Are Macroeconomic Variables Useful for Forecasting the Distribution of U.S. Inflation?

Transcription

1 Are Macroeconomic Variables Useful for Forecasting the Distribution of U.S. Inflation? Sebastiano Manzan a and Dawit Zerom b a Department of Economics & Finance, Baruch College, CUNY b California State University at Fullerton Abstract Much of the inflation forecasting literature examines the ability of macroeconomic indicators to accurately predict mean inflation. For the period after 1984, existing empirical evidence largely suggests that the likelihood of accurately predicting inflation using macroeconomic indicators is no better than a random walk model. We expand the scope of inflation predictability by exploring whether macroeconomic indicators are useful in predicting the distribution of inflation. We consider six commonly used macro indicators and core/non-core versions of the Consumer Price Index (CPI) and the Personal Consumption Expenditure (PCE) deflator as measures of inflation. Based on monthly data and for the forecast period after 1984, we find that some of the macro indicators, such as unemployment rate, housing starts and term spread, provide significant out-of-sample predictability for the distribution of core inflation. The analysis of the quantiles of the predictive distribution reveals interesting patterns which otherwise would be ignored by existing inflation forecasting approaches that rely only on forecasting the mean. We also illustrate the importance of inflation distribution forecasting in evaluating some events of policy interest by focusing on predicting the likelihood of deflation. JEL Classification: C22, C53, E31, E52 Keywords: Distribution; Economic indicators; Inflation; Predictability; Quantiles. Acknowledgements: We are grateful to Barbara Rossi, seminar participants at Baruch College, Rutgers University, SNDE 2009 Symposium, the 2009 NBER Summer Institute, as well as three anonymous referees and the Editor for helpful comments that significantly improved the paper.

2 1 Introduction Forecasting the behavior of inflation plays a central role in the conduct of monetary policy due to the lagged impact of the central bank actions on economic activity. It is thus important to accurately predict the effect of the many shocks that hit the economy on the future dynamics of inflation. The standard approach for forecasting inflation has been the Phillips curve (PC) model that, in its expectation-augmented version, assumes a trade-off between unexpected inflation and unemployment, or more generally, indicators of real economic activity. Despite its long-time success, recent empirical evidence on the effectiveness of the PC model is far from unanimous. Stock and Watson (1999) provide a detailed study on the out-of-sample forecast accuracy of the PC by using an extensive set of macroeconomic variables. Using the forecast evaluation period January September 1996, their conclusion is that PC models have better forecasting performances (compared to univariate time series models) using the unemployment rate as well as other leading indicators of economic activity (e.g., output gap and capacity utilization). They also find that combining information or models might provide better results than simply relying on few indicators. However, Atkenson and Ohanian (2001) provide an opposite empirical evidence, albeit a different forecast evaluation period January November 1999, where they report that PC models are no better than the naïve model, which assumes that the expected inflation over the next 12 months is equal to inflation over the previous 12 months. Fisher et al. (2002) conduct a systematic comparison of the forecasting accuracy at the one-year horizon in different sub-periods and find that the PC forecasts outperformed the naiïve forecasts only in the window. There is evidence that the declining predictive power of macroeconomic indicators is a characteristic also of other variables (see D Agostino et al., 2006, and Rossi and Sekhposyan, 2010). For a comprehensive survey as well as discussion of the outstanding issues in inflation forecasting, see Stock and Watson (2008). In this paper, departing from the existing focus on conditional mean forecasting, we explore whether leading indicators of economic activity are useful in predicting the distribution of future inflation. Despite the availability of extensive literature on inflation forecasting, little or no attention has been paid to examining whether indicators of economic activity carry useful information about the dynamics of higher moments, beyond the mean. For example, having some idea on the conditional second-order moment of future inflation can be vital in assessing the risk to inflation stability due to macroeconomic shocks. Greenspan (2004) discusses this issue in the following terms: Given our inevitably incomplete knowledge about key structural aspects of an ever-changing economy and the sometimes asymmetric costs or benefits of particular outcomes, a central bank needs to consider 2

