Forecasting models
Zeng Yao
5/9/2021
#INTRODUCTION This week i will be working on the same data, tesla inc dataset to forecast the closing stock using ETS, ARIMA and GARCH models.
#removing all the variables in the working space
rm(list = ls())
#loading the libraries
library(fpp2)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## -- Attaching packages ---------------------------------------------- fpp2 2.4 --## v ggplot2 3.3.3 v fma 2.4
## v forecast 8.14 v expsmooth 2.3## library(tidyverse)## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --## v tibble 3.0.3 v dplyr 1.0.1
## v tidyr 1.1.2 v stringr 1.4.0
## v readr 1.4.0 v forcats 0.5.0
## v purrr 0.3.4## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(readr)
library(psych)##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphalibrary(caret)## Loading required package: lattice##
## Attaching package: 'caret'## The following object is masked from 'package:purrr':
##
## liftlibrary(readr)
library(lubridate)##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, unionlibrary(rugarch)## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:purrr':
##
## reduce## The following object is masked from 'package:stats':
##
## sigma#Loading the tesla dataset
TSLA <- read_csv("C:/Users/HP/Desktop/TSLA.csv")##
## -- Column specification --------------------------------------------------------
## cols(
## Date = col_date(format = ""),
## Open = col_double(),
## High = col_double(),
## Low = col_double(),
## Close = col_double(),
## `Adj Close` = col_double(),
## Volume = col_double()
## )View(TSLA)
##attaching the variables in the TSLA dataset
attach(TSLA)
##describing the closing prices
describe(Close)## vars n mean sd median trimmed mad min max range skew kurtosis
## X1 1 121 95.58 167.49 48.01 49.38 23.76 4.88 793.53 788.65 2.99 7.92
## se
## X1 15.23#selecting only important variables to be used in forecasting tesla closing stock prices
tesla <- select(TSLA, Date, Close)
tesla## # A tibble: 121 x 2
## Date Close
## <date> <dbl>
## 1 2011-05-01 6.03
## 2 2011-06-01 5.83
## 3 2011-07-01 5.63
## 4 2011-08-01 4.95
## 5 2011-09-01 4.88
## 6 2011-10-01 5.87
## 7 2011-11-01 6.55
## 8 2011-12-01 5.71
## 9 2012-01-01 5.81
## 10 2012-02-01 6.68
## # ... with 111 more rowsview(tesla)#Declare the tesla dataset as a time series data
tesla_data <- ts(tesla [, 2], start = c(2011,1), frequency = 12)
tesla_data## Jan Feb Mar Apr May Jun Jul Aug Sep
## 2011 6.028 5.826 5.634 4.948 4.878 5.874 6.548 5.712 5.814
## 2012 5.900 6.258 5.484 5.704 5.856 5.626 6.764 6.774 7.502
## 2013 19.552 21.472 26.856 33.800 38.674 31.988 25.456 30.086 36.282
## 2014 41.554 48.012 44.660 53.940 48.536 48.340 48.904 44.482 40.720
## 2015 50.160 53.652 53.230 49.812 49.680 41.386 46.052 48.002 38.240
## 2016 44.646 42.456 46.958 42.402 40.806 39.546 37.880 42.738 50.386
## 2017 68.202 72.322 64.694 71.180 68.220 66.306 61.770 62.270 70.862
## 2018 56.946 68.590 59.628 60.332 52.954 67.464 70.096 66.560 61.404
## 2019 37.032 44.692 48.322 45.122 48.174 62.984 65.988 83.666 130.114
## 2020 167.000 215.962 286.152 498.320 429.010 388.040 567.600 705.670 793.530
## 2021 739.780
## Oct Nov Dec
## 2011 6.682 7.448 6.626
## 2012 6.966 7.578 10.798
## 2013 48.962 41.690 41.578
## 2014 40.668 37.754 45.210
## 2015 38.386 45.954 48.152
## 2016 49.998 55.660 62.814
## 2017 68.612 53.226 58.780
## 2018 63.976 55.972 47.738
## 2019 133.598 104.800 156.376
## 2020 675.500 667.930 739.780
## 2021#time series plot
autoplot(tesla_data)+
ggtitle("Tesla Closing Prices")+
ylab("Closing prices")+
xlab("Time") +
theme_grey()
#Training and testing sets for modelling
tesla_train <- ts(tesla_data[1:97], start =2011 , frequency = 12)
tesla_train## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 2011 6.028 5.826 5.634 4.948 4.878 5.874 6.548 5.712 5.814 6.682
