The method of undetermined coefficients, described by Uhlig [1995], can be used to ob-tain a reduced-form recursive solution of the linear rational expectations system formed from equations (A.12), (A.13), (A.13), (A.14), (A.16), (A.17), (A.18), the total number of which is Md, and the equations for exogenous variables, (A.2) and (A.3), of which there are Mein total. Applying the method determines the values of matrices A, B,C in the reduced form solution
ζt = Aζt−1+ Bzt, ζt∈ RMd, A ∈ RMd×Md, B ∈ RMd×Me (A.19) zt = Czt−1+ ηt, C∈ RMe×Me, ηt, zt ∈ RMe. (A.20)
Where, ζt = [ygt, yt, πt, rnt, it, nt]T and zt = [at, vt]T. In order to facilitate Bayesian parameter estimation, when fitting the model to economic data, a state-space represen-tation of the reduced-form recursive solution is used.
γt = Φγt−1+ Θηt, γt= [ζt, zt]T. (A.21)
The matrices Φ, Θ ∈ RM×M (for M = Md+ Me) are functions of the matrices A, B,C above. As an illustration, the model is fitted to two observable economic time series, together with shocks to production (at) and technology (vt):
• πt: UK Consumer Price Inflation (UK CPI).
• it: UK Sterling Overnight Index Average (SONIA).
Quarterly data from March 1997 to June 2013, publicly available from Office of Na-tional Statistics-UK [2013], is used. Once a parameter fit is obtained plots of IRFs are produced that visualise how an isolated one standard deviation shock in an exogenous variable (either at or vt) evolve the endogenous variables through time.
-0.9
Figure A.1. An input-response function diagram showing the relations to policy shock.
-0.20
Figure A.2. An input-response function diagram showing the relations to technology shock.
Appendix B
Pulse-coupled model details and proofs - for chapter 4
Description of the stochastic model
During the diffusion phase, at time t, the state variables θu(t) update according to a simple unbiased continuous-time random walk between nearest neighbour states, satis-fying
1
2 = P{θu(t + δu) = s + 1 | θu(t) = s}
= 1 − P{θu(t + δu) = s − 1 | θu(t) = s}, (B.1)
where s ∈ {1, . . . , 2K − 1}, θu(t) is the current oscillator state and δu are independent exponentially distributed random variables, Exp(Λ) with mean 1/Λ, representing the passage of time until the next state transition. Without loss of generality, throughout this study we set Λ = 1. The first oscillator to transition to either of the firing states occurs at the boundary hitting time,
τ = min
u {t : θu(t) = 0, or θu(t) = 2K}, (B.2)
at which time the diffusion process ends and the cascade phase begins. The existence of finite hitting times, for this random walk between two boundaries, is guaranteed by standard results Redner [2001]. The cascade process continues as described for the original one sided model DeVille et al. [2010]; DeVille and Peskin [2008], with the difference being oscillators reset to state K after firing and the cascade ending, before the diffusion phase restarts.
Mean field model.
Let vvvi ∈ R2K+1+ denote the i-th standard basis vector, with 1 in position i and 0 else-where, and let S0, S+, S−be subsets of phase space, defined by
S0= {yyy ∈ R2K+1+ : hyyy, vvv0i < 1, hyyy, vvv2Ki < 1}, S+= {yyy ∈ R2K+1+ : hyyy, vvv2Ki ≥ 1},
S−= {yyy ∈ R2K+1+ : hyyy, vvv0i ≥ 1},
(B.3)
where R+= {r ∈ R : r ≥ 0}, and h. . .i denotes the standard inner product on R2K+1. The set S0represents the system state during the integrate (or diffusion) phase - that is, between firing (pulse-coupling) events. The sets S+ and S− represent the state of the system during a cascade phase originating from either the positive pulse-coupling firing state (S+), or the negative pulse-coupling firing state (S−). Throughout this section, all vectors and matrices are indexed with component labels ranging from 0 to 2K.
The basis of the mean field model, is the vector of expected state occupation, which encodes the macroscopic state of the system. Let xs(t) ≥ 0 be the expected number of oscillators in state s at time t, then xxx(t) is given by
xxx(t) = (x0(t), . . . , x2K(t)) ∈ R2K+1+ . (B.4)
Our aim is to use the MF system to solve for the vector xxx(t), in specific cases. For instance, equation (4.5) shows the solution for xxx(t) when the MF system produces singleton firings that alternating indefinitely between the upper and lower boundaries.
Although the mean field model is deterministic, the dynamics still occur in two phases:
a continuous-time diffusion phase and instantaneous cascade phase. Since the diffusion of each oscillator state evolves according to equation (B.1), during the diffusion phase xs(t) evolves according to
dx0(t)
with all other entries zero. Recall that the diffusion phase ceases as soon as an oscillator transitions to either of the firing states. For the non-normalised mean field system, this condition is encoded as x0(t) ≥ 1 or x2K(t) ≥ 1, or equivalently as
hxxx(t), vvv0i ≥ 1 (negative pulse condition), hxxx(t), vvv2Ki ≥ 1 (positive pulse condition),
(B.7)
We say the equations hxxx(t), vvv0i = 1 and hxxx(t), vvv2Ki = 1 define discontinuity boundaries Casini and Vestroni [2004]; di Bernardo et al. [2001], in the context of piecewise-smooth dynamical systems, of which the mean field model is a simple example. As
soon as one of the conditions in equation (B.7) is satisfied, the cascade phase begins, with the appropriate pulse-coupling.
