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Data_explore.R
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# This script explore economic and financial data and does exploratory modeling
#**********************************************************************
# Notes ####
#**********************************************************************
# A discussion of ARIMA transfer function models:
# https://robjhyndman.com/hyndsight/arimax/
## Issues:
# Simulate ARIMA process with initial values and regressors
#**********************************************************************
# Packages ####
#**********************************************************************
library(tidyverse)
library(broom)
library(readxl)
library(magrittr)
library(stringr)
library(forcats)
library(grid)
library(gridExtra)
library(scales)
library(knitr)
# packages for time series modeling
library(astsa) # companion package
library(TSA) # companion package; arimax: flexible transfer function model
library(tseries) #
library(forecast) # Arima
# packages for
library(zoo)
library(xts)
library(timetk)
library(tidyquant)
library(lubridate)
# check tidyquant, timetk, sweep (broom ), tibbletime
# Intro to zoo cran.r-project.org/web/packages/zoo/vignettes/zoo-quickref.pdf
# sweep: http://www.business-science.io/code-tools/2017/07/09/sweep-0-1-0.html
#**********************************************************************
# Global settings ####
#**********************************************************************
dir_data_raw <- "data_raw/"
dir_data_out <- "data_out/"
#**********************************************************************
# Loading Data ####
#**********************************************************************
# Loading saved data
load(paste0(dir_data_out, "dataAll.RData"))
#**********************************************************************
# Selecting data for Wilkie-like models ####
#**********************************************************************
# Data needed:
# 1. Inflation:
# - SBBI inflation index
# - FRED CPI-U, seasonally adjusted
# - FRED CPI_U, not seasonally adjusted
# 2. Equity total return index (large cap)
# - SBBI total return index
# - SP500TR index
# 3. Equity price index
# - SBBI capital appreciation index
# - SP500 index
# 4. Long-term bond yield index
# - x SBBI long-term corb bond yield
# - x SBBI long-term gov bond yield
# - Moody's long-term bond yield
# 5. Long-term bond total return
# - SBBI long-term corb bond total return index
# - SBBI long-term gov bond total return index
# 6. GDP
# - FRED quarterly GDP
# - x Stock-Watson monthly GDP
# - x MA monthly GDP
vars_wilkie <- c("Inflation_Index", # SBBI price index, based on CPI-U, not seasonally adjusted
"CPIU_SA_FRED", # CPI-U, seasonally adjusted
"CPIU_NA_FRED", # CPI-U, not seasonally adjusted
"LCap_TRI", # SBBI Large Cap total return
"LCap_CAI", # SBBI Large Cap capital appreciation
"CBond_TRI", # SBBI corporate bond total return index
"LTGBond_TRI", # SBBI government bond total return index
"CBond_Yield_AAA", # Moody's AAA long-term bond yield
"GDP_FRED" # GDP
)
# df_dataWilkie <- df_dataAll %>% select(year, month, yearMon, one_of(vars_wilkie))
#**********************************************************************
# GDP growth ####
#**********************************************************************
# Quarterly GDP
df_GDP_q <-
df_dataAll_q %>%
select(yearMon,
year, month,
GDP = GDP_FRED) %>%
mutate(dl_GDP = log(GDP/lag(GDP)),
l_GDP = log(GDP)) %>%
tk_xts(date_var = yearMon)
# Annual GDP
df_GDP_y <-
df_dataAll_y %>%
select(yearMon,
year, month,
GDP = GDP_FRED) %>%
mutate(l_GDP = log(GDP),
dl_GDP = log(GDP/lag(GDP))) %>%
tk_xts(date_var = yearMon)
df_GDP_q["1945/", "l_GDP"] %>% plot
df_GDP_q["1945/", "dl_GDP"] %>% plot
df_GDP_y["1945/", "l_GDP"] %>% plot
df_GDP_y["1945/", "dl_GDP"] %>% plot
df_GDP_q["1945/", "dl_GDP"] %>% mean(na.rm = TRUE)
df_GDP_y["1945/", "dl_GDP"] %>% mean(na.rm = TRUE)
## ACF and PACF
acf2(df_GDP_q["1945/2015", "dl_GDP"]) # Looks like AR1
acf2(df_GDP_y["1945/2015", "dl_GDP"]) # Looks like random walk
## Auto ARIMA modeling
# Quarterly data
sample_period <- "1950/"
auto.arima(df_GDP_q[sample_period, "l_GDP"], max.d = 1, seasonal = FALSE) # ARIMA(2, 1, 0) with drift
sarima(df_GDP_q[sample_period , "dl_GDP"], 1, 0, 0)
# AR1 (or AR2) looks fine for quarterly GDP growth
# Annual data
sample_period <- "1950/"
auto.arima(df_GDP_y[sample_period, "l_GDP"], max.d = 1, seasonal = FALSE) # ARIMA(0, 1, 0) with drift
sarima(df_GDP_y[sample_period , "dl_GDP"], 0, 0, 0)
# Annual data looks random walk with a mean around 3.1%
#**********************************************************************
# Inflation ####
#**********************************************************************
## Variable "Inflation_Index"
df_inflation_q <-
df_dataAll_q %>%
select(year, month, yearMon, infl_index = CPIU_SA_FRED, infl_index_NA = Inflation_Index) %>%
