Bayesian Analysis of a Vector Autoregressive Model with Stochastic Volatility and Time-Varying Parameters (BVAR-SV-TVP)

This paper mainly wants to study

  • How COVID-19 changes the central bank’s monetary policy?

  • Why is the current inflation rate in the United States so high? What are the main reasons? What is the correlation with monetary policy?

  • How could monetary policy to tame the current high inflation in US?

Abstract

After the outbreak of the COVID-19 epidemic,:

A: Decreasing unemployment rate policy:

  • FED unexpected increase the money supply is effective in reducing the unemployment rate, but it is useless by using interest rate policy.

B: Tame down inflation policy :

  • FED trying to reduce the inflation rate with an unexpected drop in the money supply has only one period and minor effect, and it is invalid by using interest rate policy to tame down inflation.

C: Policy reaction:

  • . After the outbreak of the epidemic, the central bank will not make any adjustments to its monetary policy in the face of an unexpected rise in inflation.

  • Before the outbreak, FED would not have any response to the unexpected rise in unemployment, but after the outbreak, the central bank will cut interest rates by to reduce unemployment.

  • Before the outbreak of the epidemic, the central bank would respond to the unexpected rise of the unemployment with a increasing monetary policy.

  • After the outbreak, the central bank will not respond with a monetary policy to reduce the unemployment rate.

The Background

Inflation rate after COVID-19

Inflation rate 疫情爆發以後通貨膨脹率(INF)持續攀升,目前已創下歷史新高(7.8%, 01/2022).

After the outbreak of the COVID-19 epidemic, the inflation rate (INF) has continued to climb and has now reached a record high (7.8%, 01/2022).

Fed Fund Rate 疫情爆發以後FED Fund Rate 利率幾乎維持零利率.

After the outbreak of the epidemic, the Fed Fund Rate interest rate remained almost zero.

Unemployment Rate After COVID-19 2020年4月失業率達到最高點(14.8)然後緩慢的下降.

The unemployment rate peaked in April 2020 (14.8%) and then slowly declined.

Unemployment Rate

疫情爆發以後失業率迅速上升,然後又迅速猛然下降.

Unemployment rose rapidly after the outbreak, and then fell sharply again.

M1 Money Supply

m1b: 疫情爆發以後貨幣供給迅速跳躍式且持續上升.

m1b: After the outbreak of the epidemic, the M1b money supply jumped rapidly and continued to rise.

Methododology: Bayesian TVP-SV-VAR

Bayesian VAR model with TVP(time-varying parameters) and SV(stochatic volatility)

data: UNR(unemployment rate),INF(inflation rate ),FFR(FED Fund Rate),M1G(m1b growth rate) , from 2010/02/01 to 2021/12/01, monthly data form FRED.

knitr::opts_chunk$set(echo = TRUE)

load required packages

library(bvarsv) # for time-varying parameters stochastic volatility VAR models
library(vars) # for tranditional VAR analysis
library(tseries)  # for unit root tests
library(forecast)
library(ggplot2)

plot the series

dusmacro<-readRDS('dusmacro.rds')
autoplot(as.ts(dusmacro),ylab='' ,main='Time series of differenced us macro economic data')

將資料設定為時間序列資料(time series data)

Set dusmacro as time series data.

dusmacro=ts(dusmacro,start=c(2010,02,01),end=c(2021,12,01),frequency=12)

將資料設定成疫情爆發前(2010-2019)與疫情爆發後(2016-2021)二段資料. 因為模型係數推估(inference)需要40筆資料作為訓練推估資料, 因此我們將疫情爆發後日期從2020年1月1號,往前推估是40個月, 因此從2016年1月開始作為疫情爆發以後開始的資料.