3 not only the most likely future path for the economy, but also the distribution of possible outcomes about that path. The decision-makers then need to reach a judgment about the probabilities, costs, and benefits of the various possible outcomes under alternative choices for policy (p. 37). While average future inflation may signal the direction of the economy, it cannot help policymakers to evaluate the risks of deviations from the most likely path and the cost for the economy of such deviations. In a recent paper, Kilian and Manganelli (2008) introduce a model in which the monetary policy maker is viewed as a risk manager trying to balance the risks to inflation and output stability. In this framework, if the preferences of the policy maker are assumed to be quadratic and symmetric, then the only relevant moment (of the inflation and output distributions) is the conditional mean. However, they provide evidence of departure of the preferences from such a benchmark. All the above elements point to the suggestion that forecasting the distribution of inflation represents a relevant tool in the conduct of monetary policy. In fact, the Bank of England has been publishing for many years the so-called fan charts that represent the subjective forecasts of the Bank about the future distribution of inflation. We use linear quantile regression to incorporate macroeconomic variables into the prediction of the conditional distribution of future inflation. The approach considers several conditional quantiles of future inflation, and by doing so, offers more flexibility (than, for example, the conventional PC models) in capturing the possible role of macroeconomic indicators in predicting the different parts of the inflation distribution. For instance, one may be able to investigate if some periods of low or high inflation are driven by some macroeconomic indicators. Surely such information cannot be delivered by PC-type models that deal only with predicting average inflation. We find strong empirical evidence of predictability for U.S. core monthly inflation using indicators of economic activity, in particular the unemployment rate and housing starts. Importantly, the empirical findings apply to a forecast evaluation period that is intentionally chosen to be post 1984, when the existing literature shows that macroeconomic indicators are not relevant to predict future average inflation (see Stock and Watson, 2007). We attribute this result to the ability of our approach to account for the varying predictive effect of economic indicators on core inflation at different quantiles of its distribution. For some indicator variables, we also find an asymmetric effect in the sense that an indicator is more relevant on the lower part of the forecasting distribution than the upper part (and vice versa, depending on the indicator considered). These observed quantile effects take place far away from the center of the distribution, making them difficult to be detected with approaches (like PC-type models) that solely focus on evaluating the relevance of these variables in predicting the conditional mean. 3

4 A possible explanation for these findings is that the public might form inflation expectations that are more responsive to macroeconomic news when inflation is in the tails of the distribution. This can happen if the monetary policy-maker has an (implicit or explicit) target located near the center of the distribution. When inflation fluctuates around the center of the distribution, expectations are well-anchored to the target in the sense that the public believes the central banks will be able to maintain price stability. However, when inflation deviates significantly from the target the public might distrust the ability of monetary policy to bring inflation back to the target and thus form expectations that rely more on macroeconomic variables which represent more reliable indicators of future inflation. This paper is related to an increasing number of papers that deal with distributional aspects of inflation. Cogley et al. (2005) propose a Bayesian VAR model where both the conditional mean and variance are time varying. They forecast inflation for the UK and illustrate their method by comparing interval forecasts from their model to the fan charts of the Bank of England. More recently, Clark (2011) considers a Bayesian VAR with stochastic volatility and finds that accounting for time variation in volatility is essential in producing accurate density forecasts for a a wide range of macroeconomic variables, in particular in the post-1985 sample period. A similar conclusion has also been reached by Jore et al. (2010) using AR and VAR models that allow for structural breaks. Compared to these papers, the quantile regression approach adopted in this paper models directly the possible changes in the inflation distribution and explains these changes on the economic indicators. Robertson et al. (2005) forecast the distribution of inflation based on a VAR specification and propose a methodology to twist the forecasting distribution in order to incorporate theoretical restrictions (e.g., a Taylor rule). Corradi and Swanson (2006) evaluate the performance of time series and PC models in forecasting one-month ahead inflation using different distributional assumptions for the error term, and Amisano and Giacomini (2007) use a Markov Switching model and find that the one-month ahead inflation forecasts from the nonlinear specification are more accurate compared to a linear one. The rest of the paper is organized as follows. In Section (2) we discuss econometric approaches, including our proposal, to estimate the conditional forecast distribution of future inflation. Section (3) outlines a test of predictive accuracy that is used to evaluate the conditional distribution models discussed in Section (2). In Section (4), we present (with discussion) the empirical findings of the paper. Finally, Section (5) concludes. 4

5 2 Econometric methodology We denote the annualized inflation over a h-month period by Y h t = (1200/h)[log P t log P t h ] and the one-month annualized inflation by Y t = 1200[log P t log P t 1 ] where P t is the level of the price index in month t. Also let X t be some indicator of real economic activity such as unemployment rate. A baseline specification often used in forecasting inflation is the Phillips curve (PC) model. The PC model (see Stock and Watson, 1999) postulates that changes in h-month inflation, Y h t+h, depend on recent changes in one-month inflation and past and present values of a candidate economic indicator, Y h t+h Y t = µ h 0 + β h (L) Y t + γ h (L)X t + U pc t+h (1) where µ h 0 is a constant, βh (L) and γ h (L) are lag polynomials written in terms of the lag operator L and U pc t+h is the error term. Note that in the above specification we follow Stock and Watson (2007) and assume that the inflation rate Y t has a unit root, although other papers take the opposite view and consider inflation stationary (e.g., Ang et al., 2006). In evaluating the forecasting performance of the PC model, it is often compared against two univariate models: the autoregressive (AR) model and the naïve (random walk) model. Although simple, these two time series models are very competitive benchmarks. The AR model is a special case of the PC model where no information on present and past values of X t are included, i.e., where U ar t+h Y h t+h Y t = µ h 0 + β h (L) Y t + U ar t+h (2) is the error term. The naïve model (see Atkenson and Ohanian, 2001) specifies that the expected inflation over the next h months is equal to inflation over the previous h months, i.e., where U ao t+h model. Y h t+h = Y h t + U ao t+h (3) is the error term. In the rest of the paper we will refer to the naïve model by AO 2.1 Proposed approach Note that in the PC model the economic indicator is assumed to affect the inflation rate only through the conditional mean, which implies that its error U pc t+h should be independent of the past and present values of X t (and lags of Y t ). In other words, the effect (if any) of the macroeconomic 5

