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2024_01.qmd
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2024_01.qmd
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---
title: "January 2024"
author:
- name: "Tomasz Woźniak"
url: https://github.com/donotdespair
affiliations:
- University of Melbourne
orcid: 0000-0003-2212-2378
execute:
freeze: auto
---
```{r date setup}
#| echo: false
# setup these
forecast_day = "2024-01-31" # this forecast was performed on
forecast_month = substr(forecast_day, 1, 7) # a string for the month the forecast was performed in
mfo = zoo::as.yearmon(forecast_month) # monthly forecast origin
wfo = as.Date(forecast_day) + 7 # weekly forecast origin (daily date)
```
```{r interest data}
#| echo: false
#| message: false
#| warning: false
###############################################################
# download daily interest rates
# from the RBA using package readrba
###############################################################
icr_dwnld = readrba::read_rba(series_id = "FIRMMCRTD") # Cash Rate Target
icr_tmp = xts::xts(icr_dwnld$value, icr_dwnld$date)
dates_tmp = xts::xts(as.Date(icr_dwnld$date), icr_dwnld$date)
by1m_dwnld = readrba::read_rba(series_id = "FIRMMBAB30D")
by1m_tmp = xts::xts(by1m_dwnld$value, by1m_dwnld$date)
by3m_dwnld = readrba::read_rba(series_id = "FIRMMBAB90D")
by3m_tmp = xts::xts(by3m_dwnld$value, by3m_dwnld$date)
by6m_dwnld = readrba::read_rba(series_id = "FIRMMBAB180D")
by6m_tmp = xts::xts(by6m_dwnld$value, by6m_dwnld$date)
by2y_dwnld = readrba::read_rba(series_id = "FCMYGBAG2D")
by2y_tmp = xts::xts(by2y_dwnld$value, by2y_dwnld$date)
by3y_dwnld = readrba::read_rba(series_id = "FCMYGBAG3D")
by3y_tmp = xts::xts(by3y_dwnld$value, by3y_dwnld$date)
by5y_dwnld = readrba::read_rba(series_id = "FCMYGBAG5D")
by5y_tmp = xts::xts(by5y_dwnld$value, by5y_dwnld$date)
by10y_dwnld = readrba::read_rba(series_id = "FCMYGBAG10D")
by10y_tmp = xts::xts(by10y_dwnld$value, by10y_dwnld$date)
###############################################################
# The following steps have to be performed because
# some of the downloaded rates have a gap from
# 16 May 2013 to 1 Jan 2022 as reported and discussed at:
# https://github.com/MattCowgill/readrba/issues/43
###############################################################
long_ou_tmp = na.omit(merge(by2y_tmp, by3y_tmp, by5y_tmp, by10y_tmp))
long_be = long_ou_tmp["/2013-05-16"]
long_af = long_ou_tmp["2022-01-01/"]
long_in_tmp = readxl::read_xls(path = "f02d.xls", skip = 10)
long_in = xts::xts(long_in_tmp[,2:5], as.Date(long_in_tmp$`Series ID`))
long_in = long_in["2013-05-17/2021-12-31"]
colnames(long_in) <- colnames(long_af)
short = na.omit(merge(icr_tmp, by1m_tmp, by3m_tmp, by6m_tmp))
long = rbind(long_be, long_in, long_af)
###############################################################
# construct a matrix (xts) of daily interest rates data
###############################################################
variables_all = na.omit(merge(short, long))
colnames(variables_all) = c("cash rate", "1m", "3m", "6m", "2y", "3y", "5y", "10y")
variables_all = variables_all[paste0("/",forecast_day)]
variables_weekly = xts::to.weekly(variables_all, OHLC = FALSE)
variables_monthly = xts::to.monthly(variables_all, OHLC = FALSE)
###############################################################
# construct log-volatilities
###############################################################
vol_tmp = na.omit(diff(variables_all[,2:8], lag = 1)^2)
vol_m = log(
xts::apply.monthly(
vol_tmp,
function(x){apply(x,2,sum)}
)
)
vol_w = log(
xts::apply.weekly(
vol_tmp,
function(x){apply(x,2,sum)}
)
)
vol_w = vol_w[!apply(vol_w, 1, function(x) any(x == -Inf)),]
colnames(vol_m) = colnames(vol_w) = paste0("vol",c("1m", "3m", "6m", "2y", "3y", "5y", "10y"))
###############################################################
# create a dummy for the interest raise regime
# from May 22 to June 23
# the dummies used for forecasting DO NOT include the hikes term
###############################################################
T = nrow(variables_monthly)