mutate(dl_inflation = log(infl_index / lag(infl_index)),
l_inflation = log(infl_index),
dl_inflation_NA = log(infl_index_NA / lag(infl_index_NA)),
l_inflation_NA = log(infl_index_NA)) %>%
filter(!is.na(dl_inflation_NA)) %>%
tk_xts(date_var = yearMon)
range(df_inflation_q %>% index) # 1926Q1 - 2015Q4 (SBBI data range, FRED data starts in 1948)
df_inflation_y <-
df_dataAll_y %>%
select(year, month, yearMon, infl_index = CPIU_SA_FRED, infl_index_NA = Inflation_Index) %>%
mutate(dl_inflation = log(infl_index / lag(infl_index)),
l_inflation = log(infl_index),
dl_inflation_NA = log(infl_index_NA / lag(infl_index_NA)),
l_inflation_NA = log(infl_index_NA)) %>%
filter(!is.na(dl_inflation_NA)) %>%
tk_xts(date_var = yearMon)
range(df_inflation_y %>% index) #1927 - 2015
# Plotting series
sample_period <- "1950/"
df_inflation_q[sample_period, "l_inflation"] %>% plot
df_inflation_q[sample_period, "dl_inflation"] %>% plot # potential issue: inflation rate does not look stationary
df_inflation_q[sample_period, "dl_inflation"] %>% diff %>% plot # differenced inflation rate has stable mean, but time varying variance
df_inflation_y[sample_period, "l_inflation"] %>% plot
df_inflation_y[sample_period, "dl_inflation"] %>% plot
df_inflation_y[sample_period, "dl_inflation"] %>% diff %>% plot # annual data look more like stationary
df_inflation_q[sample_period, "dl_inflation"] %>% mean(na.rm = TRUE)
df_inflation_y[sample_period, "dl_inflation"] %>% mean(na.rm = TRUE)
df_inflation_y[sample_period, "dl_inflation"] %>% diff %>% mean(na.rm = TRUE)
## ACF and PACF
acf2(df_inflation_q[sample_period, "dl_inflation"]) # Look non-stationary
acf2(df_inflation_y[sample_period, "dl_inflation"]) # Look AR(1)
acf2(df_inflation_q[sample_period, "dl_inflation"] %>% diff) # AR?
acf2(df_inflation_y[sample_period, "dl_inflation"] %>% diff) # AR?
df_inflation_q
## Auto ARIMA modeling
# Quarterly data
sample_period <- "1950/"
# Auto ARIMA
auto.arima(df_inflation_q[sample_period, "l_inflation"], max.d = 1, seasonal = FALSE) # ARIMA(5, 1, 1) with drift
auto.arima(df_inflation_q[sample_period, "l_inflation"], max.d = 2, seasonal = FALSE) # estimation issue with second order difference
# Try manually
# assume inflation rate is stationary
sarima(df_inflation_q[sample_period , "dl_inflation"], 3, 0, 0) # AR(3), or even larger AR orders, with non-zero mean
# assume differenced inflation rate is stationary (changes in quarterly inflation rate)
sarima(df_inflation_q[sample_period , "dl_inflation"], 3, 1, 1) # AR(3), or even larger AR orders, with zero mean
# Annual data
sample_period <- "1950/"
# Auto ARIMA: log inflation index
auto.arima(df_inflation_y[sample_period, "l_inflation"], max.d = 1, seasonal = FALSE) # ARIMA(1, 1, 0) with drift
auto.arima(df_inflation_y[sample_period, "l_inflation"], max.d = 2, seasonal = FALSE) # ARIMA(0, 2, 0) annual changes in infl rate looks like random walk
# Try manually: inflation rate
sarima(df_inflation_y[sample_period , "dl_inflation"], 1, 0, 0) # AR(1)
sarima(df_inflation_y[sample_period , "dl_inflation"], 0, 1, 0) # random walk in changes of infl rate, worse than AR(1)
# AR(1) looks fine for annual inflation rate, which is consistent with the Wikie approach
## Outlier at 2008Q4
df_inflation_q1 <- df_inflation_q
df_inflation_q1["200812", "l_inflation"] <-
(as.numeric(df_inflation_q["200809","l_inflation"]) + as.numeric(df_inflation_q["200903","l_inflation"]))/2
df_inflation_q1["200812", "dl_inflation"] <- as.numeric(df_inflation_q1["200812", "l_inflation"]) - as.numeric(df_inflation_q1["200809", "l_inflation"])
sample_period <- "1950/"
auto.arima(df_inflation_q1[sample_period, "l_inflation"], max.d = 2, seasonal = FALSE) # ARIMA(5, 1, 4) with drift
sarima(df_inflation_q1[sample_period , "dl_inflation"], 4, 0, 0)
## Wilkie Model: Inflation rate modeled as AR(1)
# mod_inflation_q <- Arima(df_inflation_q$dl_inflation, order = c(1, 0, 0))
Arima(df_inflation_y["1927/1989","dl_inflation" ], order = c(1, 0, 0))
Arima(df_inflation_y["1927/2015","dl_inflation" ], order = c(1, 0, 0))
Arima(df_inflation_y["1950/2015","dl_inflation" ], order = c(1, 0, 0))
mod_inflation_y <- Arima(df_inflation_y["1927/2015", "dl_inflation" ], order = c(1, 0, 0))
df_inflation_y <- cbind(df_inflation_y["1927/2015",], infl_residual = xts(mod_inflation_y$residuals, index(df_inflation_y["1927/2015",])))
## diagnostic of AR(1)
df_inflation_y[,"infl_residual"] %>% acf
df_inflation_y[,"infl_residual"] %>% pacf
# AR(1) seems not sufficient for yearly inflation
#**********************************************************************
# Wilkie Dividend Yield ####
#**********************************************************************
df_dividend_y <-
df_dataAll_y %>%
select(year, month, yearMon, LCap_TRI, LCap_CAI) %>%
mutate(LCap_DivI = (LCap_TRI/lag(LCap_TRI)) * lag(LCap_CAI) - LCap_CAI,
LCap_DivY = LCap_DivI / LCap_CAI,