6 variable on the conditional distribution of future inflation (Yt+h h ) is only limited to the conditional mean. So the PC approach ignores the possibility for economic indicators to carry useful information about the dynamics of higher moments, and hence help improve the accuracy of the forecast distribution of inflation. To circumvent this limitation of PC models, we focus instead on the forecast distribution of Yt+h h conditional on the available information set at time t. Because existing empirical evidence (for the period after 1984) largely suggests that the likelihood of accurately predicting inflation using economic indicators is no better than a random walk model, we constrain the conditional mean of the forecast distribution of Yt+h h to follow the AO model (see Equation 3). Let the past and present values of a particular economic indicator variable, X, be denoted by the vector X t = (X t,,x t q+1 ). Use of the AO model implies that X t does not carry any relevant information for predicting the mean future inflation. However, unlike the AO model, we allow X t to have an effect on higher-order moments of Y h t+h and, more generally, on the conditional distribution of Yt+h h, which is not permitted in the PC model. Under the proposed set-up, if we find (relative) predictability in the conditional distribution of Y h t+h using X t while no (relative) predictability by the PC model compared to the benchmark AO model, we can then conclude that the predictability is occurring at parts of the distribution beyond the conditional mean. The proposed approach is denoted by AO-U X where the conditional mean of Yt+h h follows the AO model and X t is used to model the conditional quantile of the AO forecast error, Ut+h ao, which can be considered stationary. We should note that the direct nature of our forecasts introduces overlapping observations in Ut+h ao which can be assumed to follow a MA(h 1) process. This serial correlation may impact the accuracy of the estimated conditional quantiles 1 and we thus decided to carry out the conditional quantile step using the residuals (denoted by Ũao t+h ) from a fitted MA(h 1) model on Ut+h ao. Denote the α (0, 1) conditional quantile of Uao t+h and Ũao t+h conditional on the available information set at time t by Q t+h t (α) and Q t+h t (α), respectively. We estimate Q t+h t (α) using the quantile regression model (see Koenker and Bassett, 1978), Q t+h t (α) = δ 0,α + q δ k,α X t k+1. (4) k=1 Although the local effect of X t k+1 on the α-quantile is assumed to be linear, the model is very flexible because each slope coefficient δ k,α is allowed to differ across quantiles. This is a useful 1 Since quantiles are affected by the variance, the MA(h 1) structure has the potential to bias the quantile forecasts, especially for large h. To the best of our knowledge, there are no methods available that deal with MA errors in a quantile regression framework. 6

7 property since it provides guidance as to where in the distribution of Y h t+h the indicator X t has a significant effect. Finally, we construct the desired conditional distribution (or density) of Yt+h h from the conditional quantile estimates ˆQ t+h t (α) where the latter is obtained by taking the product of ˆ Qt+h t (α) in Equation (4) and the standard deviation of the fitted MA model (see above). In our implementation the sequence α is chosen to be between 0.05 and 0.95 in 0.01 interval. Further, to guarantee monotonicity of the conditional quantile estimates, and hence validity of the implied conditional distribution, we apply the proposal of Chernozhukov et al. (2010) which consists of rearranging the (possibly crossing) ˆQ t+h t (α) into a monotonic rearranged quantile curve. 3 Measuring relative predictability Suppose we have a pair of models (k and j) where we consider model k as the benchmark. We are interested in measuring the relative accuracy of model j when h-step ahead out-of-sample density or quantile forecasts are available. As benchmark, we consider two univariate models, i.e., the AR model (see Equation 2) and the AO model (see Equation 3) 2. The accuracies of the two benchmark (univariate) models are compared separately against two alternative models that incorporate the effect of macroeconomic indicators. For the latter, we consider the PC model (see Equation 1) and the proposed AO-U X model. We adopt a rolling window scheme when generating out-ofsample density or quantile forecasts from the various models. The first forecast is for January 1985 and the models are estimated on the window 1959:1-1984:1 and 1959:1-1984:7 for h=12 and 6, respectively. It should be noted that the forecast evaluation period is chosen intentionally to be post 1984 where current research shows that macroeconomic indicators do not add much to the predictability of mean inflation. The next forecast is for February 1985 and the estimation window is kept constant by dropping one observation at the beginning of the sample. For example, the estimation window is updated to (1959:2-1984:2) for h=12. We do this until forecasts for December 2007 are computed 3. In this manner, we collect a total of n=276 density or quantile forecasts for each model. 2 To ensure consistency, we estimate their forecast densities and quantile forecasts in a similar fashion as that of AO-U X model, except that the quantile regression in (4) is based on regressing errors Ũao t+h or Ũar t+h only on a constant (hence their respective error distribution is assumed to be independent of economic indicators). Similarly for the PC model. 3 For the AR and PC models, we select the lag order for Y t using the Akaike Information Criterion (AIC), recursively for each rolling window sample. For both the PC model and the proposed AO-U X model, we find that including lags of X t does not provide forecast improvements over the no lag case. So all results in next section are based on conditioning on current values of economic indicators, i.e. X t. 7