dm = xts::xts(as.matrix(rep(0, T)), zoo::index(variables_monthly))
dm["2022-05/2023-06"] = 1
colnames(dm) = "dum"
dmf = as.matrix(rep(0,12)) # turn off the 2022/23 hikes dummy
colnames(dmf) = "dum"
Tw = nrow(variables_weekly)
dw = xts::xts(as.matrix(rep(0, Tw)), zoo::index(variables_weekly))
dw["2022-05/2023-06"] = 1
colnames(dw) = "dum"
dwf = as.matrix(rep(0,55)) # turn off the 2022/23 hikes dummy
colnames(dwf) = "dum"
```
```{r cointegrating rank}
#| echo: false
#| eval: false
#| message: false
#| warning: false
#| results: hide
###############################################################
# perform cointegrating rank test to specify VECM models
###############################################################
library(vars)
# Johansen's cointegrating rank test
vecm_vma = ca.jo(variables_monthly, type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vma) # r = 5, N = 8
vecm_vms = ca.jo(variables_monthly[,1:4], type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vms) # r = 3, N = 4
vecm_vml = ca.jo(variables_monthly[,c(1,5:8)], type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vml) # r = 4, N = 5
vecm_vwa = ca.jo(variables_weekly, type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vwa) # r = 7, N = 8
vecm_vws = ca.jo(variables_weekly[,1:4], type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vws) # r = 3, N = 4
vecm_vwl = ca.jo(variables_weekly[,c(1,5:8)], type = "trace", ecdet = "const", K = 5, spec = "transitory")
summary(vecm_vwl) # r = 4, N = 5
```
```{r forecasting}
#| echo: false
#| message: false
#| warning: false
###############################################################
# forecasting using VECM models
###############################################################
# idea:
# create many alternative VECM specifications with
# - various lag orders (pp below)
# - various data (all variables, only short- and long-run rates collected in vm)
# - various data sampling frequency (weekly and monthly)
# - model with the raise dummy dm and without
# the forecasts are weighted based on their capacity to model cash rate
###############################################################
library(vars)
###############################################################
# forecast with monthly data
###############################################################
vm = list(
variables_monthly,
variables_monthly[,1:4],
variables_monthly[,c(1,5:8)]
) # three types of monthly data
pp = c(3, 5, 7, 9, 11, 13, 15, 17) # selected lag orders
rr = c(5, 3, 4) # various cointegrating ranks for data systems
mm = 2 * length(pp) * length(vm) # count the number of models in this section
forecasts = array(NA, c(12, 3, mm)) # to be filled in with forecasts
loglik = rep(NA, mm) # to be filled in with value of cash rate-specific likelihood
f = 1 # it's used to fill in the outcomes
###############################################################
# the loop below estimates the models and predicts 12 months ahead
###############################################################
for (v in 1:3) {
for (p in pp) {
# estimation of models with raise dummy
###############################################################
vecm = ca.jo(vm[[v]], type = "trace", ecdet = "const", K = p, spec = "transitory", dumvar = dm)
var_cr = vec2var(vecm, r = rr[v])
var_pr = predict(var_cr, n.ahead = 12, ci = .68, dumvar = dmf)
forecasts[,,f] = var_pr$fcst$cash.rate[,1:3]
standardised_resid = var_cr$resid[,1]/sqrt(sum(var_cr$resid[,1]^2)/var_cr$obs)
loglik[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
# estimation of models without the raise dummy
###############################################################
vecm = ca.jo(vm[[v]], type = "trace", ecdet = "const", K = p, spec = "transitory")
var_cr = vec2var(vecm, r = rr[v])
var_pr = predict(var_cr, n.ahead = 12, ci = .68)
forecasts[,,f] = var_pr$fcst$cash.rate[,1:3]
standardised_resid = var_cr$resid[,1]/sqrt(sum(var_cr$resid[,1]^2)/var_cr$obs)
loglik[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
}
}
# compute the weights for forecasts from each of the models
# weights are proportional to marginal likelihood for the cash rate
###############################################################
