Section B4: Unit Roots and Cointegration Analysis 

B4.1. Cointegrated Time Series

Some pairs of economic time series (e.g., wholesale and retail prices of a single commodity) may 

be expected to follow similar patterns of change over time. Even when short-term conditions 

cause such series to diverge, economic and/or policy dynamics will eventually force them back

into equilibrium. In the economics literature, such pairs of data series are called cointegrated

series.

To analyze cointegrated time series, we conceptualize them as stochastic processes, i.e., 

processes subject to randomness, and define properties of these processes. The notation in this 

section differs from that of Sections 1 through 3.

B4.2. Definitions of Short and Long Memory Stochastic Processes

We begin by defining some properties of stochastic processes. The stochastic process {εεtt} is 

stationary (or strictly stationary) if, for every collection of time indices �ttjj�

jj=1

nn , the joint 

distribution of �εεttjj�

jj=1

nn

is the same as the joint distribution of �εεttjj+h�

jj=1

nn

for all integers h ≥ 1. 

It follows that the εεtt are identically distributed.

Because strict stationarity is defined in terms of the joint distributions of subsets of the stochastic 

process, the strict stationary of an empirical data series is often difficult to establish. Many data 

series, however, are clearly nonstationary, e.g., series that clearly display trends or cycles.

Covariance stationarity is defined in terms of the moments of the stochastic process and is thus 

easier to establish for an empirical data series. The stochastic process {εεtt} is covariance 

stationary if, for some constant cc,

 EE[εεtt] = cc,

 VVVVVV[εεtt] = σσεε

2 (constant variance), and

CCCCCC�εεtt,εεtt+h� = ff(h), for h ≥ 1.

(B4.2.1)

If a strictly stationary process has a finite second moment, it is covariance stationary. A 

covariance stationary series, however, need not be stationary.

A covariance stationary process {εεtt} is a white noise process if 

EE[εεtt] = 0,

EE[εεttεεtt−h] = 0 for h ≠ 0 (uncorrolated), and

VVVVVV[εεtt] = σσεε

2 (constant variance).

(B4.2.2)

As tt → ∞, {εεtt} will frequently cross the horizontal axis and frequently return to values near its 

mean 0. White noise is used to model unexplained short-term movements in a data series.

Let {εεtt} be a white noise process, and let
xxtt = � jjεεtt−jj

∞

jj=0

. (B4.2.3)

If jj → 0 as jj → ∞, then {xxtt} is a short memory process. When {xxtt} is a short memory process 

with constant mean and finite variance, we say that {xxtt} is integrated of order 0 and write 

{xxtt}~II(0). White noise, for example, is a short memory process. In a typical empirical time 

series generated by a short memory process, autocorrelation is present but limited. When jj ↛ 0

as jj → ∞, {xxtt} is a long memory process. 

B4.2. Analyzing Short and Long Memory Stochastic Processes

If {yytt} is a long memory process, an old shock to the system (εεtt−h, where h is large) still has an 

effect on yytt. If, however, {yytt} is a short memory process, the effect of εεtt−h on yytt diminishes 

quickly as h increases, i.e., the process “forgets” old shocks. Although short memory processes 

are usually serially correlated, the covariance of the values between two time periods diminishes 

as the distance between the time periods grows (CCCCCC[yytt−h, yytt] → 0 as h → ∞).

Differencing a Long Memory Process

In many cases, differencing a long memory process produces a short memory process. Let

yytt = yytt−1 + εεtt, (B4.2.4)

where εεtt is a white noise process. Then, although {yytt} is a long memory process, the differenced 

process

∆yytt = yytt − yytt−1 = εεtt (B4.2.5)

is a short memory process. When {yytt} is a long memory process and {∆yytt} is a short memory 

process, we say that {yytt} is integrated of order 1 and write {yytt} ~II(1).

