x_{t+1}),\omega _{t})^{\prime }$$, the system takes the form
$$\displaystyle \hat{S}_{t}$$ | $$\displaystyle =$$ | $$\displaystyle \hat{\Gamma}_{1}^{\ast \ast }\hat{S}_{t-1}+\hat{\Psi}_{S}^{\ast \ast }\hat{\varepsilon}_{t}^{x}$$ | |
$$\displaystyle =$$ | $$\displaystyle \begin{bmatrix} \frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \ 1 \ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \ \theta _{x} \ 0\end{bmatrix}E_{t-1}(x_{t})+\begin{bmatrix} 1 \ 0 \ 0 \ 0 \ 0\end{bmatrix}\varepsilon _{t}+\begin{bmatrix} \frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \ 1 \ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \ \theta _{x} \ 0\end{bmatrix}\nu _{x,t}.$$ | (93) |
Finally, to rewrite the reduced-form solution for the augmented representation that includes the non-fundamental shock,
$$\eta _{y,t}$$, in the auxiliary process, we recall equations (87) and (90) that we report below in equations (94) and (95)
$$\displaystyle \hat{w}_{2,t}$$ | $$\displaystyle =$$ | 0 $$\displaystyle \Longleftrightarrow$$ $$$\left\{ \begin{array}{l} E_{t}(y_{t+1})=-\frac{\theta _{x}}{\theta _{x}-\theta _{y}}E_{t}(x_{t+1}) \ \omega _{t}=0\end{array}\right.$$$ | (94) |
$$\displaystyle \nu _{x,t}$$ | $$\displaystyle =$$ | $$\displaystyle \frac{\theta _{x}-\theta _{y}}{\theta _{x}}\varepsilon _{t}-\frac{\theta _{x}-\theta _{y}}{\theta _{x}}\nu _{y,t}$$ | (95) |
Using the above equations, we can rewrite the system in (93) as
$$\displaystyle \begin{bmatrix}y_{t} \\ x_{t} \\ E_{t}(y_{t+1}) \\ E_{t}(x_{t+1})\end{bmatrix}=\begin{bmatrix}1 \\ \frac{\theta _{y}-\theta _{x}}{\theta _{x}} \\ \theta _{x} \\ \theta _{y}-\theta _{x}\end{bmatrix}E_{t-1}(y_{t})+\begin{bmatrix}0 \\ \frac{\theta _{x}-\theta _{y}}{\theta _{x}} \\ -\theta _{x} \\ \theta _{x}-\theta _{y}\end{bmatrix}\varepsilon _{t}+\begin{bmatrix}1 \\ -\frac{\theta _{x}-\theta _{y}}{\theta _{x}} \\ \theta _{x} \\ -(\theta _{x}-\theta _{y})\end{bmatrix}\nu _{y,t}.$$ | (96) |
In this section, we show the equivalence of the representations obtained using the two methodologies. In equation (97) below, we report the solution for the endogenous variables, $$S_{t}=(y_{t},x_{t},E_{t}(y_{t+1}),E_{t}(x_{t+1}))^{\prime }$$, using the methodology of Lubik and Schorfheide, 2003,
$$\displaystyle \begin{bmatrix} y_{t} \\\ x_{t} \\\ E_{t}(y_{t+1}) \\\ E_{t}(x_{t+1})\end{bmatrix}=\begin{bmatrix} \frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\\ 1 \\\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ \theta _{x}\end{bmatrix}E_{t-1}(x_{t})+\frac{\theta _{y}^{2}}{d^{2}}\begin{bmatrix} 1 \\\ \frac{\theta _{x}}{(\theta _{x}-\theta _{y})} \\\ -\frac{\theta _{x}^{3}}{(\theta _{x}-\theta _{y})^{2}} \\\ \frac{\theta _{x}^{2}}{(\theta _{x}-\theta _{y})}\end{bmatrix}\varepsilon _{t}+\frac{\theta _{y}}{d}\begin{bmatrix} \frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\\ 1 \\\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ \theta _{x}\end{bmatrix}\left( \overset{\backsim }{M}\varepsilon _{t}+\zeta _{t}\right) ,$$ | (97) |
