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nrate.prg
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'___________________________________________________________________________________________________________________________________________________________________________
' ESTIMATE NUETRAL INTEREST RATE FOR AUSTRALIA
' email: david.stephan@gmail.com ;
' reference: https://www.rba.gov.au/publications/bulletin/2017/sep/pdf/bu-0917-2-the-neutral-interest-rate.pdf
'___________________________________________________________________________________________________________________________________________________________________________
close @all
%path = @runpath
cd %path
logmode l
!import_data = 0 'Set to 1 if you want to load the rawdata from ABS/RBA/FRED. Set to zero if you want it to load the previously imported database
include MUE_Stage1.prg 'Stage 1 Median Unbiased Estimator
include MUE_Stage2.prg 'Stage 2 Median Unbiased Estimator
'-----------------------------------------------------------------------------
if !import_data = 1 then
logmsg "Importing Data"
'Import Rawdata from ABS/RBA
'Create Workfile to populate with ABS/RBA Data
wfcreate(wf=RSTAR, page=RSTAR) q 1959q3 2021q4
'Set Strings to Load Data using R packages readabs and readrba
%rpath = @replace(@runpath,"\","/")
%setwd = "setwd("+"""" + @left(%rpath,@len(%rpath)-1) + """)"
'Open R connection
xopen(r)
'Set wd so can access and store R dataframes
xrun {%setwd}
'Download RBA and ABS Data
xpackage readabs
xpackage readrba
xpackage readr
xpackage reshape2
xpackage lubridate
xpackage tidyverse
xpackage zoo
xpackage data.table
xon
'---------------------------------------------------------------------------------------------------------
'Download Most Recent ABS Data
'---------------------------------------------------------------------------------------------------------
'Import Data from ABS Website
abs_5206 = read_abs(series_id = c("A2304402X", "A2302915V"))
abs_6202 = read_abs(series_id = c("A84423043C", "A84423047L"))
'5206.0 Australian National Accounts: National Income, Expenditure and Product
R_5206 = abs_5206 %>% filter(series_id %in% c("A2304402X")) %>% mutate(date = zoo::as.yearqtr(date)) %>% dplyr::select(date, series_id, value)
R_5206 = distinct(R_5206,date,series_id, .keep_all= TRUE)
R_5206 = dcast(R_5206, date ~ series_id)
'6202.0 Labour Force, Australia - Monthly
R_6202 = abs_6202 %>% filter(series_id %in% c("A84423043C", "A84423047L")) %>% dplyr::select(date, series_id, value)
R_6202 = distinct(R_6202,date,series_id, .keep_all= TRUE)
R_6202 = dcast(R_6202, date ~ series_id)
R_6202 = R_6202 %>% group_by(date=floor_date(date, "quarter")) %>% summarize(A84423043C=mean(A84423043C), A84423047L=mean(A84423047L)) %>% mutate(date = zoo::as.yearqtr(date))
'---------------------------------------------------------------------------------------------------------
'Download Most Recent RBA Data
'---------------------------------------------------------------------------------------------------------
'Trimmed-Mean Inflation (Year-Ended)
rba_g1 = read_rba(series_id = c("GCPIOCPMTMYP"))
R_g1 = rba_g1 %>% filter(series_id %in% c("GCPIOCPMTMYP")) %>% mutate(date = zoo::as.yearqtr(date)) %>% dplyr::select(date, series_id, value)
R_g1 = distinct(R_g1,date,series_id, .keep_all= TRUE)
R_g1 = dcast(R_g1, date ~ series_id)
'Bond-market inflation expectations
rba_g3 = read_rba(series_id = c("GBONYLD"))
R_g3 = rba_g3 %>% filter(series_id %in% c("GBONYLD")) %>% mutate(date = zoo::as.yearqtr(date)) %>% dplyr::select(date, series_id, value)
R_g3 = distinct(R_g3,date,series_id, .keep_all= TRUE)
R_g3 = dcast(R_g3, date ~ series_id)
'Interbank Overnight Cash Rate; monthly average
rba_f1 = read_rba(series_id = c("FIRMMCRI"))
R_f1 = rba_f1 %>% arrange(date) %>% group_by(date=floor_date(date, "quarter")) %>% summarize(FIRMMCRI=mean(value)) %>% mutate(date = zoo::as.yearqtr(date))
'---------------------------------------------------------------------------------------------------------
'Download COVID Stringency Index
'---------------------------------------------------------------------------------------------------------
myfile = "https://raw.githubusercontent.com/OxCGRT/covid-policy-tracker/master/data/OxCGRT_latest.csv"
covid_raw = read_csv(myfile)
covid_raw$Date = ymd(covid_raw$Date)
covid = covid_raw %>% filter(CountryCode == "AUS") %>% rename(date=Date) %>% group_by(date=floor_date(date, "quarter")) %>% summarize(StringencyIndex=mean(StringencyIndex)) %>% mutate(date = zoo::as.yearqtr(date)) %>% select(date, StringencyIndex)
xrun RSTAR_data = list(R_5206, R_6202, R_g1, R_g3, R_f1, covid) %>% Reduce(function(dtf1,dtf2) left_join(dtf1,dtf2,by="date"), .)
