in Trend University Gary Koop # 590

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1 A Bounded Model of Time Variation in Trend Inflation, NAIRU and the Phillips Curve Joshua C.C. Chan Research School of Economics, Australian National University Gary Koop University of Strathclyde Simon M. Potter Federal Reserve Bank of New York ANU Working Papers in Economics and Econometrics # 59 October 22 ISBN:

2 A Bounded Model of Time Variation in Trend In ation, NAIRU and the Phillips Curve Joshua C.C. Chan Australian National University Gary Koop University of Strathclyde Simon M. Potter Federal Reserve Bank of New York October 22 Abstract: In this paper, we develop a bivariate unobserved components model for in ation and unemployment. The unobserved components are trend in ation and the non-accelerating in ation rate of unemployment (NAIRU). Our model also incorporates a time-varying Phillips curve and time-varying in ation persistence. What sets this paper apart from the existing literature is that we do not use unbounded random walks for the unobserved components, but rather use bounded random walks. For instance, trend in ation is assumed to evolve within bounds. Our empirical work shows the importance of bounding. We nd that our bounded bivariate model forecasts better than many alternatives, including a version of our model with unbounded unobserved components. Our model also yields sensible estimates of trend in ation, NAIRU, in ation persistence and the slope of the Phillips curve. Keywords: trend in ation, non-linear state space model, natural rate of unemployment, in ation targeting, Bayesian The views expressed in this paper are those of the authors and do not necessarily re ect the views of the Federal Reserve Bank of New York or the Federal Reserve System. Gary Koop is a Fellow of the Rimini Center for Economic Analysis.

3 Introduction Parsimonious models of in ation and unemployment, inspired by the Phillips curve, have enjoyed great popularity for modeling latent states such as trend in ation or the non-accelerating in ation rate of unemployment (NAIRU) as well as for forecasting (see, among many others, Staiger, Stock and Watson, 997 and Stella and Stock, 2). In the in ation literature, univariate models such as the unobserved components stochastic volatility (UCSV) model of Stock and Watson (27) are commonly-used. It is often found (e.g. Atkeson and Ohanian, 2, Stella and Stock, 2 and many others) that simple univariate methods forecast in ation as well as more complicated multivariate models, at least on average. However, it is also noted that multivariate models, often based on the Phillips curve relationship, can forecast better at some points in time, even if on average they do not beat univariate methods (e.g. Dotsey, Fujita, and Stark, 2, Stock and Watson, 2 and Stella and Stock, 2 and many others). Such considerations motivate interest in bivariate models of in ation and unemployment, but not conventional linear constant coe cients models which assume the same Phillips curve relationship holds at each point in time. Instead the desire is for models where structures with economic interpretation such as the Phillips curve, trend in- ation and underlying in ation can change over time. See, for instance, Clark and Doh (2) which is a recent paper which surveys various approaches to modeling trend in ation. Related to this is the large literature on modeling or forecasting macroeconomic variables using models with time-varying coe cients (see, among many others, Cogley and Sargent, 25, Primiceri, 25, Sims and Zha, 26, Cogley and Sbordone, 28, Cogley, Primiceri and Sargent, 2). These considerations motivate interest in models with three characteristics. First, they are bivariate models of in ation and unemployment. Second, some coe cients can change over time. Third, they are written in terms of latent state vectors that can be given an economic interpretation. Such a model is developed in Stella and Stock (2), which is closely related to the model developed in this paper. This is a bivariate model with latent states which can be interpreted as a time-varying NAIRU and time-varying trend in ation. In addition, their model is based on a Phillips curve relationship but the slope of the Phillips curve can change over time. The most important way that our approach di ers from papers such as Stella and Stock (2) is in its treatment of the latent states. Following most 2

4 of the existing literature, Stella and Stock (2) model trend in ation and the NAIRU as driftless random walks. Modeling trend in ation as a random walk is a component of many macroeconomic models (e.g., among many others, Smets and Wouters, 23, Cogley and Sargent, 25, Ireland, 27, Stock and Watson, 27, Cogley and Sbordone, 28 and Cogley, Primiceri and Sargent, 2), despite the fact that there are many reasons for thinking that trend in ation should not wander in an unbounded random-walk fashion. For instance, the existence of explicit or implicit in ation targets by central banks means that trend or underlying in ation will be kept within bounds and not allowed to grow in an unbounded fashion. Similar issues apply for NAIRU. In previous work, Chan, Koop and Potter (22), we discussed these points in detail in the context of various theoretical models of trend in ation. We developed and presented evidence for a model for in ation which involved trend in ation following a bounded random walk. In this paper, we extend our earlier work to allow for trend in ation, NAIRU and the time-varying Phillips curve coe cients to all be bounded. Models such as the UCSV or the one in Stella and Stock (2) are Normal linear state space models (apart from the stochastic volatility present in the errors). Standard econometric methods (e.g. involving the Kalman lter) exist for these models. Since standard econometric methods also exist for the treatment of stochastic volatility (e.g. the algorithm of Kim, Shephard and Chib, 998), estimation of such models is theoretically straightforward. However, in practice, these models can be di cult to estimate without restrictions or strong prior information. For instance, Stella and Stock (2) note that their likelihood function is at in several dimensions and set seven parameters (error variances in measurement and state equations) to xed constants. Alternatively, papers such as Cogley and Sargent (25) and Primiceri (25) use very informative priors (calibrated using a training sample of data). The priors on the error covariance matrices in their state equations are of particular importance. The fact that our latent states are bounded mean our model is not a Normal linear state space model and, accordingly, conventional econometric methods cannot be used. Accordingly, we use an algorithm which is an extension of the ones developed in Chan and Jeliazkov (29), Chan and Strachan (22) and Chan, Koop and Potter (22). However, the use of bounds does lessen the need for restrictions or tight priors on parameters. In addition, our bounding restriction is easier to justify (e.g. as being based on in ation targeting behavior) than a conventional prior (e.g. an inverted- 3

