Problem Set #4
2025-04-09
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(stargazer)##
## Please cite as:## Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables.## R package version 5.2.3. https://CRAN.R-project.org/package=stargazerlibrary(urca)
gdp_data <- read.csv("A939RX0Q048SBEA.csv")
gdp_ts <- ts(gdp_data$A939RX0Q048SBEA, start = c(1947, 1), frequency = 4)ln_gdp <- log(gdp_ts)
ar1_model <- arima(ln_gdp, order = c(1, 0, 0))
stargazer(ar1_model, type = "text", title = "AR(1) Model for Log GDP")##
## AR(1) Model for Log GDP
## =============================================
## Dependent variable:
## ---------------------------
## ln_gdp
## ---------------------------------------------
## ar1 1.000***
## (0.0002)
##
## intercept 10.413***
## (0.699)
##
## ---------------------------------------------
## Observations 302
## Log Likelihood 896.046
## sigma2 0.0002
## Akaike Inf. Crit. -1,786.092
## =============================================
## Note: *p<0.1; **p<0.05; ***p<0.01residuals_ar1 <- residuals(ar1_model)
ar1_residuals <- arima(residuals_ar1, order = c(1, 0, 0), include.mean = FALSE)
stargazer(ar1_residuals, type = "text", title = "AR(1) Model for Residuals")##
## AR(1) Model for Residuals
## =============================================
## Dependent variable:
## ---------------------------
## residuals_ar1
## ---------------------------------------------
## ar1 0.265***
## (0.055)
##
## ---------------------------------------------
## Observations 302
## Log Likelihood 911.122
## sigma2 0.0001
## Akaike Inf. Crit. -1,818.243
## =============================================
## Note: *p<0.1; **p<0.05; ***p<0.01rho_hat <- coef(ar1_residuals)["ar1"]
ln_gdp_lag <- lag(ln_gdp, -1)
transformed_var <- ln_gdp - rho_hat * ln_gdp_lag
transformed_var <- na.omit(transformed_var)final_ar1 <- arima(transformed_var, order = c(1, 0, 0))
stargazer(final_ar1, type = "text", title = "AR(1) Model for Transformed Variable")##
## AR(1) Model for Transformed Variable
## =============================================
## Dependent variable:
## ---------------------------
## transformed_var
## ---------------------------------------------
## ar1 1.000***
## (0.0004)
##
## intercept 7.652***
## (0.513)
##
## ---------------------------------------------
## Observations 301
## Log Likelihood 903.944
## sigma2 0.0001
## Akaike Inf. Crit. -1,801.888
## =============================================
## Note: *p<0.1; **p<0.05; ***p<0.01phi_bar <- coef(final_ar1)["ar1"]
alpha <- coef(final_ar1)["intercept"]
rho <- rho_hatcheckresiduals(final_ar1)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,0,0) with non-zero mean
## Q* = 11.968, df = 7, p-value = 0.1016
##
## Model df: 1. Total lags used: 8checkresiduals(final_ar1, plot = FALSE)##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,0,0) with non-zero mean
## Q* = 11.968, df = 7, p-value = 0.1016
##
## Model df: 1. Total lags used: 8library(sandwich)
library(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericgdp_var <- gdp_ts
gdp_df <- data.frame(
y = log(gdp_var),
y_lag = log(lag(gdp_var, -1)))
gdp_df <- na.omit(gdp_df)
ar_lm <- lm(y ~ y_lag, data = gdp_df)hac_results <- coeftest(ar_lm,
vcov = kernHAC(ar_lm,
prewhite = 0,
kernel = "Quadratic Spectral"))## Warning in summary.lm(x): essentially perfect fit: summary may be unreliableols_results <- coeftest(ar_lm)## Warning in summary.lm(object, ...): essentially perfect fit: summary may be
## unreliable"HAC Standard Errors:"## [1] "HAC Standard Errors:"hac_results##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.1448e-14 1.0893e-15 -1.0510e+01 < 2.2e-16 ***
## y_lag 1.0000e+00 1.0213e-16 9.7913e+15 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"OLS Standard Errors:"## [1] "OLS Standard Errors:"ols_results##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.1448e-14 6.1222e-16 -1.8700e+01 < 2.2e-16 ***
## y_lag 1.0000e+00 5.8671e-17 1.7044e+16 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Part 4: GARCH
1.
wilshire <- read.csv("WILL5000INDFC.csv")
colnames(wilshire) <- c("time", "value")2.
library(tsbox)
wilshire <- subset(wilshire, value != ".")
wilshire$value <- as.numeric(wilshire$value)
if(!require(tsbox)) install.packages("tsbox")
wilshire_ts <- ts_ts(ts_long(wilshire))3.
pacf(wilshire_ts, lag.max = 365.2425, na.action = na.pass)
### (a) The PACF shows significant spikes at lags 1, 2, and some
seasonal lags, suggesting an AR(2) or AR(3) model might be appropriate.
Yes this does seem feasable.
4. & 5.
if(!require(fGarch)) install.packages("fGarch")## Loading required package: fGarch## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
##
## If needed attach them yourself in your R script by e.g.,
## require("timeSeries")if(!require(mice)) install.packages("mice")## Loading required package: mice##
## Attaching package: 'mice'## The following object is masked from 'package:stats':
##
## filter## The following objects are masked from 'package:base':
##
## cbind, rbindlibrary(fGarch)
library(mice)
