1.7. Perturbation types for regime-switching models

A perturbation of a constant-parameter model names one thing, the scale \(\sigma\) of the shocks, and expands the policy function around the deterministic steady state, the point the economy rests at when \(\sigma = 0\). A regime-switching model adds a second source of variation, the Markov chain, and when the switching parameters move the steady state there is no longer one point to expand around. Each perturbation type in RISE is a different answer to that problem, and the answers differ in three places:

• What is perturbed. The shock scale alone ('m', 'bm', {'m','scl'}), the shock scale and the off-diagonal transition probabilities ('mw'), or the shock scale and the switching parameters themselves ('frwz').

• What the economy is at the origin of the perturbation. The switching economy without shocks ('m', {'m','scl'}, 'bm'), the regimes in isolation ('mw'), or one economy with common parameters ('frwz').

• Where the derivatives are taken. At the isolated steady states with the lead of the pair \((i,j)\) evaluated at regime \(j\)’s steady state ('m', 'mw'), at the P-weighted anchors of Barthélemy and Marx ('bm'), at the rest points of the shock-free switching economy with the lead at the transition point ({'m','scl'}), or at the steady state of the common-parameter economy ('frwz').

The type is selected with the solve_perturbation_type option of solve (or set); the syntax of each value is given in its section below. With a common steady state across regimes, or on a constant-parameter model, every type returns the same solution, the standard perturbation: the differences below are about what happens when the steady state switches.

1.7.1. The types at a glance

Value	Perturbed	Economy at the origin	Anchors	Lead of pair \((i,j)\) at	First-order constant
'm'	\(\sigma\)	switching, shock-free	isolated steady states \(\bar x_i\)	\(\bar x_j\)	none
'mw'	\(\sigma\) and \(P(\sigma)=(1-\sigma)I+\sigma P\)	regimes in isolation	isolated steady states	\(\bar x_j\)	from the tilt of \(P\)
'frwz'	\(\sigma\) and \(\theta_i(\chi)=\bar\theta+\chi(\theta_i-\bar\theta)\)	common parameters \(\bar\theta\) (ergodic mean)	one steady state \(\bar x(\bar\theta)\)	\(\bar x\)	from the parameter gap
'bm'	\(\sigma\)	switching, shock-free	P-weighted anchors (their Assumption 2)	\(\bar z_j\)	none
{'m','scl'}	\(\sigma\)	switching, shock-free	rest points \(\hat x_i\) of that economy	transition point \(T_j(\hat x_i)\)	none (certainty equivalence)
'mt' (experimental)	\(\sigma\)	switching, shock-free	isolated steady states	transition point	carries the rest-point residual
'mwt' (experimental)	\(\sigma\) and \(P(\sigma)\)	regimes in isolation	isolated steady states	transition point	from the tilt

Two relations hold between them and are checked in the test suite. With a common steady state 'm', 'bm', {'m','scl'} and 'mt' are identical at every order, and 'frwz' with an empty partition is 'm'. With P = I (regimes that never communicate) {'m','scl'} is 'mw' at first order, and {'mw',{'*none'}} is 'm'.

1.7.2. Perturbation M ('m')

The default. Only the shock scale is perturbed, the transition matrix stays active in the first-order slopes, and regime \(i\)’s policy is expanded around its own isolated steady state \(\bar x_i\), the solution of the model with the parameters of regime \(i\) held forever. The derivatives of the pair \((i,j)\) are evaluated with next period’s variables at \(\bar x_j\) and the current and lagged variables at \(\bar x_i\). The first-order solution carries no constant, and its rest points are therefore the isolated steady states whatever the switching does. The higher orders are the standard ones, computed by the hops engine or the recursive engine.

This is exactly right when the steady state does not switch, which is the case of switching volatilities, switching policy-rule coefficients and switching persistences, and it is what most of the estimation literature uses. When a switching parameter moves the steady state (a productivity level, an inflation target, a tax rate), the isolated steady state is not a rest point of the switching economy, because agents in regime \(i\) anticipate the switch: the expansion point is then a point the reference economy never visits, the first-order solution has a level error of the order of the gap between steady states, and that error is a constant that no higher order removes. The Euler-error test below shows it as a flat verdict.

'm' and 'maih' are the same value. The type takes no partition.

