In bfunovits/indeterminateR: Evaluates the Existence and Uniqueness Properties of LRE Models

knitr::opts_chunk$set(echo = TRUE)
library(indeterminateR)
library(tidyverse)
library(printr)

This vignette analyses a verison of the New Keynesian monetary model. The first section introduces the economic in more detail. The second section verifies that for the parameter values indicated in the associated paper, the uniqueness condition in Sims (2001) is satisfied even though the existence condition has multiple solutions. The internal indeterminate characteristics are thus not identifiable from the external characteristics. The last section explains the structure of the package.

Description of the New Keynesian Monetary Model

We consider the model

$$ \begin{array}{rl} \mathbb{E}{t} \left( y{t+1} \right) + \sigma \pi_{t} & = y_{t} + \sigma R_{t} \ \beta \mathbb{E}{t} \left( \pi{t+1} \right) & = \pi_{t} - \kappa y_{t}\ R_{t} & = \phi_{R} R_{t-1} + \phi_{\pi} \pi_{t} + \phi_{y} y_{t} + \varepsilon_{t}^{R} \end{array} $$

where $y_t$, $\pi_t$, and $R_t$ are log-deviations from the steady state of output, inflation and the interest rate. Furthermore, $\sigma$ denotes the intertemporal substitution elasticity, $\beta$ the time preference rate, $\kappa^{-1}$ measures the elasticity of aggregate supply with respect to inflation, $\phi_R$ denotes the interest rate smoothing coefficient, $\phi_{\pi}$ denotes the elasticity of the interest rate response with respect to inflation, and $\phi_{y}$ denotes the elasticity of the interest rate response with respect to output. Finally, $\varepsilon_t^{R}$ denotes an unanticipated policy implementation shock.

The second equation is identical to equation (2) in Lubik and Schorfheide (2003). It describes the inflation dynamics due to firms' optimal price-setting. The first equation corresponds to an intertemporal Euler equation. Compared with equation (1) in Lubik and Schorfheide (2003), i.e.

$$ \mathbb{E}{t} \left( y{t+1} \right) + \sigma \mathbb{E}{t} \left( \pi{t} \right) = y_{t} + \sigma R_{t}, $$ we replace $\mathbb{E}{t} \left( \pi{t} \right)$ with $\pi_t$. The third equation corresponds to equation (26) for $i = 0$ in Lubik and Marzo (2007) and includes an additional shock $\varepsilon_t^{R}$.

Solving the model by replacing the conditional expectations

Replacing the conditional expectations with new variables $\mathbb{E}{t} \left( y{t+1} \right) = \xi^{y}t$ and $\mathbb{E}{t} \left( \pi_{t+1} \right) = \xi^{\pi}_t$, adding equations

$$ \begin{array}{rl} y_{t} & = \xi^{y}{t-1} + \eta^{y}_t\ \pi{t} & = \xi^{\pi}_{t-1} + \eta^{\pi}_t \end{array} $$

and replacing $y_t$ and $\pi_t$ in the three equation New Keynesian monetary model, leads to the following system in the canoncial form of Sims (2001)

$$ \begin{pmatrix} 1 & 0 & -\sigma\ 0 & \beta & 0\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \xi_{t}^{y}\ \xi_{t}^{\pi}\ R_{t} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0\ -\kappa & 1 & 0\ 0 & 0 & \phi_{R} \end{pmatrix} \begin{pmatrix} \xi_{t-1}^{y}\ \xi_{t-1}^{\pi}\ R_{t-1} \end{pmatrix} + \begin{pmatrix} 0\ 0\ 1 \end{pmatrix} \varepsilon_{t}^{R} + \begin{pmatrix} 1 & -\sigma\ -\kappa & 1\ \phi_{y} & \phi_{\pi} \end{pmatrix} \begin{pmatrix} \eta_{t}^{y}\ \eta_{t}^{\pi} \end{pmatrix} $$ where the matrices correspond to $$ \left( \Gamma_0, \Gamma_1, \Psi, \Pi \right) $$ in Sims' notation.

Verification of the Statements in the Paper

Here, we verify that, as stated in the associated paper, for the parameters $\kappa = \frac{1}{\sigma}$ and $\phi_{y} = -\phi_{\pi} \kappa$ and, e.g., $\beta = 0.95$ and $\phi_R = 0.6$, the existence condition has a non-trivial kernel but the uniqueness condition is satisfied. According to Lubik and Schorfheide (2003), this would correspond to an (internally) indeterminate equilibrium.

The functions prints evaluate_my_model prints

• the linear parameters $$ \left( \Gamma_0, \Gamma_1, \Psi, \Pi \right) $$ in Sims' canonical form,
• the orthogonal matrix $Q$ whose rows describe the stable and unstable subspaces,
• the generalized eigenvalues in the QZ decomposition, and
• the matrices $$ Q_{U \bullet} \Psi, Q_{S \bullet} \Pi, Q_{U \bullet} \Pi $$ used in the existence and uniqueness condition.

results <- evaluate_my_model(deep_params = 
                               list(s = 1.5, 
                                    b = 0.99, 
                                    k = 1/1.5, 
                                    f_y = -1.3/1.5, 
                                    f_p = 1.3, 
                                    f_r = 0.6),
                             my_model = "Lubik_Marzo_26_0", 
                             verbose = TRUE)

Finally, the results for existence and uniqueness are given. The boolean values for the existence and uniqueness condition indicate that both are satisfied. Even though the dimension of the kernel in the existence condition is one, the dimension of indeterminacy is zero.

as.list(results)

The boolean values for existence and uniqueness indicate that a solution exist and that it is unique. The dimension of the kernel of the existence condition is one. Therefore, Lubik and Schorfheide would characterize this system as indeterminate. The dimension of indeterminacy, however, is zero. The last entry corresponds to the smallest singular value of the intersection of the spaces spanned by $\tilde{V}{1 \bullet}$ and $V{2 \bullet}$, i.e. the smallest singular value of $$ \tilde{V}'{\bullet1}V{\bullet2} $$ as described in the paper.

The Structure of the Package

The main function is evaluate_my_model(). Its inputs are

• deep structural parameters, given as a list, and
• a model, given as a string.

First, it calls the function linear_parameters_LM_26_0() which uses the deep structural parameters to generate the matrices in Sims' canonical form. Subsequently, some parts of Sims' gensys programme are called to obtain the ordered QZ decomposition. Last, the the existence and uniqueness condition are evaluated.

All functions are well documented and can be found with the command help(package = "indeterminateR"). In particular, a description of the parameter of the model described above can be found in the help file of linear_parameters_LM_26_0(). Moreover, the outputs of evaluate_my_model() are functions of the matrices in Sims' canonical form and are described in the help file of evaluate_my_model.

Help files

?linear_parameters_LM_26_0

?evaluate_my_model

bfunovits/indeterminateR documentation built on May 28, 2019, 7:10 p.m.