Bayesian evidence on the structure of unemployment

Economics Letters 83 (2004) 299 – 306 www.elsevier.com/locate/econbase

Bayesian evidence on the structure of unemployment Peter M. Summers * Faculty of Economics and Commerce, Melbourne Institute, The University of Melbourne, Melbourne, Victoria 3010, Australia Received 13 May 2003; accepted 30 September 2003

Abstract This paper applies Bayesian techniques for detecting multiple structural breaks in the level of a time series to a data set recently used by Papell et al. (Rev. Econ. Stat. (2000) 309). I find virtually no support for unit roots in OECD unemployment rates. D 2004 Elsevier B.V. All rights reserved. Keywords: Multiple structural breaks; Bayesian analysis; Unit root; Unemployment rate JEL classification: C220; C110; E240

1. Introduction In a recent paper, Papell et al. (2000) (PMG) examine whether the behavior of unemployment rates in the OECD is best described by so-called ‘structuralist’ theories, which imply that the unemployment rate is stationary about the natural rate, or by theories incorporating a unit root form of hysteresis. PMG apply the unit root tests of Perron and Vogelsang (1992) and Bai and Perron (1998), which allow for the presence of structural breaks under the unit root null, to the unemployment rates of 16 OECD countries. The possible presence of structural breaks is crucial here; using standard ADF tests, PMG cannot reject a unit root in any country in their sample. Allowing for structural breaks, the unit root null is rejected in 10 countries. This paper re-examines PMG’s results from a Bayesian perspective. As is well-known, Bayesian and frequentist methods can produce dramatically divergent inference about unit roots in time series models. Sims and Uhlig (1991), Uhlig (1994) and Bauwens et al. (1999) provide thorough discussions of how and why this divergence occurs. Bayesian techniques for detecting multiple structural breaks in the level, * Tel.: +61-3-8344-5313; fax: +61-3-8344-5630. E-mail address: [email protected] (P.M. Summers). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.09.032

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trend and variance of time series have recently been developed by Wang and Zivot (2000). I apply these tests to the data used by PMG, and find virtually no support for unit root hysteresis in OECD unemployment rates.

2. Bayesian inference with multiple breaks The general form of the model is AðLÞut ¼ l þ et ;

ð1Þ

where ut is the unemployment rate, A(L) = 1  a1L . . .  aqLq, and etfN(0,r2). I chose q by a procedure similar to that of PMG, starting from a maximum of six lags. Wang and Zivot (2000) developed Bayesian methods for analyzing time series with an unknown number of breaks in the mean, trend, and/or variance. In this paper, I only consider models with breaks in the intercept, l. See Wang and Zivot (2000) for extensions and details of their Gibbs sampling algorithm. Consider a (first order) version of Eq. (1) with n structural breaks in the intercept: ut ¼ l1 þ l2 þ : : : þ lnþ1 þ qut1 þ et :

ð2Þ

Defining I(A) to be the indicator function that takes the value 1 if event A is true, Wang and Zivot write Eq. (2) as a linear regression with n + 1 dummy variables: ut ¼ l1 Ið1Vt
ð3Þ

with kj being the time index of the jth break and T the last observation. Conditional on n, Bayesian inference is straightforward using natural conjugate priors on the lj and r2. 2.1. Priors The prior distributions are centred on a random walk with no drift: lf j N ð0; 1000Þbj ¼ 1; . . . ; n þ 1 af i N ð0; 1000Þbi ¼ 1; . . . ; q qf N ð1; 1000Þ r2f IGð3; 1Þ: The inverted gamma (IG) prior for r2 has a mean of 1, while the average variance of a random walk model fitted to each of the 16 countries is 0.96. Proper priors are necessary to ensure that Bayes factors and posterior odds ratios are well-defined. However, all of these priors are quite diffuse.

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Fig. 1. Prior distributions.

As pointed out by Lubrano (1995), Bauwens et al. (1999) and Uhlig (1994), specification of the prior on q requires particular care, especially regarding the treatment of explosive values. The Normal prior specified above may seem unsuitable here, as it puts more weight on the explosive region (AqA>1) than the stationary region. Nevertheless, Uhlig (1994) argues that a Normal prior centred on a random walk is ‘reasonable’ if there is no time trend in the model (as is the case here). I use this prior as a convenient base case, but also conduct inference using four other priors that are arguably more realistic. The first of these simply tightens pffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffi the base prior variance from 1000 to 0.5. The second is a b prior on the range  1 þ vVqV 1 þ v (see Lubrano, 1995 or Bauwens et al., 1999). I set v = 0.5, implying qa[  1.225, 1.225] and favoring the non-stationary region by about two-to-one. The fourth prior is a b(10,2) prior on the same range as Lubrano’s prior. This prior has a mean of 0.82, standard deviation 0.25 and a mode at 0.98. This prior favors roots near unity, but downweights explosive roots much more than Lubrano’s prior. The final prior is that of Berger and Yang (1994), extended to the non-stationary region:

f ðqÞ ¼

8 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 < ð2p 1  q2 Þ ; :

