Non-Linear Phillips Curves with Inflation Regime-Switching

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Non-Linear Phillips Curves with Inflation Regime-Switching Jeremy Nalewaik 2016-078 Please cite this paper as: Nalewaik, Jeremy (2016). “Non-Linear Phillips Curves with Inflation Regime-Switching,” FinanceandEconomicsDiscussionSeries2016-078. Washington: BoardofGovernorsofthe Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.078. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Non-Linear Phillips Curves with Inflation Regime-Switching Jeremy Nalewaik∗ Federal Reserve Board August 2016 Abstract Building on the results in Nalewaik (FEDS 2015-93), this work models wage growth and core PCE price inflation as regime-switching processes, whose characteristics in the 1970s, 1980s and early 1990s differ fundamentally from their characteristics in the 1960s and from the mid-1990s to present. The key innovation here is the addition to the models of fundamental driving variables like labor-market slack, and the evidence strongly suggests a non-linear effect of slack on wage growth and core PCE price inflation that becomes much larger after labor markets tighten beyond a certain point. The results are informative for assessing the likelihood and risks of meeting certain inflation targets on a sustained basis. ∗ Board of Governorsof the Federal Reserve System, 20th Street and Constitution Avenue NW, Stop 80, Washington, DC 20551 (e-mail: jeremy.j.nalewaik@frb.gov). Thanks to seminar participants and colleagues at the Board of Governorsfor comments. The views expressed in this paper are those of the author and do not necessarily reflect the views of the Board of Governors of the Federal Reserve System or the rest of its staff.

1 Introduction When the demand for a commodity or service is high relatively to the supply of it we expect the price to rise, the rate of rise being greater the greater the excess demand.1 Above is the first sentence of the paper that ushered in its namesake, the Phillips curve. The sentence could have been taken from any introductory economics textbook discussing price theory. Phillips (1958) then modestly suggest that it: “seems plausible that this principle should operate as one of the factors determining the rate of change of money wage rates, which are the price of labour services.” Two sentences later Phillips (1958) describes a notion that sounds much like downward nominal wage rigidity—see Akerlof, Dickens andPerry (1996)andDalyandHobijn(2014): “itappearsthatworkers arereluctant to offer their services at less than the prevailing rates when the demand for labour is low and unemployment ishighsothatwageratesfallonlyveryslowly.” Phillips(1958)thenconcludes the last sentence of his first paragraph with: “The relation between unemployment and the rate of change of wage rates is therefore likely to be highly non-linear.” Of the subsequent papers critical of Phillips (1958), one of the most prominent, Phelps (1967), found the nonlinearity assumption sensible, positing that the slope of the Phillips curve should approach minus infinity as the unemployment rate approaches zero. So, it is somewhat odd that this paper needs to qualify the term Phillips curve with “non-linear,” since non-linearity was central to the Phillips curve from the beginning. But that qualifier is indeed necessary, since most recent applied work using Phillips curves assumes linearity. This paper embeds non-linear Phillips curves into regime-switching processes for wage 1Phillips (1958), page 283. 1

growth andcore PCE price inflation, building on thework ofNalewaik (2015).2 Moderntime series models of the inflation process now routinely embed time-varying volatility and persistence; examples include Stock and Watson (2007), Cogley, Primiceri and Sargent (2010), Mertens (2011), and Clark and Doh (2014). The Markov-switching approach of Nalewaik (2015) does as well, and results here show one advantage of that approach is that a rich set of control variables—labor-market slack, a non-linear function of labor-market slack, the real dollar exchange rate, and bank lending—can be included into the model easily without the imposition of strong prior assumptions. In the Markov-switching models, the characteristics of wage growth and core PCE price inflation in the 1970s, 1980s and early 1990s differ fundamentally from their characteristics in the 1960s and from the mid-1990s to present. In the 1970s, 1980s and early 1990s, the effect of variation in the control variables on the inflation measures is cumulative, carrying over from one period to the next because the processes are non-stationary. In the 1960s and from the mid-1990s to present, the processes are stationary, so the effect of the control variables on the inflation measures does not carry over from one period to the next. In the stationary world, the terms natural rate of unemployment or NAIRU are meaningless, since level shifts up or down in the labor market slack time series are simply absorbed into the constant term of the equation, leaving model fit and forecasts unchanged. However, based on widely used estimates of the natural rate from the Congressional Budget Office (CBO), the paper shows a sharp steepening of the Phillips curve after labor-market slack becomes sufficiently negative, so the effect of slack on inflation becomes much larger after labor markets tighten beyond a certain point. These results suggest an alternative definition 2The core PCE price index is the price index for personal consumption expenditures (PCE) excluding food and energy. 2

of the natural rate of unemployment: the rate below which non-linearities in the Phillips curve begin to appear. With their rich set of control variables, the models here are designed to be useful for both forecasting and risk assessments.3 The estimated probabilities of the non-stationary inflation regime should be useful for assessing the likelihood that inflation expectations have become “unanchored,” or adaptive-causal to be more precise, meaning inflation expectations both respond strongly to past inflation and have a strong causal effect on subsequent inflation, thereby imparting non-stationarity to the inflation processes. Such a transition occurred once in the sample studied here, after the relatively high inflation in the second half of the 1960s. With the unemployment rate running well below the natural rate throughout that period, the effect of the non-linear Phillips curve likely caused that high inflation initially. However, that high inflation then appears to have entrenched elevated inflation expectations and institutional mechanisms for dealing with it like indexation of wage growth to past price changes, forces that probably kept inflation high even after labor markets slackened from 1969-70. While the models here show the regime transition occurred after that slackening of the labor market, some results suggest that an earlier slackening in 1967 or 1968 would have resulted in the same regime transition outcome. In other words, five years of elevated inflation generated by tight labor markets were enough to “unanchor” inflation expectations, but a shorter period of just a couple such years might have been sufficient to produce that outcome as well. That would be consistent with results from the models without conditioning variables in Nalewaik (2015), which show the transition into the non-stationary regime likely occurred in 1967. To the author’s knowledge, this paper is the first to embed a non-linear Phillips curve 3Out-of-sample forecast evaluation of the models, both pseudo and real, is left for future work. 3

into a Markov-switching model of the inflation process. Other recent papers on non-linear Phillips curves include Daly and Hobijn (2014), Kumar and Orrenius (2014), Fisher and Koenig (2014), and Speigner (2014). Other regime-switching models of the inflation process include Kim (1993), Evans and Wachtel (1993), Lanne (2006), and Davig and Doh (2014). 2 Model Structure The log difference of the core PCE price index from the fourth quarter of year t−1 to the fourth quarter of year t, π , and the annual average change in log wages, πw, both follow a t t two regime or two state (S = 1 or S = 2) Markov-switching process (see Hamilton (1989)), t t where: π µ+Θ X (σ )2 ρ σ σw  t S = 1,X  ∼ N  1 t−1 , 1 1 1 1  t t−1 πw µw +ΘwX ρ σ σw (σw)2  t   1 t−1   1 1 1 1        π S = 2,X , π +Θ X +θε (σ )2 ρ σ σw  t t t−1  ∼ N  t−1 2 t−1 t−1 , 2 2 2 2  πw π ,πw ,ε ,εw πw +ΘwX +θwεw ρ σ σw (σw)2  t t−1 t−1 t−1 t−1   t−1 2 t−1 t−1   2 2 2 2        The specification of thequarterly-frequency analogto this modelisdescribed inAppendix A. After conditioning on the explanatory variables X , the inflation measures are stationary t−1 in regime 1—hence the means µ and µw, while the measures follow a non-stationary process in regime 2, an MA(1) in differences with parameters θ and θw. The motivation for modeling inflation as this type of regime-switching process can be found in Nalewaik (2015), where total PCE price inflation follows an MA(2) in differences in its non-stationary regime. For parsimony, the work here uses the more traditional MA(1), the equivalent of modeling each of the inflation processes as the sum of a permanent and a transitory component—see Stock 4

and Watson (2007). The explanatory variables X include slack and a non-linear function t−1 of slack discussed in the next section. Let H = {1,π ,πw ,X ,π ,πw ,X ,...,π ,πw,X ,ε = ε = εw = εw = 0} t−1 t−1 t−1 t−1 t−2 t−2 t−2 1 1 1 0 −1 0 −1 denote the history of the inflation and explanatory variables, prob(S = 1 | H ) = t−1 t−1 p , and: prob(S = 2 | H ) = 1−p . The probabilities and likelihood funct−1|t−1 t−1 t−1 t−1|t−1 tion, a weighted average of the state-contingent likelihood functions with these probabilities as weights, is computed as in Kim (1994), with the probability of each state persisting from one period to the next governed by the transition matrix: p p 1−p p  t+1|t  =  11 22  t|t . 1−p 1−p p 1−p  t+1|t   11 22  t|t       The initial probability p is treated as an estimated parameter. 0 The key characteristics of regime 2 are: • Inflation is non-stationary, so the effect of X on inflation is cumulative. For example, t assume core PCE inflation is 2 percent in year t and slack imparts one percentage point of upward pressure, moving inflation to 3 percent in year t + 1. Through the lagged inflation term, the effect of that past pressure persists into year t + 2, and if slack continues to impart one percentage point of upward pressure each year, inflation then moves up to 4 percent in year t+ 2, and 5 percent in year t+3, and 6 percent in year t+ 4, etc. This cumulative effect provides the rationale for the term NAIRU (non-accelerating inflation rate of unemployment): as long as the unemployment rate remains below (above) the NAIRU in the non-stationary regime 2, inflation moves ever higher (lower). 5