l_LCap_DivY = log(LCap_DivY),
dl_LCap_DivY = l_LCap_DivY - lag(l_LCap_DivY),
l_LCap_DivI = log(LCap_DivI),
dl_LCap_DivI = l_LCap_DivI - lag(l_LCap_DivI)) %>%
filter(!is.na(l_LCap_DivY)) %>%
tk_xts(date_var = yearMon)
range(df_dividend_y %>% index)
df_dividend_y <- cbind(df_inflation_y, df_dividend_y)
df_dividend_q <-
df_dataAll_q %>%
select(year, month, yearMon, LCap_TRI, LCap_CAI) %>%
mutate(LCap_DivI = (LCap_TRI/lag(LCap_TRI)) * lag(LCap_CAI) - LCap_CAI,
LCap_DivY = LCap_DivI / LCap_CAI,
l_LCap_DivY = log(LCap_DivY),
dl_LCap_DivY = l_LCap_DivY - lag(l_LCap_DivY),
l_LCap_DivI = log(LCap_DivI),
dl_LCap_DivI = l_LCap_DivI - lag(l_LCap_DivI)) %>%
filter(!is.na(l_LCap_DivY)) %>%
tk_xts(date_var = yearMon)
range(df_dividend_y %>% index)
df_dividend_q <- cbind(df_inflation_q, df_dividend_q)
## Plotting series
sample_period <- "1927/"
df_dividend_q[sample_period, "l_LCap_DivY"] %>% plot
df_dividend_q[sample_period, "dl_LCap_DivY"] %>% plot
df_dividend_y[sample_period, "l_LCap_DivY"] %>% plot
df_dividend_y[sample_period, "dl_LCap_DivY"] %>% plot
df_dividend_q[sample_period, "dl_LCap_DivY"] %>% mean(na.rm = TRUE)
df_dividend_y[sample_period, "dl_LCap_DivY"] %>% mean(na.rm = TRUE)
## ACF and PACF
acf2(df_dividend_q[sample_period, "dl_LCap_DivY"]) # looks like AR1 or AR3
acf2(df_dividend_y[sample_period, "dl_LCap_DivY"]) # looks like random walk
## Auto ARIMA modeling
# Quarterly data
sample_period <- "1950/"
# Auto ARIMA
auto.arima(df_dividend_q[sample_period, "l_LCap_DivY"], max.d = 0, seasonal = FALSE) # AR(2) is selected if assuming Div Yield is stationary
auto.arima(df_dividend_q[sample_period, "l_LCap_DivY"], max.d = 1, seasonal = FALSE) # ARIMA(5, 1, 2)
# Try manually
# assume log dividend yield is stationary
sarima(df_dividend_q[sample_period , "l_LCap_DivY"], 2, 0, 2) # No satisfactory spec
# assume differenced dividend yield is stationary (change in dividend yield)
sarima(df_dividend_q[sample_period , "l_LCap_DivY"], 5, 1, 2) # ARIMA(5, 1, 2) (selected by auto arima) is not satisfactory
# Annual data
sample_period <- "1950/"
# Auto ARIMA: log inflation index
auto.arima(df_dividend_y[sample_period, "l_LCap_DivY"], max.d = 1, seasonal = FALSE) # Looks random walk
# Try manually: inflation rate
sarima(df_dividend_y[sample_period , "l_LCap_DivY"], 1, 0, 0) # Level: AR(1) or AR(2)
sarima(df_dividend_y[sample_period , "l_LCap_DivY"], 1, 1, 0) # differenced: AR(1) par is not significant
sarima(df_dividend_y[sample_period , "l_LCap_DivY"], 0, 1, 0) # differentced:Looks random walk
# Dividend yield does not look stationary
# No satisfactory model spec found for quarterly dividend yield (for neither level nor difference)
# Annual level dividend yield: AR(1) or AR(2), but the problem is that the level data do not look stationary
# Annual differenced dividend yield: look like random walk.
## Wilkie model:
# Dividend yield level is modeled as AR(1)
Arima(df_dividend_y["1927/2014", "l_LCap_DivY" ], xreg = df_dividend_y["1927/2014", "dl_inflation"], order = c(1, 0, 0))
# As in HSZ2016, parameter for inflation rate is not significant. (while we have very different point estiamte)
mod_divY_y <- Arima(df_dividend_y["1927/2014", "l_LCap_DivY" ], order = c(1, 0, 0))
mod_divY_y
mod_divY_y$sigma2^0.5
Arima(df_dividend_y["1927/1989", "l_LCap_DivY" ], order = c(1, 0, 0))
# Model AR(1)+transfer function of inflation
#**********************************************************************
# Wilkie Dividend Index ####
#**********************************************************************
## Plotting series
sample_period <- "1927/"
df_dividend_q[sample_period, "l_LCap_DivI"] %>% plot
df_dividend_q[sample_period, "dl_LCap_DivI"] %>% plot
# Great volatility before 1955. Data after 1955 look stationary
df_dividend_y[sample_period, "LCap_DivI"] %>% plot
df_dividend_y[sample_period, "dl_LCap_DivI"] %>% plot
df_dividend_q[sample_period, "dl_LCap_DivI"] %>% mean(na.rm = TRUE)
df_dividend_y[sample_period, "dl_LCap_DivI"] %>% mean(na.rm = TRUE)
## ACF and PACF
acf2(df_dividend_q[sample_period, "dl_LCap_DivI"]) # looks like MA2
acf2(df_dividend_y[sample_period, "dl_LCap_DivI"]) # looks like AR1 or MA1
## Auto ARIMA modeling
# Quarterly data
sample_period <- "1950/"
# Auto ARIMA
auto.arima(df_dividend_q[sample_period, "l_LCap_DivI"], max.d = 1, seasonal = FALSE) # ARIMA(3, 1, 3) with drift
# model spec selected by auto.arima is sensitive to sample period
# Try manually
# assume log differenced dividend yield is stationary
sarima(df_dividend_q[sample_period , "dl_LCap_DivI"], 4, 0, 3) #
# auto-selected model spec cannot pass Ljung-Box tests
# Annual data
sample_period <- "1960/"
# Auto ARIMA: log inflation index
auto.arima(df_dividend_y[sample_period, "l_LCap_DivI"], max.d = 1, seasonal = FALSE) # MA1 looks fine
# Try manually: inflation rate
sarima(df_dividend_y[sample_period , "dl_LCap_DivI"], 0, 0, 1) # MA1, but bad performance in normaltiy and L-B test.
