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Online Appendix The Usefulness of the Median CPI in Bayesian VARs used for Macroeconomic Forecasting and Policy Brent Meyer and Saeed Zaman February 2013 Revised: October 2016 This document is a companion to the paper that has the same title as this one. *Brent Meyer is an assistant policy adviser and an economist at the Federal Reserve Bank of Atlanta (brent.meyer@atl.frb.org). Saeed Zaman is an economist at the Federal Reserve Bank of Cleveland (saeed.zaman@clev.frb.org). The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Atlanta, Federal Reserve Bank of Cleveland or the Federal Reserve System. Brent Meyer thankfully acknowledges that much of the research documented in this paper was performed while he was a member of the Research Department at the Cleveland Fed. The authors thank Todd Clark, Domenico Giannone, Edward Knotek II, Ellis Tallman, and Randy Verbrugge for their helpful criticisms and guidance. They also thank conference participants at the 35th International Symposium on Forecasting in Riverside, California. SECTION A1 Name BVAR1 Name BVAR2 Name A Description of the BVAR models we employ Literature Referenced Beauchemin and Zaman (2011) Literature Referenced Banbura, Giannone, and Reichlin (2010) Literature Referenced Banbura, Giannone, and Reichlin (2010) BVAR3 Christiano, Eichenbaum, and Evans (1999) Variables (transformations) Hyperparameters Frequency Real GDP (log-level) Unemployment rate (level) Consumer Price Index (log-level) Consumer Price Index ex food and energy (loglevel) Effective federal funds rate (level) Nonfarm business compensation (log-level) Nonfarm business productivity (log-level) Real personal consumption expenditures (log-level) Real personal disposable income (log-level) Payroll employment: total nonfarm (log-level) KR-CRB spot commodity price index: all commodities (log-level) 10-year Treasury note yield at constant maturity (level) Moody’s seasoned Aaa corporate bond yield (level) S&P 500 composite stock price index (log-level) S&P 500 composite dividend yield (level) Trade weighted exchange value of the US$ vs. major currencies (log-level) λ = max(ML) µ = max(ML) Quarterly Variables (transformations) Hyperparameters Frequency Payroll employment: total nonfarm (log-level) Consumer Price Index (log-level) Consumer Price Index ex food and energy (loglevel) Effective federal funds rate (level) λ=∞ µ=∞ Monthly Variables (transformations) Hyperparameters Frequency Payroll employment: total nonfarm (log-level) Consumer Price Index (log-level) Consumer Price Index ex food and energy (loglevel) Effective federal funds rate (level) Index of sensitive materials prices (log-level) Money stock: M2 (log-level) Depository Institutions Reserves: Total (log-level) Depository Institutions Reserves: Nonborrowed (log-level) λ = 0.262 µ = 10*0.262 Monthly Name Literature Referenced Banbura, Giannone, and Reichlin (2010) BVAR4 Name BVAR5 Christiano, Eichenbaum, and Evans (1999) Literature Referenced Carriero, Clark, and Marcellino (2015) Variables (transformations) Hyperparameters Frequency Payroll employment: total nonfarm (log-level) Consumer Price Index (log-level) Consumer Price Index ex food and energy (loglevel) Effective federal funds rate (level) Unemployment rate (level) Personal income less transfer payments (log-level) Manufacturing capacity utilization (level) Industrial production (log-level) Housing starts (log-level) Producer Price Index: finished goods (log-level) Average hourly earnings (log-level) M1 (log-level) S&P 500 composite stock price index (log-level) Index of sensitive materials prices (log-level) Money stock: M2 (log-level) Depository