## _wp11121 ## Source details **Canonical URL:** [_wp11121](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11121.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11121.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11121.pdf.json) --- ### Introduction and paper approach - Objective: Estimate accelerationist Phillips curves with quarterly data for 1960-2007 using CPI and CPI less food and energy (XFE); produce dynamic forecasts for 2008-2010 using the unemployment path. - Key empirical contrast: - A simple pre-2008 Phillips curve forecasted large deflation (four-quarter moving average of core inflation falls to -4.3% in 2010Q4 in one specification; four-quarter XFE forecasts falling to -5.5% or -4.1% in other specifications). - Actual four-quarter core inflation was 0.6% in 2010Q4 (source text reports actual four-quarter core inflation fell from 2.6% in 2007Q4 to 0.4% in 2010Q4 for XFE; median and other measures differ as shown below). - Two theory-motivated modifications proposed to reconcile the Phillips curve with observed inflation: - Measure core inflation with a weighted median of price changes across industries (median inflation). - Allow the Phillips-curve slope (coefficient on unemployment) to vary over time, steepening when inflation is high and/or variable. - Expectations anchoring concepts introduced: - Shock anchoring: expectations unresponsive to supply shocks. - Level anchoring: expectations revert toward a fixed level (assumed level: 2.5% for core CPI inflation, about 2% for core PCE inflation). ### The simple accelerationist Phillips curve (specification and baseline estimates) - Baseline equation (backward-looking expectations): - πt = πet + α( u − u* )t + εt - πt = 1/4(πt−1 + πt−2 + πt−3 + πt−4) + α(u − u*)t + εt - Lag-structure implications: - A one-percentage-point increase in unemployment for one quarter changes inflation in the long run by 0.4 times α. - The long-run effect of one point of unemployment sustained for a year is 1.6 times α. - Identification and data notes: - Main activity gap series from the Congressional Budget Office (CBO); Staiger et al. (1997) technique used for robustness. - CBO natural-rate series: about 5.5% to 6.3% in 1960s–1970s, falls to 5.0% in the 1990s, remains 5.0% through 2007, rises slightly to 5.2% in 2010. - Estimation 1960:1–2007:4: - Unemployment coefficients ≈ -0.5 (t >5). - Output-gap coefficients ≈ 0.25. - Example long-run effect with XFE and output ≈ (1.6)(0.25) = 0.4 percentage points. - Sacrifice ratio implied ≈ 1/0.4 = 2.5. - The puzzle: - Dynamic forecasts using 1960–2007 Phillips curve and 2008–2010 unemployment path predict large deflation (e.g., XFE forecasts falling to -5.5% or -4.1%), inconsistent with actual inflation. ### Measuring supply shocks and core inflation - Gordon “triangle” decomposition: inflation = expected inflation + aggregate activity effect + supply shocks. - Common supply-shock measures: - Changes in relative prices of food and energy (flexible) versus other sticky prices. - Asymmetries in price-change distributions (Ball and Mankiw): skewness and tail behavior. - Empirical relations: - Food and energy measures and distribution-asymmetry measures are positively correlated, but both can be tested for independent explanatory power. - Ball and Mankiw (1949–1989 result): when both included, price-change asymmetries (not food/energy) are significant. - Defining core inflation: - If supply shocks measured by food and energy: core = XFE inflation. - If supply shocks measured by price-change asymmetries: core = weighted median of industry price changes (median inflation). ### Median inflation: construction, properties, and recent behavior - Cleveland Fed median series: monthly median inflation beginning 1968; about forty industries (rising from 36 to 45). - Data variants: - “Original” weighted median 1968-2007 uses owner’s equivalent rent (OER) as price for one large industry. - “Revised” median 1983-present includes OER for four geographic regions; revision affects monthly volatility where series overlap. - Quarterly construction for this analysis: - Convert monthly median inflation rates to monthly price levels → average three months to obtain quarterly price level → compute annualized percentage changes. - Persistence evidence (Stock and Watson 2007 decomposition, sample starting 1983Q2): - Divergences between total and permanent inflation are smaller when inflation is measured by median inflation. - In the 2000s median inflation appears to have almost no transitory component. - Interpretation: median filters out large industry-specific transitory price swings (used cars, lodging, apparel). - Median inflation during the Great Recession (four-quarter moving averages, 2007Q4 to 2010Q4): - Median inflation fell from 3.1% to 0.5%. - XFE inflation fell from 2.3% to 0.6%. - Median fell more because it started at a higher level; distributional anomalies in 2007 (large price decreases in jewelry, car rental, lodging, etc.) affected XFE more than the median. - Implication: using median inflation reduces divergence between forecasted and actual inflation over 2008–2010 but is insufficient alone to fully resolve the Phillips-curve puzzle. ### Time-varying Phillips-curve slope: theory, specification, and estimates - Theory (Ball, Mankiw, Romer 1988): higher level and variance of inflation reduce nominal price stickiness → firms adjust prices more frequently → Phillips-curve slope steepens. - Empirical time-varying specification (preserved equations): - π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α_t (u−u^*)_t + ǫ_t - α_t = α_{t−1} + η_t - ǫ_t and η_t are white noise with variances V and W; α follows a random walk. - Estimation approaches: - Version 1: assume V/W, estimate α_t with Kalman smoother (choose V/W for plausible smoothness). - Version 2: estimate V and W by maximizing likelihood via Kalman smoother. - Sample construction: use Cleveland Fed revised median 1984Q2–2010Q4, original median 1968Q2–1984Q1, XFE for 1960Q1–1968Q1; results similar using XFE for full sample. - Estimated α_t path (summary): - α falls from near zero in 1960 to around -1 in early 1970s, fluctuates until 1980, then rises sharply and levels off near -0.2. - Since mid-1980s estimated α is quite stable; no evidence against constant α over 1985-2010 given standard errors. - Regression representation with level and variance of inflation: - α_t = (a_0 + a_1 ̄π_t + a_2 σ_t) - Substitution yields: - π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + a_0 (u−u*)_t + a_1 ̄π_t (u−u*)_t + a_2 σ_t (u−u*)_t + ǫ_t - Table 2 (1960:1-2010:4): joint significance of ̄π and σ high (p <0.01); collinearity makes individual roles hard to disentangle. - Subsample constant-slope estimates (Table 3 highlights): - 1960:1-1972:4 unemployment coefficient ≈ -0.231 (median) or -0.?? (XFE as reported in table). - 1973:1-1984:4 unemployment coefficient ≈ -0.650 (median) or ≈ -0.688 (XFE). - 1985:1-2007:4 unemployment coefficient ≈ -0.202 (median) and ≈ -0.246 (XFE). - For 1985-2007 with median, output-year long-run effect example: (1.6)(0.11) = 1.76 and implied sacrifice ratio ≈ 1/(0.176) ≈ 6. - Extension 2007–2010: - For XFE, coefficients decline substantially in absolute value. - For median, coefficients fall less when activity measured by unemployment gap or not at all for output gap. - Suggests greater stability in Phillips curve when using median inflation. ### Anchored expectations: shock anchoring, level anchoring, estimation, and implications - Shock anchoring evidence: - Since the Volcker disinflation, supply shocks have not strongly fed into expectations; SPF and other measures