where $$d=\sqrt{\theta _{y}^{2}+\left( \theta _{x}\theta _{y}\right) ^{2}/(\theta _{x}-\theta _{y})^{2}}$$. We now report in equation (98) below the solution using our methodology when we include the forecast error, $$\eta _{x,t}$$, in the auxiliary process20
$$\displaystyle \begin{bmatrix}y_{t} \\ x_{t} \\ E_{t}(y_{t+1}) \\ E_{t}(x_{t+1})\end{bmatrix}=\begin{bmatrix}\frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\ 1 \\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\ \theta _{x}\end{bmatrix}E_{t-1}(x_{t})+\begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix}\varepsilon _{t}+\begin{bmatrix}\frac{\theta _{x}}{(\theta _{y}-\theta _{x})} \\ 1 \\ \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\ \theta _{x}\end{bmatrix}\nu _{x,t}.$$ | (98) |
To show the equivalence between the two representations, we need to recall the restrictions that each methodology imposed on the forecast errors, $$\eta _{t}$$, as a function of the exogenous shock, $$\varepsilon _{t}$$, and the additional sunspot shock. Following Lubik and Schorfheide, 2003, we derived that
$$\displaystyle \eta _{t}=-V_{.1}D_{11}^{-1}U_{.1}^{\prime }\tilde{\Psi}_{2}\varepsilon _{t}+V_{.2}\left( \overset{\backsim }{M}\varepsilon _{t}+M_{\zeta }\zeta _{t}\right) ,$$ |
where we know that
$$V^{\prime }=\begin{bmatrix} V_{.1}^{\prime } \ V_{.2}^{\prime }\end{bmatrix}=\begin{bmatrix} \frac{a}{d} & \frac{b}{d} \ \frac{b}{d} & -\frac{a}{d}\end{bmatrix}$$,
$$D_{11}=d=\sqrt{a^{2}+b^{2}}$$, $$U_{1}=1$$,
$$\tilde{\Psi}_{2}=-a=\theta _{y}$$ and
$$b=-\theta _{x}\theta _{y}/(\theta _{x}-\theta _{y})$$. Therefore, normalizing
$$M_{\zeta }=1$$, we obtain
$$\displaystyle \eta _{t}$$ | $$\displaystyle =$$ | $$\displaystyle \begin{bmatrix} \frac{a}{d} \ \frac{b}{d}\end{bmatrix}\frac{a}{d}\varepsilon _{t}+\begin{bmatrix} \frac{b}{d} \ -\frac{a}{d}\end{bmatrix}\left( \overset{\backsim }{M}\varepsilon _{t}+\zeta _{t}\right)$$ | |
$$\displaystyle =$$ | $$\displaystyle \left\{ \frac{\theta _{y}^{2}}{d^{2}}\begin{bmatrix} 1 \ \frac{\theta _{x}}{(\theta _{x}-\theta _{y})}\end{bmatrix}+\frac{\theta _{y}}{d}\begin{bmatrix} -\frac{\theta _{x}}{(\theta _{x}-\theta _{y})} \ 1\end{bmatrix}\overset{\backsim }{M}\right\} \varepsilon _{t}+\frac{\theta _{y}}{d}\begin{bmatrix} -\frac{\theta _{x}}{(\theta _{x}-\theta _{y})} \ 1\end{bmatrix}\zeta _{t}.$$ | (99) |
Similarly, from the derivation using our methodology, we know that
$$\displaystyle \eta _{t}=-\left( \hat{\Pi}_{2,x}^{\ast \ast }\right) ^{-1}\hat{\Psi}_{2}^{\ast \ast }\hat{\varepsilon}_{t}^{x}$$ $$\displaystyle \Longleftrightarrow$$ $$\displaystyle \eta _{t}=\begin{bmatrix}1 & -\frac{\theta _{x}}{\theta _{x}-\theta _{y}} \\ 0 & 1\end{bmatrix}\begin{bmatrix}\varepsilon _{t} \\ \nu _{x,t}\end{bmatrix}$$ | (100) |
Comparing equations (99) and (100), we also point out that the sunspot shock introduced in our representation, $$\nu _{x,t}$$, has a clear interpretation: It is always equivalent to the forecast error that is included in the auxiliary process. On the contrary, the sunspot shock, $$\zeta _{t}$$, in Lubik and Schorfheide, 2003 has a more complex interpretation and the authors provide a formal argument to consider it as a trigger of belief shocks that lead to a revision of the forecasts.