wilshire_mice <- mice(ts_df(wilshire_ts), method = "norm.nob", m = 1)##
## iter imp variable
## 1 1 value
## 2 1 value
## 3 1 value
## 4 1 value
## 5 1 valuewilshire_imp <- ts_ts(complete(wilshire_mice))6.
garch_out<-garchFit(formula = ~arma(1,1) + garch(1,1), data = wilshire_imp)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 15296
## Recursion Init: mci
## Series Scale: 1.086248
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.50567183 0.5056718 0.05057028 TRUE
## ar1 -0.99999999 1.0000000 -0.26724848 TRUE
## ma1 -0.99999999 1.0000000 0.24583815 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 beta1
## 1 2 3 4 5 7
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 20411.381: 0.0505703 -0.267248 0.245838 0.100000 0.100000 0.800000
## 1: 20395.059: 0.0505718 -0.266319 0.246744 0.0936242 0.0979293 0.797505
## 2: 20363.202: 0.0505877 -0.257249 0.255571 0.0774887 0.108586 0.815151
## 3: 20356.383: 0.0505890 -0.256829 0.255971 0.0739031 0.106403 0.814115
## 4: 20344.267: 0.0506019 -0.252794 0.259813 0.0689535 0.103737 0.829666
## 5: 20304.655: 0.0506286 -0.248613 0.263581 0.0520653 0.0857533 0.853700
## 6: 20269.455: 0.0507088 -0.262902 0.247824 0.0119184 0.0615288 0.926466
## 7: 20262.669: 0.0507090 -0.262785 0.247938 0.0137037 0.0617405 0.927219
## 8: 20259.035: 0.0507105 -0.262394 0.248303 0.0135092 0.0599109 0.926828
## 9: 20254.153: 0.0507163 -0.261317 0.249277 0.0142420 0.0577959 0.929687
## 10: 20248.873: 0.0507300 -0.259031 0.251325 0.0125308 0.0525185 0.934264
## 11: 20248.598: 0.0507804 -0.253623 0.255894 0.0108168 0.0448105 0.945727
## 12: 20242.555: 0.0509708 -0.248980 0.257424 0.0113407 0.0461802 0.941146
## 13: 20239.513: 0.0510896 -0.248862 0.255552 0.0142593 0.0498671 0.935663
## 14: 20238.635: 0.0513858 -0.248459 0.250963 0.0148300 0.0475576 0.935762
## 15: 20237.055: 0.0516397 -0.242688 0.252522 0.0151510 0.0479589 0.935890
## 16: 20237.009: 0.0519011 -0.237146 0.253612 0.0149511 0.0487503 0.934504
## 17: 20236.417: 0.0520634 -0.235603 0.252236 0.0153357 0.0484544 0.935209
## 18: 20236.141: 0.0522235 -0.234362 0.250530 0.0151749 0.0476848 0.935662
## 19: 20235.933: 0.0525522 -0.231569 0.247212 0.0155256 0.0475089 0.935716
## 20: 20235.565: 0.0532032 -0.225493 0.241049 0.0159330 0.0485139 0.933974
## 21: 20234.776: 0.0563887 -0.192275 0.212704 0.0145999 0.0452775 0.938313
## 22: 20234.680: 0.0563888 -0.192280 0.212698 0.0147397 0.0453446 0.938372
## 23: 20234.647: 0.0563892 -0.192300 0.212670 0.0148040 0.0452954 0.938231
## 24: 20234.630: 0.0564020 -0.192176 0.212533 0.0149060 0.0453340 0.938225
## 25: 20234.604: 0.0564290 -0.191900 0.212262 0.0148841 0.0453086 0.938152
## 26: 20234.571: 0.0564832 -0.191346 0.211716 0.0149820 0.0453495 0.938103
## 27: 20232.987: 0.0652931 -0.0986378 0.125659 0.0152403 0.0472434 0.935980
## 28: 20232.599: 0.0676752 -0.0757590 0.0966464 0.0155123 0.0461372 0.937039
## 29: 20232.437: 0.0676751 -0.0757652 0.0966403 0.0153487 0.0460191 0.936882
## 30: 20232.425: 0.0676646 -0.0758495 0.0967408 0.0154134 0.0460451 0.936876
## 31: 20232.414: 0.0676432 -0.0760251 0.0969339 0.0154421 0.0460250 0.936755
## 32: 20232.400: 0.0675990 -0.0763429 0.0973500 0.0155243 0.0460662 0.936716
## 33: 20232.385: 0.0675081 -0.0769379 0.0981117 0.0155646 0.0460868 0.936565
## 34: 20232.183: 0.0658722 -0.0786194 0.100056 0.0163190 0.0473043 0.934606
## 35: 20232.106: 0.0658034 -0.0575490 0.0753088 0.0164647 0.0477831 0.933894
## 36: 20232.082: 0.0649328 -0.0378682 0.0559391 0.0166506 0.0477552 0.933761
## 37: 20232.034: 0.0604449 0.0301222 -0.0107285 0.0165600 0.0481484 0.933626
## 38: 20231.981: 0.0561562 0.100489 -0.0800078 0.0166930 0.0481974 0.933307
## 39: 20231.942: 0.0507397 0.152778 -0.132319 0.0167772 0.0480100 0.933359
## 40: 20231.928: 0.0514719 0.154771 -0.134914 0.0167464 0.0478811 0.933509
## 41: 20231.927: 0.0509148 0.163713 -0.143966 0.0167191 0.0478559 0.933568
## 42: 20231.927: 0.0510069 0.161691 -0.142006 0.0167130 0.0478456 0.933583
## 43: 20231.927: 0.0509759 0.162320 -0.142612 0.0167143 0.0478483 0.933580
## 44: 20231.927: 0.0509759 0.162322 -0.142614 0.0167143 0.0478482 0.933580
##
## Final Estimate of the Negative LLH:
## LLH: 21497.36 norm LLH: 1.405424
## mu ar1 ma1 omega alpha1 beta1
## 0.05537247 0.16232206 -0.14261412 0.01972173 0.04784823 0.93358002
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -23916.66244 -1611.3770 -27.47556 979.5106 6.600338e+01
## ar1 -1611.37700 -14368.5483 -14267.08711 -1406.6139 1.553302e+02
## ma1 -27.47556 -14267.0871 -14278.75800 -1468.2747 1.622687e+02
## omega 979.51059 -1406.6139 -1468.27471 -2473313.2819 -1.665822e+06
## alpha1 66.00338 155.3302 162.26875 -1665822.3535 -1.548837e+06
## beta1 113.14297 -1210.3305 -1216.87150 -2057251.3905 -1.635710e+06
## beta1
## mu 113.143
## ar1 -1210.330
## ma1 -1216.871
## omega -2057251.391
## alpha1 -1635709.811
## beta1 -1903384.256
## attr(,"time")
## Time difference of 0.2443931 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 1.231213 secsgarch_21 <- garchFit(formula = ~arma(2,1) + garch(1,1), data = wilshire_imp)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 15296
## Recursion Init: mci
## Series Scale: 1.086248
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.50567183 0.5056718 0.05056329 TRUE
