Stationarity – Time Series Rejects Null Hypothesis for ADF Test with Drift No Trend. Is the Time Series Stationary?

augmented-dickey-fullerdifferencingstationaritytrend

TL;DR: My time series passes a ADF test with drift no trend. So, should I leave my data alone and proceed? Or do still need to differentiate it before modelling, because it has drift? Or have I made an error in my process? I am struggling to understand my ADF results. See below for details.

I have a time series of 40 observations, measuring data monthly from 2002–2005.
In R:

library(fpp2)
str(insurance[,"TV.advert"])

I want to do a test of stationarity.

Begin with visual tests.
First, I plot at the raw data with a trend line:

It looks quite stationary (lots of consistent variation), but there is a slight trend. Is this a cycle? Or a deterministic trend? I'm not sure.

So, I plot a correlogram of the data.

It appears to be stationary: significance decays sharply after lag 1. But there is some significance at lag 10, too. So, is there a trend or cycle? I think it is ambiguous.

To remove any doubt, I choose to run an Augmented Dickey-Fuller (ADF) test.

library(aTSA)
stationary.test(insurance[,"TV.advert"],
                nlag = 13)

In the ADF test, I make k = 12 to include lags for 12 months of the year (in a stationary.test(), for 12 lags I have to make k = 13).

In the results of the ADF test, the time series rejects the null hypothesis for a random walk with drift (Tμ) at lag 0, lag 1, lag 9, and at lag 10. The Tμ-values at lag 0, 1, 9 and 10 (-3.73, -3.64, -4.29, and -3.47) are more negative than -2.93 (using the “Empirical Cumulative Distribution of T” table in Box-Steffensmeier et al., 2014, p. 134) and the p-values are < 0.05.

In addition, the data also passes the ADF at lags 0, 1, 9, and 10 for “Type 3: with drift and trend,” too. The Tτ-values (-3.758, -3.801, -4.274, and -3.626) are more negative than −3.50 and the the p-values are < 0.05.

Here are the results:

Augmented Dickey-Fuller Test 
alternative: stationary 
 
Type 1: no drift no trend 
      lag     ADF p.value
 [1,]   0 -0.3395   0.541
 [2,]   1 -0.4569   0.507
 [3,]   2  0.0580   0.655
 [4,]   3  0.0886   0.663
 [5,]   4 -0.0463   0.625
 [6,]   5  0.4357   0.763
 [7,]   6  0.4857   0.777
 [8,]   7  0.6657   0.829
 [9,]   8  0.3848   0.748
[10,]   9  0.3560   0.740
[11,]  10  0.3008   0.724
[12,]  11  0.2334   0.705
[13,]  12  0.1393   0.678
Type 2: with drift no trend 
      lag   ADF p.value
 [1,]   0 -3.73  0.0100
 [2,]   1 -3.64  0.0107
 [3,]   2 -2.71  0.0860
 [4,]   3 -2.82  0.0702
 [5,]   4 -2.19  0.2587
 [6,]   5 -1.93  0.3537
 [7,]   6 -2.01  0.3274
 [8,]   7 -1.87  0.3764
 [9,]   8 -2.45  0.1633
[10,]   9 -4.29  0.0100
[11,]  10 -3.47  0.0175
[12,]  11 -1.43  0.5384
[13,]  12 -1.09  0.6563
Type 3: with drift and trend 
      lag    ADF p.value
 [1,]   0 -3.758  0.0334
 [2,]   1 -3.801  0.0300
 [3,]   2 -2.780  0.2615
 [4,]   3 -2.867  0.2285
 [5,]   4 -2.324  0.4346
 [6,]   5 -1.938  0.5866
 [7,]   6 -1.978  0.5705
 [8,]   7 -1.816  0.6362
 [9,]   8 -2.391  0.4090
[10,]   9 -4.274  0.0100
[11,]  10 -3.626  0.0437
[12,]  11 -0.953  0.9332
[13,]  12 -0.020  0.9900
---- 
Note: in fact, p.value = 0.01 means p.value <= 0.01

Now, I am unclear what to do.

In most examples I see, the time series fail on all the ADF tests, so it has to be differentiated.

Because I have passed the ADF “Type 2 with drift no trend,” do I just leave the time series alone? Or do I still have to differentiate it, because it has drift?

Best Answer

A visual inspection of your time series together with the results of the ADF tests suggest your series does not have a unit root. Therefore, it need not be differenced.

Your series may contain a deterministic time trend, and that could be accounted for by including a linear (or nonlinear) trend as a regressor in your model. However, visually it is hard to tell whether any trend is present. Aside from a higher spike in 2005, it does not seem there is much of a trend in the data.

Related Solutions

Augmented Dickey-Fuller Test – Difference Among None, Drift, and Trend

The Wikipedia page states the following:

The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model $$ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t $$

As you very well note, there are variations of the test, which involve restricting $\alpha$ and/or $\beta$ equal to 0. Imposing the restriction on $\alpha$ corresponds to omitting a constant while restricting $\beta$ corresponds to omitting a time trend.

