Tutorial: Volatility Models
by Kevin Kotzé
1 Conditional Heteroscedastic Model - S&P500
The first example considers the use of an ARCH/GARCH modelling framework for the S&P500 index. This example is contained in the file tut12a-sp500.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the fGarch and tsm packages, so we make use of the library command.
library(fGarch)
library(tsm)If you need install these packages run the following routine: install.packages("fGarch"). To install the tsm package we make use of the commands, library(devtools) and devtools::install_github("KevinKotze/tsm"). As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the S&P500 monthly excess returns between 1926 and 1991.
dat <- read.csv(file = "sp500.csv")
sp500 <- ts(dat$sp500, start = c(1926, 1), freq = 12)To ensure that this has all been completed correctly, we can plot the data.
plot(sp500)
plot(sp500^2)
The first part of the ARCH modelling process is to specify an appropriate mean equation that will be able to explain the serial dependence in the first moment of the process. To complete this task, we can make use of an autocorrelation function.
res1 <- ac(sp500)
Box.test(sp500, lag = 12, type = "Ljung")##
## Box-Ljung test
##
## data: sp500
## X-squared = 34.372, df = 12, p-value = 0.0005892In this case it is noted that there is some possible persistence and the model for the mean equation may take the form of an AR(3) or a MA(3), although the coefficients in both the autocorrelation and partial autocorrelation coefficients are small. We can also have a quick look at the degree of persistence in the volatility of the process.
res2 <- ac(abs(sp500))
res3 <- ac(sp500^2)
where we note that there is definitely some persistence. To test for the significance of a mean that differs from zero we can use a t-test, which suggests that the mean is different from zero. Therefore, we demean the time series variable and perform further tests for volatility effects. These include the Box-Ljung and Arch test of Engle (1982).
t.test(sp500)##
## One Sample t-test
##
## data: sp500
## t = 2.9573, df = 791, p-value = 0.003196
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 0.002065495 0.010220616
## sample estimates:
## mean of x
## 0.006143056at <- sp500 - mean(sp500)
Box.test(abs(at), lag = 12, type = "Ljung")##
## Box-Ljung test
##
## data: abs(at)
## X-squared = 557.08, df = 12, p-value < 2.2e-16archTest(at, 12)##
## Call:
## lm(formula = atsq ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.046462 -0.001960 -0.001089 0.000496 0.122800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0008310 0.0004186 1.985 0.04749 *
## x1 0.1112910 0.0360675 3.086 0.00210 **
## x2 0.0632316 0.0360710 1.753 0.08001 .
## x3 0.1500567 0.0361453 4.151 3.67e-05 ***
## x4 0.0344645 0.0358201 0.962 0.33628
## x5 -0.0851777 0.0347166 -2.454 0.01437 *
## x6 -0.0062502 0.0348527 -0.179 0.85772
## x7 -0.0023330 0.0348531 -0.067 0.94665
## x8 0.2463731 0.0347162 7.097 2.91e-12 ***
## x9 0.2010920 0.0358199 5.614 2.76e-08 ***
## x10 -0.0010279 0.0361446 -0.028 0.97732
## x11 0.1100626 0.0360725 3.051 0.00236 **
## x12 -0.0588513 0.0360674 -1.632 0.10315
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01012 on 767 degrees of freedom
## Multiple R-squared: 0.2457, Adjusted R-squared: 0.2339
## F-statistic: 20.82 on 12 and 767 DF, p-value: < 2.2e-16These results are similar to what was provided earlier, where the F-statistics and p-values confirm that ARCH effects are present. We can now consider the results of a few competing models, which include an AR(3), MA(3), AR(3) with GARCH(1,1), and a GARCH(1,1) with a student-t distribution.
m1 <- arima(sp500, order = c(0, 0, 3))
m1##
## Call:
## arima(x = sp500, order = c(0, 0, 3))
##
## Coefficients:
## ma1 ma2 ma3 intercept
## 0.0949 0.0096 -0.1415 0.0062
## s.e. 0.0348 0.0355 0.0357 0.0020
##
## sigma^2 estimated as 0.003321: log likelihood = 1136.3, aic = -2262.6m2 <- arima(sp500, order = c(3, 0, 0))
m2##
## Call:
## arima(x = sp500, order = c(3, 0, 0))
##
## Coefficients:
## ar1 ar2 ar3 intercept
## 0.0890 -0.0238 -0.1229 0.0062
## s.e. 0.0353 0.0355 0.0353 0.0019
##
## sigma^2 estimated as 0.00333: log likelihood = 1135.25, aic = -2260.5m3 <- garchFit(sp500 ~ arma(3, 0) + garch(1, 1), data = sp500,
trace = F)
summary(m3)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = sp500 ~ arma(3, 0) + garch(1, 1), data = sp500,
## trace = F)
##
## Mean and Variance Equation:
## data ~ arma(3, 0) + garch(1, 1)
## <environment: 0x4477fc0>
## [data = sp500]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3
## 7.7077e-03 3.1968e-02 -3.0261e-02 -1.0649e-02
## omega alpha1 beta1
## 7.9746e-05 1.2425e-01 8.5302e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 7.708e-03 1.607e-03 4.798 1.61e-06 ***
## ar1 3.197e-02 3.837e-02 0.833 0.40473
## ar2 -3.026e-02 3.841e-02 -0.788 0.43076
## ar3 -1.065e-02 3.756e-02 -0.284 0.77677
## omega 7.975e-05 2.810e-05 2.838 0.00454 **
## alpha1 1.242e-01 2.247e-02 5.529 3.22e-08 ***
## beta1 8.530e-01 2.183e-02 39.075 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 1272.179 normalized: 1.606287
##
## Description:
## Wed Jun 6 07:59:30 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 73.04843 1.110223e-16
## Shapiro-Wilk Test R W 0.9857968 5.961505e-07
## Ljung-Box Test R Q(10) 11.56744 0.3150483
## Ljung-Box Test R Q(15) 17.78746 0.2740041
## Ljung-Box Test R Q(20) 24.11916 0.2372259
## Ljung-Box Test R^2 Q(10) 10.31614 0.4132084
## Ljung-Box Test R^2 Q(15) 14.22819 0.5082978
## Ljung-Box Test R^2 Q(20) 16.79404 0.6663039
## LM Arch Test R TR^2 13.34305 0.3446074
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -3.194897 -3.153581 -3.195051 -3.179018m4 <- garchFit(sp500 ~ garch(1, 1), data = sp500, cond.dist = "std",
trace = F)
summary(m4)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = sp500 ~ garch(1, 1), data = sp500, cond.dist = "std",
## trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x318fff8>
## [data = sp500]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.00845504 0.00012485 0.11302582 0.84220210
## shape
## 7.00318063
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 8.455e-03 1.515e-03 5.581 2.39e-08 ***
## omega 1.248e-04 4.519e-05 2.763 0.00573 **
## alpha1 1.130e-01 2.693e-02 4.198 2.70e-05 ***
## beta1 8.422e-01 3.186e-02 26.432 < 2e-16 ***
## shape 7.003e+00 1.680e+00 4.169 3.06e-05 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 1283.417 normalized: 1.620476
##
## Description:
## Wed Jun 6 07:59:30 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 99.61249 0
## Shapiro-Wilk Test R W 0.9836345 9.72802e-08
## Ljung-Box Test R Q(10) 11.37961 0.3287173
## Ljung-Box Test R Q(15) 18.2163 0.2514649
## Ljung-Box Test R Q(20) 24.91842 0.2045699
## Ljung-Box Test R^2 Q(10) 10.52266 0.3958941
## Ljung-Box Test R^2 Q(15) 16.14586 0.3724248
## Ljung-Box Test R^2 Q(20) 18.93325 0.5261686
## LM Arch Test R TR^2 14.88667 0.247693
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -3.228325 -3.198814 -3.228404 -3.216983Note that either the AR(3) or MA(3) would appear to be appropriate, however, when we include a GARCH specification with the AR(3) we note that all the coefficients from the mean equation appear to be insignificant. This is due to the fact that the persistence in the second moment is also reflected by persistence in the first moment in the autocorrelation functions. As such we proceed to model a simple GARCH(1,1), where we note that all the coefficients are significant.
To ensure that there is no remaining serial correlation in the residuals of the GARCH(1,1) we make use of autocorrelation functions once again.
check1 <- ac(residuals(m4, standardize = T))
check2 <- ac((residuals(m4, standardize = T)^2)^0.5)
check3 <- ac((residuals(m4, standardize = T)^2))
par(mfrow = c(1, 1))
plot.ts(residuals(m4, standardize = T))
where we note that the mean, absolute value and variance do not display notable degrees of serial correlation. In addition, the plot of the residuals would also appear to represent white noise.
Should we wish to generate a forecast for this model then we can do so by making use of the predict command.
predict(m4, 5)## meanForecast meanError standardDeviation
## 1 0.008455044 0.05330089 0.05330089
## 2 0.008455044 0.05327886 0.05327886
## 3 0.008455044 0.05325781 0.05325781
## 4 0.008455044 0.05323769 0.05323769
## 5 0.008455044 0.05321847 0.053218472 Conditional Heteroscedastic Model - Intel Share Price
The second example considers the use of an ARCH model for the Intel share price. This example is contained in the file tut12b-intel.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the fGarch and tsm packages, so we make use of the library command.
library(fGarch)
library(tsm)If you need install these packages run the following routine: install.packages("fGarch"). To install the tsm package we make use of the commands, library(devtools) and devtools::install_github("KevinKotze/tsm"). As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the Intel log returns between 1973 and 2009.
dat <- read.csv(file = "intel.csv", sep = ";")
intel <- ts((log(dat$intc + 1)), start = c(1973, 1), freq = 12)To ensure that this has all been completed correctly, we can plot the data.
plot(intel)
plot(intel^2)
The first part of the ARCH modelling process is to specify an appropriate mean equation that will be able to explain the serial dependence in the first moment of the process. To complete this task, we can make use of an autocorrelation function.
res1 <- ac(intel)
Box.test(intel, lag = 12, type = "Ljung")##
## Box-Ljung test
##
## data: intel
## X-squared = 18.676, df = 12, p-value = 0.09665In this case it is noted that there is not much persistence in the mean equation. We can also have a quick look at the degree of persistence in the volatility of the process.
res2 <- ac(abs(intel))
res3 <- ac(intel^2)
Box.test(abs(intel), lag = 12, type = "Ljung")##
## Box-Ljung test
##
## data: abs(intel)
## X-squared = 124.91, df = 12, p-value < 2.2e-16where we note that there is definitely some persistence. To test for the significance of a mean that differs from zero we can use a t-test, which suggests that the mean is different from zero. Therefore, we demean the time series variable and perform further tests for volatility effects. These include the Box-Ljung and Arch test of Engle (1982).
t.test(intel)##
## One Sample t-test
##
## data: intel
## t = 2.3788, df = 443, p-value = 0.01779
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 0.00249032 0.02616428
## sample estimates:
## mean of x
## 0.0143273at <- intel - mean(intel)
Box.test(at^2, lag = 12, type = "Ljung")##
## Box-Ljung test
##
## data: at^2
## X-squared = 92.939, df = 12, p-value = 1.332e-14archTest(at, 12)##
## Call:
## lm(formula = atsq ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.07440 -0.01153 -0.00658 0.00395 0.35255
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.005977 0.002249 2.658 0.008162 **
## x1 0.093817 0.048147 1.949 0.052013 .
## x2 0.153085 0.048102 3.183 0.001569 **
## x3 0.146087 0.048614 3.005 0.002815 **
## x4 0.023539 0.049126 0.479 0.632075
## x5 0.007347 0.049107 0.150 0.881139
## x6 0.010342 0.047027 0.220 0.826050
## x7 0.057183 0.047027 1.216 0.224681
## x8 0.014320 0.047079 0.304 0.761149
## x9 0.007157 0.046968 0.152 0.878965
## x10 -0.019742 0.046566 -0.424 0.671810
## x11 -0.057537 0.046041 -1.250 0.212116
## x12 0.161945 0.045965 3.523 0.000473 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03365 on 419 degrees of freedom
## Multiple R-squared: 0.1248, Adjusted R-squared: 0.0997
## F-statistic: 4.978 on 12 and 419 DF, p-value: 9.742e-08These results are similar to what was provided earlier, where the F-statistics and p-values confirm that ARCH effects are present. We can now consider the results of a few competing models, which include an AR(3), MA(3), AR(3) with GARCH(1,1), and a GARCH(1,1) with a student-t distribution.