ll = exp(loglik - max(loglik))
weights = ll/sum(ll)
# reweight the forecasts
###############################################################
forecasts_w = forecasts
for (i in 1:mm) {
forecasts_w[,,i] = weights[i] * forecasts[,,i]
}
# set up forecast date sequence
# this is used to assign the forecasts monthly periods
###############################################################
ym13 = mfo
ym1 = ym13 + 1/12 # the first forecasted period
ym2 = ym1 + 11/12 # the last forecasted period
s = seq(ym1, ym2, 1/12) # create yearmon sequence
###############################################################
# compute the pooled forecasts
# for VECM forecasting using monthly data
###############################################################
pooled_forecasts_m = apply(forecasts_w, 1:2, sum)
colnames(pooled_forecasts_m) = c("forecast", "lower", "upper")
pooled_forecasts_m = xts::xts(pooled_forecasts_m, s)
###############################################################
# forecast with weekly data
###############################################################
vw = list(
variables_weekly,
variables_weekly[,1:4],
variables_weekly[,c(1,5:8)]
) # three types of weekly data
rrw = c(7, 3, 4) # various cointegrating ranks for data systems
mmw = 2 * length(pp) * length(vw) # count the number of models in this section
forecastsw = array(NA, c(55, 3, mmw)) # to be filled in with forecasts
loglikw = rep(NA, mmw) # to be filled in with value of cash rate-specific likelihood
f = 1 # it's used to fill in the outcomes
for (v in 1:3) {
for (p in pp) {
# estimation of models with raise dummy
###############################################################
vecm = ca.jo(vw[[v]], type = "trace", ecdet = "const", K = p, spec = "transitory", dumvar = dw)
var_cr = vec2var(vecm, r = rrw[v])
var_pr = predict(var_cr, n.ahead = 55, ci = .68, dumvar = dwf)
forecastsw[,,f] = var_pr$fcst$cash.rate[,1:3]
standardised_resid = var_cr$resid[,1]/sqrt(sum(var_cr$resid[,1]^2)/var_cr$obs)
loglikw[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
# estimation of models without raise dummy
###############################################################
vecm = ca.jo(vw[[v]], type = "trace", ecdet = "const", K = p, spec = "transitory")
var_cr = vec2var(vecm, r = rrw[v])
var_pr = predict(var_cr, n.ahead = 55, ci = .68)
forecastsw[,,f] = var_pr$fcst$cash.rate[,1:3]
standardised_resid = var_cr$resid[,1]/sqrt(sum(var_cr$resid[,1]^2)/var_cr$obs)
loglikw[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
}
}
# compute the weights for forecasts from each of the models
# weights are proportional to marginal likelihood for the cash rate
###############################################################
llw = exp(loglikw - max(loglikw))
weightsw = llw/sum(llw)
# reweight the forecasts
###############################################################
forecastsw_w = forecastsw
for (i in 1:mmw) {
forecastsw_w[,,i] = weightsw[i] * forecastsw[,,i]
}
# set up forecast date sequence
# this is used to assign the forecasts weekly periods
###############################################################
sw = as.Date(rep(NA, 55))
sw[1] = wfo
for (i in 2:55) {
sw[i] = sw[i - 1] + 7
}
###############################################################
# compute the pooled forecasts
# for VECM forecasting using weekly data
# and transform them to monthly forecasts (I report monthly forecasts)
###############################################################
pooled_forecasts_ww = apply(forecastsw_w, 1:2, sum)
colnames(pooled_forecasts_ww) = c("forecast", "lower", "upper")
pooled_forecasts_ww = xts::xts(pooled_forecasts_ww, sw)
pooled_forecasts_wm = xts::to.monthly(pooled_forecasts_ww, OHLC = FALSE)
```
```{r data exchange rates and foreign ir}
#| echo: false
#| eval: true
#| message: false
#| warning: false
icr_dwnld = readrba::read_rba(series_id = "FIRMMCRTD") # Cash Rate Target
icr_tmp = xts::xts(icr_dwnld$value, icr_dwnld$date)
crt = xts::to.monthly(icr_tmp, OHLC = FALSE)
dates_tmp = xts::xts(as.Date(icr_dwnld$date), icr_dwnld$date)
