Stat 145 Take Home Exam

R Codes:

install.packages("tsbox")
library(tsbox)
install.packages("tseries")
library(tseries)
install.packages("MASS")
library(MASS)
install.packages("forecast")
library(forecast)

coin <- read.csv("C:\\Users\\mbbac\\Downloads\\bitcoin_dataset_trim.csv")

#use only the variables "Date" and "btc_market_price"

coin2<-data.frame(
  Dates=coin$Date,
  btc_price=coin$btc_market_price
)

#convert data frame to time series

coin_ts<-ts_ts(ts_long(coin2))

#checking for the class
class(coin_ts)

#Step 1: Plot

plot(coin_ts)

Insights about the plot: 1. Trend (Upward) 2. Seasonality 3. Increasing variance 4. Stationary

#Step 2: Generate sample ACF and PACF of original series

acf(coin_ts)
pacf(coin_ts)

Insights about the ACF plot: 1. Autocorrelation at various lags 2. Gradual decay (trend or nonstationarity) 3. Need for differencing (gradual decay) thus we perform unit root test (formal test) to test stationarity.

### Augmented Dickey-Fuller Test

adf.test(coin_ts)

### KPSS Test for Level Stationarity

kpss.test(coin_ts)

Both test result: “the process is not stationary”

### Differencing

diff_ts <- diff(coin_ts, differences = 2)

### Original vs Differenced Series

par(mfrow = c(2, 1))
plot(coin_ts, main = "Original Time Series", ylab = "Value", col = "blue")
plot(diff_ts, main = "Differenced Time Series", ylab = "Value", col = "red")

After first differencing, the series still shows non constancy of variance (increasing variance). Same result if we do second differencing.

### Box Cox Transformation

lambda <- BoxCox.lambda(coin_ts)  # Find the optimal lambda
boxcox_ts <- BoxCox(coin_ts, lambda = lambda)  # Apply the transformation
plot(boxcox_ts, main = "Box-Cox Transformed Time Series", col = "purple")

lambda result: 1 (no transformation needed) Plot for Box-Cox transformed is still the same with the original time series data.

### Alternative: Log Transformation

log_ts <- log(coin_ts) 
plot(log_ts, main = "Log-Transformed Time Series", col = "blue")

Insights: 1. Trend 2. Seasonality 3. Increasing variance