I. Download monthly data for J.P.Morgan Chase (JPM) for the period
January, 2014 to today. Use the data for the following tests:
#Import tidyquant to get stock prices
library(tidyquant)
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#Extracting stock prices using tidyquant package
jpm_data <- tq_get('JPM',
from = '2014-01-01',
to = '2023-04-24',
get='stock.prices',
periodicity='daily')
head(jpm_data)
## # A tibble: 6 x 8
## symbol date open high low close volume adjusted
## <chr> <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 JPM 2014-01-02 58.3 58.5 58.0 58.2 15627600 45.0
## 2 JPM 2014-01-03 58.3 59.0 58.2 58.7 14214100 45.3
## 3 JPM 2014-01-06 59.2 59.5 58.8 59 17550700 45.6
## 4 JPM 2014-01-07 59.3 59.4 58.1 58.3 17851200 45.0
## 5 JPM 2014-01-08 58.5 58.9 58.3 58.9 14687400 45.5
## 6 JPM 2014-01-09 59.0 59 58.3 58.8 13242500 45.4
#Extract the close price
jpm <- tibble::tibble(date=jpm_data$date,jpm=jpm_data$close)
head(jpm)
## # A tibble: 6 x 2
## date jpm
## <date> <dbl>
## 1 2014-01-02 58.2
## 2 2014-01-03 58.7
## 3 2014-01-06 59
## 4 2014-01-07 58.3
## 5 2014-01-08 58.9
## 6 2014-01-09 58.8
1. Test for the stationarity of the closing prices for JPM.
#We will conduct several methods to check the stationarity
# Method 1: Plot the stock price
# According to the price graph, it seems that the closing price is not stationary
library(ggplot2)
jpm %>%
ggplot(aes(x=date,y=jpm))+
geom_line(color='#4373B6') +
theme_bw() +
ggtitle('JPM from 2014(M1) - 2023(M4)')+
xlab('Date')+
ylab('Close Price')
#Custome code for ACF and PACF Plot
ggplot.corr <- function(data, lag.max = 24, ci = 0.95, large.sample.size = TRUE, horizontal = TRUE,...) {
options(dplyr.banner_expression = FALSE)
require(ggplot2)
require(dplyr)
require(cowplot)
if(horizontal == TRUE) {numofrow <- 1} else {numofrow <- 2}
list.acf <- acf(data, lag.max = lag.max, type = "correlation", plot = FALSE)
N <- as.numeric(list.acf$n.used)
df1 <- data.frame(lag = list.acf$lag, acf = list.acf$acf)
df1$lag.acf <- dplyr::lag(df1$acf, default = 0)
df1$lag.acf[2] <- 0
df1$lag.acf.cumsum <- cumsum((df1$lag.acf)^2)
df1$acfstd <- sqrt(1/N * (1 + 2 * df1$lag.acf.cumsum))
df1$acfstd[1] <- 0
df1 <- dplyr::select(df1, lag, acf, acfstd)
list.pacf <- acf(data, lag.max = lag.max, type = "partial", plot = FALSE)
df2 <- data.frame(lag = list.pacf$lag,pacf = list.pacf$acf)
df2$pacfstd <- sqrt(1/N)
if(large.sample.size == TRUE) {
plot.acf <- ggplot(data = df1, aes( x = lag, y = acf)) +
geom_area(aes(x = lag, y = qnorm((1+ci)/2)*acfstd), fill = "#B9CFE7") +
geom_area(aes(x = lag, y = -qnorm((1+ci)/2)*acfstd), fill = "#B9CFE7") +
geom_col(fill = "#4373B6", width = 0.7) +
scale_x_continuous(breaks = seq(0,max(df1$lag),6)) +
scale_y_continuous(name = element_blank(),
limits = c(min(df1$acf,df2$pacf),1)) +
ggtitle("ACF") +
theme_bw() +
theme(plot.title = element_text(size=10))
plot.pacf <- ggplot(data = df2, aes(x = lag, y = pacf)) +
geom_area(aes(x = lag, y = qnorm((1+ci)/2)*pacfstd), fill = "#B9CFE7") +
geom_area(aes(x = lag, y = -qnorm((1+ci)/2)*pacfstd), fill = "#B9CFE7") +
geom_col(fill = "#4373B6", width = 0.7) +
scale_x_continuous(breaks = seq(0,max(df2$lag, na.rm = TRUE),6)) +
scale_y_continuous(name = element_blank(),
limits = c(min(df1$acf,df2$pacf),1)) +
ggtitle("PACF") +
theme_bw() +
theme(plot.title = element_text(size=10))
}
else {
plot.acf <- ggplot(data = df1, aes( x = lag, y = acf)) +
geom_col(fill = "#4373B6", width = 0.7) +
geom_hline(yintercept = qnorm((1+ci)/2)/sqrt(N),
colour = "sandybrown",
linetype = "dashed",
size=1) +
geom_hline(yintercept = - qnorm((1+ci)/2)/sqrt(N),
colour = "sandybrown",
linetype = "dashed",
size=1) +
scale_x_continuous(breaks = seq(0,max(df1$lag),6)) +
scale_y_continuous(name = element_blank(),
limits = c(min(df1$acf,df2$pacf),1)) +
ggtitle("ACF") +
theme_bw() +
theme(plot.title = element_text(size=10))
plot.pacf <- ggplot(data = df2, aes(x = lag, y = pacf)) +
geom_col(fill = "#4373B6", width = 0.7) +
geom_hline(yintercept = qnorm((1+ci)/2)/sqrt(N),
colour = "sandybrown",
linetype = "dashed",
size=1) +
geom_hline(yintercept = - qnorm((1+ci)/2)/sqrt(N),
colour = "sandybrown",
linetype = "dashed",
size=1) +
scale_x_continuous(breaks = seq(0,max(df2$lag, na.rm = TRUE),6)) +
scale_y_continuous(name = element_blank(),
limits = c(min(df1$acf,df2$pacf),1)) +
ggtitle("PACF") +
theme_bw() +
theme(plot.title = element_text(size=10))
}
cowplot::plot_grid(plot.acf, plot.pacf, nrow = numofrow)
}
# Method 2: ACF & PACF
# According to the ACF & PACF, the closing price is not stationary
ggplot.corr(jpm$jpm,large.sample.size = FALSE)
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## The following objects are masked from 'package:xts':
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## filter, lag
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## 필요한 패키지를 로딩중입니다: cowplot
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## The following object is masked from 'package:lubridate':
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## stamp
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## i Please use `linewidth` instead.
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## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
# Method 3: adf test
# The p-value for adf test is 0.28, thus we conclude it's not stationary
library(tseries)
adf.test(jpm$jpm)
##
## Augmented Dickey-Fuller Test
##
## data: jpm$jpm
## Dickey-Fuller = -2.7017, Lag order = 13, p-value = 0.2813
## alternative hypothesis: stationary
2. Test for the stationarity of the returns for JPM.
# Compute Daily return of JPM
jpm_return <- jpm_data %>%
tq_transmute(select = close,
mutate_fun = dailyReturn,
type = 'log',
col_rename = 'returns')
head(jpm_return)
## # A tibble: 6 x 2
## date returns
## <date> <dbl>
## 1 2014-01-02 0
## 2 2014-01-03 0.00770
## 3 2014-01-06 0.00578
## 4 2014-01-07 -0.0116
## 5 2014-01-08 0.00939
## 6 2014-01-09 -0.00187
# Stationarity test
# Method1: Plot the return
# We can tell from the graph that the return is stationary with conditional heteroskadasticity during COVID
jpm_return %>%
ggplot(aes(x=date,y=returns))+
geom_line(color='#4373B6') +
theme_bw() +
ggtitle('JPM Returns from 2014(M1) - 2023(M4) ')+
xlab('Date')+
ylab('Returns')+
scale_y_continuous(labels = scales::percent)
# Method 2: ACF & PACF
# According to the ACF & PACF, the return is stationary
ggplot.corr(jpm_return$returns,large.sample.size = FALSE)
# Method 3: adf test
# The p-value for adf test is lower than 0.05, thus we conclude it's stationary
adf.test(jpm_return$returns)
## Warning in adf.test(jpm_return$returns): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: jpm_return$returns
## Dickey-Fuller = -13.392, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
3. Run the best ARIMA model for JPM returns.
# Convert return data to ts format
library(tseries)
ts_jpm_return <- jpm_return$returns %>%
ts()
tail(ts_jpm_return)
## [1] 0.072794649 0.007897844 0.011165298 -0.001273746 -0.002907515
## [6] -0.001919354
#Custom code for ARIMA Model ACF PACF reisduals
library(dplyr)
library(rsample)
arima.order <- c(2,0,2)
p <- 0:arima.order[1]
d <- 0:arima.order[2] #Revise this part depending on how many differencing was conducted
q <- 0:arima.order[3]
df.arima.order <- data.frame(t(expand.grid(p=p,d=d,q=q)))
df.colnames <- data.frame(expand.grid(p=p,d=d,q=q))
df.colnames$type <- with(df.colnames,paste0
("ARIMA(",p,",",d,",",q,")"))
list.arima.model <- lapply(df.arima.order,
function(data, x, ...) {
arima(x = data, order = x, ...)