place high weight on lagged core vs lagged total inflation in 1985-2010. - Estimated weight on lagged core is 0.89 in a regression; regressing SPF one-year expected inflation (1985-2010) on average of lagged core and lagged total yields weight on lagged core = 0.86 (standard error = 0.06). - Time-varying-weight experiments show weights on lagged core near one in post-Volcker period. - Level anchoring estimation (assume level 2.5% for core CPI expectations): - Expectations model (equation preserved): - πet = δt 2.5 + (1−δt) (1/4)(πt−1 + πt−2 + πt−3 + πt−4) + εt - δt follows a random walk: δt = 0 (purely backward-looking) to δt = 1 (fully anchored at 2.5%). - SPF-based estimation with Kalman smoother: - δt near zero until early 1990s, then rises. - δt ≈ 0.6 over 2007-2010 for SPF expectations. - Actual-inflation version (equation preserved): - πt = δt 2.5 + (1−δt) (1/4)(πt−1 + πt−2 + πt−3 + πt−4) + α( u − u*t ) + εt - Estimated δt for actual inflation rose similarly from near zero in early 1990s. - Values in 2010Q4: - δ = 0.47 when Phillips curve includes unemployment gap. - δ = 0.30 when Phillips curve includes output gap. - Estimated α in equation (6): - α = -0.24 (standard error = 0.03) for the unemployment gap. - α = 0.13 (standard error = 0.02) for the output gap (sign convention in source). - Interpretation: expectations and actual inflation partially anchored toward 2.5%, but anchoring is incomplete and gradual; price setters may learn more slowly than professional forecasters. - Uncertainty and dynamics: - 95% confidence interval for δ in 2010Q4: - With unemployment in equation: roughly [0.2, 0.7]. - With output gap: roughly [0.2, 0.4]. - Anchoring may weaken if actual inflation remains well below 2.5% for extended periods. ### Forecasts and the Great Recession: 2008–2010 and 2011–2013 scenarios - Dynamic forecasts (1985-2007 regressions) for 2008–2010: - Median-inflation forecasts are close to actual inflation — no missing deflation for median inflation. - XFE forecasts fall to around -1% at end-2010, significantly below actual XFE inflation. - Time-variation in slope reduces predicted inflation fall; median starts at higher 2007 level which reduces forecasted drop. - Forecasts for 2011–2013 using CBO unemployment and output-gap paths (equation estimates 1985–2010, assumed natural rate constant at 5.2%): - CBO unemployment forecasts assumed: 9.4% in 2011, 8.4% in 2012, 7.6% in 2013. - Four-quarter-average forecast results (Figure 9 summary): - With unemployment above natural rate and output below potential, inflation falls steadily and becomes negative at end-2011. - At end-2013 inflation reaches: - -1.9% (based on unemployment forecasts). - -1.3% (based on output forecasts). - Implication: the Phillips curve that fits recent history predicts deflation will arrive soon absent other forces (e.g., anchoring). - Dynamic forecasts with anchoring (hold δ at 2010Q4 level, equation (6) estimated 1985-2010): - δ in 2010Q4 held constant: - With unemployment-gap equation: δ ≈ 0.47 in 2010Q4 (estimated); using anchoring prevents the large deflation predicted by purely backward-looking model. - With output-gap equation: δ ≈ 0.30 in 2010Q4. - Forecasts holding δ at 2010Q4 value: - Inflation does not fall into deflation; instead inflation is steady at about 0.5% and then rises to 1% at end-2013. - Mechanism: partial anchoring pulls expected inflation toward 2.5%, moderating downward pressure from unemployment gap. - Caveats: - Forecast reliability limited by uncertainty in δ and possibility that anchoring weakens if actual inflation stays low. - Long-run expectations (ten-year SPF) have been close to 2.5% since 2000, but price-setting relies on one-year expectations in most models. ### New Keynesian Phillips Curve (NKPC) evaluation - NKPC theory and formulation: - Original form: πt = Et πt+1 + λ(y − y* )t → Et πt+1 − πt = −λ(y − y* )t (λ > 0). - Gali and Gertler (1999) modify: πt = Et πt+1 + λ mc t, using real unit labor costs (labor’s share) as proxy for real marginal cost. - Empirical issues: - In data, output has a positive correlation with change in inflation during the Great Recession, producing negative λ estimates in the basic NKPC form. - Credibility of labor’s share as marginal cost proxy contested (Rudd and Whelan; Mazumder). - NKPC performance during Great Recession: - Gali-Gertler specification fits poorly for recent years: coefficient λ significant for 1960-1997 and 1985-2007 (t = 2.04), but insignificant for 1985-2010 (t = 0.92). - Adding 2008-2010 deteriorates model fit: 2009–2010 are outliers where labor’s share fell sharply while unemployment gap large. - Interpretation: substitution of mc for output gap implies expected inflation should rise when labor’s share falls, but observed inflation fell → repeated forecast errors that undermine rational-expectations NKPC interpretation. - Conclusion: NKPC variants do not help explain recent inflation behavior and can produce counterfactual predictions (e.g., rising inflation over 2008–2010). ### Main empirical conclusions and policy-relevant implications - A simple accelerationist Phillips curve can fit 1960–2010, including the Great Recession, under two conditions: - Core inflation measured with the weighted median of price changes (median inflation). - Phillips-curve slope allowed to change with level and variance of inflation. - Expectations: - Shock anchoring has been a stable feature of the post-Volcker regime. - Level anchoring toward 2.5% emerged gradually from the early 1990s and was partial by 2010 (δ ≈ 0.3–0.6 depending on series). - Implications for deflation risk and policy: - Without anchoring, basic accelerationist model implies substantial deflation risk given high unemployment. - With estimated 2010 degree of anchoring, inflation can remain positive over 2011–2013, reducing near-term deflation risk. - However, if anchoring dissipates or if long-term unemployment reduces the unemployment-inflation effect, deflation risks could re-emerge or forecasts could change. - Research and monitoring priorities (implied): - Monitor measures of median inflation alongside XFE. - Track measures of expectation anchoring (δ) and one-year SPF expectations. - Monitor the composition of unemployment (long-term vs short-term) given potential differences in inflationary impact. ### Appendix — robustness checks and selected quantitative results - Robustness extensions tried that do not materially change conclusions: - Longer lags (Gordon-style): including four lags of unemployment and twenty-four lags of inflation yields pre-2008 dynamic forecast for 2010Q4 of -5.1% (median/robustness to lag structure). - Stock-Watson unemployment gap (uSWt = ut − min(ut,..,ut−11)) produces similar Phillips-curve results. - Nonlinear Debelle-Laxton unemployment specification produces similar fits; Debelle-Laxton 1985-2007 predicts 2010Q4 XFE = 1.3% vs authors’ linear model 0.3%. - Productivity growth (Ball and Moffitt) adds modest R2; including g − ĝ yields median inflation forecast -0.32% for 2010Q4 vs -0.24% baseline. - Time-varying NAIRU (Staiger et al.; Kalman filter V/W = 400) closely matches CBO NAIRU (example: 1985:1 CBO = 6.03% vs authors’ 6.04%; end-2010 CBO = 5.20% vs authors’ 5.44%). - Total CPI regression with Shockt = total CPI − median CPI: estimated β ≈ 1 in all periods; reduces to basic core-inflation specification when β = 1. - Selected numerical estimates from tables (preserved): - Table 1 (1960:1-2007:4): α coefficients ≈ -0.507 (TOTAL CPI), -0.474 (XFE CPI); output coefficients ≈ 0.308, 0.257; R^2 range 0.703–0.746. - Table 2 (1960:1-2010:4): a_0 = -0.355 (SE (0.058), R^2 0.773) in simple model; with level/variance terms joint p-value 0.000 and R^2 up to 0.807. - Table 3 subsamples: 1973:1-1984:4 α ≈ -0.650 (median) and -0.688 (XFE); 1985:1-2007:4 α ≈ -0.202 (median) and -0.246 (XFE). - Table 5 NKPC: λ = 0.060 (SE (0.012)) for 1960:1-1997:4; λ = 0.012 (SE (0.013)) for 1985:1-2010:4 (fit deteriorates when including 2008-2010). - Anchoring δ estimates: δ ≈ 0.31 in 2007Q4 (unemployment-gap equation) and δ ≈ 0.26 in 2007Q4 (output-gap equation); δ ≈ 0.47 (2010Q4, unemployment) and δ ≈ 0.30 (2010Q4, output). - Forecast endpoints for 2013 under backward-looking Phillips curve without anchoring: end-2013 inflation = -1.9% (unemployment-based) and -1.3% (output-based). - Forecasts with anchoring held at 2010Q4: inflation steadies ~0.5% then rises to 1% by end-2013. - Figures and tables reproduced in source: dynamic forecast plots (Figures 1–14), time-varying α (Figures 3–4), labor-share versus unemployment (Figure 15), and detailed tables (Tables 1–5, Appendix A1) as summarized above. *Source: _wp11121 (extracted content).