5 Wishart prior on an error covariance matrix with particular choices for prior hyperparameters), much less a dogmatic prior which sets parameters to xed constants. After developing and justifying our model and describing relevant econometric methods, we present empirical work using US data on CPI in ation and the unemployment rate. We nd that our model forecasts better than many comparators, including an unbounded version of our model and an unrestricted VAR. Estimates of trend in ation, NAIRU, in ation persistence and the slope of the Phillips curve are found to be sensible. 2 A Bounded Trend Model for In ation and Unemployment We begin with a general bivariate model for in ation, t, and unemployment, u t of the form: ( t t ) = t t t + t (u t u t ) + " t (u t u t ) = u u t u t + u 2 u t 2 u t 2 + " u t t = t + " t u t = u t + " u t t = t + " t t = t + " t : () Both dependent variables are written as deviations from trends, t and u t. These trends are unobserved latent states which can be interpreted as trend (or underlying) in ation and the NAIRU. This model incorporates the properties that it is deviations of unemployment from NAIRU and deviations of in ation from its trend that drive the Phillips curve. These are features in common with the model of Stella and Stock (2) and, for the in ation equation, with Stock and Watson (27), Clark and Doh (2) and Chan, Koop and Potter (22). Thus, the rst equation embodies a Phillips curve. The coe cients in the Phillips curve equation are time-varying and evolve according to random walks as in, e.g., Cogley and Sargent (25), Primiceri (25) and Stella and Stock (2). Stella and Stock (2) emphasize that time variation in t is a useful extension of a conventional Phillips curve. If there are time periods when t = then the Phillips curve relationship does not exist. In general, a model where t varies over time allows for the 4

6 strength of the unemployment-in ation relation to vary over time, consistent with the episodic forecasting performance of the Phillips curve noted by Stella and Stock and others. A feature of () which is not present in Stella and Stock (2) is the time variation in t. This feature was incorporated in the univariate model of Chan, Koop and Potter (22) and was found to be empirically important, allowing for di erences in the way the Fed tolerates deviations of in ation from target. For instance, evidence in the historical study of Weise (2), suggests that the high in ation period of the 97s was not necessarily a time when the trend level of in ation increased dramatically (as would be implied by the version of our model with t = ), but was a time when deviations from the desired level of in ation were quite persistent. That is, the Fed either was temporarily more tolerant of higher-than-desired in ation or less able to quickly return in ation to the desired level. After the 97s, these characteristics were reversed. A model where t changes over time can model such features. Adding in the Fed s dual mandate also motivates the inclusion of unemployment and the Phillips curve relationship in our in ation equation. The second equation implies AR(2) behavior for the unemployment rate. The AR(2) assumption is empirically sensible and commonly-used. Note that we are assuming constant coe cients in the unemployment equation. Stella and Stock (2) also assume an AR(2) with constant coe cients for their unemployment equation. In our empirical work, we also consider a model where u and u 2 vary over time, but nd it to perform very poorly (see below). Accordingly, the main model we focus on does not have timevariation in the coe cients in the unemployment equation. We make the standard assumption that the errors in () (and the following equations) are independent with one another and at all leads and lags, but that the errors in the in ation equation exhibit stochastic volatility. Thus, " t N(; e ht ) (2) h t = h t + " h t ; " h t N(; 2 h): However, based on preliminary empirical work and following Stella and Stock (2) we assume " u t N(; 2 u). Thus far, we have speci ed a exible bivariate model for unemployment and in ation which is similar to many in the literature. In particular, it 5