},
data = ts_jpm_return) # Data has to be in ts format
df.arima.residual <- sapply(list.arima.model,
function(x) {
x$residuals
})
df.arima.residual <- data.frame(df.arima.residual)
colnames(df.arima.residual) <- df.colnames$type
df.arima.residual$time <- time(ts_jpm_return) # Data has to be in ts format
df.arima.residual.plot <- gather(data = df.arima.residual, value = "residual", key = "type",-time)
df.acf <- sapply(subset(df.arima.residual, select = -time),
function(x, lag.max,...) {
acf(x = x, lag.max = lag.max, plot = FALSE)$acf
},
lag.max = 24)
df.acf <- data.frame(df.acf)
colnames(df.acf) <- df.colnames$type
df.acf$lag <- 0:(dim(df.acf)[1]-1)
df.acf.plot <- gather(df.acf,
value = "value",
key = "Model",
-lag)
df.pacf <- sapply(subset(df.arima.residual, select = -time),
function(x, lag.max,...) {
acfvalue <- as.numeric(acf(x = x,
lag.max = lag.max,
type = "partial",
plot = FALSE)$acf)
return(c(1,acfvalue))
},
lag.max = 24)
df.pacf <- data.frame(df.pacf)
colnames(df.pacf) <- df.colnames$type
df.pacf$lag <- 0:(dim(df.pacf)[1]-1)
df.pacf.plot <- gather(df.pacf,
value = "value",
key = "Model",
-lag)
fig.acf <- ggplot(data = df.acf.plot, aes(x = lag, y = value)) +
geom_col(width = 0.4, fill = "#667fa2") +
geom_hline(yintercept = qnorm((1+0.95)/2)/sqrt(dim(df.acf.plot)[1]),
linetype = "dashed",
colour = "grey50") +
geom_hline(yintercept = -qnorm((1+0.95)/2)/sqrt(dim(df.acf.plot)[1]),
linetype = "dashed",
colour = "grey50") +
scale_y_continuous(name = element_blank()) +
scale_x_continuous(breaks = seq(0,24,4)) +
ggtitle("ACF of residuals") +
facet_wrap(~Model) +
theme_bw() +
theme(panel.grid.minor = element_blank(),
strip.background = element_rect(fill = "#f8ddc4"))
fig.pacf <- ggplot(data = df.pacf.plot, aes(x = lag, y = value)) +
geom_col(width = 0.4, fill = "#667fa2") +
geom_hline(yintercept = qnorm((1+0.95)/2)/sqrt(dim(df.pacf.plot)[1]),
linetype = "dashed",
colour = "grey50") +
geom_hline(yintercept = -qnorm((1+0.95)/2)/sqrt(dim(df.pacf.plot)[1]),
linetype = "dashed",
colour = "grey50") +
scale_y_continuous(name = element_blank()) +
scale_x_continuous(breaks = seq(0,24,4)) +
ggtitle("PACF of residuals") +
facet_wrap(~Model) +
theme_bw() +
theme(panel.grid.minor = element_blank(),
strip.background = element_rect(fill = "#f8ddc4"))
fig.residual <- ggplot(data = df.arima.residual.plot, aes(x = time, y = residual)) +
geom_line(colour = "#667fa2") +
ggtitle("Residuals") +
facet_wrap(~type) +
theme_bw() +
theme(panel.grid.minor = element_blank(),
strip.background = element_rect(fill = "#f8ddc4"))
# Plot ACF of residuals
fig.acf
# Plot PACF of residuals
fig.pacf
# Plot residuals
fig.residual
##
## Don't know how to automatically pick scale for object of type <ts>. Defaulting
## to continuous.
# Custom Code for AIC
# From AIC we can tell that ARIMA (0,0,0) has the lowest AIC.
aic <- sapply(list.arima.model, function(X) {X$aic})
aic <- round(as.numeric(aic),2)
df.aic.plot <- data.frame(aic = aic, p = df.colnames$p, q = df.colnames$q)
ggplot(data = df.aic.plot, aes(x = p, y = q)) +
geom_raster(aes(fill = aic)) +
geom_text(aes( label = aic), colour = "black") +
scale_x_continuous(expand = c(0.05,0.05)) +
scale_color_continuous( high = "#FFA800", low = "#4AC200") +
theme_bw() +
theme(panel.grid.minor = element_blank()) +
scale_fill_distiller(palette = "Spectral", direction = -1) +
ggtitle('Akaike Information Criterion (AIC)')
4. Test for the existence of heteroskedasticity on the residuals of
the JPM’s ARIMA model.
library(forecast)
# Autoarima gives ARIMA(2,0,0)
jpm_arima <- auto.arima(jpm_return$returns)
jpm_arima
## Series: jpm_return$returns
## ARIMA(2,0,0) with zero mean
##
## Coefficients:
## ar1 ar2
## -0.0897 0.0687
## s.e. 0.0206 0.0206
##
## sigma^2 = 0.0002974: log likelihood = 6186.8
## AIC=-12367.6 AICc=-12367.59 BIC=-12350.32
library(car)
## 필요한 패키지를 로딩중입니다: carData
##
## 다음의 패키지를 부착합니다: 'car'
## The following object is masked from 'package:dplyr':
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## recode
# Null (H0): Heteroskedasticity is not present.
# Since the p-value < 0.05, there is heteroskedasticity
white.test(jpm_arima$residuals)
##
## White Neural Network Test
##
## data: jpm_arima$residuals
## X-squared = 50.828, df = 2, p-value = 9.18e-12
5. Find the historical measure of the volatility of the JPM’s
returns.
# Calculate historical standard deviation
# Annualize
sd(jpm_return$returns)*sqrt(252)
## [1] 0.2755784
6. Find the Exponentially Weighted Moving Average (EWMA) measure of
the volatility of JPM’s returns.
library(timetk)
library(quantmod)
library(DT)
# Change the data format to xts to calculate rolling volaitlity
jpm_return_xts <- jpm_return %>% tk_xts()
## Warning: Non-numeric columns being dropped: date
## Using column `date` for date_var.
realized_vol <- rollapply(jpm_return_xts,width=22,FUN=sd.annualized)
realized_vol_df <- realized_vol %>% tk_tbl()
datatable(realized_vol_df)
# Plot the Rolling Volatility
realized_vol_df %>%
ggplot(aes(x=index,y=returns))+
geom_line(color='#4373B6')+
theme_bw()+
labs(x='Date',y='Volatility',title='JPM Rolling Volatility')+
scale_y_continuous(labels = scales::percent)
## Warning: Removed 21 rows containing missing values (`geom_line()`).
7. Test whether the two measures of the volatility, historical and
EWMA, are statistically the same.
# Test whether the rolling vollatility and historical volaitility is the same
# Rolling Vollatility and the historical volatility is not the same
library(stats)
t.test(realized_vol_df$returns,mu=sd(jpm_return$returns)*sqrt(252))
##
## One Sample t-test
##
## data: realized_vol_df$returns
## t = -12.797, df = 2320, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0.2755784
## 95 percent confidence interval:
## 0.2322638 0.2437746
## sample estimates:
## mean of x
## 0.2380192
8. Estimate an auto-regression model of the volatility for JPM’s
returns using r2 and log(High/low) as measures of volatility. Use the
model to forecast next three-periods conditional variance.
# Compute Error^2
error2 <- ts((jpm_arima$residuals)^2)
# Plot error^2
autoplot(error2,color='#4373B6')+
theme_bw()+
labs(x='Time',y='Volatility',title='Error^2')+
scale_y_continuous(labels = scales::percent)
# Test for Stationarity
adf.test(error2)
## Warning in adf.test(error2): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: error2
## Dickey-Fuller = -8.3374, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
# Plot acf and pacf of error^2
# Implies Autocorrelation
ggplot.corr(error2,large.sample.size = FALSE)
# Conduct ARIMA on error^2
varianceq <- auto.arima(error2)
varianceq
## Series: error2
## ARIMA(2,1,3)
##
## Coefficients:
## ar1 ar2 ma1 ma2 ma3
## -0.6526 -0.6842 -0.0155 0.0286 -0.5818
## s.e. 0.0521 0.0727 0.0525 0.0800 0.0496
##
## sigma^2 = 8.991e-07: log likelihood = 12975.69
## AIC=-25939.39 AICc=-25939.35 BIC=-25904.84
fore <- forecast(varianceq,h=50)
# Plot the error^2 forecast
autoplot(fore$x %>% tail(300))+
autolayer(fore,color='red')+
theme_bw()+
labs(y='Error2',title='Error2 Forecast ARIMA (2,1,3)')
9. Use the auto-regressive variance model to estimate the long-run
or unconditional variance of the JPM’s returns.
# 0.0004228115 is the long-run forecast
fore
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2343 0.0003097112 -0.0009054868 0.001524909 -0.001548774 0.002168196
## 2344 0.0007409376 -0.0005394588 0.002021334 -0.001217260 0.002699135
## 2345 0.0002925830 -0.0009950668 0.001580233 -0.001676707 0.002261873
## 2346 0.0002901513 -0.0010072799 0.001587583 -0.001694098 0.002274401
## 2347 0.0005984802 -0.0007395009 0.001936461 -0.001447785 0.002644746
## 2348 0.0003989313 -0.0009542008 0.001752063 -0.001670506 0.002468368
## 2349 0.0003182117 -0.0010453866 0.001681810 -0.001767232 0.002403655
## 2350 0.0005074100 -0.0008838121 0.001898632 -0.001620281 0.002635101
## 2351 0.0004391653 -0.0009707229 0.001849053 -0.001717073 0.002595403
## 2352 0.0003542611 -0.0010678559 0.001776378 -0.001820679 0.002529201
## 2353 0.0004563584 -0.0009869066 0.001899623 -0.001750925 0.002663642
## 2354 0.0004478179 -0.0010146287 0.001910264 -0.001788801 0.002684437
## 2355 0.0003835414 -0.0010927307 0.001859814 -0.001874222 0.002641305
## 2356 0.0004313306 -0.0010629034 0.001925565 -0.001853903 0.002716564
## 2357 0.0004441186 -0.0010685448 0.001956782 -0.001869300 0.002757538
## 2358 0.0004030783 -0.0011244235 0.001930580 -0.001933034 0.002739191
## 2359 0.0004211119 -0.0011227833 0.001965007 -0.001940072 0.002782296
## 2360 0.0004374211 -0.0011238533 0.001998695 -0.001950342 0.002825184
## 2361 0.0004144402 -0.0011621096 0.001990990 -0.001996685 0.002825565
## 2362 0.0004182794 -0.0011738580 0.002010417 -0.002016685 0.002853243
## 2363 0.0004314964 -0.0011770584 0.002040051 -0.002028576 0.002891569
## 2364 0.0004202445 -0.0012036129 0.002044102 -0.002063231 0.002903720
## 2365 0.0004185450 -0.0012204279 0.002057518 -0.002088048 0.002925138
## 2366 0.0004273520 -0.0012272609 0.002081965 -0.002103160 0.002957864
## 2367 0.0004227674 -0.0012469390 0.002092474 -0.002130828 0.002976363
## 2368 0.0004197339 -0.0012647443 0.002104212 -0.002156453 0.002995921
## 2369 0.0004248501 -0.0012746598 0.002124360 -0.002174326 0.003024026
## 2370 0.0004235867 -0.0012906972 0.002137870 -0.002198184 0.003045358
## 2371 0.0004209109 -0.0013078408 0.002149663 -0.002222987 0.003064808
## 2372 0.0004235215 -0.0013197788 0.002166822 -0.002242626 0.003089669
## 2373 0.0004236485 -0.0013340711 0.002181368 -0.002264552 0.003111848
## 2374 0.0004217796 -0.0013501124 0.002193672 -0.002288095 0.003131654
## 2375 0.0004229123 -0.0013631294 0.002208954 -0.002308603 0.003154427
## 2376 0.0004234517 -0.0013766576 0.002223561 -0.002329578 0.003176481
## 2377 0.0004223247 -0.0013916637 0.002236313 -0.002351931 0.003196581
## 2378 0.0004226912 -0.0014051028 0.002250485 -0.002372678 0.003218061
## 2379 0.0004232230 -0.0014183059 0.002264752 -0.002393152 0.003239598
## 2380 0.0004226253 -0.0014324942 0.002277745 -0.002414535 0.003259786
## 2381 0.0004226515 -0.0014459651 0.002291268 -0.002435151 0.003280454
## 2382 0.0004230433 -0.0014589989 0.002305086 -0.002455292 0.003301379
## 2383 0.0004227697 -0.0014725830 0.002318122 -0.002475922 0.003321461
## 2384 0.0004226802 -0.0014858861 0.002331246 -0.002496220 0.003341580
## 2385 0.0004229258 -0.0014987794 0.002344631 -0.002516069 0.003361920
## 2386 0.0004228267 -0.0015119204 0.002357574 -0.002536114 0.003381767
## 2387 0.0004227233 -0.0015249725 0.002370419 -0.002556020 0.003401467
## 2388 0.0004228586 -0.0015377095 0.002383427 -0.002575572 0.003421289
## 2389 0.0004228411 -0.0015505134 0.002396196 -0.002595144 0.003440826
## 2390 0.0004227600 -0.0015632937 0.002408814 -0.002614647 0.003460167
## 2391 0.0004228249 -0.0015758518 0.002421502 -0.002633887 0.003479537
## 2392 0.0004228380 -0.0015883828 0.002434059 -0.002653059 0.003498735
10. Test whether the volatility measures derived from alternative
methods are statistically the same.