m0 <- garchFit(intel ~ garch(3, 0), data = intel, trace = F)
summary(m0)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = intel ~ garch(3, 0), data = intel, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(3, 0)
## <environment: 0x39a91e8>
## [data = intel]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 alpha3
## 0.012567 0.010421 0.232889 0.075069 0.051994
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.012567 0.005515 2.279 0.0227 *
## omega 0.010421 0.001238 8.418 <2e-16 ***
## alpha1 0.232889 0.111541 2.088 0.0368 *
## alpha2 0.075069 0.047305 1.587 0.1125
## alpha3 0.051994 0.045139 1.152 0.2494
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 303.9607 normalized: 0.6845963
##
## Description:
## Wed Jun 6 07:59:31 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 203.362 0
## Shapiro-Wilk Test R W 0.9635971 4.898647e-09
## Ljung-Box Test R Q(10) 9.260782 0.5075463
## Ljung-Box Test R Q(15) 19.36748 0.1975619
## Ljung-Box Test R Q(20) 20.46983 0.4289059
## Ljung-Box Test R^2 Q(10) 7.322136 0.6947234
## Ljung-Box Test R^2 Q(15) 27.41532 0.02552908
## Ljung-Box Test R^2 Q(20) 28.15113 0.1058698
## LM Arch Test R TR^2 25.23347 0.01375447
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.346670 -1.300546 -1.346920 -1.328481m1 <- garchFit(intel ~ garch(1, 0), data = intel, trace = F)
summary(m1)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = intel ~ garch(1, 0), data = intel, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 0)
## <environment: 0x215b4d8>
## [data = intel]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1
## 0.013130 0.011046 0.374976
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.013130 0.005318 2.469 0.01355 *
## omega 0.011046 0.001196 9.238 < 2e-16 ***
## alpha1 0.374976 0.112620 3.330 0.00087 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 299.9247 normalized: 0.675506
##
## Description:
## Wed Jun 6 07:59:31 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 144.3783 0
## Shapiro-Wilk Test R W 0.9678175 2.670321e-08
## Ljung-Box Test R Q(10) 12.12248 0.2769429
## Ljung-Box Test R Q(15) 22.30705 0.1000019
## Ljung-Box Test R Q(20) 24.33412 0.2281016
## Ljung-Box Test R^2 Q(10) 16.57807 0.08423723
## Ljung-Box Test R^2 Q(15) 37.44349 0.001089733
## Ljung-Box Test R^2 Q(20) 38.81395 0.007031558
## LM Arch Test R TR^2 27.32897 0.006926821
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.337499 -1.309824 -1.337589 -1.326585m2 <- garchFit(intel ~ garch(1, 1), data = intel, trace = F)
summary(m2)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = intel ~ garch(1, 1), data = intel, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x2530408>
## [data = intel]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.01126568 0.00091902 0.08643831 0.85258554
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0112657 0.0053931 2.089 0.03672 *
## omega 0.0009190 0.0003888 2.364 0.01808 *
## alpha1 0.0864383 0.0265439 3.256 0.00113 **
## beta1 0.8525855 0.0394322 21.622 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 312.3307 normalized: 0.7034475
##
## Description:
## Wed Jun 6 07:59:31 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 174.904 0
## Shapiro-Wilk Test R W 0.9709615 1.030282e-07
## Ljung-Box Test R Q(10) 8.016844 0.6271916
## Ljung-Box Test R Q(15) 15.5006 0.4159946
## Ljung-Box Test R Q(20) 16.41549 0.6905368
## Ljung-Box Test R^2 Q(10) 0.8746345 0.9999072
## Ljung-Box Test R^2 Q(15) 11.35935 0.7267295
## Ljung-Box Test R^2 Q(20) 12.55994 0.8954573
## LM Arch Test R TR^2 10.51401 0.5709617
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.388877 -1.351978 -1.389037 -1.374326In terms of the notation, mu represents the constant in mean equation and alpha refers to the ARCH terms. Note that alpha2 and alpha3 appear to be insignificant in m0, while the Qstat is fairly large, which would suggest that there is no serial correlation in residual. In the second model, which represents an ARCH(1) the coefficients appear significant and the Qstat is acceptable at the 5% level of significance, while the Qstat for the volatility is relatively small. In the third model, which takes the form of a GARCH(1,1), we note that all the coefficients are significant.
To ensure that there is no remaining serial correlation in the residuals of the GARCH(1,1) we make use of autocorrelation functions once again.
check1 <- ac(residuals(m2, standardize = T))
check2 <- ac((residuals(m2, standardize = T)^2)^0.5)
check3 <- ac((residuals(m2, standardize = T)^2))
par(mfrow = c(1, 1))
plot.ts(residuals(m2, standardize = T))
where we note that the mean, absolute value and variance do not display notable degrees of serial correlation. In addition, the plot of the residuals would also appear to represent white noise.
As an alternative we could make use of a Students t-distribution or a skewed Students t-distribution.
m3 <- garchFit(intel ~ garch(1, 1), data = intel, cond.dist = "std",
trace = F)
summary(m3)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = intel ~ garch(1, 1), data = intel, cond.dist = "std",
## trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x54cebc0>
## [data = intel]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 beta1 shape
## 0.0165075 0.0011576 0.1059030 0.8171313 6.7723503
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0165075 0.0051031 3.235 0.001217 **
## omega 0.0011576 0.0005782 2.002 0.045286 *
## alpha1 0.1059030 0.0372047 2.846 0.004420 **
## beta1 0.8171313 0.0580141 14.085 < 2e-16 ***
## shape 6.7723503 1.8572388 3.646 0.000266 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 326.2264 normalized: 0.734744
##
## Description:
## Wed Jun 6 07:59:32 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 203.4933 0
## Shapiro-Wilk Test R W 0.9687607 3.970603e-08
## Ljung-Box Test R Q(10) 7.877778 0.6407741
## Ljung-Box Test R Q(15) 15.5522 0.4124197
## Ljung-Box Test R Q(20) 16.50475 0.6848581
## Ljung-Box Test R^2 Q(10) 1.066054 0.9997694
## Ljung-Box Test R^2 Q(15) 11.49875 0.7165045
## Ljung-Box Test R^2 Q(20) 12.61496 0.8932865
## LM Arch Test R TR^2 10.80739 0.5454935
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.446966 -1.400841 -1.447215 -1.428776m4 <- garchFit(intel ~ garch(1, 1), data = intel, cond.dist = "sstd",
trace = F)
summary(m4)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = intel ~ garch(1, 1), data = intel, cond.dist = "sstd",
## trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x3273fa0>
## [data = intel]
##
## Conditional Distribution:
## sstd
##
## Coefficient(s):
## mu omega alpha1 beta1 skew
## 0.0133343 0.0011621 0.1049289 0.8177875 0.8717220
## shape
## 7.2344225
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0133343 0.0053430 2.496 0.012572 *
## omega 0.0011621 0.0005587 2.080 0.037519 *
## alpha1 0.1049289 0.0358860 2.924 0.003456 **
## beta1 0.8177875 0.0559863 14.607 < 2e-16 ***
## skew 0.8717220 0.0629129 13.856 < 2e-16 ***
## shape 7.2344225 2.1018038 3.442 0.000577 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 328.0995 normalized: 0.7389628
##
## Description:
## Wed Jun 6 07:59:32 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 195.2178 0
## Shapiro-Wilk Test R W 0.9692506 4.892686e-08
## Ljung-Box Test R Q(10) 7.882126 0.6403496
## Ljung-Box Test R Q(15) 15.62496 0.4074054
## Ljung-Box Test R Q(20) 16.5774 0.6802193
## Ljung-Box Test R^2 Q(10) 1.078429 0.9997569
## Ljung-Box Test R^2 Q(15) 11.95155 0.6826923
## Ljung-Box Test R^2 Q(20) 13.03792 0.8757513
## LM Arch Test R TR^2 11.18826 0.5128574
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.450899 -1.395550 -1.451257 -1.429071where we note that all the coefficients appear significant and the Q-statistic is also acceptable.
3 Garch-in-Mean Model - S&P500
The third example considers the use of an GARCH-in-Mean model, which is applied to data for the S&P500 index. This example is contained in the file tut12c-garchM.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the tsm package, so we make use of the library command.
library(tsm)To install the tsm package we make use of the commands, library(devtools) and devtools::install_github("KevinKotze/tsm"). As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the S&P500 monthly excess returns between 1926 and 1991.
dat <- read.csv(file = "sp500.csv")
sp500 <- ts(dat$sp500, start = c(1926, 1), freq = 12)To model this data we make use of the command, where type=1 refers to the typical GARCH-in-Mean model:
GARCHmean <- garchM(as.numeric(sp500) * 100, 1)## 0: 2380.0229: 0.422452 0.00561297 0.806149 0.121976 0.854361
## 3: 2379.9525: 0.427597 0.00652845 0.806107 0.121466 0.855756
## 6: 2378.4838: 0.606825 0.0375083 0.801889 0.126193 0.846832
## 9: 2377.9587: 0.673992 0.00167148 0.798192 0.125514 0.851917
## 12: 2377.8601: 0.686589 0.0416544 0.800771 0.122248 0.854362
## 15: 2377.7967: 0.728617 0.0427501 0.787469 0.122399 0.855074
## Maximized log-likehood: 2377.797
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## mu 0.7286167 0.1540169 4.73076 2.2368e-06 ***
## gamma 0.0427501 0.1407062 0.30383 0.761261
## omega 0.7874686 0.2805923 2.80645 0.005009 **
## alpha 0.1223987 0.0221922 5.51539 3.4800e-08 ***
## beta 0.8550744 0.0216905 39.42159 < 2.22e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In this case the coefficient names are as follows: mu is the mean in the mean equation, gamma refers to the relationship between mean and the measure of volatility (which may be termed the risk-premium), omega is the constant in the volatility equation, alpha refers to the ARCH(1) coefficient and beta is the coefficient for the GARCH(1) term. Note that gamma is insiginficant, which would suggest that there is no risk-premium.
