Series Temporales ARCH & GARCH
2024-03-05
1.- INTRODUCCIÓN
Procesos estocásticos
Como hemos mencionado anteriormente, nosotros no nos centraremos en el estucio clásico de series temporales, sino que consideraremos nuestros datos como fenómenos que evolucionan en el tiempo. Para ello, los modelos adecuados son los llamados procesos estocásticos.
Procesos estacionarios
Cada serie temporal tiene su particular forma, cosa que se puede observar al hacer su respectiva gráfica. Necesitamos que los procesos estocásticos sean la representación matemática de algún tipo de regularidad, para que puedan ser utilizados como modelos de nuestras series.
Modelos ARCH
La heterocedasticidad condicional auto-regresiva (en inglés, Autoregressive conditional heteroscedasticity; ARCH) es la condición de que hay uno o más puntos de datos en una serie para los cuales la varianza del término de error actual o innovación es una función de los tamaños reales de los términos de error de los perÃodos de tiempo anteriores: se relaciona con los cuadrados de las innovaciones anteriores. En la econometrÃa, los modelos ARCH se utilizan para caracterizar y modelar series temporales. Una variedad de otras siglas se aplican a las estructuras particulares que tienen una base similar. Los modelos de ARCH se emplean comúnmente en el modelado de series de tiempo financieras que presentan agrupaciones de volatilidad variables en el tiempo, es decir, perÃodos de oscilaciones entremezclados con perÃodos de relativa calma. A veces se considera que los modelos de tipo ARCH pertenecen a la familia de modelos de volatilidad estocástica , aunque esto es estrictamente incorrecto ya que en el tiempo t la volatilidad es completamente predeterminada (determinista) dados los valores anteriores.
2.- EJERCICIO
# Bibliotecas
#---------------------
library(tidyquant)## Loading required package: lubridate##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, union## Loading required package: PerformanceAnalytics## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
##
## legend## Loading required package: quantmod## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(TTR)
library(ggplot2)
library(knitr)
library(nortest)
library(MASS)
library(stats)
library(moments)##
## Attaching package: 'moments'## The following objects are masked from 'package:PerformanceAnalytics':
##
## kurtosis, skewnesslibrary(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")##
## Attaching package: 'fGarch'## The following object is masked from 'package:TTR':
##
## volatilitylibrary(fBasics)##
## Attaching package: 'fBasics'## The following objects are masked from 'package:moments':
##
## kurtosis, skewness## The following object is masked from 'package:TTR':
##
## volatility## The following objects are masked from 'package:PerformanceAnalytics':
##
## kurtosis, skewness# Datos
# -------------------------
AAPL <- tq_get("AAPL", get = "stock.prices", from = "2020-01-01", to = "2024-01-01")
AAPL <- data.frame(AAPL)# Plot
#------------------------
plot(AAPL$adjusted)
precio <- AAPL$adjustedn <- length(precio)rendimiento <- c()
for (i in 1:n) {
rendimiento[i] <- (precio[i+1]-precio[i])/(precio[i])
}
rendimiento<-na.omit(rendimiento)summary(rendimiento)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.1286470 -0.0091270 0.0009882 0.0011866 0.0127428 0.1198085Modelo de heterocedasticidad condicional
modelo <- garchFit(~garch(1,0), data = rendimiento)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 0)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 0
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 1005
