Finance and Economics Discussion Series: Accessible versions of figures for 2020-013

# Inflation at Risk

## Accessible version of figures

Figure 1: Inflation-at-Risk.
The figure displays simulated distributions. In the left panel, the probability of the average future inflation rate falling below 0% is 4 percent. In the right panel the probability of average future inflation exceeding 3% is 15 percent.

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Figure 2: Quantile Regression Slopes.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2). The lines illustrate the slopes associated with the median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Circles indicate scatterplots of average future inflation against a given inflation determinant. Grey circles indicate scatterplots of average future inflation against a given financial variable prior to 1999:Q4 whereas black circles indicate the scatterplot for the period starting in 2000:Q1.

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Figure 3: Time Evolution of 10th Inflation Quantile Across Models.
The figure displays the time evolution of the 10th inflation quantile estimated from the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). The credit spread (purple dashed) is also reported. Shaded bars indicate NBER-dated recessions.

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Figure 4: Rossi and Sekhposyan, 2019 Test for Correct Calibration of Predictive Density.
The figure illustrates the CDF of a uniform distribution along with the empirical CDFs of out-of-sample PITs obtained fromthe quantile regressions model (2), in its baseline version (blue) and in two versions which either do not condition on financial variables in estimation (green) or in the construction of the inflation quantiles (black). The 5% critical values for each model (dashed-dotted), are bootstrapped following the Rossi and Sekhposyan, 2019 procedure for multi-step-ahead forecasts. As in Adrian et al., 2019, the PITs are constructed via an expanding rolling windows estimation initiallyusing 20 years of data. Confidence bands should thus be taken as general guidance since Rossi and Sekhposyan, 2019 derive them for PITs computed using a fixed rolling window scheme.

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Figure 5: Conditional Predictive Inflation Densities at Selected Time Episodes.
The figures shows for selected time episodes the estimated skewed-$$t$$ conditional densities of average four-quarter-ahead core CPI inflation associated with the quantile regressions model (2). The inset box reports the values of average future inflation across quantiles at the same selected periods. More formally, it depicts the estimated skewed-$$t$$ inverse CDF associated with the conditional densities in the main panel.

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Figure 6: Quantile Regression Slopes Across Subsamples.
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2). Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure 7: "Counterfactual" Predictive Densities (left) and Inflation Probabilities (right).
The left panels show "counterfactual" skewed $$t$$-Student conditional densities of average four-quarter-ahead core CPI inflation computed using "counterfactual" conditional quantiles over 2000-2019 which were obtained using different subsample estimates of the quantile regressions model (2). The right panels show "counterfactual" inflation probabilities for different cutoffs. These probabilities are computed from the "counterfactual" skewed $$t$$-Student conditional densities shown in the left panels. Shaded bars indicate NBER-dated recessions.

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Figure 8: Quantile Regression Slopes and Confidence Intervals.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead euro area core HICP (left) and United States Core CPI inflation (right) defined in (2). The black squares correspond to the point estimates whereas the vertical lines to the 68% confidence intervals computed via "blocks-of-blocks" bootstrap using 10,000 replications (see Appendix B) for the 10th quantile (blue), median (red) and 90th quantile (yellow). The estimation period is 1999:Q1 to 2017:Q4. The OLS estimates and their 95% confidence intervals are represented by the red lines.

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Figure 9: Time Evolution of Selected Conditional Inflation Quantiles.
The figure displays the time evolution of the conditional quantiles of euro area core HICP inflation (top) and of Unites States core CPI inflation (bottom) estimated from the quantile regression defined in (2). Shaded bars indicate NBER-dated recessions for the United States and OECD-based recession indicators for the euro area.

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Figure 10: Historical Contributions of Economic and Financial Factors.
The figure shows historical contributions of average four-quarter-ahead euro area core HICP (left) and United States core CPI inflation (right) associated with the quantile regressions model (2). The contribution of a given inflation determinant is obtained by multiplying its time series with its estimated slope. Its relative share is then obtained by weighting the contribution with its relative magnitude vis-à-vis the sum of all contributions (the share of the constant term is distributed across the inflation determinants based on their relative share so as to not distort results). Shaded bars indicate NBER-dated recessions for the United States and OECD-based recession indicators for the euro area.

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Figure 11: Partial Effect of Credit Spread on 10th Inflation Quantile.
The figure displays the time evolution of the 10th conditional inflation quantile of euro area core HICP inflation (left) and of Unites States core CPI inflation (right) estimated from the quantile regressions model (2), in its baseline version (blue straight) and in its version in which the effect of credit spreads is set to zero (black dash-dotted). Shaded bars indicate NBER-dated recessions for the United States and OECD-based recession indicators for the euro area.