4 GJR-Garch Model - Intel and IBM Share Price
The fourth example considers the use of a threshold GARCH model, which is applied to data for the Intel share price. This example is contained in the file tut12d-GJR.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the tsm package, so we make use of the library command.
library(fGarch)As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the Intel logarithmic returns.
dat <- read.csv(file = "intel.csv", sep = ";")
intel <- ts((log(dat$intc + 1)), start = c(1973, 1), freq = 12)To model this data we make use of an aparch model, where delta=2 would refer to the GJR-GARCH model. In this case we make use of both a normal and students-t distribution for the errors.
mod.intel.n <- garchFit(~aparch(1, 1), data = intel, trace = F,
delta = 2, include.delta = F)
summary(mod.intel.n)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~aparch(1, 1), data = intel, delta = 2, include.delta = F,
## trace = F)
##
## Mean and Variance Equation:
## data ~ aparch(1, 1)
## <environment: 0x44f9050>
## [data = intel]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 gamma1
## 0.01205856 0.00081818 0.08965604 -0.09613466
## beta1
## 0.85673669
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0120586 0.0054562 2.210 0.027100 *
## omega 0.0008182 0.0003679 2.224 0.026160 *
## alpha1 0.0896560 0.0270726 3.312 0.000927 ***
## gamma1 -0.0961347 0.1198266 -0.802 0.422390
## beta1 0.8567367 0.0377736 22.681 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 312.6275 normalized: 0.7041161
##
## Description:
## Wed Jun 6 07:59:34 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 146.8649 0
## Shapiro-Wilk Test R W 0.9733289 3.011759e-07
## Ljung-Box Test R Q(10) 8.103169 0.6187601
## Ljung-Box Test R Q(15) 15.73172 0.4001025
## Ljung-Box Test R Q(20) 16.62711 0.6770373
## Ljung-Box Test R^2 Q(10) 0.7591979 0.9999521
## Ljung-Box Test R^2 Q(15) 11.67257 0.7036244
## Ljung-Box Test R^2 Q(20) 12.93901 0.8799855
## LM Arch Test R TR^2 10.69651 0.5550912
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.385710 -1.339585 -1.385959 -1.367520mod.intel.s <- garchFit(~aparch(1, 1), data = intel, trace = F,
delta = 2, include.delta = F, cond.dist = "std")
summary(mod.intel.s)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~aparch(1, 1), data = intel, delta = 2, cond.dist = "std",
## include.delta = F, trace = F)
##
## Mean and Variance Equation:
## data ~ aparch(1, 1)
## <environment: 0x573ec60>
## [data = intel]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 gamma1 beta1
## 0.0162315 0.0012378 0.1029181 0.0692277 0.8127180
## shape
## 6.7174202
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0162315 0.0051496 3.152 0.001622 **
## omega 0.0012378 0.0006262 1.977 0.048070 *
## alpha1 0.1029181 0.0374525 2.748 0.005997 **
## gamma1 0.0692277 0.1569563 0.441 0.659167
## beta1 0.8127180 0.0594966 13.660 < 2e-16 ***
## shape 6.7174202 1.8262046 3.678 0.000235 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 326.3315 normalized: 0.7349808
##
## Description:
## Wed Jun 6 07:59:34 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 228.2265 0
## Shapiro-Wilk Test R W 0.9669724 1.882266e-08
## Ljung-Box Test R Q(10) 7.796908 0.648667
## Ljung-Box Test R Q(15) 15.382 0.4242672
## Ljung-Box Test R Q(20) 16.37239 0.6932698
## Ljung-Box Test R^2 Q(10) 1.169358 0.999649
## Ljung-Box Test R^2 Q(15) 11.13286 0.7431206
## Ljung-Box Test R^2 Q(20) 12.22266 0.908199
## LM Arch Test R TR^2 10.55077 0.5677586
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.442935 -1.387586 -1.443293 -1.421107Note that as gamma is not significant it suggests that there are no significant leverage effects in this particular variable.
As an alternative we can make use of data for the monthly returns from the IBM share price.
rm(list = ls())
graphics.off()We can then load the data.
dat <- read.csv(file = "ibm.csv")
ibm <- ts((log(dat$ibm + 1) * 100), start = c(1926, 1),
freq = 12)
plot(ibm)
Thereafter, we can make use of similar model structures.
mod.ibm.n <- garchFit(~aparch(1, 1), data = ibm, trace = F,
delta = 2, include.delta = F)
summary(mod.ibm.n)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~aparch(1, 1), data = ibm, delta = 2, include.delta = F,
## trace = F)
##
## Mean and Variance Equation:
## data ~ aparch(1, 1)
## <environment: 0x3117368>
## [data = ibm]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 gamma1 beta1
## 1.18659 4.33663 0.10767 0.22732 0.79468
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.18659 0.20019 5.927 3.08e-09 ***
## omega 4.33663 1.34161 3.232 0.00123 **
## alpha1 0.10767 0.02548 4.225 2.39e-05 ***
## gamma1 0.22732 0.10018 2.269 0.02326 *
## beta1 0.79468 0.04554 17.449 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -3329.177 normalized: -3.302755
##
## Description:
## Wed Jun 6 07:59:35 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 67.07416 2.775558e-15
## Shapiro-Wilk Test R W 0.9870135 8.590932e-08
## Ljung-Box Test R Q(10) 16.90603 0.07646941
## Ljung-Box Test R Q(15) 24.19033 0.06193097
## Ljung-Box Test R Q(20) 31.89097 0.04447406
## Ljung-Box Test R^2 Q(10) 4.591691 0.9167342
## Ljung-Box Test R^2 Q(15) 11.98464 0.6801912
## Ljung-Box Test R^2 Q(20) 14.79531 0.7879979
## LM Arch Test R TR^2 7.162971 0.8466584
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 6.615430 6.639814 6.615381 6.624694mod.ibm.s <- garchFit(~aparch(1, 1), data = ibm, trace = F,
delta = 2, include.delta = F, cond.dist = "std")
summary(mod.ibm.s)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~aparch(1, 1), data = ibm, delta = 2, cond.dist = "std",
## include.delta = F, trace = F)
##
## Mean and Variance Equation:
## data ~ aparch(1, 1)
## <environment: 0x53a98b8>
## [data = ibm]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 gamma1 beta1 shape
## 1.20476 3.98975 0.10468 0.22366 0.80711 6.67329
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.20476 0.18715 6.437 1.22e-10 ***
## omega 3.98975 1.45331 2.745 0.006046 **
## alpha1 0.10468 0.02793 3.747 0.000179 ***
## gamma1 0.22366 0.11595 1.929 0.053738 .
## beta1 0.80711 0.04825 16.727 < 2e-16 ***
## shape 6.67329 1.32779 5.026 5.01e-07 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -3310.21 normalized: -3.283938
##
## Description:
## Wed Jun 6 07:59:35 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 67.82336 1.887379e-15
## Shapiro-Wilk Test R W 0.9869701 8.215318e-08
## Ljung-Box Test R Q(10) 16.91352 0.07629961
## Ljung-Box Test R Q(15) 24.08691 0.06363224
## Ljung-Box Test R Q(20) 31.75305 0.04600187
## Ljung-Box Test R^2 Q(10) 4.553248 0.9189583
## Ljung-Box Test R^2 Q(15) 11.66891 0.7038973
## Ljung-Box Test R^2 Q(20) 14.18533 0.8209764
## LM Arch Test R TR^2 6.771675 0.872326
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 6.579782 6.609042 6.579711 6.590898In this case the results suggest that there could be leverage effects in the data.
5 NGarch Model - USD/EU exchange rate data
The sixth example considers the use of an NGARCH model, which is applied to data for the USD/EU exchange rate. This example is contained in the file tut12f-Ngarch.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the tsm package, so we make use of the library command.
library(tsm)As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the monthly returns of the USD/EU exchange rate.
dat <- read.csv(file = "ex_usd_eu.csv", sep = ";")
fx <- log(dat$rate)
eu <- diff(fx) * 100To model this data we make use of the Ngarch command.
m1 <- Ngarch(eu)##
## Estimation results of NGARCH(1,1) model:
## estimates: -0.001094043 0.002366721 0.9618047 0.02118565 0.7309616
## std.errors: 0.01080893 0.000580552 0.006045803 0.003604727 0.2501548
## t-ratio: -0.1012166 4.076674 159.0863 5.877186 2.922037names(m1)## [1] "residuals" "volatility"res <- m1$residuals
vol <- m1$volatility
resi <- res/volWe can then compare the residuals to the actual data, with the aid of the following commands:
par(mfcol = c(2, 1), mar = c(2.2, 2.2, 1, 2.2), cex = 0.8)
plot.ts(eu)
plot.ts(resi)
Thereafter, we can check for remaining serial correlation in the residuals.
Box.test(resi, lag = 10, type = "Ljung")##
## Box-Ljung test
##
## data: resi
## X-squared = 14.776, df = 10, p-value = 0.1404Box.test(abs(resi), lag = 10, type = "Ljung")##
## Box-Ljung test
##
## data: abs(resi)
## X-squared = 19.632, df = 10, p-value = 0.03293While there would appear to be no serial correlation in the mean, the results for the volatility equation could be better.
6 IGarch Model - Intel Share Price
The sixth example considers the use of an IGARCH model, which is applied to data for the Intel share price. This example is contained in the file tut12g-Igarch.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the tsm package, so we make use of the library command.
library(tsm)As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the monthly logarithmic returns of the Intel share price.
dat <- read.csv(file = "intel.csv", sep = ";")
intel <- log(dat$intc + 1)
plot.ts(intel)
To model this data we make use of the Igarch command.
mm <- Igarch(intel, include.mean = T)## Estimates: 0.00939309 0.9269988
## Maximized log-likehood: -302.8558
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## mu 0.00939309 0.00555180 1.6919 0.090665 .
## beta 0.92699884 0.01436914 64.5131 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1mm <- Igarch(intel)## Estimates: 0.9217433
## Maximized log-likehood: -301.412
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## beta 0.9217433 0.0155534 59.2633 < 2.22e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We can then inspect the residuals with the aid of the command:
par(mfcol = c(1, 1), mar = c(2.2, 2.2, 1, 2.2), cex = 0.8)
plot(mm$volatility)
7 Multivariate Garch Model
The seventh example considers the use of a multivariate GARCH model, which is applied to data for the S&P500 and FTSE indices. This example is contained in the file tut12h-MVgarch.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the ccgarch package, so we make use of the library command.
library(ccgarch)As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the monthly logarithmic returns of the Intel share price.
dat <- read.csv(file = "sp_ftse.csv")
mv.dat <- cbind(dat$SP500, dat$FTSE)
plot.ts(mv.dat[, 1])
plot.ts(mv.dat[, 2])
To model a diagnol vech model we would firstly need to specify a number of starting values. In this case we use:
c <- c(0.003, 0.001)
A <- diag(c(0.2, 0.15))
B <- diag(c(0.75, 0.8))
dcc.para <- c(0.01, 0.98)where c refers to the initial values for constants, A refers to the initial values for ARCH parameters, B refers to the initial values for GARCH parameters and dcc.para refers to the initial values for conditional correlation parameters. To estimate the model and consider the values for the coefficients, we would then make use of the following commands.
dcc.results <- dcc.estimation(inia = c, iniA = A, iniB = B,
ini.dcc = dcc.para, dvar = mv.dat, model = "diagonal")## ****************************************************************
## * Estimation has been completed. *
## * The outputs are saved in a list with components: *
## * out : the estimates and their standard errors *
## * loglik : the value of the log-likelihood at the estimates *
## * h : a matrix of estimated conditional variances *
## * DCC : a matrix of DCC estimates *
## * std.resid : a matrix of the standardised residuals *
## * first : the results of the first stage estimation *
## * second : the results of the second stage estimation *
## ****************************************************************dcc.results$out## a1 a2 A11
## estimates 0.011966947 0.01248153 0.07779838
## std.err 0.005765007 0.01293381 0.01174175
## A22 B11 B22 dcc alpha
## estimates 0.109739228 0.9140172 0.88518077 0.008492336
## std.err 0.003979286 0.0149225 0.01369105 0.003112054
## dcc beta
## estimates 0.989691605
## std.err 0.004649934Note that all the parameters are positive and the standard errors are relatively small. We can also extract the respective measures of volatility, where h11_t is the standardized volatility for the S&P500, h22_t is the standardized volatility for the FTSE, and h12.dcc refers to the cross correlation.
h11 <- sqrt(252 * dcc.results$h[, 1]) # h11_t series which is standardized
h22 <- sqrt(252 * dcc.results$h[, 2]) # h22_t series which is standardized
h12.dcc <- (dcc.results$DCC[, 2]/(sqrt(dcc.results$h[, 1])) *
(sqrt(dcc.results$h[, 2])))We can then plot these results, where we note that the correlation in volatility has increased form observation 1000. In addition, the plot of the residual would suggest that the residuals represent white noise.
plot.ts(cbind(h11, h22, h12.dcc), ylab = c("sp500", "ftse",
"dcc"))
plot.ts(dcc.results$std.resid)
8 Stochastic Volatility Model
The last example considers the use of an Stochastic Volatility model, which is applied to data for the USD/ZAR exchange rate. This example is contained in the file tut12i-SV.R. To start off we can clear all the variables from the current environment and close all the plots.
rm(list = ls())
graphics.off()Thereafter, we will need to make use of the stochvol package, so we make use of the library command.
library(stochvol)As this data is contained in a .csv file we need to set the directory to tell R where to find the datafile. You will need to change the following extension to ensure that the correct path is specified.
setwd("C:\\Users\\image")This allows us to load the data and create the time series objects for the monthly returns of the USD/ZAR exchange rate. In this case we are going to use the model to forecast future exchange rate, so we make use of a reduced insample period.
