Structural changes in inflation dynamics: multiple breaks at different dates for different parameters

2.1 Model specification

I consider a pth-order autoregressive model for an analysis of inflation dynamics and the AR(p) model allows for structural breaks in three parameter groups such as (i) the unconditional mean μ, (ii) persistence (ϕ₁, …, ϕ_t) and (iii) the residual variance σ². I focus on changes in the unconditional mean rather than the intercept because changes in the intercept may contain mixed information on changes in level and persistence. Sims (2001) and Stock (2001) note that allowing for changes in the residual variance helps avoid any potential distortions of identifying structural breaks in coefficients.^[6] The AR(p) model with structural breaks is given by

where S_1,t∈{1, …, M₁+1}, S_2,t∈{1, …, M₂+1}, and S_3,t∈{1, …, M₃+1} represent the regimes for the unconditional mean parameter with M₁ breaks, the persistence parameter group with M₂ breaks and the residual variance with M₃ breaks, respectively. The nature of the structural breaks in each parameter group is independent of one another in terms of the number and timing of structural changes.

To make inferences about multiple-group changepoint models, I modify Chib’s (1998) approach in which the structural breaks are interpreted as regime transitions. Chib (1998) assumes that all the parameters which undergo the structural changes have the structural shifts at the same dates.^[7] Thus, I propose an efficient Bayesian MCMC method that allows for a number of possibilities for the nature of structural breaks. This modified approach is developed to have the following attractive features: (i) model specification of considering multiple structural changes in multiple parameters; (ii) model flexibility in allowing the multiple structural breaks to occur mutually independently at different dates; and (iii) model selection procedure by comparing various potentially non-nested structural break models.

Suppose a multiple-group changepoint model which allows parameters to change at different dates with the different number of breaks.^[8] For example, in (1) the model for inflation dynamics may have one break in the unconditional mean, two breaks in persistence, and three breaks in the residual variance. In this case, the modified approach would require only augmentations with three independent latent regime indicator variables (S˜1,T, S˜2,T, S˜3,T), where S˜i,T=[Si,1 Si,2 … Si,T−1 Si,T]′ for each parameter group i=1, 2, 3, and three transition probability matrices (P₁, P₂, P₃) corresponding to three parameter groups while the single-group changepoint model is augmented with only one regime indicator variable and one transition probability matrix. Since all the regime indicator variables are mutually independent in the multiple-group changepoint model, the date of regime transition in a parameter is allowed to occur close to that of regime transition in other parameters without any necessary minimum distance unlike the restriction in Levin and Piger (2007).

For a parameter group i, the latent state variable S_t follows a first-order Markov process with the transition probabilities constrained: for k=1, …, M_t

and

where pk,ki indicates the probability that a regime S_t for the parameter group i stays in the current regime k. A transition probability matrixP_t for the parameter group i can then be formed as a (M_t+1)-by-(M_t+1) matrix with elements containing information about the first-order Markov process in (2) and (3).^[9]

A joint posterior density can be obtained as being proportional to a product of a prior density and a likelihood function of Y_t=[y₁ … y_t]′ such as

π(θ, P|YT)∝π(θ, P)f(YT|θ, P)

where π(·) denotes a density function; θ=(μ˜, ϕ˜, σ˜2) is a collection of model parameters; μ˜=(μ1, …, μM1+1), ϕ˜=(ϕ1,1, …, ϕp,1, …, ϕ1,M2+1, …, ϕp,M2+1), and σ˜2=(σ12, …, σM3+12); and P=(P₁, P₂, P₃) is a collection of transition probability matrices. The model parameters, θ=(μ˜, ϕ˜, σ˜2), are then augmented with the transition probability matrices, P=(P₁, P₂, P₃), and the latent regime indicators (S˜1,T, S˜2,T, S˜3,T). The MCMC sampler is developed through a hierarchical specification in which one draws the model parameters conditional on the regime indicators and the observed data; the regime indicators conditional on the model parameters and the observed data; and finally the transition probabilities conditional on the regime indicators via Gibbs sampling. More details for the MCMC sampling algorithm are explained in the Appendix.

2.2 Model selection procedures

I consider two different model selection procedures: (i) the Bayes factor comparison using the marginal likelihoods and (ii) the posterior probability for the number of structural breaks in each individual parameter group by integration. Thus, I find not only the most preferred model based on the Bayes factor but also the marginal probability for the number of breaks in each parameter group for robustness to the model selection results. In this analysis, the maximum number of structural breaks is specified to four for each parameter group and in total, 125 different models for each inflation measure are considered including a model with no break (125=5³).