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Figure 12: Selected Time Episodes of Predictive Densities (Left) and Skewness (Right). Euro Area Core HICP Inflation.
The left panels show the estimated skewed $$t$$-Student densities of average four-quarter-ahead euro area core HICP inflation for alternative specifications of the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). The right panels show the estimated skewed $$t$$-inverse cumulative associated with the conditional densities in the left panels.

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Figure 13: Selected Time Episodes of Predictive Densities (Left) and Skewness (Right). United States Core CPI Inflation.
The left panels show the estimated skewed $$t$$-Student densities of average four-quarter-ahead United States core CPI inflation for alternative specifications of the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). The right panels show the estimated skewed $$t$$-inverse cumulative associated with the conditional densities in the left panels.

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Figure 14: Inflation Probabilities for Alternative Cutoff Values.
The figure shows the time evolution of inflation probabilities of euro area core HICP inflation (left) and United States core CPI inflation (right) for different cutoffs. These probabilities are computed from the skewed $$t$$-Student conditional densities of the average four-quarter-ahead inflation measures which were fitted on the estimated conditional quantiles for alternative specifications of the quantile regression model (2). Both panels are reported for the specification without and with the credit spread (in blue straight and black dash-dotted lines, respectively). Shaded bars indicate NBER-dated recessions for the United States and OECD-based recession indicators for the euro area.

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Figure 15: Inflation Probabilities, Quantile Model vs. Financial Markets.
The top panel shows the options-implied inflation probabilities of United States headline CPI inflation next year being above 1% along with the 3-months- and 6-months-ahead oil price surprises computed using the oil market price expectations that Baumeister and Kilian, 2016 recovered from oil futures prices (top panel). The bottom panel shows the probability of average one-year-ahead core CPI inflation being below 1% coming from thequantile regression model as well as the probability of one-year-ahead headline CPI being below 1% implied from inflation caps and floors contracts as in Kitsul and Wright, 2013, purified from oil, energy and food price effects and transformed to quarterly frequency.

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Figure 16: Credit Spread Coefficients, Quantile Regression vs. Financial Markets.
The left panel reports the slopes of separate regressions of inflation probabilities on the credit spread (at monthly frequency), along with their 95% confidence interval. The coefficient for the probability of future inflation being below 1% is rescaled to match the slope estimated on the lowest inflation quantile which arises from the quantile Phillips curve model (right panel, taken from Figure 8). The coefficients are transformed from positive to negative for the probability of inflation being below 1% - as a positive correlation between the credit spread and this probability is equivalent to a negative relationship between the credit spread and the lowest inflation quantile.

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Figure A-1: Inflation Measures, United States.

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Figure A-2: Regressors, United States.

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Figure A-3: Inflation Measures, Euro Area.

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Figure A-4: Regressors, Euro Area.

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Figure B-1: Quantile Regression Slopes and Confidence Intervals.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead United States Core CPI inflation defined in (2). The black squares correspond to the point estimates whereas the vertical lines to the 68% confidence intervals computed via "blocks-of-blocks" bootstrap using 10,000 replications for the 10th quantile (blue), median (red) and 90th quantile (yellow). The estimation period is 1973:Q1 to 2019:Q1. The OLS estimates and their 95% confidence intervals are represented by the red lines.

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Figure C-1: Quantile Regression Slopes, Core PCE.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead core PCE inflation defined in expression (2). The lines illustrate the slopes associated with the median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Circles indicate scatterplots of average future inflation against a given inflation determinant. Grey circles indicate scatterplots of average future inflation against a given financial variable prior to 1999:Q4 whereas black circles indicate the scatterplot for the period starting in 2000:Q1.

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Figure C-2: Quantile Regression Slopes, Stock and Watson, 2019 CSI.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead Stock and Watson,2019 Cyclically Sensitive Inflation defined in expression (2). The lines illustrate the slopes associated with the median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Circles indicate scatterplots of average future inflation against a given inflation determinant. Grey circles indicate scatterplots of average future inflation against a given financial variable prior to 1999:Q4 whereas black circles indicate the scatterplot for the period starting in 2000:Q1.

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Figure C-3: Quantile Regression Slopes Across Subsamples, Core PCE.
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core PCE inflation defined in expression (2). Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure C-4: Quantile Regression Slopes Across Subsamples, Stock and Watson, 2019 CSI.
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead Stockand Watson, 2019 Cyclically Sensitive Inflation defined in expression (2). Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure C-5: Quantile Regression Slopes and Confidence Intervals.
The figure displays the slope coefficients of the quantile regression of average four-quarter-ahead of core PCE inflation (left) and Stock and Watson, 2019Cyclically Sensitive Inflation (right) defined in (2). The black squares correspond to the point estimates whereas the vertical lines to the 68% confidence intervals computed via "blocks-of-blocks" bootstrap (see Appendix B) using 10,000 replications for the 10th quantile (blue), median (red) and 90th quantile (yellow). The estimation period is 1999:Q1 to 2017:Q4.