## Recursion Init: mci
## Series Scale: 0.02114593
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.56113285 0.5611329 0.05611329 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
## 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 omega alpha1
## 1 2 3
## Persistence: 0.1
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3006.7556: 0.0561133 0.100000 0.100000
## 1: 1412.0659: 0.0561624 1.06199 0.373067
## 2: 1408.4169: 0.0563679 1.08020 0.284060
## 3: 1388.8408: 0.0564495 0.918388 0.201101
## 4: 1387.9049: 0.0566401 0.697330 0.146242
## 5: 1382.1955: 0.0567111 0.765909 0.237166
## 6: 1381.9079: 0.0629987 0.745913 0.241227
## 7: 1381.7314: 0.0782486 0.750404 0.240933
## 8: 1381.7102: 0.0820689 0.736148 0.258305
## 9: 1381.6984: 0.0815047 0.739781 0.249762
## 10: 1381.6983: 0.0813011 0.740303 0.250176
## 11: 1381.6983: 0.0813450 0.740163 0.250153
## 12: 1381.6983: 0.0813447 0.740164 0.250150
##
## Final Estimate of the Negative LLH:
## LLH: -2493.891 norm LLH: -2.481484
## mu omega alpha1
## 0.0017201098 0.0003309645 0.2501501114
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1
## mu -2745212.135 -2709482.3 2700.2551
## omega -2709482.309 -3397897076.7 -609158.9362
## alpha1 2700.255 -609158.9 -470.5157
## attr(,"time")
## Time difference of 0.005178928 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.02600312 secs# Coeficientes
#------------
coef(modelo)## mu omega alpha1
## 0.0017201098 0.0003309645 0.2501501114h^2_t = 0.00033+0.25015u^2_(t-1)
predict(modelo,n.ahead=5)## meanForecast meanError standardDeviation
## 1 0.00172011 0.01854001 0.01854001
## 2 0.00172011 0.02041933 0.02041933
## 3 0.00172011 0.02086299 0.02086299
## 4 0.00172011 0.02097250 0.02097250
## 5 0.00172011 0.02099981 0.02099981predict(modelo,n.ahead=30, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.00172011 0.01854001 0.01854001 -0.03461765 0.03805787
## 2 0.00172011 0.02041933 0.02041933 -0.03830104 0.04174126
## 3 0.00172011 0.02086299 0.02086299 -0.03917060 0.04261082
## 4 0.00172011 0.02097250 0.02097250 -0.03938524 0.04282546
## 5 0.00172011 0.02099981 0.02099981 -0.03943876 0.04287898
## 6 0.00172011 0.02100663 0.02100663 -0.03945214 0.04289236
## 7 0.00172011 0.02100834 0.02100834 -0.03945548 0.04289570
## 8 0.00172011 0.02100877 0.02100877 -0.03945632 0.04289654
## 9 0.00172011 0.02100887 0.02100887 -0.03945653 0.04289675
## 10 0.00172011 0.02100890 0.02100890 -0.03945658 0.04289680
## 11 0.00172011 0.02100891 0.02100891 -0.03945659 0.04289681
## 12 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 13 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 14 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 15 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 16 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 17 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 18 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 19 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 20 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 21 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 22 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 23 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 24 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 25 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 26 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 27 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 28 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 29 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682
## 30 0.00172011 0.02100891 0.02100891 -0.03945660 0.04289682modelo2 <- garchFit(~garch(2,0), data=rendimiento)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 0)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 2 0
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 1005