1.7.3. The Maih–Waggoner perturbation ('mw')

Maih and Waggoner (2018) perturb the transition matrix along with the shocks: \(P(\sigma)=(1-\sigma)I+\sigma P\), so that at the origin the regimes are isolated and each has its own deterministic steady state, a valid perturbation in the classical sense. The first-order slopes are then those of the isolated regimes (decoupled across regimes), and the switching enters the first order through a constant that comes from the derivative of the policy with respect to the tilt of \(P\), which RISE computes from the \(\sigma\)-equation of the first-order system. Syntax:

m = solve(m,'solve_perturbation_type','mw');              % tilt every chain
m = solve(m,'solve_perturbation_type',{'mw',{'pol'}});    % tilt the chain 'pol' only
m = solve(m,'solve_perturbation_type',{'mw',{'*none'}});  % no tilt: identical to 'm'

The cell form names the Markov chains whose off-diagonal probabilities are tilted; the others keep their probabilities at \(\sigma=0\), as under 'm'. Plain 'mw' tilts every chain. 'mw' and 'maih_waggoner' are the same value.

What to expect. The scheme is the natural one when the regimes are distinct economies that last long, since each regime’s steady state is genuinely a point the economy rests at for a long time, and it is the scheme whose reference economy is deterministic. Two things follow from the decoupling. The first-order slopes of a regime do not depend on the transition probabilities, so a regime that is indeterminate in isolation (a passive monetary policy) has several bounded first-order solutions under 'mw' even when the switching model as a whole is determinate. With the default solver setting RISE detects the decoupled system, uses the eigenvalue solver regime by regime and reports the indeterminacy (return code 21, as on the FRWZ NK model whose second regime is passive); naming 'mfi' explicitly returns one of the bounded solutions, the one the iteration selects. And along the path \(P(\sigma)\) the rest points of the switching economy move like \(p/(p+\kappa)\) in the switching probability \(p\), a saturating function whose Taylor series in \(\sigma\) converges only when \(p\) is small relative to the speed \(\kappa\) at which the state adjusts: the higher orders of 'mw' reconnect with the exact rest points when the regimes last hundreds of periods and diverge when they last tens. The Euler-error test shows this as a non-monotone verdict that does not respond to the shock scale.

1.7.4. The Foerster–Rubio-Ramírez–Waggoner–Zha perturbation ('frwz')

Foerster, Rubio-Ramírez, Waggoner and Zha (2016) perturb the switching parameters along with the shocks: \(\theta_i(\chi)=\bar\theta+ \chi(\theta_i-\bar\theta)\), where \(\bar\theta\) is the ergodic mean of the parameters, so that at the origin the regimes are identical and the economy has one steady state \(\bar x(\bar\theta)\). The expansion is a joint Taylor series in \((\sigma,\chi)\) around that point, evaluated at \((1,1)\). Syntax:

m = solve(m,'solve_perturbation_type',{'frwz',{'mu'}});   % perturb mu only
m = solve(m,'solve_perturbation_type','frwz');            % perturb every switching parameter

The cell form is the version the paper recommends: the partition names the switching parameters that move the steady state, which are perturbed; the other switching parameters keep their regime dependence, as under 'm'. Plain 'frwz' perturbs every switching parameter, which the paper calls the naive perturbation. The distinction matters: a persistence or a policy coefficient that switches by a large amount is a poor perturbation variable, and on a model whose persistence switches from 0.9 to 0.5 with a common steady state the naive form is non-monotone in the order while the partitioned form is 'm' and converges. The partition may be {'*all'}, {'*none'} or {'*all_but', 'v1', ...}.

What to expect. FRWZ is a valid perturbation whose series converges on every model tested where the gap between steady states is moderate, and at higher orders it is the most accurate scheme on the growth model of the test suite. Its properties come with four consequences to know about.

• The first order is regime-blind in the slopes. The parameters in the partition enter the first-order solution only through the regime-specific constants (the terms of order \(\chi\)); the slopes are those of the common-parameter economy and are the same in every regime. Regime-specific dynamics from a switching level, target or tax appear at order 2, through the cross terms in \(\chi x\). Estimation at first order, the common practice, therefore uses a solution in which the regimes differ by their intercepts only. On the growth model with a 5 percent productivity gap the two regimes’ slopes are identical under FRWZ to the last digit, while they differ by 0.13 ('m'), 0.13 ('mt'), 0.17 ('mw') and 0.26 ({'m','scl'}) in their largest entry.

• The expansion point is a point no regime visits. The steady state of the ergodic-mean economy is between the regimes’ rest points, and the polynomial has to reach each regime’s rest point through the series in \(\chi\). The rest points are reproduced with an error of order \(g^{k+1}\) in the gap \(g\) at order \(k\), so they are only as good as the order.