The Bayes factor is presented in favor of the alternative model, M=(M₁, M₂, M₃) versus the most preferred model, ℳ∗=(M1∗, M2∗, M3∗) by

BF=m(YT|ℳ=(M1, M2, M3))m(YT|ℳ∗=(M1∗, M2∗, M3∗))

where m(Y_t|ℳ=(M₁, M₂, M₃)) is the marginal likelihood for the model with structural breaks of (M₁, M₂, M₃). I calculate the marginal likelihood at the posterior mean. The algorithm to compute the marginal likelihood is described in the Appendix.

I also calculate the posterior probability for the number of structural breaks in each individual parameter group by integrating out the number of breaks in other parameters. For example, the posterior probability for l structural breaks in the parameter μ˜, denoted by M₁=l, is given by

where

m(YT|M1=l)=∑j=04∑k=04m(YT|M1=l, M2=j, M3=k)π(M2=j, M3=k|M1=l)

is the integrated likelihood when M₁=l; π(M₁=l) is the prior probability for M₁=l; and π(M₂=j, M₃=k|M₁=l) is the joint prior probability for M₂=j and M₃=k conditional on M₁=l. Because all the models are considered a priori equally likely as well as independent, π(M₁=l) is equal to 1/5 and π(M₂=j, M₃=k|M₁=l) is equal to 1/25 when the maximum number of structural breaks is specified to four as in this analysis. In fact, the posterior probability of l structural breaks in the parameter μ˜ is simply the sum of posterior probabilities for all the models such that M₁=l. For other parameters, ϕ˜ and σ˜2, Pr(M2=j|YT) and Pr(M₃=k/Y_t) can be easily obtained by using the same approach in (5). Note that all the terms in (5) are readily available when the marginal likelihood calculations are completed.

3.1 Data and prior

I consider two different quarterly US inflation measures based on the consumer price index (CPI) and the GDP deflator.^[10] Each inflation is defined as 100 times the log change in the corresponding price index for the period of 1953:Q1–2013:Q4.

Figure 1 depicts the two inflation series. Both inflation measures appear to be less volatile for the “Great Moderation” period since the early 1980s but CPI inflation has been recently more volatile. Moreover, CPI inflation appears to be less persistent than GDP deflator inflation. Table 1 presents summary statistics over the sample period. CPI inflation has a slightly higher mean, is less persistent, and more volatile than for GDP deflator inflation. In addition, the correlation between the two inflation measures is 0.78. Thus, the timing and the number of structural changes might be different across model parameters such as the unconditional mean, persistence, and the residual variance as well as across the different measures of inflation. In this case, the multiple-group changepoint model can effectively detect the structural changes in individual parameter groups.

Figure 1:

US inflation rates: CPI and GDP deflator (quarterly percentage change): 1953:Q1–2013:Q4.

In this analysis, the lag order p is set to two for both series by the Bayesian information criterion. The diffuse and same prior distributions across different regimes are chosen in order to avoid any distortions from the choice of specific prior distributions when estimating different structural break models. The priors for regression coefficients are distributed with mean zero and variance one (i.e., μ, ϕ_t~N(0, 1)) and the priors of variance parameters follow an Inverse Gamma distribution such as σ2∼IG(5.02, 1.52) for CPI inflation and σ2∼IG(4.22, 0.52) for GDP deflator inflation. The priors for the variance parameters are set differently reflecting the fact that GDP deflator inflation is more persistent and less volatile than CPI inflation so that the residual variance for the GDP inflation regression is much smaller than for the CPI inflation regression. See the summary statistics in Table 1. The prior of the transition probability that the current regime k stays in the same regime k in the next period is distributed as p_t~Beta(10, 0.1). The prior expected duration of a given regime is about 101 quarters and the prior expected number of breaks is 2.4 given the sample size of 241. All the estimations are based on 3000 Gibbs simulations after discarding 3000 burn-ins.^[11]

3.2 Empirical findings for CPI inflation

I first discuss model selection results for CPI inflation. Table 2 presents the ten best models based on marginal likelihood calculations among 125 models. The comparison of the marginal likelihoods shows that the most preferred model has one structural break in persistence, three structural breaks in the residual variance, and no structural break in the unconditional mean. The most preferred model clearly dominates the other models in the sense that the Bayes factor is lower than 1/8 in favor of any alternative model.

