WP8: Occasionally binding constraints in DSGE models

Tom Holden (et al.)

School of Economics, University of Surrey

Promised deliverables

• “D8.4: Computational Paper: Particle filter estimation of DSGE models paper with occasionally binding inequality constraints”.• 24 months delivery.

• “D8.8 Theoretical/Empirical Paper. An NK-Banking Model with an Occasionally Binding Incentive Constraint”.• 30 months delivery.

• We also promised contributions to software resources for the solution and estimation of non-linear models, particularly those with OBC.

Introduction to “dynareOBC”:Our key computational contribution

• dynareOBC is an open source toolkit for Dynare, available from:

https://github.com/tholden/OBCToolkit

• It embeds the algorithms for the solution of models with occasionally binding constraints of Holden (2010), Holden and Paetz (2012) and Holden (2015).

• It also contains early code for estimation of models with occasionally binding constraints.

• Its abilities go beyond occasionally binding constraints however, and its tools can be used for increased accuracy in both non-linear simulation, and non-linear estimation.

• Its use is as simple as “dynareOBC ModFile.mod”, where ModFile.mod may contain “max”, “min” and “abs” within its model block.

The core algorithms for simulation of models with OBC in dynareOBC

• The basic algorithm of Holden (2010) and Holden and Paetz (2012) was based on adding “news”-type shocks to the bounded equation to capture information about the likelihood of being pushed away from the bound in future.• At a first order approximation to the underlying model, this is just a perfect foresight

approximation

• dynareOBC extends this to higher order pruned perturbation approximations, using the NLMA toolkit of Lan and Meyer-Gohde (2014).• Using higher orders will almost always deliver significantly improved accuracy away

from the bound (for example, the mean interest rate will include a risk premium term).

• Near the bound, this will still generally be more accurate than the first order approximation, but it still misses the effects of the risk of hitting the bound, e.g. additional precautionary saving.

The first improved algorithm in dynareOBC

• The first improvement we make to the algorithm is inspired by the stochastic extended path (S.E.P.) method of Adjemian and Juillard (2013).• To improve accuracy close to the bound, we need to integrate over future

uncertainty, just as in the S.E.P..

• Our base algorithm requires the expected future path of the bounded variable, ignoring the bound.

• To allow for future uncertainty, we need the distribution of the future path of the bounded variable, ignoring the bound.

• But, by the nice properties of pruned perturbation solutions, we can obtain a Gaussian approximation to this distribution without any Monte-Carlo simulation.

• Thus, we can integrate over 𝐾 periods of future uncertainty solving the underlying “perfect foresight” problem polynomial in 𝐾 times, irrespective of the number of shocks (𝑁) in the model.• Obtaining a similar accuracy in the more general S.E.P. would require exponential in 𝐾 and polynomial in 𝑁 solutions of the perfect foresight problem.

• The Adjemian and Juillard (2013) shock pruning algorithm may reduce the expense at the cost of some accuracy.

Accuracy (1/2)

• As an illustration of the procedure’s accuracy, consider a very simple closed economy model, with no capital and inelastic unit labour supply.

• Households maximise:

𝔼0

𝑡=0

∞

𝛽𝑡𝐶𝑡1−𝛾− 1

1 − 𝛾

• Subject to the budget constraint:𝐴𝑡 + 𝑅𝑡−1𝐵𝑡−1 = 𝐶𝑡 + 𝐵𝑡

• 𝐴𝑡 is productivity.• 𝑅𝑡 is the real interest rate.• 𝐵𝑡 is the household’s holdings of zero net supply bonds.

• Define 𝑔𝑡 ≔ log𝐴𝑡 − log𝐴𝑡−1.

• Productivity evolves according to:𝑔𝑡 = max 0, 1 − 𝜌 𝑔 + 𝜌𝑔𝑡−1 + 𝜎휀𝑡 ,

• 휀𝑡~NIID 0,1 , 𝛽 ≔ 0.99, 𝛾 ≔ 5, 𝑔 ≔ 0.05, 𝜌 ≔ 0.95, 𝜎 ≔ 0.07.

Accuracy (2/2)

• In equilibrium, 𝑅𝑡 is a function of the expectation of a truncated log-normal distribution, so has a closed form expression.• Thus we can examine the departure from the true value.

• As may be seen, even using a very fast degree 3 monomial cubature rule produces good accuracy.

• Furthermore, when using general sparse cubature rules, we can exploit their nested structure to make the cubature degree dynamic, greatly increasing the simulation speed while the model is away from the bound.