7 incorporates the same features as Stella and Stock (2) and is a bivariate generalization of unobserved components models used in Stock and Watson (27), Clark and Doh (2) and many others. However, the conventional literature would next assume that all the errors in the state equations for t, u t ; t and t have normal distributions. It is with this assumption that we part with the existing literature. As discussed in Chan, Koop and Potter (22), unbounded random walk behavior for trend in ation is not plausible when there are in ation targets or anchored or constrained in ation expectations. Similarly, unbounded random walk behavior of the NAIRU is not consistent with much macroeconomic theory or the Fed s dual mandate. Thus, we model trend in ation and the NAIRU as bounded random walks. This can be done through the assumption that " t and " u t have truncated normal distributions. Formally, we assume " " u t T N(a t ; b t ; ; 2 ) t T N(a u u t ; b u u t ; ; 2 u) where T N(a; b; ; 2 ) denotes the normal distribution with mean and variance 2 truncated to the interval (a; b). This speci cation ensures that t lies in the interval (a ; b ) and u t lies in the interval (a u ; b u ). These bounds can either be set to particular values suggested by the underlying economics (e.g. if a central bank has an o cial in ation target or target interval) or estimated from the data. In our empirical work, we estimate the bounds a ; b ; a u and b u. Similarly, we bound the time-varying coe cients of the Phillips curve t and t. As mentioned in the introduction, in time-varying parameter models it is common to either use tight priors (as done, e.g., in Cogley and Sargent, 25 and Primiceri, 25) or to restrict parameters to particular values (as in Stella and Stock, 2) to surmount problems caused by at likelihoods in these parameter rich models. But the problems are often due to parameters wandering according to unbounded random walks that sometimes move into undesirable regions of the parameter space. We argue that this problem can be avoided by directly bounding the parameter spaces, thus avoiding the need for tight priors or xing parameters. Thus, we bound t so that this coe cient is constrained to the interval (; ) and never wanders into the explosive region of the parameter space. Similarly we bound t to ( ; ) to avoid non-stable behavior and ensure that the Phillips curve has a negative (3) 6

8 slope. To be precise, we assume " t T N( t ; t ; ; 2 ) " t T N( t ; t ; ; 2 ): (4) We also impose the stationary condition on the unemployment equation and assume u + u 2 <, u 2 u < and j u 2j < : In summary, we have a exible model which allows for the estimation of trend in ation, the NAIRU, in ation persistence and the slope of the Phillips curve. All of these can vary over time. However, inspired by the idea that it is not reasonable to assume trend in ation and the NAIRU follow unbounded random walks, our model constrains these to lie in intervals. Furthermore, for the time-varying coe cients in the Phillips curve, we also bound them to lie in stable regions, thus ensuring more precise inference. The Technical Appendix to this paper describes our Bayesian estimation methods. In particular, given priors for the initial conditions and the other parameters, we derive a Markov chain Monte Carlo (MCMC) algorithm for simulating from the posterior. Most of the blocks in this algorithm are standard. The non-standard blocks are those for drawing the latent states. 3 Empirical Work 3. Overview We divide our empirical work into two sub-sections. The rst is a forecasting exercise, comparing our bounded trend model of in ation and unemployment described in the preceding section to a range of alternative models for in ation and unemployment. We will use Bi-UC as an acronym for this bivariate unobserved components model. In the second, we present estimates of the trend in ation, the NAIRU and the other latent states produced by the Bi- UC model. Our data consist of quarterly CPI in ation rates and (civilian seasonally adjusted) unemployment rates from 948Q to 2Q3. The data are plotted in Figure. 7

9 π t u t Figure : CPI in ation and the unemployment rate. 3.2 Forecasting Results We begin by comparing our Bi-UC model to a range of alternative bivariate models for unemployment and in ation. We measure forecast performance using the standard metrics of root mean squared forecast errors (RMSFEs) and the sums of log predictive likelihoods. Note that the latter is a common Bayesian model selection device. RMSFEs evaluate the performance only of the point forecasts whereas predictive likelihoods evaluate the quality of the entire predictive density. We compare the Bi-UC model to various alternatives, each designed to investigate some aspect of our speci cation. These models, along with their acronyms, are as follows:. Bi-UC: the model described in Section Bi-UC-const- t : this is the restricted version of Bi-UC where t is time-invariant. 3. Bi-UC-const- t : the restricted version of Bi-UC where t is timeinvariant. 4. Bi-UC-const- t - t : the restricted version of Bi-UC where both t and t are time-invariant. 8