## Recursion Init: mci
## Series Scale: 0.02114593
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.56113285 0.5611329 0.05611329 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.05000000 TRUE
## alpha2 0.00000001 1.0000000 0.05000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## gamma2 -0.99999999 1.0000000 0.10000000 FALSE
## 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 omega alpha1 alpha2
## 1 2 3 4
## Persistence: 0.1
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 2610.3924: 0.0561133 0.100000 0.0500000 0.0500000
## 1: 1417.0998: 0.0561402 0.927139 0.436566 0.457930
## 2: 1416.0398: 0.0590162 1.16464 0.0115681 0.356221
## 3: 1410.5148: 0.0648157 1.13385 0.0576209 1.00000e-08
## 4: 1369.2830: 0.0713194 0.799310 0.191323 0.326758
## 5: 1351.4299: 0.0713323 0.670094 0.133769 0.260115
## 6: 1350.5955: 0.0713662 0.527033 0.144175 0.197862
## 7: 1347.6678: 0.0757933 0.526817 0.245121 0.287505
## 8: 1347.0655: 0.0755089 0.537257 0.223162 0.262900
## 9: 1346.8973: 0.0765494 0.548343 0.204386 0.242826
## 10: 1346.8969: 0.0765520 0.549064 0.204558 0.243111
## 11: 1346.8969: 0.0765627 0.549305 0.203866 0.242869
## 12: 1346.8968: 0.0766056 0.549452 0.204005 0.242959
## 13: 1346.8966: 0.0770283 0.549287 0.204009 0.243028
## 14: 1346.8965: 0.0772244 0.549488 0.204143 0.242658
## 15: 1346.8965: 0.0772264 0.549362 0.204119 0.242962
## 16: 1346.8965: 0.0771960 0.549377 0.204107 0.242910
## 17: 1346.8965: 0.0772016 0.549378 0.204110 0.242910
##
## Final Estimate of the Negative LLH:
## LLH: -2528.693 norm LLH: -2.516112
## mu omega alpha1 alpha2
## 0.0016325002 0.0002456544 0.2041101569 0.2429101343
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 alpha2
## mu -3208472.4513 -1261073.4 1416.2322 -768.6621
## omega -1261073.4481 -4448270059.8 -796212.3152 -700590.6077
## alpha1 1416.2322 -796212.3 -568.3013 -151.2328
## alpha2 -768.6621 -700590.6 -151.2328 -541.3589
## attr(,"time")
## Time difference of 0.0109868 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.03432512 secscoef(modelo2)## mu omega alpha1 alpha2
## 0.0016325002 0.0002456544 0.2041101569 0.2429101343h^2_t = 0.00024 + 0.2041u^2_(t-1) + 0.2491u^2_(t-2)
predict(modelo2, n.ahead=30, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.0016325 0.01599700 0.01599700 -0.02972104 0.03298604
## 2 0.0016325 0.01760633 0.01760633 -0.03287528 0.03614028
## 3 0.0016325 0.01926361 0.01926361 -0.03612349 0.03938849
## 4 0.0016325 0.01991720 0.01991720 -0.03740450 0.04066950
## 5 0.0016325 0.02041481 0.02041481 -0.03837980 0.04164480
## 6 0.0016325 0.02066595 0.02066595 -0.03887202 0.04213702
## 7 0.0016325 0.02083417 0.02083417 -0.03920172 0.04246672
## 8 0.0016325 0.02092829 0.02092829 -0.03938620 0.04265120
## 9 0.0016325 0.02098789 0.02098789 -0.03950301 0.04276801
## 10 0.0016325 0.02102276 0.02102276 -0.03957135 0.04283635
## 11 0.0016325 0.02104429 0.02104429 -0.03961355 0.04287855
## 12 0.0016325 0.02105713 0.02105713 -0.03963872 0.04290372
## 13 0.0016325 0.02106497 0.02106497 -0.03965409 0.04291909
## 14 0.0016325 0.02106969 0.02106969 -0.03966334 0.04292834
## 15 0.0016325 0.02107256 0.02107256 -0.03966896 0.04293396
## 16 0.0016325 0.02107429 0.02107429 -0.03967235 0.04293735
## 17 0.0016325 0.02107534 0.02107534 -0.03967441 0.04293941
## 18 0.0016325 0.02107597 0.02107597 -0.03967565 0.04294065
## 19 0.0016325 0.02107636 