## ar1 -0.99999999 1.0000000 0.13287262 TRUE
## ar2 -0.99999999 1.0000000 0.01510075 TRUE
## ma1 -0.99999999 1.0000000 -0.15335324 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 beta1
## 1 2 3 4 5 6 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 20405.827: 0.0505633 0.132873 0.0151007 -0.153353 0.100000 0.100000 0.800000
## 1: 20389.422: 0.0505635 0.133727 0.0149829 -0.152506 0.0935931 0.0979311 0.797484
## 2: 20357.677: 0.0505656 0.142181 0.0136991 -0.144131 0.0771058 0.108807 0.815277
## 3: 20351.049: 0.0505658 0.142579 0.0135645 -0.143738 0.0735746 0.106634 0.814282
## 4: 20339.349: 0.0505671 0.146346 0.0122469 -0.140007 0.0689097 0.103865 0.829673
## 5: 20299.648: 0.0505692 0.150671 0.00940542 -0.135723 0.0522265 0.0862843 0.853170
## 6: 20263.248: 0.0505730 0.137101 0.00231670 -0.149181 0.0105597 0.0582482 0.930799
## 7: 20255.090: 0.0505730 0.137193 0.00233455 -0.149090 0.0123752 0.0585694 0.931576
## 8: 20251.257: 0.0505732 0.137678 0.00239045 -0.148607 0.0121347 0.0567033 0.931487
## 9: 20246.686: 0.0505738 0.138914 0.00252876 -0.147379 0.0128907 0.0544936 0.934239
## 10: 20242.172: 0.0505752 0.141554 0.00289746 -0.144760 0.0112814 0.0492353 0.938725
## 11: 20240.931: 0.0505781 0.144804 0.00368962 -0.141587 0.0111557 0.0456269 0.944205
## 12: 20235.552: 0.0505888 0.146368 0.00484512 -0.140439 0.0129083 0.0489988 0.937516
## 13: 20234.840: 0.0505959 0.146622 -0.00146574 -0.140576 0.0144927 0.0460852 0.939199
## 14: 20234.354: 0.0506040 0.149781 0.00202873 -0.137788 0.0129987 0.0436770 0.941008
## 15: 20232.299: 0.0506104 0.151851 0.00713406 -0.135991 0.0138633 0.0457849 0.939422
## 16: 20232.015: 0.0506166 0.151433 0.00300707 -0.136569 0.0146934 0.0485003 0.935462
## 17: 20231.838: 0.0506166 0.151448 0.00301198 -0.136555 0.0149641 0.0484848 0.935531
## 18: 20231.697: 0.0506166 0.151468 0.00301895 -0.136536 0.0149861 0.0482559 0.935373
## 19: 20231.572: 0.0506182 0.151575 0.00345912 -0.136499 0.0153039 0.0481951 0.935420
## 20: 20231.416: 0.0506231 0.152079 0.00438022 -0.136208 0.0152292 0.0479937 0.935296
## 21: 20231.325: 0.0506387 0.153799 0.00368894 -0.135106 0.0153178 0.0477309 0.935752
## 22: 20231.221: 0.0506800 0.152310 0.00622126 -0.138288 0.0157790 0.0470945 0.935464
## 23: 20231.167: 0.0506801 0.152503 0.00615928 -0.138098 0.0161845 0.0468804 0.935261
## 24: 20231.148: 0.0506801 0.152517 0.00615469 -0.138085 0.0161085 0.0468349 0.935191
## 25: 20231.138: 0.0506810 0.152571 0.00606764 -0.138066 0.0161464 0.0468470 0.935208
## 26: 20231.125: 0.0506827 0.152705 0.00590877 -0.137999 0.0161389 0.0468334 0.935150
## 27: 20231.098: 0.0506887 0.153003 0.00559733 -0.137949 0.0162158 0.0468128 0.935161
## 28: 20231.070: 0.0506993 0.153531 0.00626272 -0.137887 0.0162767 0.0469099 0.934927
## 29: 20231.035: 0.0507220 0.154172 0.00464376 -0.138200 0.0163733 0.0468555 0.934891
## 30: 20231.019: 0.0507546 0.155263 0.00503866 -0.138733 0.0162069 0.0477480 0.934201
## 31: 20230.997: 0.0507546 0.155270 0.00503665 -0.138727 0.0162863 0.0477737 0.934241
## 32: 20230.990: 0.0507546 0.155301 0.00501234 -0.138712 0.0162976 0.0477005 0.934204
## 33: 20230.980: 0.0507556 0.155422 0.00492135 -0.138743 0.0163480 0.0476494 0.934276
## 34: 20230.972: 0.0507605 0.155741 0.00490728 -0.138889 0.0163412 0.0475814 0.934274
## 35: 20230.963: 0.0507706 0.156331 0.00504821 -0.139253 0.0164069 0.0476333 0.934206
## 36: 20230.953: 0.0507787 0.157421 0.00495146 -0.140225 0.0164754 0.0476666 0.934022
## 37: 20230.949: 0.0507381 0.158225 0.00441469 -0.140937 0.0164819 0.0474328 0.934276
## 38: 20230.942: 0.0506974 0.159244 0.00465660 -0.141615 0.0164532 0.0474506 0.934240
## 39: 20230.937: 0.0506664 0.160234 0.00481213 -0.142496 0.0165337 0.0475913 0.934076
## 40: 20230.935: 0.0506358 0.161127 0.00476552 -0.143488 0.0165860 0.0476910 0.933864
## 41: 20230.932: 0.0505984 0.162081 0.00476819 -0.144353 0.0165963 0.0476910 0.933882
## 42: 20230.930: 0.0505020 0.163816 0.00480570 -0.145828 0.0165515 0.0476201 0.933977
## 43: 20230.920: 0.0499448 0.172195 0.00416582 -0.154262 0.0166994 0.0477118 0.933693
## 44: 20230.919: 0.0499448 0.172198 0.00416465 -0.154260 0.0167083 0.0477226 0.933702
## 45: 20230.918: 0.0499448 0.172216 0.00415696 -0.154242 0.0166834 0.0477382 0.933696
## 46: 20230.918: 0.0499444 0.172274 0.00415419 -0.154294 0.0166920 0.0477471 0.933703
## 47: 20230.917: 0.0499436 0.172394 0.00414693 -0.154397 0.0166814 0.0477506 0.933697
## 48: 20230.917: 0.0499417 0.172620 0.00413894 -0.154618 0.0166871 0.0477574 0.933701
## 49: 20230.916: 0.0499368 0.173073 0.00412207 -0.155056 0.0166792 0.0477601 0.933693
## 50: 20230.916: 0.0499190 0.173941 0.00409655 -0.155920 0.0166865 0.0477677 0.933693
## 51: 20230.887: 0.0459617 0.249998 0.00240166 -0.231147 0.0168572 0.0479695 0.933270
## 52: 20230.880: 0.0445618 0.265253 0.00207769 -0.245995 0.0168786 0.0479878 0.933263
## 53: 20230.874: 0.0434861 0.284829 0.00103103 -0.265552 0.0167216 0.0476852 0.933696
## 54: 20230.871: 0.0423445 0.303483 0.000538829 -0.284573 0.0166941 0.0478068 0.933651
## 55: 20230.871: 0.0423445 0.303484 0.000538505 -0.284572 0.0166885 0.0478051 0.933647
## 56: 20230.870: 0.0423444 0.303481 0.000538146 -0.284568 0.0166909 0.0478078 0.933648
## 57: 20230.870: 0.0423441 0.303481 0.000536159 -0.284557 0.0166863 0.0478134 0.933644
## 58: 20230.870: 0.0423436 0.303463 0.000535111 -0.284536 0.0166889 0.0478164 0.933643
## 59: 20230.870: 0.0423270 0.302396 0.000483001 -0.283321 0.0167106 0.0478693 0.933568
## 60: 20230.870: 0.0423868 0.301599 0.000471091 -0.282484 0.0167098 0.0478637 0.933574
## 61: 20230.870: 0.0423628 0.302079 0.000449654 -0.282905 0.0167145 0.0478703 0.933563
## 62: 20230.870: 0.0423902 0.301629 0.000453415 -0.282463 0.0167139 0.0478697 0.933564
##
## Final Estimate of the Negative LLH:
## LLH: 21496.3 norm LLH: 1.405354
## mu ar1 ar2 ma1 omega
## 0.0460462558 0.3016287965 0.0004534149 -0.2824626429 0.0197212578
## alpha1 beta1
## 0.0478697266 0.9335641234
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -34162.47278 -2316.3254 -2384.05207 -55.14521 1154.1153
