8 Time series

Almost all the data we use here is associated with a particular point in time, like the price of a stock on a given day. That is called a time series. However, it isn’t easy to work with time series as one has to keep track of the day, month and year and know about leap years, time zones, holidays and other market closures. Consequently, it is best to work with data as if it were not a time series and only turn it into a time series when needed, typically for plotting, reporting and aggregation.

Most financial applications involve working with dates. These can be monthly, weekly, daily, or even intraday data. Storing data as text is not helpful since we cannot easily order or subset it.

R has a specific data type called Date. In this section, we will explore some packages that help us work with Date objects.

8.1 Libraries

library(lubridate)
library(zoo)
library(reshape2)
source("common/functions.r",chdir=TRUE)

8.2 Loading data

data=ProcessRawData()
sp500=data$sp500
sp500tr=data$sp500tr
Price=data$Price
Return=data$Return
UnAdjustedPrice=data$UnAdjustedPrice
Ticker=data$Ticker

8.3 Plotting time series

Start by plotting the SP-500 with the improvements from Section Section 7.3.2:

par(mar=c(3,4.2,1,0))
plot(sp500$price,
 type='l',
 lwd=2,
 col='blue',
 las=1,
 bty='l',
 xlab="day",
 ylab='price',
 main="The SP-500 index"
)

8.4 `lubridate`

We use the lubridate package to convert numbers and strings into dates. ymd()stands for year-month-day. If we have data with the American date convention, we can use mdy(), and in some cases we have ymd() formatted dates.

lubridate handles both string labels, JAN and integer 01.

ymd(20200110)

[1] "2020-01-10"

ymd("20200110")

[1] "2020-01-10"

class(ymd("20200110"))

[1] "Date"

ymd("2015JAN11")

[1] "2015-01-11"

class(ymd("20200110"))

[1] "Date"

ymd("04-MAR-5")

[1] "2004-03-05"

class(ymd("04MAR5"))

[1] "Date"

dmy("1/june/2019")

[1] "2019-06-01"

class(dmy("1/june/2019"))

[1] "Date"

dmy("28-december-14")

[1] "2014-12-28"

class(dmy("28-december-14"))

[1] "Date"

We can use lubridate to make a proper date column for the SP-500.

sp500$date.ts=ymd(sp500$date)
tail(sp500,2)

	date	price	y	date.ts
5034	20221229	3849.28	0.0173106	2022-12-29
5035	20221230	3839.50	-0.0025440	2022-12-30

8.5 Plotting with dates

Now, we can make a time series plot. Note that in sp500$date.ts,sp500$price, we use sp500$date.ts for the x-axis and sp500$price for the y-axis.

par(mar=c(2,4,1,0))
plot(sp500$date.ts,sp500$price,
 type='l',
 lwd=2,
 col='blue',
 las=1,
 bty='l',
 xlab="Day",
 ylab='Price',
 main="The SP-500 index"
)

You can make it a log plot

par(mar=c(2,4,1,0))
plot(sp500$date.ts,sp500$price,
 type='l',
 lwd=2,
 col='blue',
 las=1,
 bty='l',
 xlab="day",
 ylab='price',
 main="The SP-500 index",
 log='y'
)

We can customise this a bit more and sub-tickmarks.

par(mar=c(2,4,1,0))
plot(sp500$date.ts,sp500$price,
 type='l',
 lwd=2,
 col='blue',
 las=1,
 bty='l',
 xlab="Day",
 ylab='Price',
 main="The SP-500 index"
)
w=seq(ymd(20000101),ymd(20300101),by='year')
axis(1,at=w,label=FALSE,tcl=-0.3)

8.6 The `zoo` package

What we did above was plot a price vector against a date vector. We can also directly associate dates to prices with the zoo package, which allows us to work with ordered date-indexed observations. That allows many useful operations.

8.6.1 Make a `zoo`

sp500$y.ts = zoo(sp500$y, order.by = sp500$date.ts)
sp500$price.ts = zoo(sp500$price, order.by = sp500$date.ts)
class(sp500$y.ts)

[1] "zoo"

head(sp500$y.ts)

   2003-01-03    2003-01-06    2003-01-07    2003-01-08    2003-01-09 
-0.0004841496  0.0222255557 -0.0065661110 -0.0141857185  0.0192005899 
   2003-01-10 
 0.0000000000

Then, we can plot it directly as a time series.

plot(sp500$y.ts, 
 main="S&P500 Daily Return", 
 ylab="Return"
)

We can do useful things with zoo data.