## Wilkie model:
dd <- 0.38
wd <- 1
get_ewa <- function(x, d){
# compute exponentially weighted average
# x: time series in xts
# d: weight of current period
#x <- df_inflation_y[, "dl_inflation"]
#d <- 0.38
n <- length(x)
ewa <- numeric(n)
ewa[1] <- x[1]
for(i in 2:n) ewa[i] <- (1 - d) * ewa[i - 1] + d * x[i]
ewa <- xts(ewa, index(x))
}
# add exponentially weighted inflation
df_dividend_y <- cbind(df_dividend_y, dl_inflation_NA_ewa = get_ewa(df_inflation_y[, "dl_inflation_NA"], dd))
df_dividend_y[sample_period, "dl_inflation_NA_ewa"] %>% plot
# reduced Wilkie specification: MA(1) + exponentially weighted inflation
sample_period <- "1926/1989"
mod_divI_y <- Arima(df_dividend_y[sample_period, "dl_LCap_DivI"], xreg = df_dividend_y[sample_period, "dl_inflation_NA_ewa"], order = c(0, 0, 1),
fixed = c(NA, NA, wd))
mod_divI_y
mod_divI_y$sigma2^0.5
sample_period <- "1926/2014"
mod_divI_y <- Arima(df_dividend_y[sample_period, "dl_LCap_DivI"], xreg = df_dividend_y[sample_period, "dl_inflation_NA_ewa"], order = c(0, 0, 1),
fixed = c(NA, NA, wd))
mod_divI_y
mod_divI_y$sigma2^0.5
# Pure MA1
Arima(df_dividend_y[sample_period, "dl_LCap_DivI"], order = c(0, 0, 1),
fixed = c(NA, NA))
## Issues:
# Model parameter estimates are sensitive to sample period. Including data after 2010 significantly increases the estiamte of intercept
## Further examine the diffrence between dividend growth and inflation rate
dfx <- df_dividend_y[sample_period, "dl_LCap_DivI"] - df_dividend_y[sample_period, "dl_inflation_ewa"]
dfx %>% plot
dfx %>% acf2
auto.arima(dfx) # MA(1) model selected
#**********************************************************************
# Long term corporate bond yield ####
#**********************************************************************
dc <- 0.058
wc <- 1
c_min <- 0.005
df_cbond_y <-
df_dataAll_y %>%
select(year, month, yearMon,
infl_index = CPIU_SA_FRED,
infl_index_NA = Inflation_Index,
CBond_Yield_AAA) %>%
mutate(CBond_Yield_AAA = CBond_Yield_AAA/100,
dl_inflation = log(infl_index / lag(infl_index)),
dl_inflation_NA = log(infl_index_NA / lag(infl_index_NA))
#CBond_Yield_real = CBond_Yield_AAA - dl_inflation,
#l_CBond_Yield_real = log(CBond_Yield_AAA),
#dl_CBond_Yield_real = l_CBond_Yield_real - lag(l_CBond_Yield_real)
) %>% tk_xts(date_var = yearMon)
range(df_cbond_y %>% index)
Index_cbond_y <- index(df_cbond_y)
df_cbond_q <-
df_dataAll_q %>%
select(year, month, yearMon,
infl_index = CPIU_SA_FRED,
infl_index_NA = Inflation_Index,
CBond_Yield_AAA) %>%
mutate(CBond_Yield_AAA = CBond_Yield_AAA/100,
dl_inflation = log(infl_index / lag(infl_index)),
dl_inflation_NA = log(infl_index_NA / lag(infl_index_NA))
#CBond_Yield_real = CBond_Yield_AAA - dl_inflation,
#l_CBond_Yield_real = log(CBond_Yield_AAA),
#dl_CBond_Yield_real = l_CBond_Yield_real - lag(l_CBond_Yield_real)
) %>% tk_xts(date_var = yearMon)
range(df_cbond_y %>% index)
Index_cbond_q <- index(df_cbond_q)