Cointegration

Before presenting a rigorous definition of cointegration, we state some facts about linear 

combinations of II(0) and II(1) processes. Let and bb be constants.

a) If {xxtt}~II(0), then + bbxxtt is II(0).

b) If {xxtt}~II(1), then + bbxxtt is II(1).

c) A linear combination of short memory processes is a short memory process, i.e., if 

{xxtt}~II(0), and {yytt}~II(0), then xxtt + bbyytt is II(0). 

d) A linear combination of a short memory process and a long memory process is a long 

memory process, i.e., if {xxtt}~II(0), and {yytt}~II(1), then xxtt + bbyytt is II(1). 
Definition of Cointegration

A linear combination of two long memory processes is a long memory process unless the two 

processes “share” the same long-term memory. Suppose {xxtt}~II(1) and {yytt}~II(1). If there 

exist constants m, a, and b such that

1. mm + xxtt + bbyytt is II(0) and

2. EE[mm + xxtt + bbyytt] = 0, 

then {xxtt} and {yytt} are cointegrated. 

Intuitively, because cointegrated long memory series share the same long-term memory, their 

long-term memories cancel out in the linear combination, leaving a short memory series. In the 

econometrics literature, the term “cointegrated” is commonly used even when the relationship 

between the two series is not strictly linear (e.g., it may be log-linear).

Problem of Spurious Regression

In time series regression analysis, we assume that one time series can be expressed as a linear 

combination of other time series, modulo an error term. When the time series have long 

memories, we are assuming cointegration.

Traditional regression diagnostics can be deceptive in the presence of long memory series. In 

particular, we may see high values of RR2 and low standard errors, leading to inflated t-statistics. 

There is a high probability of regression diagnostics indicating a relationship between two 

independent, randomly generated II(1) series (Newbold and Granger 1974). Differencing the 

series may eliminate this problem but allows only analysis of short-run changes. Use of 

additional statistical tests and diagnostics enables us to analyze long-run relationships between 

cointegrated series.

B4.3. Testing for Unit Roots

Definition of a Unit Root Process in the Linear Case

If 

yytt = αα + ρρyytt−1 + εεtt, (B4.3.1)

where EE(εεtt|yytt−1, yytt−2, ... , yy0) = 0, then {yytt} has a unit root if and only if |ρρ| = 1.

When ρρ = 1 and αα = 0, {yytt} is a random walk without drift. When ρρ = 1 and αα ≠ 0, {yytt} is a 

random walk with drift (Figure 5). Unit root processes are long memory processes.

Testing HH0: |ρρ| = 1

Observe that 

• When ρρ < 0, {yytt} is negatively autocorrelated, which is rare in economic time series.
• When |ρρ| > 1, {yytt} is blowing up (positive or negative).

So we are usually interested in testing 

HH0: ρρ = 1 vs. HH1: 0 < ρρ < 1. (B4.3.2)

Dickey-Fuller Test for a Unit Root

We set θθ = ρρ − 1. Then

∆yytt = αα + ρρyytt−1 − yytt−1 + εεtt

(B4.3.3)

= αα + θθyytt−1 + εεtt

Now we can test

HH0: θθ = 0 vs. HH1: θθ < 0. (B4.3.4)

We can compute the usual t-statistic for the hypothesis test. However, under HH0, the t-statistic is 

not asymptotically normal. Dickey and Fuller (1979) tabulated the critical values for the 

asymptotic distribution, which is known as the Dickey-Fuller (DF) distribution. 

Asymptotic Critical Values for the Dickey-Fuller (DF) Unit Root Test

Significance Level 1% 2.5% 5% 10%

Critical Value -3.43 -3.12 -2.86 -2.57

We reject HH0: θθ = 0 against the alternative HH1: θθ < 0 if the t-statistic ttθθ� < cc, where c is one of 

the critical values above.

If ttθθ� is strongly negative, we have evidence for rejecting the hypothesis that {yytt}~II(1) in favor 

of the hypothesis that {yytt}~II(0). Otherwise, we suspect that {yytt} has a unit root, and spurious 

regression is a concern.

Application of the asymptotic critical values for the DF test requires a large sample, i.e., a long 

time series. Critical values for small samples have also been tabulated and are available in 

statistical software packages such as SAS and R. 

The power of a hypothesis test is the probability of not committing a Type II error. Type II error 

is failing to reject a false null hypothesis. Unit root tests have been criticized for their low 
power. There is a relatively high probability that these tests may indicate a unit root in a series 

with no unit roots. In addition to the DF test, other tests for autocorrelation and unit roots are 

discussed in the literature (e.g., Ljung-Box, Durbin-Watson). The DF test is popular because it is 

simple and robust. 