8 We use scoring rules to compare the pair of density or quantile forecasts. A scoring rule is a loss function whose arguments are the density/quantile forecast and the realization of the future observation. In this paper, we consider two such score functions: the Logarithmic Score (LS) function and the Weighted Quantile Score (WQS) function. Let ˆf j t+h t ( ) denotes the forecast density of Yt+h h conditional on the available information set at time t obtained from model j. The LS function is defined as LS j t+h t = log ˆf j t+h t (Y t+h h ). Instead, the WQS is defined from quantile forecasts and hence allows the forecaster to compare the performance of competing models on specific areas of the forecast distribution. Let ˆQ j t+h t (α) denote the α-quantile forecast of Y h t+h conditional on the available information set at time t obtained from model j. Instead, the WQS function (Gneiting and Raftery, 2007, and Gneiting and Ranjan, 2011) is given by WQS j t+h t = 1 0 QSj t+h t (α)ω(α)dα4, where QS j t+h t (α) = [ ˆQ j t+h t (α) Y t+h h ][1(Y t+h h ˆQ j t+h t (α)) α] is the quantile loss and 1( ) denotes the indicator function. Four weight functions are of interest, (1) ω(α) = 1 which is a uniform weight and provides an overall evaluation of the forecast distribution (can be viewed as a direct alternative to the LS), (2) ω(α) = α(1 α) which concentrates the weight in the middle of the distribution, (3) ω(α) = (1 α) 2 that assigns more weight to the left tail of the distribution, and (4) ω(α) = α 2 that puts more weight to the right tail. The latter two weights help us examine possible asymmetry in the predictive power of an economic indicator X. In order to put the contribution of the paper in the context of existing empirical evidence on U.S. inflation forecasting, we also evaluate the models in terms of their accuracy for predicting the conditional mean. We denote the score function by SS and is defined as the square of the forecast error, i.e. SS j t+h t = (ǫj t+h )2. For example, when j=ao model, the forecast error is simply U ao t+h. Now, given two competing models (k and j) where model k is considered the benchmark (AR or AO) and model j is the alternative model (PC or AO-U X), model j is said to be relatively more accurate in terms of a particular metric S {LS, WQS-unif, WQS-center, WQS-left, WQS-right, SS} if its average score S j n = 1 S j n t+h t t is higher than that of model k, S k n. Note that in our rolling setting, t (1984 : : 6) for h=6 and t (1984 : : 12) for h=12. We consider the approach proposed in Giacomini and White (2006) and Amisano and Giacomini (2007) to evaluate the hypothesis of equal forecast 4 In implementation, we replace this by a discrete version i=1 ω(αi)qsj t+h t (αi) where αi [0.05, 0.95] in 0.01 interval. 8

9 accuracy based on the test statistics t n (S) = n Sk n S j n ˆσ n (5) where ˆσ n is a heteroscedasticity and autocorrelation consistent (HAC) standard error estimator of the difference in scores S k t+h t Sj t+h t. Assuming suitable regularity conditions, the statistic t n(s) is asymptotically standard normal under the null hypothesis of vanishing expected score differentials. In the case of rejection, model j is preferred if t n (S) is negative and model k is preferred if t n (S) is positive. 4 U.S. inflation forecasts We use four measures of the monthly price index (P t ): Consumer Price Index for all items (CPI), CPI excluding food and energy (core-cpi), Personal Consumption Expenditure deflator (PCE), and the PCE excluding food and energy (core-pce). We follow the recent inflation forecasting literature (see Stock and Watson, 1999, and Ang et al., 2006) and include six of the indicators of economic activity that are often considered as predictors of inflation, i.e., the civilian unemployment rate (UNEM), the index of industrial production (IP), real personal consumption expenditure (INC), employees on non-farm payrolls (WORK), housing starts (HS), and the term spread (SPREAD) defined as the yield on the 5-year Treasury bond minus the 3-month Treasury bill. Thus, we have X {UNEM, IP, WORK, HS, INC, SPREAD}. All the data (on P t and X t ) were gathered from the Federal Reserve Bank of Saint Louis database FRED and the sample period spans from January 1959 until December Some of the leading indicators (i.e., IP, INC, and WORK) are not stationary. We thus consider these variables in gap form where the long-run trend is modeled using a Hodrick and Prescott (1997) filter (HP) with parameter equal to (typically used for monthly data) 6. The trend is estimated only on information available at the time the forecast is made. 4.1 Summary of t n (S) test results Results for h = 6 and h = 12 are given in Tables (1) and (2), respectively. For both h, the PC model is either significantly worse or no better than the AO model in terms of conditional mean predictive accuracy for all X and all inflation measures. In fact, for h = 12, all t n (SS) values 5 The macroeconomic series consists of revised data available at the January 2008 vintage due to the lack of a comprehensive real-time dataset at the monthly frequency. 6 We also considered a quadratic trend as in Ang et al. (2006) and the results are very similar to the HP filter. To conserve space we decided to report only the results of the HP filter. 9