To understand what you're doing when using the adf.test() function from the tseries package in R, we should first consult the documentation provided by the package authors. To do this, we execute ?adf.test in the R console. Doing this will provide us details about the function; what it does, how we can use it, etc. For present purposes, we just need to be aware that the documentation states:

The general regression equation which incorporates a constant and a linear trend is used and the t-statistic for a first order autoregressive coefficient equals one is computed.

(Do we need more information than that?)

Coupled with that fact, if we look at the usage of the function; namely,

adf.test(x, alternative = c("stationary", "explosive"),
         k = trunc((length(x)-1)^(1/3)))

one begins to think that the function has limited capabilities with regard to the restricted variations of the ADF test. Reading all of the documentation seems to make it clear that the function only runs one variation of the test; the unrestricted version, which includes both a constant and a trend.

(Do we need more information than that?)

Since you're using R, we don't have to be left wondering if the function somehow imposes the restrictions internally without us knowing! To really be sure what's going on behind the scenes, we can look at the source code of the adf.test() function. Below, I step through the code, which I have shortened, and I hope it's instructive to you.

# Import some toy data
data(sunspots)

# Set arguments that are normally function inputs
x           <- sunspots
alternative <- "stationary"
k           <- trunc((length(x) - 1)^(1/3))

# Let the function go to work! (short version)
k <- k + 1          # Number of lagged differenced terms
y <- diff(x)        # First differences
n <- length(y)      # Length of first differenced series
z <- embed(y, k)    # Used for creating lagged series

# Things get interesting here as variables are prepared for the regression
yt  <- z[, 1]       # First differences
xt1 <- x[k:n]       # Series in levels - the first k-1 observations are dropped
tt  <- k:n          # Time-trend
yt1 <- z[, 2:k]     # Lagged differenced series - there are k-1 of them

# Next, the key pieces of code.

# Regression 1: if k > 0
# The augmented Dickey-Fuller test (with constant and time-trend)
res <- lm(yt ~ xt1 + 1 + tt + yt1) 

# Regression 2: if k = 0
# The standard Dickey-Fuller test (with constant and time-trend)
res <- lm(yt ~ xt1 + 1 + tt)

By my count, the adf.test() function is, in fact, made up of 57 lines of code, which I encourage you to inspect. The rest of the function code is not important in the context of this question. All that needs to be known is that the function does do what it says on the tin. Importantly, there does not seem to be a high level way of using the function to run a restricted variation of the ADF test and retrieve the associated critical values.

What to do? Your first instinct should be to check out the CRAN Task View: Time Series Analysis page. In doing so, you'll learn that the urca package provides an alternative implementation of the ADF test. Indeed, as I mentioned in the comments, the ur.df() function should be able to meet your needs. Inspecting the function usage is quite informative!

ur.df(y, type = c("none", "drift", "trend"), lags = 1, 
      selectlags = c("Fixed", "AIC", "BIC"))

The urca package can be found here and I recommend consulting the package documentation and the source code if you need to. I suspect that you should be able to use the function and not worry about issues regarding critical values; the authors of the package will have taken care of that so you can concentrate on using it as a high-level function and doing your research.

In terms of applying the ADF test (knowing which tests to run and in which order), I would suggest the Dolado et al. procedure. The reference is:

Dolado, J. J., Jenkinson, T., and Sosvilla-Rivera, S. (1990). Cointegration and unit roots, Journal of Economic Surveys, 4, 249-273.

Final note on matching the R code to the mathematical equation. You can basically think of it as follows (strictly speaking, the parameters should be omitted, but...):

yt = $\Delta y_{t}$

xt = $\gamma y_{t-1}$

+ 1 = $\alpha$

tt = $\beta t$

yt1 = $\delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1}$

Solved – How many lags to use for ADF test if several values reject null hypothesis

You typically use the longest lag that is statistically significant. You can do that easily by using an ACF graph and looking at any column that crosses through the confidence interval lines denoting statistical significance of autocorrelation given a specific lag. Of course don't test for any more lags than the frequency of your data calls for. If you have quarterly data, test up to 4 lags. If you have monthly data test up to 12 lags.

If the ADF test comes up with a high tau value and a resulting low p-value, you can reject the null hypothesis that the variable is non-stationary. In plain English, in such a situation your variable is deemed stationary (because you reject the null hypothesis that the variable is non-stationary).

In your specific situation, if you have quarterly data and you tested up to 4 lags, you are good. The ADF test has demonstrated that your variable is stationary. If you have monthly data, you may have to use more lags if the longest lag that has a statistically significant autocorrelation is longer than 4. Very often the lag 12 mth has stat. sign. autocorrelation because of seasonality. In that case you should use lag 12 within your ADF test.

Best Answer

Related Solutions

Augmented Dickey-Fuller Test – Difference Among None, Drift, and Trend

Solved – How many lags to use for ADF test if several values reject null hypothesis

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