Institutions Reserves: Total (log-level) Depository Institutions Reserves: Nonborrowed (log-level) 10-year Treasury note yield at constant maturity (level) Trade weighted exchange value of the US$ vs. major currencies (log-level) λ = 0.108 µ = 10*0.108 Monthly Variables (transformations) Hyperparameters Frequency Unemployment rate (level) Consumer Price Index (log-change, annualized rate) Consumer Price Index ex food and energy (logchange, annualized rate) Payroll employment: total nonfarm (log change, annualized rate) Weekly hours worked (level) Initial claims for unemployment insurance (level) Nominal retail sales (log change, annualized rate) UM Index of Consumer Sentiment (level) Single-family housing starts (log change) Industrial production (log change, annualized rate) Manufacturing capacity utilization (level) ISM PMI: Index of supplier delivery times (level) ISM PMI: Index of new orders (level) West Texas Intermediate spot price (log change) Effective federal funds rate (level) S&P 500 (log change) 10-year Treasury note yield at constant maturity (level) Trade weighted exchange value of the US$ vs. major currencies (log-level) λ = 0.2 µ=1 τ=1 Monthly Name BVAR6 Name BVAR7 Literature Referenced Carriero, Clark, and Marcellino (2015) Literature Referenced Beauchemin and Zaman (2011) Variables (transformations) Hyperparameters Frequency Unemployment rate (level) PCE Price Index (log-change, annualized rate) PCE Price Index ex food and energy (logchange, annualized rate) Payroll employment: total nonfarm (log change, annualized rate) Weekly hours worked (level) Initial claims for unemployment insurance (level) Nominal retail sales (log change, annualized rate) UM Index of Consumer Sentiment (level) Single-family housing starts (log change) Industrial production (log change, annualized rate) Manufacturing capacity utilization (level) ISM PMI: Index of supplier delivery times (level) ISM PMI: Index of new orders (level) West Texas Intermediate spot price (log change) Effective federal funds rate (level) S&P 500 (log change) 10-year Treasury note yield at constant maturity (level) Trade weighted exchange value of the US$ vs. major currencies (log-level) λ = 0.2 µ=1 τ=1 Monthly Variables (transformations) Hyperparameters Frequency Real GDP (log-level) Unemployment rate (level) PCE Price Index (log-level) PCE Price Index ex food and energy (log-level) Effective federal funds rate (level) Nonfarm business compensation (log-level) Nonfarm business productivity (log-level) Real personal consumption expenditures (log-level) Real personal disposable income (log-level) Payroll employment: total nonfarm (log-level) KR-CRB spot commodity price index: all commodities (log-level) 10-year Treasury note yield at constant maturity (level) Moody’s seasoned Aaa corporate bond yield (level) S&P 500 composite stock price index (log-level) S&P 500 composite dividend yield (level) Trade weighted exchange value of the US$ vs. major currencies (log-level) λ = max(ML) µ = max(ML) Quarterly Table A2: Forecast Comparison of BVAR in Beauchemin and Zaman (2011) --- Pre-Crisis Relative Mean Squared Error (RMSE) --- relative to BVAR1 A2a: BVAR1 vs. BVAR1 with median CPI inclusion Real GDP growth h=1Q 1.002 h=2Q 0.999 h=3Q 1.011 h=4Q 1.007 h=5Q 1.002 h=6Q 1.011 h=7Q 1.011 h=8Q 1.005 Core CPI inflation 0.946 0.946 0.931 0.938 0.935 0.933 0.945 0.953 Headline CPI inflation 0.968 0.968 0.967 0.941 0.940 0.950 0.949* 0.950** Unemployment Rate 1.007 1.006 1.012 1.016 1.017 1.018 1.018 1.019 Fed Funds Rate 0.999 0.985 0.975 0.967 0.962 0.960* 0.956* 0.958 A2b: BVAR1 vs. BVAR1 with median CPI replacing the core CPI Real GDP growth h=1Q 1.023 h=2Q 1.011 