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A *unit root* process is a data-generating
process whose first difference is stationary. In other words, a unit
root process *y _{t}* has the
form

*y _{t}* =

A unit root test attempts to determine whether a given time series is consistent with a unit root process.

The next section gives more details of unit root processes, and suggests why it is important to detect them.

There are two basic models for economic data with linear growth characteristics:

Trend-stationary process (TSP):

*y*=_{t}*c*+*δt*+ stationary processUnit root process, also called a difference-stationary process (DSP): Δ

*y*=_{t}*δ*+ stationary process

Here Δ is the differencing operator, Δ*y _{t}* =

The processes are indistinguishable for finite data. In other
words, there are both a TSP and a DSP that fit a finite data set arbitrarily
well. However, the processes are distinguishable when restricted to
a particular subclass of data-generating processes, such as AR(*p*)
processes. After fitting a model to data, a unit root test checks
if the AR(1) coefficient is 1.

There are two main reasons to distinguish between these types of processes:

A TSP and a DSP produce different forecasts. Basically, shocks
to a TSP return to the trend line *c* + *δt* as time increases. In contrast,
shocks to a DSP might be persistent over time.

For example, consider the simple trend-stationary model

*y*_{1,t} = 0.9*y*_{1,t –
1} + 0.02*t* + *ε*_{1,t}

and the difference-stationary model

*y*_{2,t} = 0.2 + *y*_{2,t –
1} + *ε*_{2,t}.

In these models, *ε*_{1,t} and *ε*_{2,t} are
independent innovation processes. For this example, the innovations
are independent and distributed *N*(0,1).

Both processes grow at rate 0.2. To calculate the growth rate
for the TSP, which has a linear term 0.02*t*, set *ε*_{1}(*t*) = 0. Then solve
the model *y*_{1}(*t*) = *c* + *δt* for *c* and *δ*:

*c* + *δt* =
0.9(*c* + *δ*(*t*–1))
+ 0.02*t*.

The solution is *c* = –1.8, *δ* = 0.2.

A plot for *t* = 1:1000
shows the TSP stays very close to the trend line, while the DSP has
persistent deviations away from the trend line.

T = 1000; % Sample size t = (1:T)'; % Period vector rng(5); % For reproducibility randm = randn(T,2); % Innovations y = zeros(T,2); % Columns of y are data series % Build trend stationary series y(:,1) = .02*t + randm(:,1); for ii = 2:T y(ii,1) = y(ii,1) + y(ii-1,1)*.9; end % Build difference stationary series y(:,2) = .2 + randm(:,2); y(:,2) = cumsum(y(:,2)); figure plot(y(:,1),'b') hold on plot(y(:,2),'g') plot((1:T)*0.2,'k--') legend('Trend Stationary','Difference Stationary',... 'Trend Line','Location','NorthWest') hold off

Forecasts based on the two series are different. To see this
difference, plot the predicted behavior of the two series using `varm`

, `estimate`

, and `forecast`

. The following
plot shows the last 100 data points in the two series and predictions
of the next 100 points, including confidence bounds.