AqA < 1

pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2pAqA q2  1Þ1 ; AqAz1:

See Bauwens et al. (1999), for more details. Fig. 1 graphs the priors on the interval qa[0.5, 1.2] (the base prior is indistinguishable from the x-axis). I investigate the role of these prior distributions on unit root inference by using the reweighting method of Geweke (1999). Given a sequence of K draws of q, {qi}iK= 1, from the posterior fN (qjy) implied by the base prior fN (q), draws from the posterior fj (qjy) implied by a different prior fj (q) are obtained by multiplying the qi by the ratio of the two prior densities:  n oK i i fj ðqÞ ¼ q qj : i¼1 fN ðqÞ

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2.2. Inference on the number of breaks The results in Wang and Zivot (2000) are conditional on the number of break points, n. Those authors use the Bayesian Information Criterion (BIC) to choose among models with different values of n. Here, I draw on recent results of Carlin and Chib (1995), Han and Carlin (2001), Godsill (2001), Dellaportas et al. (2002) and Lopes and West (2000), and treat n as an additional parameter to be estimated. These papers describe Markov Chain Monte Carlo (MCMC) methods for conducting inference in situations involving competing models of varying dimension. My approach in this paper is closest to that of Lopes and West (2000). The algorithm proceeds as follows: 1. Specify an initial number of breaks n0a[0, nmax], where nmax is the maximum number of breaks under consideration. 2. Draw a parameter vector h0=(k1, . . ., kn0, l1, . . ., ln0 + 1, q, r2) corresponding to the n0-break model. 3. Propose a new number of breaks n1, with the probability of moving from a model with n0 breaks to one with n1 breaks denoted by J(n1,n0). 4. Draw the corresponding parameter vector h1 from a proposal distribution q(h1). 5. Accept the proposed move with probability  l1 ðyAn1 ; h1 Þ pðh1 An1 Þpðn1 Þ qðh0 Þ J ðn1 ; n0 Þ    a ¼ min 1; l0 ðyAn0 ; h0 Þ pðh0 An0 Þpðn0 Þ qðh1 Þ J ðn0 ; n1 Þ (this expression is discussed below). 6. If the move is accepted, set n0 = n1; otherwise, keep n0 unchanged, 7. Go to 2.

Table 1 Posterior model probabilities Number of breaks Australia Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands Norway Spain Sweden UK US

0

1

2

3

4

5

31.69 10.28 28.04 11.89 3.33 40.34 30.64 17.2 28.85 19.4 14.05 29.7 10.01 2.21 7.8 25.97

32.09 12.46 27.49 19.48 5.93 37.29 25.13 29 34.71 25.95 18.25 28.67 26.22 5.81 12.9 28.6

16.45 15.8 11.99 14.31 7.51 13.36 11.3 12.33 12.51 18.95 5.82 17.22 16.73 7.12 9.33 16.15

9.35 8.17 10.17 16.06 12.01 6.25 5.77 10.17 7.17 18.06 4.02 14.99 14.39 13.71 16.7 11.27

5.49 22.91 15.2 22.71 33.65 1.11 14.64 23.26 6.64 12.99 39.78 8.86 19.24 39.76 31.45 11.41

4.93 30.38 7.11 15.55 37.57 1.65 12.52 8.04 10.12 4.65 18.07 0.56 13.41 31.39 21.82 6.6

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Table 2 Quantiles and tail areas of posterior distributions of q, various priors Quantile

Prior

Prior

N(1, 1000) N(1, 0.5) Lubrano b(10,2) B – Y N(1, 1000) N(1, 0.5) Lubrano b(10,2) B – Y Mean S.D. Median 0.90 0.95 0.975 1