• The gap is a perturbation variable, with a radius of convergence. A large gap, in the parameter or in the steady state, puts \(\chi=1\) outside the radius and the series stops improving; the Euler-error test shows it as a non-monotone verdict that does not respond to the shock scale.

• The partition is a modeling choice. The user has to say which parameters are perturbed, and there is a version of the scheme for each choice; the steady-state-moving parameters are the recommended one.

One thing the scheme does that none of the others can: it does not need the isolated steady states to exist. On the FRWZ RBC model with a negative growth rate in one regime (\(\mu=-0.0337\)), that regime has no steady state in isolation: the net return the Euler equation requires, \(\beta^{-1}e^{(1-\upsilon)\mu}-1+\delta\), is \(-0.038\), the capital stock that would deliver it is complex, and 'm', 'mw', 'bm', {'m','scl'} and 'mt' stop at the steady-state stage with return code 1. {'frwz',{'mu'}} solves from the steady state of the ergodic-mean economy (\(\bar\mu=0.0070\)), which exists. The switching economy itself does have a rest point in that regime, since agents expect the return of positive growth, and {'m','scl'} reaches it through its Start option, a homotopy in the parameter gap from the ergodic-mean steady state: at first order it converges to rest points of 9.24 and 47.88 for the capital stock. But with probabilities of staying of 0.75 and 0.5 the regimes last one to four periods, the economy never gets near a rest point 40 units from where it lives, and the SCL’s Euler error, \(10^{-2.5}\), is behind FRWZ’s \(10^{-3.2}\); its higher orders do not converge there. Fast switching is FRWZ’s domain, persistent regimes with a switching steady state the SCL’s.

M with the self-consistent linearization ({'m','scl'})

Perturbation M scales only the future shocks and keeps the transition matrix active in the first-order slopes. Its reference economy is the shock-free switching economy: parameters switch, shocks are off. When the steady state differs across regimes, that economy does not rest at the isolated steady states: after a switch the state moves toward the new steady state at the rate given by the policy slopes, and agents anticipate further switches. Expanding around the isolated steady states, with the lead of the pair \((i,j)\) evaluated at regime \(j\)’s steady state, is then an expansion around points the reference economy never visits. The first-order solution of 'm' carries no constant, so its rest points are the isolated steady states whatever the switching does, and its forward-looking residual does not improve with the order of approximation.

The self-consistent linearization (SCL) is an option of perturbation M, requested as {'m','scl'}. It keeps the same perturbation principle and moves the expansion configuration: the anchors are the rest points \(\hat x_i\) of the shock-free switching economy, and the derivatives of the pair \((i,j)\) are evaluated at the transition point, the value of regime \(j\)’s policy at regime \(i\)’s rest point. Rest points and slopes are determined jointly by a fixed point (Anderson-accelerated functional iteration); the inner problem at each iteration is the usual coupled quadratic, so every regime-switching solver applies. The \(\sigma\)-constant is zero by construction (certainty equivalence at first order). With a common steady state the option returns exactly plain 'm', at every order and at the cost of 'm': the rest configuration is the steady state and every transition point is that same point, so the solve hands over to the standard perturbation without an outer iteration (report: outer 0, method 'common steady state'). With P = I it returns 'mw' at first order. On a constant-parameter model there is nothing to re-anchor: the request is accepted and solve returns the 'm' solution. Optimal-policy models are re-anchored like any other (see Optimal-policy models below).

At solve_order 2 and above the default higher orders (HigherOrders 'curvature', described below) are one pass of the standard higher-order engine from the first-order configuration: the rest points stay at their first-order values, and an order-K solve costs the first-order stage plus one standard order-K solve, the cost of 'm'’s higher orders. The refinement 'joint' re-expands regime \(j\)’s polynomial at regime \(i\)’s rest point (the Taylor shift) and solves rest points, slopes and all higher-order coefficients as one fixed point, through the toolbox’s recursive engine; under it the first-order block of an order-2 solution is not the order-1 solution, because the rest points and slopes are outputs of the closure and improve with the order. The higher-order stages run only if the first-order block is mean-square stable (return code 25 otherwise), and the returned solution passes the same test at every order.

The options of the method live in one object, rise.options.scl_options, and nowhere else: they are not fields of the model options and they are refused with any other perturbation type. The object is built with name-value pairs (tab completion works) and travels inside the perturbation type, as the option of 'm':

m = solve(m,'solve_perturbation_type',{'m','scl'});           % defaults
o = rise.options.scl_options('MaxIter',500,'Display','iter');
m = solve(m,'solve_perturbation_type',{'m',o});
m = solve(m,'solve_perturbation_type',o);                     % same thing

The stored form is {'m',o}. The earlier spellings 'scl' and {'scl',o} are accepted and mean the same thing.