dat <- read.csv(file = "ex_data.csv")
plot.ts(dat$ZAR)
zar.logret <- diff(log(dat$ZAR))
zar.dat <- zar.logret - mean(zar.logret)
zar.ins <- zar.dat[1:3622]
zar.out <- zar.dat[3623:3652]To model this data we make use of the svsample command, which executes the MCMC sampler for the SV model.
res.sv <- svsample(zar.ins)We can then consider the description of volatility and the posterior densities for the parameters:
volplot(res.sv)
paradensplot(res.sv)
summary(res.sv)##
## Summary of 10000 MCMC draws after a burn-in of 1000.
##
## Prior distributions:
## mu ~ Normal(mean = 0, sd = 100)
## (phi+1)/2 ~ Beta(a0 = 5, b0 = 1.5)
## sigma^2 ~ 1 * Chisq(df = 1)
##
## Posterior draws of parameters (thinning = 1):
## mean sd 5% 50% 95% ESS
## mu -9.4841 0.19536 -9.7964 -9.4845 -9.168 5514
## phi 0.9820 0.00450 0.9743 0.9823 0.989 269
## sigma 0.1987 0.01893 0.1698 0.1976 0.231 148
## exp(mu/2) 0.0088 0.00086 0.0075 0.0087 0.010 5514
## sigma^2 0.0398 0.00771 0.0288 0.0391 0.053 148
##
## Posterior draws of initial and contemporaneous latents (thinning = 1):
## mean sd 5% 50% 95% mean(exp(h_t/2))
## h_0 -10.6 0.53 -11.4 -10.6 -9.7 0.0052
## h_1 -10.6 0.50 -11.4 -10.6 -9.8 0.0051
## h_2 -10.6 0.46 -11.4 -10.6 -9.8 0.0051
## h_3 -10.6 0.43 -11.3 -10.6 -9.9 0.0051
## h_4 -10.6 0.41 -11.3 -10.6 -9.9 0.0051
## h_5 -10.6 0.40 -11.3 -10.7 -10.0 0.0050
## h_6 -10.7 0.39 -11.3 -10.7 -10.0 0.0049
## h_7 -10.7 0.38 -11.3 -10.7 -10.0 0.0049
## h_8 -10.7 0.36 -11.3 -10.7 -10.1 0.0048
## h_9 -10.8 0.36 -11.3 -10.8 -10.2 0.0047
## h_10 -10.9 0.36 -11.4 -10.9 -10.3 0.0045
## h_11 -10.9 0.37 -11.5 -11.0 -10.3 0.0043
## h_12 -11.1 0.38 -11.7 -11.1 -10.4 0.0040
## h_13 -11.2 0.39 -11.8 -11.2 -10.5 0.0038
## h_14 -11.2 0.39 -11.9 -11.2 -10.6 0.0037
## h_15 -11.3 0.38 -11.9 -11.3 -10.7 0.0036
## h_16 -11.4 0.38 -12.0 -11.4 -10.7 0.0035
## h_17 -11.4 0.37 -12.0 -11.4 -10.8 0.0034
## h_18 -11.5 0.37 -12.1 -11.5 -10.9 0.0032
## h_19 -11.6 0.37 -12.2 -11.6 -11.0 0.0031
## h_20 -11.7 0.38 -12.3 -11.7 -11.0 0.0030
## h_21 -11.8 0.39 -12.4 -11.8 -11.1 0.0028
## h_22 -11.9 0.39 -12.5 -11.9 -11.2 0.0027
## h_23 -12.0 0.40 -12.6 -12.0 -11.3 0.0026
## h_24 -12.0 0.40 -12.7 -12.0 -11.4 0.0025
## h_25 -12.1 0.40 -12.7 -12.1 -11.4 0.0024
## h_26 -12.1 0.40 -12.8 -12.2 -11.5 0.0024
## h_27 -12.2 0.41 -12.9 -12.2 -11.5 0.0023
## h_28 -12.3 0.41 -12.9 -12.3 -11.6 0.0022
## h_29 -12.4 0.42 -13.0 -12.4 -11.7 0.0021
## h_30 -12.4 0.43 -13.1 -12.4 -11.7 0.0020
## h_31 -12.5 0.44 -13.2 -12.5 -11.8 0.0020
## h_32 -12.5 0.45 -13.3 -12.5 -11.8 0.0019
## h_33 -12.6 0.45 -13.3 -12.6 -11.8 0.0019
## h_34 -12.6 0.45 -13.3 -12.6 -11.8 0.0019
## h_35 -12.6 0.45 -13.3 -12.6 -11.9 0.0019
## h_36 -12.6 0.45 -13.3 -12.6 -11.8 0.0019
## h_37 -12.5 0.43 -13.2 -12.5 -11.8 0.0019
## h_38 -12.5 0.42 -13.2 -12.5 -11.8 0.0020
## h_39 -12.5 0.42 -13.1 -12.5 -11.8 0.0020
## h_40 -12.4 0.40 -13.1 -12.4 -11.7 0.0021
## h_41 -12.4 0.40 -13.0 -12.4 -11.7 0.0021
## h_42 -12.3 0.38 -12.9 -12.3 -11.6 0.0022
## h_43 -12.2 0.37 -12.8 -12.2 -11.6 0.0023
## h_44 -12.2 0.36 -12.8 -12.2 -11.6 0.0023
## h_45 -12.1 0.35 -12.7 -12.1 -11.5 0.0024
## h_46 -12.0 0.34 -12.6 -12.0 -11.5 0.0025
## h_47 -12.1 0.35 -12.6 -12.1 -11.5 0.0024
## h_48 -12.1 0.37 -12.7 -12.2 -11.5 0.0023
## h_49 -12.2 0.39 -12.8 -12.2 -11.6 0.0023
## h_50 -12.3 0.40 -12.9 -12.3 -11.6 0.0022
## h_51 -12.3 0.42 -13.0 -12.3 -11.6 0.0022
## h_52 -12.4 0.43 -13.0 -12.4 -11.7 0.0021
## h_53 -12.4 0.44 -13.1 -12.4 -11.7 0.0021
## h_54 -12.4 0.45 -13.1 -12.4 -11.7 0.0021
## h_55 -12.4 0.45 -13.1 -12.4 -11.7 0.0021
## h_56 -12.4 0.45 -13.1 -12.4 -11.6 0.0021
## h_57 -12.3 0.44 -13.0 -12.3 -11.6 0.0022
## h_58 -12.3 0.43 -13.0 -12.3 -11.6 0.0022
## h_59 -12.2 0.41 -12.9 -12.2 -11.5 0.0023
## h_60 -12.1 0.40 -12.7 -12.1 -11.4 0.0024
## h_61 -12.0 0.37 -12.6 -12.0 -11.4 0.0025
## h_62 -11.9 0.34 -12.4 -11.9 -11.3 0.0027
## h_63 -11.7 0.30 -12.2 -11.7 -11.2 0.0029
## h_64 -11.8 0.33 -12.3 -11.8 -11.3 0.0028
## h_65 -11.9 0.35 -12.5 -11.9 -11.3 0.0027
## h_66 -12.0 0.36 -12.6 -12.0 -11.4 0.0025
## h_67 -12.1 0.38 -12.7 -12.1 -11.5 0.0024
## h_68 -12.2 0.39 -12.8 -12.2 -11.5 0.0023
## h_69 -12.3 0.40 -12.9 -12.3 -11.6 0.0022
## h_70 -12.3 0.40 -13.0 -12.3 -11.6 0.0022
## h_71 -12.4 0.41 -13.1 -12.4 -11.7 0.0021
## h_72 -12.5 0.43 -13.2 -12.5 -11.8 0.0020
## h_73 -12.5 0.43 -13.2 -12.5 -11.8 0.0019
## h_74 -12.6 0.43 -13.3 -12.6 -11.8 0.0019
## h_75 -12.6 0.43 -13.3 -12.6 -11.9 0.0019
## h_76 -12.6 0.42 -13.3 -12.6 -11.9 0.0019
## h_77 -12.6 0.41 -13.2 -12.6 -11.9 0.0019
## h_78 -12.6 0.41 -13.2 -12.6 -11.9 0.0019
## h_79 -12.6 0.42 -13.3 -12.6 -11.9 0.0019
## h_80 -12.6 0.42 -13.3 -12.6 -11.9 0.0019
## h_81 -12.6 0.42 -13.3 -12.6 -11.9 0.0019
## h_82 -12.6 0.44 -13.3 -12.6 -11.9 0.0018
## h_83 -12.6 0.45 -13.4 -12.7 -11.9 0.0018
## h_84 -12.6 0.46 -13.4 -12.7 -11.9 0.0018
## h_85 -12.6 0.47 -13.4 -12.6 -11.9 0.0019
## h_86 -12.6 0.47 -13.4 -12.6 -11.8 0.0019
## h_87 -12.6 0.47 -13.3 -12.6 -11.8 0.0019
## h_88 -12.5 0.47 -13.3 -12.5 -11.8 0.0020
## h_89 -12.5 0.48 -13.3 -12.5 -11.7 0.0020
## h_90 -12.4 0.48 -13.2 -12.4 -11.6 0.0021
## h_91 -12.4 0.49 -13.1 -12.3 -11.6 0.0021
## h_92 -12.3 0.48 -13.0 -12.3 -11.5 0.0022
## h_93 -12.1 0.47 -12.9 -12.1 -11.4 0.0024
## h_94 -12.0 0.47 -12.8 -12.0 -11.2 0.0025
## h_95 -11.9 0.47 -12.7 -11.9 -11.1 0.0027
## h_96 -11.7 0.46 -12.5 -11.7 -11.0 0.0029
## h_97 -11.6 0.45 -12.3 -11.6 -10.8 0.0032
## h_98 -11.4 0.43 -12.1 -11.4 -10.7 0.0035
## h_99 -11.2 0.41 -11.9 -11.2 -10.5 0.0038
## h_100 -11.0 0.39 -11.6 -11.0 -10.3 0.0042
## h_101 -10.7 0.36 -11.3 -10.7 -10.1 0.0047
## h_102 -10.5 0.33 -11.0 -10.5 -9.9 0.0053
## h_103 -10.4 0.35 -11.0 -10.4 -9.8 0.0055
## h_104 -10.3 0.36 -10.9 -10.4 -9.7 0.0058
## h_105 -10.2 0.36 -10.8 -10.2 -9.6 0.0061
## h_106 -10.1 0.36 -10.7 -10.1 -9.5 0.0064
## h_107 -10.0 0.36 -10.6 -10.0 -9.4 0.0068
## h_108 -9.9 0.36 -10.5 -9.9 -9.3 0.0072
## h_109 -9.8 0.36 -10.4 -9.8 -9.2 0.0075
## h_110 -9.8 0.38 -10.4 -9.8 -9.1 0.0077
## h_111 -9.7 0.39 -10.4 -9.7 -9.1 0.0079
## h_112 -9.7 0.40 -10.3 -9.7 -9.0 0.0082
## h_113 -9.6 0.40 -10.2 -9.6 -8.9 0.0085
## h_114 -9.5 0.40 -10.1 -9.5 -8.8 0.0089
## h_115 -9.4 0.39 -10.0 -9.4 -8.7 0.0095
## h_116 -9.2 0.39 -9.9 -9.3 -8.6 0.0100
## h_117 -9.1 0.39 -9.7 -9.1 -8.5 0.0106
## h_118 -9.0 0.38 -9.6 -9.0 -8.4 0.0112
## h_119 -8.9 0.39 -9.5 -8.9 -8.3 0.0118
## h_120 -8.8 0.38 -9.4 -8.8 -8.2 0.0125
## h_121 -8.7 0.38 -9.3 -8.7 -8.0 0.0134
## h_122 -8.6 0.38 -9.2 -8.6 -7.9 0.0141
## h_123 -8.4 0.38 -9.1 -8.4 -7.8 0.0150
## h_124 -8.3 0.37 -8.9 -8.3 -7.7 0.0160
## h_125 -8.2 0.36 -8.7 -8.2 -7.6 0.0172
## h_126 -8.0 0.34 -8.5 -8.0 -7.4 0.0187
## h_127 -7.9 0.33 -8.4 -7.9 -7.3 0.0197
## h_128 -7.8 0.35 -8.4 -7.8 -7.3 0.0202
## h_129 -7.8 0.34 -8.3 -7.8 -7.2 0.0207
## h_130 -7.7 0.34 -8.3 -7.7 -7.1 0.0214
## h_131 -7.6 0.33 -8.1 -7.6 -7.1 0.0223
## h_132 -7.6 0.34 -8.2 -7.6 -7.1 0.0222
## h_133 -7.6 0.34 -8.2 -7.6 -7.1 0.0223
## h_134 -7.6 0.34 -8.1 -7.6 -7.0 0.0225
## h_135 -7.7 0.34 -8.2 -7.7 -7.1 0.0220
## h_136 -7.7 0.35 -8.3 -7.7 -7.1 0.0216
## h_137 -7.7 0.35 -8.3 -7.8 -7.2 0.0211
## h_138 -7.8 0.35 -8.3 -7.8 -7.2 0.0208
## h_139 -7.8 0.35 -8.4 -7.9 -7.2 0.0201
## h_140 -7.9 0.36 -8.5 -7.9 -7.3 0.0192
## h_141 -8.0 0.36 -8.6 -8.0 -7.4 0.0185
## h_142 -8.1 0.36 -8.7 -8.1 -7.5 0.0178
## h_143 -8.2 0.36 -8.8 -8.2 -7.6 0.0171
## h_144 -8.3 0.36 -8.8 -8.3 -7.6 0.0164
## h_145 -8.3 0.37 -8.9 -8.4 -7.7 0.0156
## h_146 -8.5 0.38 -9.1 -8.5 -7.8 0.0149
## h_147 -8.5 0.39 -9.2 -8.5 -7.9 0.0143
## h_148 -8.6 0.39 -9.3 -8.6 -8.0 0.0137
## h_149 -8.7 0.40 -9.4 -8.7 -8.0 0.0130
## h_150 -8.8 0.41 -9.5 -8.8 -8.1 0.0125
## h_151 -8.9 0.42 -9.6 -8.9 -8.2 0.0120
## h_152 -9.0 0.43 -9.7 -9.0 -8.2 0.0116
## h_153 -9.0 0.44 -9.7 -9.0 -8.3 0.0113
## h_154 -9.1 0.44 -9.8 -9.1 -8.3 0.0111
## h_155 -9.1 0.44 -9.8 -9.1 -8.3 0.0110
## h_156 -9.1 0.43 -9.8 -9.1 -8.3 0.0109
## h_157 -9.1 0.43 -9.8 -9.1 -8.4 0.0110
## h_158 -9.1 0.43 -9.8 -9.1 -8.4 0.0109
## h_159 -9.1 0.43 -9.8 -9.1 -8.4 0.0110
## h_160 -9.1 0.43 -9.8 -9.1 -8.3 0.0110
## h_161 -9.0 0.42 -9.7 -9.0 -8.3 0.0112
## h_162 -9.0 0.44 -9.7 -9.0 -8.3 0.0112
## h_163 -9.0 0.44 -9.7 -9.0 -8.3 0.0114
## h_164 -9.0 0.43 -9.7 -9.0 -8.2 0.0116
## h_165 -8.9 0.43 -9.6 -8.9 -8.2 0.0120
## h_166 -8.8 0.42 -9.5 -8.8 -8.1 0.0123
## h_167 -8.7 0.41 -9.4 -8.8 -8.1 0.0129
## h_168 -8.6 0.39 -9.3 -8.7 -8.0 0.0135
## h_169 -8.5 0.38 -9.1 -8.5 -7.9 0.0143
## h_170 -8.4 0.35 -9.0 -8.4 -7.8 0.0151
## h_171 -8.3 0.32 -8.8 -8.3 -7.8 0.0160
## h_172 -8.3 0.33 -8.8 -8.3 -7.7 0.0161
## h_173 -8.3 0.33 -8.8 -8.3 -7.7 0.0160
## h_174 -8.3 0.33 -8.8 -8.3 -7.7 0.0160
## h_175 -8.4 0.36 -9.0 -8.4 -7.8 0.0152
## h_176 -8.5 0.38 -9.1 -8.5 -7.9 0.0145
## h_177 -8.6 0.40 -9.2 -8.6 -7.9 0.0140
## h_178 -8.6 0.40 -9.3 -8.6 -8.0 0.0136
## h_179 -8.7 0.41 -9.3 -8.7 -8.0 0.0134
## h_180 -8.7 0.42 -9.4 -8.7 -8.0 0.0131
## h_181 -8.7 0.42 -9.4 -8.7 -8.0 0.0130
## h_182 -8.7 0.42 -9.4 -8.7 -8.0 0.0129
## h_183 -8.7 0.42 -9.4 -8.7 -8.0 0.0129
## h_184 -8.7 0.42 -9.4 -8.7 -8.0 0.0130
## h_185 -8.7 0.41 -9.4 -8.7 -8.0 0.0131
## h_186 -8.7 0.41 -9.4 -8.7 -8.0 0.0132
## h_187 -8.7 0.41 -9.3 -8.7 -8.0 0.0135
## h_188 -8.6 0.39 -9.2 -8.6 -7.9 0.0139
## h_189 -8.5 0.37 -9.1 -8.5 -7.9 0.0144
## h_190 -8.5 0.38 -9.1 -8.5 -7.9 0.0144
## h_191 -8.5 0.38 -9.1 -8.5 -7.9 0.0144
## h_192 -8.5 0.38 -9.1 -8.5 -7.9 0.0144
## h_193 -8.5 0.38 -9.1 -8.5 -7.9 0.0146
## h_194 -8.4 0.38 -9.1 -8.5 -7.8 0.0149
## h_195 -8.4 0.36 -9.0 -8.4 -7.8 0.0154
## h_196 -8.3 0.35 -8.9 -8.3 -7.8 0.0157
## h_197 -8.4 0.36 -8.9 -8.4 -7.7 0.0156
## h_198 -8.4 0.35 -8.9 -8.4 -7.8 0.0156
## h_199 -8.3 0.34 -8.9 -8.3 -7.8 0.0157
## h_200 -8.3 0.33 -8.8 -8.3 -7.8 0.0159
## h_201 -8.4 0.35 -9.0 -8.4 -7.8 0.0150
## h_202 -8.5 0.37 -9.1 -8.5 -7.9 0.0143
## h_203 -8.6 0.37 -9.2 -8.6 -8.0 0.0138
## h_204 -8.7 0.38 -9.3 -8.7 -8.0 0.0134
## h_205 -8.7 0.37 -9.3 -8.7 -8.1 0.0131
## h_206 -8.7 0.37 -9.3 -8.8 -8.1 0.0128
## h_207 -8.8 0.37 -9.4 -8.8 -8.2 0.0124
## h_208 -8.9 0.39 -9.5 -8.9 -8.3 0.0119
## h_209 -9.0 0.40 -9.6 -9.0 -8.3 0.0114
## h_210 -9.1 0.42 -9.8 -9.1 -8.4 0.0109
## h_211 -9.1 0.43 -9.8 -9.1 -8.4 0.0106
## h_212 -9.2 0.43 -9.9 -9.2 -8.5 0.0103
## h_213 -9.2 0.43 -9.9 -9.2 -8.5 0.0102
## h_214 -9.2 0.42 -9.9 -9.2 -8.5 0.0101
## h_215 -9.2 0.41 -9.9 -9.2 -8.6 0.0101
## h_216 -9.2 0.39 -9.8 -9.2 -8.6 0.0102
## h_217 -9.2 0.36 -9.7 -9.2 -8.5 0.0105
## h_218 -9.2 0.37 -9.8 -9.2 -8.6 0.0103
## h_219 -9.2 0.38 -9.8 -9.2 -8.6 0.0101
## h_220 -9.3 0.38 -9.9 -9.3 -8.6 0.0100
## h_221 -9.3 0.38 -9.9 -9.3 -8.6 0.0099
## h_222 -9.3 0.37 -9.8 -9.3 -8.6 0.0100
## h_223 -9.3 0.36 -9.9 -9.3 -8.7 0.0099
## h_224 -9.3 0.37 -9.9 -9.3 -8.7 0.0097
## h_225 -9.4 0.39 -10.0 -9.4 -8.7 0.0094
## h_226 -9.4 0.40 -10.1 -9.5 -8.8 0.0091
## h_227 -9.5 0.41 -10.2 -9.5 -8.8 0.0088
## h_228 -9.5 0.42 -10.2 -9.5 -8.8 0.0087
## h_229 -9.6 0.42 -10.3 -9.6 -8.9 0.0085
## h_230 -9.6 0.43 -10.3 -9.6 -8.9 0.0085
## h_231 -9.6 0.43 -10.3 -9.6 -8.9 0.0084
## h_232 -9.6 0.43 -10.3 -9.6 -8.9 0.0084
## h_233 -9.6 0.43 -10.3 -9.6 -8.9 0.0085
## h_234 -9.6 0.43 -10.3 -9.6 -8.9 0.0084
## h_235 -9.6 0.44 -10.3 -9.6 -8.9 0.0084
## h_236 -9.6 0.44 -10.3 -9.6 -8.9 0.0085
## h_237 -9.5 0.43 -10.2 -9.5 -8.8 0.0087
## h_238 -9.5 0.42 -10.2 -9.5 -8.8 0.0089
## h_239 -9.4 0.40 -10.0 -9.4 -8.7 0.0093
## h_240 -9.3 0.38 -9.9 -9.3 -8.7 0.0097
## h_241 -9.2 0.36 -9.8 -9.2 -8.6 0.0102
## h_242 -9.1 0.33 -9.6 -9.1 -8.5 0.0108
## h_243 -9.1 0.34 -9.6 -9.1 -8.5 0.0107
## h_244 -9.1 0.35 -9.7 -9.1 -8.5 0.0107
## h_245 -9.2 0.36 -9.7 -9.2 -8.5 0.0104
## h_246 -9.2 0.38 -9.8 -9.2 -8.6 0.0102
## h_247 -9.2 0.38 -9.8 -9.3 -8.6 0.0100
## h_248 -9.3 0.38 -9.9 -9.3 -8.6 0.0100
## h_249 -9.3 0.38 -9.9 -9.3 -8.6 0.0099
## h_250 -9.3 0.38 -9.9 -9.3 -8.6 0.0098
## h_251 -9.3 0.37 -9.9 -9.3 -8.6 0.0099
## h_252 -9.4 0.40 -10.0 -9.4 -8.7 0.0095
## h_253 -9.4 0.42 -10.1 -9.4 -8.7 0.0092
## h_254 -9.5 0.44 -10.2 -9.5 -8.7 0.0090
## h_255 -9.5 0.45 -10.2 -9.5 -8.7 0.0089
## h_256 -9.5 0.46 -10.3 -9.5 -8.7 0.0089
## h_257 -9.5 0.47 -10.2 -9.5 -8.7 0.0090
## h_258 -9.4 0.47 -10.2 -9.5 -8.7 0.0091
## h_259 -9.4 0.46 -10.2 -9.4 -8.6 0.0093
## h_260 -9.3 0.46 -10.1 -9.3 -8.6 0.0096
## h_261 -9.3 0.45 -10.0 -9.2 -8.5 0.0101
## h_262 -9.1 0.44 -9.9 -9.1 -8.4 0.0106
## h_263 -9.0 0.42 -9.7 -9.0 -8.3 0.0112
## h_264 -8.9 0.39 -9.5 -8.9 -8.2 0.0120
## h_265 -8.8 0.38 -9.4 -8.8 -8.1 0.0127
## h_266 -8.7 0.37 -9.2 -8.7 -8.0 0.0134
## h_267 -8.6 0.36 -9.1 -8.6 -7.9 0.0141
## h_268 -8.5 0.34 -9.0 -8.5 -7.9 0.0148
## h_269 -8.4 0.34 -8.9 -8.4 -7.8 0.0154
## h_270 -8.3 0.33 -8.8 -8.3 -7.7 0.0160
## h_271 -8.2 0.31 -8.7 -8.2 -7.7 0.0166
## h_272 -8.2 0.29 -8.6 -8.2 -7.7 0.0170
## h_273 -8.4 0.33 -8.9 -8.4 -7.8 0.0154
## h_274 -8.6 0.36 -9.1 -8.6 -8.0 0.0141
## h_275 -8.7 0.38 -9.3 -8.7 -8.1 0.0130
## h_276 -8.9 0.39 -9.5 -8.9 -8.2 0.0121
## h_277 -9.0 0.41 -9.7 -9.0 -8.3 0.0113
## h_278 -9.1 0.43 -9.8 -9.1 -8.4 0.0106
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## h_1938 -8.7 0.35 -9.3 -8.7 -8.1 0.0130
## h_1939 -8.7 0.36 -9.3 -8.8 -8.2 0.0128
## h_1940 -8.8 0.36 -9.4 -8.8 -8.2 0.0127
## h_1941 -8.8 0.37 -9.4 -8.8 -8.2 0.0123
## h_1942 -8.9 0.37 -9.5 -8.9 -8.2 0.0121
## h_1943 -8.9 0.37 -9.5 -8.9 -8.3 0.0119
## h_1944 -8.9 0.37 -9.5 -8.9 -8.3 0.0117
## h_1945 -9.0 0.38 -9.6 -9.0 -8.3 0.0115
## h_1946 -9.0 0.39 -9.7 -9.0 -8.4 0.0111
## h_1947 -9.1 0.40 -9.8 -9.1 -8.4 0.0107
## h_1948 -9.2 0.41 -9.8 -9.2 -8.5 0.0105
## h_1949 -9.2 0.42 -9.9 -9.2 -8.5 0.0103
## h_1950 -9.2 0.42 -9.9 -9.2 -8.5 0.0102
## h_1951 -9.2 0.41 -9.9 -9.2 -8.5 0.0103
## h_1952 -9.2 0.40 -9.8 -9.2 -8.5 0.0104
## h_1953 -9.2 0.40 -9.8 -9.2 -8.5 0.0105
## h_1954 -9.1 0.40 -9.8 -9.1 -8.5 0.0106
## h_1955 -9.1 0.39 -9.7 -9.1 -8.4 0.0107
## h_1956 -9.1 0.38 -9.7 -9.1 -8.4 0.0110
## h_1957 -9.0 0.35 -9.6 -9.0 -8.4 0.0114
## h_1958 -9.0 0.37 -9.6 -9.0 -8.4 0.0113
## h_1959 -9.0 0.38 -9.6 -9.0 -8.4 0.0112
## h_1960 -9.0 0.38 -9.6 -9.0 -8.4 0.0113
## h_1961 -9.0 0.37 -9.6 -9.0 -8.3 0.0115
## h_1962 -8.9 0.36 -9.5 -8.9 -8.3 0.0118
## h_1963 -8.9 0.34 -9.4 -8.9 -8.3 0.0121
## h_1964 -8.8 0.33 -9.4 -8.9 -8.3 0.0122
## h_1965 -8.9 0.34 -9.4 -8.9 -8.3 0.0118
## h_1966 -9.0 0.36 -9.6 -9.0 -8.4 0.0112
## h_1967 -9.1 0.37 -9.7 -9.1 -8.5 0.0107
## h_1968 -9.2 0.38 -9.8 -9.2 -8.5 0.0103
## h_1969 -9.2 0.39 -9.9 -9.2 -8.6 0.0100