10 5. Bi-UC-NoBound: an unbounded version of Bi-UC, where all the states follow random walks without bounds. 6. Bi-UC-NoSV: the restricted version of Bi-UC where the measurement equation for t is homoskedastic (no stochastic volatility). 7. Bi-UC-TVP- u : an extension of Bi-UC where the AR coe cients in the unemployment equation ( u and u 2) are time-varying. 8. VAR(2): A standard, homoskedastic, VAR(2). 9. VAR(2)-SV: A VAR(2) with heteroskedastic errors modeled using the stochastic volatility speci cation of Cogley and Sargent (25). The relatively non-informative prior we use for Bi-UC is given in the Technical Appendix. The priors for the models which restrict one or more latent states in Bi-UC to be constant are the same as those used in Bi-UC, except for restricted state(s). For the latter, the prior for the initial condition for the states used in Bi-UC becomes the prior for the constant coe cient. The model with unbounded states, Bi-UC-NoBound, simply removes the bounds. It can be interpreted as a restricted version of Bi-UC which sets lower/upper bounds such as a =b to += in nity. For Bi-UC-TVP- u, we use the same priors for the initial conditions as we used for the constant coe cients, u and u 2, in Bi-UC. The priors for VAR(2) and VAR(2)-SV are given in the Technical Appendix. Tables, 2, 3 and 4 report RMSFEs and sums of log predictive likelihoods for unemployment and in ation individually. We also present, in Table 5, sums of log predictive likelihoods based on the joint predictive density for in ation and unemployment. Results are presented relative to the forecast performance of the VAR(2). For RMSFEs, we take a ratio so that a number less than unity indicates a model is forecasting better than the VAR(2). For sums of log predictive likelihoods (relative to VAR(2)) we take di erences, so that a positive number indicates a model is forecasting better than the VAR(2). All of the tables are divided into two panels for a long forecast evaluation period (since 975) and a shorter post-great Moderation forecast period (since 985). We consider forecast horizons of k = ; 4; 8; 2; 6 quarters. Overall, these tables indicate that our Bi-UC is forecasting well. Usually (for any forecast metric, time period or forecast horizon) it is the best model. 9

11 For cases where it is not the best, it is forecasting almost as well as the best model. Furthermore, most of the other candidates forecast very poorly in at least one case. We elaborate on these points in detail below. Relative to the model without bounds, a clear pattern emerges. At longer forecast horizons, the forecast performance of Bi-UC-NoBound deteriorates substantially. The unbounded random walk behavior of the latent states in this model is clearly leading to unreasonable long run forecasts. However, even with short run forecasts, Bi-UC-NoBound is almost always beaten by Bi-UC. In our introduction, we tried to argue that bounding latent states like trend in ation was economically sensible and more consistent with central bank behavior than assuming unbounded random walk behavior. Here we have also established empirically that bounding improves forecast performance. Similarly, Bi-UC-TVP- u, the extension of Bi-UC which allows for time-variation in the coe cients in the unemployment equation, forecasts poorly, particularly at longer forecast horizons. In fact, it typically exhibits the worst forecast performance at medium and long horizons. However, even with short run forecasts, Bi-UC-TVP- u occasionally forecasts poorly and almost never beats Bi-UC. It is for this reason that in () we assumed constant coe cients in the unemployment equation. Empirically, allowing for time-varying coe cients in the unemployment equation is not warranted and contaminates forecasts. Our Bi-UC is leading to smaller forecasting improvements relative to the VAR(2). Indeed, for the unemployment equation, subsequent to the Great Moderation, RMSFEs and predictive likelihoods are indicating that the Bi-UC and VAR(2) are forecasting roughly the same as each other. If we consider the longer forecast performance since 975, the relative performance of our Bi-UC model improves somewhat (particularly when k = 6). However, for forecasting in ation, the improvements of Bi-UC relative to the VAR(2) are more substantial, particularly for the post-975 forecast evaluation period. Including time-variation in parameters does seem to be important in the less stable period. The bene ts of using the Bi-UC model for forecasting in ation come through stronger when using predictive likelihoods than when using RMSFE. This suggests that bounding, although useful for getting good point forecasts, is particularly useful for getting the dispersion and tails of the predictive distribution correct. With regards to all the restricted versions of our Bi-UC model, none of them improves forecast performance. Restricting t to be constant is the

12 best of these restricted models, although even the Bi-UC-const- t model is typically beaten by the Bi-UC model, albeit by small amounts. Restricting t to be constant has a more substantive, negative, impact on forecast performance. Nevertheless, it does seem that the largest forecasting bene ts in our Bi-UC model are due to allowing trend in ation and the NAIRU to follow bounded random walks. Allowing for the coe cients t and t to have bounded random walks leads to only small improvements in forecasting performance. Finally, including stochastic volatility in the in ation equation is important since its inclusion leads to substantial improvements in in ation forecasts. Table : Relative RMSFEs (against VAR(2)) for forecasting in ation k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u Figures 2 through 4 allow us to look at forecast performance of several of

13 Table 2: Relative RMSFEs (against VAR(2)) for forecasting unemployment k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u

14 Table 3: Sum of log predictive likelihoods for forecasting in ation (relative to VAR(2)) k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u

15 Table 4: Sum of log predictive likelihoods for forecasting unemployment (relative to VAR(2)) k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u