h=3Q 1.006 h=4Q 1.005 h=5Q 0.996 h=6Q 1.007 h=7Q 1.006 h=8Q 0.996 Core CPI inflation Headline CPI inflation ----0.966 ----0.960 ----0.964 ----0.938 ----0.938 ----0.954 ----0.959 ----0.966 Unemployment Rate 1.018 1.011 1.030 1.027 1.022 1.018 1.014 1.011 Fed Funds Rate 0.986 0.978 0.974 0.966 0.958 0.955 0.947* 0.946* Notes for the table: The table A2a lists the mean squared forecast error (MSFE) of the modified Bayesian VAR with Median CPI added to it relative to the mean squared forecast error of the modified BVAR1. The table 1b lists the mean squared forecast error (MSFE) of the modified BVAR1 in which core CPI is replaced with Median CPI relative to the mean squared forecast error of the modified BVAR1. The reported RMSFEs are for the real GDP growth (quarterly at annual rate), core CPI inflation (quarterly at annual rate), headline CPI inflation (quarterly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…8 step head forecasts (i.e. 2 years out) for the evaluation period 1987Q1 – 2007Q3. Numbers in the bold indicate that the MSFE of the modified BVAR1 with Median CPI is less than the MSFE from the modified BVAR1. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A3: Forecast Comparison of BVARs in Banbura et al. 2010 --- Pre-Crisis Relative Mean Squared Error --- relative to BVARs in Banbura at al 2010 SMALL BVAR (BVAR2) in Banbura et al. 2010 Payroll growth A3a: BVAR2 vs. BVAR2 with median CPI inclusion h=1M h=6M h=9M h=12M h=15M h=18M 1.036 0.968 0.984 0.966 0.973 1.065 h=21M 1.126 h=24M 1.187 Headline CPI inflation 1.008 1.048 0.969 0.943 0.967 0.901 0.881 0.878 Fed Funds Rate 0.961 1.043 1.068 1.074 0.997 0.884 0.819 0.793 A3b: BVAR2 with Core CPI vs. BVAR2 with median CPI h=1M h=6M h=9M h=12M h=15M h=18M 1.014 1.072 1.066 0.943 0.854 0.808 h=21M 0.805 h=24M 0.840 Payroll growth Headline CPI inflation 0.958 0.929 0.924 0.920 0.940 0.925 0.962 0.982 Fed Funds Rate 0.922 0.836 0.898 1.039 1.049 1.018 0.974 0.947 A3c: BVAR3 vs. BVAR3 with median CPI inclusion h=1M h=6M h=9M h=12M h=15M h=18M 0.980 1.000 1.024 1.015 1.023 1.039 h=21M 1.061 h=24M 1.096 CEE BVAR (BVAR3) in Banbura et al. 2010 Payroll growth Headline CPI inflation 1.026 1.026 1.016 0.962 0.995 0.978 0.960 0.967 Fed Funds Rate 0.940 0.994 1.017 0.990 0.972 0.954 0.922 0.901 A3d: BVAR3 with Core CPI vs. BVAR3 with median CPI h=1M h=6M h=9M h=12M h=15M h=18M 1.031 1.060 1.048 1.033 1.026 1.035 h=21M 1.072 h=24M 1.093 Payroll growth Headline CPI inflation 1.000 1.011 1.014 0.969 1.007 0.998 0.986 0.993 Fed Funds Rate 0.977 1.037 1.053 1.058 1.042 1.022 0.985 0.957 Medium BVAR (BVAR4) in Banbura et al. 2010 Payroll growth A3e: BVAR4 vs. BVAR4 with median CPI inclusion h=1M h=6M h=9M h=12M h=15M h=18M 0.990 1.001 1.006 1.006 1.005 1.003 h=21M 1.008 h=24M 1.018 Headline CPI inflation 0.997 0.990 0.996 0.960** 0.982 0.979* 0.973** 0.982 UR 0.999 0.986 0.995 1.004 1.010 1.013 1.014 1.015 Fed Funds Rate 0.960 0.952* 0.970 0.968 0.967 0.962 0.947 0.933* A3f: BVAR4 with Core CPI vs. BVAR4 with median CPI h=1M h=6M h=9M h=12M h=15M h=18M 1.008 1.013 1.004 1.003 1.008 1.014 h=21M 1.042 h=24M 1.045 Payroll growth Headline CPI inflation 0.996 0.991 0.998 0.960** 0.981 0.982 0.973* 0.984 UR 1.005 1.030 1.054 1.052 1.046 1.039 1.031 1.027 Fed Funds Rate 1.006 0.994 0.988 0.981 0.975 0.969 0.951 0.933 Notes for the table: The tables A3a-A3f lists the mean squared forecast error (MSFE) of the BVAR with Median CPI added to it relative to the mean squared forecast error of the BVAR without it. The reported RMSFEs are for the payroll growth (monthly at annual rate), core CPI inflation (monthly at annual rate), headline CPI inflation (monthly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…24 step head forecasts (i.e. 2 years out) for the evaluation period 1987M1 – 2007M09. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A4: Forecast Comparison of Benchmark BVAR in Carriero et al (2011) --- Pre-Crisis Relative Mean Squared Error --- relative to modified BVAR5 A4a: BVAR5 vs. BVAR5 with median CPI inclusion h=1M 0.995 h=6M 0.978 h=9M 1.010 h=12M 1.016 h=15M 1.017 h=18M 1.012 h=21M 1.008 h=24M 1.022 Core CPI 0.912*** 0.961 0.938 0.948 0.939 0.939 0.933 0.918 Headline CPI 0.978 0.982 1.006 0.947 0.981 0.989 0.988 0.976 UR 1.004 0.981 0.988 0.987 0.984 0.992 0.997 0.998 Fed Funds Rate 1.026 0.935 0.940 0.924 0.908 0.898 0.892 0.878 Payroll growth A4b: BVAR5 vs. BVAR5 with median CPI replacing core CPI h=1M 1.008 h=6M 1.023 h=9M 1.017 h=12M 0.999 h=15M 0.991 h=18M 1.001 h=21M 1.009 h=24M 1.015 Core CPI Headline CPI ----0.975 ----0.969 ----0.988 ----0.935* ----0.975 ----0.967 ----0.978 ----0.961* UR 1.005 0.992 1.048 1.034 1.011 1.002 1.004 1.011 Fed Funds Rate 1.068 0.946 0.967 0.957 0.927 0.918 0.912 0.900 Payroll growth Notes for the table: The tables A4a-A4b lists the mean squared forecast error (MSFE) of the BVAR with Median CPI added to it relative to the mean squared forecast error of the BVAR without it. The reported RMSFEs are for the payroll growth (monthly at annual rate), core CPI inflation (monthly at annual rate), headline CPI inflation (monthly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…24 step head forecasts (i.e. 2 years out) for the evaluation period 1987M1 – 2007M09. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A5: Exercise using the median CPI to forecast PCE-based inflation using Carriero et al (2011) Monthly BVAR--- Pre-Crisis Relative Mean Squared Error --- relative to BVAR6 A5a: BVAR6 vs. BVAR6 with median CPI inclusion Payroll growth h=1M 1.003 h=6M 1.017 h=9M 1.049 h=12M 1.015 h=15M 0.987 h=18M 0.992 h=21M 1.010 h=24M 1.032 Core PCE inflation 0.965 0.972 0.959 0.968 0.931* 0.907** 0.894** 0.872** Headline PCE inflation 0.961 0.955 0.972 0.949 0.943 0.923** 0.934* 0.911** UR 1.005 0.988 1.023 1.026 1.003 0.987 0.982 0.982 Fed Funds Rate 1.010 0.918** 0.970 0.960 0.927 0.904 0.885 0.862* A5b: BVAR6 vs. BVAR6 with median CPI replacing core PCE Payroll growth h=1M 0.987 h=6M 1.003 h=9M 1.043 h=12M 1.005 h=15M 0.984 h=18M 1.006 h=21M 1.019 h=24M 1.030 Core PCE inflation Headline PCE inflation ----0.982 ----0.951 ----0.967 ----0.943 ----0.939 ----0.917** ----0.932* ----0.916** UR 1.017 0.961 1.001 1.000 0.968 0.955** 0.964 0.979 Fed Funds Rate 1.007 0.902* 0.953 0.933 0.889 0.867 0.851* 0.832* Notes for the table: The tables A5a-A5b lists the mean squared forecast error (MSFE) of the BVAR with Median CPI added to it relative to the mean squared forecast error of the BVAR without it. The reported RMSFEs are for the payroll growth (monthly at annual rate), core PCE inflation (monthly at annual rate), headline PCE inflation (monthly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…24 step head forecasts (i.e. 2 years out) for the evaluation period 1987M1 – 2007M09. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A6: Exercise using the median CPI to forecast PCE-based inflation using Beauchemin and Zaman (2011) Quarterly BVAR--- Pre-Crisis Relative Mean Squared Error (RMSE) --- relative to BVAR7 A6a: BVAR7 vs. BVAR7 with median CPI inclusion Real GDP growth h=1Q 0.997 h=2Q 1.000 h=3Q 1.018 h=4Q 1.009 h=5Q 1.004 h=6Q 1.010 h=7Q 1.011 h=8Q 1.008 Core PCE inflation 0.973 0.960 0.962 0.953 0.946* 0.942* 0.945* 0.953 Headline PCE inflation 0.996 0.982 0.978 0.957 0.957 0.960* 0.957* 0.961 Unemployment Rate 1.001 0.999 1.004 1.009 1.008 1.008 1.013 1.019 Fed Funds Rate 