## ar1 -2316.32542 -15422.1434 -4718.00288 -15150.17485 -1404.9683
## ar2 -2384.05207 -4718.0029 -15362.11134 -4165.04624 230.7877
## ma1 -55.14521 -15150.1748 -4165.04624 -15043.10196 -1478.0505
## omega 1154.11530 -1404.9683 230.78768 -1478.05046 -2472414.8242
## alpha1 51.75001 145.1154 -9.57813 141.67306 -1664861.9947
## beta1 127.66010 -1185.7886 290.76717 -1197.45548 -2056329.0484
## alpha1 beta1
## mu 5.175001e+01 127.6601
## ar1 1.451154e+02 -1185.7886
## ar2 -9.578130e+00 290.7672
## ma1 1.416731e+02 -1197.4555
## omega -1.664862e+06 -2056329.0484
## alpha1 -1.547624e+06 -1634682.1830
## beta1 -1.634682e+06 -1902431.0622
## attr(,"time")
## Time difference of 0.3225029 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 2.502192 secsgarch_12 <- garchFit(formula = ~arma(1,2) + garch(1,1), data = wilshire_imp)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 2)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 1 2
## Max ARMA Order: 2
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 15296
## Recursion Init: mci
## Series Scale: 1.086248
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.50567183 0.5056718 0.05054327 TRUE
## ar1 -0.99999999 1.0000000 0.13143031 TRUE
## ma1 -0.99999999 1.0000000 -0.15199348 TRUE
## ma2 -0.99999999 1.0000000 0.01563575 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 ma2 omega alpha1 beta1
## 1 2 3 4 5 6 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 20405.944: 0.0505433 0.131430 -0.151993 0.0156357 0.100000 0.100000 0.800000
## 1: 20389.538: 0.0505436 0.132289 -0.151145 0.0155065 0.0935939 0.0979306 0.797484
## 2: 20357.763: 0.0505469 0.140778 -0.142762 0.0141126 0.0771215 0.108796 0.815269
## 3: 20351.128: 0.0505471 0.141178 -0.142368 0.0139729 0.0735889 0.106624 0.814273
## 4: 20339.386: 0.0505493 0.144968 -0.138634 0.0126067 0.0689147 0.103861 0.829659
## 5: 20299.699: 0.0505532 0.149317 -0.134366 0.00970515 0.0522393 0.0862666 0.853162
## 6: 20263.098: 0.0505604 0.135742 -0.147918 0.00267225 0.0106327 0.0583546 0.930628
## 7: 20255.060: 0.0505604 0.135835 -0.147826 0.00268893 0.0124419 0.0586699 0.931403
## 8: 20251.252: 0.0505607 0.136317 -0.147348 0.00273379 0.0122019 0.0568097 0.931304
## 9: 20246.692: 0.0505614 0.137545 -0.146132 0.00284203 0.0129550 0.0546102 0.934053
## 10: 20242.164: 0.0505632 0.140170 -0.143538 0.00314352 0.0113450 0.0493679 0.938529
## 11: 20240.882: 0.0505670 0.143400 -0.140403 0.00381703 0.0112091 0.0457624 0.944015
## 12: 20235.490: 0.0505835 0.145360 -0.138997 0.00486843 0.0129185 0.0490427 0.937448
## 13: 20234.743: 0.0505971 0.145137 -0.139764 -0.00109587 0.0144572 0.0462373 0.939031
## 14: 20234.310: 0.0506077 0.148168 -0.137162 0.00234589 0.0130293 0.0438100 0.940894
## 15: 20232.347: 0.0506155 0.150398 -0.135239 0.00720852 0.0138543 0.0458168 0.939409
## 16: 20232.066: 0.0506267 0.150495 -0.135436 0.00314590 0.0146230 0.0484913 0.935548
## 17: 20232.001: 0.0507149 0.148950 -0.140121 0.00605475 0.0161777 0.0476994 0.934949
## 18: 20231.644: 0.0507511 0.151497 -0.138875 0.00581215 0.0157198 0.0467516 0.935252
## 19: 20231.151: 0.0507711 0.153291 -0.137769 0.00363496 0.0158974 0.0465976 0.935900
## 20: 20231.036: 0.0508073 0.155685 -0.136803 0.00524308 0.0158362 0.0466498 0.935686
## 21: 20231.007: 0.0508551 0.156931 -0.137521 0.00298269 0.0160235 0.0476165 0.934809
## 22: 20231.007: 0.0508696 0.157027 -0.138138 0.00426582 0.0160598 0.0477240 0.934357
## 23: 20230.971: 0.0508764 0.157095 -0.138404 0.00493223 0.0162276 0.0478474 0.934300
## 24: 20230.952: 0.0508808 0.157314 -0.138380 0.00423513 0.0161925 0.0476749 0.934404
## 25: 20230.950: 0.0508808 0.157317 -0.138378 0.00423057 0.0163381 0.0476254 0.934392
## 26: 20230.936: 0.0508828 0.157356 -0.138446 0.00432228 0.0163035 0.0475878 0.934320
## 27: 20230.933: 0.0508857 0.157421 -0.138540 0.00439461 0.0163458 0.0475990 0.934303
## 28: 20230.931: 0.0508925 0.157593 -0.138745 0.00444145 0.0163589 0.0475848 0.934249
## 29: 20230.928: 0.0509051 0.157973 -0.139106 0.00423150 0.0163846 0.0475876 0.934258
## 30: 20230.924: 0.0509279 0.158732 -0.139961 0.00426098 0.0164182 0.0476234 0.934144
## 31: 20230.923: 0.0509336 0.159395 -0.140705 0.00452323 0.0164894 0.0476876 0.934049
## 32: 20230.919: 0.0509318 0.160128 -0.141262 0.00403299 0.0164666 0.0476425 0.934090
## 33: 20230.918: 0.0509318 0.160134 -0.141257 0.00402963 0.0164975 0.0476079 0.934049
## 34: 20230.917: 0.0509311 0.160172 -0.141288 0.00400730 0.0165222 0.0476172 0.934056
## 35: 20230.917: 0.0509303 0.160213 -0.141326 0.00399058 0.0165173 0.0476123 0.934043
## 36: 20230.916: 0.0509287 0.160296 -0.141399 0.00396101 0.0165476 0.0476173 0.934019
## 37: 20230.915: 0.0509254 0.160463 -0.141566 0.00392620 0.0165451 0.0476243 0.933992
## 38: 20230.913: 0.0509028 0.161702 -0.142847 0.00386107 0.0165817 0.0477374 0.933860
## 39: 20230.912: 0.0508802 0.163045 -0.144027 0.00374358 0.0165868 0.0477132 0.933845
## 40: 20230.911: 0.0508406 0.164287 -0.145144 0.00368397 0.0166076 0.0477169 0.933855
## 41: 20230.910: 0.0507757 0.165245 -0.146062 0.00375807 0.0166014 0.0477189 0.933840
## 42: 20230.907: 0.0501415 0.172317 -0.153285 0.00318750 0.0166381 0.0476936 0.933802
## 43: 20230.899: 0.0488860 0.190965 -0.172158 0.00313856 0.0166865 