10 are positive and the majority are also significant. This evidence confirms earlier results in the literature (e.g., Atkenson and Ohanian, 2001) of the difficulty of outperforming the random walk benchmark based on Phillips-curve models in the post-1984 period. The tests of relative accuracy of density/quantile forecasts of the PC models against the AO naïve model lead, in most cases, to the same conclusion reached above for the comparison of the mean forecasts. This is not surprising since the models differ in their conditional mean specification and thus evaluating their mean or distribution forecasts is very likely to lead to similar outcomes. It should be noted that no t n (SS) values are reported for AO - U X as its conditional mean forecast is the same as AO model by construction. The comparison of the density/quantile forecasts of the AO-U X models against the AO benchmark results in several significant rejections (negative t n (S) that are shown in bold) for the core measures of inflation, but only in few cases for PCE and CPI at both h = 6 and h = 12. The WQS tests show that the most benefit from including the macroeconomic indicators occurs on the left tail of the forecast distribution of inflation (see t n (WQS-left)). Although mainly limited to coreinflation, UNEM, HS and SPREAD indicators also show improvement over the AO model (see t n (WQS-center)) for predicting the middle part of the forecast distribution of inflation. For example, considering core-pce (for both h), the SPREAD indicator adds forecast improvement over AO model at all parts of the forecast distribution. Notice that even for the headline inflation measures, most of the economic indicators lead to better quantile forecasts (where t n (S) is negative), although the improvements are not statistically significant. 4.2 Is there episodic predictability? The relative forecast performance results reported in Table (1) and (2) are based on test statistics that average (aggregate) performances over a period of 23 years (1985:1-2007:12). It is possible that the tests may have overlooked differences in performance between models that occur at certain times during the forecast period. Here, we investigate the possibility of episodic predictability in the sense that economic indicators might provide more accurate forecasts only during certain times of the forecasting period. To do so, we conduct a fluctuation test that evaluates the (relative) performance of two competing forecast models and signals whether one of the forecasts is significantly (given appropriate critical values) more accurate at any point in time. Using similar notations as in Section 10

11 (3) and for a window size m, we define average score for model j at time t and for h = 12 by S j t = 1 m t+m/2 i=t m/2 S j i+h i, t (1990 : : 12) In this application we set m equal to 120 such that the average score for each t is based on a two-sided rolling window of 10 years. We follow Giacomini and Rossi (2010) and test the equal forecast performance of model k and j based on the test statistics F t (S) = m Sk t S j t ˆσ t (6) where ˆσ t is a heteroscedasticity and autocorrelation consistent (HAC) standard error estimator of the difference in scores S k i+h i Sj i+h i. The asymptotic distribution of F t(s) under the null hypothesis of equal performance is non standard. Critical values for various µ = m/n and two significance levels are provided in Giacomini and Rossi (2010). In our case, we have n = 276 and m = 120 resulting in µ Rejection occur at the 5% (10%) level against a two-sided alternative when max t F t (S) > (2.626), while the one-sided critical value is ±2.624 (±2.334). Note that the time variation of F t (S) also contains valuable information. Model j is more accurate at time t if F t (S) crosses the lower bound at time t and model k is more accurate F t (S) crosses the upper bound. To save space, we consider only the score function S = LS that compares density forecasts of the AO-U X model (j) against the AO model (k). The results for CPI and PCE are given in Figure (1) and those for core-cpi and core-pce are shown in Figure (2). Focusing on Figure (1a) which corresponds to CPI, observe that there is evidence of predictability by most economic indicators (F t (LS) takes a negative value) in the period , although none is statistically significant at conventional levels. Note that for INC GAP and HS, F t (LS) is negative (although not significant) during most part of For PCE (see Figure 1b), the pattern of the F t (LS) is similar to CPI with better predictability (compared to the AO model) after 1997 for most economic indicators. The best performance is achieved by HS for which F t (LS) is negative and significant at 10% in the periods and between mid-1995 and Figure (2a) gives the time path of F t (LS) of core-cpi. The fluctuation test for core-cpi shows a much better performance of the AO-U X compared to the case of CPI inflation, especially before Comparing Figure (2a) with Figure (1a), the most striking difference is observed for UNEM and HS where these economic indicators become relevant in forecasting core-cpi but are irrelevant to forecast CPI. Notice from Figure (2a) that the F t (LS) for both UNEM and HS are always 11