0.02107636 -0.03967640 0.04294140
## 20 0.0016325 0.02107659 0.02107659 -0.03967686 0.04294186
## 21 0.0016325 0.02107673 0.02107673 -0.03967713 0.04294213
## 22 0.0016325 0.02107682 0.02107682 -0.03967730 0.04294230
## 23 0.0016325 0.02107687 0.02107687 -0.03967740 0.04294240
## 24 0.0016325 0.02107690 0.02107690 -0.03967746 0.04294246
## 25 0.0016325 0.02107692 0.02107692 -0.03967750 0.04294250
## 26 0.0016325 0.02107693 0.02107693 -0.03967752 0.04294252
## 27 0.0016325 0.02107694 0.02107694 -0.03967754 0.04294254
## 28 0.0016325 0.02107694 0.02107694 -0.03967755 0.04294255
## 29 0.0016325 0.02107694 0.02107694 -0.03967755 0.04294255
## 30 0.0016325 0.02107695 0.02107695 -0.03967755 0.04294255modelo3 <- garchFit(~garch(3,0), data=rendimiento)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(3, 0)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 3 0
## Max GARCH Order: 3
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 1005
## Recursion Init: mci
## Series Scale: 0.02114593
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.56113285 0.5611329 0.05611329 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.03333333 TRUE
## alpha2 0.00000001 1.0000000 0.03333333 TRUE
## alpha3 0.00000001 1.0000000 0.03333333 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## gamma2 -0.99999999 1.0000000 0.10000000 FALSE
## gamma3 -0.99999999 1.0000000 0.10000000 FALSE
## 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 omega alpha1 alpha2 alpha3
## 1 2 3 4 5
## Persistence: 0.1
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 2503.8299: 0.0561133 0.100000 0.0333333 0.0333333 0.0333333
## 1: 1435.0575: 0.0561416 0.802406 0.442460 0.452097 0.438151
## 2: 1429.9499: 0.0611083 0.454254 0.645210 0.715825 0.539621
## 3: 1413.0418: 0.101433 0.377225 0.899510 0.815030 1.00000e-08
## 4: 1392.2585: 0.101313 0.289345 0.767193 0.675706 0.0563440
## 5: 1376.3048: 0.101242 0.378746 0.638832 0.543140 0.132349
## 6: 1363.9229: 0.100392 0.286200 0.490063 0.428129 0.0717916
## 7: 1349.5369: 0.0970964 0.410114 0.335915 0.422858 0.144097
## 8: 1345.6688: 0.0939061 0.363362 0.171001 0.305754 0.105455
## 9: 1338.1959: 0.0917166 0.454763 0.292651 0.163775 0.160117
## 10: 1336.0887: 0.0913317 0.443119 0.219164 0.195502 0.165619
## 11: 1335.7694: 0.0912271 0.457722 0.185604 0.186574 0.178194
## 12: 1335.7620: 0.0891443 0.466506 0.180782 0.182940 0.168736
## 13: 1335.7558: 0.0890660 0.464138 0.184378 0.188418 0.176099
## 14: 1335.7548: 0.0889203 0.455278 0.184333 0.184155 0.177237
## 15: 1335.7408: 0.0887611 0.458925 0.185309 0.186050 0.177779
## 16: 1335.7397: 0.0885805 0.459519 0.183136 0.185564 0.174538
## 17: 1335.7362: 0.0883334 0.461476 0.184090 0.186707 0.175394
## 18: 1335.7122: 0.0837498 0.461183 0.183532 0.187875 0.173271
## 19: 1335.7121: 0.0835007 0.459391 0.184574 0.184837 0.176664
## 20: 1335.7108: 0.0833335 0.460397 0.184303 0.185844 0.176208
## 21: 1335.7103: 0.0831580 0.460854 0.183560 0.185705 0.175417
## 22: 1335.7100: 0.0829735 0.460664 0.183671 0.186240 0.175248
## 23: 1335.7098: 0.0825297 0.460913 0.183722 0.186287 0.174935
## 24: 1335.7098: 0.0824564 0.460758 0.183509 0.186378 0.175086
## 25: 1335.7098: 0.0824885 0.460825 0.183605 0.186341 0.175013
## 26: 1335.7098: 0.0824888 0.460826 0.183603 0.186341 0.175014
##
## Final Estimate of the Negative LLH:
## LLH: -2539.88 norm LLH: -2.527243
## mu omega alpha1 alpha2 alpha3
## 0.0017443017 0.0002060584 0.1836034994 0.1863414510 0.1750137965
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 alpha2 alpha3