## h_1970 -9.3 0.38 -9.9 -9.3 -8.7 0.0098
## h_1971 -9.4 0.40 -10.0 -9.4 -8.7 0.0094
## h_1972 -9.4 0.40 -10.1 -9.4 -8.8 0.0092
## h_1973 -9.5 0.41 -10.1 -9.5 -8.8 0.0089
## h_1974 -9.5 0.41 -10.2 -9.5 -8.8 0.0087
## h_1975 -9.6 0.40 -10.2 -9.6 -8.9 0.0086
## h_1976 -9.6 0.41 -10.2 -9.6 -8.9 0.0085
## h_1977 -9.6 0.40 -10.3 -9.6 -8.9 0.0084
## h_1978 -9.6 0.40 -10.3 -9.6 -8.9 0.0083
## h_1979 -9.6 0.40 -10.3 -9.6 -8.9 0.0083
## h_1980 -9.6 0.39 -10.2 -9.6 -9.0 0.0083
## h_1981 -9.7 0.40 -10.3 -9.7 -9.0 0.0082
## h_1982 -9.7 0.41 -10.3 -9.7 -9.0 0.0081
## h_1983 -9.7 0.41 -10.3 -9.7 -9.0 0.0081
## h_1984 -9.7 0.40 -10.3 -9.7 -9.0 0.0081
## h_1985 -9.6 0.38 -10.2 -9.6 -9.0 0.0083
## h_1986 -9.6 0.38 -10.2 -9.6 -9.0 0.0083
## h_1987 -9.6 0.38 -10.2 -9.6 -9.0 0.0083
## h_1988 -9.6 0.38 -10.2 -9.6 -9.0 0.0083
## h_1989 -9.6 0.39 -10.3 -9.7 -9.0 0.0082
## h_1990 -9.6 0.39 -10.3 -9.7 -9.0 0.0082
## h_1991 -9.7 0.39 -10.3 -9.7 -9.0 0.0082
## h_1992 -9.7 0.40 -10.3 -9.7 -9.0 0.0081
## h_1993 -9.6 0.40 -10.3 -9.7 -9.0 0.0082
## h_1994 -9.6 0.40 -10.3 -9.6 -9.0 0.0082
## h_1995 -9.6 0.41 -10.3 -9.6 -9.0 0.0083
## h_1996 -9.6 0.40 -10.2 -9.6 -8.9 0.0085
## h_1997 -9.5 0.39 -10.2 -9.5 -8.9 0.0087
## h_1998 -9.4 0.37 -10.1 -9.4 -8.8 0.0090
## h_1999 -9.3 0.35 -9.9 -9.4 -8.8 0.0095
## h_2000 -9.3 0.32 -9.8 -9.3 -8.7 0.0099
## h_2001 -9.3 0.33 -9.8 -9.3 -8.7 0.0097
## h_2002 -9.4 0.34 -9.9 -9.4 -8.8 0.0094
## h_2003 -9.4 0.35 -10.0 -9.4 -8.8 0.0091
## h_2004 -9.5 0.36 -10.1 -9.5 -8.9 0.0089
## h_2005 -9.5 0.37 -10.1 -9.6 -8.9 0.0086
## h_2006 -9.6 0.37 -10.2 -9.6 -9.0 0.0084
## h_2007 -9.6 0.37 -10.2 -9.6 -9.0 0.0083
## h_2008 -9.6 0.37 -10.2 -9.7 -9.0 0.0082
## h_2009 -9.7 0.39 -10.4 -9.7 -9.1 0.0079
## h_2010 -9.8 0.40 -10.4 -9.8 -9.1 0.0076
## h_2011 -9.9 0.40 -10.5 -9.9 -9.2 0.0074
## h_2012 -9.9 0.40 -10.5 -9.9 -9.2 0.0073
## h_2013 -9.9 0.40 -10.5 -9.9 -9.2 0.0073
## h_2014 -9.9 0.40 -10.6 -9.9 -9.2 0.0072
## h_2015 -9.9 0.39 -10.6 -9.9 -9.3 0.0072
## h_2016 -9.9 0.39 -10.5 -9.9 -9.3 0.0072
## h_2017 -9.9 0.38 -10.5 -9.9 -9.3 0.0072
## h_2018 -9.9 0.37 -10.5 -9.9 -9.3 0.0072
## h_2019 -9.9 0.36 -10.4 -9.9 -9.3 0.0073
## h_2020 -9.9 0.36 -10.5 -9.9 -9.3 0.0073
## h_2021 -9.9 0.37 -10.5 -9.9 -9.3 0.0071
## h_2022 -9.9 0.37 -10.5 -9.9 -9.3 0.0071
## h_2023 -9.9 0.36 -10.5 -9.9 -9.3 0.0071
## h_2024 -10.0 0.37 -10.5 -10.0 -9.3 0.0070
## h_2025 -10.0 0.37 -10.5 -10.0 -9.3 0.0070
## h_2026 -10.0 0.37 -10.6 -10.0 -9.4 0.0068
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## h_2028 -10.1 0.38 -10.7 -10.1 -9.4 0.0066
## h_2029 -10.1 0.38 -10.7 -10.1 -9.5 0.0065
## h_2030 -10.1 0.38 -10.8 -10.2 -9.5 0.0064
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## h_2032 -10.2 0.39 -10.8 -10.2 -9.5 0.0062
## h_2033 -10.2 0.40 -10.9 -10.2 -9.6 0.0061
## h_2034 -10.2 0.39 -10.9 -10.3 -9.6 0.0061
## h_2035 -10.3 0.39 -10.9 -10.3 -9.6 0.0060
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## h_2038 -10.3 0.39 -10.9 -10.3 -9.6 0.0059
## h_2039 -10.3 0.38 -10.9 -10.3 -9.6 0.0059
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## h_2041 -10.3 0.38 -11.0 -10.4 -9.7 0.0058
## h_2042 -10.4 0.40 -11.0 -10.4 -9.7 0.0056
## h_2043 -10.4 0.40 -11.1 -10.4 -9.8 0.0056
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## h_2045 -10.4 0.39 -11.1 -10.4 -9.8 0.0055
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## h_2059 -10.3 0.38 -10.9 -10.3 -9.7 0.0060
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## h_3571 -8.6 0.42 -9.3 -8.6 -7.9 0.0141
## h_3572 -8.5 0.40 -9.1 -8.5 -7.8 0.0148
## h_3573 -8.4 0.40 -9.0 -8.4 -7.7 0.0155
## h_3574 -8.3 0.39 -8.9 -8.3 -7.6 0.0161
## h_3575 -8.2 0.39 -8.8 -8.2 -7.6 0.0169
## h_3576 -8.1 0.38 -8.7 -8.1 -7.5 0.0177
## h_3577 -8.0 0.37 -8.6 -8.0 -7.4 0.0187
## h_3578 -7.9 0.35 -8.4 -7.9 -7.3 0.0198
## h_3579 -7.8 0.33 -8.3 -7.8 -7.2 0.0209
## h_3580 -7.6 0.30 -8.1 -7.6 -7.1 0.0222
## h_3581 -7.7 0.32 -8.2 -7.7 -7.1 0.0219
## h_3582 -7.7 0.33 -8.2 -7.7 -7.2 0.0215
## h_3583 -7.8 0.35 -8.3 -7.8 -7.2 0.0207
## h_3584 -7.9 0.36 -8.4 -7.9 -7.3 0.0200
## h_3585 -7.9 0.38 -8.5 -7.9 -7.3 0.0194
## h_3586 -8.0 0.38 -8.6 -8.0 -7.3 0.0189
## h_3587 -8.0 0.38 -8.6 -8.0 -7.4 0.0186
## h_3588 -8.0 0.37 -8.6 -8.0 -7.4 0.0183
## h_3589 -8.1 0.38 -8.7 -8.1 -7.5 0.0178
## h_3590 -8.2 0.39 -8.8 -8.2 -7.5 0.0173
## h_3591 -8.2 0.40 -8.9 -8.2 -7.6 0.0168
## h_3592 -8.3 0.40 -8.9 -8.3 -7.6 0.0164
## h_3593 -8.3 0.40 -8.9 -8.3 -7.6 0.0161
## h_3594 -8.3 0.41 -9.0 -8.3 -7.7 0.0159
## h_3595 -8.4 0.41 -9.0 -8.4 -7.7 0.0157
## h_3596 -8.4 0.42 -9.1 -8.4 -7.7 0.0155
## h_3597 -8.4 0.41 -9.0 -8.4 -7.7 0.0155
## h_3598 -8.4 0.40 -9.0 -8.4 -7.7 0.0156
## h_3599 -8.4 0.40 -9.0 -8.4 -7.7 0.0156
## h_3600 -8.3 0.39 -9.0 -8.3 -7.7 0.0158
## h_3601 -8.3 0.38 -8.9 -8.3 -7.7 0.0160
## h_3602 -8.3 0.38 -8.9 -8.3 -7.6 0.0162
## h_3603 -8.3 0.37 -8.8 -8.3 -7.6 0.0164
## h_3604 -8.3 0.38 -8.9 -8.3 -7.6 0.0164
## h_3605 -8.2 0.37 -8.8 -8.2 -7.6 0.0166
## h_3606 -8.2 0.35 -8.8 -8.2 -7.6 0.0169
## h_3607 -8.2 0.36 -8.8 -8.2 -7.6 0.0168
## h_3608 -8.2 0.35 -8.7 -8.2 -7.6 0.0169
## h_3609 -8.2 0.36 -8.8 -8.2 -7.6 0.0167
## h_3610 -8.3 0.36 -8.8 -8.3 -7.7 0.0163
## h_3611 -8.3 0.37 -8.9 -8.3 -7.7 0.0161
## h_3612 -8.3 0.38 -8.9 -8.3 -7.7 0.0158
## h_3613 -8.4 0.38 -9.0 -8.4 -7.7 0.0156
## h_3614 -8.4 0.38 -9.0 -8.4 -7.7 0.0156
## h_3615 -8.4 0.37 -9.0 -8.4 -7.7 0.0156
## h_3616 -8.4 0.39 -9.0 -8.4 -7.7 0.0152
## h_3617 -8.5 0.41 -9.1 -8.5 -7.8 0.0148
## h_3618 -8.5 0.42 -9.2 -8.5 -7.8 0.0146
## h_3619 -8.5 0.44 -9.3 -8.6 -7.8 0.0143
## h_3620 -8.6 0.48 -9.4 -8.6 -7.8 0.0138
## h_3621 -8.7 0.51 -9.5 -8.7 -7.8 0.0135
## h_3622 -8.7 0.54 -9.6 -8.7 -7.8 0.0133
## sd(exp(h_t/2))
## h_0 0.00141
## h_1 0.00131
## h_2 0.00121
## h_3 0.00112
## h_4 0.00108
## h_5 0.00103
## h_6 0.00098
## h_7 0.00094
## h_8 0.00089
## h_9 0.00086
## h_10 0.00083
## h_11 0.00080
## h_12 0.00078
## h_13 0.00076
## h_14 0.00073
## h_15 0.00070
## h_16 0.00067
## h_17 0.00064
## h_18 0.00061
## h_19 0.00059
## h_20 0.00058
## h_21 0.00056
## h_22 0.00054
## h_23 0.00053
## h_24 0.00051
## h_25 0.00049
## h_26 0.00048
## h_27 0.00047
## h_28 0.00046
## h_29 0.00045
## h_30 0.00045
## h_31 0.00044
## h_32 0.00044
## h_33 0.00043
## h_34 0.00043
## h_35 0.00043
## h_36 0.00043
## h_37 0.00042
## h_38 0.00042
## h_39 0.00043
## h_40 0.00042
## h_41 0.00043
## h_42 0.00043
## h_43 0.00042
## h_44 0.00043
## h_45 0.00043
## h_46 0.00043
## h_47 0.00044
## h_48 0.00044
## h_49 0.00045
## h_50 0.00046
## h_51 0.00046
## h_52 0.00046
## h_53 0.00047
## h_54 0.00047
## h_55 0.00047
## h_56 0.00048
## h_57 0.00048
## h_58 0.00048
## h_59 0.00049
## h_60 0.00048
## h_61 0.00048
## h_62 0.00047
## h_63 0.00045
## h_64 0.00046
## h_65 0.00046
## h_66 0.00046
## h_67 0.00046
## h_68 0.00045
## h_69 0.00045
## h_70 0.00044
## h_71 0.00044
## h_72 0.00043
## h_73 0.00043
## h_74 0.00042
## h_75 0.00041
## h_76 0.00041
## h_77 0.00040
## h_78 0.00040
## h_79 0.00040
## h_80 0.00040
## h_81 0.00040
## h_82 0.00041
## h_83 0.00042
## h_84 0.00043
## h_85 0.00044
## h_86 0.00045
## h_87 0.00046
## h_88 0.00047
## h_89 0.00048
## h_90 0.00050
## h_91 0.00053
## h_92 0.00055
## h_93 