8.6.2 `lag` function

This function allows us to take the lag or leads of a time series object. The syntax is: lag(x, k, na.pad = F), where:

x, a time series object to lag
k, number of lags (in units of observations); could be positive or negative (if negative, k is the number of forward lags)
na.pad, adds NAs for missing observations if TRUE

head(sp500$y.ts)

   2003-01-03    2003-01-06    2003-01-07    2003-01-08    2003-01-09 
-0.0004841496  0.0222255557 -0.0065661110 -0.0141857185  0.0192005899 
   2003-01-10 
 0.0000000000

head(lag(sp500$y.ts, k = 2))

  2003-01-03   2003-01-06   2003-01-07   2003-01-08   2003-01-09   2003-01-10 
-0.006566111 -0.014185718  0.019200590  0.000000000 -0.001413291  0.005812968

8.6.3 `diff` function

Takes the lagged difference of a time series. Syntax: diff(x, lag, differences, na.pad = F), where:

x = a time series object
lag = number of lags(in unit of observations)
differences = the order of the difference

head(diff(sp500$y.ts, lag = 1, na.pad = TRUE))

  2003-01-03   2003-01-06   2003-01-07   2003-01-08   2003-01-09   2003-01-10 
          NA  0.022709705 -0.028791667 -0.007619607  0.033386308 -0.019200590

8.6.4 The `window` function

We can use the window() function to subset a zoo object to a given time period. For example, let’s say we are interested in the returns during the Covid-19 crisis:

par(mar=c(2,4,1,0))
sub_y.ts = window(sp500$y.ts, start = ymd("20200201"), end = ymd("20200401"))
plot(sub_y.ts, 
 main = "Returns in Covid",
 xlab = "Date", 
 ylab = "Returns", 
 col = "mediumblue",
 lwd=2
)

8.6.5 Aggregate

We often need to aggregate time series data. For example, we may want to calculate end-of-month prices or realised monthly variance. The aggregate function makes that easy.

p.monthly=aggregate(sp500$price.ts,as.yearmon,tail,1)
head(p.monthly,5)

Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 
  855.70   841.15   848.18   916.92   963.59

realized.variance=aggregate(sp500$y.ts,as.yearmon,sd)
head(realized.variance,5)

  Jan 2003   Feb 2003   Mar 2003   Apr 2003   May 2003 
0.01435287 0.01188719 0.01747653 0.01169788 0.01026536

p.monthly.mean=aggregate(sp500$y.ts,as.yearmon,mean)

par(mar=c(4,4,1,0.6))
plot(p.monthly.mean,realized.variance,
 bty='l',
 main="SP-500 monthly mean and volatility",
 col='red',
 pch=16,
 xlab="mean",
 ylab="volatility",
 xaxt='n',
 yaxt='n'
 )
w=pretty(p.monthly.mean)
axis(1,w,label=paste0(100*w,"%"))
w=pretty(realized.variance)
axis(2,w,label=paste0(100*w,"%"),las=1)
regression_line=lm(realized.variance ~ p.monthly.mean)
abline(regression_line,col='green',lwd=3)

8.7 Multivariate plots

We use the matplot command for many assets. Call the list of assets Assets:

par(mar=c(2,4,0,0))
matplot(Price[,Ticker])

This is quite ugly and can be made to look better.

par(mar=c(2,4,0,0))
matplot(
 Price[,Ticker],
 type='l',
 lty=1,
 ylab='Price'
)

We can add a date to it the same way we did before.

