Analysis of BitCoin Closing prices
Anurag Sangwan(s3701261) Arjun Balaji(s3629999) Kriti Kumar(s3675682) Tanyya Varshney(s3690009)
09 June 2019
rm(list = ls())
library(easypackages)
checkForPack = c("knitr","car","jsonlite","lattice","stringr","sp","tidyr","forecast", "dplyr",
"readr","expsmooth","TSA","dynlm","Hmisc","car","AER","tseries","lmtest",
"FitAR","fUnitRoots","FSAdata", "kableExtra", "truncnorm", "rugarch", "lubridate")
checkNInst = checkForPack %in% rownames(installed.packages())
if ( any(!checkNInst) ) { install.packages( checkForPack[!checkNInst] ) }
libraries(checkForPack)
source('C:/Arjun/RMIT/7. Time Series/Project/ts-bitcoin/residual.analysis.R')
source('C:/Arjun/RMIT/7. Time Series/Project/ts-bitcoin/sort.score.R')
source('C:/Arjun/RMIT/7. Time Series/Project/ts-bitcoin/MASE.R')
load("C:\\Arjun\\RMIT\\7. Time Series\\Project\\ts-bitcoin\\Report_10_06_19.RData")
#source('C:/Arjun/RMIT/7. Time Series/Functions/residual.analysis.R')
set.seed(738)Problem
Bitcoin is decentralized currency which is borderless, secure and enables unaltered transactions at low cast. Everything buyer needs to know from bull market surge to losing its value assessing the volatility of values of bitcoin prices. Tracking daily closing price of bitcoin (27th of April 2013 to the 24th of February 2019). Source: coinmarketcap.com.
Our task is to analyze the data by using the analysis methods covered in MATH1318 Time Series Analysis course in this semester, fit the best model that accurately predict the value of bitcoin for the next 10 days, and has a minimum MASE.
forecasting_period = 10
testSet = c(3882.7, 3854.36, 3851.05, 3854.79, 3859.58, 3864.42, 3847.18, 3761.56, 3896.38, 3903.94)
bitCoin_csv = read.csv("bitCoin_Historical_Price.csv")
bitCoin_csv$Close = as.numeric(as.character(gsub(",","",bitCoin_csv$Close))) # Will convert factor to a numeric value by stripping the commas to avoid NA's
bitCoin_ts = ts(as.numeric(as.vector(bitCoin_csv$Close)), start=c(2013, 117), frequency = 365)any(!is.finite(bitCoin_ts)) #checking for NA's## [1] FALSEData Description
Bit coing closing prices from 27th April 2013 to 24th February 2019.
Descriptive Statistics
par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
plot(bitCoin_ts,
#axes = FALSE,
ylab='Closing Price in $',
xlab='Year',
type='l',
col = c("#003366"),
lwd=2,
main = "Time Series Plot of BitCoin Closing Prices (in $)")
legend("topleft",lty=1, bty = "n" ,text.width = 15, col=c("#003366"),lwd=1.5,
c("Closing Price in $"))
grid()
Fig 1.: Time Series Plot of BitCoin Closing Prices
From the above time series plot(Fig 1.) we can comment on the fundamental concepts mentioned below:
- Trend: No visible evidence of trend until mid August of 2017.
- Changing variance: No apparent change in the variance until the end of the series.
- Seasonality: No repeating pattern seen, signifying no apparent indication of seasonality effect. This might indicate the volatility in crypto currencies.
- Intervention: Start of the series does not show any Intervention effect. However, there might be a possible Intervention point at mid August 2017. Regulation changes and the hard fork splitting might be a reason for rising crypto currency prices but we need to research on the whether it was actually an intervention or it is still because of its volatile nature.
bitCoin_split = bitCoin_csv
bitCoin_split$Date <- as.Date(bitCoin_split$Date, format = "%d/%m/%y")
class(bitCoin_split$Date)## [1] "Date"bitCoin_split_17 = bitCoin_split[bitCoin_split$Date >= as.Date("2017-04-01"),]
bitcoints = ts(bitCoin_split_17$Close,
start = c(2017, as.numeric(format(as.Date("2017-04-01"), "%j"))),
frequency = 365)par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
plot(bitcoints,
# axes = FALSE,
ylab='Closing Price in $',
xlab='Year',
type='l',
col = c("#003366"),
lwd=2,
main = "Time Series Plot of BitCoin Closing Prices after 2017 split (in $)")
legend("topleft",lty=1, bty = "n" ,text.width = 15, col=c("#003366"),lwd=2,
c("Closing Price in $ after split"))
grid()
Fig 2.: Time Series Plot of BitCoin Closing Prices after split
Upon observation from the above time series plot(Fig 2.) we can notice the effect of Trend & Changing variance clearly. Might be worth considering the split series for modelling.
