The purpose of this in-class lab is to use R to practice with time series forecasting. The lab should be completed in your group. To get credit, upload your .R script to the appropriate place on Canvas.

For starters

Open up a new R script (named ICL14_XYZ.R, where XYZ are your initials) and add the usual “preamble” to the top:

# Add names of group members HERE
library(lubridate) # This package converts dates to a special date numbering system
library(fable)     # This one may take awhile to install
library(feasts)    # You also need to install this one
library(urca)      # and this one

Load the data

First we’ll look at the return on a 3-month treasury bill over the period of 1960q1–1990q4. Second, we’ll read in Google’s and Apple’s stock price data from January 3, 2005 until April 4, 2022.

# T-bill rates by quarter
df1 <- as_tibble(intqrt)
df1 %<>% mutate(quarter = seq(yq('1960:Q1'),yq('1990:Q4'), by = 'quarters')) # create quarter
df1 %<>% select(r3,quarter)

# Stock prices
df2 <- pdfetch_YAHOO(c("goog","aapl"), fields = c("adjclose"), 
                     from = as.Date("2005-01-01"),
                     to = as.Date("2022-04-04"),
                     interval = "1d") %>% %>% rownames_to_column(var="date") %>%
       as_tibble %>% mutate(date=ymd(date)) # create date variable

Declare as time series objects

df1 %<>% as_tsibble(index=quarter)
df2 %<>% as_tsibble(index=date) %>% # aggregate from daily to weekly
           index_by(year_week = yearweek(date)) %>%
          summarize(goog = mean(goog, na.rm=TRUE),
                    aapl = mean(aapl, na.rm=TRUE))

Plot time series data

Let’s have a look at the 3-month T-bill return for the US over the period 1960–1990:

autoplot(df1) + xlab("Year") + ylab("T-bill return")
## Plot variable not specified, automatically selected `.vars = r3`

And now the Google adjusted closing price:

autoplot(df2) + xlab("Year") + ylab("Price")
## Plot variable not specified, automatically selected `.vars = goog`

Testing for a unit root

Let’s test for a unit root in each of the time series. The way to do this is the Augmented Dickey-Fuller (ADF) test, which is available as adf.test() in the tseries package.

The function tests \(H_0: \text{Unit Root}, H_a: \text{Stationary}\).

adf.test(df1$r3, k=1)
adf.test(df2$goog, k=1)
## Warning in adf.test(df2$goog, k = 1): p-value greater than printed p-value
adf.test(df2$aapl, k=1)
## Warning in adf.test(df2$aapl, k = 1): p-value greater than printed p-value
  1. Which of these time series has a unit root, according to the ADF test? Explain what the consequences are of analyzing a time series that contains a unit root.

Estimating AR(1) models

To alternatively examine the unit root, we can estimate AR(1) models for each series:

est.tbill <- lm(r3 ~ lag(r3,1), data=df1)  <- lm(goog ~ lag(goog,1), data=df2)
est.aapl  <- lm(aapl ~ lag(aapl,1), data=df2)

  1. Are the \(R^2\) values from these estimates meaningful?


Now let’s use our time series data to forecast future stock prices. First, we should create a shortened version of the time series so we can compare our forecast to actual data:

df2.short    <- df2 %>% filter(year_week<yearweek("2022-01-01"))

Estimating simple ARIMA(1,1,0) models  <- lm(difference(goog) ~ lag(difference(goog)), data=df2)
simple.aapl  <- lm(difference(aapl) ~ lag(difference(aapl)), data=df2)

which is estimating \[ \Delta goog_t = \rho \Delta goog_{t-1} + u_t \]

Estimating ARIMA models

We can also use the ARIMA function in the fable package to allow the computer to choose the best ARIMA model: <- ARIMA(df2.short$goog)
auto.aapl <- ARIMA(df2.short$aapl)

Plotting forecasts

We can compare the 90-day-ahead (12-week-ahead) forecasts of each model by looking at their plots:

df2 %>% 
    arima = ARIMA(goog),
    snaive = SNAIVE(goog)
  ) %>%
  forecast(h=12) %>% autoplot(filter(df2, year(year_week)>2020),level = NULL)