12 negative throughout the forecast period. Figure (2b) shows the time path of F t (LS) of core-pce. This measure of inflation is the one with the most evidence of predictability in the sense that all the economic indicators contribute to outperform the benchmark throughout the forecast period (F t (LS) is negative). Notice that HS provides significant predictability throughout the forecast period. In summary, there is ample evidence of predictability for all versions of inflation, although the evidence is stronger for the core versions of inflation, i.e. core-cpi and core-pce. In particular, the core-pce measure of inflation shows the most significant evidence of predictability notably when using UNEM and HS as predictors. The fluctuation test is a useful tool in uncovering relevant predictability information that is overlooked by t n (LS) test. Relying only on the t n (LS) test gives the impression that all the considered economic indicators are uninformative in predicting CPI and PCE versions of inflation. But the more detailed predictability picture provided by the fluctuation test shows that many of the indicators are indeed relevant, although higher performance seem to be overwhelmed by periods of poor performance which, on aggregate, results in statistically insignificant t n (LS) values. 4.3 Distribution forecasts of inflation In this paper we ask the question: are macroeconomic variables useful in forecasting the distribution (beyond the mean) of U.S. inflation in the post 1984 period? Several relative accuracy tests of density/quantile forecasts confirm that some economic indicators provide significant predictability of the distribution of inflation for large part of the post 1984 period. Further, most of the evidence of predictability occurs at the left tail of the inflation forecast distribution. To offer more insights on these results, we examine selected quantiles (α=5%, 50% and 95%) of the forecast distribution of inflation by focusing on core-pce which shows the most predictability. We summarize results for UNEM and HS indicators and focus on h = 12. Unemployment rate (UNEM) In the top portion of Figure (3), we display the quantile (at 5%, 50% and 95% ) forecasts of core- PCE for the period 1985:1-2007:12 based on the benchmark AO model (shown with broken lines) and the AO-U X (shown with solid lines) where X=UNEM. In the lower panel, we present the time series plot of UNEM. Note that it is shifted forward by 12 months so that it becomes aligned with the target date. For example, the UNEM value in January 1985 actually refers to that of 12

13 January 1984 which represents the value of UNEM used to produce the distribution forecasts for the target date (January 1985). Until about 1995, no noticeable differences are observed in the quantile forecasts of the two models. But, after 1995, the forecast distribution from AO-U X model shifts upward from the the AO model and the most gap between the quantile forecasts occurs at the 5% quantile level. Although not shown in the plot, large gaps also occur at other tail quantile levels. This observation may be attributed to the upward pressure on inflation derived from the persistent decrease in the unemployment rate in the late 1990 s as observed from the time series plot of UNEM (see the lower portion of Figure 3). In the late 1990 s and early 2000 unemployment was at historically low levels approaching 4%. Consistent with the existence of a trade-off between unemployment and inflation rates, the low unemployment rate shifted the whole forecast distribution of inflation to higher levels (compared to the AO model distribution forecasts), with a much larger shift at the lower tail quantile of the distribution. In addition, there appear to be a tendency for the forecast distributions of core PCE to gradually become narrower toward the end of the sample. This characteristic is common to both forecasting methods and can be attributed to the decline in inflation volatility captured by the rolling window estimation (see Jore et al., 2010, and Clark, 2011, for similar findings). By comparing the AO and AO-U X models both estimated on a rolling window we account for the post-1984 decline in inflation volatility, but we are also able to show that macroeconomic variables introduce additional time-variation in the quantile forecasts which, as the previous Section has shown, delivers more accurate forecasts compared to the AO model. Housing Starts (HS) Figure (4) displays quantile (5%, 50% and 95%) forecasts of core-pce for the period 1985:1-2007:12 from the AO model (shown with broken lines) and the AO-U X (shown with solid lines) where X=HS. Before 1995, the quantile forecasts (at both 50% and 5% levels) of the AO-U X closely track those of the AO model. On the other hand, the two quantile forecasts at the 95% level appear to differ appreciably. Although not shown, quantile forecasts at other right tail quantile levels behave similarly. This observation may be due to the strongest downward pressure on high levels of inflation resulting from the slow and steady decrease in housing starts (from mid-1980 s to beginning of 1990) as noticed from the time plot of HS (see the lower portion of Figure 4). Between 1995 and 1998, both models generate quantile forecasts that are closely comparable. After 1998, quantile forecasts start to diverge again while the most gap between the quantile forecasts 13