AR = {[NaN 0; 0 NaN]}; % Independenct response series trend = [NaN; 0]; % Linear trend in first series only Mdl = varm('AR',AR,'Trend',trend); EstMdl = estimate(Mdl,y); EstMdl.SeriesNames = ["Trend stationary" "Difference stationary"]; [ynew,ycov] = forecast(EstMdl,100,y); % This generates predictions for 100 time steps seY = sqrt(diag(EstMdl.Covariance))'; % Extract standard deviations of y CIY = zeros([size(y) 2]); % In-sample intervals CIY(:,:,1) = y - seY; CIY(:,:,2) = y + seY; extractFSE = cellfun(@(x)sqrt(diag(x))',ycov,'UniformOutput',false); seYNew = cell2mat(extractFSE); CIYNew = zeros([size(ynew) 2]); % Forecast intervals CIYNew(:,:,1) = ynew - seYNew; CIYNew(:,:,2) = ynew + seYNew; tx = (T-100:T+100); hs = 1:2; figure; for j = 1:Mdl.NumSeries hs(j) = subplot(2,1,j); hold on; h1 = plot(tx,tx*0.2,'k--'); axis tight; ha = gca; h2 = plot(tx,[y(end-100:end,j); ynew(:,j)]); h3 = plot(tx(1:101),squeeze(CIY(end-100:end,j,:)),'r:'); plot(tx(102:end),squeeze(CIYNew(:,j,:)),'r:'); h4 = fill([tx(102) ha.XLim([2 2]) tx(102)],ha.YLim([1 1 2 2]),[0.7 0.7 0.7],... 'FaceAlpha',0.1,'EdgeColor','none'); title(EstMdl.SeriesNames{j}); hold off; end legend(hs(1),[h1 h2 h3(1) h4],... {'Trend','Process','Interval estimate','Forecast horizon'},'Location','Best');

Examine the fitted parameters by executing `summarize(EstMdl)`

and you find
`estimate`

did an excellent job.

The TSP has confidence intervals that do not grow with time,
whereas the DSP has confidence intervals that grow. Furthermore, the
TSP goes to the trend line quickly, while the DSP does not tend towards
the trend line *y* = 0.2*t* asymptotically.

The presence of unit roots can lead to false inferences in regressions between time series.

Suppose *x _{t}* and

*x _{t}* =

where *ε _{i}*(

If both processes have trends (*c _{i}* ≠ 0), there
is a correlation between

There are four Econometrics
Toolbox™ tests for unit roots.
These functions test for the existence of a *single* unit
root. When there are two or more unit roots, the results of these
tests might not be valid.

`adftest`

performs the augmented
Dickey-Fuller test. `pptest`

performs
the Phillips-Perron test. These two classes of tests have a null hypothesis
of a unit root process of the form

*y _{t}* =

which the functions test against an alternative model

*y _{t}* =

where *γ* < 1. The null and alternative
models for a Dickey-Fuller test are like those for a Phillips-Perron
test. The difference is `adftest`

extends the model
with extra parameters accounting for serial correlation among the
innovations:

*y _{t}* =

where

*L*is the lag operator:*Ly*=_{t}*y*_{t–1}.Δ = 1 –

*L*, so Δ*y*=_{t}*y*–_{t}*y*_{t–1}.*ε*_{t}is the innovations process.

Phillips-Perron adjusts the test statistics to account for serial correlation.

There are three variants of both `adftest`

and `pptest`

,
corresponding to the following values of the `'model'`

parameter:

`'AR'`

assumes*c*and*δ*, which appear in the preceding equations, are both`0`

; the`'AR'`

alternative has mean 0.`'ARD'`

assumes*δ*is`0`

. The`'ARD'`

alternative has mean*c*/(1–*γ*).`'TS'`

makes no assumption about*c*and*δ*.

For information on how to choose the appropriate value of `'model'`

,
see Choose Models to Test.

The KPSS test, `kpsstest`

,
is an inverse of the Phillips-Perron test: it reverses the null and
alternative hypotheses. The KPSS test uses the model:

*y _{t}* =

Here *u _{t}* is a stationary
process, and

The variance ratio test, `vratiotest`

,
is based on the fact that the variance of a random walk increases
linearly with time. `vratiotest`

can also take
into account heteroscedasticity, where the variance increases at a
variable rate with time. The test has a null hypotheses of a random
walk:

Δ*y _{t}* =

Transform your time series to be approximately linear before testing for a unit root. If a series has exponential growth, take its logarithm. For example, GDP and consumer prices typically have exponential growth, so test their logarithms for unit roots.

If you want to transform your data to be stationary instead of approximately linear, unit root tests can help you determine whether to difference your data, or to subtract a linear trend. For a discussion of this topic, see What Is a Unit Root Test?