The action of a single oscillator firing is encoded using the pulse-coupling matrix, LC, and the map Fpgiven by
Fp(xxx(t)) = (I + pLC)xxx(t) − vvv, (B.8)
where the term −vvv removes the firing oscillator from the system, after it has fired, to satisfy the requirement that it enters a refractory state. The matrix LC describes the effect of pulse-coupling on the remaining oscillators in the system, and can take one of two values. For an initial positive pulse LC = LC,+ and vvv = vvv2K are used, while for an initial negative pulse LC= LC,− and vvv = vvv0, where
with all other entries zero, for both matrices. The cascade, refractory and resetting pro-cesses continue in the same way as for the original model DeVille et al. [2010]; DeVille and Peskin [2008], with the exception that oscillators reset to state K after firing In order to correctly encode the cascade procedure involving multiple oscillators, the map given by equation (B.8) must be applied to the state vector xxx(t) each time an os-cillator fires. To do this, we use functional composition defined as follows: for an arbitrary function f , and arbitrary integer a, the a-fold composition is denoted via an exponent
appro-priate values of LC and vvv, and S0 defined by equation (B.3). Finally, the m oscillators that fired during the cascade, and subsequently removed from the system, are added back in and reset to level K. Hence, we can define a map
φ : S+∪ S−→ S0 (B.11)
φ (xxx(t)) = Fpm(xxx(t))+ m(xxx(t))vvvK,
where S+, S− are defined by equations (B.3).
Using the above definitions, we can state the dynamics of the mean field system as
˙xxx(t) = LDxxx(t) for xxx(t) ∈ S0, xxx(t) 7→ φ (xxx(t)) for xxx(t) ∈ S+∪ S−.
(B.12)
B.1 Construction of the map G
0In the asynchronous state, isolated cascades (of size 1) occur in an alternating pattern originating from the two firing states 0 and 2K. Therefore, we construct a map that takes the system state vector initially in set S0 (defined by Eqn. (B.3)) and describes the system undergoing an isolated (size 1) cascade originating from state 2K (when the system state vector is in set S+ defined by Eqn. (B.3)), followed by a second diffusion and an isolated cascade originating from state 0 (when the system state vector is in set S−defined by Eqn. (B.3)).
The physical actions in detail are as follows:
1. Initially the system has normalised state vector:
xxx∈ S0
2. The system diffuses while in set S0 (under the relevant action given by Eqn (B.12)) for time τ1 at which point a the state vector is now in the set S+. The state vector is now:
eτ1LDxxx∈ S+.
3. The system undergoes an isolated cascade given by φ in Eqn. (B.11). Because
the cascade is of size 1, φ is given by Fpdefined in Eqn. (B.8), with p, LD and vvv replaced with εKq, LC,+and εvvv2K respectively. The state vector is now:
(I + εKqLC,+)eτ1LDxxx− εvvv2K.
4. After this cascade, the 2K-th component of the state vector is reset (mapped) back to the K-th component of the state vector. Because we are considering an isolated (size 1) cascade, the 2K-th component is ε, and so we must add εvvvKback to the system state vector. The state vector is now:
(I + εKqLC,+)eτ1LDxxx− ε(vvv2K− vvvK) ∈ S0.
5. The system diffuses while in set S0 (under the relevant action given by Eqn (B.12)) for a time τ2 at which point a the state vector is now in the set S−. The state vector is now:
eτ2LD[(I + εKqLC,+)eτ1LDxxx− ε(vvv2K− vvvK)] ∈ S−.
6. The system undergoes an isolated cascade given by φ in Eqn. (B.11). Because the cascade is of size 1, φ is given by Fpdefined in Eqn. (B.8), with p, LD and vvv replaced with εKq, LC,−and εvvv0respectively. The state vector is now:
(I + εKqLC,−)eτ2LD[(I + εKqLC,+)eτ1LDxxx− ε(vvv2K− vvvK)] − εvvv0.
7. After this cascade, the 0-th component of the state vector is reset (mapped) back to the K-th component of the state vector. Because we are considering an isolated cascade (of size 1), the 0-th component is ε and therefore add εvvvK back to the system state vector. The state vector is now:
(I + εKqLC,−)eτ2LD[(I + εKqLC,+)eτ1LDxxx− ε(vvv2K− vvvK)] − ε(vvv0− vvvK) ∈ S0.
The map G0: S0→ S0is defined as
G0(xxx) = (B.13)
(I + εKqLC,−)eτ2LD[(I + εKqLC,+)eτ1LDxxx− ε(vvv2K− vvvK)] − ε(vvv0− vvvK).
When computing the solution, up to O(ε), of the fixed point equation G0(xxx) = xxx, it is
noted the times τ1and τ2are of order ε, and linearise the exponential matrix as
eτ LD≈ I + τLD. (B.14)
Furthermore, the matrix multiplications are performed while keeping careful track of simplifications arising from the kernel of the matrices LD, LC,+and LC,−.