14 at the 5% quantile level. Notice from the time plot of HS (shown in the lower portion of Figure 4) that the slow and steady increase in housing starts (from 2001 to 2007) exerts an upward pressure on the whole distribution of inflation, with pressure seems to be more pronounced at the lower tail quantile. 4.4 Probability of deflation In the beginning of 1998 a debate started on the possibility of the U.S. economy entering a period of deflation that was also the topic of a speech given by the Federal Reserve Board Chairman on January 3rd, 1998 (see Greenspan, 1998). Using this historical fact as a motivation, we present a complementary (to relative density/quantile forecast evaluation tests) approach to assess forecast distributions of inflation by focusing on the accuracy of the methods in predicting the likelihood of deflation (negative inflation). We provide a brief illustration using core-pce and the unemployment rate with focus on h = 12. In Figure (5) we display probability of deflation for the period 1996:7-2007:12 as predicted by the benchmark AO model and AO-U X model where X=UNEM. From the Figure it can be observed that a forecaster using the AO model would have predicted the probability of deflation to be larger than 5% for several months over the years with the highest probability reaching 18.2% for the June 1999 target date. In contrast, for the same June 1999, AO- U X would have predicted only a 1.18% chance of deflation. Overall, the use of the unemployment rate leads to a likelihood of deflation close to zero throughout the 1996:7-2007:12 period, and thus appears to provide a more appropriate prediction of the event. 5 Conclusion Forecasting the behavior of inflation plays a central role in economic policy-making due to the inherently forward-looking nature of economic decisions. Typically, inflation forecasting focuses on modeling the conditional mean or the most likely outcome. While relevant, relying only on the dynamics of the conditional mean leaves out other interesting aspects of the inflation process, such as the dynamics occurring at higher moments of the inflation distribution. For example, the central bank may be interested in evaluating unattractive outcomes for the economy such as deflation or high inflation, and those models that rely only on the conditional mean will not offer the tools to do such evaluations. Accurate characterization of the complete distribution of future inflation, beyond the conditional mean, is needed. This paper examines whether indicators of economic activity carry relevant information about the 14

15 dynamics of higher moments of inflation, and hence help improve the accuracy of density forecasts. Our findings indicate that, in particular for the core inflation measures, conditioning the dynamics of the inflation distribution on the leading indicators provides more accurate forecasts relative to the random walk model. This is due to the relevance of the activity indicators in forecasting quantile effects that take place far away from the center of the core inflation distribution. We also investigate the possibility of episodic predictability, in the sense that economic indicators might provide more accurate density forecasts during limited periods of the forecasting sample. The results indicate that some variables (in particular housing starts and the unemployment rate) have this characteristic, in particular when forecasting PCE and core-cpi, while they provide consistent evidence of predictability for core-pce throughout the period Overall, our results indicate that economic variables are more useful indicators of the dynamics of the tails of the inflation distribution, rather than its center. This finding can be of particular interest for policy makers when evaluating the likelihood of certain events, such as whether inflation will be above or below a certain level in the future. 15

16 References Amisano, G. and Giacomini, R. (2007). Comparing density forecasts via Weighted Likelihood Ratio Tests. Journal of Business & Economic Statistics, 25, Ang, A., Bekaert, G. and Wei, M. (2006). Do macro variables, asset markets, or surveys forecast inflation better? Journal of Monetary Economics, 54, Atkenson, A. and Ohanian, L. (2001). Are Phillips Curves useful for forecasting inflation? Federal Reserve Bank of Minneapolis Quarterly Review, 25, Chernozhukov, V., Fernandez-Val, I. and Galichon, A. (2010). Quantile and probability curves without crossing. Econometrica, 78, Clark, T.E. (2011). Real-time density forecasts from Bayesian Vector Autoregressions with stochastic volatility. Journal of Business and Economic Statistics, 29, Cogley, T., Morozov, S. and Sargent, T.J. (2005). Bayesian fan charts for U.K. inflation: forecasting and sources of uncertainty in an evolving monetary system. Journal of Economic Dynamics and Control, 29, Corradi, V. and Swanson, N.R. (2006). Predictive density and conditional confidence interval accuracy tests. Journal of Econometrics, 135, D Agostino, A., Giannone, D. and Surico, P. (2006). (Un)Predictability and macroeconomic stability. European Central Bank, working paper No Fisher, J.D.M., Liu, C.T. and Zhou, R. (2002). When can we forecast inflation? Federal Reserve Bank of Chicago Economic Perspectives Giacomini, R. and Rossi, B. (2010). Forecast comparisons in unstable environments. Journal of Applied Econometrics,, 25, Giacomini, R. and White, H. (2006). Tests of conditional predictive ability. Econometrica, 74, Gneiting, T. and Raftery, A.E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102, Gneiting, T. and Ranjan, R. (2011). Comparing density forecasts using threshold and quantile weighted scoring rules. Journal of Business and Economic Statistics, 29, Greenspan, A. (1998). Problems of price measurement. Federal Reserve Board, Testimony and Speeches (January 3rd). Greenspan, A. (2004). Risk and uncertainty in monetary policy. American Economic Review, 94, Hodrick, R.J. and Prescott, E.C. (1997). Postwar U.S. business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29, Jore, A.S., Mitchell, J. and Vahey, S.P. (2010). Combining forecast densities from VARs with uncertain instabilities. Journal of Applied Econometrics, 25,

