Identifying Stationary versus Non-Stationary Data

The first thing you have to establish in any time-series analysis is whether your data are stationary or not because it essentially determines whether you should be working with levels of the data or first differences of the data.

Stationary Data

In the figure below I simulate and plot a stationary series. The series comes from 500 draws of a normal distribution with mean = 300 and standard deviation = 0.25. Notice that it is basically flat, with random ups and downs.

That is basically what stationarity means, all the data come from a single probability distribution.

# If you are following along, un-comment the next lines and run once to install the required packages 
# install.packages('ggplot2')
# install.packages('xts')
# install.packages("stargazer")
# install.packages('quantmod')
# install.packages('sde')
# install.packages('broom')
# install.packages('tseries')
# install.packages("kableExtra")
# install.packages("knitr")


# These lines 'load' the required packages, something you have to do every time you start an R session, even if the package is installed on your machine.  
library(ggplot2)
library(xts)

# Making the fake price series
## Dates
end     <- Sys.Date()
start   <- end - 499 
t       <- as.Date(start:end)

## 500 fake prices from a normal distribution
x       <- rnorm(n = 500, mean = 300, sd = .25)

## Put dates and prices together in a time-series object
p       <- as.xts(x, order.by = t)

# Plot It
autoplot(p, ts.colour = "dark blue") + 
  labs(title = "Fake Stationary Price Series Centered around P = $300", x = "", y = "Price") + 
  theme_bw() + 
  scale_x_date(date_labels = "%m-%d-%Y", date_breaks = "3 month")

When you look at this plot it is not hard to imagine that it came from draws from a single probability distribution, namely:

\[p_t \sim N(\mu = 300, \sigma^2 = (.25)^2)\]

We need our data to be stationary in order to be able to do statistical tests and modelling on it. So if the data is not stationary, we have to transform it so that it is stationary. The next sub-section explains why.

Non-Stationary or Unit Root Data

Below I plot SPY, the exchange traded fund that tracks the S&P 500 Index, from 1990 to 2017. By visual inspection you should have a sense that this series is non-stationary in that the prices appear to be trending upward.

library(quantmod)
getSymbols(c('SPY', 'GS'))

## [1] "SPY" "GS"

# In Yahoo Finance data, the 'Adjusted' column accounts for any stock splits that may have taken place.
autoplot(SPY$SPY.Adjusted) + 
  theme_bw() + 
  geom_line(color = "dark blue") + 
  labs(title = "SPY Prices from 2007 to 2017", x = "")

So for example, if you wanted to write down the probability distribution that generates the prices, you would have to know where you are in time. Since the probability distribution that generated the prices keeps changing over time, the distribution is not stationary. This fact will mess up our statistical analysis.

\[p_t \sim N(\mu_t, \sigma^2_t)\]

Spurious Regression

One of the worst things non-stationary data does is set you up to believe that you have found a really strong relationship between two variables, when you really have just found a ‘spurious relationship’. In the figure below I plot SPY and Goldman Sachs (GS) prices from 2007 to 2017.

To illustrate a ‘spurious regression’ let’s regress GS prices on lagged SPY prices. That means we are going to try to use yesterday’s prices of SPY to try to predict today’s GS prices.

\[GS_t = \beta_0 + \beta_1SPY_{t-1} + \epsilon_t\]

The results of the regression are printed below. We find that lagged SPY prices are highly statistically significant predictors of GS prices. If you take these results to the stock market and think you can predict tomorrows GS price with today’s SPY price you will probably be very poor very soon.

The problem is that the ‘statistical significance’ comes from the fact that we just ran a super-consistent regression. That basically means since the variables we used in the regression are (probably, we didn’t prove it yet) non-stationary pretty much any regression we run will conclude statistical significance in a typical t-test. In other words, the regression results below aren’t valid because the statistic doesn’t follow the t distribution in the first place, but when you go along as if it does follow the t distribution, you will get ‘significant’ results almost every time.

library(stargazer)
GS <- GS$GS.Adjusted
SPY <- SPY$SPY.Adjusted
linMod <- lm(GS ~  lag(SPY))
stargazer(linMod, type  = "html", title = "GS Prices Regressed on Lagged GS and Lagged SPY Prices", align = TRUE)

You may be skeptical at this point. I mean, after all, it isn’t that crazy to think that today’s GS prices would be related to yesterday’s SPY prices. So to illustrate, now we’ll simulate two series so we are sure they are not related. Then we will regress one on the lag of the other and show statistical ‘significance’.