We then combine equations (99) and (100) to establish the following relationship
$$\displaystyle \nu _{x,t}=\left[ \frac{\theta _{y}^{2}}{d^{2}}\frac{\theta _{x}}{(\theta _{x}-\theta _{y})}+\frac{\theta _{y}}{d}\overset{\backsim }{M}\right] \varepsilon _{t}+\frac{\theta _{y}}{d}\zeta _{t}.$$ | (101) |
Plugging this relationship in the solution in equation (98) obtained using our methodology, we derive the solution in (97) derived using the methodology of Lubik and Schorfheide, 2004. This result shows that anyparametrization in Lubik and Schorfheide, 2004 has a unique mapping to our representation. In particular, we now consider the parametrization $$\overset{\backsim }{M}=M^{\ast }(\theta )+M$$, where $$M$$ is centered at 0 and $$M^{\ast }(\theta )$$ is found by minimizing the distance between the impulse response functions under determinacy and indeterminacy at the boundary of the determinacy region. We can therefore write equation (101) as
$$\displaystyle \nu _{x,t}=\gamma _{\varepsilon }(M^{\ast }(\theta ))\varepsilon _{t}+\gamma _{\zeta }\zeta _{t},$$ | (102) |
$$\displaystyle \sigma _{\nu _{x}}^{2}(M^{\ast }(\theta ))=\gamma _{\varepsilon }^{2}(M^{\ast }(\theta ))\sigma _{\varepsilon }^{2}+\gamma _{\zeta }^{2}\sigma _{\zeta }^{2}$$ | (103) |
$$\displaystyle \sigma _{\varepsilon ,\nu _{x}}(M^{\ast }(\theta ))=\gamma _{\varepsilon }(M^{\ast }(\theta ))\sigma _{\varepsilon }^{2}$$ | (104) |
The variance-covariance matrix of the shocks $$\hat{\varepsilon}_{t}^{x}=\{\varepsilon _{t},\nu _{x,t}\}^{\prime }$$ can be written as
$$\displaystyle \Omega _{\hat{\varepsilon}^{x}}(M^{\ast }(\theta ))\equiv \begin{bmatrix}\sigma _{\varepsilon }^{2} & \sigma _{\varepsilon ,\nu _{x}}(M^{\ast }(\theta )) \\ \sigma _{\varepsilon ,\nu _{x}}(M^{\ast }(\theta )) & \sigma _{\nu _{x}}^{2}(M^{\ast }(\theta ))\end{bmatrix}.$$ | (105) |
$$\displaystyle \hat{\varepsilon}_{t}^{x}=\begin{bmatrix}\varepsilon _{t} \\ \nu _{x,t}\end{bmatrix}=L(M^{\ast }(\theta ))u_{t}\equiv \begin{bmatrix}\sigma _{\varepsilon } & 0 \\ \frac{\sigma _{\varepsilon ,\nu _{x}}(M^{\ast }(\theta ))}{\sigma _{\varepsilon }} & \sqrt{\sigma _{\nu _{x}}^{2}(M^{\ast }(\theta ))-\left( \frac{\sigma _{\varepsilon ,\nu _{x}}(M^{\ast }(\theta ))}{\sigma _{\varepsilon }}\right) ^{2}}\end{bmatrix}\begin{bmatrix}u_{1,t} \\ u_{2,t}\end{bmatrix},$$ | (106) |