* ### Introduction ........................................................................................................... ### Introduction ### Background and motivation - Friedman (1968) framed short-run inflation as depending on expected inflation and the unemployment gap; with expected inflation proxied by past inflation this yields an accelerationist Phillips curve relating the change in inflation to the unemployment gap. - Successive refinements include Gordon (1982, 1990)’s supply shocks and Staiger et al. (1997)’s time-varying natural rate. - Debate intensified after the U.S. slump beginning in 2007: some view recent inflation behavior as a puzzle because inflation fell less than a traditional Phillips curve predicts given high unemployment. - Representative quote: John Williams (September 2010) — “Based on the experience of past severe recessions, I would have expected inflation to fall by twice as much as it has.” - Bernanke (2010) on deflation risk: “Falling into deflation is not a significant risk for the United States at this time, but that is true in part because the public understands that the Federal Reserve will be vigilant and proactive in addressing significant further disinflation.” ### Paper’s contributions and approach - Estimate accelerationist Phillips curves with quarterly data for 1960-2007 using CPI and CPI less food and energy (XFE) as inflation measures; produce dynamic forecasts for 2008-2010 using the unemployment path. - Show that a simple pre-2008 Phillips curve forecasted a large deflation (four-quarter moving average of core inflation falls to -4.3% in 2010Q4) whereas actual four-quarter core inflation was 0.6% in 2010Q4. - Propose two theory-motivated modifications that reconcile the Phillips curve with observed inflation: - Measure core inflation with a weighted median of price changes across industries (median inflation). - Allow the Phillips-curve slope (coefficient on unemployment) to vary over time, steepening when inflation is high and/or variable. - Distinguish between “shock anchoring” (expectations unresponsive to supply shocks) and “level anchoring” (expectations revert toward a fixed level). Assume a level of 2.5% for core CPI inflation (about 2% for core PCE inflation). - Evaluate New Keynesian Phillips curve specifications (Calvo-based); find they do not help explain recent inflation behavior and can produce counterfactual predictions (e.g., rising inflation over 2008-2010). ### Key empirical findings (overview) - Pre-2008 simple Phillips-curve dynamic forecasts imply a large deflation for 2008-2010 (four-quarter XFE forecast falls to -5.5% for the unemployment equation and -4.1% for the output equation), inconsistent with actual inflation. - Combining median inflation and a time-varying slope produces a fitted model that explains the 2008–2010 period; using these modifications forecasted four-quarter inflation in 2010Q4 is 0.3%, reported as close to the actual level of 0.5% in the source text. - Expectations have been fully shock-anchored since the 1980s; level anchoring has been gradual and partial. The fraction of a change in core inflation passed into expectations fell from roughly one in 1985 to between 0.4 and 0.7 in 2010. - Forecasts for 2011–2013 depend critically on whether expectation anchoring is incorporated: the basic accelerationist specification predicts deflation is on the way, but the estimated degree of anchoring in 2010 is high enough to keep inflation positive (forecast reliability limited if anchoring dissipates). --- ### The simple Phillips curve and the puzzle - Baseline specification (accelerationist Phillips curve, backward-looking expectations): - πt = πet + α( u − u* )t + εt - With πet taken as the average of the past four quarters: πt = 1/4(πt−1 + πt−2 + πt−3 + πt−4) + α(u − u*)t + εt - Implications of the lag structure: - A one-percentage-point increase in unemployment for one quarter changes inflation in the long run by 0.4 times the coefficient α. - The long-run effect of one point of unemployment sustained for a year is 1.6 times α. - Identification and data notes: - Main activity gap series from the Congressional Budget Office (CBO); robustness includes Staiger et al. (1997) technique for natural rate estimation. - CBO natural-rate series behavior: rises modestly in the 1960s and 1970s from about 5.5% to 6.3%, then falls to 5.0% in the 1990s, remains at 5.0% through 2007, then rises slightly to 5.2% in 2010. - Estimation (1960:1–2007:4): - Unemployment coefficients ≈ -0.5 and highly significant (t >5). - Output-gap coefficients ≈ 0.25. - Example long-run effect: with XFE inflation and output, effect ≈ (1.6)(0.25) = 0.4 percentage points. - Sacrifice ratio implied ≈ 1/0.4 = 2.5. - The puzzle: - Dynamic forecasts using the 1960–2007 Phillips curve and the 2008–2010 unemployment path predict a large deflation (e.g., XFE inflation forecasts falling to -5.5% or -4.1% depending on specification), but actual XFE inflation fell from 2.6% in 2007Q4 to 0.4% in 2010Q4 (four-quarter series). The pre-2008 Phillips curve predicts a deflation that did not occur. --- ### Measuring core inflation and supply shocks - Many empirical Phillips-curve implementations include supply shocks or use core inflation measures that strip out relative-price changes (commonly food and energy). - Median inflation (weighted median of industry price changes) is motivated by price-adjustment models where unusually large relative-price changes influence aggregate inflation when nominal prices are sticky. - Empirical observation: median inflation fell by more than XFE inflation from 2007 to 2010, reflecting a higher initial level (in 2007 median inflation ≈ 3% and XFE inflation ≈ 2%). The relatively large fall in median inflation reduces the gap between forecasted and actual inflation. --- ### Time-varying Phillips-curve slope - Theory (Ball, Mankiw, Romer, 1988): the Phillips-curve slope steepens when inflation is high and/or variable because those conditions reduce nominal price stickiness. - U.S. time-series evidence supports a time-varying slope: the Phillips curve has been relatively flat in the low-inflation period since the mid-1980s. - Incorporating a time-varying slope reduces the pre-2008 forecasted fall in inflation; together with median inflation measurement, the modified Phillips curve forecasts four-quarter inflation in 2010Q4 of 0.3% (reported as close to actual 0.5%). --- ### Anchored expectations - Definitions: - Shock anchoring: expectations do not respond to supply shocks. - Level anchoring: expectations stay fixed at a certain level