# Trade-weighted Index May 1970 = 100
twi_dwnld = readrba::read_rba(series_id = "FXRTWI")
twi_tmp = xts::xts(twi_dwnld$value, twi_dwnld$date)
twi = xts::to.monthly(twi_tmp, OHLC = FALSE)
# US dollar trade-weighted index
twius_dwnld = readrba::read_rba(series_id = "FUSXRTWI")
twius_tmp = xts::xts(twius_dwnld$value, twius_dwnld$date, tclass = 'yearmon')
twius = xts::to.monthly(twius_tmp, OHLC = FALSE)
# Canada Target Rate
rcan_dwnld = readrba::read_rba(series_id = "FOOIRCTR")
rcan_tmp = xts::xts(rcan_dwnld$value, rcan_dwnld$date, tclass = 'yearmon')
rcan = xts::to.monthly(rcan_tmp, OHLC = FALSE)
# Euro Area Refinancing Rate
reur_dwnld = readrba::read_rba(series_id = "FOOIREARR")
reur_tmp = xts::xts(reur_dwnld$value, reur_dwnld$date, tclass = 'yearmon')
reur = xts::to.monthly(reur_tmp, OHLC = FALSE)
# Japan Policy Rate
rjap_dwnld = readrba::read_rba(series_id = "FOOIRJTCR")
rjap_tmp = xts::xts(rjap_dwnld$value, rjap_dwnld$date, tclass = 'yearmon')
rjap = xts::to.monthly(rjap_tmp, OHLC = FALSE)
# United Kingdom Bank Rate
rgbt_dwnld = readrba::read_rba(series_id = "FOOIRUKOBR")
rgbt_tmp = xts::xts(rgbt_dwnld$value, rgbt_dwnld$date, tclass = 'yearmon')
rgbt = xts::to.monthly(rgbt_tmp, OHLC = FALSE)
# United States Federal Funds Minimum Target Rate
rusa_dwnld = readrba::read_rba(series_id = "FOOIRUSFFTRMX")
rusa_tmp = xts::xts(rusa_dwnld$value, rusa_dwnld$date, tclass = 'yearmon')
rusa = xts::to.monthly(rusa_tmp, OHLC = FALSE)
foreign = na.omit(merge(
crt, twi, twius, rcan, reur, rjap, rgbt, rusa
))
library(vars)
vecm_f = ca.jo(foreign, type = "trace", ecdet = "const", K = 5, spec = "transitory")
# summary(vecm_f) # r = 1, N = 8
vecm_fr = ca.jo(foreign[,c(1,4:8)], type = "trace", ecdet = "const", K = 5, spec = "transitory")
# summary(vecm_fr) # r = 1, N = 8
vecm_feus = ca.jo(foreign[,c(1:3,8)], type = "trace", ecdet = "const", K = 5, spec = "transitory")
# summary(vecm_feus) # r = 1, N = 8
###############################################################
# forecast with monthly data
###############################################################
vm = list(
foreign,
foreign[,c(1,4:8)],
foreign[,c(1:3,8)]
) # three types of monthly data
pp = c(3, 5, 7, 9, 11, 13, 15, 17) # selected lag orders
rr = c(1, 1, 1) # various cointegrating ranks for data systems
mm = length(pp) * length(vm) # count the number of models in this section
forecasts = array(NA, c(12, 3, mm)) # to be filled in with forecasts
loglik = rep(NA, mm) # to be filled in with value of cash rate-specific likelihood
f = 1 # it's used to fill in the outcomes
###############################################################
# the loop below estimates the models and predicts 12 months ahead
###############################################################
for (v in 1:3) {
for (p in pp) {
# estimation of models without the raise dummy
###############################################################
vecm = ca.jo(vm[[v]], type = "trace", ecdet = "const", K = p, spec = "transitory")
var_cr = vec2var(vecm, r = rr[v])
var_pr = predict(var_cr, n.ahead = 12, ci = .68)
forecasts[,,f] = var_pr$fcst$crt[,1:3]
standardised_resid = var_cr$resid[,1]/sqrt(sum(var_cr$resid[,1]^2)/var_cr$obs)
loglik[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
}
}
# compute the weights for forecasts from each of the models
# weights are proportional to marginal likelihood for the cash rate
###############################################################
ll = exp(loglik - max(loglik))
weights = ll/sum(ll)
# reweight the forecasts
###############################################################
forecasts_f = forecasts
for (i in 1:mm) {
forecasts_f[,,i] = weights[i] * forecasts[,,i]
}
# set up forecast date sequence
# this is used to assign the forecasts monthly periods
###############################################################
ym13 = mfo
ym1 = ym13 + 1/12 # the first forecasted period
ym2 = ym1 + 11/12 # the last forecasted period
s = seq(ym1, ym2, 1/12) # create yearmon sequence
###############################################################