par(mar=c(5,4,4.5,1),mfrow=c(1,2),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
acf(bitCoin_ts, main = "ACF plot for BitCoin Prices")
pacf(bitCoin_ts, main = "PACF plot for BitCoin Prices")
Fig 3.: ACF and PACF Plots of BitCoin Closing Prices
par(mar=c(5,4,4.5,1),mfrow=c(1,2),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
acf(bitcoints, main = "ACF plot for BitCoin Prices post split")
pacf(bitcoints, main = "PACF plot for BitCoin Prices post split")
Fig 4.: ACF and PACF Plots of BitCoin Closing Prices
#adf_test_model <- adf.test(bitcoin_ts)The above (Fig 3.) shows the ACF and PACF plots for the original series. There is a strong decaying pattern in ACF which shows that the series has a AUto Regressive behaviour and it is expected as from the (Fig 1.) time series plot we noticed the successive data points in the series. (Fig 4.) ACF & PACF plots of the split series shows the same AR behaviour. But since the ACF & PACF plots shows a strong declining trend which shows the evidence of non-stationarity, we must first deal with trend & changing variance'before we comment on the behaviour of the series*.
par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
y = bitCoin_ts
x = zlag(bitCoin_ts)
index = 2:length(x)
cor = cor(y[index],x[index])
plot(y=bitCoin_ts,
x=zlag(bitCoin_ts),
ylab='Closing Price in $ after split',
xlab='Previous Year Closing Price in $',
main = "Relationship between successive data points",
col = c("#003366"))
legend("topleft",lty=3, bty = "n" ,text.width = 15, col=c("#003366"),lwd=4,
c(paste0("Correlation: ", round(cor,4))))
grid()
Fig 5.: Relationship between successive data points
From (Fig 5.), we see the time series has a strong Auto Regressive behaviour with high correlation value [* 0.9976471 *] between the succeeding data points.
Model Selection
Considering above (Fig 3.) & (Fig 4.) & (Fig 5.); we see a strong trend component in the series. So we can proceed with basic Linear Trend model to see how well it fits our series. Further we can also go for higher order ARMA models with AR components of possible values [p = c(1,2,3,4,5)] & q = 0[purely based on (Fig 3.) & (Fig 4.).
Linear Trend Model
lmModel = lm(bitCoin_ts~ time(bitCoin_ts))
summary(lmModel)##
## Call:
## lm(formula = bitCoin_ts ~ time(bitCoin_ts))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2907.7 -1865.4 -371.2 1212.0 14773.3
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.778e+06 6.383e+04 -43.52 <2e-16 ***
## time(bitCoin_ts) 1.379e+03 3.166e+01 43.56 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2461 on 2128 degrees of freedom
## Multiple R-squared: 0.4714, Adjusted R-squared: 0.4711
## F-statistic: 1898 on 1 and 2128 DF, p-value: < 2.2e-16The linear trend model is significant with p-value very negligible[close to 0]. The coefficients and intercept terms are significant. Residual errors are higher but can be analysed further in our residual analysis. However, the R2 values are pretty much low which questions whether the model is a good fit for this data set.
Let us fit the model.
Model Fitting
par(mar=c(5,4,4,3),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
plot(bitCoin_ts,
ylab='Closing Price in $ after split',
xlab='Year',
type='l',
col = c("#003366"),
lwd = 2,
main = "Fitted Linear Curve to the BitCoin Closing Price Series")
#lines(as.vector(fitted(lmModel)), col="black", lwd = 1)
abline(lmModel, col = "red", lwd = 2)
legend("topleft",lty=1, text.width = 20, col=c("#003366","red"), bty = "n",
c("BitCoin Series", "Fitted Linear Curve"))
Fig 6.: Fitted Curves to the BitCoin Closing Proce Series
(Fig 6.) confirms visually that the linear Trend model is not a good fit for the time series data. We can further perform a Residual check and confirm that as well.