• Thispersistence intheinflationprocess likely works, inlargepart, throughinflationexpectations. Nalewaik (2016) shows that expectations appear adaptive-causal in periods classified as belonging to the non-stationary regime 2, meaning that inflation expectations both (1) respond strongly to lagged inflation, and (2) appear to cause subsequent inflation. The evidence for the second condition is the strong approximately one-forone predictive power of expectations for subsequent inflation. Widespread awareness of high inflation by the general public and institutional mechanisms for dealing with it like indexation of wage growth to past price change likely explain the adaptive-causal features of inflation expectations in the non-stationary regime 2. In other words, high price inflation leads to high wage demands by the general public, which then facilitates higher price inflation asbusinesses pass onhigher wage costs—i.e., a wage-price spiral.4 • The most extreme tail risks to the inflation process, like the steep price declines in the U.S. Great Depression, or the hyperinflations seen in other counties, are possible in the non-stationary regime 2. The key characteristics of regime 1 are: • Inflation is stationary, so the effect of X on inflation is not cumulative. In the above t example, assume core PCE inflation is at its hypothetical mean of 2 percent in year t, and slack then imparts one percentage point of upward pressure, moving inflation to 3 percent in year t+1. If slack continues to impart the same degree of upward pressure each year, inflation simply remains at 3 percent in year t+2 and all subsequent years. 4SuchfeedbackeffectscouldoperatethroughthecommonlaggedexplanatoryvariablesX t−1inthemodel, particularlyslack. Specificationsallowingwageinflationtodependonlaggedpriceinflationaswellaslagged wage inflation show statistically significant evidence for such dependence, as well as the reverse, with price inflation depending on lagged wage inflation. However, allowing for these effects produced only modest improvements in model fit, so the paper reports results from the more parsimonious version of the model. 6

Absent non-linearities in the Phillips curve, the term NAIRU is meaningless in the stationary regime 1, as shifting UGAP up or down by any arbitrary constant has no effect on model fit or forecasts, since those shifts are simply offset one-for-one in µ. • Nalewaik (2016) shows that inflation expectations appear less adaptive and non-causal in periods classified as belonging to the stationary regime 1. So inflation expectations respond less strongly to lagged inflation in those periods, and, importantly, there is no evidence inflation expectations cause subsequent inflation. This lack of pass-though of inflation to inflation expectations and lack of a causal effect of inflation expectations on subsequent inflation likely explains the relatively-low inflation persistence in the stationary regime 1. Indifference of the general public to inflation when inflation is low and stable—see Akerlof, Dickens, and Perry (2000)—is one explanation for these results. • The probability of the most extreme tail risks to the inflation process materializing in the stationary regime 1 is negligible, vanishingly small. 3 Data Themodelinthispaperisreducedformratherthanstructural, buttheoreticalconsiderations guide the choice of explanatory variables X , which are labor-market slack in the fourth t−1 quarter of the prior year UGAP , a quadratic function ofslack when the gapturns negative t−1 −(min(UGAP ,0))2, the two-year change in the real dollar exchange rate lagged one year t−1 ∆XR (i.e. the log change in the real exchange rate from the fourth quarter of year t−1,t−3 t-3 to the fourth quarter of year t-1), and the fourth quarter to fourth quarter (Q4/Q4) log 7

difference inbanklending laggedoneyear ∆B , normalized byits 10-year moving average.5 t−1 The exchange rate explains only price inflation, not wage inflation, and bank lending drives inflation only in the non-stationary regime 2: Θ X = β ∆XR +β UGAP −γ (min(UGAP ,0))2 1 t−1 d,1 t−1,t−3 u,1 t−1 u t−1 Θ X = β ∆B +β ∆XR +β UGAP −γ (min(UGAP ,0))2 2 t−1 b,2 t−1 d,2 t−1,t−3 u,2 t−1 u t−1 ΘwX = βw UGAP −γw(min(UGAP ,0))2 1 t−1 u,1 t−1 u t−1 ΘwX = βw ∆B +βw UGAP −γw(min(UGAP ,0))2. 2 t−1 b,2 t−1 u,2 t−1 u t−1 The coefficients are free to vary across states except the coefficient on the quadratic function ofslack, whichiscommonacrossstates. Theremainderofthissectiondiscusses thesechoices. 3.1 Price and Wage Inflation Working with the four-quarter log change in the core PCE price index has the benefit of averaging away some of the higher-frequency noise in the monthly or quarterly changes, driven by residual seasonality, sampling error, and other reasons.6 After taking lags, the availablesampleextendsfrom1961Q1to2016Q1. Thewagegrowthmeasureiscompensation per hour in the non-farm business sector, the only well-known, publicly available U.S. wage 5These explanatory variables are lagged to facilitate the usefulness of the model for forecasting,but also becausewageandpriceinflationtendtolagmanyothereconomicindicators. Thelaggingnatureofinflation helps motivate the choice of frequency and forecast horizon; with inflation, looking a year ahead or more is prudent. 6U.S. Bureau of Economic Analysis, Personal Consumption Expenditures Excluding Food and Energy (Chain-Type Price Index) [PCEPILFE], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/PCEPILFE. 8

growth measure that is benchmarked to administrative tax records.7 Irregular payments like bonuses add considerable volatility to compensation per hour, and this paper works with annual average growth rates to smooth through some of this volatility. The initial estimates of both of these variables are subject to potentially important subsequent revisions. This paper uses the vintage of both time series available in June 2016.8 These annual growth rates are shown in Figures 9 and 10. The paper estimates the quarterly analog to the annual Markov-switching model on the annualized quarterly logchanges in core PCE price inflation, shown in Figure 12. 3.2 Labor-Market Slack The measure of slack UGAP used in the paper is the difference between the quarterly t unemployment rate from the Bureau of Labor Statistics and estimates of the natural rate of unemployment available from the CBO in March 2016. These two components, along with the non-linear term −(min(UGAP ,0))2 are plotted in Figure 1. The lack of variation in the t non-linear term over most of the sample motivates the decision to estimate γ using the full u sample, although the γ estimates in Tables 4-5 use only periods classified as belonging to u the stationary regime 1. Of course, the natural rate of unemployment is unobserved, so the CBO estimates are just that, estimates. As emphasized by Orphanides (2002, 2003), gap estimates available in real time can be revised in important ways later. For example, the natural rate estimates in Figure 1 range from 51 to 6 percent from the early 1960s to early 1970s, but in real time, 2 7The2015wageandsalaryestimateshavebeenbenchmarkedtotaxrecordsbutarestillsubjecttofurther adjustments, as are the supplements to wages and salaries in prior years. 8Examination of how the results in the paper might change if the data available in real time were used for estimation is left for future research. 9

economic policy advisers thought that “full employment” was 4 or 41 percent; see Barber 2 (1975)andDeMarchi (1975). This isimportant tokeep in mindwhen applying theresults in the paper to real time forecasting and policy analysis, but Tables 4 and 5 show that the main results in the paper hold when the raw unemployment rate U is substituted for UGAP . t t Providing a complete theoretical justification for the relation between slack and inflation is beyond the scope of the paper, but in applied empirical work, the wage Phillips curve is often motivated by the same type of supply/demand considerations as discussed by Phillips (1958). Speaking loosely, the assumption is that “tight” labor markets with low unemployment rates typically coincide with relatively strong labor demand that bids up wage growth. The price Phillips curve is typically motivated either by pass through of wage inflation to price inflation, appealing to the large share of labor compensation in production costs, or by making similar arguments about product markets “tightening” at low unemployment rates. Again speaking loosely, the assumption is that tight labor markets boost demand for some products, leading to scarcity and price inflation. For some markets like those for apartment and house rentals, direct measures of “tightness” such as rental vacancy rates are observable. 3.3 Bank Lending This variable is loans and leases in bank credit, all commercial banks, seasonally adjusted.9 The four quarter log change in this variable, lagged one year and normalized by its 10-year moving average to make it approximated mean zero, is plotted in Figures 2 and 3 with four- 9Board of Governors of the Federal Reserve System (US), Loans and Leases in Bank Credit, All Commercial Banks [LOANS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/LOANS. 10

quarter core PCE price inflation and annual-average compensation-per-hour growth.10 Even lagged one year, it is clear that bank lending growth had some predictive power for wage and price inflation from the mid-1970s to early 1980s, when both measures surged, came back down, and then surged again. The idea that rapid bank lending growth might be inflationary is not new. For example, struggling to bring down inflation in the aftermath of World War II, president Truman and his economic advisers sent a “Special Message to the Congress” in November 1947, and its first policy proposal was: “1. To restore consumer credit controls and to restrain the creation of inflationary bank credit.”11 The effect of bank lending on inflation could work through a number of potential theoretical channels; a few are listed below: • Bank lending could be a proxy for effects of the rate of change in real economic activity on inflation, sometimes called “speed” effects. As discussed in the original Phillips (1958) and also Gordon (1980), these effects are distinct from the effect of slack on inflation, since slack measures the level of economic activity (relative to its trend) instead of its rate of change. Romer (1996) suggests “speed” effects stem largely from the behavior of raw materials prices, which could move with the rate of change of economic activity because their supply is relatively inelastic. • In the “debt-deflation” theory of Fisher (1933), private sector attempts to liquidate aggregate debt put downward pressure on the aggregate price level. Movements in bank lending should reflect this dynamic, which could provide another explanation for rate of change or “speed” effects. Indeed, these effects appeared prominently in the 10The approximate meanzero property is important in the non-stationaryregime 2, giventhe cumulative property of the X t−1 variables on π t . 11See Goodwin and Herren (1975), page 31. 11

dynamics of the Great Depression, when PCE price inflation showing remarkably high correlation with real GDP growth—see Figure 4. • Bank lending could be a proxy for the effect of money growth on price inflation, due to the importance of bank lending for the money multiplier. Meltzer (2005) proposes money growth as an explanation for the behavior of inflation in the 1970s and early 1980s. More recently, during the first ofseveral quantitative easing programs, members of the Federal Open Market Committee (FOMC) discussed the role of bank lending in determining whether the large increase in bank reserves generated by the program would result in inflationary pressures: CHAIRMANBERNANKE. Thankyou. Just acouple ofcomments. Saying that reserves facilitate deposits implies that the banks have to lend the money to the consumers. MR. KOCHERLAKOTA. Yes. CHAIRMAN BERNANKE. So they are at the moment not lending, as you know, and if they were to lend, it would show up in inflation but also show up in aggregate demand.12 After conditioning on the other explanatory variables, the annual-frequency Markovswitching model estimates showed nosignificant positive effect oflaggedbanklending growth onpriceorwageinflationinthestationaryregime1, sothemodelrestrictsthelaggedeffectof bank lending to appear only in the non-stationary regime 2. However, bank lending growth has remained contemporaneously positively correlated with both price and wage inflation over the past 20 year (see Figures 5 and 6). 12Transcripts of November 11th, 2009 meeting of the Federal Open Market Committee, 12