# compute the pooled forecasts
# for VECM forecasting using monthly data
###############################################################
pooled_forecasts_f = apply(forecasts_f, 1:2, sum)
colnames(pooled_forecasts_f) = c("forecast", "lower", "upper")
pooled_forecasts_f = xts::xts(pooled_forecasts_f, s)
```
```{r univariate monthly forecasting}
#| echo: false
#| message: false
#| warning: false
###############################################################
# forecast cash rate using univariate models
###############################################################
# idea:
# forecast using various models that differ by:
# - sampling data frequency (weekly and monthly cash rate)
# - ARMA model lag order (p_max and q_max) for conditional mean
# - various specifications of the conditional volatility equations:
# - GARCH model
# - GARCH-IN-MEAN model (time-varying risk premium)
# - GJR-GARCH model (leverage effect)
# - GJR-GARCH-IN-MEAN model
# The forecasts are pooled using weights
# based on their capacity to model cash rate
# Note: rugarch package requires bootstraping the forecasts to
# compute the forecasting standard errors
###############################################################
# monthly cash rate
###############################################################
crm = variables_monthly[,1]
# ARMA models maximum lag orders
###############################################################
p_max = 3
q_max = 3
# matrices to be filled in by forecasts
# and cash rate-specific likelihood
###############################################################
garchfm = array(NA, c(12, 3, 4, (p_max ) * (q_max + 1)))
loglik = matrix(NA, 4, (p_max ) * (q_max + 1))
loglik_sr = matrix(NA, 4, (p_max ) * (q_max + 1))
for (i in 1:p_max) {
for (j in 0:q_max) {
# arma-garch forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "sGARCH"),
mean.model = list(armaOrder = c(i, j)),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crm, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 12,
method = "Partial", n.bootpred = 5000)
garchfm[,,1, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[1, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-garch-in-mean forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "sGARCH"),
mean.model = list(armaOrder = c(i, j), archm = TRUE),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crm, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 12,
method = "Partial", n.bootpred = 5000)
garchfm[,,2, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[2, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-gjr-garch forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "gjrGARCH"),
mean.model = list(armaOrder = c(i, j)),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crm, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 12,
method = "Partial", n.bootpred = 5000)
garchfm[,,3, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[3, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-gjr-garch-in-mean forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "gjrGARCH"),
mean.model = list(armaOrder = c(i, j), archm = TRUE),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crm, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 12,
method = "Partial", n.bootpred = 5000)
garchfm[,,4, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[4, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
}
}
# models that failed to estimate obtain -Inf log-likelihood
###############################################################
loglik[is.na(loglik)] = -1000000
# compute the weights
###############################################################
llm = exp(loglik - max(loglik))
weightsm = llm/sum(llm)
# reweight the forecasts
###############################################################
forecastsm = garchfm
forecastsm[is.na(forecastsm)] = 0
for (i in 1:dim(garchfm)[3]) {
for (j in 1:dim(garchfm)[4]) {
forecastsm[,,i,j] = weightsm[i,j] * forecastsm[,,i,j]
}
}
###############################################################
# pool the weighted monthly forecasts using univariate models
###############################################################