16 Table 5: Sum of log predictive likelihoods for jointly forecasting in ation and unemployment (relative to VAR(2)) k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u k = k = 4 k = 8 k = 2 k = 6 VAR(2)..... VAR(2)-SV Bi-UC Bi-UC-const- t Bi-UC-const- t Bi-UC-const- t - t Bi-UC-NoBound Bi-UC-NoSV Bi-UC-TVP- u our main approaches in more detail by plotting cumulative sums of log predictive likelihoods for in ation, unemployment and the two variables jointly, respectively. Overall, it can be seen that Bi-UC model is consistently forecasting well. Furthermore, a few particular patterns are worth noting. For forecasting in ation for k =, it is clear that unobserved components models (including unbounded and bounded versions of them), did better at capturing the abrupt shock of the nancial crisis than the VARs. For unemployment, this statement also holds true, but to a lesser extent. The poor performance of the unbounded trend model, Bi-UC-NoBound, in forecasting in ation at longer horizons mostly occurs in the relatively stable period from the early 98s through early 99s. However, for unem- 5

17 ployment, this period of poor forecasting occurred later, from the mid 99s through 2. In general, this unbounded model is not forecasting poorly during unstable times, when its unbounded random walk states allow for rapid change, but in more stable times. 3 k = 3 k = k = 2 3 k = π VAR(2) SV Bi UC Bi UC const λ ρ t t Bi UC NoBound Figure 2: Cumulative sums of log predictive likelihoods for forecasting in ation (relative to VAR(2)). 6

18 4 k = k = k = 2 3 k = π VAR(2) SV Bi UC Bi UC const λ ρ t t Bi UC NoBound Figure : Figure 3: Cumulative sums of log predictive likelihoods for forecasting unemployment (relative to VAR(2)). 4 k = 3 k = k = k = VAR(2) SV Bi UC Bi UC const λ t ρ t π Bi UC NoBound 7 Figure 4: Cumulative sums of log predictive likelihoods for jointly forecasting in ation and unemployment (relative to VAR(2)).

19 3.3 Estimates of Trend In ation, NAIRU and Other Latent States In previous work with a simple univariate model, Chan, Koop and Potter (22), we showed the bene ts of bounding in producing reasonable estimates of trend in ation. If trend in ation is left to evolve according to an unbounded random walk, it can track actual in ation too closely, leading to erratic (and, in the last 97s very high) estimates of trend in ation. But by keeping trend in ation bounded in an interval consistent with beliefs about the behavior of central bankers, smoother and much more sensible behavior is obtained. In the present case, we obtain similar results. Figure 5 plots the four main latent states, or t, u t ; t and t, estimated using the full sample. That is, Figure 5 contains smoothed estimates, based on information available at time T (as opposed to ltered estimates, to be presented shortly). With regards to the observed increase in in ation in the late 97s, it can be seen that our bounded model of trend in ation chooses to estimate it as a slight increase in trend in ation, but a much larger increase in persistence (i.e. t increases substantially at this time). This is consistent with a story where the Fed has kept a fairly low implicit in ation target, but was more willing to tolerate (or less able to correct) deviations from target in the 97s than subsequently. Our model also implies a smoothly evolving NAIRU. Despite large uctuations in unemployment (see Figure ), Figure 5 suggests NAIRU increases from roughly 5 to 7% the end of the 98s before subsequently falling to the region of 6%. These numbers are consistent with the existing literature. For instance, Staiger, Stock and Watson (997) present NAIRU estimates in the 5.5% to 5.9% range. The unemployment equation is very persistent, as the posterior means for u and u 2 are estimated to be :634 and :696, respectively. The coe cient controlling the slope of the Phillips curve is, as expected, a negative number and tends to be around :4. Consistent with the evidence of the preceding sub-section, there is less evidence that it varies over time. However, there is some tendency for it to fall (become more negative) in the late 97s and subsequently tend to move towards zero, indicating a weakening of the Phillips curve relationship. This is consistent with the nding of Stella and Stock (2) that the Phillips curve was steeper in the 97s than the 99s. However, they also found a steepening after 28, which we do not nd to any great extent. 8

20 4 τ π 8 τ u ρ π λ Figure 5: Posterior estimates for the latent states using the full sample. The solid lines represent the posterior means while the dotted lines are the 6 and 84 percentiles. Figure 5 contains smoothed estimates of the states. These are useful for a retrospective analysis using all available information. However, for some purposes, ltered estimates are useful. That is, it is also useful to consider a real time analysis, estimating the states at time t using data which were available at time t. These are presented in Figure 6. The broad patterns in Figure 6 are similar to Figure 5, suggesting that our bivariate unobserved components model can provide sensible real-time estimates of NAIRU and the trend in ation. As one would expect of ltered estimates, they tend to be slightly more erratic than smoothed estimates. In some ways, this is sensible. For instance, the smoothed estimates have the NAIRU rising as of shortly after 2. This is due to a slight increase in unemployment after the 2 slowdown and the much larger increase in unemployment in the recent recessions. Figure 5 smooths these two together as a gradual rise in NAIRU throughout the 2-2 period. But the ltered estimates which, in 27 do not know the nancial crisis which is about to occur, are still low in 27 9