0.0477020 0.933773
## 44: 20230.892: 0.0480170 0.213617 -0.194771 0.00241804 0.0166688 0.0475794 0.933872
## 45: 20230.888: 0.0467047 0.232162 -0.212162 0.00175673 0.0165840 0.0478524 0.933755
## 46: 20230.883: 0.0454685 0.250837 -0.231499 0.00167975 0.0165753 0.0478700 0.933667
## 47: 20230.872: 0.0443155 0.270796 -0.251545 0.00128362 0.0166770 0.0478512 0.933656
## 48: 20230.870: 0.0439017 0.277251 -0.257859 0.00111702 0.0167065 0.0478117 0.933615
## 49: 20230.870: 0.0435395 0.284100 -0.264949 0.00100476 0.0166843 0.0478345 0.933619
## 50: 20230.869: 0.0431369 0.290614 -0.271478 0.000863630 0.0167063 0.0478431 0.933607
## 51: 20230.869: 0.0427018 0.296720 -0.277572 0.000718489 0.0167142 0.0478743 0.933560
## 52: 20230.869: 0.0427888 0.295635 -0.276501 0.000705372 0.0167173 0.0478708 0.933558
## 53: 20230.869: 0.0427915 0.295447 -0.276310 0.000728209 0.0167105 0.0478636 0.933574
## 54: 20230.869: 0.0427981 0.295363 -0.276225 0.000738160 0.0167147 0.0478714 0.933562
## 55: 20230.869: 0.0427823 0.295631 -0.276493 0.000724189 0.0167137 0.0478683 0.933566
##
## Final Estimate of the Negative LLH:
## LLH: 21496.3 norm LLH: 1.405354
## mu ar1 ma1 ma2 omega
## 0.0464721394 0.2956305608 -0.2764933057 0.0007241893 0.0197210433
## alpha1 beta1
## 0.0478683223 0.9335655452
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 ma2 omega
## mu -33534.17641 -2273.1670 -53.55374 -121.42405 1143.6507
## ar1 -2273.16699 -15358.9552 -15092.53951 -4356.28838 -1403.1306
## ma1 -53.55374 -15092.5395 -14987.99053 -3974.25975 -1475.5392
## ma2 -121.42405 -4356.2884 -3974.25975 -14578.82438 191.5599
## omega 1143.65069 -1403.1306 -1475.53922 191.55988 -2472509.3888
## alpha1 51.51849 147.1725 143.60193 6.07595 -1664941.8334
## beta1 126.88106 -1185.6213 -1197.25953 302.16323 -2056413.3347
## alpha1 beta1
## mu 5.151849e+01 126.8811
## ar1 1.471725e+02 -1185.6213
## ma1 1.436019e+02 -1197.2595
## ma2 6.075950e+00 302.1632
## omega -1.664942e+06 -2056413.3347
## alpha1 -1.547707e+06 -1634758.5137
## beta1 -1.634759e+06 -1902507.8437
## attr(,"time")
## Time difference of 0.3276031 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 1.919876 secsgarch_22 <- garchFit(formula = ~arma(2,2) + garch(1,1), data = wilshire_imp)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 2)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 2 2
## Max ARMA Order: 2
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 15296
## Recursion Init: mci
## Series Scale: 1.086248## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.50567183 0.5056718 0.05140263 TRUE
## ar1 -0.99999999 1.0000000 0.20705614 TRUE
## ar2 -0.99999999 1.0000000 -0.60822176 TRUE
## ma1 -0.99999999 1.0000000 -0.22409936 TRUE
## ma2 -0.99999999 1.0000000 0.62684493 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 ma2 omega alpha1 beta1
## 1 2 3 4 5 6 7 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 20412.535: 0.0514026 0.207056 -0.608222 -0.224099 0.626845 0.100000 0.100000 0.800000
## 1: 20395.900: 0.0514042 0.208114 -0.608463 -0.223060 0.626555 0.0936065 0.0979235 0.797505
## 2: 20364.281: 0.0514198 0.217861 -0.610838 -0.213508 0.623725 0.0779684 0.108443 0.814968
## 3: 20362.658: 0.0514223 0.218349 -0.611076 -0.213060 0.623426 0.0708581 0.104162 0.812811
## 4: 20354.065: 0.0514234 0.218537 -0.611170 -0.212890 0.623307 0.0731141 0.105597 0.816172
## 5: 20320.629: 0.0514509 0.223118 -0.613425 -0.208755 0.620386 0.0580530 0.0972570 0.845113
## 6: 20291.364: 0.0514912 0.219834 -0.608154 -0.212861 0.624227 0.0417851 0.0798488 0.870026
## 7: 20267.262: 0.0515502 0.214546 -0.605384 -0.219347 0.624653 0.0333098 0.0665462 0.900541
## 8: 20259.513: 0.0515765 0.216646 -0.609608 -0.217725 0.619476 0.0304973 0.0578965 0.904565
## 9: 20251.643: 0.0516214 0.213056 -0.607624 -0.222213 0.620189 0.0241020 0.0613456 0.912300
## 10: 20251.383: 0.0516227 0.213780 -0.607683 -0.221507 0.620098 0.0247157 0.0610586 0.913343
## 11: 20249.537: 0.0516285 0.214389 -0.607167 -0.221005 0.620456 0.0239538 0.0601007 0.913573
## 12: 20247.795: 0.0516432 0.215812 -0.606289 -0.219852 0.620940 0.0238875 0.0588767 0.915665
## 13: 20244.618: 0.0516885 0.217190 -0.607573 -0.219339 0.618406 0.0221968 0.0552314 0.919421
## 14: 20242.143: 0.0517794 0.218806 -0.607134 -0.219395 0.616727 0.0205650 0.0533998 0.924561
## 15: 20241.519: 0.0518366 0.220885 -0.602660 -0.218343 0.620010 0.0191035 0.0528995 0.925748
## 16: 20240.037: 0.0519904 0.223340 -0.603992 -0.218597 0.615122 0.0195812 0.0526690 0.925695
## 17: 20239.876: 0.0519925 0.223746 -0.604118 -0.218225 0.614947 0.0196197 0.0524280 0.926287
## 18: 20239.617: 0.0520105 0.223851 -0.604206 -0.218440 0.614441 0.0192936 0.0520524 0.926543
## 19: 20239.398: 0.0520553 0.224167 -0.604170 -0.218921 0.613426 0.0191137 0.0516576 0.927384
## 20: 20239.110: 0.0521738 0.225558 -0.602741 -0.219675 0.612085 0.0188576 0.0512541 0.927709
## 21: 20238.904: 0.0522911 0.227088 -0.601039 -0.220290 0.611013 0.0189436 0.0512950 0.927838
## 22: 20237.806: 0.0537505 0.240826 -0.583809 -0.233565 0.592534 0.0163075 0.0506970 0.931608
## 23: 20237.075: 0.0552111 0.255723 -0.566523 -0.245928 0.574165 0.0172479 0.0481095 0.933837
## 24: 20235.691: 0.0566710 0.271379 -0.548707 -0.257700 0.556273 0.0164904 0.0465526 0.934065
## 25: 20234.224: 0.0580982 0.287732 -0.531575 -0.269175 0.537181 0.0175803 0.0476700 0.932779
## 26: 20233.426: 0.0594774 0.300153 -0.512268 -0.284986 0.519056 0.0169313 0.0459783 0.935029
## 27: 20233.376: 0.0594775 0.300151 -0.512269 -0.284991 0.519050 0.0167814 