pooled_forecasts_garchm = apply(forecastsm, 1:2, sum)
colnames(pooled_forecasts_garchm) = c("forecast", "lower", "upper")
pooled_forecasts_garchm = xts::xts(pooled_forecasts_garchm, s)
```
```{r univariate weekly forecasting}
#| echo: false
#| message: false
#| warning: false
# weekly cash rate series
###############################################################
crw = variables_weekly[,1]
# ARMA maximum lag orders for weekly arma-garch forecasting
###############################################################
p_max = 4
q_max = 1
# to be filled in with forecasts and
# cash rate-specific likelihoods
###############################################################
garchfw = array(NA, c(55, 3, 4, (p_max ) * (q_max + 1)))
loglik = matrix(NA, 4, (p_max ) * (q_max + 1))
for (i in 1:p_max) {
for (j in 0:q_max) {
# arma-garch forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "sGARCH"),
mean.model = list(armaOrder = c(i, j)),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crw, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 55,
method = "Partial", n.bootpred = 5000)
garchfw[,,1, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[1, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-garch-in-mean forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "sGARCH"),
mean.model = list(armaOrder = c(i, j), archm = TRUE),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crw, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 55,
method = "Partial", n.bootpred = 5000)
garchfw[,,2, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[2, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-gjr-garch forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "gjrGARCH"),
mean.model = list(armaOrder = c(i, j)),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crw, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 55,
method = "Partial", n.bootpred = 5000)
garchfw[,,3, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[3, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
# arma-gjr-garch-in-mean forecasting
###############################################################
garch_ms = rugarch::ugarchspec(
variance.model = list(model = "gjrGARCH"),
mean.model = list(armaOrder = c(i, j), archm = TRUE),
distribution.model = "norm"
)
garch_mf = rugarch::ugarchfit(garch_ms, crw, solver = "hybrid")
garch_mfor = rugarch::ugarchboot(garch_mf, n.ahead = 55,
method = "Partial", n.bootpred = 5000)
garchfw[,,4, (q_max + 1)*(i - 1) + j + 1] = cbind(garch_mfor@forc@forecast$seriesFor,
t(apply(garch_mfor@fseries, 2, HDInterval::hdi , credMass = .68)))
loglik[4, (q_max + 1)*(i - 1) + j + 1] = -sum(garch_mf@fit$log.likelihoods)
}
}
# models that fail to estimate correctly obtain -Inf log-likelihood
###############################################################
loglik[is.na(loglik)] = -1000000
# compute the weights
###############################################################
llm = exp(loglik - max(loglik))
weightsw = llm/sum(llm)
# reweight the forecasts
###############################################################
forecastsw = garchfw
forecastsw[is.na(forecastsw)] = 0
for (i in 1:dim(garchfw)[3]) {
for (j in 1:dim(garchfw)[4]) {
forecastsw[,,i,j] = weightsw[i,j] * forecastsw[,,i,j]
}
}
###############################################################
# pool the weighted weekly forecasts from univariate models
###############################################################
pooled_forecasts_garchw = apply(forecastsw, 1:2, sum)
colnames(pooled_forecasts_garchw) = c("forecast", "lower", "upper")
pooled_forecasts_garchw = xts::xts(pooled_forecasts_garchw, sw)
pooled_forecasts_garchwm = xts::to.monthly(pooled_forecasts_garchw, OHLC = FALSE)
```
```{r forecasting with vol}
#| echo: false
#| message: false
#| warning: false
###############################################################
# forecasting using VAR models and realised volatility
###############################################################