regardless of movements in actual inflation. - Assumed level: 2.5% for core CPI inflation (corresponds to about 2% for core PCE inflation). - Empirical findings on anchoring: - Expectations have been fully shock-anchored since the 1980s (i.e., SPF and other measures do not incorporate supply shocks). - Level anchoring has been gradual and partial: fraction of a change in core inflation passed into expectations fell from roughly one in 1985 to between 0.4 and 0.7 in 2010. - Implications for forecasts: - Without anchoring, the basic accelerationist model implies substantial deflation risk. - With the estimated 2010 degree of anchoring, inflation can remain positive over 2011–2013; confidence in this forecast is limited because it assumes continued anchoring at 2.5% while actual inflation may remain below 1%. --- ### New Keynesian Phillips curve evaluation - Examine the New Keynesian Phillips curve (Calvo, rational expectations) and the Gali and Gertler (1999) specification where marginal cost is proxied by labor’s share of income. - Empirical verdict: the New Keynesian Phillips curve does not help to explain recent inflation behavior and, in some specifications, produces counterfactual predictions (e.g., rising inflation over 2008–2010). --- ### Forecasting and robustness notes - Dynamic forecasts are generated by starting with actual inflation through 2007 (or through 2010 when appropriate) and feeding in unemployment/output paths for 2008–2010 or 2011–2013 (CBO forecasts used for natural rate and output gaps). - Robustness checks include: - Alternative lag structures and longer lags. - Alternative measures of the unemployment gap (e.g., Stock-Watson). - Nonlinear Phillips-curve specifications. - Variations in productivity growth and alternative NAIRU estimation techniques. - Comparison to other recent work (Fuhrer et al., Fuhrer and Olivei, Stock and Watson): overlaps exist, but this paper uses information not available in real time (e.g., 2011 CBO natural-rate series) to understand inflation behavior rather than focus exclusively on real-time forecasting performance. *Source: Introduction, _wp11121 - Introduction* ### 3.1 Measuring Supply Shocks ### 3.1 Measuring Supply Shocks ### Theoretical rationale for supply-shock measures - The Phillips curve in much applied work is Gordon’s “triangle model,” explaining total inflation with three factors: expected inflation, aggregate activity, and supply shocks. - Common empirical measures of supply shocks are changes in the relative prices of food and energy; since the 1970s these variables have added greatly to the R^2s of estimated Phillips curves. - Theoretical question (Friedman 1975): “Why should the average level of prices be affected significantly by changes in the price of some things relative to others?” - Sticky-price explanations: - Conventional approach: assume food and energy prices are flexible and other prices are sticky; shocks that raise relative food and energy prices increase their nominal prices while other prices stay constant, implying an increase in the aggregate price level (cited texts include Dornbusch and Fischer (1990) and Blanchard and Gali (2008)). - Ball and Mankiw (1995) approach: price adjustment is endogenous. Firms receive shocks to equilibrium relative prices and decide whether to pay a menu cost to adjust; those receiving the largest shocks adjust. Inflation depends on the distribution of price changes across industries—if the distribution is skewed to the right, aggregate price level rises. Ball and Mankiw measure supply shocks with skewness of relative price changes and other asymmetry measures. ### Empirical relationships among supply-shock measures - In practice, measures based on food and energy prices and measures based on asymmetries in price-change distributions are positively correlated because large changes in food and energy prices commonly create large tails in price distributions. - There is enough independent variation to test which measures relate most closely to inflation. - Ball and Mankiw’s finding for 1949 to 1989: when both types of measures are included in a Phillips curve, only price-change asymmetries (not changes in food and energy prices) are significant. ### From supply shocks to core inflation (section 3.2) - Core inflation is defined as the part of inflation not explained by supply shocks, but rather by expected inflation and economic activity (the other two parts of the triangle). - Operational measures: - When supply shocks are measured by food and energy price changes, core inflation is measured with XFE inflation (strips away direct effects of food and energy). - When supply shocks are asymmetries in price-change distributions, a core measure should eliminate those asymmetries; a simple measure is the weighted median of price changes across industries (median inflation, Bryan and Cecchetti (1994)). - Forecasting considerations: - In theory, core inflation so defined might not forecast future inflation well if supply shocks feed into expected inflation. - Empirical findings: papers such as Sommer (2004) and Hooker (2002) find that, since the 1980s, supply shocks have not fed strongly into future inflation; thus core inflation is a good forecaster. - Smith (2004) finds that over 1984-1997, forecasts based on median inflation are more accurate than those based on XFE inflation. ### Measuring median inflation (section 3.3) - The Federal Reserve Bank of Cleveland maintains a monthly series for median inflation beginning in 1968; the economy is disaggregated into about forty industries (the number rises from 36 to 45 over time). - Core inflation is measured by the weighted median of industry inflation rates, using industries’ weights in the CPI. - Data variants: - “Original” weighted median for 1968-2007 includes owner’s equivalent rent (OER) as the price for one large industry. - “Revised” median for 1983 to present includes OER for four geographic regions. - The revision matters because the change in OER (original) or one regional change (revised) is the median price change for around half of observations; the original series shows somewhat greater monthly volatility where the series overlap. - For documentation, see Bryan and Pike (1991) and Bryan et al. (1997). - Quarterly construction used in this analysis: - Convert monthly median inflation rates to a monthly series for price levels. - Average three months to obtain a quarterly price level. - Compute annualized percentage changes in that quarterly price-level variable. - Time-aggregation caveat: - The median is not linear in industry price changes; the median of quarterly changes across industries could differ from the quarterly variable constructed from monthly medians. - Future research might compare median measures based on different data frequencies. ### Some new evidence on persistence (section 3.4) - Core inflation (determined by expected inflation and activity gap) should be persistent; transitory quarterly movements in core inflation should be limited. - Methodology: - Use Stock and Watson (2007) procedure to decompose inflation into permanent (random-walk) and transitory (white-noise) components; aggregate inflation follows an IMA(1,1) process. - Allow variances of permanent and transitory shocks to change over time; estimate permanent component series and variances. - Application: - Apply procedure to XFE inflation and median inflation using sample starting in 1983Q2 (when the revised median begins). - Finding: divergences between total and permanent inflation (the transitory shocks) are smaller when inflation is measured by median inflation. This is especially