0.00057
## h_94 0.00060
## h_95 0.00064
## h_96 0.00067
## h_97 0.00071
## h_98 0.00076
## h_99 0.00079
## h_100 0.00083
## h_101 0.00086
## h_102 0.00089
## h_103 0.00098
## h_104 0.00106
## h_105 0.00112
## h_106 0.00120
## h_107 0.00126
## h_108 0.00131
## h_109 0.00138
## h_110 0.00148
## h_111 0.00159
## h_112 0.00167
## h_113 0.00175
## h_114 0.00183
## h_115 0.00189
## h_116 0.00199
## h_117 0.00208
## h_118 0.00219
## h_119 0.00233
## h_120 0.00244
## h_121 0.00256
## h_122 0.00271
## h_123 0.00286
## h_124 0.00301
## h_125 0.00312
## h_126 0.00324
## h_127 0.00335
## h_128 0.00357
## h_129 0.00362
## h_130 0.00373
## h_131 0.00373
## h_132 0.00383
## h_133 0.00387
## h_134 0.00385
## h_135 0.00387
## h_136 0.00382
## h_137 0.00379
## h_138 0.00370
## h_139 0.00361
## h_140 0.00355
## h_141 0.00341
## h_142 0.00325
## h_143 0.00315
## h_144 0.00306
## h_145 0.00296
## h_146 0.00293
## h_147 0.00283
## h_148 0.00274
## h_149 0.00268
## h_150 0.00262
## h_151 0.00258
## h_152 0.00256
## h_153 0.00251
## h_154 0.00248
## h_155 0.00245
## h_156 0.00242
## h_157 0.00239
## h_158 0.00240
## h_159 0.00241
## h_160 0.00240
## h_161 0.00242
## h_162 0.00249
## h_163 0.00252
## h_164 0.00256
## h_165 0.00261
## h_166 0.00264
## h_167 0.00270
## h_168 0.00271
## h_169 0.00272
## h_170 0.00270
## h_171 0.00264
## h_172 0.00271
## h_173 0.00272
## h_174 0.00269
## h_175 0.00275
## h_176 0.00281
## h_177 0.00281
## h_178 0.00277
## h_179 0.00278
## h_180 0.00279
## h_181 0.00278
## h_182 0.00277
## h_183 0.00275
## h_184 0.00276
## h_185 0.00275
## h_186 0.00275
## h_187 0.00279
## h_188 0.00276
## h_189 0.00271
## h_190 0.00275
## h_191 0.00278
## h_192 0.00280
## h_193 0.00283
## h_194 0.00285
## h_195 0.00285
## h_196 0.00281
## h_197 0.00284
## h_198 0.00279
## h_199 0.00275
## h_200 0.00265
## h_201 0.00266
## h_202 0.00268
## h_203 0.00262
## h_204 0.00256
## h_205 0.00247
## h_206 0.00240
## h_207 0.00237
## h_208 0.00236
## h_209 0.00234
## h_210 0.00229
## h_211 0.00229
## h_212 0.00225
## h_213 0.00218
## h_214 0.00215
## h_215 0.00209
## h_216 0.00202
## h_217 0.00194
## h_218 0.00196
## h_219 0.00196
## h_220 0.00196
## h_221 0.00192
## h_222 0.00188
## h_223 0.00184
## h_224 0.00184
## h_225 0.00186
## h_226 0.00186
## h_227 0.00184
## h_228 0.00185
## h_229 0.00184
## h_230 0.00184
## h_231 0.00185
## h_232 0.00186
## h_233 0.00185
## h_234 0.00186
## h_235 0.00188
## h_236 0.00189
## h_237 0.00189
## h_238 0.00190
## h_239 0.00189
## h_240 0.00188
## h_241 0.00186
## h_242 0.00183
## h_243 0.00188
## h_244 0.00190
## h_245 0.00193
## h_246 0.00196
## h_247 0.00194
## h_248 0.00195
## h_249 0.00193
## h_250 0.00191
## h_251 0.00189
## h_252 0.00193
## h_253 0.00197
## h_254 0.00202
## h_255 0.00204
## h_256 0.00208
## h_257 0.00212
## h_258 0.00216
## h_259 0.00219
## h_260 0.00224
## h_261 0.00229
## h_262 0.00235
## h_263 0.00239
## h_264 0.00240
## h_265 0.00244
## h_266 0.00251
## h_267 0.00259
## h_268 0.00261
## h_269 0.00267
## h_270 0.00267
## h_271 0.00262
## h_272 0.00253
## h_273 0.00258
## h_274 0.00254
## h_275 0.00249
## h_276 0.00241
## h_277 0.00233
## h_278 0.00227
## h_279 0.00221
## h_280 0.00214
## h_281 0.00205
## h_282 0.00198
## h_283 0.00192
## h_284 0.00186
## h_285 0.00180
## h_286 0.00173
## h_287 0.00164
## h_288 0.00162
## h_289 0.00159
## h_290 0.00160
## h_291 0.00157
## h_292 0.00156
## h_293 0.00154
## h_294 0.00149
## h_295 0.00143
## h_296 0.00140
## h_297 0.00132
## h_298 0.00129
## h_299 0.00130
## h_300 0.00130
## h_301 0.00129
## h_302 0.00128
## h_303 0.00126
## h_304 0.00125
## h_305 0.00123
## h_306 0.00122
## h_307 0.00119
## h_308 0.00117
## h_309 0.00116
## h_310 0.00114
## h_311 0.00111
## h_312 0.00108
## h_313 0.00107
## h_314 0.00106
## h_315 0.00105
## h_316 0.00106
## h_317 0.00104
## h_318 0.00102
## h_319 0.00100
## h_320 0.00100
## h_321 0.00100
## h_322 0.00099
## h_323 0.00099
## h_324 0.00097
## h_325 0.00097
## h_326 0.00096
## h_327 0.00095
## h_328 0.00093
## h_329 0.00092
## h_330 0.00091
## h_331 0.00089
## h_332 0.00088
## h_333 0.00086
## h_334 0.00085
## h_335 0.00085
## h_336 0.00084
## h_337 0.00084
## h_338 0.00084
## h_339 0.00084
## h_340 0.00085
## h_341 0.00085
## h_342 0.00084
## h_343 0.00084
## h_344 0.00083
## h_345 0.00083
## h_346 0.00083
## h_347 0.00083
## h_348 0.00083
## h_349 0.00082
## h_350 0.00081
## h_351 0.00081
## h_352 0.00082
## h_353 0.00083
## h_354 0.00084
## h_355 0.00085
## h_356 0.00086
## h_357 0.00087
## h_358 0.00088
## h_359 0.00087
## h_360 0.00088
## h_361 0.00089
## h_362 0.00090
## h_363 0.00090
## h_364 0.00089
## h_365 0.00091
## h_366 0.00092
## h_367 0.00092
## h_368 0.00093
## h_369 0.00095
## h_370 0.00096
## h_371 0.00096
## h_372 0.00096
## h_373 0.00096
## h_374 0.00097
## h_375 0.00099
## h_376 0.00099
## h_377 0.00098
## h_378 0.00099
## h_379 0.00099
## h_380 0.00098
## h_381 0.00097
## h_382 0.00097
## h_383 0.00096
## h_384 0.00094
## h_385 0.00094
## h_386 0.00094
## h_387 0.00094
## h_388 0.00094
## h_389 0.00092
## h_390 0.00091
## h_391 0.00089
## h_392 0.00088
## h_393 0.00087
## h_394 0.00085
## h_395 0.00084
## h_396 0.00082
## h_397 0.00081
## h_398 0.00079
## h_399 0.00079
## h_400 0.00078
## h_401 0.00077
## h_402 0.00077
## h_403 0.00075
## h_404 0.00073
## h_405 0.00072
## h_406 0.00071
## h_407 0.00070
## h_408 0.00069
## h_409 0.00068
## h_410 0.00068
## h_411 0.00068
## h_412 0.00067
## h_413 0.00066
## h_414 0.00066
## h_415 0.00064
## h_416 0.00065
## h_417 0.00066
## h_418 0.00066
## h_419 0.00065
## h_420 0.00063
## h_421 0.00064
## h_422 0.00064
## h_423 0.00065
## h_424 0.00065
## h_425 0.00064
## h_426 0.00064
## h_427 0.00063
## h_428 0.00065
## h_429 0.00066
## h_430 0.00067
## h_431 0.00067
## h_432 0.00068
## h_433 0.00068
## h_434 0.00069
## h_435 0.00068
## h_436 0.00070
## h_437 0.00072
## h_438 0.00074
## h_439 0.00075
## h_440 0.00075
## h_441 0.00077
## h_442 0.00078
## h_443 0.00079
## h_444 0.00080
## h_445 0.00081
## h_446 0.00082
## h_447 0.00083
## h_448 0.00083
## h_449 0.00084
## h_450 0.00085
## h_451 0.00085
## h_452 0.00084
## h_453 0.00083
## h_454 0.00082
## h_455 0.00081
## h_456 0.00081
## h_457 0.00080
## h_458 0.00079
## h_459 0.00078
## h_460 0.00076
## h_461 0.00076
## h_462 0.00074
## h_463 0.00072
## h_464 0.00070
## h_465 0.00067
## h_466 0.00065
## h_467 0.00063
## h_468 0.00060
## h_469 0.00058
## h_470 0.00057
## h_471 0.00056
## h_472 0.00054
## h_473 0.00051
## h_474 0.00049
## h_475 0.00048
## h_476 0.00048
## h_477 0.00047
## h_478 0.00047
## h_479 0.00046
## h_480 0.00045
## h_481 0.00044
## h_482 0.00044
## h_483 0.00044
## h_484 0.00044
## h_485 0.00044
## h_486 0.00044
## h_487 0.00044
## h_488 0.00044
## h_489 0.00045
## h_490 0.00045
## h_491 0.00046
## h_492 0.00047
## h_493 0.00047
## h_494 0.00048
## h_495 0.00048
## h_496 0.00049
## h_497 0.00049
## h_498 0.00048
## h_499 0.00049
## h_500 0.00051
## h_501 0.00051
## h_502 0.00052
## h_503 0.00052
## h_504 0.00052
## h_505 0.00052
## h_506 0.00052
## h_507 0.00051
## h_508 0.00051
## h_509 0.00051
## h_510 0.00050
## h_511 0.00051
## h_512 0.00052
## h_513 0.00052