0.0460136 0.934977
## 28: 20233.336: 0.0594776 0.300149 -0.512271 -0.284998 0.519043 0.0167908 0.0461597 0.935047
## 29: 20233.298: 0.0594867 0.300244 -0.512156 -0.285099 0.518905 0.0166581 0.0461891 0.934987
## 30: 20233.259: 0.0595066 0.300459 -0.511895 -0.285317 0.518605 0.0166865 0.0463089 0.935030
## 31: 20233.213: 0.0595472 0.300901 -0.511362 -0.285752 0.518001 0.0165900 0.0463503 0.934957
## 32: 20233.149: 0.0596288 0.301798 -0.510284 -0.286619 0.516799 0.0166336 0.0464715 0.934946
## 33: 20231.749: 0.0645632 0.356621 -0.444298 -0.338523 0.444794 0.0180202 0.0479519 0.931594
## 34: 20231.692: 0.0645632 0.356616 -0.444280 -0.338529 0.444810 0.0180613 0.0480879 0.931708
## 35: 20231.651: 0.0645641 0.356626 -0.444234 -0.338550 0.444818 0.0178903 0.0481132 0.931681
## 36: 20231.612: 0.0645724 0.356752 -0.444022 -0.338672 0.444712 0.0179174 0.0482315 0.931763
## 37: 20231.574: 0.0645902 0.357030 -0.443584 -0.338928 0.444464 0.0177910 0.0482573 0.931716
## 38: 20231.534: 0.0646259 0.357609 -0.442714 -0.339442 0.443928 0.0178293 0.0483735 0.931747
## 39: 20231.491: 0.0646930 0.358795 -0.440961 -0.340486 0.442797 0.0177600 0.0484173 0.931624
## 40: 20231.464: 0.0647651 0.361294 -0.437223 -0.342799 0.440185 0.0179285 0.0486019 0.931467
## 41: 20231.422: 0.0645004 0.362000 -0.435707 -0.344077 0.438567 0.0178351 0.0487517 0.931260
## 42: 20231.391: 0.0639689 0.363984 -0.433098 -0.346216 0.435078 0.0177116 0.0487897 0.931563
## 43: 20231.345: 0.0636566 0.368731 -0.427513 -0.349870 0.429378 0.0174808 0.0484532 0.931999
## 44: 20231.309: 0.0633427 0.372855 -0.421475 -0.353883 0.423904 0.0175658 0.0485649 0.931947
## 45: 20231.272: 0.0629253 0.375705 -0.416302 -0.357488 0.419242 0.0176165 0.0488407 0.931474
## 46: 20231.241: 0.0624727 0.378719 -0.411977 -0.360470 0.414634 0.0175985 0.0488541 0.931613
## 47: 20231.188: 0.0615505 0.386396 -0.402058 -0.367242 0.403970 0.0173594 0.0485204 0.932123
## 48: 20230.994: 0.0541913 0.444843 -0.319821 -0.426578 0.322043 0.0175687 0.0490853 0.931154
## 49: 20230.751: 0.0462850 0.498728 -0.246186 -0.480142 0.244897 0.0172970 0.0489364 0.932025
## 50: 20230.710: 0.0453863 0.508830 -0.238663 -0.489591 0.237664 0.0172540 0.0487332 0.932149
## 51: 20230.706: 0.0447832 0.513273 -0.238394 -0.494376 0.236950 0.0171384 0.0487157 0.932166
## 52: 20230.682: 0.0442802 0.519834 -0.237847 -0.500910 0.236287 0.0171588 0.0486978 0.932277
## 53: 20230.605: 0.0378079 0.636326 -0.247942 -0.616500 0.245100 0.0167962 0.0479036 0.933427
## 54: 20230.580: 0.0315549 0.750443 -0.280089 -0.731908 0.273894 0.0164234 0.0475990 0.934190
## 55: 20230.566: 0.0324833 0.766889 -0.304096 -0.748590 0.297718 0.0164087 0.0475678 0.934240
## 56: 20230.549: 0.0350879 0.715725 -0.295009 -0.696590 0.289902 0.0167807 0.0478179 0.933522
## 57: 20230.547: 0.0332544 0.745763 -0.290415 -0.726564 0.284314 0.0167662 0.0479233 0.933452
## 58: 20230.545: 0.0336162 0.748931 -0.303048 -0.729893 0.297157 0.0166749 0.0478622 0.933624
## 59: 20230.545: 0.0336588 0.744868 -0.299387 -0.725770 0.293468 0.0167253 0.0478634 0.933556
## 60: 20230.545: 0.0336264 0.745515 -0.299287 -0.726409 0.293375 0.0167162 0.0478718 0.933561
## 61: 20230.545: 0.0336116 0.746046 -0.299638 -0.726945 0.293714 0.0167160 0.0478694 0.933563
## 62: 20230.545: 0.0336192 0.745820 -0.299538 -0.726718 0.293617 0.0167162 0.0478699 0.933562
##
## Final Estimate of the Negative LLH:
## LLH: 21495.98 norm LLH: 1.405333
## mu ar1 ar2 ma1 ma2 omega
## 0.03651878 0.74581989 -0.29953765 -0.72671761 0.29361692 0.01972408
## alpha1 beta1
## 0.04786988 0.93356216
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 ma2
## mu -54715.26981 -3747.6342 -3843.7607 -124.5819 -221.9342
## ar1 -3747.63423 -22942.1318 -13189.1836 -22420.9856 -12877.8406
## ar2 -3843.76073 -13189.1836 -22991.8243 -12364.5596 -22366.5587
## ma1 -124.58195 -22420.9856 -12364.5596 -22164.1872 -12315.8195
## ma2 -221.93416 -12877.8406 -22366.5587 -12315.8195 -22020.4561
## omega 1449.48319 -1353.1497 -536.8078 -1414.7656 -586.2912
## alpha1 54.14917 152.0862 -349.5990 168.0418 -327.6197
## beta1 145.26516 -1063.4659 -166.0546 -1054.2157 -142.8460
## omega alpha1 beta1
## mu 1449.4832 5.414917e+01 145.2652
## ar1 -1353.1497 1.520862e+02 -1063.4659
## ar2 -536.8078 -3.495990e+02 -166.0546
## ma1 -1414.7656 1.680418e+02 -1054.2157
## ma2 -586.2912 -3.276197e+02 -142.8460
## omega -2472228.4064 -1.664739e+06 -2056235.2735
## alpha1 -1664738.8727 -1.547425e+06 -1634519.7414
## beta1 -2056235.2735 -1.634520e+06 -1902303.0988
## attr(,"time")
## Time difference of 0.5152261 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 3.448922 secs(b) The ARMA (1,1) model seems to fit the best with the highest log-likelihood.
stargazer(garch_out, title = "GARCH(1,1) Model Results", type = "text")##
## GARCH(1,1) Model Results
## ===============================================
## Dependent variable:
## ---------------------------
## wilshire_imp
## -----------------------------------------------
## mu 0.055**
## (0.024)
##
## ar1 0.162
## (0.349)
##
## ma1 -0.143
## (0.348)
##
## omega 0.020***
## (0.002)
##
## alpha1 0.048***
## (0.003)
##
## beta1 0.934***
## (0.005)
##
## -----------------------------------------------
## Observations 15,296
## Log Likelihood 21,497.360
## Akaike Inf. Crit. 2.812
## Bayesian Inf. Crit. 2.815
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01plot(garch_out, which=2)
### (d) I notice that there are 3 significant spikes.
pred_out <- predict(garch_out,n.ahead=365,plot=TRUE,nx=365)