residual.analysis(lmModel)##
## Shapiro-Wilk normality test
##
## data: res.model
## W = 0.82362, p-value < 2.2e-16## Warning in (ra^2)/(n - (1:lag.max)): longer object length is not a multiple
## of shorter object length
Fig 7.: Residual Analysis of Linear Trend Model
Residual Analysis (Fig 7.) results proves that the linear trend model is not a good fit for this series. This might be because of the Auto Regressive behaviour we see on the ACF & PACF plots from (Fig 3.) & (Fig 4.), the residual analysis also confirms the AR effect. Hence we might have to go for different models such as ARIMA & ARIMA-GARCH models.
ARIMA Model
Before we proceed further with ARIMA models, the requirement is that the time series must be stationary. Let us perform a ADF Test and see whether the series pass that requirement.
ADF Test
- Assumtions:
- H0: The given series is non-stationary
- Ha: Otherwise; stationary
testAdf = adf.test(bitCoin_ts)
testAdf##
## Augmented Dickey-Fuller Test
##
## data: bitCoin_ts
## Dickey-Fuller = -2.6913, Lag order = 12, p-value = 0.2856
## alternative hypothesis: stationaryWith * 0.2856399. * > 0.05 we infer that there is no enough statistical evidence & we fail to reject the H0:. Hence the series is non-stationary.
Transformations
As we mentioned above, before concluding about the behaviour of the series we must first deal with trend followed by changing variance.
Log Transformation
Let us go with log transformation with lambda value = 0, since we see a changing variance in the series.
lambda = 0
bitCoin.log = log(bitCoin_ts)Box-Cox Tranformation
round(min(bitCoin_ts),2) #checking for any negative values before## [1] 68.43bitCoin.BoxCox = BoxCox.ar(bitCoin_ts, seq(-1,1,0.01), method = "ols")
lambda <- (as.numeric(bitCoin.BoxCox$ci[1]) + as.numeric(bitCoin.BoxCox$ci[2])) / 2
lambda## [1] -0.1par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
plot(bitCoin.log,
# axes = FALSE,
ylab='Closing Price in $',
xlab='Year',
type='l',
col = c("#003366"),
lwd=2,
main = "Time Series Plot of Transformed BitCoin Closing Prices (in $)")
legend("topleft",lty=1, bty = "n" ,text.width = 15, col=c("#003366"),lwd=2,
c("log transformed series"))
Fig 5.: Time Series Plot of Transformed BitCoin Closing Prices
par(mar=c(5,4,4.5,1),mfrow=c(1,2),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
acf(bitCoin.log, main = "ACF plot for tranformed BitCoin Prices")
pacf(bitCoin.log, main = "PACF plot for transformed BitCoin Prices")
Fig 6.: ACF and PACF Plots of log tranformed BitCoin Closing Prices
From (Fig 5.) the log transformed series still has the incremental trend and the ACF & PACF plots (Fig 6.) still shows prevalance of non-stationarity with some changing variance. We can perform adf test again to confirm if the log transformed series is stationary.
testAdf = adf.test(bitCoin.log)
testAdf##
## Augmented Dickey-Fuller Test
##
## data: bitCoin.log
## Dickey-Fuller = -1.6272, Lag order = 12, p-value = 0.7362
## alternative hypothesis: stationaryWith * 0.7361506. * > 0.05 we infer that there is no enough statistical evidence & we fail to reject the H0:. Hence the series is non-stationary.
To de-rend the series we will perform 1st differencing.
Differencing
diff.bitCoin.log = diff(bitCoin.log, differences = 1)
testAdf = adf.test(diff.bitCoin.log)## Warning in adf.test(diff.bitCoin.log): p-value smaller than printed p-valuetestAdf##
## Augmented Dickey-Fuller Test
##
## data: diff.bitCoin.log
## Dickey-Fuller = -11.171, Lag order = 12, p-value = 0.01
## alternative hypothesis: stationaryWith * 0.01. * < 0.05 we infer that there is enough statistical evidence & we reject the H0:. Hence the Ha: : series is stationary. Now that we have a stationary series, let us take a visual inspection of the differenced series and comment on the behaviour of the series.
par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
plot(diff.bitCoin.log,
# axes = FALSE,
ylab='Closing Price in $',
xlab='Year',
type='l',
col = c("#003366"),
lwd=1,
main = "Time Series Plot of Transformed BitCoin Closing Prices (in $)")
legend("topleft",lty=1, bty = "n" ,text.width = 15, col=c("#003366"),lwd=1,
c("Log-tranformed Differenced Series"))
Fig 7.: Time Series Plot of Differenced Series
From (Fig 7.), we can interpret as the series does not have any trend compared to the earlier non- differenced series.