## 2012 5.900 6.258 5.484 5.704 5.856 5.626 6.764 6.774 7.502 6.966
## 2013 19.552 21.472 26.856 33.800 38.674 31.988 25.456 30.086 36.282 48.962
## 2014 41.554 48.012 44.660 53.940 48.536 48.340 48.904 44.482 40.720 40.668
## 2015 50.160 53.652 53.230 49.812 49.680 41.386 46.052 48.002 38.240 38.386
## 2016 44.646 42.456 46.958 42.402 40.806 39.546 37.880 42.738 50.386 49.998
## 2017 68.202 72.322 64.694 71.180 68.220 66.306 61.770 62.270 70.862 68.612
## 2018 56.946 68.590 59.628 60.332 52.954 67.464 70.096 66.560 61.404 63.976
## 2019 37.032
## Nov Dec
## 2011 7.448 6.626
## 2012 7.578 10.798
## 2013 41.690 41.578
## 2014 37.754 45.210
## 2015 45.954 48.152
## 2016 55.660 62.814
## 2017 53.226 58.780
## 2018 55.972 47.738
## 2019tesla_test <- ts(tesla_data[98: 121], start = 2019 , frequency = 12)
tesla_test## Jan Feb Mar Apr May Jun Jul Aug Sep
## 2019 44.692 48.322 45.122 48.174 62.984 65.988 83.666 130.114 133.598
## 2020 215.962 286.152 498.320 429.010 388.040 567.600 705.670 793.530 675.500
## Oct Nov Dec
## 2019 104.800 156.376 167.000
## 2020 667.930 739.780 739.780#Modelling
####ETS Model
ets_tesla_model <- ets(tesla_train)
ets_tesla_model## ETS(M,A,N)
##
## Call:
## ets(y = tesla_train)
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 0.1633
##
## Initial states:
## l = 5.3083
## b = 0.0628
##
## sigma: 0.1504
##
## AIC AICc BIC
## 733.9797 734.6391 746.8533#modelling
###auto.arima model
arima_tesla_model <- auto.arima(tesla_train)
arima_tesla_model## Series: tesla_train
## ARIMA(0,1,0)
##
## sigma^2 estimated as 27.79: log likelihood=-295.81
## AIC=593.62 AICc=593.66 BIC=596.18#Forecasting the closing prices for the next 6 months using the ETS model
forecast_ets <- forecast(ets_tesla_model)
forecast_ets## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2019 33.33149479 26.906975 39.75601 2.350604e+01 43.15695
## Mar 2019 29.63015128 20.158002 39.10230 1.514375e+01 44.11655
## Apr 2019 25.92880778 13.841162 38.01645 7.442348e+00 44.41527
## May 2019 22.22746427 7.693368 36.76156 -5.184806e-04 44.45545
## Jun 2019 18.52612076 1.614210 35.43803 -7.338415e+00 44.39066
## Jul 2019 14.82477725 -4.449993 34.09955 -1.465344e+01 44.30300
## Aug 2019 11.12343375 -10.535298 32.78217 -2.200074e+01 44.24761
## Sep 2019 7.42209024 -16.670606 31.51479 -2.942451e+01 44.26869
## Oct 2019 3.72074673 -22.882217 30.32371 -3.696498e+01 44.40647
## Nov 2019 0.01940322 -29.196030 29.23484 -4.466175e+01 44.70055
## Dec 2019 -3.68194029 -35.638649 28.27477 -5.255551e+01 45.19163
## Jan 2020 -7.38328379 -42.237920 27.47135 -6.068885e+01 45.92228
## Feb 2020 -11.08462730 -49.023143 26.85389 -6.910658e+01 46.93732
## Mar 2020 -14.78597081 -56.025102 26.45316 -7.785578e+01 48.28384
## Apr 2020 -18.48731432 -63.275978 26.30135 -8.698566e+01 50.01104
## May 2020 -22.18865782 -70.809214 26.43190 -9.654738e+01 52.17007
## Jun 2020 -25.89000133 -78.659359 26.87936 -1.065938e+02 54.81377
## Jul 2020 -29.59134484 -86.861936 27.67925 -1.171792e+02 57.99647
## Aug 2020 -33.29268835 -95.453348 28.86797 -1.283592e+02 61.77384
## Sep 2020 -36.99403186 -104.470828 30.48276 -1.401909e+02 66.20282
## Oct 2020 -40.69537536 -113.952457 32.56171 -1.527324e+02 71.34166
## Nov 2020 -44.39671887 -123.937238 35.14380 -1.660434e+02 77.25001
## Dec 2020 -48.09806238 -134.465223 38.26910 -1.801852e+02 83.98911
## Jan 2021 -51.79940589 -145.577690 41.97888 -1.952209e+02 91.62210autoplot(forecast_ets,
main = "Tesla ETS Model",
ylab = " Closing prices")
#Forecasting the closing prices for the next 6 months using the ARIMA model
forecast_arima <- forecast(arima_tesla_model)
forecast_arima## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2019 37.032 30.275676 43.78833 26.6990939 47.36491