# Part 5: Cochrane-Orcutt GDP ## 1.
ar1_model <- arima(ln_gdp, order = c(1, 0, 0))
residuals_co <- residuals(ar1_model)
library(fGarch)
garch_resid <- garchFit(~garch(1,1), data = residuals_co)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 302
## Recursion Init: mci
## Series Scale: 0.01133152
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -4.22683757 4.226838 0.4226838 TRUE
## omega 0.00000100 100.000000 0.1000000 TRUE
## alpha1 0.00000001 1.000000 0.1000000 TRUE
## gamma1 -0.99999999 1.000000 0.1000000 FALSE
## beta1 0.00000001 1.000000 0.8000000 TRUE
## delta 0.00000000 2.000000 2.0000000 FALSE
## skew 0.10000000 10.000000 1.0000000 FALSE
## shape 1.00000000 10.000000 4.0000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1
## 1 2 3 5
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 401.14756: 0.422684 0.100000 0.100000 0.800000
## 1: 397.77302: 0.423119 0.0784712 0.117250 0.776461
## 2: 394.22730: 0.423811 0.0853960 0.152746 0.774080
## 3: 388.90123: 0.425496 0.0475681 0.196634 0.730584
## 4: 378.53914: 0.431994 0.0632545 0.326203 0.669030
## 5: 378.04144: 0.433101 0.0535171 0.329334 0.660068
## 6: 377.70020: 0.434848 0.0618176 0.339495 0.658468
## 7: 376.83904: 0.442165 0.0573264 0.348791 0.639471
## 8: 375.50496: 0.459347 0.0769041 0.368587 0.614154
## 9: 373.46936: 0.490439 0.0585965 0.440132 0.576606
## 10: 371.50581: 0.493940 0.113706 0.449046 0.481269
## 11: 370.55594: 0.505591 0.134587 0.549549 0.450000
## 12: 368.77601: 0.510938 0.140612 0.603692 0.354362
## 13: 368.59687: 0.536438 0.184854 0.653015 0.289210
## 14: 368.44464: 0.519798 0.180188 0.679111 0.269492
## 15: 368.37530: 0.508456 0.163333 0.682108 0.309846
## 16: 368.36729: 0.512157 0.153097 0.678884 0.311686
## 17: 368.31149: 0.517743 0.154667 0.681063 0.315338
## 18: 368.29293: 0.518128 0.160774 0.689855 0.306405
## 19: 368.26638: 0.523796 0.160757 0.705025 0.298825
## 20: 368.26601: 0.527002 0.157467 0.713185 0.303780
## 21: 368.26243: 0.525675 0.158003 0.716938 0.299825
## 22: 368.26215: 0.524830 0.158616 0.714638 0.299424
## 23: 368.26213: 0.524940 0.158488 0.715163 0.299433
## 24: 368.26213: 0.524940 0.158486 0.715157 0.299439
##
## Final Estimate of the Negative LLH:
## LLH: -984.7483 norm LLH: -3.260756
## mu omega alpha1 beta1
## 5.948372e-03 2.035019e-05 7.151574e-01 2.994392e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1
## mu -4.702902e+06 -107748876 -149.7197 -5490.6245
## omega -1.077489e+08 -96880063296 -1887593.7246 -6832264.8140
## alpha1 -1.497197e+02 -1887594 -108.3218 -189.8692
## beta1 -5.490625e+03 -6832265 -189.8692 -674.7677
## attr(,"time")
## Time difference of 0.005463839 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.02631712 secslibrary(stargazer)
stargazer_out <- capture.output(stargazer(garch_resid, type = "text"))
plot(garch_resid, which = 2)
predict(garch_resid, n.ahead = 20, plot = TRUE, nx = 40)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.005948372 0.009434011 0.009434011 -0.01254195 0.02443869
## 2 0.005948372 0.010519024 0.010519024 -0.01466854 0.02656528
## 3 0.005948372 0.011515866 0.011515866 -0.01662231 0.02851905
## 4 0.005948372 0.012445926 0.012445926 -0.01844520 0.03034194
## 5 0.005948372 0.013323374 0.013323374 -0.02016496 0.03206171
## 6 0.005948372 0.014158163 0.014158163 -0.02180112 0.03369786
## 7 0.005948372 0.014957597 0.014957597 -0.02336798 0.03526472
## 8 0.005948372 0.015727224 0.015727224 -0.02487642 0.03677316
## 9 0.005948372 0.016471375 0.016471375 -0.02633493 0.03823167
## 10 0.005948372 0.017193502 0.017193502 -0.02775027 0.03964702
## 11 0.005948372 0.017896416 0.017896416 -0.02912796 0.04102470
## 12 0.005948372 0.018582436 0.018582436 -0.03047253 0.04236928
## 13 0.005948372 0.019253504 0.019253504 -0.03178780 0.04368455
## 14 0.005948372 0.019911267 0.019911267 -0.03307699 0.04497374
## 15 0.005948372 0.020557132 0.020557132 -0.03434287 0.04623961
## 16 0.005948372 0.021192318 0.021192318 -0.03558781 0.04748455
## 17 0.005948372 0.021817884 0.021817884 -0.03681390 0.04871064
## 18 0.005948372 0.022434762 0.022434762 -0.03802295 0.04991970
## 19 0.005948372 0.023043773 0.023043773 -0.03921659 0.05111334
## 20 0.005948372 0.023645648 0.023645648 -0.04039625 0.052292992.
garch_gdp <- garchFit(~arma(1,0) + garch(1,1), data = ln_gdp)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 1 0
## Max ARMA Order: 1
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 302
## Recursion Init: mci
## Series Scale: 0.4392655
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -237.34255950 237.3426 23.7046605 TRUE
## ar1 -0.99999999 1.0000 0.9998477 TRUE
## omega 0.00000100 100.0000 0.1000000 TRUE
## alpha1 0.00000001 1.0000 0.1000000 TRUE
## gamma1 -0.99999999 1.0000 0.1000000 FALSE
## beta1 0.00000001 1.0000 0.8000000 TRUE
## delta 0.00000000 2.0000 2.0000000 FALSE
## skew 0.10000000 10.0000 1.0000000 FALSE
## shape 1.00000000 10.0000 4.0000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 omega alpha1 beta1
## 1 2 3 4 6
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 1428.9346: 23.7047 0.999848 0.100000 0.100000 0.800000
## 1: 1387.6785: 23.0299 0.971387 0.100245 0.237131 0.869366
## 2: 1193.0956: 18.1405 0.765579 0.100277 0.273371 0.699642
## 3: 775.40674: 10.3310 0.436741 0.100349 0.315291 0.498140
## 4: 195.56443: 11.8920 0.501962 0.101128 0.321865 0.502404
## 5: 195.12381: 11.9253 0.507994 0.0944124 0.321417 0.500743
## 6: 189.56217: 11.8425 0.505179 0.0932191 0.321365 0.500483
## 7: -18.006333: 6.69093 0.724859 1.00000e-06 0.365012 0.370452
## 8: -30.564181: 1.48376 0.941613 0.0933016 0.431251 0.173628
## 9: -44.460553: 1.43546 0.939628 0.0929145 0.431247 0.173582
## 10: -46.226195: 1.41034 0.941565 0.0921316 0.431748 0.172001
## 11: -74.550148: -0.0186306 1.00000 0.0905112 0.473104 0.0459620
## 12: -83.930454: 0.0644286 1.00000 0.0864127 0.383944 1.00000e-08
## 13: -91.453697: 0.921239 0.963757 0.0812300 0.294772 1.00000e-08
## 14: -94.964185: 0.877495 0.961966 0.0806434 0.294770 1.00000e-08
## 15: -98.991341: 0.906721 0.962434 0.0801383 0.292543 1.00000e-08
## 16: -103.69211: 0.902547 0.961056 0.0761174 0.289428 1.00000e-08
## 17: -108.56575: 0.976595 0.959264 0.0748041 0.279602 