pronounced in the 2000s, where median inflation appears to have almost no transitory component. - Interpretation: results support measuring core inflation with the median. - Reason for differences: - Large industry-specific price changes (annualized monthly changes of 20% or more) occur frequently in industries besides food and energy (examples: used cars, lodging away from home, women’s apparel). - Such large changes cause transitory movements in XFE inflation but are filtered out by the Cleveland Fed median. ### Median inflation during the Great Recession (section 3.5) - Over 2007Q4 to 2010Q4, the four-quarter moving average statistics are: - Median inflation fell from 3.1% to 0.5%. - XFE inflation fell from 2.3% to 0.6%. - Median inflation fell more primarily because it started at a higher level. - Causes of high median inflation in 2007: - The distribution of price changes was left-skewed during many months of 2007, driven by large price decreases in various industries (examples from March 2007: jewelry and watches fell at an annualized rate of 30%, car and truck rental fell 22%, lodging away from home fell 13%). - These large price decreases reduced XFE inflation but not median inflation. - Implication: - The larger fall in median inflation reduces divergence between actual and forecasted inflation over 2008-2010, but changing the definition of core inflation alone is insufficient to resolve the broader Phillips-curve puzzle. ### A Phillips curve with a time-varying slope (section 4 and subsections) - Sticky-price models imply time variation in the Phillips-curve slope: firms adjust more frequently when inflation level and variance are higher, making the aggregate price level more flexible and steepening the Phillips curve (Ball, Mankiw, and Romer (1988)). - Empirical specification (equations preserved exactly): - π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α_t (u−u^*)_t + ǫ_t - α_t = α_{t−1} + η_t - ǫ_t and η_t are white noise errors with variances V and W respectively; α follows a random walk. - Estimation approaches (section 4.1): - Version 1: assume a value for the ratio V/W and estimate α_t path with the Kalman smoother; choose V/W to create a plausible degree of smoothness in α_t. - Version 2: estimate V and W along with α_t by maximizing likelihood produced by the Kalman smoother (choosing variances to maximize likelihood is roughly equivalent to minimizing one-step-ahead forecast errors). - Sample: estimate equation (3) for 1960-2010. For 1984Q2-2010Q4 use Cleveland Fed revised median; 1968Q2-1984Q1 use original median; 1960Q1-1968Q1 use XFE inflation. Using XFE for the entire sample yields similar results. - Estimated α_t path (summary of Figure 3 results): - Both estimation panels show broad trends: α falls from near zero in 1960 to around -1 in the early 1970s, fluctuates around this level until 1980, then rises sharply and levels off near -0.2. - Since the mid-1980s the estimated α is quite stable; given standard errors, there is no evidence against a constant α over 1985-2010. - Determinants of the slope (section 4.2): - Theory predicts α is determined by the level and variance of inflation. - Comparison (Figure 4): estimated α (smoothed with V/W = 100, presented on an inverted scale) closely follows two series from the Stock-Watson IMA(1,1) model: the level of permanent inflation and the standard deviation of the sum of permanent and transitory shocks. - Interpretation: high and variable inflation of the 1970s and early 80s created a steep Phillips curve; the curve was flatter before 1973 and after the Volcker disinflation when inflation was relatively low and stable. - Regression representation: - Assume α_t = (a_0 + a_1 ̄π_t + a_2 σ_t) where ̄π and σ are the level of permanent inflation and the standard deviation of shocks. - Substitution yields equation (4) (preserved exactly): - π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + a_0 (u−u^*)_t + a_1 ̄π_t (u−u^*)_t + a_2 σ_t (u−u^*)_t + ǫ_t - Table 2 results: for 1960-2010 the joint significance of ̄π and σ is high (p <0.01) whether these are measured quarterly or with four-quarter moving averages; collinearity between ̄π and σ makes their individual roles hard to distinguish. - Alternative explanations and critique: - Other authors (Roberts 2006, Mishkin 2007) attribute a flatter Phillips curve since the 1980s to anchoring of inflation expectations and greater policy credibility; authors here question that view on two grounds: - Micro foundations imply the unemployment coefficient is determined by slope of marginal cost and frequency of price adjustment; anchoring affects expected-inflation term not the unemployment coefficient. - Anchoring/credibility does not explain why the Phillips curve was flat in the 1960s as well as the post-Volcker era; the Ball-Mankiw-Romer model explains this pattern. - Estimating constant slopes for subsamples (section 4.3): - Data suggest α has been close to constant since the early 1980s; assume constant α starting in 1985Q1. - Examine periods ending in 2007Q4 and 2010Q4 to check Great Recession effects; also estimate constant α for 1960-1972 and 1973-1984 for comparison. - Table 3 highlights: - For 1960-1972 and 1985-2007 the unemployment coefficient is around -0.2 or -0.25. - For 1973-1984 (high and volatile inflation) the unemployment coefficient is around -0.7. - Output coefficients are about -0.5 times unemployment coefficients, as suggested by Okun’s Law. - Multiplying the output coefficient by 1.6 yields the long-run effect on inflation of a one percent output gap for a year. Example: for 1985-2007 with inflation measured by the median, this effect is (1.6)(0.11) = 1.76. The sacrifice ratio is 1/(0.176), or about 6. - Extending the final sample from 2007 to 2010: - For XFE inflation, coefficients decline substantially in absolute value. - For median inflation, coefficients fall by less when activity is measured by the unemployment gap or not at all for the output gap. - This suggests greater stability in the Phillips curve when inflation is measured by the median; dynamic forecasts will further confirm this result. *Source: _wp11121 - 3.1 Measuring Supply Shocks (extracted text).* ### 4.4 The Great Recession and the Risk of Deflation ### 4.4 The Great Recession and the Risk of Deflation ### Forecasts for 2008–2010 and explanation of the "missing deflation" - Dynamic forecasts of quarterly inflation are based on unemployment and output gaps and estimated Phillips curves for 1985-2007. - Inflation measures used: - Median inflation (Figures 5 and 7). - XFE inflation (Figures 6 and 8). - Key findings: - Forecasts for median inflation are close to actual inflation over 2008-2010 — there is no missing deflation for median inflation. - Forecasts of XFE inflation fall to around -1% at the end of 2010, significantly below actual inflation. - Reason for difference: allowance for time-variation in the Phillips curve slope. Output and unemployment coefficients for 1985-2007 are less than half as large as estimates for the entire 1960-2007 period (which includes 1973-84). Smaller coefficients imply a smaller predicted fall in inflation. - Forecasted XFE inflation falls farther than forecasted median inflation because XFE inflation starts at a lower level in 2007 and estimated coefficients on unemployment and output are somewhat larger for XFE over 1985-2007. ### Forecasts for 2011–2013 using CBO unemployment and output-gap paths - Assumptions for dynamic forecasts (estimates of the equation from 1985 through 2010): - Unemployment (CBO forecasts): 9.4% in 2011, 8.4% in 2012, and 7.6% in 2013. - Natural rate is constant at 