For

`adftest`

or`pptest`

, choose`model`

in as follows:If your data shows a linear trend, set

`model`

to`'TS'`

.If your data shows no trend, but seem to have a nonzero mean, set

`model`

to`'ARD'`

.If your data shows no trend and seem to have a zero mean, set

`model`

to`'AR'`

(the default).

For

`kpsstest`

, set`trend`

to`true`

(default) if the data shows a linear trend. Otherwise, set`trend`

to`false`

.For

`vratiotest`

, set`IID`

to`true`

if you want to test for independent, identically distributed innovations (no heteroscedasticity). Otherwise, leave`IID`

at the default value,`false`

. Linear trends do not affect`vratiotest`

.

Setting appropriate lags depends on the test you use:

`adftest`

— One method is to begin with a maximum lag, such as the one recommended by Schwert [2]. Then, test down by assessing the significance of the coefficient of the term at lag*p*_{max}. Schwert recommends a maximum lag of$${p}_{\mathrm{max}}=\text{maximumlag}=\lfloor 12{\left(T/100\right)}^{1/4}\rfloor ,$$

where $$\lfloor x\rfloor $$ is the integer part of

*x*. The usual*t*statistic is appropriate for testing the significance of coefficients, as reported in the`reg`

output structure.Another method is to combine a measure of fit, such as SSR, with information criteria such as AIC, BIC, and HQC. These statistics also appear in the

`reg`

output structure. Ng and Perron [3] provide further guidelines.`kpsstest`

— One method is to begin with few lags, and then evaluate the sensitivity of the results by adding more lags. For consistency of the Newey-West estimator, the number of lags must go to infinity as the sample size increases. Kwiatkowski et al. [4] suggest using a number of lags on the order of*T*^{1/2}, where*T*is the sample size.For an example of choosing lags for

`kpsstest`

, see Test Time Series Data for a Unit Root.`pptest`

— One method is to begin with few lags, and then evaluate the sensitivity of the results by adding more lags. Another method is to look at sample autocorrelations of*y*–_{t}*y*_{t–1}; slow rates of decay require more lags. The Newey-West estimator is consistent if the number of lags is O(*T*^{1/4}), where*T*is the effective sample size, adjusted for lag and missing values. White and Domowitz [5] and Perron [6] provide further guidelines.For an example of choosing lags for

`pptest`

, see Test Time Series Data for a Unit Root.`vratiotest`

does not use lags.

Run multiple tests simultaneously by entering a vector of parameters
for `lags`

, `alpha`

, `model`

,
or `test`

. All vector parameters must have the same
length. The test expands any scalar parameter to the length of a vector
parameter. For an example using this technique, see Test Time Series Data for a Unit Root.

[1] Granger,
C. W. J., and P. Newbold. “Spurious Regressions in Econometrics.” *Journal
of Econometrics*. Vol 2, 1974, pp. 111–120.

[2] Schwert,
W. “Tests for Unit Roots: A Monte Carlo Investigation.” *Journal
of Business and Economic Statistics*. Vol. 7, 1989, pp.
147–159.

[3] Ng, S.,
and P. Perron. “Unit Root Tests in ARMA Models with Data-Dependent
Methods for the Selection of the Truncation Lag.” *Journal
of the American Statistical Association*. Vol. 90, 1995,
pp. 268–281.

[4] Kwiatkowski,
D., P. C. B. Phillips, P. Schmidt and Y. Shin. “Testing the
Null Hypothesis of Stationarity against the Alternative of a Unit
Root.” *Journal of Econometrics*. Vol.
54, 1992, pp. 159–178.

[5] White,
H., and I. Domowitz. “Nonlinear Regression with Dependent Observations.” *Econometrica*.
Vol. 52, 1984, pp. 143–162.

[6] Perron,
P. “Trends and Random Walks in Macroeconomic Time Series: Further
Evidence from a New Approach.” *Journal of Economic
Dynamics and Control*. Vol. 12, 1988, pp. 297–332.

`adftest`

| `forecast`

| `kpsstest`

| `pptest`

| `vratiotest`

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