## mu -3352659.431 -2646659.5 1022.5125 -1049.3707 1130.8191
## omega -2646659.454 -4996717261.7 -872340.8088 -766413.3188 -886211.4623
## alpha1 1022.513 -872340.8 -632.2097 -162.2344 -185.4468
## alpha2 -1049.371 -766413.3 -162.2344 -697.5366 -228.9762
## alpha3 1130.819 -886211.5 -185.4468 -228.9762 -590.2607
## attr(,"time")
## Time difference of 0.02141404 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.06105399 secscoef(modelo3)## mu omega alpha1 alpha2 alpha3
## 0.0017443017 0.0002060584 0.1836034994 0.1863414510 0.1750137965h^2_t = 0.000206 + 0.1836u^2_(t-1) + 0.1836u^2_(t-2) + 0.175u^2_(t-3)
predict(modelo3, n.ahead=30, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.001744302 0.01469012 0.01469012 -0.02704780 0.03053640
## 2 0.001744302 0.01597798 0.01597798 -0.02957197 0.03306057
## 3 0.001744302 0.01738210 0.01738210 -0.03232398 0.03581259
## 4 0.001744302 0.01862450 0.01862450 -0.03475905 0.03824765
## 5 0.001744302 0.01925425 0.01925425 -0.03599334 0.03948195
## 6 0.001744302 0.01978989 0.01978989 -0.03704318 0.04053178
## 7 0.001744302 0.02019292 0.02019292 -0.03783309 0.04132169
## 8 0.001744302 0.02046422 0.02046422 -0.03836484 0.04185344
## 9 0.001744302 0.02067541 0.02067541 -0.03877875 0.04226736
## 10 0.001744302 0.02083131 0.02083131 -0.03908431 0.04257291
## 11 0.001744302 0.02094471 0.02094471 -0.03930657 0.04279518
## 12 0.001744302 0.02103038 0.02103038 -0.03947449 0.04296309
## 13 0.001744302 0.02109390 0.02109390 -0.03959898 0.04308758
## 14 0.001744302 0.02114103 0.02114103 -0.03969135 0.04317995
## 15 0.001744302 0.02117632 0.02117632 -0.03976051 0.04324912
## 16 0.001744302 0.02120259 0.02120259 -0.03981200 0.04330061
## 17 0.001744302 0.02122218 0.02122218 -0.03985040 0.04333900
## 18 0.001744302 0.02123681 0.02123681 -0.03987908 0.04336769
## 19 0.001744302 0.02124773 0.02124773 -0.03990048 0.04338908
## 20 0.001744302 0.02125588 0.02125588 -0.03991645 0.04340506
## 21 0.001744302 0.02126196 0.02126196 -0.03992838 0.04341699
## 22 0.001744302 0.02126651 0.02126651 -0.03993729 0.04342589
## 23 0.001744302 0.02126990 0.02126990 -0.03994394 0.04343254
## 24 0.001744302 0.02127244 0.02127244 -0.03994891 0.04343751
## 25 0.001744302 0.02127433 0.02127433 -0.03995261 0.04344122
## 26 0.001744302 0.02127574 0.02127574 -0.03995538 0.04344399
## 27 0.001744302 0.02127680 0.02127680 -0.03995745 0.04344606
## 28 0.001744302 0.02127758 0.02127758 -0.03995900 0.04344760
## 29 0.001744302 0.02127817 0.02127817 -0.03996015 0.04344875
## 30 0.001744302 0.02127861 0.02127861 -0.03996101 0.04344962EJERCICIO GARCH
#Bibliotecas
#----------------------
library(tidyquant)
library(TTR)
library(ggplot2)
library(knitr)
library(nortest)
library(MASS)
library(stats)
library(moments)
library(fGarch)
library(fBasics)AAPL <- tq_get("AAPL", get = "stock.prices", from = "2021-07-01", to = "2021-07-31")
AAPL <- data.frame(AAPL)precio <- AAPL$close
n<-length(precio)
rendimiento <- c()
for (i in 1:n) {
rendimiento[i] <- (precio[i+1]-precio[i])/(precio[i])
}
rendimiento<-na.omit(rendimiento)modelo
modeloo<-garchFit(~garch(1,2), data = rendimiento)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 2)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 2
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 20
## Recursion Init: mci
## Series Scale: 0.01437149
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -2.18314624 2.183146 0.2183146 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 0.4000000 TRUE
## beta2 0.00000001 1.000000 0.4000000 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 beta2
## 1 2 3 5 6
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 27.974431: 0.218315 0.100000 0.100000 0.400000 0.400000