The Cointegrating Parameter

Suppose we have determined that {yytt} and {xxtt} both have unit roots. 

• If {yytt} and {xxtt} are not cointegrated, regression of one series on the other may be 

“spurious.” 

• If {yytt} and {xxtt} are cointegrated, we are interested in the cointegrating parameter ββ such 

that yytt − ββxxtt is an II(0) process.

When we know or hypothesize the cointegrating parameter ββ, we can set zztt = yytt − ββxxtt and 

perform the standard DF test on the series {zztt}. If we find evidence that {zztt} has a unit root, we 

may conclude that {yytt} and {xxtt} are not cointegrated. (In practice, we may also consider that our 

hypothesized value of ββ may be incorrect.) The null hypothesis in the DF test is that {zztt} has a 

unit root, i.e., that {yytt} and {xxtt} are not cointegrated.

When {yytt} and {xxtt} are cointegrated, the OLS estimator ββ̂ from the regression 

yytt = αα� + ββ̂xxtt + εεtt (B4.3.5)

is a consistent estimator of ββ. However, under the null hypotheses that zztt = yytt − ββxxtt has a unit 

root, we must run a spurious regression to obtain ββ̂.

To perform the DF test with an estimated value of ββ, we set 

uu�tt = yytt − ββ̂xxtt, (B4.3.6)

where ββ̂ is the OLS estimator of ββ. Davidson and McKinnon (1993) tabulated critical values for 

the DF test for the presence of a unit root in the series {uu�tt}. 

Asymptotic Critical Values for the Dickey-Fuller (DF) Unit Root Test with Estimated ββ

Significance Level 1% 2.5% 5% 10%

Critical Value -3.90 -3.59 -3.34 -3.04
To perform a cointegration test with an estimated value of ββ, we run the regression

∆uu�tt = ττ + θθuu�tt−1 + εεtt (B4.3.7)

and test HH0: θθ = 0 against the alternative HH1: θθ < 0. If the t-statistic is below the critical value, 

we have evidence that {yytt} and {xxtt} are cointegrated. When ββ is estimated, we must get a tstatistic more strongly negative to show cointegration. This is because OLS tends to produce 

residuals that look like an II(0) sequence even when {yytt} and {xxtt} are not cointegrated.

Cointegration Test for Series with Linear Time Trends

Suppose {yytt} and {xxtt} display linear time trends. The trends need not be the same, e.g, {yytt} may 

be increasing faster than {xxtt}. Let yytt = γγtt + ggtt, and let xxtt = ζζ + zztt. The stochastic portions of 

the series, {ggtt} and {zztt}, may be cointegrated. We may remove the trends from xxtt and yytt and 

then test for unit roots and the cointegration of {ggtt} with {zztt}. Alternatively, we may run the 

regression

yytt = αα� + ηη̂tt + ββ̂xxtt (B4.3.8)

and apply the DF test to the residuals eê

tt. In this case, we note the following:

• The asymptotic critical values differ from those of the standard DF test and are given in 

the table below.

• If we find that {ggtt} and {zztt} are cointegrated with cointegration parameter ββ, this means 

that yytt − ββxxtt is not an II(1) process.

• It’s possible that yytt − ββxxtt will still have a linear time trend.

Asymptotic Critical Values for the Cointegration Test with Linear Time Trends

Significance Level 1% 2.5% 5% 10%

Critical Value -4.32 -4.03 -3.78 -3.50

B4.4. Series with Complex Autocorrelation Structures
If we suspect that {yytt}~II(1) and that {yytt} has a complex autocorrelation structure, we may 

include lags of ∆yytt in the autoregressive model on which we base the unit root test. In this case, 

we apply the augmented Dickey-Fuller (ADF) test.

We use the general model

∆yytt = αα + θθyytt−1 + �γγii∆yytt−ii

pp

ii=1

+ εεtt. (B4.4.1)

To find the appropriate value of p, we may set p = 0 and then increase p until γγpp is statistically 

insignificant. We can use an ordinary t- test to test HH0: γγpp = 0 (or an F test for the joint 

significance of the γγii). As in the DF test, we test

HH0: θθ = 0 vs. HH1: θθ < 0. (B4.4.2)

The asymptotic critical values for the ADF test are the same as those for the DF test. Under HH0, 

{∆yytt} follows an autoregressive model of order p; under HH1, {∆yytt} follows an autoregressive 

model of order pp + 1.