Its properties follow MATLAB’s optimization options where a name exists: MaxIter (300) and TolFun (1e-8) for the outer fixed point, Algorithm ('anderson', the default, or 'picard'), InitDamping (initial step of the Picard iteration), Display ('off', 'final', 'iter'), Continuation, Start and HigherOrders (see below). The previous solve’s rest configuration and policy jets are always reused as the starting point when their dimensions match, which is what estimation needs; there is no option for it. The solver report is in model_data.state_space{1}.scl: outer iterations, method, residual of the rest-point equations, displacement of the anchors, whether the warm start was used, and the rest configuration xhat.

Because the steady state is the rest configuration, an output of the solve, a steady-state-only request (solve_order 0 or solve_derivatives_only) is refused under the SCL; solve at order 1 or above, or use plain 'm' for the isolated steady states. After a solve, get(m,'sstate') returns the rest points.

Limits known at the time of writing: with very large gaps between rest points (a quarter of the capital stock in the test model), or with transition probabilities whose curvature is on the scale of the gap, the outer fixed point may fail to converge from a cold start at order 3 and above. The Continuation option (default true) then walks the displacement carried by the higher-order feedback from 0 to 1 with adaptive steps; if the fixed point disappears before the full shift the solve returns retcode 29 (SCL_OUTER_NOT_CONVERGED) and the report records the displacement fraction reached (mu_reached). Order 2 converges without continuation in all cases tested. Anticipated shocks are supported at every order.

Where the rest-point iteration starts is the Start option: 'auto' (the default) starts from the isolated steady states and, when a regime has none and the steady-state stage fails, from the unique steady state of the ergodic-mean economy, the point FRWZ expand around; 'isolated' and 'ergodic' force one or the other. From the ergodic-mean point the rest points can be far away, so the solve walks the switching parameters from their ergodic mean, where the SCL is the standard perturbation, to their regime values, warm-starting the outer fixed point at each step (a homotopy in the parameter gap, with the same adaptive steps as the displacement continuation). The report records the start (start), whether the homotopy ran (gap_continuation) and the fraction of the gap reached (mu_gap_reached). Start also accepts a numeric matrix (endogenous variables by regimes, in the order of the endogenous variables): an explicit starting rest configuration, which takes precedence over the warm start and is reported as 'given'. It is the device for exploring the branches of the rest-point equation, which can have several solutions on a model with a peg regime (two, both mean-square stable at first order, on the ZLB model of the SCL note with two-quarter spells); a start of the wrong size is refused. On the growth model the two starts reach the same fixed point to \(10^{-9}\); on FRWZ’s RBC model, where the contraction regime has no isolated steady state, the first order converges to rest points of 9.24 and 47.88 for the capital stock, 40 units from the start, while the higher orders do not converge.

What the solve does once and what it repeats: the isolated steady states are solved once, in the steady-state stage every perturbation type runs, and never again; they start the outer iteration, and the rest points move inside it by one Newton step per iteration at a negligible cost. What every outer iteration recomputes is the derivatives of the model at the transition configurations, the first-order block and, at order 2 and above, the higher-order engine. Profiled on the growth and NK models, the derivatives are 16 to 24 percent of a first-order solve (the inner first-order solves 23 to 38 percent), a third at order 2, and 15 to 20 percent at orders 3 and 4, where the engine’s right-hand sides and Sylvester solves are 50 to 75 percent. The cost is the number of outer iterations, 7 to 80, times the coefficient solves inside each.