## 1: 27.963262: 0.218306 0.0984128 0.0925684 0.394741 0.395780
## 2: 27.949471: 0.218248 0.107334 0.0894441 0.396070 0.399248
## 3: 27.940618: 0.218211 0.110453 0.0808451 0.392095 0.397322
## 4: 27.928170: 0.218156 0.119269 0.0774193 0.393413 0.400781
## 5: 27.919535: 0.218119 0.123056 0.0688020 0.389868 0.399361
## 6: 27.909087: 0.218070 0.131684 0.0646854 0.390955 0.402615
## 7: 27.901178: 0.218039 0.136025 0.0560397 0.387975 0.401763
## 8: 27.892706: 0.218006 0.144265 0.0510329 0.388924 0.404818
## 9: 27.886067: 0.217995 0.148984 0.0422444 0.386999 0.404815
## 10: 27.865705: 0.218103 0.180245 1.00000e-08 0.391414 0.417108
## 11: 27.865680: 0.217653 0.178890 1.00000e-08 0.390705 0.419596
## 12: 27.865657: 0.217347 0.176312 1.00000e-08 0.389171 0.423840
## 13: 27.865303: 0.215217 0.149752 1.00000e-08 0.373197 0.467314
## 14: 27.858681: 0.207000 0.0431420 1.00000e-08 0.309068 0.641202
## 15: 27.848277: 0.198671 1.00000e-06 1.00000e-08 0.282134 0.709992
## 16: 27.847857: 0.200997 1.00000e-06 1.00000e-08 0.284222 0.707978
## 17: 27.846795: 0.208486 1.00000e-06 1.00000e-08 0.295922 0.696009
## 18: 27.846438: 0.209728 1.00000e-06 1.00000e-08 0.303411 0.687955
## 19: 27.846178: 0.208801 1.00000e-06 1.00000e-08 0.313022 0.677416
## 20: 27.846156: 0.207771 1.00000e-06 1.00000e-08 0.314588 0.675702
## 21: 27.846150: 0.207330 1.00000e-06 1.00000e-08 0.315936 0.674337
## 22: 27.846141: 0.206868 1.00000e-06 1.00000e-08 0.319768 0.670502
## 23: 27.846108: 0.205879 1.00000e-06 1.00000e-08 0.335296 0.654995
## 24: 27.846012: 0.203901 1.00000e-06 1.00000e-08 0.385385 0.605019
## 25: 27.845645: 0.197609 1.00000e-06 1.00000e-08 0.598791 0.392167
## 26: 27.844462: 0.189820 1.00000e-06 1.00000e-08 0.992277 1.00000e-08
## 27: 27.843174: 0.195193 1.00000e-06 1.00000e-08 0.993031 1.00000e-08
## 28: 27.842168: 0.206263 1.00000e-06 1.00000e-08 0.992926 1.00000e-08
## 29: 27.842136: 0.205194 1.00000e-06 1.00000e-08 0.993082 1.00000e-08
## 30: 27.842136: 0.205169 1.00000e-06 1.00000e-08 0.993072 1.00000e-08
## 31: 27.842136: 0.205171 1.00000e-06 1.00000e-08 0.993072 1.00000e-08
##
## Final Estimate of the Negative LLH:
## LLH: -57.00804 norm LLH: -2.850402
## mu omega alpha1 beta1 beta2
## 2.948617e-03 2.065397e-10 1.000000e-08 9.930725e-01 1.000000e-08
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1 beta2
## mu -102523.279 15878301 2606.8527 2951.804 2971.167
## omega 15878301.125 -26388621834 -4828637.5637 -5420415.441 -5442427.330
## alpha1 2606.853 -4828638 -910.9959 -1027.417 -1020.912
## beta1 2951.804 -5420415 -1027.4172 -1102.926 -1108.068
## beta2 2971.167 -5442427 -1020.9117 -1108.068 -1113.228
## attr(,"time")
## Time difference of 0.003675938 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): Se han producido NaNs##
## Time to Estimate Parameters:
## Time difference of 0.01845503 secscoef(modeloo)## mu omega alpha1 beta1 beta2
## 2.948617e-03 2.065397e-10 1.000000e-08 9.930725e-01 1.000000e-08varianza(t) = 2.06e-10 + 9.93R^2_(t-1) + 1e-0.8R^2(t-2) + 1e-0.8u^2_(t-1)
predict(modeloo, n.ahead=5, plot=TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.002948617 0.01306825 0.01306825 -0.02266468 0.02856192
## 2 0.002948617 0.01302291 0.01302291 -0.02257582 0.02847306
## 3 0.002948617 0.01297773 0.01297773 -0.02248728 0.02838451
## 4 0.002948617 0.01293271 0.01293271 -0.02239903 0.02829627
## 5 0.002948617 0.01288785 0.01288785 -0.02231110 0.02820833modeloo2 <- garchFit(~garch(2,2), data=rendimiento)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 2)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 2 2
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 20
