Skip to content

[IrisToolbox] for Macroeconomic Modeling

Simulating nonlinear models

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

  • Conditional expectations of shocks: \(\E{t-1} \left[x_t\right] = \E{t-2} \left[x_t\right] = \cdots = 0\)

  • Conditional higher moments: \(\E{t-1}\left[ \epsilon_t\, \epsilon_t{}' \right] = \E{t-2}\left[ \epsilon_t\, \epsilon_t{}' \right] = \cdots = \Omega, \dots\)

Methods for nonlinear simulations

Characteristics Local approximation Global approximation Stacked time
Solution form Function Function Sequence
Explicit terminal ✖︎ ✖︎ ✔︎
Global nonlinearities ✖︎ ✔︎ ✔︎
Stochastic nonlinearities ✔︎ ✔︎ ✖︎
Automated design ✔︎ ✖︎ ✔︎
Large scale models ✔︎ ✖︎ ✔︎
Computational load Increasing Large Manageable

Local approximation methods

Non-stochastic steady state
  • A "fixed point" calculated under the following "non-stochastic" assumptions
\[ \begin{gathered} \epsilon_t = 0 \\[5pt] \mathrm E_{t-k}\! \left[ \epsilon_t \right] = 0 \\[5pt] \mathrm E_{t-k}\! \left[ \epsilon_t\, \epsilon_t{}' \right] = 0 \\[5pt] k = 1, \dots, \infty \end{gathered} \]
  • Stationary steady state: characterized by a single number, \(\bar x\)
\[ \bar x_t = \bar x \]
  • Steady growth path with a constant difference: characterized by two numbers, \(\bar x_t\) and \(\Delta \bar x\), at a particular yet arbitrary snapshot along the path
\[ \bar x_t = \bar x_{t-1} + \Delta \bar x \]
  • Steady growth path with a constant rate of change: characterized by \(\bar x_t\) and \(\delta \bar x\), after logarithm, conceptually the same as the constant difference case
\[ \bar x_t = \bar x_{t-1} \cdot \delta \bar x \]
\[ \log \bar x_t = \log \bar x_{t-1} + \log \delta \bar x \]
  • A noteworth special case: unit root process with zero difference/rate of change – flat in steady state but not stationary (not pinned down to a fixed number)
\[ x_t = x_{t-1} \]

Local approximation methods

Deviations from non-stochastic steady state

Vector of deviations from steady path

\[ \hat x_t = \left[ \hat x_t^1, \dots, \hat x_t^n \right]' \]
\[ \hat x^i_t = x_t^i - \bar x^i_t \qquad \text{or} \qquad \hat x_t^i = \log x_t^i - \log \bar x_t^i \]

Find a function approximated around the nonstochastic steady state by terms up to a desired order, with coefficient matrices (solution matrices) \(A_0\), \(A_1\), \(A_2\), \(\dots\), \(B\)

\[ \hat x_t = A_0 + A_1\ \hat x_{t-1} + \hat x_{t-1}' \ A_2\ \hat x_{t-1} + \ \cdots\ + B_1 \ \epsilon_t + \epsilon_t' \ B_2\ \epsilon_t + \ \cdots \]

that are consistent with the original system of equations up to a desired order



The coefficient matrices \(A_0, A_1, \ A_2,\ A_3, \ \dots, B_1, B_2,\ \dots\) dependent on

  • the 1st, 2nd, ..., \(k\)-th order Taylor expansions of the original functions \(f_1,\ \dots,\ f_k\)
  • model parameters \(\theta\)

The higher-order coefficient matrices \(A_2, A_3, \dots, B_2, B_3 \ \dots\) also dependent on

  • the higher moments of shocks \(\Omega, \dots\)

Sequential calculation of local approximate solutions

  1. Calculate non-stochastic steady state

  2. Use generalized Schur decomposition to determine the first-order solution matrices

  3. Based on steps 1 and 2, calculate second-order solution matrices

  4. Based on steps 1, 2, and 3, calculate third-order solution matrices

Global approximation

Find a parametric policy ("solution") function \(g\)

\[ x_t = g\left(x_{t-1}, \epsilon_t \ \middle|\ \theta, \Omega, \dots \right) \]

consistent with the original system taking into account the expectations operator

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

The function \(g\) is a parameterized global approximation of the true function, e.g. parameterized sum of polynominals, function over a discrete grid of points, etc.

Policy function method versus parametrized expectations method

Stacked time

Find a sequence of numbers, \(x_1, \dots, x_T\) that comply with the original system of equations stacked \(T\) times underneath each other dropping the expectations operator

\[ \begin{gathered} f_1\left( x_{-1}, x_1, x_{2}, \epsilon_1 \ \middle| \ \theta \right) = 0 \\[10pt] \vdots \\[10pt] f_k\left( x_{t-1}, x_t, x_{2}, \epsilon_1 \ \middle| \ \theta \right) = 0 \\[10pt] \vdots \\[10pt] \vdots \\[10pt] f_1\left( x_{T-1}, x_T, x_{T+1}, \epsilon_T \ \middle| \ \theta \right) = 0 \\[10pt] \vdots \\[10pt] f_k\left( x_{T-1}, x_T, x_{T+1}, \epsilon_T \ \middle| \ \theta \right) = 0 \end{gathered} \]

Initial condition \(x_{-1}\) given

Terminal condition \(x_{T+1}\) needs to be determined

Combining anticipated and unanticipated shocks in stacked time

By design, all shocks included within one particular simulation run are known/seen/anticipated throughout the simulation range

Simulating a combination of anticipated and unanticipated shocks means

  • split the simulation range into sub-ranges by the occurrence of unanticipated shocks
  • run each sub-range as a separate simulation, taking the end-points of the previous sub-range simulation as initial condition
  • make sure you run a sufficient number of periods in each sub-simulation