3.4 The Real Dollar Exchange Rate Figure 7 plots the two-year change in the quarterly real broad dollar exchange rate, lagged one year and on an inverted scale, against core PCE price inflation.13 Some predictive power for price inflation appears from the mid-1970s to mid-1980s. Figure 8 examines the last twenty years in isolation, showing the continued predictive power of the exchange rate for inflation fluctuations. Clearly, an increase in the exchange value of the dollar tends to hold down the dollar price of imported consumer goods and thus core PCE price inflation. While much of this effect may be quite rapid, as in the model in Yellen (2015), the long lagged effect shown here suggests some ofthe pass throughof exchange rate movements toconsumer prices is delayed. Or the lagged effect may represent a combination of other factors: • In recent times, movements in the dollar exchange rate often have been negatively correlated with movements in the price of crude oil, an important input into the production of many goods and services. These movements in production costs may take time to pass through to consumer prices. • Dollar exchange rate movements have been similarly negatively correlated with movementsinarangeofothercommoditypricesthatcouldhavelaggedeffectsonproduction costs and consumer prices. • One explanation for the relatively low core PCE inflation in the second half of the 1990s was rapid productivity growth, which may have put downward pressure on price inflation even though wage increases were quite robust. At the same time, that surpris- 13Board of Governors of the Federal Reserve System (US), Real Trade Weighted U.S. Dollar Index, Broad [TWEXBPA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/TWEXBPA. Prior to 1973, the exchange rate change is set to zero, reflecting adherence to the Bretton-Woods system of fixed exchange rates. 13

ingly fast productivity growth may have produced upward pressure on the exchange value of the dollar, contributing to the correlation shown in Figure 8. Due in part to that experience in the second half of the 1990s, the Markov switching models found no strong relation between lagged exchange rate movements and wage growth, so the exchange rate was not included in the set of explanatory variables for wage growth.14 4 Empirical Results 4.1 Prologue: Interpreting the 1960s The 1960sprovide much (but not all) of the identification of the Phillips curve non-linearities in the 1961Q1 to 2016Q1sample studied here. A vast body of work has studied why inflation took off in the 1960s, both contemporaneously and retrospectively—see Cochrane (1975), De Marchi (1975), Meltzer (2005), andLevin andTaylor (2010)forreviews. Muchof itexamines monetary policy, fiscal policy, and their interaction, showing how policy errors may have led to the 1960s inflation. But those are secondary causes, and this paper focuses on the list of possible primary causes of inflation—i.e. the channels through which the policy errors had their effect on inflation. The list includes: 1. Real activity effects, which work through slack and perhaps the other variables capturing speed effects. 2. Expectations effects, which are likely operable through lagged inflation in the nonstationary inflation regime. 14TheeffectofimportedgoodspriceinflationonwagegrowthwasdiscussedintheoriginalPhillips(1958), which concluded its effect could be ignored except under exceptional circumstances. 14

3. Money supply effects, which are potentially operable through bank lending in the nonstationary inflation regime. For example, if government spending increases were the secondary cause of the 1960s inflation, they may have operated through the primary causes of either real activity effects driving down the unemployment rate or perhaps higher inflation expectations that had a positive causal effect on subsequent inflation. Similarly, a secondary cause of excessively loose monetary policy could have operated through any of the above three primary causes, so disentangling them is useful. Some other potential explanations for the rise of inflation in the 1960s, like the power of labor unions to secure outsized increases in compensation per hour, might have operated outside of the three channels listed above. But as shown by the leftmost column of data in Table1,compensationperhourgrewarelativelymodest3to4percentperyearthrough1965. The “moral suasion” of government officials urging adherence to “guideposts” for wage and price setting might have restrained wage growth over this period, but that restraint largely disappeared in 1966 after the unemployment rate fell below 4 percent, which surely improved the bargaining position of labor.15 Those are precisely the type of developments that are supposed to be captured by Phillips curves, even if the mechanics of how labor contracts are set may have changed over time.16 Regarding expectations effects, Table 1 shows a proxy for “inflation expectations”, forecasts from the semi-annual Livingston survey of professional forecasters of CPI inflation over 15See Barber (1975) and Cochrane (1975). 16As discussed below, since the Phillips curve model residuals for compensation-per-hour growth are not systematically positive in the 1960s,they do not suggest extra-model factors led to excessive wage gains. 15

the next 12 months.17 As discussed in Nalewaik (2016), movements in inflation expectations are most likely to have a strong positive causal effect on subsequent inflation when they are widespread throughout the general public, driving demands that wage growth keep up with price inflation and allowing businesses to pass along price increases expected by their customers. As such, data on the inflation expectations of the general public would be more informative for whether inflation expectations drove inflation in the 1960s, but such data are unavailable. With surveys of professional forecasters, any positive correlation between inflation forecasts and inflation realizations may simply reflect predictability in the inflation process known to the forecasters, rather than a causal effect of their forecasts on subsequent inflation. For example, Table 1 shows the CPI inflation and unemployment rate readings that prevailed at the time each Livingston inflation forecast was made; the upward drift in the inflation forecasts might have been a passive response to prior downward movements in the unemployment rate, as many forecasters were doubtless aware of the results in Phillips (1958). Indeed, the timing of those unemployment rate declines lines up well with the wage and price accelerations from 1966 to 1968, suggesting a tightening labor market may have been the key driving factor. The RVAC column shows that the rental vacancy rate t followed the unemployment rate down over this period, lending credence to the Phillips curve mechanism of tightening labor markets boosting demand for some products, leading to scarcity and price inflation. In contrast, inflation expectations appear to have lagged behind CPI inflation through most of the 1960s. For example, CPI inflation moved up to 1.6 percent and then well above 2 percent six months before the Livingston forecasts reached 17Federal Reserve Bank of Philadelphia, Livingston Survey, https://www.philadelphiafed.org/researchand-data/real-time-center/livingston-survey. 16

those levels in December 1965 and December 1966, not suggestive of a strong causal effect of inflation expectations on subsequent inflation. When did inflation expectations begin to have a strong causal effect on subsequent inflation? The estimated probabilities of the non-stationary inflation regime provide an answer, assuming the non-stationarity is driven by adaptive-causal inflation expectations. The onesided probabilities in the last column of Table 1 (from the unrestricted model estimates in Table 3 below) show these effects appear after the 1969-70 recession. Before then, the model explains and predicts inflation with just the non-linear Phillips curve, but after, something else is necessary to explain the difference between 1 to 2 percent inflation in June 1963 and 4 to 5 percent inflation in June 1971 with similar unemployment rates. That additional ingredient is non-stationarity in the inflation process. By the early 1970s, multiple years of high inflation likely entrenched high inflation expectations, consistent with the Livingston forecasts of continued elevated inflation, and those expectations were likely an important causal determinant of subsequent inflation by that point.18 Core PCE price inflation was between 3 and 5 percent from 1966 to 1970, so five years of core inflation in that range certainly appears to have been sufficient to entrench high inflation expectations and produce a transition to the non-stationary inflation regime. Would a shorter period of relatively high inflation have produced the same outcome? A period as short as a single year of core inflation around 3 percent seems unlikely to produce a regime transition, sincetheinstitutional mechanisms thatmakeexpectationscausalforactualsubsequent inflation, mechanisms like indexation of wage growth to past price changes, probably 18Some contemporaneous narrative evidence corroborates that notion. For example, economists at the CouncilofEconomicAdvisers(CEA) wereforecastinginflationwithastationary(andnon-linear,ofcourse) Phillips curve in 1970, but by 1971, they concluded that the model had broken down because inflation expectations had become a more important driving force in the inflation process—see De Marchi (1975). 17

take time to develop and become widespread. Further, the regime probabilities from the annual models estimated below tend to smooth through transitory fluctuations. However, it would be too sanguine to conclude that core PCE inflation of 3 percent or higher for a two-to-four-year period does not increase the risk of a regime transition substantially, for at least three reasons. First, the unrestricted model attributes the pick up in core PCE price inflation from 3 percent from 1966-7 to between 4 and 5 percent from 1968-9 to the decline in the unemployment rate from just under 4 percent in 1967 to around 31 percent in 1968. While the 2 statistical evidence favors that interpretation, it is possible that the Phillips curve is not so non-linear. Inthatcase, thestepupincoreinflationin1968mayhave reflected slack starting to have a cumulative effect on inflation after a transition to the non-stationary regime—see section 2. The Markov switching models estimated with constraints below do have Phillips curves that are more linear, and the one-sided probabilities from such a model shown in Table 1 do interpret the move up in inflation in 1968 as marking a regime transition. So, it is possible that the regime transition occurred after just two years of 3 percent core PCE inflation.19 Second, while the unrestricted, highly non-linear effect of slack on inflation does not appear to be cumulative in the 1960s, that cumulative effect may have been present but masked by offsetting shocks or movements in other conditioning variables. An obvious candidate is the 1969-1970 recession itself, as the unemployment rate began to edge up in the second half of 1969 and rose throughout 1970. The rate of change or “speed” effects in the models are lagged one year rather than contemporaneous, but contemporaneous “speed” 19The two-sided probabilities from this model show the parameters are estimated restricting the entire 1960stobegovernedbythenon-stationaryregime. However,theone-sidedprobabilitiesshowthestationary regime fits the data better up until 1968. 18