## Recursion Init: mci
## Series Scale: 0.01437149
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -2.18314624 2.183146 0.2183146 TRUE
## omega 0.00000100 100.000000 0.1000000 TRUE
## alpha1 0.00000001 1.000000 0.0500000 TRUE
## alpha2 0.00000001 1.000000 0.0500000 TRUE
## gamma1 -0.99999999 1.000000 0.1000000 FALSE
## gamma2 -0.99999999 1.000000 0.1000000 FALSE
## beta1 0.00000001 1.000000 0.4000000 TRUE
## beta2 0.00000001 1.000000 0.4000000 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 alpha2 beta1 beta2
## 1 2 3 4 7 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 28.170183: 0.218315 0.100000 0.0500000 0.0500000 0.400000 0.400000
## 1: 27.924754: 0.211097 0.481954 1.00000e-08 1.00000e-08 0.222692 0.329487
## 2: 27.922363: 0.211343 0.445889 1.00000e-08 1.00000e-08 0.184822 0.291770
## 3: 27.814952: 0.211349 0.457380 0.0267217 1.00000e-08 0.194658 0.301610
## 4: 27.703492: 0.212334 0.427057 0.138508 1.00000e-08 0.152600 0.263799
## 5: 27.691258: 0.212541 0.422030 0.134272 1.00000e-08 0.140332 0.254599
## 6: 27.679154: 0.213447 0.442141 0.156872 1.00000e-08 0.124708 0.253676
## 7: 27.654597: 0.213564 0.454349 0.158592 1.00000e-08 0.0972009 0.237290
## 8: 27.615387: 0.216835 0.519803 0.208701 1.00000e-08 0.0334769 0.228921
## 9: 27.562178: 0.218340 0.560548 0.205485 1.00000e-08 1.00000e-08 0.154663
## 10: 27.474363: 0.259219 0.729793 0.215570 1.00000e-08 1.00000e-08 1.00000e-08
## 11: 27.463940: 0.282105 0.706564 0.252274 1.00000e-08 1.00000e-08 1.00000e-08
## 12: 27.461109: 0.276170 0.692816 0.265528 1.00000e-08 1.00000e-08 1.00000e-08
## 13: 27.460400: 0.275302 0.682440 0.269024 1.00000e-08 1.00000e-08 1.00000e-08
## 14: 27.460358: 0.275302 0.681351 0.267558 1.00000e-08 1.00000e-08 1.00000e-08
## 15: 27.460357: 0.275366 0.681484 0.267124 1.00000e-08 1.00000e-08 1.00000e-08
## 16: 27.460357: 0.275371 0.681500 0.267113 1.00000e-08 1.00000e-08 1.00000e-08
##
## Final Estimate of the Negative LLH:
## LLH: -57.38982 norm LLH: -2.869491
## mu omega alpha1 alpha2 beta1 beta2
## 0.0039574883 0.0001407567 0.2671125192 0.0000000100 0.0000000100 0.0000000100
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 alpha2 beta1
## mu -109295.61079 -387096.35 116.902458 -138.231690 -86.293074
## omega -387096.35164 -317763896.55 -37126.647129 -34668.561113 -53439.972054
## alpha1 116.90246 -37126.65 -12.791437 -3.714648 -8.755150
## alpha2 -138.23169 -34668.56 -3.714648 -5.585587 -8.966858
## beta1 -86.29307 -53439.97 -8.755150 -8.966858 -10.527384
## beta2 -386.92494 -51715.58 -10.580606 -7.666944 -10.172590
## beta2
## mu -386.924942
## omega -51715.581804
## alpha1 -10.580606
## alpha2 -7.666944
## beta1 -10.172590
## beta2 -8.903082
## attr(,"time")
## Time difference of 0.00468421 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): Se han producido NaNs##
## Time to Estimate Parameters:
## Time difference of 0.01228309 secscoef(modeloo2)## mu omega alpha1 alpha2 beta1 beta2
## 0.0039574883 0.0001407567 0.2671125192 0.0000000100 0.0000000100 0.0000000100varianza(t) = 0.00014 + 0.0000000001R^2_(t-1) + 0.000000001R^2_(t-1) + 0.2671u^2_(t-1) + 0.000000001u^2(t-2)
predict(modeloo2, n.ahead=5, plot=TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 0.003957488 0.01193131 0.01193131 -0.01942744 0.02734242
## 2 0.003957488 0.01337093 0.01337093 -0.02224906 0.03016403
## 3 0.003957488 0.01372995 0.01372995 -0.02295273 0.03086770
## 4 0.003957488 0.01382427 0.01382427 -0.02313759 0.03105257
## 5 0.003957488 0.01384936 0.01384936 -0.02318676 0.03110173