Table 3 lists the marginal posterior probability of the number of structural breaks in individual parameter groups using (5). The highest probability of the number of breaks for each parameter is calculated as follows: 0.89 for no break in the unconditional mean, 0.97 for one break in persistence, and 0.91 for three breaks in the residual variance. This finding from the marginal probability calculations is completely consistent with the model selection of ℳ∗=(M1∗=0, M2∗=1, M3∗=3) based on the Bayes factors.

Table 4 summarizes the posterior distributions with mean and standard deviation and Figure 2 plots the posterior mean and 90% credible band for each parameter group over the sample period. The sum of autoregressive coefficients is used for the measure of persistence. The posterior distributions clearly show that persistence sharply decreased from 0.89 to 0.10 (posterior mean) in the early 1980s while the residual variance switched from a low volatility regime to a high volatility regime around the early 1970s, returned to another low volatility regime around the early 1990s, and then has increased again since the early 2000s.^[12] The changes in the residual variance also appear to be abrupt. It is evident that the break point for persistence is different from the break points for the residual variance. Thus, these structural changes could be caused by different sources.

Figure 2:

US CPI inflation: posterior distribution of parameters over time.

(A) Unconditional mean, (B) persistence (sum of AR parameters), (C) residual variance.

Posterior mean and 90% band are plotted over time. The estimated break dates (posterior mode) are 1981:Q3 for persistence and 1972:Q4, 1991:Q1, and 2001:Q2 for the residual variance.

3.3 Empirical findings for GDP deflator inflation

Now, I apply the modified approach to an AR(2) model for GDP deflator inflation. Table 5 presents calculations for the marginal likelihood and Bayes factor. The marginal likelihood calculations select the model with two breaks in the residual variance only. The Bayes factor between ℳ^*=(0, 0, 2) and ℳ=(0, 0, 1), the second preferred model, is 0.61. Table 6 shows the marginal probability for the number of breaks in each parameter group. The posterior probability for two breaks in the residual variance (σ²) is 0.60 and for one break is 0.37 while the posterior probabilities for no break in the unconditional mean (μ) and in persistence (ϕ) are 0.99 and 0.95, respectively. Thus, not only the marginal likelihood comparisons but also the posterior probability calculations produce very strong evidence that the autoregressive model for GDP deflator inflation has no break in the unconditional mean and persistence but the specification for two breaks in the residual variance is slightly more preferred than for one break.^[13] This result is also consistent with the finding that evidence for shifts in persistence for GDP deflator inflation is not statistically significant, particularly once allowing for shifts in the residual variance as in Pivetta and Reis (2007) and Stock (2002).

Table 7 summarizes posterior distributions for the parameters in the most preferred model and Figure 3 plots posterior mean and 90% credible band for each parameter group. The residual variance switched from a low volatility regime to a high volatility regime around the early 1970s and then returned to another low volatility regime around the early 1980s. In addition, the residual variance for the first regime appears to be bigger than that for the third regime. The changes in the residual variance are also abrupt and this implies that the multiple-group changepoint model provides precise information about the timing of the structural breaks in the residual variance. Note also that GDP deflator inflation is highly persistent in the sense that the sum of the autoregressive coefficients (posterior mean) is 0.88. This finding on high inflation persistence is consistent with that in the literature (e.g. Fuhrer and Moore (1995)). However, the 90% credible band does not cover the unit root.^[14]

Figure 3:

GDP deflator inflation: posterior distribution of parameters over time.

(A) Unconditional mean, (B) persistence (Sum of AR parameters), (C) residual variance.

Posterior mean and 90% band are plotted over time. The estimated break dates (posterior mode) are 1970:Q2 and 1981:Q2 for the residual variance.

4.1 Comparison with Chib’s (1998) approach

For robustness, I consider Chib’s approach, a single-group changepoint model, in which all the parameters including the residual variance undergo breaks at the same time. The models are estimated using the same priors as in the multiple-group changepoint models. The results for the Bayes factor calculations are summarized for both measures of inflation in Table 8. Based on Chib’s approach, the values of the highest (log) marginal likelihood are equal to –208.54 (three breaks) for CPI inflation and –31.43 (one break) for GDP deflator inflation, respectively. However, these marginal likelihood values are significantly lower than those based on the multiple-group changepoint approach.

