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A non-linear Keynesian Goodwin-type endogenous model of the cycle: Bayesian evidence for the USA

Theodore Mariolis, Konstantinos N. Konstantakis, Panayotis G. Michaelides and Efthymios G. Tsionas

Abstract

This paper incorporates the so-called Bhaduri-Marglin accumulation function in Goodwin’s original growth cycle model and econometrically estimates the proposed model for the case of the US economy in the time period 1960–2012, using a modern Bayesian sequential Monte Carlo method. Based on our findings, the US economy follows an exhilarationist regime throughout our investigation period with the sole exception of an underconsumption regime for the time period 1974–1978. In general, the results suggest that the proposed approach is an appropriate vehicle for expanding and improving traditional Goodwin-type models.

JEL Classification: B51; C11; C62; E32

Technical Appendix

Sequential Monte Carlo

Chopin (2002) proposed a sequential PF for static models. Given a target posterior p(θ | Y): = p(θ | Y1:T), a particle system is a sequence {θj, wj} such that E(h(θ)|Y):=h(θ)p(θ|Y)dθlimJj=1Jwjh(θj)j=1Jwj, almost surely, for any measurable function h, provided the expectation exists. We consider the sequence of posterior distributions pt: = p(θ | Yt). The PF algorithm is as follows:

  1. Reweight: update the weights wjwjpt+1(θj)pt(θj),j=1,...,J.

  2. Resampling: resample {θj,wjJj=1H{θjr,1}j=1J}

  3. Move: draw θjmKt+1(θjr),j=1,...,J, where Kt+1 is any transition kernel whose stationary distribution is pt+1.

  4. Loop: tt+1,{θj,wj}j=1J{θjm,1}j=1J and return to Step 1.

Chopin (2002) recommends the independence Metropolis algorithm to select the kernel, which requires a source distribution. A possible choice, if we sampled from pn (n < T), with respect to pn+s is N(E^n+s,V^n+s) where

E^n+s=j=1Jwjθjj=1Jwj,V^n+s=j=1Jwj(θjEn+p)(θjEn+p)j=1Jwj.

The strategy can be parallelized easily. If K processors are available, we can partition the particle system into K subsets, say Sk, k = 1, …, K), and implement computations for particles of Sk in processor k. The algorithm can deal with new data at a nearly geometric rate and, therefore, the frequency of exchanging information between processors (after reweighting) decreases at a rate exponential to n, which is highly efficient.

Resampling, according to θjmKt(θjr,.), reduces particle degeneracy (Gilks and Berzuini 2001) since identical replicates of a single particle are replaced by new ones, without altering the stationary distribution. For this application, using J = 212 particles gave a mean squared error in posterior means of 10–5 over 100 runs.

Chopin (2004) introduces a variation of MSC in which the observation dates at which each cycle terminates (say t1, …, tL) and the parameters involved in specifying the Metropolis updates (say λ1, …, λL) are specified. Therefore, 0 = t0 < t1 < … < tL = T and we have the following scheme (we rely heavily on Durham and Geweke 2014).

  1. Initialize l = 0 and θjn(l)p(θ),jJ, nN.

  2. For l = 1, …, L:

    1. Correction phase:

      1. wjn(tl–1) = 1, jJ, nN

      2. For s = tl–1 + 1, …, tl

        wjn(s)=wjn(s1)p(ys|y1:s1,θjn(l1)),jJ,nN.
      3. wjn(l1):=wjn(tl),jJ,nN.

    2. Selection phase, applied independently to each group jJ : Using multinomial or residual sampling based on {wjn(l),nN}, select

      {θjn(l,0),nN}

      from {θjn(l1),nN}.

    3. Mutation phase, applied independently across jJ, nN :

      θjn(l)p(θ|y1:t,θjn(0),λl)

      where the drawings are independent and the pdf above satisfies the invariance condition:

      Θp(θ|y1:tl,θ,λl)p(θ|y1:tl)dν(θ)=p(θ|y1:tl).
  3. θjn:=θjn(l),jJ,nN.

At the end of every cycle, the particles θjn(l) have the same distribution p(θ|y1:tl). The amount of dependence, within each group, depends upon the success of the Mutation phase, which avoids degeneracy.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2016-0137).


Published Online: 2018-07-03

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