The data are set as two sections of data before the outbreak (2010-2019) and after the outbreak (2016-2021). Because the model coefficient estimation (inference) requires 40 months of data as training estimation data, we estimate the post-epidemic date from January 1, 2020 to 40 months forward, so we start from January 2016 as Data starting after the outbreak.

before covid

dY1=ts(dmacro,end=c(2019,12,01))

after covid

dY2=window(dmacro,start=c(2016,01))

找出最適遞延期數(optimal VAR lags)

Find optimal VAR lags

select number of lags based on information criteria on VAR model on levels of your time series

VARselect(dY1[,-1])$selection

#AIC(n) HQ(n) SC(n) FPE(n) # 4 2 2 4

VARselect(dY2[,-1])$selection

#AIC(n) HQ(n) SC(n) FPE(n) # 4 2 2 4

Based on Parsimonious models principle which simple models with great explanatory predictive power, we choose TVP-SV-VAR(2) as our model.

run bvar.sv.tvp

set-up the burn-in draws: Number of MCMC draws used to initialize the sampler. These draws do not enter the computation of posterior moments, forecasts etc.

nburn <- 10000

MCMC with 50000 draws excluding burn-in(10000) (i.e., total with 60000 draws)

nrep<- 50000

set up quantile computing.

quan<- function(z) c(mean(z), quantile(z, c(0.16, 0.84)))[c(2,1,3)]

Estimate Bayesian TVP-SV-VAR model: Before COVID-19

dy1.bvar.fit <- bvar.sv.tvp(dY1, p = 2, nburn = 10000, nrep = 50000)

Estimate Bayesian TVP-SV-VAR model: After COVID-19

start=2010-end:2021/12/01

dy2.bvar.fit <- bvar.sv.tvp(dY2, p = 2,nburn = nburn, nrep = nrep)

IRF analysis: Impulse-Response Analysis

Unexpected shocks of m1b to unr

dy1.bvar.fit<-readRDS('dy1.bvar.fit.rds')
dy2.bvar.fit <-readRDS('dy2.bvar.fit.rds')
########################### irf41: unexpected shocks of m1b to unr ########################### 
par(mfrow=c(1,2))

# before covid
#unexpected shocks of m1b to unr
befor_irf41<-impulse.responses(dy1.bvar.fit, impulse.variable = 4, 
                               response.variable = 1,nhor = 6, draw.plot =T)

#after covid
#unexpected shocks of m1b to unr
after_irf41<-impulse.responses(dy2.bvar.fit, impulse.variable = 4, 
                          response.variable = 1,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
irf41b<-apply(befor_irf41$irf,2,mean)
irf41a<-apply(after_irf41$irf,2,mean)
autoplot(as.ts(cbind(after=irf41a,before=irf41b)),ylab='unr',
         main='Fig.1: Impulse of m1b  on unr')

上圖灰色區域代表 25及 75 percent quantiles ;黑色區域代表 5及 95 percent quantiles. 黑色實線為平均數估計值. 以下我們以25及 75 percent quantiles信心區間(confidence interval) 來判斷政策顯著性及有效性.

疫情爆發前,貨幣供給非預期性的增加,失業率會下降小但不明顯.

疫情爆發後,貨幣供給非預期性的增加,失業率會明顯下降,而且持續時間很長.

因此疫情爆發後,貨幣供給的非預期性增加衝擊,對失業率的降低有效,而且效果期間比較長.

The gray area above represents 25 and 75 percent quantiles; the black area represents 5 and 95 percent quantiles. The solid black line is the mean estimate. Below we use the 25 and 75 percent quantiles as confidence intervals to judge the significance and effectiveness of FED policy.

Before the outbreak, an unexpectedly increase the money supply, the unemployment rate will decrease slightly and insignificantly.

After the outbreak of the epidemic, the unanticipated increase in money supply will significantly reduce the unemployment rate, and it will last for a long time.

Therefore, after the outbreak of the epidemic, the shock of the unexpected increase in the money supply is effective in reducing the unemployment rate, and the effect period is relatively long.