# Inflationary and real parts of long-term corp bond rate.
# Notes:
# set c_min, so that
# if the real component of long-term bond yield is smaller than c_min, then
# forces the real component to be c_min, and the inflationary component to be
# the total rate minus c_min, where the total rate is the sum of unadjusted real
# and inflationary components
# This ensures that simulated the real part is always positive and can take log.
decomp_cbondY <- function(infl, yield_tot, dc, c_min){
## compute inflationary and real components of corp bond yield based on p9 HSZ2016
# infl: inflation in xts
# yield_tot: weight of current period
# dc: weight
# c_min: min of real component of yield
# infl <- df_cbond_y[, "dl_inflation"]
# yield_tot <- df_cbond_y[, "CBond_Yield_AAA"]
# dc <- 0.058
# c_min <- 0.005
Index <- index(infl)
non_NA <- (!is.na(infl) & !is.na(yield_tot)) %>% as.vector
infl <- infl[non_NA, ]
yield_tot <- yield_tot[non_NA, ]
n <- length(infl)
yield_infl <- numeric(n)
yield_real <- numeric(n)
yield_infl[1] <- min(infl[1], yield_tot[1] - c_min)
for(i in 2:n) yield_infl[i] <- min((1 - dc) * yield_infl[i - 1] + dc * infl[i], yield_tot[i] - c_min)
yield_infl <- xts(yield_infl, index(yield_tot))
yield_real <- yield_tot - yield_infl
colnames(yield_real) <- "yield_real"
yield_decomp <- merge(merge(yield_real, yield_infl), Index)
yield_decomp
}
cbondY_comp_y <- decomp_cbondY(df_cbond_y[, "dl_inflation"], df_cbond_y[, "CBond_Yield_AAA"], dc, c_min)
cbondY_comp_y_NA <- decomp_cbondY(df_cbond_y[, "dl_inflation_NA"], df_cbond_y[, "CBond_Yield_AAA"], dc, c_min)
colnames(cbondY_comp_y_NA) <- paste(colnames(cbondY_comp_y_NA), "NA", sep = "_")
cbondY_comp_q <- decomp_cbondY(df_cbond_q[, "dl_inflation"], df_cbond_q[, "CBond_Yield_AAA"], dc, c_min)
cbondY_comp_q_NA <- decomp_cbondY(df_cbond_q[, "dl_inflation_NA"], df_cbond_q[, "CBond_Yield_AAA"], dc, c_min)
colnames(cbondY_comp_q_NA) <- paste(colnames(cbondY_comp_q_NA), "NA", sep = "_")
df_cbond_y <- merge(df_cbond_y, cbondY_comp_y, cbondY_comp_y_NA)
df_cbond_q <- merge(df_cbond_q, cbondY_comp_q, cbondY_comp_q_NA)
cbondY_comp_q_NA %>% plot
cbondY_comp_y_NA %>% plot
df_cbond_y[, c("dl_inflation_NA", "CBond_Yield_AAA")] %>% plot
df_cbond_y %<>%
as.data.frame() %>%
mutate(l_yield_real = log(yield_real),
l_yield_real_NA = log(yield_real_NA),
dl_yield_real = l_yield_real - lag(l_yield_real),
dl_yield_real_NA = l_yield_real_NA - lag(l_yield_real_NA)) %>%
tk_xts(order.by = Index_cbond_y)
df_cbond_q %<>%
as.data.frame() %>%
mutate(l_yield_real = log(yield_real),
l_yield_real_NA = log(yield_real_NA),
dl_yield_real = l_yield_real - lag(l_yield_real),
dl_yield_real_NA = l_yield_real_NA - lag(l_yield_real_NA)) %>%
tk_xts(order.by = Index_cbond_q)
df_cbond_y %>% head
## Plotting series
sample_period <- "1927/"
df_cbond_q[sample_period, "CBond_Yield_AAA"] %>% plot
df_cbond_q[sample_period, "yield_real_NA"] %>% plot
df_cbond_q[sample_period, "dl_yield_real_NA"] %>% plot
df_cbond_y[sample_period, "CBond_Yield_AAA"] %>% plot
df_cbond_y[sample_period, "yield_real_NA"] %>% plot
df_cbond_y[sample_period, "yield_infl_NA"] %>% plot
df_cbond_y[sample_period, "dl_inflation_NA"] %>% plot
df_cbond_y[sample_period, "dl_yield_real_NA"] %>% plot
df_cbond_q[sample_period, "yield_real"] %>% mean(na.rm = TRUE)
df_cbond_y[sample_period, "yield_real"] %>% mean(na.rm = TRUE)
## ACF and PACF
acf2(df_cbond_q[sample_period, "dl_yield_real"]) #
acf2(df_cbond_y[sample_period, "dl_yield_real"]) #
## ARIMA modeling
# Quarterly data
sample_period <- "1950/"
# Auto ARIMA
auto.arima(df_cbond_q["1927/2014", "l_yield_real_NA"], max.d = 1, seasonal = FALSE) # ARIMA(0, 1, 0) starting from 1927 ?
auto.arima(df_cbond_q["1950/2014", "l_yield_real_NA"], max.d = 1, seasonal = FALSE) # ARIMA(0, 1, 0) starting from 1950 ?
