intro=BULLET_SLIDE("Can a Dual Mandate Support Price Stability?","The paper's headline result comes from a calibrated nonlinear model.","RainbowStats does not yet solve nonlinear DSGE models.","We test the mechanism indirectly with a VAR and COVID-break regressions.")

design=BULLET_SLIDE("Empirical Design","The VAR uses quarterly data from 1995Q1 through 2019Q4.","Core PCE inflation and productivity growth are annualized log changes.","The system contains inflation, unemployment, the federal funds rate, and productivity growth.","Four quarterly lags allow the entire system to interact dynamically.")

u=SET_NAME(SERIES_BETWEEN(TO_QUARTERLY(UNRATE),19950101,20191231),"Unemployment")
pi=SET_NAME(SERIES_BETWEEN(400*LOGDIFF(TO_QUARTERLY(PCEPILFE)),19950101,20191231),"Core PCE Inflation")
rate=SET_NAME(SERIES_BETWEEN(TO_QUARTERLY(FEDFUNDS),19950101,20191231),"Federal Funds Rate")
prod=SET_NAME(SERIES_BETWEEN(400*LOGDIFF(OPHNFB),19950101,20191231),"Labor Productivity Growth")
system=SAME_DATE_RANGE(LIST(pi,u,rate,prod))

data_chart=SET_TITLE(LINE_CHART(SAME_DATE_RANGE(LIST(pi,u,rate))),"Inflation, Unemployment, and the Policy Rate")
corr_slide=BULLET_SLIDE("Observed Correlations","Unemployment and core inflation: -0.142.","Unemployment and the federal funds rate: -0.602.","The federal funds rate and core inflation: 0.190.","These raw correlations are not the paper's HP-filtered moments.")

vm=VAR_MODEL(system,4)
u_to_rate=SET_TITLE(EXTRACT(vm,shockdiff:2:1:1),"Policy-Rate Response to a 1-Point Unemployment Shock")
u_to_pi=SET_TITLE(EXTRACT(vm,shockdiff:0:1:1),"Inflation Response to a 1-Point Unemployment Shock")
rate_to_pi=SET_TITLE(EXTRACT(vm,shockdiff:0:2:1),"Inflation Response to a 1-Point Policy-Rate Shock")

strict_eq=REGRESSION(SAME_DATE_RANGE(LIST(pi,LAG(pi,1),LAG(pi,2),LAG(pi,3),LAG(pi,4),LAG(rate,1),LAG(rate,2),LAG(rate,3),LAG(rate,4),LAG(prod,1),LAG(prod,2),LAG(prod,3),LAG(prod,4))))

dual_eq=REGRESSION(SAME_DATE_RANGE(LIST(pi,LAG(pi,1),LAG(pi,2),LAG(pi,3),LAG(pi,4),LAG(u,1),LAG(u,2),LAG(u,3),LAG(u,4),LAG(rate,1),LAG(rate,2),LAG(rate,3),LAG(rate,4),LAG(prod,1),LAG(prod,2),LAG(prod,3),LAG(prod,4))))

strict_resid=SET_NAME(EXTRACT(strict_eq,residuals),"Inflation Error: No Employment")
dual_resid=SET_NAME(EXTRACT(dual_eq,residuals),"Inflation Error: With Employment")
resid_chart=SET_TITLE(LINE_CHART(SAME_DATE_RANGE(LIST(strict_resid,dual_resid))),"Inflation Errors With and Without Employment")

comparison=BULLET_SLIDE("What Changes When Employment Is Added?","Inflation residual volatility falls from 0.669 to 0.661.","That is a reduction of about 1.1 percent, not the paper's simulated 26 percent.","The larger equation adds four regressors, so the in-sample improvement should be interpreted cautiously.")

u_full=SET_NAME(SERIES_BETWEEN(TO_QUARTERLY(UNRATE),19950101,20251231),"Unemployment")
pi_full=SET_NAME(SERIES_BETWEEN(400*LOGDIFF(TO_QUARTERLY(PCEPILFE)),19950101,20251231),"Core PCE Inflation")
rate_full=SET_NAME(SERIES_BETWEEN(TO_QUARTERLY(FEDFUNDS),19950101,20251231),"Federal Funds Rate")
prod_full=SET_NAME(SERIES_BETWEEN(400*LOGDIFF(OPHNFB),19950101,20251231),"Labor Productivity Growth")

covid_u=REGRESSION_COVID(SAME_DATE_RANGE(LIST(pi_full,u_full)))
covid_prod=REGRESSION_COVID(SAME_DATE_RANGE(LIST(pi_full,prod_full)))
covid_rate=REGRESSION_COVID(SAME_DATE_RANGE(LIST(pi_full,rate_full)))

covid_results=BULLET_SLIDE("COVID Changes the Interpretation","Controlling for unemployment, the COVID shift is +1.678 inflation points with t=7.903.","Controlling for productivity, the shift is +1.784 points with t=8.676.","Controlling for the funds rate, the shift is +1.794 points with t=8.398.","The contemporaneous funds-rate coefficient is statistically weak.")

results=BULLET_SLIDE("What the VAR Shows","Higher unemployment produces a persistent decline in the modeled policy rate.","Inflation responds unevenly, reflecting long and interacting lags.","A rate shock initially raises inflation: the familiar VAR price puzzle.","These are dynamic associations, not structurally identified causal effects.")

conclusion=BULLET_SLIDE("Conclusion","The data support the existence of a dual-mandate feedback channel.","They do not reproduce the paper's quantitative stabilization result.","COVID reveals a large regime shift not absorbed by ordinary policy or labor-market controls.","The mechanism survives, but the magnitude does not.")

appendix_intro=BULLET_SLIDE("Appendix: Full VAR Model","The following table contains every coefficient from all four VAR equations.","Actual-versus-fitted charts follow the table.","The appendix is intentionally complete for reproducibility.")

SLIDESHOW(intro,design,data_chart,corr_slide,u_to_rate,u_to_pi,rate_to_pi,resid_chart,comparison,covid_results,covid_u,results,conclusion,appendix_intro,vm)