5.2%. - Also compute forecasts using CBO forecasts of the output gap over 2011-2013. - Four-quarter-average forecast results (Figure 9): - Because unemployment remains above the natural rate and output is below potential, inflation falls steadily. - Inflation becomes negative at the end of 2011. - At the end of 2013 inflation reaches: - -1.9% (based on unemployment forecasts). - -1.3% (based on output forecasts). - Implication: the Phillips curve that fits recent history (and explains why deflation hadn’t occurred yet) also predicts that deflation will arrive soon absent other forces. ### Robustness checks (summary) - Extensions tried that do not significantly affect conclusions: - Add lags of unemployment and longer lags of inflation (as suggested by Gordon (2011)). - Use Stock and Watson (2010)’s unemployment-gap variable (difference between current unemployment and minimum unemployment over the current and previous eleven quarters). - Use Debelle and Laxton (1997)’s nonlinear transformation of unemployment as the activity variable. - Add Ball and Moffitt (2001)’s measure of the acceleration of productivity growth to the Phillips curve. - Estimate the path of the natural rate u* jointly with the Phillips curve coefficient, rather than relying on CBO estimates of u*. - Estimate an equation for total inflation that includes a measure of supply shocks (difference between total inflation and median inflation), rather than estimating an equation for core inflation. - Conclusion: None of these extensions has a significant impact on the paper’s conclusions. ### Anchored Expectations — concepts and evidence - Two kinds of anchoring distinguished: - Shock anchoring: transitory shocks to inflation are not passed into expectations or future inflation. - Level anchoring: expectations are tied to a particular level of inflation (e.g., 2%). - Shock anchoring: - Post-Volcker monetary regime exhibits shock anchoring: supply shocks do not strongly affect expectations or future inflation. - Empirical evidence: - For 1960-1972 and 1973-1984, including lagged total inflation raises R2 of the Phillips curve; lagged total dominates lagged core. - For 1985-2010, results reverse: estimated weight on lagged core is 0.89. - Regressing SPF one-year expected inflation (1985-2010) on average of lagged core and lagged total inflation yields a weight on lagged core of 0.86 (standard error = 0.06). - Time-varying-weight experiments show weights on lagged core inflation consistently close to one for both actual and expected inflation in the post-Volcker period. - Conclusion: shock anchoring is a stable feature of the post-Volcker monetary regime. - Level anchoring: - Hypothesis: expectations of core PCE inflation are anchored at 2%; because core CPI has exceeded core PCE by about 0.5% on average since 1980, expectations of core inflation might be anchored at 2.5%. - Estimation strategy for expectations (equation (5)): - πet = δt 2.5 + (1−δt) (1/4)(πt−1 + πt−2 + πt−3 + πt−4) + εt - δt follows a random walk; δt = 0 means purely backward-looking expectations; δt = 1 means fully anchored at 2.5%. - Measured πe with SPF four-quarter-ahead forecasts; past inflation measured with Cleveland Fed median; path of δt estimated with Kalman smoother assuming variance of ε is 100 times variance of innovations in δ. - Estimated path for δt: - δt near zero until the early 1990s, then rises. - δt is around 0.6 over 2007-2010 (for SPF expectations). - Behavior of actual inflation (equation (6)): - πt = δt 2.5 + (1−δt) (1/4)(πt−1 + πt−2 + πt−3 + πt−4) + α( u − u*t ) + εt - Estimated δt for actual inflation rises similarly from near zero in early 1990s. - Values in 2010Q4: - δ = 0.47 when the Phillips curve includes the unemployment gap. - δ = 0.30 when the Phillips curve includes the output gap. - Possible interpretation: price setters are less sophisticated than professional forecasters and learn more slowly about the Fed’s commitment to 2.5% inflation. - Confidence intervals for the δs overlap; differences among Figures 10–12 should not be over-interpreted. - Estimated coefficients α in equation (6): - α = -0.24 (standard error = 0.03) for the unemployment gap. - α = 0.13 (standard error = 0.02) for the output gap. - Interpretation: as expectations become anchored, so has actual inflation, but anchoring is partial and incomplete. ### Dynamic forecasts with anchoring and implications for deflation risk - Forecasts of four-quarter inflation over 2008-2010 based on equation (6) estimated for 1985-2007, assuming δ remains at 2007Q4 level: - δ in 2007Q4 = 0.31 when equation includes the unemployment gap. - δ in 2007Q4 = 0.26 when equation includes the output gap. - Result: forecasted inflation falls less than actual inflation over 2008-2010. Difference from purely backward-looking model is modest and statistically insignificant because estimated degree of anchoring in 2007Q4 is fairly small and actual inflation was near 2.5%. - Forecasts for 2011-2013 using equation (6) estimated for 1985-2010 and holding δ at the 2010Q4 level: - Assume δ stays at its 2010Q4 value. - With anchoring, deflation predicted by the basic Phillips curve does not occur. - Instead, inflation is: - Steady at about 0.5% and then rises to 1% at the end of 2013. - Mechanism: partial anchoring pulls expected inflation up toward 2.5%, causing actual inflation to bottom out rather than fall in response to high unemployment. - Caveats and uncertainties: - Considerable uncertainty about the degree of anchoring in the Phillips curve. - With unemployment in the equation, the 95% confidence interval for δ in 2010Q4 is roughly [0.2, 0.7]. - With the output gap, the 95% confidence interval is [0.2, 0.4]. - Anchoring may weaken in the near future: the weight on 2.5% rose when actual inflation was near that level; forecasts that actual inflation will stay below 1% for several years may lead price setters to learn that inflation is stuck below 2.5% and adjust expectations downward. - Long-run inflation expectations (e.g., ten-year SPF forecasts) have been close to 2.5% since 2000, but price-setting depends on one-year expectations in most Phillips-curve theories and in recent empirical work. ### Comparison with other forecasts - Forecasts with anchoring are fairly close to others: - At the end of 2010, the CBO was forecasting core CPI inflation rates of 0.9%, 1.0%, and 1.4% over 2011– (text truncated in source). *Source: _wp11121 - 4.4 The Great Recession and the Risk of Deflation* ### 2013. These forecasts are 0.3 to 0.5 percentage points above ours. These differences might be ### _wp11121 - 2013. These forecasts are 0.3 to 0.5 percentage points above ours. These differences might be ### Comparisons of external forecasts and inflation concepts - SPF median forecast for XFE inflation is 1.3% for 2011 and 1.7% for 2012 (and unavailable for 2013). - The SPF forecast for 2012 is a full percentage point above ours. - Only 44 percent of SPF forecasters say they use the concept of the natural rate of unemployment. - The authors measure core inflation with the weighted median of price changes; the CBO and other groups use XFE or alternative measures, which can produce systematic differences in forecasts. ### The New Keynesian Phillips Curve (NKPC) — theory and empirical issues - Original NKPC (Roberts 1995) specification: - πt = Et πt+1 + λ(y − y* )t - Et πt+1 is this quarter’s rational forecast of next quarter’s inflation and y − y* is the output gap. - Rearranged implication: - Et πt+1 − πt = −λ(y − y* )t - Theory implies λ is positive; thus output gap should have a negative effect on the expected change in inflation from t to