11. Run an ARCH and/or GARCH model on JPM’s returns data.
library(FinTS)
##
## 다음의 패키지를 부착합니다: 'FinTS'
## The following object is masked from 'package:forecast':
##
## Acf
# Arch Test reveals that there is an ARCH(1) effect
ArchTest(jpm_return$returns,lags=1)
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: jpm_return$returns
## Chi-squared = 473.98, df = 1, p-value < 2.2e-16
library(rugarch)
## 필요한 패키지를 로딩중입니다: parallel
##
## 다음의 패키지를 부착합니다: 'rugarch'
## The following object is masked from 'package:stats':
##
## sigma
garch1 <- ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1, 0)), mean.model=list(armaOrder=c(0, 0)), distribution.model="std")
ugarchfit(spec=garch1,data=jpm_return_xts)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,0)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000644 0.000253 2.5452 0.01092
## omega 0.000171 0.000013 13.4880 0.00000
## alpha1 0.454146 0.061317 7.4066 0.00000
## shape 4.110240 0.390873 10.5155 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000644 0.000230 2.8013 0.005091
## omega 0.000171 0.000013 13.0294 0.000000
## alpha1 0.454146 0.106729 4.2551 0.000021
## shape 4.110240 0.374248 10.9827 0.000000
##
## LogLikelihood : 6597.133
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.6303
## Bayes -5.6205
## Shibata -5.6303
## Hannan-Quinn -5.6268
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.001147 0.9730
## Lag[2*(p+q)+(p+q)-1][2] 0.678463 0.6154
## Lag[4*(p+q)+(p+q)-1][5] 1.858697 0.6522
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.33 1.269e-01
## Lag[2*(p+q)+(p+q)-1][2] 6.09 2.082e-02
## Lag[4*(p+q)+(p+q)-1][5] 17.69 8.304e-05
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 7.507 0.500 2.000 6.148e-03
## ARCH Lag[4] 16.830 1.397 1.611 8.600e-05
## ARCH Lag[6] 28.129 2.222 1.500 2.467e-07
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.3756
## Individual Statistics:
## mu 0.03711
## omega 4.32872
## alpha1 2.49304
## shape 3.94609
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.0614 0.28860
## Negative Sign Bias 0.9777 0.32835
## Positive Sign Bias 0.4514 0.65174
## Joint Effect 6.5382 0.08817 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 15.49 0.6910
## 2 30 22.67 0.7914
## 3 40 32.72 0.7506
## 4 50 38.27 0.8656
##
##
## Elapsed time : 0.253016
garch2 <- ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1, 1)), mean.model=list(armaOrder=c(0, 0)), distribution.model="std")
ugarchfit(spec=garch2,data=jpm_return_xts)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000722 0.000243 2.9735 0.002944
## omega 0.000010 0.000002 5.7214 0.000000
## alpha1 0.133806 0.010263 13.0382 0.000000
## beta1 0.834375 0.014624 57.0567 0.000000
## shape 4.955312 0.366224 13.5308 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000722 0.000213 3.3831 0.000717
## omega 0.000010 0.000003 3.5481 0.000388
## alpha1 0.133806 0.016427 8.1454 0.000000
## beta1 0.834375 0.022798 36.5986 0.000000
## shape 4.955312 0.669793 7.3983 0.000000
##
## LogLikelihood : 6675.427
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.6964
## Bayes -5.6841
## Shibata -5.6964
## Hannan-Quinn -5.6919
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2807 0.5962
## Lag[2*(p+q)+(p+q)-1][2] 1.3119 0.4072
## Lag[4*(p+q)+(p+q)-1][5] 2.6393 0.4769
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.016 0.3134
## Lag[2*(p+q)+(p+q)-1][5] 1.829 0.6594
## Lag[4*(p+q)+(p+q)-1][9] 3.264 0.7146
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.6072 0.500 2.000 0.4358
## ARCH Lag[5] 1.7379 1.440 1.667 0.5321
## ARCH Lag[7] 2.8438 2.315 1.543 0.5435
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 28.0299
## Individual Statistics:
## mu 0.03278
## omega 2.37494
## alpha1 1.21644
## beta1 1.19288
## shape 1.44379
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7637 0.44510
## Negative Sign Bias 1.9526 0.05099 *
## Positive Sign Bias 0.3858 0.69969
## Joint Effect 10.1030 0.01771 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 11.36 0.9113
## 2 30 16.96 0.9628
## 3 40 28.42 0.8944
## 4 50 36.18 0.9131
##
##
## Elapsed time : 0.2624319
garch3 <- ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(2, 1)), mean.model=list(armaOrder=c(0, 0)), distribution.model="std")
ugarchfit(spec=garch3,data=jpm_return_xts)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000722 0.000243 2.972965 0.002949
## omega 0.000010 0.000002 6.051201 0.000000
## alpha1 0.133412 0.031566 4.226422 0.000024
## alpha2 0.000001 0.032798 0.000027 0.999979
## beta1 0.834975 0.016608 50.276500 0.000000
## shape 4.949455 0.374397 13.219820 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000722 0.000213 3.390875 0.000697
## omega 0.000010 0.000003 3.723701 0.000196
## alpha1 0.133412 0.033253 4.012064 0.000060
## alpha2 0.000001 0.033740 0.000026 0.999979
## beta1 0.834975 0.022275 37.484223 0.000000
## shape 4.949455 0.669373 7.394168 0.000000
##
## LogLikelihood : 6675.332
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.6954
## Bayes -5.6807
## Shibata -5.6954
## Hannan-Quinn -5.6900
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2822 0.5953
## Lag[2*(p+q)+(p+q)-1][2] 1.3234 0.4043
## Lag[4*(p+q)+(p+q)-1][5] 2.6600 0.4726
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.049 0.3057
## Lag[2*(p+q)+(p+q)-1][8] 2.993 0.6868
## Lag[4*(p+q)+(p+q)-1][14] 4.457 0.8332
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.5009 0.500 2.000 0.4791
## ARCH Lag[6] 1.9684 1.461 1.711 0.4978
## ARCH Lag[8] 2.8781 2.368 1.583 0.5665
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 28.0603
## Individual Statistics:
## mu 0.03298
## omega 2.31977
## alpha1 1.21783
## alpha2 1.64630
## beta1 1.19158
## shape 1.44956
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7579 0.44858
## Negative Sign Bias 1.9661 0.04940 **
## Positive Sign Bias 0.3807 0.70344
## Joint Effect 10.1385 0.01742 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 11.34 0.9119
## 2 30 17.01 0.9620
## 3 40 28.11 0.9021
## 4 50 36.82 0.8999
##
##
## Elapsed time : 0.228688
# GARCH(1,1) with ARMA(0,0) had the lowest AIC
jpm_garch <- ugarchfit(spec=garch2,data=jpm_return_xts)
jpm_garch@fit$coef
## mu omega alpha1 beta1 shape
## 7.222633e-04 1.042146e-05 1.338062e-01 8.343754e-01 4.955312e+00
12. Do a three-period ahead forecast of the conditional
variance.
jpm_var <- jpm_garch@fit$var %>% ts()
jpm_var_fore <- ugarchforecast(jpm_garch,n.ahead = 30)
jpm_var_f <- jpm_var_fore@forecast$sigmaFor %>%ts()
jpm_var_t <- c(tail(jpm_var,50),rep(NA,50)) %>% ts()
jpm_var_f <- c(rep(NA,50),(jpm_var_f)^2) %>% ts()
autoplot(jpm_var_t)+
autolayer(jpm_var_f,series = 'Forecast')+
autolayer(jpm_var_t,series = 'Actual Variance')+
scale_colour_manual(name='Series', values=c('#4373B6','red'))+
theme_bw()+
labs(x='Period',title='JPM Conditional Variance and Forecast')+
theme(legend.position=c(0.98, 0.98),
legend.justification=c('right', 'top'))
## Warning: Removed 50 rows containing missing values (`geom_line()`).
## Removed 50 rows containing missing values (`geom_line()`).
Download monthly data from January 3, 2004 to today for S&P 500,
DOW, Hang Seng Index(^HIS), Brazil (^BVSP), FTS 100, and10-year bond
(^TNX).
# Aggregate symbols
symbols <- c("^GSPC", "^DJI", "^HSI", "^BVSP", "^FTSE", "^TNX")
start_date <- "2004-01-03"
end_date <- '2023-03-31'
data <- sp500_data <- tq_get(symbols,
from = start_date,
to = end_date,
periodicity = "monthly")
all <- data$adjusted
sp500_data <- tq_get('^GSPC',
from = start_date,
to = end_date,
periodicity = "monthly")
sp500 <- sp500_data$adjusted
dji_data <- tq_get('^DJI',
from = start_date,
to = end_date,
periodicity = "monthly")
dji <- dji_data$adjusted
hsi_data <- tq_get('^HSI',
from = start_date,
to = end_date,
periodicity = "monthly")
hsi <- hsi_data$adjusted
bvsp_data <- tq_get('^BVSP',
from = start_date,
to = end_date,
periodicity = "monthly")
bvsp <- bvsp_data$adjusted
ftse_data <- tq_get('^FTSE',
from = start_date,
to = end_date,
periodicity = "monthly")
ftse <- ftse_data$adjusted
tnx_data <- tq_get('^TNX',
from = start_date,
to = end_date,
periodicity = "monthly")
tnx <- tnx_data$adjusted
head(data)
## # A tibble: 6 x 8
## symbol date open high low close volume adjusted
## <chr> <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 ^GSPC 2004-02-01 1131. 1159. 1124. 1145. 27985600000 1145.