$$\displaystyle \begin{bmatrix} y_{t} \\\ x_{t} \\\ E_{t}(y_{t+1}) \\\ E_{t}(x_{t+1})\end{bmatrix}=\begin{bmatrix} \frac{\theta_{x}}{(\theta _{y}-\theta _{x})} \\\ 1 \\\ \frac{\theta_{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ \theta _{x}\end{bmatrix}E_{t-1}(x_{t})+\begin{bmatrix} 1 & \frac{\theta_{x}}{(\theta _{y}-\theta_{x})} \\\ 0 & 1 \\\ 0 & \frac{\theta _{x}^{2}}{(\theta _{y}-\theta _{x})} \\\ 0 & \theta _{x}\end{bmatrix}\begin{bmatrix} \sigma _{\varepsilon } & 0 \\\ \frac{\sigma_{\varepsilon ,\nu_{x}}(M^{\ast }(\theta ))}{\sigma_{\varepsilon }} & \sqrt{\sigma_{\nu_{x}}^{2}(M^{\ast }(\theta ))-\left( \frac{\sigma_{\varepsilon ,\nu_{x}}(M^{\ast }(\theta ))}{\sigma_{\varepsilon }}\right) ^{2}}\end{bmatrix}\begin{bmatrix} u_{1,t} \\\ u_{2,t}\end{bmatrix}.$$ | (107) |
Table 6: Posterior distribution for model parameters (2-degrees of indeterminacy)
Posterior distributions of the estimated model with two degrees of indeterminacy.
$$\left\{ \nu _{1}{\normalsize =}\nu _{\pi }{\normalsize ,}\nu _{2}=\nu _{y}\right\} $$ Mean | $$\left\{ \nu _{1}{\normalsize =}\nu _{\pi }{\normalsize ,}\nu _{2}=\nu _{y}\right\} $$ 90% prob. int. | $$\left\{\nu _{1}{\normalsize =}\nu _{\pi }{\normalsize ,}\nu _{2}=\nu _{b}\right\} $$ Mean | $$\left\{\nu _{1}{\normalsize =}\nu _{\pi }{\normalsize ,}\nu _{2}=\nu _{b}\right\} $$ 90% prob. int. | $$\left\{ \nu _{1}{\normalsize =}\nu _{y}{\normalsize ,}\nu _{2}=\nu _{b}\right\} $$ Mean | $$\left\{ \nu _{1}{\normalsize =}\nu _{y}{\normalsize ,}\nu _{2}=\nu _{b}\right\} $$ 90% prob. int. | |
---|---|---|---|---|---|---|
$${ g}$$ | 0.49 | [ 0.45,0.54] | 0.47 | [0.42,0.52] | 0.46 | [0.42,0.51] |
$${ \lambda }$$ | 0.96 | [0.93,0.99] | 0.80 | [0.65,0.94] | 0.80 | [0.67,0.94] |
$${ \alpha }$$ | 0.66 | [0.60,0.72] | 0.60 | [0.52,0.69] | 0.60 | [0.52,0.68] |
$${ 100(\gamma }^{-1}{ -1)}$$ | 0.50 | [ 0.34,0.65] | 0.46 | [0.31,0.61] | 0.45 | [0.31,0.59] |
$${ \kappa }$$ | 0.040 | [0.032,0.048] | 0.042 | [0.034,0.051] | 0.039 | [0.032,0.047] |
$$\pi ^{\ast }$$ | 0.68 | [0.36,1.00] | 0.70 | [0.38,1.01] | 0.72 | [0.37,1.01] |
$${ i}^{\ast }$$ | 1.43 | [1.09,1.75] | 1.41 | [1.09,1.74] | 1.42 | [1.08,1.74] |