# Try manually
# assume log differenced interest yield is stationary
sarima(df_cbond_q["1927/" , "l_yield_real_NA"], 0, 1, 0) # Not so good
sarima(df_cbond_q["1950/" , "dl_yield_real_NA"], 1, 0, 0) # higher orders are needed for quarterly data
sarima(df_cbond_q["1950/" , "dl_yield_real_NA"], 4, 0, 0) #
# residuals do not look normally distributed
# annual data:
sample_period <- "1927/"
# Auto ARIMA: log inflation index
auto.arima(df_cbond_y["1927/2014", "l_yield_real_NA"], max.d = 1, seasonal = FALSE) # ARIMA(1, 0, 2)
auto.arima(df_cbond_y["1951/2014", "l_yield_real_NA"], max.d = 1, seasonal = FALSE) # ARIMA(1, 1, 0)
auto.arima(df_cbond_y["1951/2014", "l_yield_real_NA"], max.d = 0, seasonal = FALSE) # ARIMA(2, 0, 0)
# Try manually: inflation rate
sarima(df_cbond_y["1927/2014" , "l_yield_real_NA"], 1, 0, 1) #
sarima(df_cbond_y["1951/2014" , "l_yield_real_NA"], 1, 0, 1) #
## Wilkie model for real part: log yield modeled as AR(1) (reduced model)
mod_cbond_real <- Arima(df_cbond_y["1927/1989", "l_yield_real_NA"], order = c(1, 0, 0))
mod_cbond_real
mod_cbond_real$coef[2] %>% exp; mod_cbond_real$sigma2^0.5
mod_cbond_real <- Arima(df_cbond_y["1926/2014", "l_yield_real_NA"], order = c(1, 0, 0))
mod_cbond_real
mod_cbond_real$coef[2] %>% exp; mod_cbond_real$sigma2^0.5
mod_cbond_real <- Arima(df_cbond_y["1951/2014", "l_yield_real_NA"], order = c(1, 0, 0))
mod_cbond_real
mod_cbond_real$coef[2] %>% exp; mod_cbond_real$sigma2^0.5
mod_cbond_real <- Arima(df_cbond_y["1991/2014", "l_yield_real_NA"], order = c(1, 0, 0))
mod_cbond_real
mod_cbond_real$coef[2] %>% exp; mod_cbond_real$sigma2^0.5 # much lower than the estimates in HSZ2016
#**********************************************************************
# Try Simulatio: Inflation ####
#**********************************************************************
# Models:
#1a. AR(1) of inflation rate on 1951-2014
#1b. AR(1) of inflation rate on 1991-2014
#2a. HSZ2016 (Hardy, Saunders, Zhang, 2016) estimates on 1951-2014
#2b. HSZ2016 (Hardy, Saunders, Zhang, 2016) estimates on 1991-2014
mdl_infl_y1a <- Arima(df_inflation_y["1951/2014" , "dl_inflation"], c(1, 0, 0)) # AR(1)
mdl_infl_y1b <- Arima(df_inflation_y["1991/2014" , "dl_inflation"], c(1, 0, 0)) # AR(1)
mdl_infl_y1a
mdl_infl_y1b
# HSZ2016 estimate on 1951-2014
mdl_infl_y2a <- mdl_infl_y1a
mdl_infl_y2a$coef <- c(ar1 = 0.7575, intercept = 0.0338)
mdl_infl_y2a$sigma2 <- 0.0171^2 # 0.00029241
# HSZ2016 estimate on 1991-2014
mdl_infl_y2b <- mdl_infl_y1a
mdl_infl_y2b$coef <- c(ar1 = 0, intercept = 0.0244 )
mdl_infl_y2b$sigma2 <- 0.0111^2 # 0.00035721
# Compare estimates
extract_est <- function(fit, mdl){
#names_col <- c(names(mdl_infl_y2b$coef), "sigma")
est <- c(fit$coef, fit$sigma2^0.5) %>% matrix( nrow = 1) %>% as.data.frame
names(est) <- c(names(fit$coef), "sigma")
est$mdl <- mdl
est %<>% select(mdl, everything())
est
}
bind_rows(
extract_est(mdl_infl_y1a, "y1a"),
extract_est(mdl_infl_y1b, "y1b"), # AR1 estimate is -0.1011, forced to 0 in simulation
extract_est(mdl_infl_y2a, "y2a"),
extract_est(mdl_infl_y2b, "y2b")
) %>% kable(digits = 4)
# Check simulated series
Arima(simulate(mdl_infl_y2a, 100), order = c(1, 0, 0))
Arima(simulate(mdl_infl_y2b, 100), order = c(0, 0, 0))
# OK
# Simulation
nsim <- 2000
nyear_sim <- 100
set.seed(1234); sim_infl_y1a <- replicate(nsim,simulate(mdl_infl_y1a, nyear_sim, future = TRUE)) %>% as.tibble() %>% mutate(mdl = "y1a", year = seq_len(nyear_sim))
set.seed(1234); sim_infl_y1b <- replicate(nsim,simulate(mdl_infl_y1b, nyear_sim, future = TRUE)) %>% as.tibble() %>% mutate(mdl = "y1b", year = seq_len(nyear_sim))
set.seed(1234); sim_infl_y2a <- replicate(nsim,simulate(mdl_infl_y2a, nyear_sim, future = TRUE)) %>% as.tibble() %>% mutate(mdl = "y2a", year = seq_len(nyear_sim))
set.seed(1234); sim_infl_y2b <- replicate(nsim,simulate(mdl_infl_y2b, nyear_sim, future = TRUE)) %>% as.tibble() %>% mutate(mdl = "y2b", year = seq_len(nyear_sim))
fn_plot <- function(data){
df <- data %>%
gather(sim, value, -year, -mdl) %>%
group_by(year) %>%
summarize(
q10 = quantile(value, 0.10),
q25 = quantile(value, 0.25),
q50 = quantile(value, 0.50),
q75 = quantile(value, 0.75),
q90 = quantile(value, 0.90),
mdl = unique(mdl))
# df %>% head()
fig <-
df %>% gather(percentile, value, -year, - mdl) %>%
mutate(percentile = as.factor(percentile) %>% fct_rev) %>%
ggplot(aes(x = year, y = value, color = percentile)) + theme_bw() +
geom_line() +
geom_point() +
coord_cartesian(ylim = c(0, 0.08)) +
scale_y_continuous(breaks = seq(0, 1, 0.01))
}
fig_infl_y1a <- fn_plot(sim_infl_y1a)
fig_infl_y1b <- fn_plot(sim_infl_y1b)
fig_infl_y2a <- fn_plot(sim_infl_y2a)
fig_infl_y2b <- fn_plot(sim_infl_y2b)
fig_infl_y1a
fig_infl_y1b
fig_infl_y2a
fig_infl_y2b
sim_infl <- bind_rows(sim_infl_mdl_infl_y1a,
sim_infl_mdl_infl_y1b,
sim_infl_mdl_infl_y2a,
sim_infl_mdl_infl_y2b)
df <- sim_infl %>%
gather(sim, value, -year, -mdl) %>%
group_by(year, mdl) %>%
summarize(
q10 = quantile(value, 0.10),
q25 = quantile(value, 0.25),
q50 = quantile(value, 0.50),
q75 = quantile(value, 0.75),
q90 = quantile(value, 0.90))
# df %>% head()
fig_infl <-
df %>% gather(percentile, value, -year, -mdl) %>%
mutate(percentile = as.factor(percentile) %>% fct_rev,
mdl = factor(mdl, labels = c("AR(1) 1951-2014", "AR(1) 1991-2014",
"AR(1) HSZ 1951-2014", "AR(1) HSZ 1991-2014"))) %>%
ggplot(aes(x = year, y = value, color = percentile)) + theme_bw() +
facet_grid(. ~ mdl) +
geom_line() +
geom_point() +
coord_cartesian(ylim = c(0, 0.08)) +
scale_y_continuous(breaks = seq(0, 1, 0.01))