xoff
pageselect RSTAR
xget(type=series) RSTAR_data
xclose
'-----------------------------------------------------------------------------
'Data Manipulations
smpl @all
'Rename Variables
'Real GDP
rename A2304402X RGDP
'Employment
rename a84423043c LE
'Labor Force
rename A84423047L LF
'Trimmed Mean
rename GCPIOCPMTMYP DLPTM
'Bond Yield Price Expectations
rename gbonyld pie_bond
'Interbank Overnight Cash Rate
rename FIRMMCRI CASH
'Unemployment Rate
series LUR = 100*(1-LE/LF)
'Qtly Real GDP Log Level and Growth
series lrgdp = log(rgdp)*100
series dlrgdp = d(lrgdp)
'Real Cash Rate (delfated with trimmed mean inflation)
series rcash = cash - dlptm
'Stringency Index of COVID
rename StringencyIndex covid
smpl @first 2019q4
covid = 0
smpl @all
wfsave importeddata
logmsg "Finished Data Import"
else
wfopen importeddata
endif
smpl @all
'-----------------------------------------------------------------------------
'Creat initial series for initialzing state space
'Set Estimation Period
sample ssest 1986q3 2019q4
smpl ssest
lrgdp.hpf(lambda=1600) ypot_ini
lur.hpf(lambda=1600) nairu_ini
rcash.hpf(lambda=1600) nrate_ini
series ygap_ini = lrgdp - ypot_ini
series g_ini = d(ypot_ini)
series z_ini = nrate_ini - 4*g_ini
'-----------------------------------------------------------------------------
'Create Parameters/Starting Values for State Space Model
' Setup coefficient vectors
coef(10) delta
coef(10) alpha
coef(10) beta
coef(10) theta
coef(10) gamma
coef(10) sigma
'-----------------------------------------------------------------------------
' Estimate initial coefficients
'***********************************************
' Output Gap
smpl ssest
equation eq_ygap.ls ygap_ini = alpha(1)*ygap_ini(-1) + alpha(2)*ygap_ini(-2) + _
alpha(3)/2*(rcash(-1) - nrate_ini(-1) + rcash(-2) - nrate_ini(-2))
' Store estimates for later use
!alpha1=alpha(1)
!alpha2=alpha(2)
!alpha3=alpha(3)
!sigma1 = eq_ygap.@se
'***********************************************
'***********************************************
' Okun's Law
smpl ssest
equation eq_okun.ls LUR = nairu_ini + _
beta(1)*(0.4*ygap_ini + 0.3*ygap_ini(-1) + 0.2*ygap_ini(-2) + 0.1*ygap_ini(-3))
' Store estimates for later use
!beta1=beta(1)
!sigma2 = eq_okun.@se
'***********************************************
'***********************************************
' Phillips Curve
smpl ssest
equation eq_ptm.ls dlptm = (1-gamma(1))*pie_bond + _
gamma(1)/3*(dlptm(-1) + dlptm(-2) + dlptm(-3)) + _
gamma(2)*ygap_ini(-1)
' Store estimates for later use
!gamma1=gamma(1)
!gamma2=gamma(2)
!sigma3 = eq_ptm.@se
'***********************************************
'***********************************************
' GDP Trend
smpl ssest
equation eq_ypot.ls ypot_ini =delta(1)+ypot_ini(-1)
' Store estimates for later use
!delta1=delta(1)
!sigma4=eq_ypot.@se
'***********************************************
'***********************************************
' Trend Growth Rate
smpl ssest
equation eq_g.ls d(g_ini) = delta(2)
!delta2=delta(2)
' Store estimates for later use
!sigma5=eq_g.@se
'***********************************************
'***********************************************
' Unexplained Neutral Rate Movements
smpl ssest
equation eq_z.ls d(z_ini) = delta(3)
!delta3=delta(3)
' Store estimates for later use
!sigma6=eq_z.@se
'***********************************************