$$\phi _{\pi }$$ | 0.31 | [0.14,0.48] | 0.33 | [0.17,0.50] | 0.30 | [0.14,0.46] |
$$\phi _{q}$$ | 0.08 | [0.04,0.12] | 0.10 | [0.05,0.15] | 0.16 | [0.08,0.23] |
$${ \rho }_{i}$$ | 0.75 | [0.61,0.88] | 0.67 | [0.51,0.82] | 0.62 | [0.45,0.79] |
$${ \sigma }_{q}$$ | 0.29 | [0.15,0.44] | 0.26 | [0.13,0.38] | 0.51 | [0.19,0.85] |
$${ \sigma }_{s}$$ | 0.11 | [0.09,0.13] | 0.11 | [0.09,0.12] | 0.11 | [0.09,0.12] |
$${ \sigma }_{i}$$ | 0.10 | [0.08,0.12] | 0.10 | [0.09,0.12] | 0.11 | [0.09,0.13] |
$${ \rho }_{q}$$ | 0.94 | [0.91,0.97] | 0.68 | [0.51,0.84] | 0.68 | [0.54,0.81] |
$${ \rho }_{s}$$ | 0.89 | [0.83,0.94] | 0.90 | [0.85,0.94] | 0.87 | [0.81,0.92] |
$${ \sigma }_{\nu _{1}}$$ | 0.28 | [0.24,0.32] | 0.27 | [0.23,0.31] | 0.70 | [0.61,0.78] |
$${ \sigma }_{\nu _{2}}$$ | 0.69 | [0.60,0.78] | 2.59 | [1.12,4.14] | 1.74 | [0.73,2.76] |
$${ \varphi }_{\nu _{1},i}$$ | - 0.42 | [-0.67,-0.16] | -0.27 | [-0.55,0.01] | 0.08 | [-0.15,0.34] |
$${ \varphi }_{\nu _{1},q}$$ | 0.07 | [-0.43,0.59] | 0.11 | [-0.45,0.67] | 0.60 | [0.40,0.81] |
$${ \varphi }_{\nu _{1},s}$$ | 0.61 | [0.48,0.73] | 0.61 | [0.48,0.73] | -0.58 | [-0.71,-0.44] |
$${ \varphi }_{\nu _{2},i}$$ | - 0.14 | [-0.40,0.13] | -0.53 | [-0.75,-0.31] | -0.72 | [-0.93,-0.51] |
$${ \varphi }_{\nu _{2},q}$$ | - 0.01 | [-0.52,0.55] | -0.11 | [-0.51,0.27] | -0.30 | [-0.63,0.03] |
$${ \varphi }_{\nu _{2},s}$$ | - 0.68 | [-0.77,-0.59] | -0.67 | [-0.80,-0.55] | -0.39 | [-0.63,-0.18] |
MDD | - 72.3 | - 73.0 | - 75.1 |
Table 7: Posterior distribution for model parameters (1-degree of indeterminacy)
Posterior distributions of the estimated model with one degree of indeterminacy.
$$\left\{ \nu _{1}{\normalsize =}\nu _{b}\right\} $$ Mean | $$\left\{ \nu _{1}{\normalsize =}\nu _{b}\right\} $$ 90% prob. int. | $$\left\{ \nu _{2}=\nu _{\pi }\right\} $$ Mean | $$\left\{ \nu _{2}=\nu _{\pi }\right\} $$ 90% prob. int. | $$\left\{ \nu _{1}{\normalsize =}\nu _{y}\right\} $$ Mean | $$\left\{ \nu _{1}{\normalsize =}\nu _{y}\right\} $$ 90% prob. int. | |
---|---|---|---|---|---|---|
$${ g}$$ | 0.49 | [ 0.45,0.54] | 0.50 | [0.45,0.54] | 0.50 | [0.45,0.54] |
$${ \lambda }$$ | 0.96 | [0.93,0.99] | 0.97 | [0.94,0.99] | 0.97 | [0.94,0.99] |