effects could have been present, offsetting the cumulative effect of slack after a transition to the non-stationary inflation regime in 1969 or 1970. If that is the case, three to four years of relatively high inflation may have been sufficient to cause a regime transition.20 Third, had the 1969-1970 increase in the unemployment rate occurred earlier, core inflation still might have remained high rather than settling back down between 1 and 2 percent. While they are occurring, multiple years of high inflation do not increase the probability of a regime transition in the models if they are well explained by a Phillips curve with a low unemployment rateanda stablemean. However, theymay markedly increase theprobability of a regime transition later by entrenching high inflation expectations that keep inflation elevated even after the unemployment rate moves up, detaching inflation from its stable mean. Formally, this can be modeled by making the transition matrix parameters functions of the length and intensity of non-linear Phillips curves effects over the recent past, interacted with a variable measuring whether slack is still putting upward pressure on inflation. One quarterly model below includes these time-varying transition probabilities, and finds that after the unemployment rate fell sufficiently far below the natural rate in the mid-1960s, a regime transition may have been inevitable when the upward pressure from slack abated. 4.2 Annual-Frequency Markov-Switching Model Estimates The paper considers two different sets of model parameter estimates. To force the non-linear Phillips curve to compete with expectations effects and money supply effects in explaining the 1960s, the first set of estimates imposes non-stationarity throughout that period. By constraining p = 1 and p = 0, inflation in this version of the model must start in the non- 11 0 20Contemporaneous “speed” effects are not included in the models because they are designedto be useful for forecasting,but the negative residuals fromthe model in recessionyears like 2008and 2009suggestthey do exist. 19

stationary regime in 1961 and cannot return to the non-stationary regime after a transition out of it; the model then imposes non-stationarity on the 1960s because it is necessary to adequately fit subsequent inflation variation in the 1970s and 1980s. The top panels of Table 2 report parameter estimates for Q4/Q4 core PCE price inflation and annual average compensation per hour growth. The two-sided probabilities from this version of the model, not shown, show the single possible transition to the stationary regime occurs in 1996. The bottom panels show the unrestricted parameter estimates of the model. As can be seen in the rightmost column, dropping the restrictions increases the maximized likelihood function substantially, easily passing a likelihood ratio test.21 The largest change is in the γ u and γw estimates, showing the size of the non-linearity in the Phillips curve approximately u triples when moving from the restricted to the unrestricted specification. Standard errors computed using numerical derivative outer product estimates are reported in parentheses below the parameters. The standard errors around some parameter estimates are quite small and might overstate the statistical significance of some of the unconstrained parameter estimates (see the previous footnote), but the non-linearities are also highly statistically 21Onacomputationalnote,thenegativeoftheseloglikelihoodfunctionswereminimizedusingtheinterior point algorithm of the fmincon function in Matlab R2016a. For the unrestricted version of the model (but not the restricted), the minimization results depended on the vector of parameters chosen to initialize the algorithm, and the results reported in table 2 are the global minimum across minima from a large array of initializations. The differences across these many minima were relatively minor. For example, considering 253 sets of initializations, over 90 percent of the minima yielded negative likelihoods less than 60, relatively close to the 55.4 shown in table 2, and of these, the mean value of γ was -0.51 with a standard deviation u across the set of minima of 0.02. The main results of the paper would not change materially if any of these local minima were reported instead of the global minimum in table 2. However, these computational issues suggest two points. First, it is always possible that examination of an even wider array of initializations couldhaveproducedalowerglobalminimum, in whichcasethe gapbetweenthe restrictedandunrestricted likelihoods would be larger than in Table 2. Second, small changes in data or rounding issues may cause jumps acrosstheserelativelycloselocalminima,andthatpossibilitysuggeststhe standarddeviationsacross the globalminimamightgiveabetter senseofthe uncertaintyaroundthe parameterestimatesthansomeof the standarderrorsin table 2. Example include µ, where the standarddeviationacrossthe set of relativelyclose minima was 0.04 arounda mean of 1.66; µw, with a standard deviationof 0.07 arounda mean of 3.86, and γw, with a standard deviation of 0.04 around a mean of -0.28. u 20

significant in the OLS regression results reported in the next subsection. The relatively low core PCE inflation in the 1960s holds down the unrestricted estimate of µ, at 1.61, compared to the restricted 1.74 estimate fit to 1996-2015 only. The restricted estimate is probably closer to the current long-run mean of core PCE price inflation in the stationary regime with zero slack and a stable real dollar exchange rate.22 The two estimates of the mean growth rate of compensation per hour, µw, are close to identical, not suggestive of special factors producing outsized wage growth in the 1960s. One- and two-sided probabilities from the unrestricted model, with data and year-ahead forecasts, are shown in Figures 9 and 10. As discussed in the previous subsection, the nonstationary regime probabilities are low through the 1960s as the non-linear Phillips curve explains the pickup in both core PCE price inflation and compensation-per-hour growth. Given the low non-stationary regime probability, the model predicts a sharp drop in both inflation measures in 1971 after the increase in the unemployment rate in 1970; when that does not occur, the probability of the non-stationary regime shoots up close to 100 percent. In the non-stationary regime, the model predicts well the disinflations in the early 1980s and early 1990s, aided by the addition of lagged bank lending growth to the model. When the two inflation measures stabilize around their stationary regime means in the mid-1990s, the probabilities show the model then infers a transition to that regime. Nalewaik (2016) discusses potential explanations for this change in the inflation regime in the 1990s, includ- 22The slightly lower mean in the 1960s could have been the result of a number of factors, including the effortsbytheKennedyandJohnsonadministrationstoactivelyholdpriceinflationclosetozero. Forexample, presidentKennedyusedthreatsofantitrustactiontoforcemajorsteelcompaniestorollbackpriceincreases in 1962—see Barber (1975). Other than holding down µ, it is not clear what effect these interventions had on model parameters like the slope of the Phillips curve. Such interventions are probably rarer today, but some major sectors of the economy such as health care are subject to more government controls than in the 1960s, and the relatively small increases in Medicare and Medicaid reimbursement rates to doctors and hospitals were an important factor holding down core PCE price inflation from 2011 to 2015—see Clemens, Gottlieb, and Shapiro (2016). 21

ing (1) the stabilization of inflation expectations and (2) the disappearance of a one-for-one causal effect of inflation expectations on subsequent inflation. Supporting the second explanation, available measures of inflation expectations became either uncorrelated with or negatively correlated with subsequent inflation afterthe 1990s. This mayhave been theresult of economic decision-makers becoming “rationally inattentive” to aggregate inflation after it stabilized at low levels—see Akerlof, Dickens, and Perry (2000). In the stationary inflation regime starting in the mid-1990s, most of the variation identifying the Phillips curve non-linearity occurs from 1998 to 2001, when the unemployment rate fell about a percentage point below the current natural rate estimates. Figure 10 shows compensation-per-hour growth was actually higher than predicted by the non-linear Phillips curve over that period, by an average of 3 percentage point. In contrast, core PCE price 4 inflationwas a littlebelow themodel forecasts, but not by much, asthe laggedeffect ofdollar exchange rateappreciation—see Figure8—roughlyoffsetstheeffect ofthenon-linearPhillips curve over much of that period. The experience of the late 1990s, then, is broadly consistent with non-linear Phillips curve parameters estimated assuming the 1960s and 1996-2015 were governed by the same stationary inflation process. For core PCE price inflation, the two standard deviation range around the mean in the stationary regime is quite narrow, extending from 0.9 to 2.3 percent conditional on zero slack and a stable real dollar exchange rate. That range is considerably tighter than the 0.4 to 3.3 percent range Nalewaik (2015) found for total PCE price inflation using a longer sample and no conditioning variables. Core PCE price inflation occupied a narrow range from 1996-2015, accounting for part of this difference; total PCE price inflation moved more due to large swings in energy prices. Conditioningonexplanatoryvariablesexplainspartofthedifferencebetweentheseranges 22

aswell, andthenon-linearPhillipscurveinFigure11illustrateshowcorePCEinflationvaries more widely with slack in the stationary inflation regime. With the unrestricted mean of 1.61 percent for core PCE price inflation, an unemployment rate 0.8 percentage point below the natural rate generates 2 percent core PCE price inflation. The curve then steepens sharply as the unemployment rate falls further below the natural rate, producing a wide range of inflation outcomes over a relatively small range of unemployment gap values. For example, unemployment rates 1.5, 2.0, and 2.5 percentage points below the natural rate generate 3, 4, and 5 percent core PCE price inflation, respectively.23 For reference, the CBO estimates show the unemployment rate was 2.0 percentage points below the natural rate at the end of 1966 and 2.5 percentage points below it at the end of 1969. The next subsection section examines a specification with time-varying transition probabilities where years of high inflation generated by such large negative unemployment gaps entrench high inflation expectations and increase the probability that inflation does not come back down after the gap normalizes. 4.3 Quarterly-Frequency Markov-Switching Model Estimates The first two sets of results in Table 3 show restricted and unrestricted parameter estimates from the quarterly version of the model described in Appendix A. The purpose of this model is not to forecast inflation one quarter ahead, but rather to allow the model to be updated more frequently than once per year. The model is structured to predict annualized quarterly core PCE price inflation four quarters ahead, and these realizations and four-quarter ahead forecasts are shown with one- and two-sided probabilities from the unrestricted model in 23With the milder non-linearities from the restricted set of parameter estimates in the top panels of table 2, a unemployment rate 1.0 percentage point below the natural rate generates 2 percent core PCE price inflation, while a rate 2.5 percentage points below generates 3 percent inflation. 23