1.007 0.994 0.996 0.995 0.987 0.980 0.972* 0.970* A6b: BVAR7 vs. BVAR7 with median CPI replace core PCE Real GDP growth h=1Q 0.992 h=2Q 1.000 h=3Q 1.016 h=4Q 1.010 h=5Q 1.003 h=6Q 1.004 h=7Q 0.999 h=8Q 0.994 Core PCE inflation Headline PCE inflation ----0.998 ----0.987 ----0.983 ----0.961 ----0.965 ----0.974 ----0.972 ----0.981 Unemployment Rate 0.983 0.977 0.987 0.993 0.995 0.997 1.003 1.009 Fed Funds Rate 0.998 0.974 0.971 0.966 0.960* 0.959* 0.956* 0.958 Notes for the table: The tables A6a-A6b lists the mean squared forecast error (MSFE) of the BVAR with Median CPI added to it relative to the mean squared forecast error of the BVAR without it. The reported RMSFEs are for the real GDP growth (quarterly at annual rate), core PCE inflation (quarterly at annual rate), headline PCE inflation (quarterly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…8 step head forecasts (i.e. 2 years out) for the evaluation period 1987Q1 – 2007Q3. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A7: Exercise using the forecasted values of the median CPI as the forecast for core CPI with Beauchemin and Zaman (2011) Quarterly BVAR--- Pre-Crisis Relative Mean Squared Error (RMSE) --- relative to BVAR1 A7a: BVAR1 vs. BVAR1 in which we use Median CPI to predict Core CPI Real GDP growth h=1Q 1.023 h=2Q 1.011 h=3Q 1.006 h=4Q 1.006 h=5Q 0.996 h=6Q 1.007 h=7Q 1.006 h=8Q 0.996 Core CPI inflation 1.114 1.095 0.996 0.937 0.915 0.896 0.903 0.913 Headline CPI inflation 0.966 0.960 0.964 0.938 0.938 0.954 0.959 0.966 Unemployment Rate 1.018 1.011 1.030 1.027 1.022 1.018 1.014 1.011 Fed Funds Rate 0.986 0.978 0.974 0.966 0.958 0.955 0.947* 0.946* Notes for the table: The table A7a lists the mean squared forecast error (MSFE) of the BVAR (which is estimated using Median CPI and uses the forecasts of the Median CPI to predict core CPI) relative to the mean squared forecast error of the BVAR without Median CPI. The reported RMSFEs are for the real GDP growth (quarterly at annual rate), core CPI inflation (quarterly at annual rate), headline CPI inflation (quarterly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…8 step head forecasts (i.e. 2 years out) for the evaluation period 1987Q1 – 2007Q3. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test) Table A8: Exercise using the forecasted values of the median CPI as the forecast for core CPI with Carriero et al (2011) Monthly BVAR--- Pre-Crisis Relative Mean Squared Error --- relative to BVAR5 A8a: BVAR5 vs. BVAR5 in which Median CPI is used to predict core CPI Payroll growth h=1M 0.973 h=6M 1.020 h=9M 1.015 h=12M 0.996 h=15M 0.988 h=18M 0.997 h=21M 1.006 h=24M 1.013 Core CPI 0.907 0.954 0.904 0.984 0.996 0.954 0.900 0.848 Headline CPI 0.983 0.969 0.994 0.937* 0.969 0.964 0.976 0.965 UR 1.001 0.989 1.046 1.033 1.012 1.003 1.005 1.012 Fed Funds Rate 1.052 0.945 0.966 0.955 0.927 0.920 0.915 0.903 Notes for the table: The table A8a lists the mean squared forecast error (MSFE) of the BVAR (which is estimated using Median CPI and uses the forecasts of the Median CPI to predict core CPI) relative to the mean squared forecast error of the BVAR without Median CPI. The reported RMSFEs are for the payroll growth (monthly at annual rate), core CPI inflation (monthly at annual rate), headline CPI inflation (monthly at annual rate), the unemployment rate, and the federal funds rate for h=1,2,…24 step head forecasts (i.e. 2 years out) for the evaluation period 1987M1 – 2007M09. Numbers in the bold indicate that the MSFE of the BVAR with Median CPI is less than the MSFE from the BVAR without Median CPI. *denotes significance at 5% level **denotes significance at 10% level (based on modified Diebold-Mariano test)