$${ \alpha }$$ | 0.66 | [0.60,0.72] | 0.66 | [0.60,0.72] | 0.67 | [0.61,0.73] |
$${ 100(\gamma }^{-1}{ -1)}$$ | 0.50 | [ 0.34,0.65] | 0.47 | [0.34,0.60] | 0.50 | [0.34,0.66] |
$${ \kappa }$$ | 0.040 | [0.032,0.048] | 0.043 | [0.033,0.049] | 0.040 | [0.032,0.048] |
$$\pi ^{\ast }$$ | 0.68 | [0.36,1.00] | 0.71 | [0.39,1.04] | 0.70 | [0.38,1.00] |
$${ i}^{\ast }$$ | 1.43 | [1.09,1.75] | 1.44 | [1.13,1.77] | 1.43 | [1.11,1.74] |
$$\phi _{\pi }$$ | 0.31 | [0.14,0.48] | 0.33 | [0.14,0.50] | 0.30 | [0.12,0.45] |
$$\phi _{q}$$ | 0.08 | [0.04,0.12] | 0.09 | [0.04,0.14] | 0.10 | [0.04,0.16] |
$${ \rho }_{R}$$ | 0.75 | [0.61,0.88] | 0.74 | [0.61,0.88] | 0.79 | [0.66,0.92] |
$$\sigma _{q}$$ | 0.29 | [0.15,0.44] | 0.30 | [0.14,0.46] | 0.32 | [0.14,0.48] |
$${ \sigma }_{s}$$ | 0.11 | [0.09,0.13] | 0.11 | [0.09,0.14] | 0.11 | [0.09,0.13] |
$${ \sigma }_{i}$$ | 0.10 | [0.08,0.12] | 0.10 | [0.09,0.12] | 0.10 | [0.08,0.12] |
$${ \rho }_{q}$$ | 0.94 | [0.91,0.97] | 0.93 | [0.90,0.97] | 0.92 | [0.88,0.96] |
$${ \rho }_{s}$$ | 0.89 | [0.83,0.94] | 0.89 | [0.84,0.94] | 0.89 | [0.84,0.94] |
$${ \sigma }_{\nu _{1}}$$ | 6.11 | [3.44,9.38] | 0.28 | [0.23,0.31] | 0.74 | [0.63,0.85] |
$${ \varphi }_{\nu _{1},i}$$ | -0.41 | [-0.65,-0.19] | -0.47 | [-0.81,-0.12] | -0.46 | [-0.78,-0.10] |
$${ \varphi }_{\nu _{1},q}$$ | - 0.70 | [-0.84,-0.56] | - 0.52 | [-0.70,-0.33] | 0.01 | [-0.27,0.29] |
$${ \varphi }_{\nu _{1},s}$$ | - 0.46 | [-0.57,-0.35] | 0.46 | [0.26,0.64] | -0.64 | [-0.76,-0.53] |
MDD | -83.2 | -84.2 | -83.0 |
Table 8: Posterior distribution for model parameters (determinacy)
Mean | 90% prob. int. | |
---|---|---|
$${ g}$$ | 0.40 | [0.35,0.44] |
$${ \lambda }$$ | 0.98 | [0.97,0.99] |
$${ \alpha }$$ | 0.69 | [0.66,0.71] |
$${ 100(\gamma }^{-1}{ -1)}$$ | 0.46 | [0.41,0.51] |
$${ \kappa }$$ | 0.050 | [0.042,0.058] |
$$\pi ^{\ast }$$ | 0.60 | [0.34,0.85] |
$${ i}^{\ast }$$ | 1.38 | [1.12,1.65] |
$$\phi _{\pi }$$ | 1.58 | [1.46,1.74] |
$$\phi _{q}$$ | 0.03 | [0.01,0.04] |
$${ \rho }_{i}$$ | 0.78 | [0.74,0.82] |
$$\sigma _{q}$$ | 0.23 | [0.13,0.33] |
$${ \sigma }_{s}$$ | 0.08 | [0.07,0.09] |
$${ \sigma }_{i}$$ | 0.20 | [0.17,0.23] |
$${ \rho }_{q}$$ | 0.95 | [0.94,0.97] |
$${ \rho }_{s}$$ | 0.94 | [0.90,0.97] |
MDD | -158.3 |