Unexpected shocks of m1b to inf

########################### irf42: unexpected shocks of m1b to inf ########################### 
par(mfrow=c(1,2))

#before covid-19
#unexpected shocks of m1b to inf
befor_irf42<-impulse.responses(dy1.bvar.fit, impulse.variable = 4, 
                           response.variable = 2,nhor = 6, draw.plot =T)

#after covid-19
#unexpected shocks of m1b to inf
after_irf42<-impulse.responses(dy2.bvar.fit, impulse.variable = 4, 
                           response.variable = 2,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf42 impulse of m1b on inf
irf42b<-apply(befor_irf42$irf,2,mean)
irf42a<-apply(after_irf42$irf,2,mean)
autoplot(as.ts(cbind(after=irf42a,before=irf42b)),ylab='inf',
         main='Fig.2: Impulse of m1b  on inf')

貨幣供給的非預期性增加會使得通貨膨脹上升.

由圖2可知,疫情爆發前,貨幣供給非預期性的增加,對於通貨膨脹只有在第二期有顯著效果 (上升0.036%),其他期間都沒有效果; 疫情爆發後,在第一期會有顯著效果(上升0.036%),其他期間都沒有效果.

因此想要以貨幣供給的非預期性下降,以減少通貨膨脹率效果短而且小.

An unexpected increases in the money supply can cause inflation to rise. As can be seen from Figure 2, before the outbreak of the epidemic, the unexpected increase in money supply had a significant effect on inflation only in the second period. (up 0.036%), no effect during other periods.

After the outbreak, there will be a significant effect in the first period (increasing 0.036%), and there will be no effect in other periods.

So trying to reduce the inflation rate with an unexpected drop in the money supply has only one period and minor effect.

Unexpected shocks of ffr to unr

########################### irf31: unexpected shocks of ffr to unr ########################### 
par(mfrow=c(1,2))

# before covid
#unexpected shocks of ffr to unr
befor_irf31<-impulse.responses(dy1.bvar.fit, impulse.variable = 3, 
                               response.variable = 1,nhor = 6, draw.plot =T)
#after covid
#unexpected shocks of m1g to unr
after_irf31<-impulse.responses(dy2.bvar.fit, impulse.variable = 3, 
                               response.variable = 1,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on unr
irf31b<-apply(befor_irf31$irf,2,mean)
irf31a<-apply(after_irf31$irf,2,mean)
autoplot(as.ts(cbind(after=irf31a,before=irf31b)),ylab='unr',
         main='Fig.3: Impulse of ffr  on unr')

圖3: 疫情爆發前,FFR 利率非預期性的上升對失業率在第一期有顯著的下降0.2 %,其他期間沒有任何顯著性影響. 疫情爆發後,FFR 利率非預期性的上升對失業率沒有任何顯著性影響.

雖然疫情爆發後, 利率非預期性的上升,對失業率的影響比疫情爆發前來得嚴重(數值比較大),但是在統計上卻是都不顯著的.

Fig 3: After the outbreak, the unexpected rise in FFR rates did not have any significant impact on the unemployment rate.

Before the outbreak, the unexpected rise in FFR interest rate had a significant drop of 0.2% in the unemployment rate in the first period, and had no significant effect in other periods.

Although the unexpected rise in interest rates after the outbreak of the epidemic has a more serious impact on the unemployment rate than before the outbreak (the value is relatively large), however,it is not statistically significant.

Unexpected shocks of ffr to inf

########################### irf31: unexpected shocks of ffr to inf #####################
par(mfrow=c(1,2))

befor_irf32<-impulse.responses(dy1.bvar.fit, impulse.variable = 3, 
                               response.variable = 2,nhor = 6, draw.plot =T)


after_irf32<-impulse.responses(dy2.bvar.fit, impulse.variable = 3, 
                               response.variable = 2,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on inf
irf32b<-apply(befor_irf32$irf,2,mean)
irf32a<-apply(after_irf32$irf,2,mean)
autoplot(as.ts(cbind(after=irf32a,before=irf32b)),ylab='inf',
         main='Fig.4: Impulse of ffr  on inf')

由圖4可知, 疫情爆發前,利率非預期性的增加,通貨膨脹會在第一期顯著性(significant)上升0.02 %左右,其他各期沒有顯著性影響.

疫情爆發後,利率非預期性的增加對通貨膨脹沒有顯著性影響.

因此FED在疫情爆發後,想要透過升息以降低通貨膨脹率是不會有什麼顯著性效果的.