21 and only start rising after the nancial crisis has hit. It is also interesting to note that the ltered estimates of trend in ation and t indicate that the 98s are a time of decreasing in ation persistence and it is only later that declines in trend in ation occurred. 4 τ π 7 τ u ρ π λ Figure 6: Estimates of the ltered states E( t j Data t ); where t = t ; u t ; t ; t and Data t is the data from the beginning of the sample to time t. 4 Conclusions In this paper, we have developed a bivariate unobserved components model for in ation and unemployment based on the Phillips curve. The model is written in terms of deviations of in ation from its trend and deviations of unemployment from its natural rate. One result of our paper is that such unobserved components models are attractive since they directly provide estimates of trend in ation and the NAIRU and can also forecast as well or better as reduced form models such as VARs. However, the literature contains 2

22 papers with similar unobserved components models making similar points about their advantages (e.g. Stella and Stock, 2). Relative to this literature, the main innovation of the present paper is the use of bounding of latent states such as trend in ation and NAIRU. The existing literature assumes these states evolve according to unbounded random walks, despite the fact that this assumption is inconsistent with much underlying economics (e.g. the fact that central bankers have implicit or explicit in ation targets). This paper develops a model which incorporates bounded random walks (where the bounds are estimated from the data). We nd that this addition not only leads to more sensible estimates of trend in ation and NAIRU, but also forecasts better than a range of other approaches. Technical Appendix In this appendix we describe the prior and MCMC algorithm used to estimate the bivariate model of in ation and unemployment given in (), (2), (3) and (4). It also provides additional estimation details about the other models used for comparison with our model. We use notation where = ( ; : : : ; T ) ; u = (u ; : : : ; u T ) ; y = ( ; u ) ; = ( ; : : : ; T ) ; u = ( u ; : : : ; u T ) ; = ( ; : : : ; T ) ; = ( ; : : : ; T ) ; h = (h ; : : : ; h T ), and de ne " ; " u ; " ; " u ; " h ; " ; " similarly. In addition, let denote the model parameters, i.e., = ( 2 u; 2 ; 2 u; 2 h ; 2 ; 2 ; a ; b ; a u ; b u ; u ; u 2) : 4. The Prior We require a prior for the initial condition in every state equation and these are: T N(a ; b ; ;! 2 ); u T N(a u ; b u ; u ;! 2 u); T N(; ; ;! 2 ); T N( h T N(h ;! 2 h); ; ; ;! 2 ); where,! 2, u, u,! 2 u,,! 2,,! 2 u, h, and! 2 h are known constants. In particular we choose the relatively non-informative values of = 3, u = u = 5, h = = =,! 2 =! 2 u =! 2 h = 5 and!2 =! 2 u =! 2 =. 2

23 Note that the need for u arises from the use of an AR(2) speci cation in the unemployment equation. The prior for the model parameters is speci ed as p() = p( 2 u)p( 2 h)p( 2 )p( 2 u)p( 2 )p( 2 )p (a ) p (b ) p (a u ) p (b u ) p ( u ) p ( u 2) ; where 2 u IG( u ; S u ), 2 h IG( h; S h ), 2 IG( ; S ), 2 u IG( u ; S u ), 2 IG( ; S ), 2 IG( ; S ), and IG(; ) denotes the inverse-gamma distribution. We choose relatively small values for the degrees of freedom parameters, which imply large prior variances, i.e., the priors are relatively non-informative. Speci cally, we set u = h = = u = = u = =. We then choose values for the scale parameters so that the parameters have the desired prior means. We set S u = S h = :9; which imply prior means E( 2 u) = E( 2 h ) = :5. Next, we set S = :8 and S u = :9; which imply E( 2 ) = :2 and E( 2 u) = :. These values are chosen to re ect the desired smoothness of the corresponding state transition. For example, the prior mean for 2 implies that with high probability the di erence between consecutive trend in ation, t t, lies within the values :3 and :3. We set S = S = :8; which imply prior means E( 2 ) = E( 2 u) = E( 2 ) = :2. These values imply a relatively smooth transition for the relevant states. For the bounds we use uniform priors: a U(; 2); b U(3; 5); a u U(3; 5); b u U(6; 8). The priors for u and u 2 are both N(; 5): 4.2 MCMC Algorithm We extend the MCMC sampler developed in Chan, Koop and Potter (22) which in turn is an adaptation of the algorithm introduced in Chan and Strachan (22). Speci cally, we sequentially draw from (we suppress the dependence on, u and u ):. p( j y; u ; ; ; h; ) 2. p( u j y; ; ; ; h; ) 3. p( j y; ; u ; ; h; ) 4. p( j y; ; u ; ; h; ) 22