'***********************************************
' LUR Trend Rate
smpl ssest
equation eq_NRATE.ls d(nairu_ini) = c(1)
!sigma7=eq_NRATE.@se
!theta1 = @mean(g_ini)
'-----------------------------------------------------------------------------------------------------------------------------------------------
'Set constraints on parameters
'Upper bound on a_3 parameter (slope of the IS curve)
' !a_r_constraint_stage1 = NA
' !a_r_constraint_stage2 = -0.0025
' !a_r_constraint_stage3 = -0.0025
'Lower bound on b_2 parameter (slope of the Phillips curve)
' !b_y_constraint_stage1 = NA
' !b_y_constraint_stage2 = 0.025
' !b_y_constraint_stage3 = 0.025
'-----------------------------------------------------------------------------------------------------------------------------------------------
'Stage 1 - No Real Rate term in IS Curve and Constant Trend GDP growth
'-----------------------------------------------------------------------------------------------------------------------------------------------
'State Space System of GDP/ UR Rate and Part Rate
sspace ss_stage1
ss_stage1.append @param alpha(1) !alpha1 alpha(2) !alpha2 alpha(3) !alpha3
ss_stage1.append @param beta(1) !beta1
ss_stage1.append @param gamma(1) !gamma1 gamma(2) !gamma2
ss_stage1.append @param sigma(1) !sigma1 sigma(2) !sigma2 sigma(3) !sigma3 sigma(4) !sigma4 sigma(6) !sigma6 sigma(7) !sigma7
ss_stage1.append @param theta(1) !theta1
ss_stage1.append @signal lrgdp = ygap + ypot
ss_stage1.append @signal LUR = NAIRU + _
beta(1)*(0.4*ygap + 0.3*ygapL1 + 0.2*ygapL2 + 0.1*ygapL3) + _
[ename = e2, var = (sigma(2)^2)]
ss_stage1.append @signal dlptm = (1-gamma(1))*pie_bond + _
gamma(1)/3*(dlptm(-1) + dlptm(-2) + dlptm(-3)) + _
gamma(2)*ygapL1 + _
[ename = e3, var = (sigma(3)^2)]
ss_stage1.append @state ygap = alpha(1)*ygap(-1) + alpha(2)*ygapL1(-1) + _
[ename = e1, var = (sigma(4)^2)]
ss_stage1.append @state ygapL1 = ygap(-1)
ss_stage1.append @state ygapL2 = ygapL1(-1)
ss_stage1.append @state ygapL3 = ygapL2(-1)
ss_stage1.append @state NAIRU = NAIRU(-1) + [ename = e4, var = (sigma(4)^2)]
ss_stage1.append @state NAIRUL1 = NAIRU(-1)
ss_stage1.append @state ypot = ypot(-1) + theta(1) + [ename = e5, var = (sigma(5)^2)]
vector(7) mprior = 0
mprior(1) = ygap_ini(@ifirst(ygap_ini))
mprior(2) = ygap_ini(@ifirst(ygap_ini))
mprior(3) = ygap_ini(@ifirst(ygap_ini))
mprior(4) = ygap_ini(@ifirst(ygap_ini))
mprior(5) = 5
mprior(6) = 5
mprior(7) = ypot_ini(@ifirst(ypot_ini))
'From RBA Paper Posterior Modes
sym(7) vprior = 0
vprior(1,1) = 0.38
vprior(2,2) = 0.38
vprior(3,3) = 0.38
vprior(4,4) = 0.38
vprior(5,5) = 0.15
vprior(6,6) = 0.15
vprior(7,7) = 0.54
ss_stage1.append @mprior mprior
ss_stage1.append @vprior vprior
logmsg "Estimating Stage 1 State Space"
smpl ssest
ss_stage1.ml(optmethod=legacy)
ss_stage1.ml(optmethod=opg)
ss_stage1.makestates(t=smooth) *_stage1
'***********************************************************************************************************************************
'Median Unbiased Estimator of Lambda
'***********************************************************************************************************************************
logmsg "Estimating First Stage Median Unbiased Estimator"
call MUE_Stage1(YPOT_stage1)
'***********************************************************************************************************************************