## 2 ^GSPC 2004-03-01 1145. 1163. 1087. 1126. 33597900000 1126.
## 3 ^GSPC 2004-04-01 1126. 1151. 1107. 1107. 31611900000 1107.
## 4 ^GSPC 2004-05-01 1107. 1128. 1076. 1121. 29326400000 1121.
## 5 ^GSPC 2004-06-01 1121. 1146. 1113. 1141. 27529500000 1141.
## 6 ^GSPC 2004-07-01 1141. 1141. 1079. 1102. 29285600000 1102.
# Plot all the data in one grid
# Need to test for stationarity using adf.test but already proved that they are all non-stationary
data %>%
ggplot(aes(x = date, y = adjusted)) +
geom_line(color='#4373B6') +
facet_wrap(~symbol, scales = "free_y") + # facet_wrap is used to make diff frames
theme_bw() + # using a new theme
labs(x = "Date", y = "Price") +
ggtitle("Price chart for stocks")
1. U.S. Indexes: Use Engel-Granger cointegration test to test for
cointegration between S&P 500 and DOW.
library(lmtest)
# residual is non-stationary, no cointegration
reg1 <- lm(sp500~dji)
summary(reg1)
##
## Call:
## lm(formula = sp500 ~ dji)
##
## Residuals:
## Min 1Q Median 3Q Max
## -293.18 -47.71 14.53 56.39 400.70
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.153e+02 1.834e+01 -11.73 <2e-16 ***
## dji 1.271e-01 9.370e-04 135.60 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 112.4 on 228 degrees of freedom
## Multiple R-squared: 0.9878, Adjusted R-squared: 0.9877
## F-statistic: 1.839e+04 on 1 and 228 DF, p-value: < 2.2e-16
adf.test(reg1$residuals)
##
## Augmented Dickey-Fuller Test
##
## data: reg1$residuals
## Dickey-Fuller = -2.5808, Lag order = 6, p-value = 0.3319
## alternative hypothesis: stationary
2. U.S. and Europe: Use Engel-Granger cointegration test to test for
cointegration between S&P 500 and FTSE 100.
# Residual is non-stationary
reg2 <- lm(sp500~ftse)
summary(reg2)
##
## Call:
## lm(formula = sp500 ~ ftse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -914.1 -508.4 -207.6 286.3 1831.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.958e+03 2.979e+02 -9.929 <2e-16 ***
## ftse 8.066e-01 4.735e-02 17.034 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 673.5 on 228 degrees of freedom
## Multiple R-squared: 0.56, Adjusted R-squared: 0.5581
## F-statistic: 290.2 on 1 and 228 DF, p-value: < 2.2e-16
adf.test(reg2$residuals)
##
## Augmented Dickey-Fuller Test
##
## data: reg2$residuals
## Dickey-Fuller = -2.5763, Lag order = 6, p-value = 0.3338
## alternative hypothesis: stationary
3. Emerging Nations: Use Engel-Granger cointegration test to test
for cointegration between HIS and BVSP.
# Non-stationary
reg3 <- lm(hsi~bvsp)
summary(reg3)
##
## Call:
## lm(formula = hsi ~ bvsp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13078.5 -2436.4 215.4 2100.8 9372.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.453e+04 6.130e+02 23.70 <2e-16 ***
## bvsp 1.141e-01 8.721e-03 13.08 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3484 on 228 degrees of freedom
## Multiple R-squared: 0.4287, Adjusted R-squared: 0.4262
## F-statistic: 171.1 on 1 and 228 DF, p-value: < 2.2e-16
adf.test(reg3$residuals)
##
## Augmented Dickey-Fuller Test
##
## data: reg3$residuals
## Dickey-Fuller = -2.0995, Lag order = 6, p-value = 0.5341
## alternative hypothesis: stationary
4. Use Engel-Granger cointegration test to test for cointegration
among all indices above.
# Non-stationary
reg4 <- lm(sp500~dji+hsi+bvsp+hsi+tnx)
summary(reg4)
##
## Call:
## lm(formula = sp500 ~ dji + hsi + bvsp + hsi + tnx)
##
## Residuals:
## Min 1Q Median 3Q Max
## -251.49 -59.19 18.56 52.78 333.67
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 75.6739767 49.3713683 1.533 0.127
## dji 0.1320781 0.0017118 77.159 < 2e-16 ***
## hsi -0.0157094 0.0019215 -8.176 2.14e-14 ***
## bvsp -0.0001439 0.0005496 -0.262 0.794
## tnx -9.3135423 7.0053060 -1.329 0.185
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.12 on 225 degrees of freedom
## Multiple R-squared: 0.991, Adjusted R-squared: 0.9908
## F-statistic: 6173 on 4 and 225 DF, p-value: < 2.2e-16
adf.test(reg4$residuals)
##
## Augmented Dickey-Fuller Test
##
## data: reg4$residuals
## Dickey-Fuller = -3.1632, Lag order = 6, p-value = 0.09494
## alternative hypothesis: stationary
5. Repeat 1-4 above using Johansen & Juselius cointegration
methodology.
library(vars)
## 필요한 패키지를 로딩중입니다: MASS
##
## 다음의 패키지를 부착합니다: 'MASS'
## The following object is masked from 'package:dplyr':
##
## select
## 필요한 패키지를 로딩중입니다: strucchange
## 필요한 패키지를 로딩중입니다: sandwich
## 필요한 패키지를 로딩중입니다: urca
##
## 다음의 패키지를 부착합니다: 'vars'
## The following object is masked from 'package:tidyquant':
##
## VAR
library(urca)
# If test > critical value reject null, cointegration exist
# No cointegration
jo1 <- ca.jo(cbind(sp500,dji),type='trace',ecdet='const')
summary(jo1)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 3.745250e-02 1.559686e-02 2.081668e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 3.58 7.52 9.24 12.97
## r = 0 | 12.29 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## sp500.l2 dji.l2 constant
## sp500.l2 1.0000000 1.0000000 1.00000000
## dji.l2 -0.1336403 -0.1280365 -0.08752148
## constant 262.3364876 416.2129121 -541.23694094
##
## Weights W:
## (This is the loading matrix)
##
## sp500.l2 dji.l2 constant
## sp500.d -0.09343496 0.04729566 5.029518e-16
## dji.d -0.42175489 0.47404556 -1.897130e-15
# If test > critical value reject null, cointegration exist
# No cointegration
jo2 <- ca.jo(cbind(sp500,ftse),type='trace',ecdet='const')
summary(jo2)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 3.316519e-02 2.602277e-02 2.775558e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 6.01 7.52 9.24 12.97
## r = 0 | 13.70 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## sp500.l2 ftse.l2 constant
## sp500.l2 1.000000 1.0000000 1.0000000
## ftse.l2 -1.851815 0.4946558 -0.3981879
## constant 9715.170642 -2256.4918921 114.5972949
##
## Weights W:
## (This is the loading matrix)
##
## sp500.l2 ftse.l2 constant
## sp500.d 0.008350039 0.004610306 -7.600404e-17
## ftse.d 0.034075609 0.001824970 -2.601655e-16
# If test > critical value reject null, cointegration exist
# No cointegration
jo3 <- ca.jo(cbind(hsi,bvsp),type='trace',ecdet='const')
summary(jo3)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 3.267944e-02 1.428419e-02 6.938894e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 3.28 7.52 9.24 12.97
## r = 0 | 10.86 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## hsi.l2 bvsp.l2 constant
## hsi.l2 1.000000e+00 1.0000000 1.0000000
## bvsp.l2 -2.501033e-02 -0.2875398 -0.3424396
## constant -2.060711e+04 543.2906867 -9657.4947437
##
## Weights W:
## (This is the loading matrix)
##
## hsi.l2 bvsp.l2 constant
## hsi.d -0.05679140 0.003756743 8.346140e-18
## bvsp.d -0.06443431 0.076617327 2.602571e-17
# If test > critical value reject null, cointegration exist
# No cointegration
jo4 <- ca.jo(cbind(sp500,dji,bvsp,tnx,hsi),type='trace',ecdet='const')
summary(jo4)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 7.076518e-02 6.057222e-02 5.586582e-02 2.951695e-02 1.261789e-02
## [6] 2.775558e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 2.90 7.52 9.24 12.97
## r <= 3 | 9.73 17.85 19.96 24.60
## r <= 2 | 22.83 32.00 34.91 41.07
## r <= 1 | 37.08 49.65 53.12 60.16
## r = 0 | 53.81 71.86 76.07 84.45
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## sp500.l2 dji.l2 bvsp.l2 tnx.l2 hsi.l2
## sp500.l2 1.000000e+00 1.00000000 1.000000e+00 1.00000000 1.000000e+00
## dji.l2 -1.300651e-01 -0.09116724 -1.327631e-01 -0.26272386 -1.823078e-01
## bvsp.l2 -9.653227e-04 -0.01153499 5.220252e-04 0.06539474 5.641564e-03
## tnx.l2 -9.866922e+01 -141.48535224 7.059068e+01 -229.79200407 -4.292124e+01
## hsi.l2 3.465135e-02 -0.03686229 2.424940e-02 -0.15853576 4.551314e-03
## constant -7.362176e+01 1495.07214266 -4.975603e+02 2348.21505235 1.046397e+03
## constant
## sp500.l2 1.000000e+00
## dji.l2 -3.972568e-02
## bvsp.l2 5.257231e-03
## tnx.l2 5.233085e+02
## hsi.l2 -5.834642e-02
## constant -1.368342e+03
##
## Weights W:
## (This is the loading matrix)
##
## sp500.l2 dji.l2 bvsp.l2 tnx.l2 hsi.l2
## sp500.d 0.073831772 0.0045963490 -1.458907e-01 -2.806556e-04 1.588974e-02
## dji.d 0.566699640 0.0937051140 -8.293700e-01 -2.279338e-03 1.957688e-01
## bvsp.d 0.930725134 2.0758267589 -3.477402e+00 -5.929877e-01 4.375427e-01
## tnx.d 0.000108652 0.0001672563 9.367085e-05 1.649583e-05 -5.535620e-06
## hsi.d -0.709177243 0.4873733990 -1.671248e+00 3.923067e-02 1.694572e-01
## constant
## sp500.d -5.080127e-17
## dji.d -2.407774e-16
## bvsp.d 2.136949e-15
## tnx.d 9.314697e-20
## hsi.d -1.033257e-15
6. Write a bi-variate VAR model using S&P 500 and FTSE and
estimate the model. Graph the impulse response functions showing the
effects of one standard deviation shock to S&P 500 on FTSE and vise
versa.