## h_514 0.00053
## h_515 0.00054
## h_516 0.00055
## h_517 0.00056
## h_518 0.00057
## h_519 0.00057
## h_520 0.00058
## h_521 0.00059
## h_522 0.00059
## h_523 0.00059
## h_524 0.00059
## h_525 0.00061
## h_526 0.00064
## h_527 0.00066
## h_528 0.00068
## h_529 0.00071
## h_530 0.00073
## h_531 0.00076
## h_532 0.00080
## h_533 0.00082
## h_534 0.00084
## h_535 0.00087
## h_536 0.00089
## h_537 0.00091
## h_538 0.00093
## h_539 0.00095
## h_540 0.00096
## h_541 0.00097
## h_542 0.00096
## h_543 0.00101
## h_544 0.00102
## h_545 0.00103
## h_546 0.00102
## h_547 0.00102
## h_548 0.00101
## h_549 0.00102
## h_550 0.00101
## h_551 0.00102
## h_552 0.00103
## h_553 0.00104
## h_554 0.00104
## h_555 0.00104
## h_556 0.00106
## h_557 0.00108
## h_558 0.00109
## h_559 0.00110
## h_560 0.00112
## h_561 0.00112
## h_562 0.00112
## h_563 0.00112
## h_564 0.00114
## h_565 0.00115
## h_566 0.00116
## h_567 0.00116
## h_568 0.00116
## h_569 0.00117
## h_570 0.00116
## h_571 0.00114
## h_572 0.00114
## h_573 0.00114
## h_574 0.00113
## h_575 0.00111
## h_576 0.00110
## h_577 0.00109
## h_578 0.00109
## h_579 0.00108
## h_580 0.00107
## h_581 0.00107
## h_582 0.00108
## h_583 0.00108
## h_584 0.00107
## h_585 0.00106
## h_586 0.00106
## h_587 0.00107
## h_588 0.00110
## h_589 0.00112
## h_590 0.00113
## h_591 0.00113
## h_592 0.00114
## h_593 0.00113
## h_594 0.00113
## h_595 0.00115
## h_596 0.00116
## h_597 0.00119
## h_598 0.00120
## h_599 0.00120
## h_600 0.00120
## h_601 0.00124
## h_602 0.00128
## h_603 0.00132
## h_604 0.00135
## h_605 0.00136
## h_606 0.00140
## h_607 0.00143
## h_608 0.00145
## h_609 0.00145
## h_610 0.00143
## h_611 0.00143
## h_612 0.00147
## h_613 0.00146
## h_614 0.00146
## h_615 0.00149
## h_616 0.00150
## h_617 0.00148
## h_618 0.00147
## h_619 0.00147
## h_620 0.00146
## h_621 0.00145
## h_622 0.00144
## h_623 0.00143
## h_624 0.00142
## h_625 0.00141
## h_626 0.00139
## h_627 0.00138
## h_628 0.00137
## h_629 0.00134
## h_630 0.00135
## h_631 0.00135
## h_632 0.00134
## h_633 0.00134
## h_634 0.00132
## h_635 0.00132
## h_636 0.00130
## h_637 0.00130
## h_638 0.00129
## h_639 0.00126
## h_640 0.00124
## h_641 0.00122
## h_642 0.00121
## h_643 0.00123
## h_644 0.00121
## h_645 0.00121
## h_646 0.00121
## h_647 0.00118
## h_648 0.00116
## h_649 0.00112
## h_650 0.00111
## h_651 0.00112
## h_652 0.00111
## h_653 0.00112
## h_654 0.00110
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## h_2923 0.00332
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## h_2926 0.00316
## h_2927 0.00315
## h_2928 0.00311
## h_2929 0.00309
## h_2930 0.00303
## h_2931 0.00297
## h_2932 0.00291
## h_2933 0.00289
## h_2934 0.00285
## h_2935 0.00277
## h_2936 0.00272
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## h_2938 0.00276
## h_2939 0.00275
## h_2940 0.00271
## h_2941 0.00269
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## h_2943 0.00260
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## h_3515 0.00165
## h_3516 0.00166
## h_3517 0.00166
## h_3518 0.00167
## h_3519 0.00168
## h_3520 0.00168
## h_3521 0.00167
## h_3522 0.00169
## h_3523 0.00172
## h_3524 0.00173
## h_3525 0.00176
## h_3526 0.00176
## h_3527 0.00178
## h_3528 0.00181
## h_3529 0.00187
## h_3530 0.00190
## h_3531 0.00193
## h_3532 0.00194
## h_3533 0.00196
## h_3534 0.00201
## h_3535 0.00205
## h_3536 0.00210
## h_3537 0.00216
## h_3538 0.00220
## h_3539 0.00222
## h_3540 0.00228
## h_3541 0.00233
## h_3542 0.00239
## h_3543 0.00243
## h_3544 0.00246
## h_3545 0.00248
## h_3546 0.00248
## h_3547 0.00247
## h_3548 0.00245
## h_3549 0.00251
## h_3550 0.00255
## h_3551 0.00256
## h_3552 0.00253
## h_3553 0.00251
## h_3554 0.00251
## h_3555 0.00253
## h_3556 0.00252
## h_3557 0.00256
## h_3558 0.00256
## h_3559 0.00255
## h_3560 0.00253
## h_3561 0.00254
## h_3562 0.00256
## h_3563 0.00257
## h_3564 0.00262
## h_3565 0.00267
## h_3566 0.00270
## h_3567 0.00275
## h_3568 0.00281
## h_3569 0.00287
## h_3570 0.00292
## h_3571 0.00299
## h_3572 0.00304
## h_3573 0.00311
## h_3574 0.00318
## h_3575 0.00333
## h_3576 0.00342
## h_3577 0.00352
## h_3578 0.00353
## h_3579 0.00351
## h_3580 0.00348
## h_3581 0.00356
## h_3582 0.00361
## h_3583 0.00365
## h_3584 0.00370
## h_3585 0.00371
## h_3586 0.00368
## h_3587 0.00358
## h_3588 0.00350
## h_3589 0.00344
## h_3590 0.00341
## h_3591 0.00340
## h_3592 0.00336
## h_3593 0.00332
## h_3594 0.00330
## h_3595 0.00330
## h_3596 0.00330
## h_3597 0.00325
## h_3598 0.00321
## h_3599 0.00320
## h_3600 0.00317
## h_3601 0.00313
## h_3602 0.00315
## h_3603 0.00315
## h_3604 0.00317
## h_3605 0.00312
## h_3606 0.00307
## h_3607 0.00306
## h_3608 0.00301
## h_3609 0.00303
## h_3610 0.00304
## h_3611 0.00303
## h_3612 0.00304
## h_3613 0.00305
## h_3614 0.00302
## h_3615 0.00300
## h_3616 0.00308
## h_3617 0.00312
## h_3618 0.00315
## h_3619 0.00324
## h_3620 0.00338
## h_3621 0.00352
## h_3622 0.00368To forecast forward over thirty steps, we make use of the commands:
sv.pred <- predict(res.sv, 30)
sv.fore <- t(matrix(apply(100 * exp(sv.pred/2), 2, quantile,
c(0.05, 0.5, 0.95)), nrow = 3))To compare these forecasting results to those of the GARCH(1,1) model we could make use of the following code:
library(fGarch)
res.garch <- garchFit(zar.ins ~ garch(1, 1), data = zar.ins,
trace = F)
summary(res.garch)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = zar.ins ~ garch(1, 1), data = zar.ins, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x1341c408>
## [data = zar.ins]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 5.0531e-06 4.9620e-07 1.1217e-01 8.9495e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 5.053e-06 1.236e-04 0.041 0.967
## omega 4.962e-07 1.262e-07 3.933 8.38e-05 ***
## alpha1 1.122e-01 9.738e-03 11.518 < 2e-16 ***
## beta1 8.950e-01 8.031e-03 111.435 < 2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 11710.94 normalized: 3.23328
##
## Description:
## Wed Jun 6 08:00:12 2018 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 710.2091 0
## Shapiro-Wilk Test R W 0.9810006 0
## Ljung-Box Test R Q(10) 14.77983 0.1402975
## Ljung-Box Test R Q(15) 23.66422 0.07101947
## Ljung-Box Test R Q(20) 26.06154 0.1637891
## Ljung-Box Test R^2 Q(10) 4.012296 0.9467906
## Ljung-Box Test R^2 Q(15) 7.031697 0.9567642
## Ljung-Box Test R^2 Q(20) 8.924805 0.9837619
## LM Arch Test R TR^2 4.722974 0.9665976
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -6.464352 -6.457510 -6.464354 -6.461914garch.h_t <- 100 * (res.garch@sigma.t)
sv.h_t <- (100 * exp(res.sv$summary$latent[, 3:5]/2))
cond.vol <- cbind(garch.h_t, sv.h_t[, 2])
plot.ts(cond.vol, plot.type = "single", col = c("blue",
"green"))
legend("topleft", legend = c("garch", "stochvol"), lty = 1,
col = c("blue", "green"))
garch.pred <- predict(res.garch, n.ahead = 30, crit_val = 2)
garch.fore <- 100 * (garch.pred)
results <- cbind((100 * zar.out)^2, garch.fore[, 3], sv.fore[,
2])
plot.ts(results, plot.type = "single", col = c("red", "blue",
"green"))
legend("topright", legend = c("actual", "garch", "stochvol"),
lty = 1, col = c("red", "blue", "green"))
Where we note that in both cases, the models provide a poor result.