'Stage 2 - Impose Lamdag and Add R to Output gap Equation - Constant z
'***********************************************************************************************************************************
sspace ss_stage2
ss_stage2.append @param alpha(1) !alpha1 alpha(2) !alpha2 alpha(3) !alpha3
ss_stage2.append @param beta(1) !beta1
ss_stage2.append @param gamma(1) !gamma1 gamma(2) !gamma2
ss_stage2.append @param sigma(1) !sigma1 sigma(2) !sigma2 sigma(3) !sigma3 sigma(4) !sigma4 sigma(6) !sigma6 sigma(7) !sigma7
ss_stage2.append @param theta(1) !theta1
ss_stage2.append @signal lrgdp = ygap + ypot
ss_stage2.append @state ygap = alpha(1)*ygap(-1) + alpha(2)*ygapL1(-1) + _
alpha(3)*(rcash(-1) + rcash(-2))/2 _
+ theta(1)*g(-1) + _
theta(2) + _
[ename = e1, var = (sigma(1)^2)]
ss_stage2.append @state ygapL1 = ygap(-1)
ss_stage2.append @state ygapL2 = ygapL1(-1)
ss_stage2.append @state ygapL3 = ygapL2(-1)
ss_stage2.append @signal LUR = NAIRU + _
beta(1)*(0.4*ygap + 0.3*ygapL1 + 0.2*ygapL2 + 0.1*ygapL3) + _
[ename = e2, var = (sigma(2)^2)]
ss_stage2.append @signal dlptm = (1-gamma(1))*pie_bond + _
gamma(1)/3*(dlptm(-1) + dlptm(-2) + dlptm(-3)) + _
gamma(2)*ygapL1 + _
[ename = e3, var = (sigma(3)^2)]
ss_stage2.append @state NAIRU = NAIRU(-1) + [ename = e4, var = (sigma(4)^2)]
ss_stage2.append @state NAIRUL1 = NAIRU(-1)
ss_stage2.append @state ypot = ypot(-1) + g(-1) + [ename = e5, var = (sigma(5)^2)]
ss_stage2.append @state g = g(-1) + [ename = e6, var = (lambda_g*sigma(5))^2]
ss_stage2.append @state gL1 = g(-1)
vector(9) mprior = 0
mprior(1) = ygap_ini(@ifirst(ygap_ini))
mprior(2) = ygap_ini(@ifirst(ygap_ini))
mprior(3) = ygap_ini(@ifirst(ygap_ini))
mprior(4) = ygap_ini(@ifirst(ygap_ini))
mprior(5) = 5
mprior(6) = 5
mprior(7) = ypot_ini(@ifirst(ypot_ini))
mprior(8) = g_ini(@ifirst(g_ini))
mprior(9) = g_ini(@ifirst(g_ini))
'From RBA Paper Posterior Modes
sym(9) vprior = 0
vprior(1,1) = 0.38
vprior(2,2) = 0.38
vprior(3,3) = 0.38
vprior(4,4) = 0.38
vprior(5,5) = 0.15
vprior(6,6) = 0.15
vprior(7,7) = 0.54
vprior(8,8) = 0.05
vprior(9,9) = 0.05
ss_stage2.append @mprior mprior
ss_stage2.append @vprior vprior
logmsg "Estimating Stage 2 State Space"
smpl ssest
ss_stage2.ml(optmethod=legacy)
ss_stage2.ml(optmethod=opg)
ss_stage2.makestates(t=smooth) *_stage2
'Save parameter from IS Curve
!ar = alpha(3)
'Inputs for median unbiased estimator stage2
group g_mue2 ygap_stage2 ygap_stage2(-1) ygap_stage2(-2) (rcash(-1)+rcash(-2))/2 4*(g_stage2(-1)+g_stage2(-2))/2 1
stom(g_mue2,x_mue2)
'***********************************************************************************************************************************
'Median Unbiased Estimator of Lambda
'***********************************************************************************************************************************
'Estimating Median Unbiased Estimate of Lambda Z
call MUE_Stage2(x_mue2)
'***********************************************************************************************************************************
'Stage 3
'***********************************************************************************************************************************
sspace ss_stage3
ss_stage3.append @param alpha(1) !alpha1 alpha(2) !alpha2 alpha(3) !alpha3
ss_stage3.append @param beta(1) !beta1
ss_stage3.append @param gamma(1) !gamma1 gamma(2) !gamma2