Model Selection
par(mar=c(5,4,4.5,1),mfrow=c(1,2),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
acf(diff.bitCoin.log, main = "ACF plot for differenced series")
pacf(diff.bitCoin.log, main = "PACF plot for differenced series")
Fig 8.: ACF and PACF Plots of Differenced series
Fig 10.: BIC Plot of Differenced series
ARIMA(6,1,6)
Model Fitting
Function
.GlobalEnv$.__C__Arima## Virtual Class "Arima" [in ".GlobalEnv"]
##
## Slots:
##
## Name: .S3Class
## Class: character
##
## Extends: "oldClass".GlobalEnv$.__C__ARIMA## Virtual Class "ARIMA" [in ".GlobalEnv"]
##
## Slots:
##
## Name: .S3Class
## Class: character
##
## Extends: "oldClass".GlobalEnv$.__C__forecast## Virtual Class "forecast" [in ".GlobalEnv"]
##
## Slots:
##
## Name: .S3Class
## Class: character
##
## Extends: "oldClass"#setOldClass(c("Arima", "ARIMA", "forecast"))
# flag = FALSE
forecasting_period = 10
# if (flag == TRUE) max_order = 2 else {
# max_order = 6
# smaller_max_order = 3
# }
arma_garch_model_list = list()
arima_model_list <- list()
#
# first_arima_model_list = list()
# last_arima_model_list = list()
# first_arma_garch_model_list = list()
# last_arma_garch_model_list = list()
setClass(Class="ArimaClass",
representation(
key="character",
model_name="character",
ar="numeric",
ma="numeric",
d="numeric",
est_method="character",
aic="numeric",
bic="numeric",
fitted_mase="numeric",
forc_mase="numeric",
fitted_val="numeric",
forc_val="numeric",
model_forc="forecast",
model_fit="ARIMA",
model="Arima"
)
)
setClass(Class="ARMA_GARCH_Models",
representation(
key="character",
model_name="character",
mean_flag="logical",
ar="numeric",
ma="numeric",
alpha_order="numeric",
beta_order="numeric",
d="numeric",
est_method = "character",
aic="numeric",
bic="numeric",
fitted_mase="numeric",
forc_mase="numeric",
fitted_val="numeric",
forc_val = "numeric",
model_forc="uGARCHforecast",
model_fit= "uGARCHfit"
)
)Execution
arima_model_list = list()
for (nar in 1:8) {
for (nma in 1:8) {
arima_model_list <- generateARIMAModels(diff.bitCoin.log,
ar=nar, d=1, ma=nma,
arima_model_list, lambda,testSet,"Modelling")
}
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## [1] "Processing :ARIMA(8 , 1 , 7) - CSS"
## [1] "Processing :ARIMA(8 , 1 , 8) - ML"
## [1] "Processing :ARIMA(8 , 1 , 8) - CSS"results_df = data.frame(model_name=character(),
ar_order=numeric(), d_order=numeric(),ma_order=numeric(),
alpha_order = numeric(), beta_order = numeric(),
est_method = character(), mase_forecast = numeric(), mase_fitted = numeric(), aic = numeric(),
bic = numeric())
names_vec = colnames(results_df)
names(results_df) <- names_vec
len <- length(arima_model_list)
for(il in 1:len) {
index <- (gregexpr(pattern ='\\(', arima_model_list[[il]]@key)[[1]][1]) -1
temp <- data.frame(substring(text = arima_model_list[[il]]@key,1, index),
arima_model_list[[il]]@ar,
arima_model_list[[il]]@d,
arima_model_list[[il]]@ma,
NA,
NA,
arima_model_list[[il]]@est_method,
arima_model_list[[il]]@forc_mase,
arima_model_list[[il]]@fitted_mase,
arima_model_list[[il]]@aic,
arima_model_list[[il]]@bic)
names(temp)<-names_vec
results_df <- rbind(results_df,temp)
}Interpretation - Best MASEs
newDf <- arrange(results_df, mase_forecast, mase_fitted)
newDf <- subset(newDf, select = c(-alpha_order, -beta_order))kable(newDf[1:5,], caption = "Best MASE from the ARIMA models - Top 5") %>%
