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[IrisToolbox] for Macroeconomic Modeling

Nonlinear equations solver

jaromir.benes@iris-toolbox.com

Use of the solver

  • Calculate steady state

  • Run nonlinear stacked-time simulations

  • Run nonlinear period-by-period simulations

IrisT implementation

  • Augmented Quasi-Newton-steepest-descent (QNSD) algorithms

  • Analytical (symbolic) or numerical Jacobian

  • Sequential block pre-analysis

  • Detection of sparse Jacobian pattern

  • Detection of simulation-invariant Jacobian elements

Steepest descent (Cauchy) step

  • Flavor of global optimization

  • Robustness

  • Underdetermined systems, aka singularity in Jacobian: critical for growth models

Curse of dimensionality in stacked-time simulations

Dimension of the problem: \(K =\) # equations \(\times\) # periods

Dimensions of the Jacobian: \(K \times K\)

The conventional ways of handling terminal condition require a larger number of periods to be simulated (to discount the effect of the "wrong" terminal condition)

Reduce the actual dimensionality and accelerate:

  • Base terminal condition upon the first order solution dramatically reduces the number of periods needed

  • Detect the sparse Jacobian pattern to avoid the evaluation of zero points

  • Detect the Jacobian elements that need to be evaluated only once (at the beginning)

QNSD algorithm

Quasi-Newton-steepest-descent where the Jacobian is regularized using the steepest descent method for underdetermined systems (steady state solver for growth models with "independent" degrees of freedeom)

\[ \begin{gathered} x_k = x_{k-1} - s_k \, D_k \\[5pt] D_k = \left( J_{k-1}^T\, J_{k-1} + \lambda_k \right)^{-1}\ J_{k-1} \ F_{k-1} \end{gathered} \]

where

  • function evaluation \(F_k = F(x_k)\)

  • Jacobian evaluation \(J_k = J(x_k)\)

  • step direction \(D_k\)

  • step length \(s_k\)

  • steepest descent (Cauchy step) mixin parameter \(\lambda_k\)

Special cases

  • Quasi-Newton with variables step length: \(\lambda_k=0\)

  • Plain vanilla Newton \(\lambda_k=0\), \(s_k=1\)

The simulate function

[s, info, frameDb] = simulate( ...
    m, d, range, ...
    "deviation",    true | false, ...
    "anticipate",   true | false, ...
    "plan",         empty | Plan, ...
    "method",       "stacked", ...
    "blocks",       true | false, ...
    "terminal",     "firstOrder" | "data", ...
    "startIter",    "firstOrder" | "data", ...
    "successOnly",  false | true, ...
    "window",       @auto | numeric, ...
    "solver",       @auto | {...}, ...
);

The solver options:

{ solverName, ...
    "skipJacobUpdate",          0 | numeric, ...
    "lastJacobUpdate",          Inf | numeric, ...
    "functionNorm",             2 | Inf, ...
    "maxIterations",            5000 | numeric, ...
    "maxFunctionEvaluations",   @auto | numeric, ...
    "functionTolerance",        1e-12 | numeric, ...
    "stepTolerance",            1e-12 | numeric, ...
}

Solver names:

solverName = "newton" | "quickNewton" | "qnsd"