Vector Autoregressive (VAR) Models

If {xxtt} and {yytt} are autoregressive series of order p, we may fit the autoregressive models

yytt = δδ0 + �ααiiyytt−ii

pp

ii=1

+ �ββiixxtt−ii

pp

ii=1

+ εεtt,

and (B4.4.3)

xxtt = γγ0 + �ττiiyytt−ii

pp

ii=1

+�φφiixxtt−ii

pp

ii=1

+ uutt.

The inclusion of the lagged values helps ensure that the residuals are not strongly autocorrelated. 

VAR models are often used for short-term forecasting.

Error Correction Models

Suppose we have established that {yytt} and {xxtt} are cointegrated with

yytt = ββxxtt + εεtt. (B4.4.4)

We can use an error correction model to study the short term dynamics of the relationship 

between {yytt} and {xxtt}. For example, consider the model

∆yytt = αα0 + γγ0∆xxtt + δδ(yytt−1 − ββxxtt−1) + εεtt, (B4.4.5)

where δδ < 0. The term δδ(yytt−1 − ββxxtt−1) is called the error correction term. The parameter δδ

indicates the speed at which {yytt} and {xxtt} return to their equilibrium relationship after a “shock” 
creates a short-term disturbance in this relationship. Note that, if yytt−1 < ββxxtt−1, then xxtt−1 is 

higher than its equilibrium value. In this case, the inclusion of the δδ term will increase the 

model-based estimate of ∆yytt. Conversely, if yytt−1 > ββxxtt−1, the error correction term 

δδ(yytt−1 − ββxxtt−1) will decrease the estimated value of ∆yytt.

When the cointegrating parameter ββ is unknown, we may estimate it first and then fit the error 

correction model using the OLS estimator ββ̂ in place of ββ. This is called the Engle-Granger twostep procedure.

B4.5. Multivariate Cointegration Analysis

In the multivariate case, we consider the k x T matrix

YY = �

yy11 ⋯ yy1TT

⋮ ⋱ ⋮

yykk1 ⋯ yykk

�.

Suppose each row yyii = [yyii1. . .yyii ] of Y represents an II(1) process, but certain linear 

combinations of the rows are II(0). We are interested in identifying cointegrating relationships 

among the k time series.

The multivariate extension of the error correction model may be expressed as

∆yytt = ππ00 + Πyytt−11 + �Φii∆yytt−ii

pp−1

ii=1

+ εεtt,

where Π = αα ′, αα and ββ are k x r matrices, and Φii is a k x k matrix. The Johansen multivariate 

formulation assumes that the error terms are independent and identically distributed with 

εεtt~NNkk(0, Σ). The r columns of ββ are called the cointegration vectors, and the rank r of ββ is 

called the cointegration rank. The parameters in αα (which must be negative, like δδ in the twovariable case) indicate corrections of short-run deviations from the long-run equilibrium 

relationships defined by the columns of ββ.

Johansen Lambda-Max and Trace Tests

Based on the normality assumption, the Johansen tests are applied sequentially to help identify 

the cointegration rank r. If we find that < kk, it may be appropriate to remove one or more time 

series from the system. (If one variable is not cointegrated with any of the others, all values of 

the ββ matrix corresponding to it will be near 0.)

• Lambda-Max Test

Using a maximum generalized eigenvalue as a test statistic, we test null hypothesis that 

the cointegration rank r is equal ii ∈ {0,1, ... , kk − 1} against the alternative that = ii + 1. 

We apply the test sequentially starting with ii = 0.

• Trace Test
Using the trace (sum of diagonal elements) of a diagonal matrix of generalized 

eigenvalues as a test statistic, we test null hypothesis that the cointegration rank r is equal 

ii ∈ {0,1, ... , kk − 1} against the alternative that = ii + 1.

Applying Multivariate Cointegration Tests to Energy Data

Johansen’s proof of the validity of the lambda-max and trace tests relies on maximum likelihood 

theory, assuming that the differenced data follow a Gaussian (or normal) distribution. Because 

this assumption is often problematic for energy data, especially energy price data, Johansen’s 

technique is not used in many analyses performed within EIA. Seasonality also complicates the 

analysis of monthly data by the vector error correction model.