t+1. - Empirical contradiction: - In the data, output has a positive correlation with the change in inflation (both before and during the Great Recession), leading estimates of λ to be consistently negative and contradicting the theory. - Gali and Gertler modification: - Replace output gap with real marginal cost: - πt = Et πt+1 + λ mc t - They measure real marginal cost with real unit labor costs (labor’s share of income) and obtain a positive estimate of λ. - Critiques of the Gali-Gertler approach: - Rudd and Whelan (2005, 2007) and Mazumder (2010) argue labor’s share is not a credible measure of real marginal cost; labor’s share is generally countercyclical while marginal cost is argued to be procyclical. - Mazumder uses a procyclical measure of marginal cost based on overtime and obtains negative estimates of λ—the same result that discredited the original NKPC. ### NKPC performance during the Great Recession (empirical results) - The Gali-Gertler specification does not fit recent data even if labor’s share is accepted as marginal cost. - Estimation approach: - Equation estimated by GMM using orthogonality condition: - Et{(πt − λ mc t − πt+1)} zt = 0 - Instruments: four lags each of inflation, labor’s share, the output gap, a long-short interest rate spread, nominal wage inflation, and commodity price inflation. - Median CPI inflation used (results similar for other inflation measures including GDP deflator). - Time variation in coefficient λ: - Gali and Gertler report significantly positive λ for 1960-1997 (replicated by authors). - Coefficient on labor’s share is significantly positive for 1985-2007 (t = 2.04), but insignificant for 1985-2010 (t = 0.92). - The model’s fit deteriorates when 2008-2010 are added to the sample. - 2009 and 2010 are big outliers: labor’s share experienced a sharp fall while unemployment gap was large. - Historically labor’s share was positively correlated with the unemployment gap (countercyclical), producing positive λ estimates. - During 2009-2010 labor’s share fell sharply—productivity growth was strong and real wages did not keep up—changing cyclicality and the Phillips curve coefficient. - Interpretation issue: - Substituting mc for the output gap in the acceleration form: - Et πt+1 − πt = −λ mc t - During 2009-2010 falling labor’s share implies the equation would predict inflation to be expected to rise, but inflation fell; this implies repeated forecast errors in the same direction and undermines rational expectations interpretation. ### Main empirical conclusions on Phillips-curve specifications - A simple accelerationist Phillips curve fits 1960–2010, including the Great Recession, under two conditions: - Core inflation measured with the weighted median of price changes. - The Phillips-curve slope is allowed to change with the level and variance of inflation. - Evidence of a change since the 1990s: - Expectations of inflation, and hence actual inflation, have become partially anchored at a level of 2.5%. - If anchoring persists, the United States is likely to avoid deflation in the near future despite high unemployment. - Deflation may occur if low inflation leads to a de-anchoring of expectations. - Future research topic: - Effect of unemployment duration on the Phillips curve: long-term unemployment (fraction unemployed for more than 26 weeks) may put less downward pressure on inflation than short-term unemployment (Llaudes 2005). - Long-term and short-term unemployment are highly collinear in U.S. data, but collinearity is diminishing because long-term unemployment has risen much more than short-term unemployment since 2008. - If long-term unemployment has less effect on inflation, inflation will fall by less than expected based on aggregate unemployment; the shift toward long-term unemployment, along with expectations anchoring, could prevent deflation. ### Appendix — robustness variations and findings - A.1 Longer lags (Gordon-style specification): - Modify basic Phillips curve by including four lags of unemployment and twenty-four lags of inflation (sum of coefficients on inflation lags set to one). - Using the Cleveland Fed median; for 1960-2007 the sum of unemployment coefficients is -0.26. - Dynamic forecast for 2010Q4 is -5.1%; pre-2008 Phillips curve incorrectly predicts deflation is robust to lag structure. - A.2 Stock-Watson unemployment gap: - Stock-Watson gap: uSWt = ut − min(ut, .., ut−11) - Replacing or augmenting CBO gap with uSW produces very similar Phillips curve results. - Joint estimation with both gaps shows the Stock-Watson gap weight is not statistically significant and does not add explanatory power for these purposes. - A.3 Nonlinear Phillips curve (Debelle-Laxton specification): - Replace (u − u*) with (u − u*) / u. - Fit (R2s) is very close to linear specification; collinearity arises when including both forms. - Debelle-Laxton model (1985-2007 estimates) predicts four-quarter inflation of 1.3% in 2010Q4 vs actual 0.5%; linear model predicts 0.3%, closer to actual. - A.4 Productivity growth (Ball and Moffitt): - Add g − ĝ (change in productivity growth relative to a weighted average ĝ) with partial adjustment parameter ϕ = 0.9875 (quarterly analog). - g − ĝ insignificant for 1960-1972 and 1973-1984; significant for 1985-2010 and adds modestly to R2. - Including g − ĝ in 1985-2007 dynamic forecasts yields median inflation forecast of -0.32% for 2010Q4, similar to -0.24% from basic specification. - A.5 Estimating a time-varying NAIRU (Staiger et al. approach; Kalman filter with V/W = 400): - Time-varying NAIRU series closely matches CBO NAIRU. - Example: 1985:1 CBO NAIRU = 6.03% vs authors’ estimate 6.04%; end-2010 CBO = 5.20% vs authors’ 5.44%. - Deviation between series from 1985 to 2010 never exceeds 0.32%. - Estimating basic specification with the time-varying NAIRU yields unemployment gap coefficient of -0.178 vs -0.168 under CBO gap. - Conclusion: results robust to alternative NAIRU measures. - A.6 Supply shocks and total CPI inflation (Ball and Mankiw approach): - Define supply shock Shockt = total CPI inflation − median CPI inflation for 1972–2010. - Estimate πTt = 1/4(πt−1 + πt−2 + πt−3 + πt−4) + α(u − u*)t + β Shockt + εt - Estimated β ≈ 1 in all periods, particularly 1985-2007 and 1985-2010. - If β = 1, model reduces to πt = 1/4(πt−1 + πt−2 + πt−3 + πt−4) + α(u − u*)t, which is the basic core-inflation Phillips-curve specification. - Therefore behavior of total inflation with supply shocks is consistent with behavior of core inflation; 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(2010): “Kalman Filtering,” Econometrics study notes. *Note: The list above reproduces the bibliography entries as presented in the source.