kable_styling("striped", full_width = F, font_size = 7, position = "center") %>%
column_spec(c(6,7), bold = T, color = "white", background = "#ed9121") #"#FF4040") | model_name | ar_order | d_order | ma_order | est_method | mase_forecast | mase_fitted | aic | bic |
|---|---|---|---|---|---|---|---|---|
| ARIMA | 1 | 1 | 1 | CSS | 0.9169127 | 5.820370 | -7289.603 | -7272.613 |
| ARIMA | 5 | 1 | 2 | CSS | 2.1153031 | 5.739614 | -7292.512 | -7247.204 |
| ARIMA | 2 | 1 | 8 | CSS | 2.2483379 | 4.843105 | -7288.489 | -7226.191 |
| ARIMA | 2 | 1 | 1 | CSS | 2.4957636 | 3.125585 | -7288.641 | -7265.987 |
| ARIMA | 2 | 1 | 3 | CSS | 2.6213102 | 4.869749 | -7284.240 | -7250.259 |
Significance test for coefficients
print("Results for ARIMA(1,1,1)")## [1] "Results for ARIMA(1,1,1)"model_111_css = arima(diff.bitCoin.log,order=c(1,1,1),method='CSS')
coeftest(model_111_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -0.0287102 0.0214228 -1.3402 0.1802
## ma1 -0.9746954 0.0054057 -180.3085 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Results for ARIMA(5,1,2)")## [1] "Results for ARIMA(5,1,2)"model_512_css = arima(diff.bitCoin.log,order=c(5,1,2),method='CSS')
coeftest(model_512_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -0.7733735 0.0710130 -10.8906 < 2.2e-16 ***
## ar2 -0.0889116 0.0262470 -3.3875 0.0007053 ***
## ar3 -0.0800373 0.0257986 -3.1024 0.0019196 **
## ar4 -0.0276754 0.0279020 -0.9919 0.3212562
## ar5 -0.0065301 0.0227844 -0.2866 0.7744164
## ma1 -0.2344971 0.0677574 -3.4608 0.0005385 ***
## ma2 -0.7119915 0.0724533 -9.8269 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Results for ARIMA(2,1,8)")## [1] "Results for ARIMA(2,1,8)"model_218_css = arima(diff.bitCoin.log,order=c(2,1,8),method='CSS')
coeftest(model_218_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -1.162404 0.079720 -14.5810 < 2.2e-16 ***
## ar2 -0.665243 0.102700 -6.4775 9.325e-11 ***
## ma1 0.179438 0.080442 2.2307 0.02570 *
## ma2 -0.490501 0.092863 -5.2820 1.278e-07 ***
## ma3 -0.641776 0.105011 -6.1115 9.871e-10 ***
## ma4 0.041857 0.028625 1.4623 0.14367
## ma5 0.055687 0.027447 2.0288 0.04247 *
## ma6 0.030577 0.023817 1.2838 0.19921
## ma7 -0.039534 0.023047 -1.7154 0.08627 .
## ma8 -0.039701 0.021779 -1.8229 0.06832 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Results for ARIMA(2,1,1)")## [1] "Results for ARIMA(2,1,1)"model_211_css = arima(diff.bitCoin.log,order=c(2,1,1),method='CSS')
coeftest(model_211_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -0.064008 0.023260 -2.7519 0.005926 **
## ar2 -0.097333 0.022157 -4.3929 1.118e-05 ***
## ma1 -0.898976 0.011434 -78.6206 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Results for ARIMA(2,1,3)")## [1] "Results for ARIMA(2,1,3)"model_213_css = arima(diff.bitCoin.log,order=c(2,1,3),method='CSS')
coeftest(model_213_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -1.180745 0.078714 -15.0005 < 2.2e-16 ***
## ar2 -0.680335 0.086561 -7.8596 3.853e-15 ***
## ma1 0.217337 0.081118 2.6793 0.007379 **
## ma2 -0.473441 0.081981 -5.7750 7.696e-09 ***
## ma3 -0.646717 0.086064 -7.5144 5.718e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the above models tried, we witness that only ARIMA(2,1,1) & ARIMA(2,1,3) are having all significant coefficients. Also from (Fig 11.) above we notice that ARIMA(2,1,1) has the best MASE compared to its latter.