How the higher orders are computed is the HigherOrders option; 'curvature' is the default and 'joint' the refinement. 'joint' is the Taylor-shift construction above: the rest points, the first order and every higher order are one fixed point, each order re-expanded at the transition points with the feedback of the orders above it, which is what makes the residual fall with the order (growth model with a 5 percent gap: \(-4.8\), \(-6.1\), \(-7.1\), \(-8.1\) at orders 1 to 4, in \(\log_{10}\) of the Euler error). Its cost is the number of outer iterations, 20 to 70 on the models tested, each one a full order-K solve; the outer tolerance is a modest lever (1e-6 instead of 1e-8 saves 10 to 30 percent of the iterations at no cost in accuracy; at 1e-4 the growth model needs 48 instead of 72 iterations at order 4 and loses 0.7 of a decade). 'recursive' solves the rest points and the first order as at order 1, then computes the higher orders once, conditional on them, each order using the orders below it re-expanded at the transition points and no feedback from above: one order-K solve (order 4 of the NK model with capital: 18 seconds against about an hour and a half). What it buys is the accuracy of order 1 with curvature added, not convergence in the order: on the growth model \(-4.8\), \(-4.5\), \(-4.5\), \(-4.5\); on FRWZ’s NK model \(-2.5\), \(-3.3\), \(-3.7\), \(-3.6\) against \(-2.5\), \(-3.4\), \(-4.6\), \(-6.4\) joint; on FRWZ’s RBC model its higher orders are not finite; on the NK model with capital and a switching target it matches the joint scheme at order 2 (\(-4.07\) against \(-4.08\)) and falls behind from order 3 (\(-4.6\) against \(-6.0\) at order 4), and with a switching capital tax (a 40 percent gap in the capital stock) it is behind from order 2 (\(-3.2\) against \(-3.9\)). The feedback of the higher orders into the rest points, about one percent of the capital stock on the growth model, is what the joint fixed point pays for. The report records the choice (higher_orders) and, for the recursive variant, stages [1 K] with a single evaluation at the second stage; the accuracy report labels the variant m{scl,recursive}. 'curvature' (the default) keeps that feedback and still costs one order-K solve: it computes the higher orders as a perturbation in the curvature of the model around the first-order SCL configuration. The rest points and the first order are solved as at order 1 (first derivatives only), the derivatives up to the solve order are evaluated once at the converged transition points, and one pass of the standard higher-order engine computes the orders 2 to K with the displacement between the rest points carried by the \(\sigma\) column of the pair’s state transition for the orders above 1, the first-order lead slope being composed without it since its shift is part of the transition point (hops by default, the recursive engine above order 5 or on request through solve_perturbation_engine; the two agree to \(10^{-8}\)). The feedback of each order on the constant and the slopes then comes out in the \(\sigma\) slots of that order, as a variance correction does in the standard perturbation, and the anchors stay at the first-order rest points: the held fixed point of the polynomial moves where the joint fixed point moves its rest points. The construction is a perturbation in the strict sense (the model linearized at the SCL configuration is solved exactly by the first-order SCL, and the corrections are linear at every order), it has the same consistency order as the joint fixed point, it nests the standard perturbation with a common steady state, and it has no fixed point to fold; what can fail is the series, and the report gives the largest coefficient by order (coefficient_norm), which falls with the order when the series converges. On the growth model with a 5 percent gap it gives \(-4.8\), \(-6.2\), \(-7.1\), \(-8.0\) at orders 1 to 4 against \(-4.8\), \(-6.1\), \(-7.1\), \(-8.1\) joint, in about one second at every order instead of 74 outer iterations at order 4; with a 20 percent gap it is monotone through order 4 (\(-5.2\)) where the joint fixed point folds; on FRWZ’s NK model it trails the joint scheme by 0.3 to 0.9 of a decade at orders 2 to 4 (\(-5.5\) against \(-6.4\) at order 4) and stays three decades ahead of FRWZ’s own partitioned expansion; on FRWZ’s RBC model, whose regimes last two to four periods, its series diverges at order 4, where the joint scheme does not solve above order 1 and FRWZ’s is the scheme to use. The report records higher_orders 'curvature', stages 1 and the outer iterations of the first-order stage; the accuracy report labels the variant m{scl}.

Note

Under the default 'maih' perturbation, RISE auto-declares several auxiliary parameters (iota_frwz_*, ss_frwz_*, perturbator, …) that are not used in the model. Their absence from your calibration triggers a benign "1(incomplete!)" warning at parse time. This warning is not an error; do not chase it. Chase the retcode.

The Barthélemy–Marx perturbation ('bm')

Barthélemy and Marx (2016, Solving endogenous regime switching models) perturb the shock scale only, as 'm' does, and define a regime-dependent steady state by the P-weighted condition of their Assumption 2: for every regime \(i\),

\[\sum_j p_{ij}(\bar z_i)\, f_i(\bar z_j, \bar z_i, \bar z_k, 0) = 0,\]

the lead of the pair \((i,j)\) at regime \(j\)’s own steady state, the current variables at regime \(i\)’s, and the lag at the steady state of any regime \(k\). Their expansion is then the ordinary perturbation at that configuration: pairwise derivatives at \((\bar z_j,\bar z_i,\bar z_i)\), the coupled first-order solve without a constant, the standard higher orders. In RISE, 'bm' is exactly that: a steady-state stage that solves the condition by Newton from the isolated steady states (with the lag at the current regime’s anchor), writes the anchors into the steady-state slot, and hands over to the 'm' machinery. The self-consistent linearization is not involved, there is no Taylor shift, and 'bm' takes no partition.