#Conver to ts format
library(timetk)
df <- tibble(date=sp500_data$date,sp500=sp500,ftse=ftse,dji=dji,bvsp=bvsp,hsi=hsi,tnx=tnx) %>% tk_xts()
## Warning: Non-numeric columns being dropped: date
## Using column `date` for date_var.
head(df)
## sp500 ftse dji bvsp hsi tnx
## 2004-02-01 1144.94 4492.2 10583.92 21755 13907.03 3.984
## 2004-03-01 1126.21 4385.7 10357.70 22142 12681.67 3.837
## 2004-04-01 1107.30 4489.7 10225.57 19607 11942.96 4.501
## 2004-05-01 1120.68 4430.7 10188.45 19545 12198.24 4.655
## 2004-06-01 1140.84 4464.1 10435.48 21149 12285.75 4.617
## 2004-07-01 1101.72 4413.1 10139.71 22337 12238.03 4.475
q6 <- df[,c(1,2)] %>% ts(start=c(2004,2),frequency = 12)
VARselect(q6, lag.max=8,type="const")$selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 1 2
q6var1 <- VAR(q6,p=1,type='const')
summary(q6var1)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: sp500, ftse
## Deterministic variables: const
## Sample size: 229
## Log Likelihood: -2898.018
## Roots of the characteristic polynomial:
## 1.003 0.9457
## Call:
## VAR(y = q6, p = 1, type = "const")
##
##
## Estimation results for equation sp500:
## ======================================
## sp500 = sp500.l1 + ftse.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 1.01027 0.01051 96.145 <2e-16 ***
## ftse.l1 -0.01229 0.01128 -1.090 0.277
## const 68.26952 56.36099 1.211 0.227
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 106.4 on 226 degrees of freedom
## Multiple R-Squared: 0.9891, Adjusted R-squared: 0.989
## F-statistic: 1.023e+04 on 2 and 226 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation ftse:
## =====================================
## ftse = sp500.l1 + ftse.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 0.03675 0.02221 1.655 0.09938 .
## ftse.l1 0.93868 0.02384 39.370 < 2e-16 ***
## const 319.47516 119.13291 2.682 0.00787 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 224.9 on 226 degrees of freedom
## Multiple R-Squared: 0.9426, Adjusted R-squared: 0.9421
## F-statistic: 1856 on 2 and 226 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sp500 ftse
## sp500 11325 15081
## ftse 15081 50600
##
## Correlation matrix of residuals:
## sp500 ftse
## sp500 1.00 0.63
## ftse 0.63 1.00
serial.test(q6var1, lags.pt=10, type="PT.asymptotic")
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object q6var1
## Chi-squared = 60.782, df = 36, p-value = 0.006049
q6fore <- forecast(q6var1)
autoplot(q6fore,color='red')+
theme_bw()+
labs(x='Date',title='FTSE and S&P500 VAR Forecast')
q6irf <- irf(q6var1, impulse.variable = 1, response.variable = 2, t = NULL, nhor = 20, scenario = 2, draw.plot = TRUE)
q6irf
##
## Impulse response coefficients
## $sp500
## sp500 ftse
## [1,] 106.4196 141.7133
## [2,] 105.7709 136.9341
## [3,] 105.1743 132.4242
## [4,] 104.6270 128.1688
## [5,] 104.1264 124.1544
## [6,] 103.6700 120.3677
## [7,] 103.2554 116.7964
## [8,] 102.8805 113.4289
## [9,] 102.5431 110.2542
## [10,] 102.2413 107.2617
## [11,] 101.9731 104.4416
##
## $ftse
## sp500 ftse
## [1,] 0.000000 174.69172
## [2,] -2.147149 163.97909
## [3,] -4.184684 153.84449
## [4,] -6.118584 144.25648
## [5,] -7.954501 135.18537
## [6,] -9.697782 126.60305
## [7,] -11.353485 118.48296
## [8,] -12.926390 110.79997
## [9,] -14.421020 103.53031
## [10,] -15.841650 96.65152
## [11,] -17.192326 90.14235
##
##
## Lower Band, CI= 0.95
## $sp500
## sp500 ftse
## [1,] 91.88232 107.66483
## [2,] 87.64094 102.08921
## [3,] 84.51505 97.54345
## [4,] 80.99761 92.79794
## [5,] 76.02615 85.44468
## [6,] 71.45041 76.98597
## [7,] 66.40387 72.60299
## [8,] 61.90716 67.60419
## [9,] 57.75767 61.57217
## [10,] 53.92498 56.05671
## [11,] 50.38163 50.33667
##
## $ftse
## sp500 ftse
## [1,] 0.000000 157.60661
## [2,] -8.191781 138.63553
## [3,] -15.366374 116.00250
## [4,] -21.655085 95.69602
## [5,] -27.168590 78.20649
## [6,] -32.004179 63.14467
## [7,] -36.440322 50.17517
## [8,] -40.417696 39.00900
## [9,] -43.961166 29.39520
## [10,] -47.117912 21.09782
## [11,] -49.929980 13.49718
##
##
## Upper Band, CI= 0.95
## $sp500
## sp500 ftse
## [1,] 122.3030 171.9154
## [2,] 120.0786 164.2598
## [3,] 118.0337 158.5180
## [4,] 117.4533 154.1863
## [5,] 117.3091 148.3791
## [6,] 116.7536 144.5951
## [7,] 115.8756 140.9774
## [8,] 115.7651 138.1844
## [9,] 115.9826 135.5754
## [10,] 116.2195 133.0724
## [11,] 116.5170 131.1907
##
## $ftse
## sp500 ftse
## [1,] 0.000000 187.2824
## [2,] 2.820378 174.5713
## [3,] 5.416809 166.9521
## [4,] 7.805030 160.5668
## [5,] 9.999705 156.3306
## [6,] 12.014494 151.7798
## [7,] 13.862129 147.3719
## [8,] 15.554468 143.1019
## [9,] 17.102559 138.9654
## [10,] 18.516696 134.9580
## [11,] 19.813848 131.0755
plot(q6irf)
7. Write a bi-variate VAR model using S&P 500 and 10-year bond
and estimate the model. Graph the impulse response functions showing the
effects of one standard deviation shock to S&P 500 on FTSE and vise
versa.
q7 <- df[,c(1,6)] %>% ts(start=c(2004,2),frequency = 12)
VARselect(q7, lag.max=8,type="const")$selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 1 2
q7var1 <- VAR(q7,p=1,type='const')
summary(q7var1)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: sp500, tnx
## Deterministic variables: const
## Sample size: 229
## Log Likelihood: -1392.094
## Roots of the characteristic polynomial:
## 0.9872 0.9872
## Call:
## VAR(y = q7, p = 1, type = "const")
##
##
## Estimation results for equation sp500:
## ======================================
## sp500 = sp500.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 0.993344 0.008085 122.864 <2e-16 ***
## tnx.l1 -14.747935 7.230327 -2.040 0.0425 *
## const 68.307900 32.972904 2.072 0.0394 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 105.7 on 226 degrees of freedom
## Multiple R-Squared: 0.9892, Adjusted R-squared: 0.9891
## F-statistic: 1.036e+04 on 2 and 226 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation tnx:
## ====================================
## tnx = sp500.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 1.370e-05 1.873e-05 0.731 0.465
## tnx.l1 9.808e-01 1.675e-02 58.542 <2e-16 ***
## const 2.404e-02 7.640e-02 0.315 0.753
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.245 on 226 degrees of freedom
## Multiple R-Squared: 0.9529, Adjusted R-squared: 0.9524
## F-statistic: 2284 on 2 and 226 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sp500 tnx
## sp500 1.118e+04 0.23171
## tnx 2.317e-01 0.06002
##
## Correlation matrix of residuals:
## sp500 tnx
## sp500 1.000000 0.008945
## tnx 0.008945 1.000000
serial.test(q7var1, lags.pt=10, type="PT.asymptotic")
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object q7var1
## Chi-squared = 57.542, df = 36, p-value = 0.01275
q7fore <- forecast(q7var1)
autoplot(q7fore,color='red')+
theme_bw()+
labs(x='Date',title='TNX and S&P500 VAR Forecast')
q7irf <- irf(q7var1, impulse.variable = 1, response.variable = 2, t = NULL, nhor = 20, scenario = 2, draw.plot = TRUE)
q7irf
##
## Impulse response coefficients
## $sp500
## sp500 tnx
## [1,] 105.72998 0.002191497
## [2,] 104.99392 0.003597965
## [3,] 104.24202 0.004967354
## [4,] 103.47492 0.006300159
## [5,] 102.69328 0.007596873
## [6,] 101.89771 0.008857992
## [7,] 101.08884 0.010084006
## [8,] 100.26728 0.011275407
## [9,] 99.43361 0.012432686
## [10,] 98.58842 0.013556331
## [11,] 97.73228 0.014646832
##
## $tnx
## sp500 tnx
## [1,] 0.000000 0.2449833
## [2,] -3.612998 0.2402813
## [3,] -7.132603 0.2356201
## [4,] -10.560038 0.2310001
## [5,] -13.896524 0.2264218
## [6,] -17.143283 0.2218857
## [7,] -20.301533 0.2173922
## [8,] -23.372491 0.2129416
## [9,] -26.357372 0.2085344
## [10,] -29.257388 0.2041709
## [11,] -32.073749 0.1998514
##
##
## Lower Band, CI= 0.95
## $sp500
## sp500 tnx
## [1,] 89.23732 -0.03450441
## [2,] 87.88507 -0.03361104
## [3,] 85.28329 -0.03086265
## [4,] 83.39621 -0.02823806
## [5,] 81.16519 -0.02939165
## [6,] 77.38986 -0.03159370
## [7,] 73.62559 -0.03357528
## [8,] 69.97972 -0.03555865
## [9,] 65.96889 -0.03540385
## [10,] 62.10992 -0.03518621
## [11,] 58.39953 -0.03491499
##
## $tnx
## sp500 tnx
## [1,] 0.000000 0.21349398
## [2,] -8.402169 0.19830912
## [3,] -16.033887 0.18339944
## [4,] -23.072317 0.16836720
## [5,] -29.521811 0.15348271
## [6,] -35.426021 0.13996092
## [7,] -40.123665 0.12539651
## [8,] -44.632582 0.11187012
## [9,] -48.822746 0.10082838
## [10,] -53.308638 0.09130210
## [11,] -58.353670 0.08284054
##
##
## Upper Band, CI= 0.95
## $sp500
## sp500 tnx
## [1,] 117.1695 0.04632685
## [2,] 115.5315 0.04717272
## [3,] 114.1114 0.04797549
## [4,] 113.8140 0.04989845
## [5,] 113.5348 0.05220921
## [6,] 113.2059 0.05602182
## [7,] 112.8320 0.06000680
## [8,] 111.8925 0.06321491
## [9,] 110.9128 0.06715177
## [10,] 109.9586 0.07075707
## [11,] 109.0230 0.07404482
##
## $tnx
## sp500 tnx
## [1,] 0.000000 0.2666460
## [2,] -1.226235 0.2577423
## [3,] -2.347785 0.2523163
## [4,] -3.375712 0.2471897
## [5,] -4.304407 0.2409386
## [6,] -5.099592 0.2358753
## [7,] -5.803790 0.2308190
## [8,] -6.425801 0.2258300
## [9,] -6.973529 0.2217289
## [10,] -7.454083 0.2188695
## [11,] -7.873859 0.2169686
plot(q7irf)
- Write a bi-variate VAR model using S&P 500 and HIS and estimate
the model. Graph the impulse response functions showing the effects of
one standard deviation shock to S&P 500 on HIS and vise versa.