I also consider another modified approach in which the unconditional mean and persistence parameters (i.e. conditional mean) undergo co-breaks at the same time, which are independent of the breaks in the residual variance. I estimate the model with one break in the conditional mean and three breaks in the residual variance in accordance with the empirical finding in the previous section. The log marginal likelihood for this modified model is calculated to be –204.81, which is lower than that for the most preferred model (–203.55) based on the multiple-group changepoint approach. The Bayes factor is then equal to 0.28 (alternatively 3.53 in favor of the most preferred model, ℳ^*=(0, 1, 3)), which shows substantial evidence for one break in persistence and three breaks in the residual variance when adopting Jeffreys’s (1961) interpretation.^[15] Note that the most preferred model for GDP deflator inflation has no break in the unconditional mean and persistence. Therefore it is not required to estimate the modified model additionally.

4.2 Unobserved components models with stochastic volatility

The empirical results reported in Section 3 show that there have been large changes in the variance, but two measures of inflation exhibit the different evolution of the residual variance in terms of the number and timing of the breaks. Following Stock and Watson (2007), I consider an unobserved components model with stochastic volatility (UC-SV) and further examine whether these empirical findings on the shifts in the residual variance are robust.^[16]

Stock and Watson (2007) estimate an UC-SV model for GDP deflator inflation from 1953:Q1 to 2004:Q4. They find that there have been substantial movements over time in the standard deviation of the permanent component while there is little change in the standard deviation of the transitory innovation. I examine CPI inflation in addition to GDP deflator inflation using the longer sample period of 1953:Q1–2013:Q4, which is used in Section 3, in comparison to Stock and Watson (2007). The UC-SV model specification is exactly the same as in Stock and Watson (2007):

πt=τt+ηt, where ηt=ση,tξη,tτt=τt−1+ϵt, where ϵt=σϵ,tξϵ,tlnση,t2=lnση,t−12+vη,tlnσϵ,t2=lnσϵ,t−12+vϵ,t

where ξ_t=(ξ_t, ξ_ε,t)′ is i.i.d. N(0, I₂); v_t=(v_η,t, v_ε,t)′ is i.i.d. N(0, γI₂); ξ_t and v_t are independently distributed; and γ determines the smoothness of the stochastic volatility process. The scalar parameter γ is set to 0.2 as in Stock and Watson (2007) and the posterior distributions of stochastic volatility are estimated by MCMC. Figures 4 and 5 plot the smoothed estimates (posterior mean and 67% band) of σϵ,t2 and ση,t2 from the UC-SV models for GDP deflator inflation and CPI inflation, respectively. The empirical results for GDP deflator inflation are similar to Stock and Watson’s (2007) results. The standard deviation of the permanent disturbance was moderate from the mid 1950s through the early 1970s, it was large during the 1970s through the mid 1980s, and it declined sharply in the mid-1980s, whereas the volatility of the transitory disturbance appears to have remained stable. This low-high-low volatility pattern on the permanent innovations is consistent with the regime changes in the residual variance based on the multiple-group changepoint model in Section 3. Note also that the changepoint model analysis finds that GDP inflation has been highly persistent for the whole sample period and this high persistence for GDP deflator inflation implies that the evolution of inflation volatility would be mostly attributed to the permanent innovations. This conjecture is confirmed by the substantial changes in the standard deviations of the permanent innovations and the relatively stable volatility of the transitory innovations. In contrast to GDP deflator inflation, the volatility of the transitory innovations for CPI inflation shows substantial movements while the permanent innovations for CPI inflation exhibit a similar pattern of the low-high-low volatility shifts for GDP deflator inflation. The high volatility of the transitory innovations appears to contribute to the large fluctuations since the early 2000s because the volatility of the permanent innovations is moderate during this period. This empirical finding is consistent with the result based on the multi-group changepoint model in the sense that CPI inflation experienced a dramatic drop in persistence around the early 1980s and the permanent innovations would not play an important role in generating high levels of CPI inflation volatility since the early 2000s.

Figure 4:

CPI inflation: stochastic volatility in UC-SV model.

(A) Permanent innovations, (B) transitory innovations.

Posterior mean and 67% band are plotted over time.

Figure 5:

GDP deflator inflation: stochastic volatility in UC-SV model.

(A) Permanent innovations, (B) transitory innovations.

Posterior mean and 67% band are plotted over time.