24 5. p(h j y; ; u ; ; ; ) 6. p (jy; ; u ; ; ; h) Step : To derive the conditional distribution p( j y; u ; ; ; h; ), we rst rewrite the in ation equation as K = + K + " ; " N(; ); where = diag(e h ; : : : ; e h T ) and ( ) + (u u ) 2 (u 2 u 2) = 3 (u 3 u 3) ; K B C A B T (u T u T :.. C T A Since jk j = for any, K is invertible. Therefore, we have i.e., ( j u; u ; ; ; h; ) N(K + ; (K K ) ); log p( j u; u ; ; ; h; ) / 2 T h 2 ( K ) K K ( K ); (5) where T is a T column of ones. Similarly, rewrite the state equation for as H = + " ; where = ; H = B A B... :. A 23

25 That is, the prior density for is given by log p( j 2 ) / 2 ( H ) H H( H ) + g ( ; 2 ); (6) where a < t < b for t = ; : : : ; T, = diag(! 2 ; 2 ; : : : ; 2 ) and g ( ; 2 b a ) = log TX log t=2! Combining (5) and (6), we obtain log p( j y; u ; ; u ; ; h; )! b t a t : / 2 ( K ) K K ( K ) 2 ( H ) H H( / 2 ( ^ ) D ( where a < t < b for t = ; : : : ; T, and H ) + g ( ; 2 ); ^ ) + g ( ; 2 ); D = H H + K K ; ^ = D (H + K K ( K )): Since this conditional density is non-standard, we sample via an independencechain Metropolis-Hastings (MH) step. Speci cally, candidate draws are rst obtained from the N(^ ; D ) distribution with the precision-based algorithm discussed in Chan and Jeliazkov (29), and they are accepted or rejected via an acceptance-rejection Metropolis-Hastings (ARMH) step. Step 2: To implement Step 2, note that information about u comes from three sources: the two measurement equations for in ation and unemployment and the state equation for u. We derive an expression for each component in turn. First, write the in ation equation as: z = u + " ; " N(; ); 24

26 where z t = ( t t ) t ( t t ) t u t, z = (z ; : : : ; z T ), and = diag( ; : : : ; T ). Hence, ignoring any terms not involving u, we have log p( j u; u ; ; ; h; ) / 2 (z u ) (z u ): (7) The second component comes from the unemployment equation, which can be written as: K u u = u + K u u + " u ; " u N(; u ); where u = I T 2 u and u (u u ) + u 2(u u ) u 2(u u ) u = ; K u = B A u u 2 u u 2 u : C u 2 u A Thus, ignoring any terms not involving u, we have log p(u j u ; ) / 2 (u K u u u ) K u u K u (u K u u u ): (8) The third component is contributed by the state equation for u : log p( u j 2 u) / 2 ( u H u ) H u H( u H u ) + g u ( u ; 2 u); (9) where u = ( u ; ; : : : ; ), a u < u t < b u for t = ; : : : ; T, u = diag(! 2 u; 2 u; : : : ; 2 u) and g u ( u ; 2 bu au u) = log TX log t=2! u! u bu u t u au u t : u 25

27 Combining (7), (8) and (9), we obtain log p( u j y; ; u ; u 2; ; h; ) / 2 (z u ) (z u ) 2 (u K u u u ) Ku u K u (u Ku u u ) 2 ( u H u ) H u H( u H u ) + g u ( u ; 2 u); / 2 ( u ^ u ) D u ( u where a u < u t < b u for t = ; : : : ; T, and ^ u ) + g u ( u ; 2 u); D u = H u H + Ku u K u + ; ^ u = D u (H u u + Ku u K u (u Ku u ) + z): Again, we sample u via an ARMH step with candidate draws obtained from N(^ u ; D u ). Step 3: Next, we derive an expression for p( j y; ; u ; ; h; ). First, let t = t t, u t = u t u t, = ( ; : : : ; T ), and u = (u ; : : : ; u T ). Then the measurement equation for in ation can be rewritten as + u = X + " ; " N(; ); where X = diag( ; : : : ; T ) and = diag( ; : : : ; T ): From the state equation for we also have H = " : Therefore, using a similar argument as before, we have p( j y; ; u ; u ; ; h; ) / 2 ( ^ ) D ( where < t < for t = ; : : : ; T, g ( ; 2 ) = TX log t t=2 ^ ) + g ( ; 2 ); t ; D = H H + X X ; ^ = D X ( + u ); 26