As can be seen from Figure 4, before the outbreak of the epidemic, the unanticipated increase in interest rates caused inflation to rise significantly by about 0.02% in the first period, and there was no significant impact in other periods.

After the outbreak, the unexpected increase in FFR had no significant impact on inflation.

Therefore, after the outbreak of the epidemic, it is invalid by using interest rate policy to tame down inflation.

Unexpected shocks of inf to ffr

########################### irf23: unexpected shocks of inf to ffr #####################
par(mfrow=c(1,2))
befor_irf23<-impulse.responses(dy1.bvar.fit, impulse.variable = 2,
                               response.variable = 3,nhor = 6, draw.plot =T)


after_irf23<-impulse.responses(dy2.bvar.fit, impulse.variable = 2, 
                               response.variable = 3,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on inf
irf23b<-apply(befor_irf23$irf,2,mean)
irf23a<-apply(after_irf23$irf,2,mean)
autoplot(as.ts(cbind(after=irf23a,before=irf23b)),ylab='ffr',
         main='Fig.5: Impulse of inf to ffr')

由圖5可知,疫情爆發前,通貨膨脹非預期的上升,央行會調降利率1個基點(basis points)左右; 疫情爆發後,通貨膨脹非預期性的上升,央行會在第4個月後才調升利率 5個基點(basis points), 然後再在第6個月後再次調升利率15個基點(basis points).

因此疫情爆發後,央行面對通貨膨脹非預期性的上升, 央行的利率政策反應幅度比較大,期間比較久.

As can be seen from Figure 5, before the outbreak of the epidemic, the central bank will reduce the interest rate by about 1 basis point if inflation unexpectedly rises.

After the outbreak of the epidemic, if inflation rose unexpectedly, the central bank would only raise interest rates by 5 basis points after the fourth month, then raised again by 15 basis points after the sixth month.

Therefore, after the outbreak of the epidemic, the central bank faced an unexpected rise in inflation, it’s interest rate policy will response more relatively large and with more longer periods reactions.

Unexpected shocks of inf to m1b

########################### irf24: unexpected shocks of inf to m1b #####################
par(mfrow=c(1,2))
befor_irf24<-impulse.responses(dy1.bvar.fit, impulse.variable = 2,
                               response.variable = 4,nhor = 6, draw.plot =T)


after_irf24<-impulse.responses(dy2.bvar.fit, impulse.variable = 2, 
                               response.variable = 4,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on inf
(irf24b<-apply(befor_irf24$irf,2,mean))
## [1]  0.100732935 -0.164687134 -0.040244158 -0.035782213  0.003198350
## [6]  0.003785345
irf24a<-apply(after_irf24$irf,2,mean)
autoplot(as.ts(cbind(after=irf24a,before=irf24b)),ylab='m1g',
         main='Fig.6: Impulse of inf to m1b')

由圖6 可知,疫情爆發前,通貨膨脹非預期的上升,央行會在第一期增加貨幣供給, 第二期開始會減少貨幣供給,效果會持續到第四期.

疫情爆發後,央行面對通貨膨脹非預期的上升,貨幣供給政策不會做任何的調整反應.

因此疫情爆發後,央行面對通貨膨脹非預期性的上升, 央行會透過利率政策來降低通貨膨脹率, 不會使用貨幣供給政策來降低通貨膨脹率.

As can be seen from Figure 6, before the outbreak of the epidemic, when inflation rose unexpectedly, the central bank would increase the money supply in the first period. The money supply will be reduced at the beginning of the second period, and the effect will last until the fourth period.

After the outbreak of the epidemic, the central bank will not make any adjustments to its monetary policy in the face of an unexpected rise in inflation.

Therefore, after the outbreak of the epidemic, when the central bank faces an unexpected rise in inflation, the central bank will reduce the FFR through interest rate policy, and will not use money supply policy to reduce the inflation rate.