fig_infl
# Inflation sims based on more recent sample period (1991-2014)
# have much lower mean and variance
# There are differences between our estimate and HSZ estimate, but the
# the magnitude is not large.
#**********************************************************************
# Try Simulation: Dividend Yield ####
#**********************************************************************
# Models
# 1a. AR(1) of log div yield on 1951-1984, with inflation as regressor
# 1b. AR(1) of log div yield on 1991-2014, with inflation as regressor
# 2a. HSZ2016 par values: AR(1) of log div yield on 1951-2014, with inflation as regressor
# 2b. HSZ2016 par values: AR(1) of log div yield on 1991-2014, with inflation as regressor
df_dividend_y %>% head
mdl_divY_y1a <- Arima(df_dividend_y["1951/2014", "l_LCap_DivY"], xreg = df_dividend_y["1951/2014", "dl_inflation"], order = c(1, 0, 0))
mdl_divY_y1a
mdl_divY_y1b <- Arima(df_dividend_y["1991/2014", "l_LCap_DivY"], xreg = df_dividend_y["1991/2014", "dl_inflation"], order = c(1, 0, 0))
mdl_divY_y1b
mdl_divY_y2a <- mdl_divY_y1a
mdl_divY_y2a$coef <- c(ar1 = 0.9582, intercept = log(0.0331), dl_inflation = 0.0504)
mdl_divY_y2a$sigma2 <- 0.131^2
mdl_divY_y2b <- mdl_divY_y1a
mdl_divY_y2b$coef <- c(ar1 = 0.9112, intercept = log(0.0252), dl_inflation = -4.5762 )
mdl_divY_y2b$sigma2 <- 0.1159^2
extract_est <- function(fit, mdl){
#names_col <- c(names(mdl_infl_y2b$coef), "sigma")
est <- c(fit$coef, fit$sigma2^0.5) %>% matrix( nrow = 1) %>% as.data.frame
names(est) <- c(names(fit$coef), "sigma")
est$mdl <- mdl
est %<>% select(mdl, everything())
est
}
bind_rows(
extract_est(mdl_divY_y1a, "y1a"),
extract_est(mdl_divY_y1b, "y1b"), # AR1 estimate is -0.1011, forced to 0 in simulation
extract_est(mdl_divY_y2a, "y2a"),
extract_est(mdl_divY_y2b, "y2b")
) %>% kable(digits = 4)
# Notes
# 1. log dividend yield does not look stationary over time.
# 2. Our estimates of AR term, intercept, and sd are not very different from HSZ
# 3. Estimation on inflation does not match HSZ well
# - 0.62 vs 0.05 for 1951-2014
# - -3.4 vs -4.5 for 1991-2014
# 4. inflation parameter for 1991-2014 is negative, which contradicts theoretical relationship between div yield and infl