Australia 0.75 0.05 0.76 0.00 0.00 0.00 0.00

0.75 0.05 0.76 0.00 0.00 0.00 0.00

0.75 0.05 0.76 0.00 0.00 0.00 0.00

0.76 0.05 0.76 0.00 0.00 0.00 0.00

0.76 0.05 0.77 0.00 0.00 0.00 0.00

Finland 0.64 0.14 0.66 0.04 0.02 0.01 0.01

0.65 0.13 0.67 0.04 0.02 0.01 0.01

0.65 0.14 0.68 0.05 0.02 0.02 0.01

0.69 0.13 0.70 0.06 0.03 0.02 0.01

0.68 0.15 0.67 0.10 0.06 0.04 0.03

Mean S.D. Median 0.90 0.95 0.975 1

Belgium 0.67 0.04 0.68 0.00 0.00 0.00 0.00

0.67 0.04 0.68 0.00 0.00 0.00 0.00

0.67 0.04 0.68 0.00 0.00 0.00 0.00

0.68 0.04 0.68 0.00 0.00 0.00 0.00

0.67 0.04 0.68 0.00 0.00 0.00 0.00

France 0.98 0.03 0.98 0.99 0.76 0.48 0.24

0.98 0.03 0.98 0.99 0.76 0.48 0.24

0.98 0.03 0.99 0.99 0.77 0.50 0.26

0.98 0.03 0.98 0.99 0.76 0.48 0.24

0.99 0.03 0.98 1.00 0.87 0.67 0.32

Mean S.D. Median 0.90 0.95 0.975 1

Canada 0.78 0.05 0.78 0.01 0.00 0.00 0.00

0.78 0.05 0.79 0.01 0.00 0.00 0.00

0.78 0.05 0.79 0.01 0.00 0.00 0.00

0.78 0.05 0.79 0.01 0.00 0.00 0.00

0.78 0.05 0.79 0.01 0.00 0.00 0.00

Germany 0.80 0.05 0.79 0.01 0.00 0.00 0.00

0.80 0.05 0.79 0.01 0.00 0.00 0.00

0.80 0.05 0.79 0.02 0.00 0.00 0.00

0.80 0.05 0.79 0.02 0.00 0.00 0.00

0.80 0.05 0.79 0.02 0.00 0.00 0.00

Mean S.D. Median 0.90 0.95 0.975 1

Denmark 0.73 0.07 0.72 0.01 0.00 0.00 0.00

0.73 0.07 0.72 0.01 0.00 0.00 0.00

0.73 0.07 0.73 0.01 0.00 0.00 0.00

0.74 0.07 0.74 0.01 0.00 0.00 0.00

0.74 0.07 0.74 0.01 0.00 0.00 0.00

Ireland 0.76 0.05 0.77 0.00 0.00 0.00 0.00

0.76 0.05 0.77 0.00 0.00 0.00 0.00

0.76 0.05 0.78 0.00 0.00 0.00 0.00

0.76 0.05 0.78 0.00 0.00 0.00 0.00

0.76 0.05 0.78 0.01 0.00 0.00 0.00

Mean S.D. Median 0.90 0.95 0.975 1

Italy 0.90 0.04 0.89 0.50 0.12 0.04 0.01

0.90 0.04 0.90 0.51 0.13 0.04 0.01

0.90 0.04 0.90 0.52 0.14 0.04 0.01

0.90 0.04 0.90 0.52 0.13 0.04 0.01

0.91 0.04 0.90 0.60 0.21 0.09 0.03

Spain 0.87 0.04 0.90 0.25 0.06 0.03 0.01

0.87 0.04 0.90 0.25 0.06 0.03 0.01

0.88 0.04 0.90 0.27 0.07 0.03 0.01

0.88 0.04 0.90 0.26 0.07 0.03 0.01

0.89 0.05 0.90 0.35 0.14 0.08 0.04

(continued on next page)

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Table 2 (continued) Quantile

Prior

Prior

N(1, 1000) N(1, 0.5) Lubrano b(10,2) B – Y N(1, 1000) N(1, 0.5) Lubrano b(10,2) B – Y Mean S.D. Median 0.90 0.95 0.975 1