• We further improve accuracy in this case by taking a statistical model of the cubature errors, which enables us to escape oscillation in the cubature error as the degree increases.

(Simulation) error:log𝑹𝒕 − log𝑹𝒕

𝑻𝑹𝑼𝑬No integration Degree 3 monomial

cubature

Root mean squared 0.0059 0.00061

Maximum 0.014 0.0014

The second improved algorithm in development for dynareOBC

• Due to the necessity of only integrating over uncertainty a limited number of periods into the future, even our first improved algorithm will fail to capture the long-run risk effects of the bound.

• Capturing this requires a semi-global solution procedure.

• We adopt a two-step solution procedure.• First, we add a polynomial in the states and shocks of the model to each bounded

equation, where the polynomial coefficients are initially set to zero.• By repeated solution and simulation, we iterate until the error in the (fully nonlinear)

bounded equation is orthogonal to each of the included monomials.• This delivers a global polynomial approximation to the bounded equation, which well

captures the long-run risk effects of the bound.

• We then simulate the resultant model using a variant of the first improved algorithm.• The principal change is that whereas previously “news” was always positive in the

presence of a lower bound, now the “news” innovations may be both positive and negative, since the polynomial approximation to the bounded variable will tend to be above the correct value when away from the bound.

• dynareOBC currently contains a less-efficient semi-global solution procedure, but this new one should be incorporated soon.

Other dynareOBC features

• A convenient interface around the code of Lan and Meyer-Gohde for pruned perturbation approximation, and an interface around Meyer-Gohde’s code for taking a first order approximation around the mean or risky steady state.

• Evaluation of average IRFs without any Monte Carlo simulation.• Thanks to the nice properties of pruned-perturbation solutions.

• Exact simulation of “model local variables” (MLVs), given the approximated path followed by standard endogenous variables (states and controls).• May be used to increase both simulation accuracy and speed.

• Simply write the .mod file so that there are as few endogenous variables as possible.

• Then define other variables of interest as MLVs.

• MLVs may be included after the stoch_simul command.

• Simulation via cubature or Monte Carlo integration of forward looking MLVs.• So to generate Euler equation errors, you just add a MLV for the error.

Estimation of models with OBC

• We promised code for particle filter estimation of models with OBC.

• At present, dynareOBC contains code for estimation of models with or without OBC (at first, second or third order), using the (Square-Root) Cubature Kalman Filter (CKF) of Arasaratnam and Haykin (2009).• This algorithm is based on approximating the distribution of the state by a Gaussian.• Idea: If last period’s state is Gaussian, and the shock is Gaussian, then the integrals

required for the predict and update steps are all Gaussian integrals, and so may be efficiently evaluated through Gaussian cubature.

• The CKF algorithm of Arasaratnam and Haykin (2009) uses the standard degree 3 monomial cubature rule, though higher order cubature is also possible (only the former is implemented in dynareOBC at present).

• The CKF is exact for linear models. For models approximated to third order, it correctly propagates the mean, but approximates the covariance.

• Our implementation:• Uses the augmented state space created by our OBC algorithm.• Requires observation equations to be MLVs, so no perturbation approximation is taken to

these.• Only performs maximum likelihood, though users may easily “hack-in” a prior to perform

MAP estimation.

• We will use the CKF as a proposal distribution for a particle filter.• Unclear if this will be fast enough to be practical. Maybe it is not needed?

Occasionally binding financial constraints (OBFC):An application of these techniques

• Holden, Levine and Swarbrick are using the dynareOBC toolkit in the paper: “Endogenous financial crises in a DSGE model of bank dividend and equity issuance”.• This will fulfil D8.8.

• By way of motivation, consider the BAA-AAA spread:

Background to the OBFC paper

• The spikes we see in the spread during recessions suggest something drastically different is happening in this time.• Away from a financial crisis, the level and volatility of the spread is consistent with

what we would expect from time varying risk premia.

• Financial frictions are not “normal” phenomena.• If we follow Bernanke-Gertler-Gilchrist (BGG) (1999), or Gertler-Kiyotaki (GK) (2010)

in assuming that the economy is always financially constrained, and then take a local approximation, then we are ruling out a priori any chance of matching the data.

• This idea of OBFC has been pursued by e.g. Dewachter and Wouters (2014), Brunnermeier and Yuliy Sannikov (2010) and Li (2012). But:• Our banking model is novel.