q8 <- df[,c(1,5)] %>% ts(start=c(2004,2),frequency = 12)
VARselect(q8, lag.max=1,type="const")$selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 1 1 1 1
q8var1 <- VAR(q8,p=1,type='const')
summary(q8var1)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: sp500, hsi
## Deterministic variables: const
## Sample size: 229
## Log Likelihood: -3334.614
## Roots of the characteristic polynomial:
## 1 0.9528
## Call:
## VAR(y = q8, p = 1, type = "const")
##
##
## Estimation results for equation sp500:
## ======================================
## sp500 = sp500.l1 + hsi.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 1.0001962 0.0083206 120.208 <2e-16 ***
## hsi.l1 0.0006271 0.0018168 0.345 0.730
## const -1.2353304 34.6854912 -0.036 0.972
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 106.7 on 226 degrees of freedom
## Multiple R-Squared: 0.989, Adjusted R-squared: 0.9889
## F-statistic: 1.018e+04 on 2 and 226 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation hsi:
## ====================================
## hsi = sp500.l1 + hsi.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 -4.025e-03 1.022e-01 -0.039 0.9686
## hsi.l1 9.528e-01 2.232e-02 42.679 <2e-16 ***
## const 1.075e+03 4.262e+02 2.521 0.0124 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1311 on 226 degrees of freedom
## Multiple R-Squared: 0.9187, Adjusted R-squared: 0.918
## F-statistic: 1278 on 2 and 226 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sp500 hsi
## sp500 11379 62418
## hsi 62418 1717994
##
## Correlation matrix of residuals:
## sp500 hsi
## sp500 1.0000 0.4464
## hsi 0.4464 1.0000
serial.test(q8var1, lags.pt=10, type="PT.asymptotic")
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object q8var1
## Chi-squared = 74.36, df = 36, p-value = 0.0001767
q8fore <- forecast(q8var1)
autoplot(q8fore,color='red')+
theme_bw()+
labs(x='Date',title='HSI and S&P500 VAR Forecast')
q8irf <- irf(q8var1, impulse.variable = 1, response.variable = 2, t = NULL, nhor = 20, scenario = 2, draw.plot = TRUE)
q8irf
##
## Impulse response coefficients
## $sp500
## sp500 hsi
## [1,] 106.6706 585.1501
## [2,] 107.0585 557.0750
## [3,] 107.4289 530.3247
## [4,] 107.7825 504.8368
## [5,] 108.1203 480.5517
## [6,] 108.4428 457.4126
## [7,] 108.7509 435.3654
## [8,] 109.0453 414.3586
## [9,] 109.3265 394.3431
## [10,] 109.5953 375.2721
## [11,] 109.8521 357.1010
##
## $hsi
## sp500 hsi
## [1,] 0.0000000 1172.8571
## [2,] 0.7355119 1117.4447
## [3,] 1.4364184 1064.6474
## [4,] 2.1043526 1014.3416
## [5,] 2.7408705 966.4099
## [6,] 3.3474548 920.7402
## [7,] 3.9255181 877.2258
## [8,] 4.4764064 835.7649
## [9,] 5.0014022 796.2606
## [10,] 5.5017274 758.6207
## [11,] 5.9785463 722.7570
##
##
## Lower Band, CI= 0.95
## $sp500
## sp500 hsi
## [1,] 92.89031 406.77292
## [2,] 92.12394 364.03098
## [3,] 91.34103 324.31801
## [4,] 89.41860 289.91000
## [5,] 85.46231 258.46919
## [6,] 82.23606 217.66047
## [7,] 78.69942 175.16423
## [8,] 75.28345 140.73877
## [9,] 72.04074 108.34626
## [10,] 68.96900 76.27547
## [11,] 66.05677 42.57278
##
## $hsi
## sp500 hsi
## [1,] 0.000000 1006.2140
## [2,] -3.948155 891.3505
## [3,] -7.451760 808.5091
## [4,] -10.557468 729.3172
## [5,] -13.304990 641.3936
## [6,] -15.730504 546.5949
## [7,] -17.866506 468.6671
## [8,] -19.780602 408.0196
## [9,] -21.495445 355.1580
## [10,] -22.959719 309.0521
## [11,] -24.248420 269.0140
##
##
## Upper Band, CI= 0.95
## $sp500
## sp500 hsi
## [1,] 121.4450 755.7184
## [2,] 121.3326 713.0292
## [3,] 121.6013 672.0646
## [4,] 121.8085 629.8078
## [5,] 121.3764 599.7017
## [6,] 121.4718 568.9401
## [7,] 122.6888 549.7225
## [8,] 123.4472 532.7077
## [9,] 124.1428 518.0021
## [10,] 124.7790 505.4093
## [11,] 125.3588 491.3933
##
## $hsi
## sp500 hsi
## [1,] 0.000000 1355.1015
## [2,] 4.876778 1230.6609
## [3,] 9.307043 1153.2169
## [4,] 13.324943 1092.3192
## [5,] 16.974321 1042.1166
## [6,] 20.282280 994.0624
## [7,] 23.357748 950.0936
## [8,] 26.353819 912.4781
## [9,] 29.127336 875.1275
## [10,] 31.689587 841.3840
## [11,] 34.051336 808.9051
plot(q8irf)
- Write a bi-variate VAR model using all market indices and estimate
the model. Graph the impulse response functions showing the effects of
one standard deviation shock to S&P 500 on other indices.
q9 <- df[,] %>% ts(start=c(2004,2),frequency = 12)
VARselect(q9, lag.max=8,type="const")$selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 1 1 2
q9var1 <- VAR(q9,p=1,type='const')
summary(q9var1)
##
## VAR Estimation Results:
## =========================
## Endogenous variables: sp500, ftse, dji, bvsp, hsi, tnx
## Deterministic variables: const
## Sample size: 229
## Log Likelihood: -8590.422
## Roots of the characteristic polynomial:
## 0.9798 0.9798 0.9556 0.9556 0.8887 0.8887
## Call:
## VAR(y = q9, p = 1, type = "const")
##
##
## Estimation results for equation sp500:
## ======================================
## sp500 = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 9.161e-01 7.361e-02 12.445 <2e-16 ***
## ftse.l1 -2.624e-02 1.751e-02 -1.498 0.1356
## dji.l1 1.355e-02 1.037e-02 1.306 0.1928
## bvsp.l1 -5.168e-04 6.636e-04 -0.779 0.4370
## hsi.l1 1.417e-03 3.136e-03 0.452 0.6519
## tnx.l1 -1.164e+01 8.241e+00 -1.412 0.1593
## const 1.418e+02 6.546e+01 2.166 0.0314 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 105.9 on 222 degrees of freedom
## Multiple R-Squared: 0.9894, Adjusted R-squared: 0.9891
## F-statistic: 3441 on 6 and 222 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation ftse:
## =====================================
## ftse = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 -0.029389 0.156844 -0.187 0.85154
## ftse.l1 0.941825 0.037319 25.237 < 2e-16 ***
## dji.l1 0.011982 0.022093 0.542 0.58812
## bvsp.l1 -0.001438 0.001414 -1.017 0.31016
## hsi.l1 -0.002399 0.006682 -0.359 0.71991
## tnx.l1 -15.940464 17.559983 -0.908 0.36498
## const 413.095142 139.482367 2.962 0.00339 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 225.7 on 222 degrees of freedom
## Multiple R-Squared: 0.9432, Adjusted R-squared: 0.9417
## F-statistic: 614.9 on 6 and 222 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation dji:
## ====================================
## dji = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 -0.125629 0.608943 -0.206 0.837
## ftse.l1 -0.100152 0.144890 -0.691 0.490
## dji.l1 1.023399 0.085777 11.931 <2e-16 ***
## bvsp.l1 -0.001942 0.005489 -0.354 0.724
## hsi.l1 0.002830 0.025944 0.109 0.913
## tnx.l1 -93.108255 68.176226 -1.366 0.173
## const 889.243431 541.537058 1.642 0.102
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 876.4 on 222 degrees of freedom
## Multiple R-Squared: 0.9881, Adjusted R-squared: 0.9878
## F-statistic: 3072 on 6 and 222 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation bvsp:
## =====================================
## bvsp = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 -3.47977 3.17932 -1.095 0.2749
## ftse.l1 -1.23513 0.75648 -1.633 0.1039
## dji.l1 0.71505 0.44785 1.597 0.1118
## bvsp.l1 0.92369 0.02866 32.229 <2e-16 ***
## hsi.l1 0.08565 0.13546 0.632 0.5278
## tnx.l1 -227.91190 355.95142 -0.640 0.5226
## const 6131.21375 2827.39156 2.169 0.0312 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 4576 on 222 degrees of freedom
## Multiple R-Squared: 0.9705, Adjusted R-squared: 0.9697
## F-statistic: 1218 on 6 and 222 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation hsi:
## ====================================
## hsi = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 -2.142e+00 9.011e-01 -2.377 0.0183 *
## ftse.l1 2.067e-01 2.144e-01 0.964 0.3361
## dji.l1 2.597e-01 1.269e-01 2.046 0.0420 *
## bvsp.l1 2.461e-03 8.123e-03 0.303 0.7622
## hsi.l1 8.816e-01 3.839e-02 22.963 <2e-16 ***
## tnx.l1 -1.647e+02 1.009e+02 -1.633 0.1039
## const 1.412e+03 8.013e+02 1.762 0.0795 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1297 on 222 degrees of freedom
## Multiple R-Squared: 0.9219, Adjusted R-squared: 0.9198
## F-statistic: 436.6 on 6 and 222 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation tnx:
## ====================================
## tnx = sp500.l1 + ftse.l1 + dji.l1 + bvsp.l1 + hsi.l1 + tnx.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sp500.l1 4.798e-04 1.671e-04 2.872 0.00448 **
## ftse.l1 6.313e-05 3.975e-05 1.588 0.11371
## dji.l1 -6.616e-05 2.353e-05 -2.811 0.00538 **
## bvsp.l1 5.801e-07 1.506e-06 0.385 0.70046
## hsi.l1 -7.999e-06 7.118e-06 -1.124 0.26233
## tnx.l1 9.610e-01 1.870e-02 51.375 < 2e-16 ***
## const 5.012e-02 1.486e-01 0.337 0.73619
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2404 on 222 degrees of freedom