Figure 12. This quarterly version of the model is univariate; the annualized quarterly growth rates of compensation per hour are too volatile to include in the model. Further, this quarterly version of the model should be used more cautiously than the annual version for forecasting and risk assessments, since it is estimated on latest-vintage data that could be quite different from that observed in real time because of revisions to seasonal factors. As in Table 2, the likelihood improves substantially when the restrictions on P and 11 P are relaxed, showing the stationary regime fits much of the 1960s better than the non- 0 stationary regime. The γ estimates governing the non-linearity in the Phillips curve are virtually identical to those in the annual-frequency specification, with the largest differences in the non-stationary regime coefficients.24 The timing of the regime shifts is similar: the one- and two-sided probabilities of the non-stationary regime jump from negligible to close to 100 percent in 1971Q1 and 1970Q4, respectively. The non-stationary regime probabilities then remain close to 100 percent until 1994-1995, when they begin to move down just a bit sooner than in the annual-frequency version of the model. The two-standard deviation range around the mean of the annualized quarterly growth rates in the stationary regime extends from 0.6 to 2.6 percent, conditional on zero slack and a stable real dollar exchange rate. For these quarterly estimates, this range could be wider for the initial estimates available in real time if subsequent data revisions smooth some volatility out of them. Further, inflation readings must be far outside of that range or persist outside of that range for multiple periods for the non-stationary regime probability to rise appreciably, given the very high regime persistence probability p . A single transitory 11 24The step-down in inflation from 1971Q3 to 1973Q1 was likely due to the price controls imposed by the Nixon administration in late 1971. These controls had little effect on the parameter estimates reported in table 3; replacing inflation in those quarters with its average over the prior year (1970Q3 to 1971Q2) produced similar estimates. 24

reading or two just outside the range would only increase the estimated standard deviation in the stationary regime, widening the range. Given the similarity of the µ, β and γ parameters across the annual- and quarterlyu,1 u frequency specifications, the non-linear Phillips curve derived from the quarterly-frequency parametersisverysimilartotheoneplottedinFigure11, andshowsthatsufficiently negative values for slack can generate core PCE price inflation well above 2.6 percent in the stationary regime. However, when such high inflation readings appeared in the 1960s, they may have entrenched high inflation expectations and increased the probability of a regime transition once the upward pressure from slack abated by making it unlikely that inflation would move back down to µ. The last specification of Table 3 examines this possibility by allowing for time variation in the stationary regime persistence probability, modeling p as a logistic 11,t function of the average non-linear Phillips curves effect over the past five years multiplied by a term measuring the extent to which the non-linear effect has diminished relative to its peak value over that period. The details of the specification are in Appendix A. With only one transition out of the stationary inflation regime in the sample, identifying time variation in the transition probability is somewhat tenuous. However, Table 3 shows allowing for such time variation does improve the maximized value of the likelihood. Given the model parameters, p falls from very close to 1.0 to around 0.2 in the second half of 11,t 1970 when the upward pressure on inflation from slack begins to diminish. What if that upward pressure had diminished sooner? By the end of 1966, the unemployment rate had fallen two percentage points below the natural rate, and had been more than a percentage point below the natural rate for a year and a half. In this specification, the upward pressure on inflation in 1966 and 1967 generated by those low unemployment rates is enough to guarantee a transition out of the stationary inflation regime, as p would have been close 11,t 25

to zero had the non-linear Phillips curve pressure disappeared in any quarter after 1967Q3. So the inevitability of the regime transition may have been decided in 1967, a result that agrees with the smoothed probabilities from the unconditional model of Nalewaik (2015), which show the transition out of the stationary inflation regime occurred that year. 4.4 Regression-Based Tests Tables 4 and 5 examine the robustness of the non-linearities in the Phillips curve using OLS regressions estimated on annual-frequency data over periods classified by the Markovswitching models as belonging to the stationary inflation regime, 1961-1970and 1995-2015.25 The last specification of the top panel of each table shows regression results explaining inflation with slack, the quadratic function of slack when the gap turns negative, a lagged dependent variable, and the dollar exchange rate for core PCE price inflation. Newey-West standard errors with two lags are below each coefficient in parentheses. The estimates are similar to the stationary-regime coefficients in Tables 2 and 3, with the non-linear slack coefficient a bit smaller for core PCE price inflation and a bit larger for compensation-perhour growth. The remainder of the specifications in the top panels of Tables 4 and 5 run linear regressions of the inflation measures on slack, with or without controls, either allowing for breaks in the slack coefficient or not. The break points κ are estimated as in Hansen and Seo (2002), and the heteroskedasticity-robust sup-LM statistic from that paper tests the significance of the breaks, computing asymptotic p-values with their fixed regressor bootstrap. This alternative approach to modeling non-linearity in the Phillips curve produces a fit very similar to 25The two-sidedprobabilitiesfromthe quarterly-frequencymodel include 1994and1995in the stationary regime, while those from the annual-frequency model do not; the sample chosen here splits the difference. 26

that producedby thequadratic, withbreak pointsforcorePCE price inflationmimicking the quadratic which steepens sharply when the slack is less than minus one percent. The break points for compensation-per-hour growth are higher, showing no negative relation between slack and compensation growth when slack is above two percent. For slack below that cutoff, the Phillips curve slope is steeper than minus one. The non-linearities are all statistically significant according to the sup-LM tests. Amajor issuewithunemployment gapmeasuresisthatthenaturalrateofunemployment isunobserved. However, thebottompanelsofTables4and5showthatsimilarnon-linearities are evident in the relation between the inflation measures and the raw unemployment rate. The specification explaining core PCE price inflation with controls shows a Phillips curve slope about ten times the slope from the linear specification when the unemployment rate is below 4.8 percent. For compensation-per-hour growth, when the unemployment rate is below 5.0 percent, the Phillips curve slope is about four to five times the slope from the linear specification. These non-linearities are statistically significant. The original Phillips (1958) curve used a function similar to the inverse of the unemployment rate to explain wage growth, and the regressions labeled (U −κ)−1 explore a similar specificationsubtractingaconstantfromtheunemployment ratebeforetheinverseistaken.26 This approach fits the data about as well as the approach allowing for discrete breaks in the coefficients, especially for compensation per hour growth. These results show the nonlinearity in the Phillips curve can be modeled in a number of ways that yield broadly similar results, suggesting the precise form of the non-linearity is not particularly important. The last specifications of table 5 show results for compensation per hour growth using 26These specifications are obviouslynot defined for unemployment rates below these κ values, 2.6 percent for core PCE price inflation and 1.6 percent for compensation-per-hour growth. 27

only 1995 to 2015. Statistically significant Phillips curve non-linearities exist even in this recent shortsample period. The formresembles thetoppanelofthetable, withaflatPhillips curve when the unemployment rate is above 6.9 percent, and a slope steeper than minus one when the unemployment rate is below that cutoff. 4.5 Informal “Out-of-Sample” Evaluation While the model samples used here are constrained by the quarterly core PCE price index extending back to only 1959, other price indexes extend back farther at the annual frequency. Annual estimates of the unemployment rate from Lebergott (1957) extend back quite far as well, showing very low values at certain points in U.S. history. It is instructive to examine, at least informally, to what extent those periods corroborate non-linearities in the Phillips curve. Moving backwards in time, the total PCE price inflation of close to 3 percent and the compensation-per-hour growthof6 percent observed in1956and1957areaboutwhat Figure 11 would predict from the observed unemployment rates around 4 percent and CBO natural rate estimates around 5.4 percent. Very low unemployment rates in the Korean War, below 3 percent, were accompanied by low inflation, but this was likely due to wartime wage and price controls, anticipation of which appears to have led to the surge in inflation in late 1950 and early 1951—see Goodwin and Herren (1975).27 From 1946-1948, the unemployment rate was low, just below 4 percent, and total PCE price inflation was quite high, consistent with a steeper non-linear Phillips curve than shown in Figure 11. The probabilities in Nalewaik (2015) suggest this period was governed by the 27AroundtheKoreanwar,thenegativePCEpriceinflationratesobservedin1949and1954wereprobably the result of strong contemporaneous “speed” effects, as the unemployment rate spiked up in both years. 28

non-stationary inflation regime, but this evidence is broadly consistent with a non-linear Phillips curve. Wartime wage and price controls also confound the interpretation of the data during World War II, but annual total PCE price inflation was generally quite high, averaging over 7 percent, and the unemployment rate was very low, often below 2 percent, again broadly consistent with a non-linear Phillips curve. Examination of the international evidence for Phillips curve non-linearities is beyond the scope of this paper, but, of course, the original Phillips (1958) provided supportive evidence from a sample of data on the United Kingdom extending back 100 years. It is also noteworthy that the unemployment rate in Japan has been relatively low through most of modern history, and its price Phillips curve has been quite steep, with a coefficient close to minus one—see Nalewaik (2016). More broadly still, through the addition of the bank lending conditioning variable in the non-stationary inflation regime, the Markov-switching models here might be capable of anticipating the most extreme inflation outcomes like the deflation in the United States in the Great Depression or the hyperinflations experienced at certain points in time by other countries. Ontheformer, FriedmanandSchwartz (1963)reportthatbanklendingcontracted by about half from 1929 to 1933, so the bank lending conditioning variable might go a long way towards explaining the 27 percent drop in the PCE price index over that time. For a model designed to gauge risks to the inflation process, it is important to have these channels through which the most extreme tail risks can appear under some circumstances. 29