• We are the first to have the tools necessary to embed these features in a model which is suitable for policy.

• We are the first to be able to estimate key parameters of an OBFC.

Our model

• Loosely based on Gertler and Kiyotaki (2010), but improved along various axes.

• Most crucially, our model allows banks to raise equity in order to escape the constraint on debt issuance.• However, there is a proportional cost of equity issuance which means banks may still

choose not to issue equity even when the debt constraint is binding.

• Rather than having an exogenous death rate of bankers, our banks pay our dividends.• It turns out, that in steady-state, with banks’ optimal rate of dividend issuance, the

debt constraint binds, but only just, whatever the parameters.

• Thus vulnerability to crises is endogenous in the model. It is not just an artefact of very specific parameter choice, as in the GK model.

• We further improve the rigour with which the GK constraint is derived, correcting the timing, and allowing for “off-equilibrium” play.

Households and firms

• Households maximize:

𝔼𝑡

𝑠=0

∞

𝛽𝑡𝐶𝑡1−𝜚1 − 𝐻𝑡

𝜚1−𝜎𝑐− 1

1 − 𝜎𝑐

• subject to 𝐶𝑡 + 𝐵𝑡 = 𝐻𝑡𝑊𝑡 + 𝑅𝑡−1𝐵𝑡−1 + 𝐷𝑡 − 𝐸𝑡 + 𝑇𝑡• 𝐷𝑡 is their period 𝑡 dividend income.

• 𝐸𝑡 is the amount they give to the bank in period 𝑡 to pay for newly issued equity.

• 𝑇𝑡 is lump sum taxes, where 𝑇𝑡 = 𝐺𝑡 = 𝑔𝑌𝑡.

• 𝐵𝑡 are riskless bonds lent to the bank for investing in firms.

• Gives the standard Euler equation 1 = 𝑅𝑡𝔼𝑡Λ𝑡+1, where Λ𝑡+1 is the s.d.f. between periods 𝑡 and 𝑡 + 1.

• Perfectly competitive firms produce using the technology:𝑌𝑡 = 𝐾𝑡−1

𝛼 𝐴𝑡𝐻𝑡1−𝛼

Banks

• Banks have two individual state variables:• 𝐾𝑡−1 the capital they own and rent to firms at a rate 𝑟𝐾,𝑡.• 𝐵𝑡−1 their borrowing from households.

• We write their value function as 𝑉𝑡 𝐾𝑡−1, 𝐵𝑡−1 . This satisfies:𝑉𝑡 𝐾𝑡−1, 𝐵𝑡−1 = max

𝐷𝑡≥0,𝐸𝑡≥0,𝐼𝑡≥0,𝐵𝑡≥0𝐷𝑡 − 𝐸𝑡 + 𝔼𝑡Λ𝑡+1𝑉𝑡+1 𝐾𝑡 , 𝐵𝑡

• where the maximisation is carried out subject to the budget constraint:𝐷𝑡 − 𝐸𝑡 = 𝑟𝐾,𝑡𝐾𝑡−1 − 𝑅𝑡−1𝐵𝑡−1 − 𝜅𝐸𝑡 + 𝐵𝑡 − 𝐼𝑡

• and the capital accumulation equation:𝐾𝑡 = 1 − 𝛿𝑡 𝐾𝑡−1 + 𝐼𝑡 .

• As in GK, in each period, 𝑡, banks have the option of choosing to exit in the nextperiod, 𝑡 + 1.• This is the timing in GK, though it is not stressed there. (Rationalised as capital is in use in

the period.)• We assume that banks may pay dividends before exiting in period 𝑡 + 1.

• When banks exit, they scrap their accumulated capital stock, converting it back to Θ𝐾𝑡 units of the consumption good that may be paid to shareholders in period 𝑡 +1, where Θ < 1.• Banks are not constrained to repay their loans from households in the period in which

they exit (𝑡 + 1), but unsurprisingly, households will not loan further to them in that period (𝑡 + 1).

The exit decision

• If a bank decided in period 𝑡, to exit in period 𝑡 + 1, then in the latter period, they will face a budget constraint of the form:

𝐷𝑡+1 − 𝐸𝑡+1 = Θ𝐾𝑡 − 𝜅𝑡+1𝐸𝑡+1• Setting 𝐸𝑡+1 = 0 is optimal, implying 𝐷𝑡+1 = 𝑉𝑡+1 = Θ𝐾𝑡.