## Multiple R-Squared: 0.9554, Adjusted R-squared: 0.9542
## F-statistic: 792.4 on 6 and 222 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sp500 ftse dji bvsp hsi tnx
## sp500 1.122e+04 15135.64 8.789e+04 2.475e+05 6.153e+04 0.66986
## ftse 1.514e+04 50952.19 1.310e+05 5.504e+05 1.734e+05 10.15755
## dji 8.789e+04 131042.16 7.680e+05 2.163e+06 5.104e+05 21.68695
## bvsp 2.475e+05 550370.71 2.163e+06 2.094e+07 2.839e+06 213.46260
## hsi 6.153e+04 173424.97 5.104e+05 2.839e+06 1.682e+06 36.25017
## tnx 6.699e-01 10.16 2.169e+01 2.135e+02 3.625e+01 0.05781
##
## Correlation matrix of residuals:
## sp500 ftse dji bvsp hsi tnx
## sp500 1.0000 0.6329 0.9467 0.5107 0.4478 0.0263
## ftse 0.6329 1.0000 0.6624 0.5329 0.5924 0.1872
## dji 0.9467 0.6624 1.0000 0.5394 0.4491 0.1029
## bvsp 0.5107 0.5329 0.5394 1.0000 0.4785 0.1940
## hsi 0.4478 0.5924 0.4491 0.4785 1.0000 0.1163
## tnx 0.0263 0.1872 0.1029 0.1940 0.1163 1.0000
serial.test(q9var1, lags.pt=10, type="PT.asymptotic")
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object q9var1
## Chi-squared = 497.44, df = 324, p-value = 1.803e-09
q9fore <- forecast(q9var1)
autoplot(q9fore,color='red')+
theme_bw()+
labs(x='Date',title='VAR Forecast')
q9irf <- irf(q9var1, impulse.variable = 1, response.variable = 2, t = NULL, nhor = 20, scenario = 2, draw.plot = TRUE)
q9irf
##
## Impulse response coefficients
## $sp500
## sp500 ftse dji bvsp hsi tnx
## [1,] 105.93978 142.87022 829.6474 2336.624 580.7686 0.006323012
## [2,] 104.08642 136.53173 817.9586 2254.752 534.7661 0.007745091
## [3,] 102.35704 130.68184 806.7603 2180.783 493.3995 0.008916103
## [4,] 100.74048 125.27587 796.0208 2113.885 456.1433 0.009871226
## [5,] 99.22653 120.27359 785.7100 2053.306 422.5332 0.010641916
## [6,] 97.80584 115.63874 775.7999 1998.370 392.1583 0.011256216
## [7,] 96.46986 111.33863 766.2642 1948.466 364.6554 0.011739058
## [8,] 95.21079 107.34374 757.0780 1903.046 339.7033 0.012112530
## [9,] 94.02146 103.62741 748.2180 1861.617 317.0183 0.012396139
## [10,] 92.89533 100.16556 739.6622 1823.735 296.3494 0.012607043
## [11,] 91.82642 96.93639 731.3898 1789.002 277.4748 0.012760275
##
## $ftse
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 174.75782 71.586181 1239.06774 517.5759 0.05295433
## [2,] -4.138568 161.58109 49.886320 1012.11616 505.3216 0.05376161
## [3,] -7.787749 149.37547 29.849285 816.40906 494.3325 0.05312193
## [4,] -10.989054 138.06509 11.432862 648.28698 484.3585 0.05128587
## [5,] -13.780473 127.58110 -5.410271 504.49912 475.1907 0.04847221
## [6,] -16.196810 117.86088 -20.731520 382.15843 466.6563 0.04487150
## [7,] -18.269970 108.84739 -34.585475 278.70159 458.6123 0.04064935
## [8,] -20.029234 100.48857 -47.029125 191.85337 450.9415 0.03594924
## [9,] -21.501491 92.73686 -58.121174 119.59482 443.5487 0.03089510
## [10,] -22.711457 85.54870 -67.921445 60.13501 436.3569 0.02559361
## [11,] -23.681868 78.88411 -76.490363 11.88582 429.3046 0.02013618
##
## $dji
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 273.1206 496.9534 -30.93781 0.046317564
## [2,] 2.860088 1.893837 274.1460 641.1211 37.23575 0.026975476
## [3,] 5.691700 3.543150 276.3601 772.9764 95.40967 0.009350743
## [4,] 8.491952 4.991570 279.6547 893.4621 144.77400 -0.006658581
## [5,] 11.257973 6.276388 283.9258 1003.4530 186.38401 -0.021150894
## [6,] 13.986935 7.429344 289.0741 1103.7594 221.17528 -0.034220834
## [7,] 16.676079 8.477324 295.0051 1195.1313 249.97701 -0.045959145
## [8,] 19.322732 9.442976 301.6290 1278.2618 273.52392 -0.056452596
## [9,] 21.924332 10.345259 308.8609 1353.7910 292.46683 -0.065783961
## [10,] 24.478434 11.199922 316.6205 1422.3097 307.38206 -0.074032027
## [11,] 26.982730 12.019931 324.8323 1484.3623 318.77976 -0.081271643
##
## $bvsp
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 0.000000 3700.548 231.38191 0.029740572
## [2,] -1.930584 -6.351303 -9.302223 3431.204 208.18873 0.028875548
## [3,] -3.542229 -11.931039 -17.405398 3188.535 187.62832 0.027361770
## [4,] -4.868190 -16.813440 -24.382770 2969.674 169.34919 0.025341383
## [5,] -5.938647 -21.065616 -30.306342 2772.063 153.04782 0.022936334
## [6,] -6.780992 -24.748290 -35.246541 2593.425 138.46235 0.020250803
## [7,] -7.420086 -27.916446 -39.271931 2431.729 125.36698 0.017373358
## [8,] -7.878494 -30.619908 -42.448994 2285.165 113.56715 0.014378876
## [9,] -8.176690 -32.903861 -44.841945 2152.119 102.89529 0.011330241
## [10,] -8.333256 -34.809308 -46.512605 2031.153 93.20722 0.008279862
## [11,] -8.365044 -36.373482 -47.520292 1920.988 84.37889 0.005271008
##
## $hsi
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 0.000000 0.00000 1010.9899 -0.0002753636
## [2,] 1.435576 -2.421164 2.886469 86.65198 891.3179 -0.0083514830
## [3,] 2.733037 -4.417849 6.147614 158.34160 784.5350 -0.0147598359
## [4,] 3.904486 -6.042189 9.677293 217.15869 689.2842 -0.0197415676
## [5,] 4.960750 -7.340822 13.385137 264.91870 604.3499 -0.0235068030
## [6,] 5.911528 -8.355414 17.194568 303.19762 528.6431 -0.0262384586
## [7,] 6.765535 -9.123153 21.041061 333.36259 461.1885 -0.0280955953
## [8,] 7.530620 -9.677179 24.870604 356.59858 401.1128 -0.0292163672
## [9,] 8.213869 -10.046992 28.638346 373.93184 347.6341 -0.0297206156
## [10,] 8.821697 -10.258809 32.307402 386.25036 300.0524 -0.0297121485
## [11,] 9.359926 -10.335897 35.847804 394.32175 257.7412 -0.0292807440
##
## $tnx
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 0.00000 0.00000 0.00000 0.2279005
## [2,] -2.652468 -3.632839 -21.21941 -51.94123 -37.54200 0.2190039
## [3,] -5.197404 -6.924056 -41.41528 -102.56264 -69.88054 0.2106267
## [4,] -7.638278 -9.907081 -60.64756 -151.70190 -97.60802 0.2027132
## [5,] -9.978482 -12.611941 -78.97066 -199.23661 -121.25217 0.1952148
## [6,] -12.221317 -15.065606 -96.43403 -245.07788 -141.28299 0.1880894
## [7,] -14.369989 -17.292299 -113.08272 -289.16478 -158.11902 0.1813002
## [8,] -16.427609 -19.313775 -128.95791 -331.45964 -172.13284 0.1748151
## [9,] -18.397186 -21.149578 -144.09729 -371.94402 -183.65609 0.1686062
## [10,] -20.281630 -22.817260 -158.53550 -410.61527 -192.98385 0.1626490
## [11,] -22.083751 -24.332592 -172.30440 -447.48362 -200.37865 0.1569224
##
##
## Lower Band, CI= 0.95
## $sp500
## sp500 ftse dji bvsp hsi tnx
## [1,] 87.67180 105.64198 686.0356 1251.9335 333.2026607 -0.02976849
## [2,] 84.65657 94.32346 639.5962 1199.1924 276.8532546 -0.03007894
## [3,] 80.07412 86.40567 600.4351 1082.0904 236.1338955 -0.03175331
## [4,] 75.28874 74.78711 547.3156 973.8835 177.6128020 -0.03225624
## [5,] 68.95142 65.46827 491.5602 889.9030 107.5109460 -0.03406852
## [6,] 63.18020 56.68092 444.5086 850.9414 49.2339478 -0.03457948
## [7,] 58.07716 48.84237 410.6664 816.4226 0.6297687 -0.03277463
## [8,] 53.55643 41.69330 379.9431 785.2380 -40.0416628 -0.03485847
## [9,] 49.67954 35.71793 351.8715 704.1735 -74.1888593 -0.03760500
## [10,] 46.80171 31.09442 326.0167 623.7618 -102.9531587 -0.04022502
## [11,] 44.27302 27.28015 306.2387 565.7693 -123.6969272 -0.04232599
##
## $ftse
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.00000 150.644697 21.64159 621.58589 350.66848 0.0182813626
## [2,] -11.20649 125.306059 -24.23746 320.92333 320.61449 0.0266761060
## [3,] -20.64604 100.722771 -98.26229 24.00224 268.76063 0.0216863929
## [4,] -28.84036 77.963711 -151.46345 -237.87090 207.59362 0.0101872814
## [5,] -36.21444 58.729783 -195.24535 -459.80948 158.28889 0.0002537744
## [6,] -42.44968 42.722374 -231.84773 -595.35187 121.86392 -0.0112019729
## [7,] -46.96749 29.416801 -264.70541 -709.70660 103.02499 -0.0227357324
## [8,] -50.89463 18.299094 -298.86763 -792.43383 71.78615 -0.0335189082
## [9,] -55.06818 7.599071 -328.99834 -875.37347 51.53981 -0.0408121122
## [10,] -58.64871 -1.463944 -355.36600 -971.61259 40.03140 -0.0478119534
## [11,] -61.68229 -7.289060 -378.23414 -1048.37657 32.25971 -0.0551742828
##
## $dji
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.00000 209.076660 -122.4822324 -190.48762 0.01799766