Two facts locate the scheme. With a common steady state it is 'm' to the last digit (and so on a constant-parameter model, like {'m','scl'}). When no lagged variable has a switching steady state, or when the lagged variables enter only through combinations invariant to the lag regime (their own model divides the lagged rate by the lagged regime’s target), the transition point of a pair is the next regime’s steady state, the depth-zero and depth-one closures coincide, and 'bm' equals {'m','scl'} (checked at orders 1 and 2 on a three-equation NK model with a switching target). Outside that class the clause “for any regime \(k\)” of Assumption 2 cannot hold, which the authors state; the report model_data.state_space{1}.bm carries the residual of that clause as an \(h\times h\) matrix (assumption2_residual, entry \((i,k)\) the largest residual of regime \(i\)’s equations with the lag at regime \(k\)’s anchor), so the distance from their class is visible. The condition with the lag at the current regime’s anchor may also have no solution: on the growth model with a switching productivity level it disappears at a gap of 2 to 3 percent, and the solve then returns retcode 30 (BM_ANCHOR_NOT_FOUND). solve_derivatives_only is refused under 'bm', since the derivatives belong at the P-weighted anchors, which the solve computes; sstate returns the isolated steady states, the starting point, and get(m,'sstate') after a solve returns the anchors.

Optimal-policy models

A perturbation type does not depend on where the equations come from. The first-order conditions that RISE derives from an @optimization_problem are equations in the variables and the multipliers like any others, and {'m','scl'} and 'bm' re-anchor them as they re-anchor any model: under commitment, under discretion and wherever the commitment parameter switches, at every order. When a switching parameter moves the steady state (an inflation target, a loss weight, a parameter of the constraints), the rest points of the multipliers move with those of the other variables, and with a common steady state both types return plain 'm'.

Discretion, and every regime in which the commitment parameter is below one, adds one object: the first-order conditions carry the policy of next period’s regime as a function of today’s state variables, and its derivatives (the policy functions written f1942 in the generated equations). Under 'm' and 'bm' the lead of the pair \((i,j)\) sits at regime \(j\)’s own anchor, and so do these policy functions. Under {'m','scl'} the lead of the pair is regime \(j\)’s policy at regime \(i\)’s rest point, and the policy functions are taken at that same point: regime \(j\)’s polynomial is re-expanded there, as every other function of the variables is evaluated at the pair’s point. At first order the slopes of a linear policy are the same everywhere and only its level moves; from order 2 its derivatives move too, by the curvature of the policy times the gap between the rest points. The replanning loop of discretion runs around the self-consistent linearization unchanged: each pass solves the rest configuration and the coefficients given the previous pass’s policy functions, and the loop stops when they reproduce themselves.

The first-order solves inside the fixed point use the solver the model is configured with (rise.engine.dsge_tools.solve.configured_solver), the one solve uses, whichever it is. A solver with requirements on the derivatives it is given applies them there as anywhere else, and says so in its own terms when they are not met: see the limits of each solver in Optimal policy.

Testing a perturbation: the Euler error against the order

A perturbation is only as good as its convergence: along a simulated path, the residual of the model’s own equations must fall as the order of approximation rises. rise.engine.dsge_tools.accuracy.perturbation_report solves a model at several orders and perturbation types, simulates one common path of regimes and shocks, measures the Euler residuals with the paradigm-free accuracy engine, and reports the forward-looking equations separately from the static ones (static identities gain curvature at every order under every scheme and would hide a floor):

m = dsge_model('mymodel', 'solve_order', 3);
rep = rise.engine.dsge_tools.accuracy.perturbation_report(m, ...
    'types', {'m','mw',{'m','scl'}}, 'orders', 1:3, 'shock_scales', [1 0.5 0.25]);

Per type and shock scale the verdict on the forward-looking column is one of three: monotone (the error falls with the order), flat (a level error that no Taylor term removes: the expansion point is not a rest point of the scheme’s own reference economy), non-monotone (the error rises at some order or the path blows up). The shock-scaling rerun simulates the same solutions with the realized shocks scaled down; a non-monotone verdict that turns monotone at a smaller scale is a radius-of-convergence problem, a flat verdict that stays flat at every scale is an expansion-point problem. rep.table holds the numbers, latex_file writes them as a tabular. The model must be constructed with a derivative order at least equal to the largest order requested.