Unexpected shocks of unr to ffr

########################### irf13: unexpected shocks of unr to ffr #####################
par(mfrow=c(1,2))
befor_irf13<-impulse.responses(dy1.bvar.fit, impulse.variable = 1,
                               response.variable =3,nhor = 6, draw.plot =T)


after_irf13<-impulse.responses(dy2.bvar.fit, impulse.variable = 1, 
                               response.variable =3,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on inf
irf13b<-apply(befor_irf13$irf,2,mean)
irf13a<-apply(after_irf13$irf,2,mean)
autoplot(as.ts(cbind(after=irf13a,before=irf13b)),ylab='ffr',
         main='Fig.7: Impulse of  unr to ffr')

由圖7 可知,疫情爆發前,央行面對失業率的非預期上升不會有任何的反應.

疫情爆發後,央行面對失業率的非預期上升,會在第二期調降1%的利率.

疫情爆發前,央行面對失業率的非預期上升不會有任何的反應, 但在疫情爆發後,央行會調降利率, 以減少失業率.

As can be seen from Figure 7, before the outbreak of the epidemic, the central bank would not have any response to the unexpected rise in unemployment.

After the outbreak, the central bank will cut interest rates by 1% in the second phase to tamb an unexpected rise in unemployment.

Before the outbreak, the central bank would not have any response to the unexpected rise in unemployment, but after the outbreak, the central bank will cut interest rates by to reduce unemployment.

Unexpected shocks of unr to m1b

########################### irf14: unexpected shocks of unr to m1b #####################
par(mfrow=c(1,2))
befor_irf14<-impulse.responses(dy1.bvar.fit, impulse.variable = 1,
                               response.variable =4,nhor = 6, draw.plot =T)


after_irf14<-impulse.responses(dy2.bvar.fit, impulse.variable = 1, 
                               response.variable =4,nhor = 6, draw.plot =T)

par(mfrow=c(1,1))
#irf impulse of ffr on inf
(irf14b<-apply(befor_irf14$irf,2,mean))
## [1] -0.0537140104  0.0974395850 -0.0330747031  0.0020010127 -0.0086165296
## [6]  0.0005453911
irf14a<-apply(after_irf14$irf,2,mean)
autoplot(as.ts(cbind(after=irf14a,before=irf14b)),ylab='m1b',
         main=' Fig.8: Impulse of  unr to m1b')

由圖8 可知,疫情爆發前,央行面對失業率的非預期上升, 貨幣供給在第二期會增加,,以減少失業率,但 其他期間不會有任何的政策反應(不顯著).

疫情爆發後,央行面對失業率非預期上升衝擊,不會有任何貨幣供給政策反應(不顯著).

疫情爆發前,央行面對失業率的非預期衝擊, 會以貨幣供給政策反應, 但在疫情爆發後,央行不會以貨幣供給政策反應,以減少失業率.

As can be seen from Figure 8, before the outbreak of the epidemic, while an unexpected rise in unemployment, the FED will increase the money supply in the second period , but there would be no policy response (insignificant) in other periods, to reduce the unemployment rate.

After the outbreak of the epidemic, the central bank will not have any monetary supply policy response to the shock of an unexpected rise in unemployment (insignificant).

Before the outbreak of the epidemic, the central bank would respond to the unexpected shock of the unemployment with a monetary policy, but after the outbreak, the central bank will not respond with a monetary policy to reduce the unemployment rate.

Conclusion

After the outbreak of the COVID-19 epidemic,:

A: Decreasing unemployment rate:

B: Tame down inflation :

C: Policy reaction:

*Before the outbreak of the epidemic, the central bank would respond to the unexpected rise of the unemployment with a increasing monetary policy.

References

Del Negro, M. and Primicerio, G.E. (2015). ‘Time Varying Structural Vector Autoregressions and Monetary Policy: A Corrigendum’, Review of Economic Studies 82, 1342-1345.

Koop, G. and D. Korobilis (2010): ‘Bayesian Multivariate Time Series Methods for Empirical Macroeconomics’, Foundations and Trends in Econometrics 3, 267-358. Accompanying Matlab code available at https://sites.google.com/site/dimitriskorobilis/matlab.

Primiceri, G.E. (2005): ‘Time Varying Structural Vector Autoregressions and Monetary Policy’, Review of Economic Studies 72, 821-852.