# idea:
# create many alternative VAR specifications with
# - various lag orders (pp below)
# - various data (all variables, only short- and long-run rates collected in vm)
# - various data sampling frequency (weekly and monthly)
# - with measures of realised volatility as exogenous and endogenous variables
# the forecasts are weighted based on their capacity to model cash rate
###############################################################
library(vars)
###############################################################
# forecast with monthly data
###############################################################
vm = list(
variables_monthly,
variables_monthly[,1:4],
variables_monthly[,c(1,5:8)]
) # three types of monthly data
volm = list(
vol_m,
vol_m[,1:3],
vol_m[,4:7]
)
pp = c(3, 5, 7, 9, 11, 13, 15, 17) # selected lag orders
mm = 3 * length(pp) # count the number of models in this section
forecasts = array(NA, c(12, 3, mm)) # to be filled in with forecasts
loglik = rep(NA, mm) # to be filled in with value of cash rate-specific likelihood
f = 1 # it's used to fill in the outcomes
###############################################################
# the loop below estimates the models and predicts 12 months ahead
###############################################################
for (v in 1:3) {
for (p in pp) {
# estimation of models with raise dummy
###############################################################
ir_m_var = VAR(
y = vm[[v]],
p = p,
type = "const",
exogen = volm[[v]]
)
vol_m_var = VAR(
y = volm[[v]],
p = p,
type = "const"
)
vol_m_for = predict(vol_m_var, n.ahead = 12, ci = .68)
vol_m_for_mat = matrix(NA, 12, 0)
for (i in 1:length(vol_m_for$fcst)) {
vol_m_for_mat = cbind(vol_m_for_mat, vol_m_for$fcst[[i]][,1])
}
ir_m_for = predict(ir_m_var, n.ahead = 12, ci = .68, dumvar = vol_m_for_mat)
forecasts[,,f] = ir_m_for$fcst$cash.rate[,1:3]
standardised_resid = residuals(ir_m_var)[,1]/sqrt(sum(residuals(ir_m_var)[,1]^2)/ir_m_var$obs)
loglik[f] = sum(dnorm(standardised_resid, log = TRUE))
f = f + 1
}
}
# compute the weights for forecasts from each of the models
# weights are proportional to marginal likelihood for the cash rate
###############################################################
ll = exp(loglik - max(loglik))
weights = ll/sum(ll)
# reweight the forecasts
###############################################################
forecasts_v = forecasts
for (i in 1:mm) {
forecasts_v[,,i] = weights[i] * forecasts[,,i]
}
###############################################################
# compute the pooled forecasts
# for VECM forecasting using monthly data
###############################################################
pooled_forecasts_v = apply(forecasts_v, 1:2, sum)
colnames(pooled_forecasts_v) = c("forecast", "lower", "upper")
pooled_forecasts_v = xts::xts(pooled_forecasts_v, s)
```
```{r pool forecasts}
#| echo: false
#| message: false
#| warning: false
###############################################################
# pool all the forecasts
# from VECM and ARMA-GARCH models for weekly and monthly data
###############################################################
# pooled_forecasts = (1/6) * (pooled_forecasts_m + pooled_forecasts_wm + pooled_forecasts_garchm + pooled_forecasts_garchwm + pooled_forecasts_f + pooled_forecasts_v)
pooled_forecasts = (1/4) * (pooled_forecasts_m + pooled_forecasts_wm + pooled_forecasts_f + pooled_forecasts_v)
# save to file
###############################################################
zoo::write.zoo(pooled_forecasts, sep = ",", file = paste0("forecasts/",forecast_month,".csv"))
```
```{bash}
#| echo: false
#| eval: false
# copy the forecast to archive
###############################################################
<!-- cp forecasts/2024-01.csv forecasts-backup/2024-01.csv -->
```
> Happy New Year! Have a look at my end-of-January forecasts prepared to answer the RBA cash rate survey by [finder.com.au](https://www.finder.com.au/rba-cash-rate).
## Cash rate forecasts
The figure below presents the monthly cash rate series starting from January 2010, with the forecasts reported from February 2024 to January 2025 as the forecast mean and the 68% forecasting intervals.