1.7.5. Experimental types: 'mt' and 'mwt'

Two types built on the SCL machinery with the anchors frozen at the isolated steady states, kept for diagnosis and not recommended for production. 'mt' is perturbation M with every derivative of the pair \((i,j)\) evaluated at the transition point, the residual of the rest-point equations carried by the \(\sigma\)-constant, and the higher orders Taylor-shifted like the SCL’s; it is the consistent all-order form of a level correction to 'm', and it converges with the order like FRWZ on the growth model but not on the NK model with a switching capital tax, where only re-anchoring works. 'mwt' is the Maih–Waggoner perturbation with the same transition-point evaluation and shift; the tilt analysis above applies to it unchanged, and it is non-monotone in the order wherever 'mw' is. Both take the same syntax as 'm' and no partition, both equal 'm' with a common steady state, and both are the standard perturbation on a constant-parameter model.

1.7.6. Which type to use

• The steady state does not switch (volatilities, policy-rule coefficients, persistences): 'm'. Every other type returns the same solution at a higher cost, or, for 'mw', a different and less accurate one.

• The steady state switches and the model is solved at first order for estimation: {'m','scl'}. It is the only scheme whose first order has regime-specific slopes, no level error and no constant, and whose rest points are those of the switching economy; on the models of the test suite it is the most accurate first-order scheme whenever the gap matters. Its cost is a dozen first-order solves, four to six times 'm' per draw along an estimation random walk (0.10 to 0.19 s warm against 0.02 to 0.04 s on the test models), the warm start being kept automatically across parameter changes; with a common steady state it costs nothing, since it hands over to 'm'.

• The steady state switches and the higher orders matter: {'m','scl'}, whose default higher orders cost what 'm'’s cost (order 2 up to about 200 variables, order 3 up to about 50, order 4 up to about 16 on a 64 GB machine, the limits of the standard engine), or {'frwz', partition}. FRWZ converges faster with the order on the growth model and wins when regimes switch every few periods; the SCL is ahead on the New Keynesian models and when the real gap is large (a switching capital tax with a 40 percent gap in the capital stock). Run both through the Euler-error test on the model at hand.

• A regime has no isolated steady state: {'frwz', partition}, or {'m','scl'} with its default Start 'auto', which starts from the ergodic-mean steady state and walks the parameter gap (first order only, so far).

• Regimes that switch every few periods: {'frwz', partition}. The rest points of the switching economy are then far from where the economy lives, and the ergodic-mean anchoring is the better one, as on FRWZ’s RBC model.

• Regimes that last decades and are meant as distinct economies: 'mw', whose first order is the isolated regimes’ and whose higher orders reconnect in that limit only.

• 'bm' reproduces the Barthélemy–Marx scheme for comparison; inside their class (no predetermined variable with a switching steady state) it equals {'m','scl'}, outside it its anchors may not exist.

The default. Plain 'm' remains the default of the toolbox. Because its SCL option returns plain 'm'’s solution at 'm'’s cost whenever the steady state does not switch, and the rest points of the switching economy with regime-specific slopes when it does, it is the candidate for the future default of 'm'. Its higher orders now cost what 'm'’s cost; what it still lacks is a cheaper first-order stage (two and a half to ten times 'm' at first order, from the outer iterations), which a joint Newton on rest points and coefficients is expected to supply. The experimental 'mt' is not a candidate: it is not a perturbation in the strict sense, it is non-monotone on the NK model with the switching tax, and its order-2 iteration fails from about 12 variables on.

1.7.7. The evidence

The verdicts of the Euler-error test on the models of the SCL project and on FRWZ’s own examples, forward-looking equations, full-size shocks, orders 1 to 3 (1 to 4 on the growth model). The numbers are \(\log_{10}\) of the mean absolute residual; the growth model has a switching productivity level (A: common level, switching persistence; B: 5 percent gap; C: 20 percent gap), the NK model with capital a switching inflation target (B) or capital tax (F), the FRWZ NK model a switching growth rate and Taylor coefficient with a passive regime, the FRWZ RBC model a switching growth rate, persistence and volatility with no isolated steady state in one regime, the BM NK model (Barthélemy and Marx, Section 3) a switching Taylor response and target with the targeted rate set by the P-weighted Fisher relation, with constant and with their endogenous, inflation-dependent probabilities.