# 5. The figure shows that inflation and dividend yield may have a long-term positive relationship, which we may want
# to capture in the model. Perhaps transfer model is not a appropriate way to model inflation and div yield jointly?
# For simulation:
# Because of the unreasonable sign of estimated parameter on inflation,
# we should just use AR(1) for simulating dividend yield
# Simulation
nsim <- 2000
nyear_sim <- 100
mdl_divY_y1a_sim <- mdl_divY_y1a
mdl_divY_y1b_sim <- mdl_divY_y1b
mdl_divY_y2a_sim <- mdl_divY_y2a
mdl_divY_y1a_sim$coef["dl_inflation"] <- 0
mdl_divY_y1b_sim$coef["dl_inflation"] <- 0
mdl_divY_y2a_sim$coef["dl_inflation"] <- 0
set.seed(1234); sim_divY_y1a <- sapply(1:2000, function(x) {simulate(mdl_divY_y1a_sim, nyear_sim, xreg = sim_infl_y1a[, x], future = TRUE) %>% exp %>% as.numeric}) %>% as.tibble %>% mutate(mdl = "y1a", year = seq_len(nyear_sim))
set.seed(1234); sim_divY_y1b <- sapply(1:2000, function(x) {simulate(mdl_divY_y1b_sim, nyear_sim, xreg = sim_infl_y1b[, x], future = TRUE) %>% exp %>% as.numeric}) %>% as.tibble %>% mutate(mdl = "y1b", year = seq_len(nyear_sim))
set.seed(1234); sim_divY_y2a <- sapply(1:2000, function(x) {simulate(mdl_divY_y2a_sim, nyear_sim, xreg = sim_infl_y2a[, x], future = TRUE) %>% exp %>% as.numeric}) %>% as.tibble %>% mutate(mdl = "y2a", year = seq_len(nyear_sim))
set.seed(1234); sim_divY_y2b <- sapply(1:2000, function(x) {simulate(mdl_divY_y2b, nyear_sim, xreg = sim_infl_y2b[, x], future = TRUE) %>% exp %>% as.numeric}) %>% as.tibble %>% mutate(mdl = "y2b", year = seq_len(nyear_sim))
sim_divY <- bind_rows(sim_divY_y1a,
sim_divY_y1b,
sim_divY_y2a,
sim_divY_y2b)
df <- sim_divY %>%
gather(sim, value, -year, -mdl) %>%
group_by(year, mdl) %>%
summarize(
q10 = quantile(value, 0.10),
q25 = quantile(value, 0.25),
q50 = quantile(value, 0.50),
q75 = quantile(value, 0.75),
q90 = quantile(value, 0.90))
# df %>% head()
fig_infl <-
df %>% gather(percentile, value, -year, -mdl) %>%
mutate(percentile = as.factor(percentile) %>% fct_rev,
mdl = factor(mdl, labels = c("AR(1) 1951-2014", "AR(1) 1991-2014",
"AR(1) HSZ 1951-2014", "AR(1) HSZ 1991-2014"))) %>%
ggplot(aes(x = year, y = value, color = percentile)) + theme_bw() +
facet_grid(. ~ mdl) +
geom_line() +
geom_point() +
coord_cartesian(ylim = c(0, 0.08)) +
scale_y_continuous(breaks = seq(0, 1, 0.01))
fig_infl
# set.seed(1234); sim_infl_y1a <- replicate(nsim,simulate(mdl_infl_y1a, nyear_sim, future = TRUE)) %>% as.tibble() %>% mutate(mdl = "y1a", year = seq_len(nyear_sim))
#**********************************************************************
# Try Simulation: Dividend growth ####
#**********************************************************************
# Models
# 1a. MA(1) of dividend growth on 1951-1984, with div yield shock as regressor
# 1b. MA(1) of dividend growth on 1991-2014, without div yield shock as regressor
# 2a. HSZ2016 par values for 1a
# 2b. HSZ2016 par values for 1b
mdl_divY_y1a$residuals
mdl_divY_y1b$residuals
## Wilkie model:
dd <- 0.38
wd <- 1
get_ewa <- function(x, d){
# compute exponentially weighted average
# x: time series in xts
# d: weight of current period
#x <- df_inflation_y[, "dl_inflation"]
#d <- 0.38
n <- length(x)
ewa <- numeric(n)
ewa[1] <- x[1]
for(i in 2:n) ewa[i] <- (1 - d) * ewa[i - 1] + d * x[i]
ewa <- xts(ewa, index(x))
}
# regressors
infl_ewa_1a <- df_dividend_y["1951/2014", "dl_inflation_NA_ewa"]
divY_res_1a <- xts(mdl_divY_y1a$residuals, index(infl_ewa)) %>% lag; colnames(divY_res_1a) <- "divY_res"
xreg_1a <- cbind(infl_ewa_1a, divY_res_1a)[-1, ]
infl_ewa_1b <- df_dividend_y["1991/2014", "dl_inflation_NA_ewa"]
divY_res_1b <- xts(mdl_divY_y1b$residuals, index(infl_ewa_1b)) %>% lag; colnames(divY_res_1b) <- "divY_res"
xreg_1b <- cbind(infl_ewa_1b, divY_res_1b)[-1, ]
# Wilkie specification: MA(1) + regressors
mdl_divI_y1a <- Arima(df_dividend_y["1952/2014", "dl_LCap_DivI"], xreg = xreg_1a, order = c(0, 0, 1),
fixed = c(NA, NA, wd, NA))
mdl_divI_y1a; mdl_divI_y1a$sigma2^0.5
mdl_divI_y1b <- Arima(df_dividend_y["1992/2014", "dl_LCap_DivI"], xreg = xreg_1a["1992/2014"], order = c(0, 0, 1),
fixed = c(NA, NA, wd, NA))
mdl_divI_y1b; mdl_divI_y1b$sigma2^0.5
mdl_divI_y2a <- mdl_divI_y1a
mdl_divI_y2a$coef <- c(ma1 = -.3468, intercept = 0.0202 , dl_inflation_NA_ewa = 1, divY_res = 0.2138 )
mdl_divI_y2a$sigma2 <- 0.1146^2
mdl_divI_y2b <- mdl_divI_y1b
mdl_divI_y2b$coef <- c(ma1 = 0.1215, intercept = 0.0306 , dl_inflation_NA_ewa = 1, divY_res = 0.1567)
mdl_divI_y2b$sigma2 <- 0.1097^2
extract_est <- function(fit, mdl){
#names_col <- c(names(mdl_infl_y2b$coef), "sigma")
est <- c(fit$coef, fit$sigma2^0.5) %>% matrix( nrow = 1) %>% as.data.frame
names(est) <- c(names(fit$coef), "sigma")
est$mdl <- mdl
est %<>% select(mdl, everything())
est
}
bind_rows(
extract_est(mdl_divI_y1a, "y1a"),
extract_est(mdl_divI_y2a, "y2a"),