5 Conclusion The generally low inflation since the end of the 2007-2009 recession has raised questions about the ability of central banks to hold inflation around their stated targets, 2 percent for total PCE price inflation in the case of the U.S. Federal Reserve.28 The Markov-switching models estimated in this paper have little trouble explaining that recent experience: the declines in labor-market slack through most of the recovery have occurred on the flat region of the Phillips curve, and so have had little effect on core PCE price inflation. The Phillips curve for compensation per hour is generally less flat, but the modest acceleration in this measure in recent years has been obscured by the volatility of the time series, some resulting from changes in tax laws in 2013. However, themodelsheresuggest itwouldbeunwise toassumethePhillipscurve remains so flat at all levels of the unemployment rate. The experience of the 1960s strongly suggests a sharp increase in the effect of slack on wage growth and core PCE price inflation after labor markets tighten beyond a certain point. Further, evidence for non-linearity in the Phillips curve is far from confined to the 1960s, and, for wage growth, includes the relatively recent experience of the late 1990s. Core PCE price inflation remained relatively subdued over that period, but likely only because the upward pressure from the non-linear Phillips curve was masked by other factors including downward pressure from a strongly appreciating dollar exchange rate. In discussing the Phillips curve, Stock and Watson (2009) ask hypothetical forecasters: “... suppose you are told that next quarter the economy would plunge into recession, with the unemployment rate jumping by 2 percentage points. Would you change your inflation 28From July 2009 to April 2016, total and core PCE price inflation in the U.S. averaged 1.4 and 1.5 percent, respectively, modestly below target. 30

forecast?” Those skeptical of a non-linear Phillips curve should consider a related question: if the unemployment rate were to fall below 4 percent on a sustained basis, as in the late 1960s, would you expect wage inflation to remain low (say, around 2 percent) and core PCE price inflation to remain below 2 percent? What if the unemployment rate were to fall below 3 percent, as in the Korean War, or below 2 percent, as may have occurred in World War II (see Lebergott, 1957)? In other words, is there no unemployment rate below which wage and price Phillips curves begin to steepen sharply? To believe not seems tantamount to denying that the forces of supply and demand, to which the original Phillips (1958) appealed, are ever operable to any appreciable degree in the labor market. The other reason core PCE price inflation has been mostly below the FOMC’s 2 percent target for total PCE price inflation since 2009 is that its mean in the stationary regime governing its behavior since the mid-1990s appears to be somewhat below that target. If price and wage inflation were certain to remain governed by that stationary regime, a lower mean simply implies that the FOMC’s 2 percent inflation target is likely to be met on a sustained basis at a lower unemployment rate (assuming total PCE price inflation runs about in line with core PCE price inflation). For example, with a mean of 1.6 percent and a stable dollar exchange rate, the results here suggest that an unemployment rate 3 percentage 4 point below the natural rate produces 2 percent core PCE price inflation. However, reaching that point necessitates moving up the steep portion of the Phillips curve, so if labor-market slack falls just an additional 3 percentage point, the result is sustained 3 percent core PCE 4 price inflation. Little in U.S. history suggests a regime of stationary, stable inflation can be maintained for long at an average level of inflation much above 3 percent. Indeed, the evidence here suggests that it was sustained inflation at or above 3 percent in the 1960s, probably caused by unemployment rates running well below the natural rate, that 31

entrenched the inflation expectations that kept inflation high even after the unemployment rate moved back up to the natural rate in 1970. At that point, inflation expectations likely hadbecomehighlyadaptiveandanimportantdrivingforceforsubsequent inflation, meaning the regime of stationary, stable inflation was over. Results here suggest that an earlier slackening of the labor market very well may have resulted in the same regime transition outcome, so an end to the era of stationary, stable inflation may have been all but inevitable by 1967 or 1968. 32

Appendix A: Quarterly Model Structure Let t index quarters in this Appendix, so the four-quarter change in PCE price inflation is π = 100∗(ln(P )−ln(P )), and let πq = 400∗(ln(P )−ln(P )) be the annualized t t t−4 t t t−1 quarterly log difference. Then, for S = 2, the annual model with t indexing quarters is t π = π +Θ X +ε +θε , estimated oneach 4th-quarter observation, T,T−4,T−8,.... t t−4 2 t−4 t t−4 Here X is the vector of controls: quarterly labor-market slack and its non-linear function t−4 lagged four quarters, the eight-quarter change in the exchange rate lagged four quarters, and the four-quarter change in bank lending lagged four quarters. Then the quarterly analog to this annual-frequency model for S = 2, estimated on the full set of quarterly observations, t is: 4 4 4 4 πq = πq +Θ X + εq +θ εq , or: t+1−i t−3−i 2 t−4 t+1−i t−3−i Xi=1 Xi=1 Xi=1 Xi=1 3 4 4 4 πq = − πq + πq +Θ X + εq +θ εq . t t−i t−3−i 2 t−4 t+1−i t−3−i Xi=1 Xi=1 Xi=1 Xi=1 The − 3 πq terms difference the quarterly-model innovations εq , εq , and εq out i=1 t−i t−1 t−2 t−3 P of the right-hand side of this equation, and all the explanatory variables are dated t − 4 or earlier, so this equation uses only information from four quarters prior to forecast πq . t In other words, the equation is essentially designed to forecast the annualized quarterly inflation rate four quarters ahead. The analog for S = 1 is similar. The transition matrix t and variance-covariance matrices are as in section 2 but at the quarterly rather than the annual frequency. In the model with the time-varying transition probability, p is a logistic function of 11,t the average non-linear Phillips curve effect over the last five years multiplied by a term 33

measuring the extent to which the non-linear effect has diminished relative to its peak value over the past five years, so: exp(ξ11 +ξ11Z ) p = 0 u t−4 where: 11,t exp(ξ11 +ξ11Z )+1 0 u t−4 20 κ Z = t−4 −min(UGAP ,0)2 and: t−4 t−4−(i+1) 20 Xi=1 κ = − min(UGAP ,0)2 −min(UGAP ,UGAP ,...,UGAP ,0)2 . t−4 t−4 t−4 t−5 t−23 (cid:0) (cid:1) Bibliography Akerlof, George; Dickens, William; and Perry, George. “The Macroeconomics of Low Inflation” Brookings Papers on Economic Activity, 1996:1, pp. 1-76. Akerlof, George; Dickens, William; and Perry, George. “Near-Rational Wage and Price Setting and the Long-Run Phillips Curve” Brookings Papers on Economic Activity, 31, 2000:1, pp. 1-60. Barber, William. “The Kennedy Years: Purposeful Pedagogy” in Exhortation and Controls: The Search for a Wage-Price Policy, 1945-1971, Craufurd Goodwin (ed.), The Brookings Institution, 1975. Clark, Todd, and Doh, Taeyoung. “Evaluating Alternative Models of Trend Inflation,” International Journal of Forecasting 30, 2014, pp. 426-448. Clemens, Jeffrey; Gottlieb, Joshua; and Shapiro, Adam. “Medicare Payment Cuts ContinuetoRestrainInflation,” FederalReserveBankofSanFranciscoEconomicLetter2016-15, May 9, 2016. Cochrane, James. “The Johnson Administration: Moral Suasion Goes to War” in Exhortation and Controls: The Search for a Wage-Price Policy, 1945-1971, Craufurd Goodwin 34

(ed.), The Brookings Institution, 1975. Cogley, Timothy; Primiceri, Giorgio; and Sargent, Thomas. “Inflation-Gap Persistence in the US” American Economic Journal: Macroeconomics, 2010, pp. 43-69. Croushore, Dean. “The Livingston Survey: Still Useful After All These Years.” Federal Reserve Bank of Philadelphia Business Review, March/April 1997. Daly, Mary and Hobijn, Bart. “Downward Nominal Wage Rigidities Bend the Phillips Curve” Federal Reserve Bank of San Francisco Working Paper 2013-08, January 2013. Davig, Troy and Doh, Taeyoung. “Monetary Policy Regime Shifts and Inflation Persistence,” The Review of Economics and Statistics, 2014, pp. 862-875. De Marchi, Neil. “The First Nixon Administration: Prelude to Controls” in Exhortation and Controls: The Search for a Wage-Price Policy, 1945-1971, Craufurd Goodwin (ed.), The Brookings Institution, 1975. Evans, Martin, and Wachtel, Paul. “Inflation Regimes and the Sources of Inflation Uncertainty” Journal of Money, Credit and Banking, 1993, pp. 475-511. Fisher, Irving. “The Debt-Deflation Theory of Great Depressions,” Econometrica , 1933, pp. 337-357. Fisher, Richard and Koenig, Evan. “Are We There Yet? Assessing Progress Towards Full Employment and Price Stability” Federal Reserve Bank of Dallas Economic Letter, Volume 9, Number 13, October 2014. Friedman, Milton, and Schwartz, Anna J. A Monetary History of the United States, 1867—1960. National Bureau of Economic Research, 1963. Goodwin, Craufurd and Herren, R. Stanley. “The Truman Administration: Problems and Policies Unfold” in Exhortation and Controls: The Search for a Wage-Price Policy, 1945-1971, Craufurd Goodwin (ed.), The Brookings Institution, 1975. 35