Model	'm'	'mw'	{'frwz',part}	'bm'	{'m','scl'}	'mt' (exp.)
growth A	monotone, −5.5 to −9.7	non-monotone	= 'm' (naive form non-monotone)	= 'm'	monotone, −5.5 to −10.1	= 'm'
growth B	flat, −3.0	blows up	monotone, −5.1 to −9.2	no anchor	monotone, −4.9 to −8.1	monotone, −4.3 to −8.5
growth C	flat, −2.6	blows up	monotone, −4.3 to −7.1	no anchor	monotone to order 3, fold at 4	monotone, −3.3 to −5.9
NK B	flat, −2.2	non-monotone	monotone, −3.4 to −5.1	flat, −2.9	monotone, −3.4 to −5.2	monotone, −3.4 to −5.2
NK F	non-monotone	blows up	monotone, −2.7 to −3.6	flat, −1.8	monotone, −3.0 to −4.7	non-monotone
FRWZ NK	flat, −1.2	fails (code 21)	monotone, −1.2 to −2.0 (naive form non-monotone, −2.2 to −1.7)	flat, −1.2	monotone, −2.5 to −4.6	monotone, −1.4 to −2.9
FRWZ RBC	no steady state	no steady state	monotone, −3.2 to −4.5	no steady state	order 1 only, −2.5 (Start homotopy)	no steady state
BM NK, constant P	flat, −2.4	non-monotone	monotone, −4.7 to −6.3	flat, −4.5 to −4.7	monotone, −4.8 to −6.5	monotone, −4.1 to −5.6
BM NK, endogenous P	flat, −2.3	explodes	fails (code 3)	monotone, −4.6 to −5.1	monotone, −4.8 to −6.1	flat, −3.2 to −3.5

Two readings. Where the steady state switches, {'m','scl'} is the most accurate first-order scheme on every model (NK F, FRWZ NK by more than a decade; growth B and NK B within a few hundredths of FRWZ), and at higher orders it converges wherever it solves, ahead of FRWZ on the NK models and behind it on the growth model. And on FRWZ’s own NK model the rest points of the interest rate under {'m','scl'} and 'mt' converge with the order to the same pair (1.0402, 1.0298), the rest configuration of the switching economy, while FRWZ’s partitioned form sits at (1.0225, 1.0114), which is why its Euler error stays two and a half decades behind at order 3. On FRWZ’s RBC model, whose regimes last one to four periods, the order is reversed: the SCL reaches the rest points of the switching economy only through the parameter-gap homotopy, sits 0.6 of a decade behind FRWZ at first order, and its higher orders fail; fast switching is FRWZ’s domain. All first-order solutions that solve are mean-square stable.

1.7.8. First-order solvers and the types

The first-order solve of a switching model is a coupled quadratic system, and the solver option chooses the algorithm: 'mfi' (functional iteration, the default), 'mafia' (the same map with Anderson acceleration), 'mnk' and 'mn' (Newton, with or without the Kronecker form), 'mfi_full', 'mnk_full' and 'mn_full' (the same on the full state vector) and 'fwz' (the Newton system of Farmer, Waggoner and Zha). Every perturbation type runs on every solver, because the types change the matrices of the coupled system and not the system itself; {'m','scl'} uses the chosen solver for its initial and final first-order solves and its own Newton for the coupled solves inside the outer iteration. What differs across solvers is which solution they land on when the system has several, and the test suite’s matrix of types by solvers (models/dsge/solving and the SCL project’s battery) shows three patterns.

• On the growth model with a 5 percent productivity gap, only the functional iterations ('mfi', 'mfi_full') return the mean-square-stable solution for 'm', {'m','scl'} and 'mt'; the Newton solvers and 'mafia' converge to a solution that fails the stability test (return code 25). Functional iteration has the contraction that selects the stable solution; Newton converges to the root nearest its start.

• On the NK model with capital, every solver except 'fwz' returns the same solution for every type (to \(10^{-7}\), the Newton solvers’ tolerance, and \(10^{-10}\) under {'m','scl'}, which tightens the inner tolerance).

• On the FRWZ NK model whose second regime is passive, the coupled system has more than one mean-square-stable solution under the transition-point types: 'fwz' returns a solution different from the other solvers’ for {'m','scl'}, 'mt' and 'mwt' (and the same one for 'm'), and the plain Newton solvers 'mn' and 'mn_full' do not converge for {'m','scl'} (return code 29). Both solutions pass the stability test. Which one is the relevant equilibrium is a question the solver cannot settle; the Continuation option of {'m','scl'} and a warm start from a known solution are the tools for keeping a solver on one branch.

The practical rule: keep 'mfi' unless it is slow, check the return code, and when a model may have several stable solutions, compare two solvers before trusting one.