Japan 0.83 0.05 0.83 0.10 0.01 0.00 0.00

0.83 0.05 0.83 0.10 0.01 0.00 0.00

0.84 0.05 0.83 0.11 0.01 0.00 0.00

0.84 0.05 0.83 0.11 0.01 0.00 0.00

0.84 0.05 0.84 0.13 0.02 0.01 0.00

Sweden 0.59 0.16 0.55 0.02 0.01 0.00 0.00

0.61 0.15 0.57 0.03 0.01 0.00 0.00

0.61 0.16 0.56 0.03 0.01 0.00 0.00

0.67 0.14 0.60 0.04 0.01 0.00 0.00

0.62 0.16 0.57 0.04 0.02 0.01 0.00

Mean S.D. Median 0.90 0.95 0.975 1

Netherlands 0.73 0.08 0.75 0.03 0.00 0.00 0.00

0.74 0.08 0.75 0.03 0.00 0.00 0.00

0.74 0.08 0.75 0.04 0.00 0.00 0.00

0.75 0.08 0.76 0.04 0.00 0.00 0.00

0.75 0.08 0.76 0.05 0.01 0.00 0.00

UK 0.60 0.07 0.63 0.00 0.00 0.00 0.00

0.61 0.07 0.64 0.00 0.00 0.00 0.00

0.61 0.07 0.64 0.00 0.00 0.00 0.00

0.62 0.07 0.65 0.00 0.00 0.00 0.00

0.61 0.07 0.64 0.00 0.00 0.00 0.00

Mean S.D. Median 0.90 0.95 0.975 1

Norway 0.63 0.05 0.63 0.00 0.00 0.00 0.00

0.64 0.05 0.63 0.00 0.00 0.00 0.00

0.64 0.05 0.63 0.00 0.00 0.00 0.00

0.64 0.05 0.64 0.00 0.00 0.00 0.00

0.64 0.05 0.63 0.00 0.00 0.00 0.00

US 0.67 0.06 0.66 0.00 0.00 0.00 0.00

0.67 0.06 0.67 0.00 0.00 0.00 0.00

0.67 0.06 0.67 0.00 0.00 0.00 0.00

0.68 0.06 0.67 0.00 0.00 0.00 0.00

0.68 0.06 0.67 0.00 0.00 0.00 0.00

This is the ‘Metropolized Carlin and Chib’ (MCC) algorithm of Dellaportas et al. (2002). The first three terms in the acceptance probability a are: the ratio of the likelihood function values for the two models; the ratio of the prior distributions for the two parameter vectors; and the ratio of the two proposal densities. The last term gives the ratio of the transition probabilities for moves between models with n0 and n1 breaks. I assume that all models with na[0, 5] are equally likely a priori, so the p(nj) terms cancel (the choice of nmax = 5 follows that of Papell et al., 2000). I also use a uniform proposal density for J(., .), so the final term also cancels. For the proposal distribution q(hj), I use the full conditional posteriors given in Wang and Zivot (2000). I analyze the same data as in Papell et al. (2000); the annual average unemployment rates for 16 OECD countries. As a preliminary step, I estimated models with zero to five breaks in the intercept term. For each of these models, I ran the Gibbs sampler of Wang and Zivot (2000) for 12,500 draws, omitting the first 2500. I then ran the MCC algorithm for 11,000 iterations, keeping the last 10,000. The posterior probabilities of the six models can then be estimated by the frequency of each model’s occurrence in the latter 10,000 draws. Finally, I base inference on the autoregressive parameter q on the output of the preliminary runs, using the estimated posterior model probabilities as weights.

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3. Results Table 1 gives the estimated posterior probabilities of the six models for each country. These results differ substantially from those of Papell et al. (2000) (see their Table 3). They find one significant structural break in Belgium, Norway and Sweden; two breaks in Canada, Denmark, Finland, Ireland, the UK and the US; and three breaks in Spain. The Bai–Perron tests are not applicable to the other countries since the unit root null cannot be rejected. The model values in Table 1 correspond to their results only for Norway. In addition, Papell et al. (2000) prefer a two-break model for Australia, although the unit root null is not rejected. My results suggest that the no-break and one-break models are about equally likely (and each about twice as likely as the two-break model). Several features of the posterior distributions of q are shown in Table 2. In addition to the posterior mean, standard deviation and median, the table gives estimates of the tail areas of each distribution to the right of q = 0.90, 0.95, 0.975 and 1. The unemployment rates in 12 of the 16 countries are clearly stationary. There is clear evidence for a unit root only in France. There is weak evidence of nonstationarity in the Italian and Spanish unemployment rates, but only with the Berger–Yang prior. In all other cases, at least 94% of the posterior distribution of q lies to the left of 0.95. 4. Conclusions This paper strengthens the results of Papell et al. (2000) by adopting Bayesian tests that allow for both model and parameter uncertainty. The results provide strong evidence against the presence of unit roots in OECD unemployment rates. Acknowledgements I am very grateful to David H. Papell for supplying me with his data set, and to Jiahui Wang and Eric Zivot for sharing their Gauss code. Computations in this paper were carried out (in part) using the Bayesian Analysis, Computation and Communications (BACC) software available at http:// www.econ.umn.edu/~bacc, and described in Geweke (1999). Any errors are solely my responsibility. This research was supported by the Australian Research Council under grant C00002131. References Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica, 47 – 78. Bauwens, L., Lubrano, M., Richard, J.-F., 1999. Bayesian Inference in Dynamic Econometric Models. Oxford Univ. Press, Oxford. Berger, J.O., Yang, R.-Y., 1994. Non-informative priors and Bayesian testing for the AR(1) model. Econometric Theory, 461 – 482. Carlin, B.P., Chib, S., 1995. Bayesian model choice via Markov chain Monte Carlo. Journal of the Royal Statistical Society. Series B, 473 – 484. Dellaportas, P., Forster, J.J., Ntzoufras, I., 2002. On Bayesian model and variable selection using MCMC. Statistics and Computing, 27 – 36. Geweke, J., 1999. Using simulation methods for Bayesian econometric models: inference, development and communication. Econometric Reviews, 1 – 73.

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