ss_stage3.append @param sigma(1) !sigma1 sigma(2) !sigma2 sigma(3) !sigma3 sigma(4) !sigma4 sigma(6) !sigma6 sigma(7) !sigma7
ss_stage3.append @param theta(1) !theta1
ss_stage3.append @signal lrgdp = ygap + ypot
ss_stage3.append @state ygap = alpha(1)*ygap(-1) + alpha(2)*ygapL1(-1) + _
alpha(3)*((rcash(-1) + rcash(-2))/2 - 4*(g(-1)+gL1(-1))/2) - _
alpha(3)*(z(-1) + zL1(-1))/2 + _
[ename = e1, var = (sigma(1)^2)]
ss_stage3.append @state ygapL1 = ygap(-1)
ss_stage3.append @state ygapL2 = ygapL1(-1)
ss_stage3.append @state ygapL3 = ygapL2(-1)
ss_stage3.append @signal LUR = NAIRU + _
beta(1)*(0.4*ygap + 0.3*ygapL1 + 0.2*ygapL2 + 0.1*ygapL3) + _
[ename = e2, var = (sigma(2)^2)]
ss_stage3.append @signal dlptm = (1-gamma(1))*pie_bond + _
gamma(1)/3*(dlptm(-1) + dlptm(-2) + dlptm(-3)) + _
gamma(2)*ygapL1 + _
[ename = e3, var = (sigma(3)^2)]
ss_stage3.append @state NAIRU = NAIRU(-1) + [ename = e4, var = (sigma(4)^2)]
ss_stage3.append @state NAIRUL1 = NAIRU(-1)
ss_stage3.append @state ypot = ypot(-1) + g(-1) + [ename = e5, var = ((1+lambda_g^2)*sigma(5)^2)]
ss_stage3.append @state g = g(-1) + [ename = e6, var = (lambda_g*sigma(5))^2]
ss_stage3.append @state gL1 = g(-1)
ss_stage3.append @state z = z(-1) + [ename = e7, var = ((lambda_z*sigma(1)/!ar)^2)]
ss_stage3.append @state zL1 = z(-1)
ss_stage3.append @state nrate = 4*(g(-1) + e7) + (z(-1) + e6)
ss_stage3.append @evar cov(e5, e6) = (lambda_g*sigma(5))^2
vector(12) mprior = 0
mprior(1) = ygap_ini(@ifirst(ygap_ini))
mprior(2) = ygap_ini(@ifirst(ygap_ini))
mprior(3) = ygap_ini(@ifirst(ygap_ini))
mprior(4) = ygap_ini(@ifirst(ygap_ini))
mprior(5) = 5
mprior(6) = 5
mprior(7) = ypot_ini(@ifirst(ypot_ini))
mprior(8) = g_ini(@ifirst(g_ini))
mprior(9) = g_ini(@ifirst(g_ini))
mprior(10) = z_ini(@ifirst(z_ini))
mprior(11) = z_ini(@ifirst(z_ini))
mprior(12) = nrate_ini(@ifirst(nrate_ini))
'From RBA Paper Posterior Modes
sym(12) vprior = 0
vprior(1,1) = 0.38
vprior(2,2) = 0.38
vprior(3,3) = 0.38
vprior(4,4) = 0.38
vprior(5,5) = 0.15
vprior(6,6) = 0.15
vprior(7,7) = 0.54
vprior(8,8) = 0.05
vprior(9,9) = 0.05
vprior(10,10) = 0.22
vprior(11,11) = 0.22
ss_stage3.append @mprior mprior
ss_stage3.append @vprior vprior
logmsg "Estimating Stage 3 State Space"
smpl ssest
ss_stage3.ml(optmethod=legacy)
ss_stage3.ml(optmethod=opg)
ss_stage3.makestates(t=smooth) *_stage3
ss_stage3.makestates(t=smoothse) *_stage3se
'-----------------------------------------------------------------------------
'Plotting Results
!bound1=@qnorm(0.85)
!bound2=@qnorm(0.95)
smpl ssest
series low70=(nrate_stage3-!bound1*nrate_stage3se)
series high70=(nrate_stage3+!bound1*nrate_stage3se)
series low90=(nrate_stage3-!bound2*nrate_stage3se)
series high90=(nrate_stage3+!bound2*nrate_stage3se)
smpl 1994q1 @last
group g_NRATE low90 high90 low70 high70 NRATE_stage3 RCASH
freeze(p_NRATE) g_NRATE.mixed band(1,2,3,4) line(5)
p_NRATE.setelem(1) fillcolor(@rgb(16, 189, 239))
p_NRATE.setelem(2) fillcolor(@rgb(14, 139, 241))
p_NRATE.setelem(1) lcolor(black)
p_NRATE.setelem(2) lcolor(red)
p_NRATE.name(1) 90 per cent confidence interval
p_NRATE.name(2)
p_NRATE.name(3) 70 per cent confidence interval
p_NRATE.name(4)
p_NRATE.name(5) Neutral Rate of Interest
p_NRATE.name(6) Real Cash Rate
p_NRATE.legend display position(0.5,2)
p_NRATE.options gridnone
show p_NRATE
p_NRATE.save(t=pdf) NeutralRate_Chart.pdf