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Figure C-6: Partial Effect of Credit Spread on 10th Inflation Quantiles. Core PCE and Stock and Watson, 2019 CSI Inflation.
The figure displays the time evolution of the 10th conditional inflation quantile of core PCEinflation (left) and Stock and Watson, 2019 Cyclically Sensitive Inflation (right) estimated from the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). Shaded bars indicate NBER-dated recessions.

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Figure C-7: Selected Time Episodes of Predictive Densities (Left) and Skewness (Right). Core PCE Inflation.
The left panels show the estimated skewed $$t$$-Student densities of average four-quarter-ahead core PCE inflation for alternative specifications of the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). The right panels show the estimated skewed $$t$$-inverse cumulative associated with the conditional densities in the left panels.

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Figure C-8: Selected Time Episodes of Predictive Densities (Left) and Skewness (Right). Stock and Watson, 2019 CSI Inflation.
The left panels show the estimated skewed $$t$$-Student densities of average four-quarter-ahead Stock and Watson, 2019 Cyclically Sensitive Inflation for alternative specifications of the quantile regressions model (2), in its baseline version (blue straight) and in its version where the effect of credit spreads is set to zero (black dash-dotted). The right panels show the estimated skewed $$t$$-inverse cumulative associated with the conditional densities in the left panels.

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Figure C-9: Inflation Probabilities for Alternative Cutoff Values. Core PCE and Stock and Watson, 2019 CSI Inflation.
The figure shows the time evolution of inflation probabilities of core PCE inflation (left) andStock and Watson, 2019 Cyclically Sensitive Inflation (right) for different cutoffs. These probabilities are computed from the skewed $$t$$-Student conditional densities of the average four-quarter-ahead inflation measures which were fitted on the estimated conditional quantiles for alternative specifications of the quantile regression model (2). Both panels are reported for the specification without and with the credit spread (in blue straight and black dash-dotted lines, respectively). Shaded bars indicate NBER-dated recessions.

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Figure D-1: Quantile Regressions Slopes Across Relative Price Measures.
The figure displays the slope coefficients on relative prices of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression(2). The lines illustrate the slopes associated with the median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Circles indicate scatterplots of average future inflation against a given inflation determinant. The left panel corresponds to the model where the relative price measure is relative import price inflation, whereas the right panel considers the model where the relative price measure is relative oil price inflation. Grey circles indicate scatterplots of average future inflation against a given financial variable prior to 1999:Q4 whereas black circles indicate the scatterplot for the period starting in 2000:Q1.

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Figure D-2: Quantile Regressions Slopes Across Financial Variables.
The figure displays the estimated coefficients of the quantile regression of average four-quarter-ahead core CPI inflation defined in (2), usingcorporate bond spreads (top, left), the Gilchrist and Zakrajšek, 2012 excess bond premium (top, right) and the National Financial Conditions Index (center, bottom) and the same sample period as the baseline. The lines illustrate the slopes associated with the median (red), the 10th (blue) and the 90th (yellow) inflation quantile. The black lines are the OLS estimates. Grey circles indicate scatterplots of average future inflation against a given financial variable prior to 1999:Q4 whereas black circles indicate the scatterplot for the period starting in 2000:Q1.

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Figure D-3: Quantile Regression Slopes Across Subsamples, Corporate Bond Spreads
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2), using corporate bond spreads and the same sample period as the baseline. Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure D-4: Quantile Regression Slopes Across Subsamples, Excess Bond Premium.
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2), using the Gilchrist and Zakrajšek, 2012 excess bond premium and the same sample period as the baseline. Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure D-5: Quantile Regression Slopes Across Subsamples, NFCI.
The figure displays the estimated slopes of the quantile regression of average four-quarter-ahead core CPI inflation defined in expression (2), using the NFCI and the same sample period as the baseline. Two different subsamples are considered: (i) 1973-1999 and (iii) 2000-2019. The bars illustrate the coefficients associated with the 10th quantile (blue), median (red) and 90th quantile (yellow).

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Figure E-1: Inflation Probabilities and The Role of Skewness.
The figure displays the three states associated with a change in an inflation determinant that causes the initial normal density (top panel) to feature a lower mean (center panel) and then also a right-skew (bottom panel).

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Figure F-1: Financial Markets' Inflation Probabilities vs. Oil, Energy and Food Price Measures.
The figure shows the monthly options-implied inflation probabilities of headline CPI inflation next year being above 1% along with the 3-months- and 6-months-ahead oil price surprises computed using the oil market price expectations that Baumeister and Kilian, 2016 recovered from oil futures prices (top panel), the negative of energy price inflation (mid panel) and the negative of food price inflation (bottom panel).

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Figure F-2: Inflation Probabilities from Financial Markets vs. Credit Spread.
The figures shows the credit spread against quarterly options-implied inflation probabilities of headline CPI inflation next year being above 1% (top panel), between 2% and 3% (mid panel) and above 4% (top panel).

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