The simulated prices are plotted below.

# The function, GBM() from the sde package simulates a geometric brownian motion. Which is the same a assuming prices follow a lognormal ditribution, and is the standard baseline model for prices. 
library(sde)
fakeSPY   <- GBM(x = 275, r = 0.15, sigma = .2, T = 1, N=650)
fakeGS    <- GBM(x = 243, r = 0.10, sigma = .3, T = 1, N=650)

library(broom)
dates     <- seq(as.Date("2018-01-01"), length = 651, by = "days")
fakeSPY   <- xts(x=fakeSPY, order.by = dates)
fakeGS    <- xts(x=fakeGS, order.by = dates)

data      <- cbind(fakeSPY, fakeGS)
colnames(data) <- c('fakeSPY', 'fakeGS')
data      <- tidy(data)

ggplot(data, aes(x = index, y = value, color = series)) + 
  geom_line() + 
  theme_bw() +
  labs(title = "Simulated SPY and Simulated GS Prices from 2007 to 2017", x = "")

Now lets run the same regression as we did with the real prices. The regression results are shown in the table.

\[fakeGS_t = \beta_0 + \beta_1fakeSPY_{t-1} + \epsilon_t\]

linMod <- lm(fakeGS ~  lag(fakeSPY) )
stargazer(linMod, type  = "html", title = "Simulated GS Prices Regressed on Lagged Simulated SPY Prices")

Huh, still crazy significant. It’s not a real relationship, but appears to be significant because we didn’t have any business doing a t-test here in the first place. We know the regression is meaningless because we simulated these prices ourselves to be independent. That is the problem with non-stationary data in a nutshell, if you regress one non-stationary variables on another non-stationary variable, you will always get spurious ‘significant’ results.

So what to do? The answer is almost always this: if your data are non-stationary, convert the series into percentage changes by creating variables of logged price differences. In the context of the examples we have been using, that would be

\[SPYReturn_t = log(SPY_t) – log(SPY_{t-1}). \]

Plotted, the price returns look like the following.

SPYRet    <- log(SPY) - log(lag(SPY))
GSRet     <- log(GS) - log(lag(GS))
data      <- cbind(SPYRet, GSRet)

colnames(data) <- c('SPY', 'GS')
data      <- tidy(data)

ggplot(data, aes(x = index, y = value, color = series)) + 
  geom_line() + 
  theme_bw() +
  labs(title = "SPY Returns and GS Returns from 2007 to 2017", x = "")

Clearly, this is not exactly white noise. We have clear periods of heightened volatility (we’ll cover what to do about that in another post). But, at least the series are not trending up or down. We have solved the problem of non-stationarity by using price percent returns instead of price levels. Just to drive this point home, let’s regress GS returns on lagged SPY returns and show they are no longer significant.

linMod <- lm(GSRet ~  lag(SPYRet) )
stargazer(linMod, type  = "html", title = "GS Returns Regressed on Lagged SPY Returns")

The statistical significance is gone! Bummer! But really this is good. It is a sign that the returns we generated are in fact stationary.

Unit Root and Stationarity Test Results for SPY and GS

unitroot[, c(1, 2, 4, 5)] %>% 
  kable("html") %>% 
  kable_styling(bootstrap_options = c("striped", "hover")) %>% 
  group_rows("SPY", 1, 3) %>% 
  group_rows("GS", 4, 6)

Notice that in these examples, the results are not controversial all tests point toward there being unit roots in SPY and GS prices. We fail to reject the null hypothesis in all tests where unit root is the null hypothesis, and we reject the null hypothesis for all tests where unit root is the alternative hypothesis. This is because both series were strongly trending over the sample period we used. If the prices were ‘flatter’ the statistics would have a harder time and be less conclusive about whether the series are stationary or not.

Books to Get You Started

Now that you know the game we play in time-series analysis, you might be interested in getting started learning more about how to actually apply these statistics and models to poke around at research ideas. I recommend the following books for self-study. They are both very approachable and hands-on. These are not affiliate links. I won’t get any money if you buy these books. I just think they are both really good for anybody getting started with time-series analysis.

That’s It!

That’s it for stationary versus non-stationary data. Stay tuned for the next posts in this series where I will explain the next basic issues in time-series econometrics.

• Determining if Variables “Move Together” or are Otherwise Correlated
• Adjusting for Residuals that may not be Perfect “White Noise”