Gordon, Robert. “A Consistent Characterization of a Near-Century of Price Behavior.” American Economic Review 70, May 1980, pp. 243-249. Hamilton, James. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle”, Econometrica, 1989, pp. 357-384. Hansen, B.E., Seo, B., 2002, “Testing for two-regine threshold cointegration in vector error-correction models,”Journal of Econometrics 110, 293-318. Kim, Chang-Jin. “Unobserved-Component Time Series Models with Markov-Switching Heteroscedasticity: Changes in Regime and the Link Between Inflation Rates and Inflation Uncertainty,” Journal of Business and Economic Statistics, (1993), pp. 341-349. Kim, Chang-Jin. “Dynamic linear models with Markov-Switching,” Journal of Econometrics, 1994, pp. 1-22. Kumar, Anil and Orrenius, Pia. “A Closer Look at the Phillips Curve Using State Level Data.” Federal Reserve Bank of Dallas Working Paper 1409, 2014. Lanne, Markku. “Nonlinear Dynamics of Interest RateandInflation,” Journal of Applied Econometrics, 2006, pp. 1157-1168. Lebergott, Stanley. “Annual Estimates of Unemployment in the United States, 1900- 1954” in The Measurement and Behavior of Unemployment, 1957, pp. 213-241. Levin, AndrewandTaylor, John. “FallingBehindtheCurve: APositiveAnalysisofStop- Start Monetary Policies and The Great Inflation” National Bureau of Economic Research working paper 15630, January 2010. Meltzer, Alan. “Origins of the Great Inflation,” Federal Reserve Bank of St. Louis Review 87 (Part 2), March/April 2005, pp. 145-175. Mertens, Elmar. “Measuring the level and uncertainty of trend inflation.” Federal Reserve Board FEDS Working Paper 2011-42, 2011. Forthcoming, Review of Economic Statis- 36

tics. Nalewaik, Jeremy. “Regime-Switching Models for Estimating Inflation Uncertainty.” Federal Reserve Board FEDS Working Paper 2015-93, 2015. Nalewaik, Jeremy. “Inflation Expectations and the Stabilization of Inflation.” Federal Reserve Board FEDS Working Paper 2016-35, 2016. Orphanides, Athanasios. “Monetary Policy Rules and the Great Inflation” American Economic Review 92, 2002, pp. 115-120. Orphanides, Athanasios. “Historical Monetary Policy Analysis and the Taylor Rule” Journal of Monetary Economics 50, 2003, pp. 983-1022. Phelps, Edmund S. “Phillips Curves, Expectations of Inflation and Optimal Unemployment over Time” Economica 34, 1967, pp. 254-281. Phillips, Edmund S. “The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957” Economica 25, 1958, pp. 283-299. Romer, Christina. “Inflation and the Growth Rate of Output” National Bureau of Economic Research working paper 5575, May 1996. Speigner, Bradley. “Long-term Unemployment and the Convexity of the Phillips Curve.” Bank of England working paper 519, December 2014. Stock, James; and Watson, Mark. “Why Has U.S. Inflation Become Harder to Forecast?” Journal of Money, Credit and Banking, 2007, pp. 3-34. Stock, James; and Watson, Mark. “Phillips Curve Inflation Forecasts” in Understanding Inflation and the Implications for Monetary Policy, Jeffrey Fuhrer, Yolanda Kodrzycki, Jane Little, and Giovanni Olivei (eds), MIT press, 2009. Yellen, Janet. “Inflation Dynamics and Monetary Policy,” speech delivered at the University of Massachusets, Amherst, September 24th, 2015. 37

Table 1: Inflation Measures, Livingston Inflation Forecasts, Unemployment, Vacancy Rates, and Non-Stationary Regime Probabilities in the 1960s Regime Probabilities (% p.p.) Date π w,CPH π πCPI ELIV πCPI U RVAC Constrained Unconstrained t t t t t+1 t t 1961Q2 3.5 1.2 1.0 1(cid:0).0 (cid:1) 7.1 8.8 98 1 1961Q4 3.3 1.2 0.7 1.1 6.1 8.5 78 0 1962Q2 3.8 1.5 1.3 1.1 5.5 8.1 18 0 1962Q4 3.9 1.2 1.3 1.05 5.7 8.1 14 0 1963Q2 3.5 1.2 1.0 1.05 5.9 8.2 2 0 1963Q4 3.4 1.5 1.3 1.0 5.7 8.3 0 0 1964Q2 3.2 1.6 1.3 1.3 5.1 8.1 0 0 1964Q4 3.1 1.3 1.3 1.2 4.8 8.3 0 0 1965Q2 3.3 1.2 1.6 0.9 4.6 8.2 0 0 1965Q4 3.3 1.3 1.6 1.6 4.1 8.5 0 0 1966Q2 4.3 2.0 2.9 1.9 3.9 7.4 0 0 1966Q4 5.7 3.0 3.8 2.2 3.6 7.7 0 0 1967Q2 5.9 2.9 2.8 2.4 3.8 6.9 0 0 1967Q4 5.7 3.1 2.7 2.8 3.9 6.2 0 0 1968Q2 6.3 4.2 3.9 2.9 3.5 6.2 100 0 1968Q4 7.3 4.5 4.7 2.85 3.4 5.4 100 0 1969Q2 7.0 4.6 5.5 3.45 3.4 5.7 100 0 1969Q4 6.6 4.6 5.9 3.4 3.5 5.1 100 0 1970Q2 6.9 4.5 6.0 3.95 4.8 5.4 100 0 1970Q4 6.8 4.7 5.6 3.8 5.9 5.2 100 0 1971Q2 6.3 4.9 4.4 4.1 5.9 5.3 100 100 Notes: The table is semi-annual to align with the timing of the Livingston survey of professional forecasters. πw,CPH is the annual average growth rate of compensation per hour in the non-farm business t sector and π is the four-quarter growth rate of the core PCE price index. ELIV πCPI is the Livingston t t t+1 survey forecast of CPI inflation over the the next 12 months, made in the last mo(cid:0)nth of(cid:1)the quarter (June andDecember). πCPI is12-monthCPIinflationandU istheunemploymentratesthatprevailedatthetime t t the forecasts were made (i.e. the May and November values known at the time of the June and December forecasts.) The timing of the Livingston Survey is conveniently aligned with the timing of the CPI; it is sentto participants the day after the releaseof the CPI data on the previousmonth—see Croushore(1997). RVAC is the quarterlyU.S.averagevacancyrateonrentalunits, fromthe U.S.census bureau. The regime t probabilities are from the quarterly model estimates in Table 3. 38

Table 2: Markov Switching Models explaining Q4/Q4 Core PCE Price Inflation and Annual Average Compensation-Per-Hour Growth Constraining Markov Transition Matrix, Stationary State Absorbing 0.029 1.000 π  (0.037)  π t|t−1 = t−1|t−1 . π = 0.000 . (cid:18) 1−π (cid:19) 0.971 (cid:18) 1−π (cid:19) 0 t|t−1  0  t−1|t−1  (0.037)  b   Stationary Regime Parameters Non-stationary Regime Parameters γ µ β β σ θ β β β σ L u u,1 d,1 1 u,2 d,2 b,2 2 -0.16 1.74 -0.12 -0.07 0.23 -0.96 -0.05 -0.06 0.16 0.83 -73.1 (0.02) (0.08) (0.04) (0.03) (0.08) (0.33) (0.06) (0.04) (0.05) (0.23) γw µw βw ρ σw θw βw ρ βw σw u u,1 1 1 u,2 2 b,2 2 -0.09 3.90 -0.65 0.00 1.02 -1.03 -0.17 0.51 0.14 0.98 (0.03) (0.51) (0.25) (1.32 (0.34) (0.47) (0.09) (0.28) (0.08) (0.40) Unconstrained Markov Transition Matrix 0.9597 0.0418 π  (0.0415) (0.0491)  π 0.90 t|t−1 = t−1|t−1 . π = . (cid:18) 1−π (cid:19) 0.0403 0.9582 (cid:18) 1−π (cid:19) 0 (1.40) t|t−1 t−1|t−1    (0.0415) (0.0491)  b   Stationary Regime Parameters Non-stationary Regime Parameters γ µ β β σ θ β β β σ L u u,1 d,1 1 u,2 d,2 b,2 2 -0.53 1.61 -0.08 -0.08 0.37 -1.47 -0.09 -0.07 0.14 0.59 -55.4 (0.01) (0.01) (0.004) (0.03) (0.07) (0.04) (0.06) (0.06) (0.04) (0.20) γw µw βw ρ σw θw βw ρ βw σw u u,1 1 1 u,2 2 b,2 2 -0.26 3.93 -0.64 0.50 0.94 -1.57 -0.30 0.39 0.11 0.70 (0.01) (0.003) (0.004) (0.68) (0.18) (0.06) (0.12) (0.88) (0.01) (0.16) 39

Table 3: Markov Switching Models Explaining Annualized Quarterly Core PCE Price Inflation One Year Ahead Constraining Markov Transition Matrix, Stationary State Absorbing Stationary Regime Parameters Non-stationary Regime Parameters γ µ β β σ θ β β β σ L u u,1 d,1 1 u,2 d,2 b,2 2 -0.17 1.74 -0.06 -0.04 0.50 -1.76 -0.71 -0.12 0.42 0.73 -296.8 (0.04) (0.05) (0.04) (0.02) (0.03) (0.25) (0.19) (0.06) (0.10) (0.10) Transition-Matrix Parameters p p p 11 22 0 1.000 0.9924 0.000 — (0.0093) — Unconstrained Markov Transition Matrix Stationary Regime Parameters Non-stationary Regime Parameters γ µ β β σ θ β β β σ L u u,1 d,1 1 u,2 d,2 b,2 2 -0.47 1.63 -0.04 -0.05 0.50 -2.96 -1.19 -0.13 0.21 0.52 -262.1 (0.01) (0.04) (0.03) (0.02) (0.03) (0.22) (0.21) (0.08) (0.01) (0.06) Transition-Matrix Parameters p p p 11 22 0 0.9993 0.9979 0.92 (0.0008) (0.0048) (1.53) Unconstrained Markov Transition Matrix, Time-Varying Transition Probabilities Stationary Regime Parameters Non-stationary Regime Parameters γ µ β β σ θ β β β σ L u u,1 d,1 1 u,2 d,2 b,2 2 -0.46 1.67 -0.06 -0.05 0.52 -2.43 -0.67 -0.19 0.18 0.61 -254.1 (0.005) (0.03) (0.02) (0.02) (0.03) (0.19) (0.21) (0.06) (0.01) (0.06) Transition-Matrix Parameters ξ11 ξ11 p p 0 u 22 0 13.93 4.60 0.9982 0.90 (7.68) (2.44) (0.0036) (1.53) 40