Hence considering both the significance of coefficients and MASE values, we can choose ARIMA(2,1,1) as the best model. However we will have to consider overfitting and residual test to confirm.
Check for Overfitting
We will perform overfitting test by incrementing the orders of p & q to 1 keeping the best model chosen as base i.e ARIMA(2,1,1).
print("Results for ARIMA(2,1,2)")## [1] "Results for ARIMA(2,1,2)"model_212_css = arima(diff.bitCoin.log,order=c(2,1,2),method='CSS')
coeftest(model_212_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -0.884826 0.059269 -14.9289 < 2e-16 ***
## ar2 -0.043438 0.022417 -1.9377 0.05265 .
## ma1 -0.095207 0.056040 -1.6989 0.08934 .
## ma2 -0.790719 0.054968 -14.3850 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Results for ARIMA(3,1,1)")## [1] "Results for ARIMA(3,1,1)"model_311_css = arima(diff.bitCoin.log,order=c(3,1,1),method='CSS')
coeftest(model_311_css)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## ar1 -0.018781 0.024891 -0.7545 0.450545
## ar2 -0.056800 0.023262 -2.4417 0.014616 *
## ar3 -0.061427 0.022153 -2.7729 0.005557 **
## ma1 -0.932636 0.011115 -83.9063 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From above results we can reject the ARIMA(3,1,1) model as the coefficients aren’t significant. In case of ARIMA(2,1,2) it has ar2 and ma1 coefficients nearly significant[90% significance level], but lower than the ARIMA(2,1,1) model and also from below (Fig 12.) we see that the error component[MASE] is lower for ARIMA(2,1,1) model. Hence we will go for that model over the ARIMA(2,1,2). But we will perform residual analysis before forecasting.
kable(newDf[ c(4,10),], caption = "Comparing MASE of ARIMA(2,1,1) and ARIMA(2,1,2)") %>%
kable_styling("striped", full_width = F, font_size = 7, position = "left") %>%
column_spec(c(7,8), bold = T, color = "white", background = "#ed9121") | model_name | ar_order | d_order | ma_order | est_method | mase_forecast | mase_fitted | aic | bic | |
|---|---|---|---|---|---|---|---|---|---|
| 4 | ARIMA | 2 | 1 | 1 | CSS | 2.495764 | 3.125585 | -7288.641 | -7265.987 |
| 10 | ARIMA | 2 | 1 | 2 | CSS | 3.767458 | 3.646530 | -7286.471 | -7258.154 |
Residual Analysis
residual.analysis(model_211_css)##
## Shapiro-Wilk normality test
##
## data: res.model
## W = 0.88488, p-value < 2.2e-16## Warning in (ra^2)/(n - (1:lag.max)): longer object length is not a multiple
## of shorter object length
Fig 13.: Residual Analysis of ARIMA(2,1,1) Model
From (Fig 13.) we see that the standardised residuals are approximately normal gicing us the bell-curve shape. Also the qq plot supports the normality check for the standardised residuals with tails showing some variance, might be possible noise or outliers. Ljung-Box Test shows the evidence of Auto regressive behaviour in the series.
Fitting plot
par(mar=c(5,4,4.5,1),cex.main=1, cex.lab=1, cex.axis=1) #c(bottom, left, top, right)
arima_key <- "ARIMA(2 , 1 , 1) - CSS"
plot(bitCoin_ts,
type="l",
lwd = 2,
col = c("#003366"),
main = "BitCoin Closing Price Series",
ylab = "Closing Price in $",
xlab = "Year")
abline(lmModel, col = "red", lwd = 2)
lines(arima_model_list[[arima_key]]@model_fit$fitted, col= c("green"), lwd = 2)
legend("topleft",lty=1, bty = "n" ,text.width = 15, col=c("#003366","red", "green"),
c("Original Series", "Fitted Linear Model", "Fitted ARIMA(2,1,1) Model"))
grid()
Fig 14.: Fitted Curves to the BitCoin Closing Price Series