```{r forecast plot}
#| echo: false
# this is used for a forecast figure
###############################################################
ym12 = zoo::as.yearmon("2010-1") # first data point for the plot - do not change
s3 = seq(ym13, ym2, 1/12) # create yearmon sequence
s2 = seq(ym12, ym2, 1/12) # create yearmon sequence
datainforecast = as.vector(vm[[2]][(dim(vm[[2]])[1] - (length(s2) - 12 - 1)):dim(vm[[2]])[1], 1])
last_point = datainforecast[length(datainforecast)]
cols = c("darkorchid4","mediumorchid1","mediumorchid2","mediumorchid3","hotpink1","hotpink2","hotpink3","hotpink4")
ci1_tmp = col2rgb(cols[2])
ci2_tmp = col2rgb(cols[4])
ci1 = rgb(ci1_tmp[1], ci1_tmp[2], ci1_tmp[3], 100, maxColorValue = 255)
ci2 = rgb(ci2_tmp[1], ci2_tmp[2], ci2_tmp[3], 100, maxColorValue = 255)
plot(x = s2, y = c(datainforecast, pooled_forecasts[,1]), main = "Cash rate forecast",
type = "l", ylab = "[%]", xlab = "time",
ylim = range(pooled_forecasts, datainforecast), bty = "n",
lwd = 1, col = cols[1]
)
polygon(x = c(s3, s3[13:1]),
y = c(last_point,as.vector(pooled_forecasts[,2]), as.vector(pooled_forecasts[,3])[12:1], last_point),
col = ci1, border = ci1)
lines(x = s2, y = c(datainforecast, pooled_forecasts[,1]), lwd = 2, col = cols[1])
abline(v = ym13, col = cols[6], lty = 3)
```
The table below makes the numerical values presented in the figure more accessible.
```{r forecast table}
#| echo: false
options(knitr.kable.NA = '')
# pooled_mq = merge(pooled_forecasts,pooled_forecasts_q)
# colnames(pooled_mq) = c("monthly", "lower", "upper", "quarterly", "lower", "upper")
# knitr::kable(as.matrix(pooled_mq), caption = "Monthly and quarterly cash rate forecasts", digits = 2)
pooled_mq = merge(pooled_forecasts)
colnames(pooled_mq) = c("monthly", "lower", "upper")
knitr::kable(as.matrix(pooled_mq), caption = "Monthly and quarterly cash rate forecasts", digits = 2)
```
## Survey answers
Based on the forecasts above, and the analysis of forecasts from individual models, I formed the following survey answers:
**When you think the RBA will change the cash rate?**
| | Feb 2024 | Mar 2024 | Apr 2024 | May 2024 | Jun 2024 | Jul 2024 | Aug 2024 | Sep 2024 | Oct 2024 or later |
|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| Increase | | | | | | | | | |
| Hold | ✓ | ✓ | | | | | | | |
| Decrease | | | | | | | | | |
**Why do you think this?**
> Happy New Year! My short-term forecasts are clearly cetred about the current cash rate value. My reading of these results is that the RBA will HOLD the cash rate at the current value to push the inflation down more. I expect the first interest rate cuts by mid-2024. My forecasts are available at https://forecasting-cash-rate.github.io/
**Do you think the cash rate has peaked?**
> YES
**When do you think the RBA will first cut the cash rate?**
> May 2024
## Forecasting system
My forecasting system for November is based on the cash rate target and government bond yields at various maturities as presented in the figure below.
```{r data plot yields}
#| echo: false
plot(
x = index(variables_weekly),
y = as.vector(variables_weekly[,2]),
main = "Australian interest rates at various maturities",
type = "l",
ylab = "yield [%]",
xlab = "time",
ylim = range(variables_weekly),
bty = "n",
lwd = 1,
col = "mediumorchid1"
)
for (i in 3:8) {
lines(
x = index(variables_weekly),
y = as.vector(variables_weekly[,i]),
col = cols[i]
)
}
lines(
x = index(variables_weekly),
y = as.vector(variables_weekly[,1]),
col = "darkorchid4",
lwd = 2
)
legend(
"topright",
legend = colnames(variables_weekly),
col = cols,
lwd = c(2, rep(1, 7)),
bty = "n"
)
```
The system consists of over two hundred of models. Half of them are are models of weekly and the other half of monthly data.
Vector Error Correction models for weekly and monthly series with different model specification parameters. Univariate models for the cash rate capture complex patterns of data persistence using autoregressive moving average equation extended by time-varying volatility equation - a GARCH model. Part of the models include the leverage effect and/or time-varying risk premium.
The forecasts are pooled in two stages. Firstly, the models are weighted in four sub-groups VECM vs. ARMA and weekly vs. monthly data models proportionally to their cash rate forecasting ability. Such four pooled forecasts are equally weighted in the second stage to provide monthly forecasts reported above.