28 and = diag(! 2 ; 2 ; : : : ; 2 ). As before, we implement an ARMH step with approximating density N(^ ; D ). Step 4: Using the same argument as before, we have p( j y; ; u ; ; h; ) / 2 ( ^) D ( ^) + g (; 2 ); where < t < for t = ; : : : ; T, g (; 2 ) = TX t=2 log t t ; D = H H + X u X u ; ^ = D X u w; X u = diag(u ; : : : ; u T ), w = ( ; : : : ; T T T ) ; and = diag(! 2 ; 2 ; : : : ; 2 ). As before, we implement an ARMH step with approximating density N(^ u ; D u ). Step 5: For Step 5, we use the algorithm in Chan and Strachan (22) to sample from p(h j y; ; u ; ; ; ). Step 6: We draw from in separate blocks, mainly using standard results for the regression model. We use notation where x for all parameters in except for x. Using standard linear regression results, it can be shown that u = ( u ; u 2) is a bivariate truncated normal: p( u j y; ; u ; ; ; h; u) / 2 (u ^ u ) D u ( u ^ u ) + g u ( u ); with the stationarity constraints u + u 2 <, u 2 D u = V u + XuX u = 2 u ; u < and j u 2j <, where u u ^ u = D u Xu = 2 u u u; X u = B A : u T u T 2 A draw from this truncated normal distribution can be obtained via acceptancerejection sampling with proposal from N(^ u ; D u ). To sample from the error variances, rst note that they are conditionally independent given the data and the states. Hence, we can sample each element one by one. 27

29 Now, both p( 2 u j y; ; u ; ; ; h; 2 u ) and p( 2 h j y; ; u ; ; ; h; 2 h ) are standard inverse-gamma densities, respectively:! ( 2 u j y; ; u ; ; ; h; 2 u ) IG u + T 2 ; S u + TX (u t u 2 u t u 2u t 2) 2 ( 2 h j y; ; u ; ; u ; ; h; 2 h ) IG h + T ; S 2 h + 2 t= Next, the log conditional density for 2 is given by! TX (h t h t ) 2 : t=2 logp( 2 j y; ; u ; ; ; h; 2 ) / ( + ) log 2 S 2 T 2 log TX ( t t ) 2 + g ( ; 2 ); t=2 which is a non-standard density. To proceed, we implement an MH step with the proposal density! IG + T ; S 2 + TX ( t t 2 ) 2 : Similarly, the log conditional density for 2 u is given by t=2 logp( 2 u j y; ; u ; ; ; h; 2 u ) / ( u + ) log 2 u S u 2 u T 2 log 2 u 2 2 u TX ( u t u t ) 2 + g u ( u ; 2 u): t=2 Again, a draw from p( 2 u j y; ; u ; ; ; h; 2 u ) is obtained via an MH step with the proposal density! IG u + T ; S 2 u + TX ( u t u t 2 ) 2 : The remaining orrery variances, 2 and 2, are sampled analogously. To draw from the bounds a, b, a u and b u, we use a Griddy-Gibbs sampler which is the same as the one used in Chan, Koop and Potter (22). The reader is referred to our earlier paper for details. 28 t=2

30 4.3 Speci cation and Estimation Details for Other Models The other models used in our forecast comparisons are mostly restricted special cases of Bi-UC and all speci cation and prior details are identical to Bi-UC except that the relevant restriction is imposed. The exceptions to this are discussed in this sub-section. The VAR(2) is speci ed as: y t = + B y t + B 2 y t 2 + " t ; where y t = ( t ; u t ) and " t = (" t ; " u t ) N(; ). We use a relatively noninformative prior. For the prior is N(; ). For the VAR coe cients, we assume each is N(,) and all are, a priori, uncorrelated with one another. :4 The prior for is IW ; so that the prior mean of the error :7 variances in the two equations are :2 and :, respectively. We also use a VAR(2) with stochastic volatility: y t = + B y t + B 2 y t 2 + A " t ; where " t N(; e ht ), " u t N(; 2 u) and A = : a The VAR coe cients have the same prior as the VAR(2) without stochastic volatility. All details (including prior) relating to h t and 2 u are exactly as in Bi-UC and a has a N(; ) prior. The VARs are estimated using MCMC methods as outlined, e.g., in Koop and Korobilis (29). For Bi-UC-TVP- u, all details are identical to Bi-UC except that we need to specify the initial conditions for u t and u 2t which are now timevarying. These are both assumed to be N(; 5). The priors for the two error variances in the two state equations are both IG(; :9), a relatively noninformative choice implying prior means of :. References Atkeson, A. and Ohanian, L. (2). Are Phillips curves useful for forecasting in ation? Federal Reserve Bank of Minneapolis Quarterly Review, 29

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32 Staiger, D., Stock, J. and Watson, M. (997). The NAIRU, unemployment and monetary policy, Journal of Economic Perspectives,, Stella, A. and Stock, J. (2). A state-dependent model for in ation forecasting, manuscript available at Stock, J. and Watson, M. (27). Why has U.S. in ation become harder to forecast?, Journal of Money, Credit and Banking 39, Stock, J. and Watson, M. (2). Modeling in ation after the crisis, manuscript prepared for the Federal Reserve Bank of Kansas City Symposium, Macroeconomic Policy: Post-Crisis and Risks Ahead, Jackson Hole, Wyoming, August Weise, C. (2). Political pressures on monetary policy during the US Great In ation, American Economic Journal: Macroeconomics, 4,