## Mar 2019 37.032 27.477115 46.58689 22.4190637 51.64494
## Apr 2019 37.032 25.329703 48.73430 19.1348810 54.92912
## May 2019 37.032 23.519351 50.54465 16.3661869 57.69782
## Jun 2019 37.032 21.924400 52.13960 13.9269184 60.13708
## Jul 2019 37.032 20.482453 53.58155 11.7216511 62.34235
## Aug 2019 37.032 19.156446 54.90756 9.6936986 64.37030
## Sep 2019 37.032 17.922229 56.14177 7.8061264 66.25788
## Oct 2019 37.032 16.763027 57.30098 6.0332798 68.03072
## Nov 2019 37.032 15.666626 58.39738 4.3564798 69.70752
## Dec 2019 37.032 14.623807 59.44020 2.7616252 71.30238
## Jan 2020 37.032 13.627405 60.43660 1.2377609 72.82624
## Feb 2020 37.032 12.671726 61.39228 -0.2238253 74.28783
## Mar 2020 37.032 11.752148 62.31185 -1.6301971 75.69420
## Apr 2020 37.032 10.864868 63.19913 -2.9871760 77.05118
## May 2020 37.032 10.006702 64.05730 -4.2996273 78.36363
## Jun 2020 37.032 9.174960 64.88904 -5.5716663 79.63567
## Jul 2020 37.032 8.367343 65.69666 -6.8068110 80.87081
## Aug 2020 37.032 7.581864 66.48214 -8.0080967 82.07210
## Sep 2020 37.032 6.816798 67.24720 -9.1781642 83.24217
## Oct 2020 37.032 6.070631 67.99337 -10.3193278 84.38333
## Nov 2020 37.032 5.342029 68.72197 -11.4336292 85.49763
## Dec 2020 37.032 4.629806 69.43420 -12.5228805 86.58688
## Jan 2021 37.032 3.932905 70.13110 -13.5886988 87.65270autoplot(forecast_arima,
main = "Tesla ARIMA Model",
ylab = " Closing prices")
#Forecasting the closing stock for the next 6 month using GARCH model
spec_garch<- ugarchspec(variance.model=list(model="sGARCH",garchOrder=c(1,1)),mean.model=list(armaOrder=c(1,1)),distribution.model = "std")
spec_garch##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : sGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(1,0,1)
## Include Mean : TRUE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : std
## Includes Skew : FALSE
## Includes Shape : TRUE
## Includes Lambda : FALSEgarch_tesla <- ugarchfit(spec=spec_garch,data=tesla_train)## Warning in .sgarchfit(spec = spec, data = data, out.sample = out.sample, :
## ugarchfit-->waring: using less than 100 data
## points for estimationgarch_tesla##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 6.044815 0.940892 6.42456 0.00000
## ar1 0.991136 0.014801 66.96407 0.00000
## ma1 -0.017795 0.127691 -0.13936 0.88916
## omega 5.388075 4.383856 1.22907 0.21904
## alpha1 0.000000 0.372543 0.00000 1.00000
## beta1 0.999000 0.016898 59.12046 0.00000
## shape 2.100000 0.029416 71.38860 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 6.044815 0.050883 118.79936 0.00000
## ar1 0.991136 0.016496 60.08382 0.00000
## ma1 -0.017795 0.216227 -0.08230 0.93441
## omega 5.388075 6.574466 0.81955 0.41247
## alpha1 0.000000 1.114629 0.00000 1.00000
## beta1 0.999000 0.130116 7.67778 0.00000
## shape 2.100000 0.071748 29.26892 0.00000
##
## LogLikelihood : -286.9664
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.0612
## Bayes 6.2470
## Shibata 6.0517
## Hannan-Quinn 6.1363
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.02865 0.8656
## Lag[2*(p+q)+(p+q)-1][5] 3.75538 0.1199
## Lag[4*(p+q)+(p+q)-1][9] 6.81016 0.1473
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.356 0.12477
## Lag[2*(p+q)+(p+q)-1][5] 6.310 0.07598
## Lag[4*(p+q)+(p+q)-1][9] 12.454 0.01428
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.572 0.500 2.000 0.058769
## ARCH Lag[5] 6.104 1.440 1.667 0.056772
## ARCH Lag[7] 11.150 2.315 1.543 0.009892
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 13.7886
## Individual Statistics:
## mu 0.18822
## ar1 0.08645
## ma1 0.10311
## omega 0.46046
## alpha1 0.14415
## beta1 0.07253
## shape 1.54094
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5595 0.5772
## Negative Sign Bias 0.9341 0.3527
## Positive Sign Bias 1.0851 0.2807
## Joint Effect 2.6923 0.4415
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 31.45 0.035976
## 2 30 55.47 0.002180
## 3 40 67.54 0.003069
## 4 50 65.49 0.057667
##
##
## Elapsed time : 0.3626781fc.garch1 <- ugarchforecast(garch_tesla,n.ahead=6,method=c("Partial","Full")[1])
fc.garch1##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: sGARCH
## Horizon: 6
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=Jan 2019]:
## Series Sigma
## T+1 36.94 22.87
## T+2 36.67 22.98
## T+3 36.40 23.09
## T+4 36.13 23.19
## T+5 35.86 23.30
## T+6 35.60 23.40#MODEL COMPARISON BASED ON PERFORMANCE The ETS, ARIMA and GARCH model shows different results in regards to forecasting the tesla inc. closing stocks. The results reveals that the ARIMA model is the best model since it has the lowest AIC value and hence it is statistically significant.