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[IrisToolbox] for Macroeconomic Modeling

Steady-state and first-order solution

jaromir.benes@iris-toolbox.com

System of nonlinear equations with model-consistent expectations

\[ \newcommand{\Et}{\mathrm{E}_t} \newcommand{\E}[1]{\mathrm{E}_{#1}\!} \]

System of \(n\) nonlinear conditional-expectations equations

\[ \begin{gathered} \E{t}\, \Bigl[ f_1\bigl( x_{t-1}, x_t, x_{t+1}, \epsilon_t \bigm| \theta \bigr) \Bigr] = 0 \\[10pt] \vdots \\[10pt] \E{t} \, \Bigl[ f_n\bigl( x_{t-1}, x_t, x_{t+1}, \epsilon_t \bigm| \theta \bigr) \Bigr] = 0 \\[40pt] \end{gathered} \]

  • Vector of \(n\) variables: \(x_t = \left[ x_t^1, \, \dots, x_t^n \right]'\)

  • Vector of \(s\) shocks: \(\epsilon_t = \left[ \epsilon_t^1, \, \dots, \epsilon_t^s \right]'\)

  • Vector of \(p\) parameters: \(\theta_t = \left[ \theta_t^1, \, \dots, \theta_t^p \right]'\)

Sequence of steps

  1. Find a nonstochastic steady state (growth path)

  2. Calculate first-order Taylor expansion around steady state

  3. Calculate state-space representation

Use generalized Schur decomposition to decouple predetermined and unpredetermined variables, create a recursive representation, and translate the expectations of endogenous variables into the expectations of exogenous shocks

Nonstochastic steady state: Case 1

  • Stationary model or model with unit roots but no deterministic growth component

  • This case is assumed by default in IrisT

  • Make the equations nonstochastic

\[ \mathrm E_t \left[ f_k \left( x_{t-1}, x_t, x_{t+1}, \epsilon_t \bigm| \theta \right) \right] =0 \]
\[ \downarrow \]
\[ f_k \left( x_{t-1}, x_t, x_{t+1}, 0 \bigm| \theta \right) =0 \]

  • Simple fixed point problem

  • Replace \(x_{t-k}\), \(x_t\), \(x_{t+k}\) with \(\bar x\) and solve the system of equations for \(\bar x\)

  • Can still be an underdetermined system (singularity in its Jacobian); need a solution algorithm able to deal with the singularity, e.g. Marquardt-Levenberg or Newton-Cauchy

Nonstochastic steady state: Case 2

  • Nonstationary models with deterministic growth component

  • No manual "stationarization" needed

  • Set growth=true to indicate deterministic growth component when creating the Model object

  • "Growth-augmented" steady point: find \(\bar x\) and \(\bar g\)

\[ \mathrm E_t \left[ f_k \left( x_{t-1}, x_t, x_{t+1}, \epsilon_t \bigm| \theta \right) \right] =0 \]
\[ \downarrow \]
\[ f_k \left( x_{t-1}, x_t, x_{t+1}, 0 \bigm| \theta \right) =0 \]

  • Replace \(x_{i,t+k} \rightarrow \bar x_i \bar g_i^{k}\) for each \(x_{i,t}\) that is a log variable, where \(\bar g_i\) is the steady-state gross rate of change for that variable

  • Replace \(x_{i,t+k} \rightarrow \bar x_i + k\, \bar g_i\) for each \(x_{i,t}\) that is a "plain" variable, where \(\bar g_i\) is the steady-state first difference for that variable

  • We have \(n\) equations but \(2n\) unknows (levels and changes): Copy the system of equations once more for another point in time

Log-status rules

  1. Variables that are growing at a constant first difference in steady state must not be declared as log-variables

  2. Variables that are growing at a constant rate of change in steady state must be declared as log-variables

  3. Variables that can be zero or negative must not be declared as log-variables

  4. Otherwise, it does not matter

  5. Variables that fall under #1 and #2 at the same time cannot be included in the model as such - use transformations, e.g. ratios over GDP etc.

Taylor expansion in first-difference form

  • Create a transition vector \(\xi_t\) from \(x_t\)

  • Transition vector \(\xi_t\) contains (i) logarithms for log-variables, (ii) auxiliary lags and leads to have a first-difference form

  • Split \(\xi_t\) into predetermined (backward looking) and non-predetermined (forward looking) parts


\[ A \, \mathrm E_t \begin{bmatrix} \xi_{b,t} - \bar\xi_{b,t} \\ \xi_{f,t+1}-\bar \xi_{f,t+1} \end{bmatrix} + B \, \begin{bmatrix} \xi_{b,t-1} - \bar \xi_{b,t-1} \\ \xi_{f,t} - \bar \xi_{f,t} \end{bmatrix} + C + D \,\epsilon_t = 0, \]

First-order triangular solution


Transition equations

\[ \begin{aligned} \begin{bmatrix} \xi_{f,t} \\ \alpha_t \end{bmatrix} =&\ \begin{bmatrix} 0 & T_f \\ 0 & T_\alpha \end{bmatrix} \begin{bmatrix} \xi_{f,t-1} \\ \alpha_{t-1} \end{bmatrix} \ \cdots \\[15pt] +&\ k + R_0\, \epsilon_t + R_1 \, \mathrm E_t \left[\epsilon_{t+1}\right] + \cdots + R_k \, \mathrm E_t \left[\epsilon_{t+k}\right] + \cdots \end{aligned} \]


Transformed vector of predetermined (backward looking) variables

\[ \xi_{b,t} = U \, \alpha_t \]


Measurement equations

\[ y_t = Z \, \alpha_t + d + H \epsilon_t \]

Triangular transition matrix

  • Upper (block) triangular (1-by-1 or 2-by-2 blocks on the main diagonal)

  • Only stable roots (eigenvalues) or unit roots; the unit roots concentrated in the top-left corner

![[transition-matrix.png]]

First-order "rectangular" solution


Equivalent to the triangular form, no \(\alpha\) vector involved

\[ \begin{aligned} \begin{bmatrix} \xi_{f,t} \\ \xi_{b,t} \end{bmatrix} =&\ \begin{bmatrix} 0 & \widetilde T_f \\ 0 & \widetilde T_{b} \end{bmatrix} \begin{bmatrix} \xi_{f,t-1} \\ \xi_{b,t-1} \end{bmatrix} \ \cdots \\[15pt] +&\ \widetilde k + \widetilde R_0\, \epsilon_t + \widetilde R_1 \, \mathrm E_t \left[\epsilon_{t+1}\right] + \cdots + \widetilde R_k \, \mathrm E_t \left[\epsilon_{t+k}\right] + \cdots \end{aligned} \]

Measurement equations

\[ y_t = \widetilde Z \, \xi_{b,t} + \widetilde d + \widetilde H \epsilon_t \]