* --- ### Tables: Phillips Curve Estimation Results (Selected Tables) ### Table 1: Phillips Curve Estimated from 1960:1 to 2007:4 Model specifications: - (a) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α(u_t − u*_t) - (b) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α(y_t − y*_t) Key reported coefficients and statistics (by π_t measure: TOTAL CPI | XFE CPI | TOTAL CPI | XFE CPI CPI): - Coefficient for α: -0.507 | -0.474 | 0.308 | 0.257 - S.E.: (0.091) | (0.077) | (0.049) | (0.042) - R^2: 0.703 | 0.746 | 0.713 | 0.744 Note: π_t is the inflation rate, u_t is the national unemployment rate, u*_t is the CBO measure of the NAIRU, y_t is log of real GDP, and y*_t is the CBO’s estimate of potential (log) real GDP. Estimation is conducted by OLS. --- ### Table 2: Phillips Curve with Level and Variance of Inflation Models: - (a) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + a_0 (u−u*)_t + ε_t - a_0 Coefficient: -0.355 - S.E.: (0.058) - R^2: 0.773 - (b) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + a_0 (u−u*)_t + a_1 ̄π_t (u−u*)_t + a_2 σ_t (u−u*)_t + ε_t - Using Quarterly Data for ̄π_t, σ_t: - a_0 Coefficient: 0.194 - a_1 Coefficient: -0.008 - a_2 Coefficient: -0.645 - S.E.: (0.115) | (0.036) | (0.187) - R^2: 0.800 - p-value for H_0: a_1 = a_2 = 0: 0.000 - Using 4-Quarter Moving Averages for ̄π_t, σ_t: - a_0 Coefficient: 0.132 - a_1 Coefficient: -0.134 - a_2 Coefficient: -0.008 - S.E.: (0.110) | (0.036) | (0.191) - R^2: 0.807 - p-value for H_0: a_1 = a_2 = 0: 0.000 Note: Time period for results in this table is 1960:1-2010:4. ̄π and σ are the level of permanent inflation and the standard deviation of shocks. --- ### Table 3: Phillips Curves for Core Inflation (by subperiods and π_t measure) Model forms as in Table 1 with π_t measures: MEDIAN CPI | XFE CPI | CPI Periods and reported results: - 1960:1-1972:4 - Coefficient for α: -0.231 | 0.135 - S.E.: (0.103) | (0.056) - R^2: 0.729 | 0.733 - S.E. of Regression: 0.992 | 0.985 - 1973:1-1984:4 - Coefficient for α: -0.650 | -0.688 | 0.365 | 0.371 - S.E.: (0.172) | (0.184) | (0.095) | (0.103) - R^2: 0.513 | 0.402 | 0.516 | 0.391 - S.E. of Regression: 2.254 | 2.408 | 2.247 | 2.429 - 1985:1-2007:4 - Coefficient for α: -0.202 | -0.246 | 0.114 | 0.136 - S.E.: (0.054) | (0.067) | (0.029) | (0.037) - R^2: 0.700 | 0.761 | 0.703 | 0.763 - S.E. of Regression: 0.425 | 0.529 | 0.423 | 0.528 - 1985:1-2010:4 - Coefficient for α: -0.168 | -0.136 | 0.114 | 0.092 - S.E.: (0.031) | (0.039) | (0.019) | (0.024) - R^2: 0.781 | 0.764 | 0.792 | 0.769 - S.E. of Regression: 0.448 | 0.570 | 0.437 | 0.563 Note: Median CPI inflation data begins in 1967:2. --- ### Table 4: Shock Anchoring in the Phillips Curve Model (a): π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α(u_t − u*_t) - Period coefficients for α: - 1960:1-1972:4: -0.231, SE (0.103), R^2 0.729 - 1973:1-1984:4: -0.650, SE (0.172), R^2 0.513 - 1985:1-2010:4: -0.168, SE (0.031), R^2 0.781 Model (b): π_t = 1/4(π^T_{t−1}+π^T_{t−2}+π^T_{t−3}+π^T_{t−4}) + α(u_t − u*_t) - Coefficients for α: - 1960:1-1972:4: -0.319, SE (0.091), R^2 0.789 - 1973:1-1984:4: -0.620, SE (0.165), R^2 0.551 - 1985:1-2010:4: -0.003, SE (0.064), R^2 0.042 Model (c): π_t = γ 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + (1−γ) 1/4(π^T_{t−1}+π^T_{t−2}+π^T_{t−3}+π^T_{t−4}) + α(u_t − u*_t) - Coefficients: - α: -0.329 | -0.630 | -0.150 (for periods 1960:1-1972:4 | 1973:1-1984:4 | 1985:1-2010:4) - SE: (0.095) | (0.165) | (0.031) - γ: -0.117 | 0.326 | 0.886 - SE: (0.295) | (0.294) | (0.046) - R^2: 0.785 | 0.553 | 0.791 Note: π_t is core inflation (XFE inflation for 1960:1-1972:4, and median inflation for 1973:1-2010:4), and π^T_t is the total CPI inflation rate. --- ### Table 5: New Keynesian Phillips Curve Results Model: π_t = E_t π_{t+1} + λ s_t - s_t is the log of the labor income share. Reported coefficients for λ and statistics: - 1960:1-1997:4 - Coefficient for λ: 0.060 - SE: (0.012) - R^2: 0.710 - 1960:1-2010:4 - Coefficient for λ: 0.040 - SE: (0.045) - R^2: 0.740 - 1960:1-1972:4 - Coefficient for λ: 0.110 - SE: (0.020) - R^2: 0.733 - 1973:1-1984:4 - Coefficient for λ: 0.129 - SE: (0.053) - R^2: 0.398 - 1985:1-2007:4 - Coefficient for λ: 0.044 - SE: (0.022) - R^2: 0.665 - 1985:1-2010:4 - Coefficient for λ: 0.012 - SE: (0.013) - R^2: 0.760 Note: For the rational expectations Phillips curve we use GMM with instruments of: four lags of inflation, the labor share, the output gap, a long-short interest rate spread, nominal wage inflation, and commodity price inflation. The combined median inflation rate described in the note is used across subperiods. --- ### Appendix Table A1: Variations on Basic Phillips Curves (Selected Panels) Models and reported coefficients (period breakdowns as indicated): (a) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α(u_t − u*_t) - Coefficient for α: - 1960:1-1972:4: -0.231, SE (0.103), R^2 0.729 - 1973:1-1984:4: -0.650, SE (0.172), R^2 0.513 - 1985:1-2010:4: -0.168, SE (0.031), R^2 0.781 (b) π_t = 1/4(π_{t−1}+π_{t−2}+π_{t−3}+π_{t−4}) + α u^{SW}_t - Coefficient for α: - 1960:1-1972:4: -0.296, SE (0.118), R^2 0.735 - 1973:1-1984:4: -0.521, SE (0.164), R^2 0.477 - 1985:1-2010:4: -0.157, SE (0.027), R^2 0.788 (c) Regime-interaction specification for 1960:1-2010:4 - β_0: -0.259, SE (0.125) - β_1: -0.258, SE (0.189) - β_2: 0.381, SE (0.141) - λ: 0.269, SE (0.340) - R^2: 0.788 (d) Nonlinear unemployment interaction specification - Coefficient for α: - 1960:1-1972:4: -0.965, SE (0.390), R^2 0.734 - 1973:1-1984:4: -5.836, SE (1.441), R^2 0.529 - 1985:1-2010:4: -1.312, SE (0.241), R^2 0.780 (e) Including growth gap (g_t − ĝ_t) - Coefficient for α: - 1960:1-1972:4: -0.224, SE (0.105), R^2 0.724 - 1973:1-1984:4: -0.636, SE (0.172), R^2 0.515 - 1985:1-2010:4: -0.158, SE (0.030), R^2 0.798 - Coefficient for β: - -0.014 | -0.096 | -0.049 - SE: (0.033) | (0.085) | (0.016) (f) Total CPI regression with Shock_t (Shock_t = total CPI inflation − median CPI inflation) - Periods and coefficients for α: - 1973:1-1984:4: -0.667, SE (0.175), R^2 0.566 - 1985:1-2007:4: -0.194, SE (0.055), R^2 0.922 - 1985:1-2010:4: -0.167, SE (0.030), R^2 0.953 - Coefficients for β: - 0.852 | 1.031 | 1.038 - SE: (0.221) | (0.034) | (0.024) Notes included in Appendix: u_t is the national unemployment rate, u*_t is the CBO measure of the NAIRU, u^{SW}_t is the difference from unemployment in quarter t from the minimum unemployment rate from quarter t to t−11. D1 = 1 for 1973:1 to 2010:4 (0 otherwise). D2 = 1 for 1985:1 to 2010:4 (0 otherwise). For various panels π_t uses XFE for 1960:1-1967:4 and median CPI for later periods as specified; a combined median inflation rate construction is described for certain panels. Shock_t is total CPI inflation minus median CPI inflation. --- ### Figures: Descriptions of Key Plots (labels and emphasis retained) - Figure 1: 4-Quarter Moving Averages of Dynamic Forecasts of Inflation for 2008 to 2010 Based on 1960:1-2007:4 Regression - Panels: (a) Total CPI Inflation; (b) XFE CPI Inflation - Series plotted: Actual Inflation; Unemployment Gap Dynamic Forecast; Output Gap Dynamic Forecast - Figure 2: Median CPI and XFE CPI Inflation Rates, 1983:2-2010:4 - Panels: (a) Median CPI Inflation (Actual Inflation and Permanent Component); (b) XFE CPI Inflation (Actual Inflation and Permanent Component) - Figure 3: Time-Varying α_t from Phillips Curve, 1960:1-2010:4 - Panels: (a) No Restriction; (b) V = 100Q - Figure 4: Permanent Component of Median CPI Inflation vs. Time-Varying α - Series: Permanent Component of Inflation; SE times 10 · alpha - Figures 5–8: Dynamic Forecasts of Median CPI and XFE CPI Inflation for 2008 to 2010 based on 1985:1-2007:4 regressions (panels for Unemployment Gap and Output Gap; Actual Inflation and Dynamic Forecast plotted) - Figures 9–14: 4-Quarter Moving Averages of Dynamic Forecasts of Median CPI Inflation for 2008–2010 and 2011–2013 under various regression specifications (1985:1-2010:4, anchoring specifications). Series: Actual Inflation; Unemployment Gap Dynamic Forecast; Output Gap Dynamic Forecast. - Figure 15: Labor Income Share vs. Unemployment Gap (Annual Averages), 1985-2010 - Series scale: Labor Share (1999=100) vs. U − U* - Figure A1: 4-Quarter Moving Averages of Dynamic Forecasts of Median CPI Inflation for 2008 to 2010 Based on Gordon (2011) - Series: Actual Inflation; Gordon (2011) Dynamic Forecast *The figures listed above reproduce the figure titles and the series/labels as presented in the source.* --- *Source: _wp11121 - References (content unit as provided).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11121.pdf_