• As a result, in the preceding period, their decisions will also be different.

• Denoting these decisions conditional on exiting with a superscript 𝐸, they maximise:𝐷𝑡𝐸 − 𝐸𝑡

𝐸 + 𝔼𝑡Λ𝑡+1Θ𝐾𝑡𝐸

= 𝑟𝑘,𝑡𝐾𝑡−1 − 𝑅𝑡−1𝐵𝑡−1 − 𝜅𝑡𝐸𝑡𝐸 + 𝐵𝑡 − 𝐼𝑡

𝐸 + 𝔼𝑡Λ𝑡+1Θ 1 − 𝛿𝑡 𝐾𝑡−1 + 𝐼𝑡𝐸

• So, 𝐼𝑡𝐸 = 0, providing 𝔼𝑡Λ𝑡+1Θ < 1.

• From comparing the value functions, we get that banks prefer (in period 𝑡) not to exit (in period 𝑡 + 1) if and only if:

𝑉𝑡 ≥ 𝑟𝑘,𝑡𝐾𝑡−1 − 𝑅𝑡−1𝐵𝑡−1 + 𝐵𝑡 + 𝔼𝑡Λ𝑡+1Θ 1 − 𝛿𝑡 𝐾𝑡−1• If this constraint is violated, banks exit with probability 1, and so households are repaid

with probability 0.• Thus households will never lend beyond the point at which constraint binds.

• Compare this constraint to that of GK (without adjustment costs): 𝑉𝑡 ≥ Θ𝐾𝑡• GK’s omission of discounting is unlikely to have a major impact, but the other terms may

do, seeing as they may tighten the constraint.

Determinacy issues

• With the model as described, when the borrowing constraint is slack, there is local indeterminacy.

• To fix this, we make a slight modification to the bank’s objective, in particular we suppose:𝑉𝑡 𝐾𝑡−1, 𝐵𝑡−1 = max

𝐷𝑡≥0,𝐸𝑡≥0,𝐼𝑡≥0,𝐵𝑡≥0𝐷𝑡 − 𝐸𝑡 + 1 − 𝛾 𝔼𝑡Λ𝑡+1𝑉𝑡+1 𝐾𝑡, 𝐵𝑡

• where 𝛾 > 0, but 𝛾 ≈ 0. (𝛾 may be made arbitrarily small.)

• One may justify 𝛾 > 0 by the difficulties in providing long term incentives to bankers, or by appealing to very rare purely exogenous bank exit shocks, as in GK.

• With this change, in steady-state, the Lagrange multiplier on the borrowing constraint is equal to

𝛾

1−𝛾.

• I.e. the borrowing constraint binds in steady-state.

• But even small shocks are enough for it to cease binding.

Parameterisation, and details of our simulation exercise

• Our parameterisation is largely standard.

• We look at simultaneous, large, shocks to productivity and depreciation. • Purely for illustrative purposes.

• Simulated under perfect foresight for now.

• We compare two values for 𝜅 in line with the evidence that:• “There has been a significant increase in fees since the onset of the financial crisis.

Average fees rose to more than 3 per cent in 2009 from around 2 to 2.5 per cent in the period from 2003 to 2007.”

• http://www.oft.gov.uk/shared_oft/market-studies/OFT1303.pdf

• Given the inevitable truncation bias, even higher calibrations may be justifiable (and indeed have some empirical support: Butler, Grullon and Weston (2005) finds a value around 5%).

𝑔 0.2 𝜌𝛿 0.75

𝜚 0.684 𝜌𝐴 0.95

𝛼 0.3 Θ 0.80381

𝛽 0.99 𝛾 0.00001

𝛿 0.025

𝜎𝑐 2 𝜅 0.01, 0.035

Simulations with 𝜅 = 0.01

Simulations with 𝜅 = 0.035

Conclusions

• dynareOBC is hopefully useful for a very wide range of applications.

• Please experiment with it, and send me bug reports!

• Papers documenting the new OBC algorithm, and the estimation algorithm are still to be written.

• I will also be experimenting to see if particle filter estimation of models with OBC can be made computationally tractable.

• The OBFC paper still needs a lot of work, not least:• Stochastic simulations, using dynareOBC.

• Global solution for comparison, to stifle some inevitable criticisms.

• Embedding in a larger policy model.

• Estimation of the OBFC parameters in that larger policy model.

• But already it suggests we will be able to explain the “spikiness” of the spread during crises, and the non-monotonic response to shocks.