## [2,] -4.855642 -15.35153 154.087529 -0.9496938 -125.33442 -0.01569596
## [3,] -8.516714 -27.10758 105.671356 43.8932063 -91.88812 -0.03966044
## [4,] -11.155434 -35.54330 66.192426 -10.6292663 -53.39123 -0.05933611
## [5,] -12.841336 -41.01861 31.642641 -64.6050234 -56.61622 -0.07742430
## [6,] -13.744752 -43.97659 6.055385 -68.5292022 -39.06965 -0.09355937
## [7,] -13.989539 -44.86775 -12.049654 -42.7866663 -23.46237 -0.10494319
## [8,] -13.680165 -44.11640 -23.949027 4.3871388 -22.23522 -0.11464937
## [9,] -12.908303 -42.10313 -30.739474 107.6133026 -21.46526 -0.12207481
## [10,] -11.754306 -40.19242 -33.422987 221.4832800 -21.06200 -0.12757322
## [11,] -11.547529 -38.11630 -32.766876 279.6991279 -20.96703 -0.13122034
##
## $bvsp
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.00000 0.00000 3006.0319 87.44999 0.0008255821
## [2,] -6.139792 -16.57176 -43.28404 2625.6217 40.75040 -0.0052158932
## [3,] -11.438883 -28.27866 -76.90476 2278.0058 -25.82050 -0.0207498270
## [4,] -16.031866 -36.02776 -104.44734 1955.0826 -73.10441 -0.0346483210
## [5,] -19.411020 -43.28577 -126.24200 1631.4859 -105.45361 -0.0458968011
## [6,] -21.724616 -50.13202 -145.95558 1357.7656 -130.94397 -0.0519318310
## [7,] -23.836671 -55.72630 -164.11999 1088.5714 -151.48915 -0.0545535975
## [8,] -25.615449 -60.21151 -179.49905 891.1746 -166.12900 -0.0569140559
## [9,] -27.219238 -62.71661 -194.29254 718.4362 -175.71426 -0.0626666259
## [10,] -29.342766 -63.70600 -209.95064 544.5980 -181.73775 -0.0699788407
## [11,] -31.244311 -64.05893 -224.04536 421.8984 -189.20839 -0.0780948774
##
## $hsi
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.00000 0.00000 0.0000 837.34163 -0.02961213
## [2,] -7.264443 -18.93141 -68.56619 -209.5260 674.11909 -0.04383044
## [3,] -12.524131 -33.50787 -113.67718 -344.6241 506.90813 -0.05580803
## [4,] -16.189541 -44.45403 -145.05706 -423.8774 355.75323 -0.06589676
## [5,] -18.753572 -52.35488 -168.22735 -486.2266 238.97778 -0.07630683
## [6,] -20.622533 -57.72762 -187.32222 -527.4898 155.45722 -0.08728362
## [7,] -21.429503 -59.87447 -200.71794 -575.1395 79.41640 -0.09361654
## [8,] -21.838922 -60.86421 -209.01606 -610.8735 25.07763 -0.09538740
## [9,] -21.973050 -61.97440 -212.44890 -637.3400 -13.12655 -0.09536783
## [10,] -21.916752 -62.24722 -211.99210 -657.0662 -42.77202 -0.09472278
## [11,] -21.729880 -61.57116 -212.24450 -660.8228 -77.18601 -0.09645192
##
## $tnx
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.00000 0.00000 0.00000 0.0000 0.0000 0.19416860
## [2,] -10.24176 -16.82601 -89.58682 -341.5191 -113.7900 0.18189941
## [3,] -18.58195 -30.08263 -163.11684 -592.8652 -199.5260 0.16594326
## [4,] -26.16732 -40.82105 -224.43977 -780.0936 -262.9469 0.14730594
## [5,] -33.68966 -49.68201 -283.15968 -960.3194 -312.3256 0.13185333
## [6,] -40.17912 -57.45079 -336.60779 -1097.6967 -351.8164 0.11941682
## [7,] -45.46118 -62.63409 -384.23320 -1176.9920 -383.1918 0.10889765
## [8,] -50.14997 -66.81902 -423.21132 -1267.4880 -405.7145 0.09986120
## [9,] -54.60663 -71.65989 -457.92083 -1345.4348 -424.9675 0.09198432
## [10,] -59.11739 -75.75507 -489.03615 -1352.9922 -438.5061 0.08413648
## [11,] -63.28548 -79.19915 -517.10747 -1378.6468 -448.3196 0.07672959
##
##
## Upper Band, CI= 0.95
## $sp500
## sp500 ftse dji bvsp hsi tnx
## [1,] 119.6597 172.6755 950.9377 3384.915 778.8430 0.04477837
## [2,] 113.3780 154.3565 914.7410 3166.184 701.8810 0.04499446
## [3,] 110.2472 142.5439 888.6201 3013.147 643.8135 0.04412766
## [4,] 108.5778 135.9250 871.3634 2890.730 593.1874 0.04656846
## [5,] 107.5361 129.8102 862.1892 2789.596 554.9868 0.05172351
## [6,] 106.5043 123.8501 851.6781 2741.828 526.1834 0.05506807
## [7,] 105.5028 118.1527 841.6447 2721.342 496.5713 0.06087521
## [8,] 104.8535 113.5311 834.6530 2662.190 469.8179 0.06592114
## [9,] 104.2277 109.7099 827.6371 2596.401 448.6633 0.07026264
## [10,] 103.6673 107.1597 820.6470 2530.055 428.4261 0.07395631
## [11,] 103.1391 104.0225 813.7158 2468.128 408.6183 0.07705751
##
## $ftse
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 192.45716 120.73556 1878.170 638.5888 0.08228283
## [2,] 1.027686 173.76879 99.49177 1571.382 638.3527 0.08318912
## [3,] 1.782284 157.88573 102.51185 1466.413 649.7811 0.08191914
## [4,] 1.463685 149.80842 99.85379 1306.137 656.0481 0.08129638
## [5,] 1.333453 139.82417 100.27234 1168.670 647.0255 0.07847607
## [6,] 1.024221 131.28990 97.03973 1105.569 631.8196 0.07524108
## [7,] 1.430829 123.14246 100.93837 1089.683 624.9321 0.07216659
## [8,] 1.837527 115.41190 106.24479 1074.484 626.4180 0.07000426
## [9,] 2.327127 108.11437 111.50082 1059.765 621.5624 0.06806073
## [10,] 2.968588 101.41890 116.78681 1045.411 616.9993 0.06616314
## [11,] 3.740237 96.19659 123.83758 1031.369 617.6860 0.06405521
##
## $dji
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.00000 312.2080 1171.889 196.9285 0.069195898
## [2,] 9.342699 15.09875 330.1115 1310.633 262.1508 0.048455129
## [3,] 17.184416 26.56466 363.1327 1447.247 355.0215 0.035212636
## [4,] 23.703731 35.33218 394.6647 1630.952 441.3061 0.024662211
## [5,] 29.481197 42.31133 414.8126 1794.397 496.1968 0.014313616
## [6,] 34.323026 46.65686 428.0566 1917.924 529.9665 0.007331119
## [7,] 38.752516 49.35030 441.3063 2102.507 547.9911 0.002222698
## [8,] 42.125456 50.92546 462.0419 2244.079 554.8507 -0.001662238
## [9,] 45.927024 51.67076 471.8063 2354.734 553.4188 -0.003617977
## [10,] 50.526274 51.81423 481.9598 2446.228 551.9142 -0.004905251
## [11,] 54.686255 51.55907 499.2104 2521.281 554.6342 -0.006549749
##
## $bvsp
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 0.00000 4185.715 375.1736 0.06134950
## [2,] 5.144479 8.982425 52.06230 3770.612 335.8491 0.05998792
## [3,] 9.397079 15.264771 94.57603 3477.562 360.7801 0.05885006
## [4,] 12.984473 19.646906 129.32680 3268.982 359.5562 0.06302035
## [5,] 15.998684 22.608441 158.24310 3088.970 357.6332 0.06630883
## [6,] 18.441661 24.488441 182.75431 2946.032 354.3004 0.06862535
## [7,] 20.351245 25.539880 204.32137 2827.779 342.5685 0.06826260
## [8,] 22.511062 26.668727 223.36565 2718.260 334.8997 0.06658588
## [9,] 24.682095 27.376960 241.63143 2632.723 329.0051 0.06518962
## [10,] 26.910232 27.733423 260.63906 2556.432 324.3251 0.06461031
## [11,] 29.259053 27.830766 277.78046 2486.717 319.3537 0.06566614
##
## $hsi
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.000000 0.000000 0.00000 0.0000 1097.0850 0.03150080
## [2,] 8.358342 7.772935 64.42824 311.8290 944.3498 0.02444410
## [3,] 15.095141 13.809914 116.82078 545.6761 841.0200 0.02162538
## [4,] 21.059418 18.341758 159.47912 745.5842 773.6054 0.02100084
## [5,] 26.164894 21.591951 194.53453 889.2416 695.7292 0.02451468
## [6,] 30.532026 24.353195 227.46910 972.2291 623.0315 0.02997827
## [7,] 33.888894 26.171850 256.76201 1042.6697 569.9917 0.03218251
## [8,] 36.391406 26.482212 282.30930 1099.7014 515.3543 0.03337670
## [9,] 39.433443 26.243715 304.60530 1134.6940 464.3783 0.03508578
## [10,] 42.126006 25.670551 324.06697 1156.9852 425.1174 0.03654166
## [11,] 44.397454 25.814348 340.72243 1167.4267 389.9914 0.03819547
##
## $tnx
## sp500 ftse dji bvsp hsi tnx
## [1,] 0.0000000 0.00000 0.000000 0.0000 0.000000 0.2484856
## [2,] 1.3331076 4.04493 5.091220 111.6228 -1.930731 0.2373161
## [3,] 2.2623472 7.35084 7.598049 177.9423 -5.956234 0.2268524
## [4,] 2.8781254 10.02740 8.431823 211.3077 -7.584207 0.2168197
## [5,] 2.6858765 12.16577 10.278922 225.4307 -8.425700 0.2110014
## [6,] 2.0557893 13.84230 11.739242 237.3949 -6.536527 0.2034759
## [7,] 1.1846028 15.12149 12.847018 243.2798 -4.083612 0.1957063
## [8,] 0.4455795 15.64742 15.072230 261.7184 -3.694883 0.1878344
## [9,] 0.1922414 15.65785 18.539219 278.0198 1.274616 0.1815783
## [10,] 0.4663877 16.20961 21.406126 285.1721 6.613124 0.1764238
## [11,] 0.7496845 16.52239 23.653635 259.0520 11.405612 0.1713822
plot(q9irf)