17 Kilian, L. and Manganelli, S. (2008). The central banker as a risk manager: estimating the Federal Reserve s preferences under Greenspan. Journal of Money, Credit, and Banking, 40, Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, Robertson, J.C., Tallman, E.W. and Whiteman, C.H. (2005). Forecasting using relative entropy. Journal of Money, Credit, and Banking, 37, Rossi, B. and Sekhposyan, T. (2010). Has models forecasting performance changed over time, and when? International Journal of Forecasting,, 26, Stock, J.H. and Watson, M.W. (1999). Forecasting inflation. Journal of Monetary Economics, 44, Stock, J.H. and Watson, M.W. (2007). Why has U.S. inflation become harder to forecast. Journal of Money, Credit, and Banking, 39, Stock, J.H. and Watson, M.W. (2008). Phillips curve inflation forecasts. NBER working paper. 17

18 Table 1: h = 6 X Method Core PCE Core CPI SS LS WQS-unif WQS-center WQS-left WQS-right SS LS WQS-unif WQS-center WQS-left WQS-right AR UNEM PC AO-U X IP PC AO-U X INC PC AO-U X WORK PC AO-U X HS PC AO-U X SPREAD PC AO-U X PCE CPI AR UNEM PC AO-U X IP PC AO-U X INC PC AO-U X WORK PC AO-U X HS PC AO-U X SPREAD PC AO-U X Test statistics t n(s) values (see Equation 5) shown in bold indicate that model AO - U X is significantly more accurate than the AO model at the 5% level.

19 Table 2: h = 12 X Method Core PCE Core CPI SS LS WQS-unif WQS-center WQS-left WQS-right SS LS WQS-unif WQS-center WQS-left WQS-right AR UNEM PC AO-U X IP PC AO-U X INC PC AO-U X WORK PC AO-U X HS PC AO-U X SPREAD PC AO-U X PCE CPI AR UNEM PC AO-U X IP PC AO-U X INC PC AO-U X WORK PC AO-U X HS PC AO-U X SPREAD PC AO-U X Test statistics t n(s) values (see Equation 5) shown in bold indicate that model AO - U X is significantly more accurate than the AO model at the 5% level.

20 (a) (b) Figure 1: Test statistics F t(ls) values where model j = AO-U X (X=UNEM, IP GAP, INC GAP, HS, SPREAD) is compared against benchmark k=ao model. The dashed horizontal lines represent the 5% (±2.890) and 10% (±2.626) critical values. The vertical lines indicate the NBER recession dates. (a) (b) Figure 2: Test statistics F t(ls) values where model j = AO-U X (X=UNEM, IP GAP, INC GAP, HS, SPREAD) is compared against benchmark k=ao model. The dashed horizontal lines represent the 5% (±2.890) and 10% (±2.626) critical values. The vertical lines indicate the NBER recession dates. 20

21 Core PCE ( h = 12, predictor= unem ) unem Figure 3: (Top plot) The (red) solid lines denote the quantile forecasts from j = AO-U X model (X=UNEM) for core-pce at 5, 50, and 95% quantile levels, the (blue) dashed lines denote same quantile levels for k = AO model and the circles show the realized core-pce inflation values. (Bottom plot) The unemployment series (shifted forward by h = 12 months). The vertical lines indicate the NBER recession dates. 21

22 Core PCE ( h = 12, predictor= hs ) hs Figure 4: (Top plot) The (red) solid lines denote the quantile forecasts from j = AO-U X model (X=HS) for core-pce at 5, 50, and 95% quantile levels, the (blue) dashed lines denote same quantile levels for k = AO model and the circles show the realized core-pce inflation values. (Bottom plot) The housing starts series (shifted forward by h = 12 months). The vertical lines indicate the NBER recession dates. 22

23 Figure 5: Estimated probability of deflation (defined as a negative inflation rate) for core-pce from k = AO model and j = AO-U X (X=UNEM) for the period 1996:7-2007:12. 23