ARIMA Time Series Analysis

1. Load Libraries

library(quantmod)

## Loading required package: xts

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Loading required package: TTR

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

library(forecast)
library(tseries)

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2. Download Data from Yahoo Finance

ticker <- "ENEL.MI"

data <- getSymbols(ticker, src="yahoo", auto.assign=FALSE)

price <- data[,6]

head(price)

##            ENEL.MI.Adjusted
## 2007-01-02         2.199554
## 2007-01-03         2.205154
## 2007-01-04         2.195354
## 2007-01-05         2.179952
## 2007-01-08         2.160351
## 2007-01-09         2.174352

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3. Compute Monthly Log Returns

monthly_price <- to.monthly(price, indexAt="lastof", drop.time=TRUE)
monthly_close <- monthly_price[,4]

returns <- diff(log(monthly_close)) * 100
returns <- na.omit(returns)

head(returns)

##            price.Close
## 2007-02-28   -2.810044
## 2007-03-31    1.446132
## 2007-04-30    4.097166
## 2007-05-31    1.309536
## 2007-06-30   -2.320569
## 2007-07-31   -5.013916

y <- ts(as.numeric(returns), frequency=12)

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4. Plot Monthly Returns

plot(y, main="Monthly Log Returns", ylab="Return (%)")

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5. Descriptive Statistics

mean(y)

## [1] 0.6538942

sd(y)

## [1] 6.39916

var(y)

## [1] 40.94925

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6. T-Test for Zero Mean

t.test(y)

## 
##  One Sample t-test
## 
## data:  y
## t = 1.5463, df = 228, p-value = 0.1234
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.1793356  1.4871239
## sample estimates:
## mean of x 
## 0.6538942

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7. Stationarity Test (ADF Test)

adf.test(y)

## Warning in adf.test(y): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  y
## Dickey-Fuller = -5.5855, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationary

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8. Fit ARIMA Model

fit <- auto.arima(y, seasonal=FALSE)
summary(fit)

## Series: y 
## ARIMA(0,0,0) with non-zero mean 
## 
## Coefficients:
##         mean
##       0.6539
## s.e.  0.4219
## 
## sigma^2 = 40.95:  log likelihood = -749.5
## AIC=1503   AICc=1503.05   BIC=1509.86
## 
## Training set error measures:
##                        ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 4.695399e-13 6.385173 5.032642 108.5337 113.9605 0.6785761
##                    ACF1
## Training set 0.05160885

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9. Residual Diagnostics

checkresiduals(fit)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,0,0) with non-zero mean
## Q* = 10.372, df = 24, p-value = 0.9928
## 
## Model df: 0.   Total lags used: 24

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10. Forecast 12 Months Ahead

fc <- forecast(fit, h=12)

plot(fc, main="ARIMA Forecast")

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11. Forecast Accuracy

accuracy(fc)

##                        ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 4.695399e-13 6.385173 5.032642 108.5337 113.9605 0.6785761
##                    ACF1
## Training set 0.05160885