1.00000e-08
## 18: -123.11338: 1.08891 0.957369 0.0618628 0.263746 1.00000e-08
## 19: -139.41882: 1.17535 0.951688 0.0594108 0.222177 1.00000e-08
## 20: -152.49651: 1.38597 0.941706 0.0535008 0.130137 1.00000e-08
## 21: -162.10711: 1.52051 0.936594 0.0495613 0.0557569 1.00000e-08
## 22: -171.83997: 1.56263 0.934267 0.0459229 0.0187979 1.00000e-08
## 23: -189.22345: 1.60638 0.933146 0.0402447 1.00000e-08 1.00000e-08
## 24: -202.79216: 1.61078 0.932798 0.0363134 1.00000e-08 1.00000e-08
## 25: -355.96298: 1.64817 0.930277 0.00484575 1.00000e-08 1.00000e-08
## 26: -367.07929: 1.65744 0.930731 0.00490971 2.49224e-06 3.18970e-07
## 27: -367.51550: 1.65355 0.930748 0.00497338 0.000578518 1.00000e-08
## 28: -368.08127: 1.65372 0.930895 0.00500825 0.00177413 1.00000e-08
## 29: -368.81189: 1.64967 0.930937 0.00504966 0.00417742 1.00000e-08
## 30: -369.91297: 1.64850 0.931142 0.00509374 0.00899278 1.00000e-08
## 31: -371.61935: 1.64203 0.931259 0.00515269 0.0186284 1.00000e-08
## 32: -374.98981: 1.63338 0.931914 0.00518324 0.0378945 1.00000e-08
## 33: -398.04532: 1.49404 0.937848 0.00278104 0.0551074 1.00000e-08
## 34: -401.24630: 1.48789 0.937723 0.00310538 0.0551144 1.26653e-06
## 35: -401.50940: 1.49218 0.938055 0.00331803 0.0551201 2.21606e-06
## 36: -403.71541: 1.48803 0.937934 0.00335836 0.0551224 2.46493e-06
## 37: -404.43095: 1.48440 0.938005 0.00349402 0.0551304 3.31699e-06
## 38: -404.68332: 1.48431 0.938170 0.00353876 0.0551358 3.73823e-06
## 39: -405.07579: 1.48161 0.938140 0.00354624 0.0552534 4.25868e-05
## 40: -405.45872: 1.48053 0.938317 0.00356487 0.0555280 0.000131010
## 41: -405.94176: 1.47758 0.938310 0.00355705 0.0561683 0.000335430
## 42: -406.58820: 1.47581 0.938514 0.00354331 0.0574569 0.000737521
## 43: -428.91979: 1.41382 0.941140 0.00240617 0.138523 0.0255593
## 44: -461.94891: 1.23928 0.948837 0.00157951 0.160800 1.00000e-08
## 45: -475.64184: 1.17152 0.951175 0.00136678 0.168171 1.00000e-08
## 46: -490.48326: 1.11875 0.953715 0.00160899 0.160651 1.00000e-08
## 47: -555.28860: 0.662632 0.973062 0.000419014 0.142130 1.00000e-08
## 48: -633.11368: 0.444702 0.981589 0.00111893 0.111625 1.00000e-08
## 49: -694.81958: 0.0177959 1.00000 0.000617734 0.109253 1.00000e-08
## 50: -706.75117: 0.0132523 0.999806 0.000475365 0.109254 1.00000e-08
## 51: -709.39412: 0.0165268 0.999854 0.000442713 0.109522 1.00000e-08
## 52: -709.95247: 0.0170213 0.999771 0.000433177 0.109817 1.00000e-08
## 53: -709.99305: 0.0178732 0.999744 0.000433870 0.109883 0.000296812
## 54: -710.46874: 0.0412022 0.998755 0.000442949 0.112002 0.00979886
## 55: -710.54885: 0.0465064 0.998558 0.000392789 0.114908 0.00616125
## 56: -711.23315: 0.0485693 0.998458 0.000428453 0.117532 0.00230498
## 57: -712.50907: 0.0626816 0.997833 0.000403541 0.132961 1.00000e-08
## 58: -714.54069: 0.0627556 0.997903 0.000401988 0.149614 1.00000e-08
## 59: -719.18815: 0.0591484 0.998039 0.000316640 0.210073 1.00000e-08
## 60: -721.53143: 0.0461660 0.998582 0.000325402 0.219099 0.0597793
## 61: -726.22563: 0.0126821 1.00000 0.000168720 0.290240 0.277194
## 62: -733.86143: 0.0144481 1.00000 7.30317e-05 0.482574 0.423814
## 63: -738.16701: 0.0686621 0.997727 0.000167877 0.639768 0.240049
## 64: -741.03725: 0.0962659 0.996519 0.000123673 0.692593 0.291232
## 65: -743.39879: 0.126816 0.995267 6.38910e-05 0.744739 0.343103
## 66: -743.95274: 0.119043 0.995630 7.31711e-05 0.748507 0.343191
## 67: -743.95887: 0.117321 0.995698 5.93080e-05 0.760053 0.370878
## 68: -744.04880: 0.117854 0.995673 6.37503e-05 0.762765 0.356126
## 69: -744.07967: 0.118576 0.995642 6.71348e-05 0.773683 0.343474
## 70: -744.08732: 0.117982 0.995666 6.82581e-05 0.782368 0.338830
## 71: -744.09091: 0.117972 0.995667 6.83629e-05 0.792382 0.335120
## 72: -744.09154: 0.117627 0.995681 6.80988e-05 0.794516 0.336229
## 73: -744.09163: 0.117731 0.995677 6.79327e-05 0.794143 0.336370
## 74: -744.09163: 0.117738 0.995676 6.79172e-05 0.794025 0.336447
## 75: -744.09163: 0.117737 0.995676 6.79157e-05 0.794022 0.336449
##
## Final Estimate of the Negative LLH:
## LLH: -992.5323 norm LLH: -3.286531
## mu ar1 omega alpha1 beta1
## 5.171777e-02 9.956764e-01 1.310462e-05 7.940217e-01 3.364494e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 omega alpha1 beta1
## mu -6.435641e+06 -1.478620e+07 -1378343163 -1293.9463 -4269.2164
## ar1 -1.478620e+07 -1.491664e+08 616702290 -6111.5270 -29750.0592
## omega -1.378343e+09 6.167023e+08 -2476528461 20209090.2424 12238640.6318
## alpha1 -1.293946e+03 -6.111527e+03 20209090 -115.3319 -227.3072
## beta1 -4.269216e+03 -2.975006e+04 12238641 -227.3072 -813.9190
## attr(,"time")
## Time difference of 0.01017094 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): NaNs produced##
## Time to Estimate Parameters:
## Time difference of 0.148809 secsgarch_phi <- garch_gdp@fit$coef["ar1"]
garch_mu <- garch_gdp@fit$coef["mu"]
garch_alpha <- garch_mu / (1 - garch_phi)
co_ar1 <- ar1_model$coef
predict(garch_gdp, n.ahead = 20, plot = TRUE, nx = 40)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 11.09556 0.008378875 0.008378875 11.07914 11.11199
## 2 11.09931 0.012730661 0.009616130 11.07436 11.12426
## 3 11.10304 0.016682641 0.010846162 11.07034 11.13573
## 4 11.10675 0.020542674 0.012086867 11.06649 11.14701
## 5 11.11045 0.024425765 0.013351323 11.06257 11.15832
## 6 11.11413 0.028391724 0.014649912 11.05848 11.16977
## 7 11.11779 0.032478621 0.015991443 11.05413 11.18145
## 8 11.12144 0.036714502 0.017383778 11.04948 11.19340
## 9 11.12507 0.041122402 0.018834229 11.04447 11.20567
## 10 11.12869 0.045722808 0.020349805 11.03907 11.21830
## 11 11.13229 0.050534994 0.021937392 11.03324 11.23134
## 12 11.13588 0.055577811 0.023603874 11.02695 11.24481
## 13 11.13945 0.060870181 0.025356233 11.02014 11.25875
## 14 11.14300 0.066431443 0.027201619 11.01280 11.27321
## 15 11.14654 0.072281599 0.029147417 11.00487 11.28821
## 16 11.15007 0.078441504 0.031201300 10.99632 11.30381
## 17 11.15358 0.084933030 0.033371276 10.98711 11.32004
## 18 11.15707 0.091779198 0.035665737 10.97719 11.33695
## 19 11.16055 0.099004312 0.038093497 10.96650 11.35459
## 20 11.16401 0.106634073 0.040663837 10.95501 11.37301plot(garch_gdp, which = 2)