Table 4: OLS Regressions Explaining Q4/Q4 Core PCE Price Inflation 1961-1970, 1995-2015 π = α+β π +β ∆XR +β UGAP +γ (min(UGAP ,κ)+u t π,1 t−1 d,1 t−1,t−3 U,1 t−1 u,1 t−1 t α β β β γ κ Adj. R2 SUP-LM π,1 d,1 u,1 u,1 p-val. 2.09 -0.39 0.39 (0.22) (0.17) -0.45 -0.06 -2.10 -1.00 0.83 0.000 (0.27) (0.05) (0.20) 0.50 0.80 -0.05 -0.18 0.77 (0.23) (0.10) (0.03) (0.08) 0.39 0.35 -0.06 -0.06 -1.11 -0.60 0.88 0.023 (0.10) (0.07) (0.02) (0.03) (0.15) −(min(UGAP,0))2 1.11 0.31 -0.06 -0.07 -0.38 0.87 (0.13) (0.08) (0.02) (0.03) (0.06) π = α+β π +β ∆XR +β U +γ (min(U ,κ)+u t π,1 t−1 d,1 t−1,t−3 U,1 t−1 u,1 t−1 t α β β β γ κ Adj. R2 SUP-LM π,1 d,1 u,1 u,1 p-val. 3.81 -0.33 0.26 (1.06) (0.17) 16.38 -0.06 -3.36 4.30 0.71 0.006 (1.61) (0.06) (0.39) 1.00 0.87 -0.05 -0.12 0.75 (0.57) (0.10) (0.04) (0.07) 6.38 0.53 -0.07 -0.03 -1.15 4.80 0.84 0.020 (2.01) (0.16) (0.03) (0.03) (0.36) (U −κ)−1 0.11 0.55 -0.07 1.84 2.60 0.81 (0.12) (0.15) (0.03) (0.56) 41

Table 5: OLS Regressions Explaining Annual Average Compensation-Per-Hour Growth 1961-1970, 1995-2015 πw = αw +βw πw +βw UGAP +γw (min(UGAP ,κw)+uw t π,1 t−1 U,1 t−1 u,1 t−1 t αw βw βw γw κw Adj. R2 SUP-LM π,1 u,1 u,1 p-val. 4.04 -0.81 0.60 (0.18) (0.15) 3.89 0.27 -1.40 2.00 0.70 0.004 (0.16) (0.14) (0.22) 3.61 0.11 -0.72 0.59 (0.83) (0.21) (0.22) 4.06 -0.05 0.14 -1.34 1.80 0.69 0.001 (0.85) (0.20) (0.15) (0.27) −(min(UGAP,0))2 3.83 -0.04 -0.58 -0.38 0.65 (0.67) (0.15) (0.19) (0.09) πw = αw +βw πw +βw U +γw (min(U ,κw)+uw t π,1 t−1 U,1 t−1 u,1 t−1 t αw βw βw γw κw Adj. R2 SUP-LM π,1 u,1 u,1 p-val. 8.03 -0.75 0.53 (1.02) (0.17) 14.09 -0.35 -1.76 5.10 0.70 0.001 (0.85) (0.07) (0.23) 6.27 0.23 -0.58 0.53 (1.90) (0.22) (0.22) 17.22 -0.20 -0.45 -2.17 5.00 0.70 0.000 (3.02) (0.22) (0.12) (0.42) (U −κ)−1 0.17 -0.20 15.18 1.60 0.70 (0.32) (0.23) (3.13) 1995-2015 only 6.63 -0.55 0.39 (0.94) (0.14) 9.76 0.14 -1.29 6.90 0.52 0.034 (1.19) (0.14) (0.34) 42

12 10 8 6 4 2 0 -2 -4 -6 -8 1691 2691 3691 4691 5691 6691 7691 8691 9691 0791 1791 2791 3791 4791 5791 6791 7791 8791 9791 0891 1891 2891 3891 4891 5891 6891 7891 8891 9891 0991 1991 2991 3991 4991 5991 6991 7991 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 Figure 1: Unemployment Rate, Slack, and Non-Linear Slack Effect Percent Unemployment Rate (U) CBO Natural Rate of Unemployment (NRU) Non-Linear Slack Effect: -(min(U-NRU,0)^2)

12 10 10 5 8 0 6 -5 4 -10 2 0 -15 1691 2691 3691 4691 5691 6691 7691 8691 9691 0791 1791 2791 3791 4791 5791 6791 7791 8791 9791 0891 1891 2891 3891 4891 5891 6891 7891 8891 9891 0991 1991 2991 3991 4991 5991 6991 7991 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 Figure 2: Core PCE Price Inflation, with Lagged Bank Lending 4-quarterpercent change, 4-quarterpercent change normalized and lagged 4 quarters Core PCE Price Inflation (left) Bank Lending (right)

12 10 10 5 8 0 6 -5 4 -10 2 0 -15 1691 2691 3691 4691 5691 6691 7691 8691 9691 0791 1791 2791 3791 4791 5791 6791 7791 8791 9791 0891 1891 2891 3891 4891 5891 6891 7891 8891 9891 0991 1991 2991 3991 4991 5991 6991 7991 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 Figure 3: Compensation-Per-Hour Growth, Non-Farm Business Sector, with Lagged Bank Lending 4-quarterpercent change, 4-quarterpercent change normalized and lagged 4 quarters CPH growth, annual average (left) Bank Lending (right)

Figure 4: Rate of Change or "Speed" Effects in the Great Depression Percent Change 15.00 PCE Price Inflation Real GDP growth 10.00 Correlation=0.87 5.00 0.00 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 -5.00 -10.00 -15.00

Figure 5: Core PCE Price Inflation, with Bank Lending 4-quarterpercent change, 4-quarterpercent change normalizedby 10-year moving average 2.6 10 2.4 5 2.2 2 0 1.8 1.6 -5 1.4 1.2 -10 Core PCE Price Inflation (left) Bank Lending (right) 1 0.8 -15 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Figure 6: Compensation-Per-Hour Growth, Non-Farm Business Sector, with Bank Lending 4-quarterpercent change, 4-quarterpercent change normalizedby 10-year moving average 8 10 7 5 6 5 0 4 -5 3 2 CPH growth, annual average (left) -10 Bank Lending (right) 1 0 -15 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

12 -15 10 -10 8 -5 6 0 4 5 2 10 0 15 4791 5791 6791 7791 8791 9791 0891 1891 2891 3891 4891 5891 6891 7891 8891 9891 0991 1991 2991 3991 4991 5991 6991 7991 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 6102 Figure 7: Core PCE Price Inflation, with Lagged Real Exchange Rate 8-quarterpercent change, 4-quarterpercent change annualized, lagged 4 quarters Core PCE Price Inflation (left) Real Dollar Exchange Rate (right, inverse scale)

Figure 8: Core PCE Price Inflation, with Lagged Real Exchange Rate 8-quarterpercent change, 4-quarterpercent change annualized, lagged 4 quarters 2.6 -8 Core PCE Price Inflation (left) 2.4 Real Dollar Exchange Rate (right, inverse scale) -6 2.2 -4 2 -2 1.8 0 1.6 2 1.4 4 1.2 6 1 8 0.8 10 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Figure 9: Q4/Q4 Core PCE Price Inflation, modelled with Markov- Switching and a Non-Linear Phillips Curve Percent Change Probability 10 1 9 0.9 8 Non-stationary regime probability, one-sided 0.8 Non-stationary regime probability, two-sided Core PCE Price Inflation 7 0.7 Model Forecast, 1 Year Ahead 6 0.6 5 0.5 4 0.4 3 0.3 2 0.2 1 0.1 0 0 1961196319651967196919711973197519771979198119831985198719891991199319951997199920012003200520072009201120132015

Figure 10: Annual Average Compensation-Per-Hour Growth, modelled with Markov-Switching and a Non-Linear Phillips Curve Percent Change Probability 11 1 Non-stationary regime probability, one-sided 10 0.9 Non-stationary regime probability, two-sided Compensation Per Hour Growth 9 0.8 Model Forecast, 1 Year Ahead 8 0.7 7 0.6 6 0.5 5 0.4 4 0.3 3 0.2 2 0.1 1 0 1961196319651967196919711973197519771979198119831985198719891991199319951997199920012003200520072009201120132015

Figure 11: Non-Linear Phillips Curves in Stationary Inflation Regime, Annual-Frequency Model Percent 9 8 Compensation Per Hour Growth Core PCE Price Inflation 7 6 5 4 3 2 1 0 -3 -2 -1 0 1 2 3 4 Unemployment Gap in Prior Year Q4

12 1 0.9 10 0.8 0.7 8 0.6 6 0.5 0.4 4 0.3 0.2 2 0.1 0 0 2691 3691 4691 5691 6691 7691 8691 9691 0791 1791 2791 3791 4791 5791 6791 7791 8791 9791 0891 1891 2891 3891 4891 5891 6891 7891 8891 9891 0991 1991 2991 3991 4991 5991 6991 7991 8991 9991 0002 1002 2002 3002 4002 5002 6002 7002 8002 9002 0102 1102 2102 3102 4102 5102 Figure 12: Annualized Quarterly Core PCE Price Inflation, modelled with Markov-Switching and a Non-Linear Phillips Curve Annualized Percent Change Probability Non-stationary regime probability, one-sided Non-stationary regime probability, two-sided Annualized Quarterly Core PCE Price Inflation Model Forecast, 4 Quarters Ahead