Price$date.ts=ymd(Price$date)
Return$date.ts=ymd(Return$date)
UnAdjustedPrice$date.ts=ymd(UnAdjustedPrice$date)

par(mar=c(2,4,0,0))
matplot(
 Price$date.ts,
 Price[,Ticker],
 type='l',
 lty=1,
 ylab='Price'
)

We can put a legend on the plot.

par(mar=c(2,4,0,0))
matplot(
 Price$date.ts,
 Price[,Ticker],
 type='l',
 lty=1,
 ylab='Price',
 col=1:6,
 las=1
)
legend("topleft",legend=Ticker,lty=1,col=1:6,bty='n',ncol=2)

In order to compare the performance of the stocks, we can re-normalise them to start at 1

par(mar=c(2,4,0,0))
pn=Price
for(i in Ticker){
 pn[[i]]=pn[[i]]/pn[[i]][1]
}
matplot(
 pn$date.ts,
 pn[,Ticker],
 type='l',
 lty=1,
 ylab='Price',
 col=1:6,
 las=1
)
legend("topleft",legend=Ticker,lty=1,col=1:6,bty='n',ncol=2)

rbind(head(pn,2),tail(pn,1))

	date	AAPL	DIS	GE	INTC	JPM	MCD	date.ts
2	20030103	1.0000	1.000000	1.0000000	1.000000	1.000000	1.000000	2003-01-03
3	20030106	1.0000	1.051263	1.0255903	1.038697	1.078639	1.032873	2003-01-06
5035	20221230	572.7678	6.279998	0.7411147	2.692354	8.998689	27.909514	2022-12-30

Log scaling the y-axis can be more informative.

par(mar=c(2,4,0.5,0))
pn=Price
for(i in Ticker){
 pn[[i]]=pn[[i]]/pn[[i]][1]
}
matplot(
 pn$date.ts,
 pn[,Ticker],
 type='l',
 lty=1,
 ylab='Price',
 col=1:6,
 las=1,
 log='y'
)
legend("topleft",legend=Ticker,lty=1,col=1:6,bty='n',ncol=2)

And put gridlines on it.

par(mar=c(2,4,0.5,0))
pn=Price
for(i in Ticker){
 pn[[i]]=pn[[i]]/pn[[i]][1]
}
matplot(
 pn$date.ts,
 pn[,Ticker],
 type='l',
 lty=1,
 ylab='Price',
 col=1:6,
 las=1,
 log='y'
)
for(i in c(0.5,1,5,10,50,100,500,1000)) 
 segments(pn$date.ts[1]-days(500),i,tail(pn$date.ts,1)+days(500),i,col="lightgray")

legend("topleft",legend=Ticker,lty=1,col=1:6,bty='n',ncol=2)

Based on this, the best-performing stock is AAPL.

8.8 `ProcessRawData()`

Even better is to put all variables into the same list and load them all with ProcessRawData(). We put that into functions.r.

ProcessRawData=function(){

    sp500=read.csv('data/sp500.csv')
    names(sp500)[2]="price"
    sp500$y = c(NA,diff(log(sp500$price)))
    sp500=sp500[!is.na(sp500$y),]
    sp500$date.ts = ymd(sp500$date)
    sp500$y.ts = zoo(sp500$y,order.by=sp500$date.ts)

    sp500tr=read.csv('data/sp500tr.csv')
    names(sp500tr)[2]="price"
    sp500tr$y = c(NA,diff(log(sp500tr$price)))
    sp500tr$date.ts = ymd(sp500tr$date)
    sp500tr$y.ts = zoo(sp500tr$y,order.by=sp500tr$date.ts)

    stocks=read.csv('data/stocks.csv')
    names(stocks)[3:4]=c("UnAdjustedPrice","price")
    Price = dcast(stocks, date ~ ticker, value.var = "price")
    head(Price,2)
    UnAdjustedPrice = dcast(stocks, date ~ ticker, value.var = "UnAdjustedPrice")

    Return=Price
    for (i in 2:dim(Price)[2]) Return[,i]=c(NA,diff(log(Price[,i])))

    Price=Price[!is.na(Return[,2]),]
    UnAdjustedPrice=UnAdjustedPrice[!is.na(Return[,2]),]
    Return=Return[!is.na(Return[,2]),]

    data=list(
        Return=Return,
        Price=Price,
        UnAdjustedPrice=UnAdjustedPrice,
        sp500=sp500,
        sp500tr=sp500tr,
        Ticker=unique(stocks$ticker)
    )

    return(data)
}

8.1 Libraries

8.2 Loading data

8.3 Plotting time series

8.4 lubridate

8.5 Plotting with dates

8.6 The zoo package

8.6.1 Make a zoo

8.6.2 lag function

8.6.3 diff function

8.6.4 The window function