Chapter 12
Regression with Time-Series Data:
Nonstationary Variables
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 1
Chapter Contents





12.1 Stationary and Nonstationary Variables
12.2 Spurious Regressions
12.3 Unit Root Tests for Nonstationarity
12.4 Cointegration
12.5 Regression When There Is No Cointegration
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 2
The aim is to describe how to estimate regression
models involving nonstationary variables
– The first step is to examine the time-series
concepts of stationarity (and nonstationarity)
and how we distinguish between them.
– Cointegration is another important related
concept that has a bearing on our choice of a
regression model
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 3
12.1
Stationary and Nonstationary
Variables
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 4
12.1
Stationary and
Nonstationary
Variables
The change in a variable is an important concept
– The change in a variable yt, also known as its
first difference, is given by Δyt = yt – yt-1.
• Δyt is the change in the value of the variable
y from period t - 1 to period t
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 5
12.1
Stationary and
Nonstationary
Variables
Principles of Econometrics, 4th Edition
FIGURE 12.1 U.S. economic time series
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 6
12.1
Stationary and
Nonstationary
Variables
Principles of Econometrics, 4th Edition
FIGURE 12.1 (Continued) U.S. economic time series
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 7
12.1
Stationary and
Nonstationary
Variables
Formally, a time series yt is stationary if its mean
and variance are constant over time, and if the
covariance between two values from the series
depends only on the length of time separating the
two values, and not on the actual times at which
the variables are observed
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 8
12.1
Stationary and
Nonstationary
Variables
That is, the time series yt is stationary if for all
values, and every time period, it is true that:
Eq. 12.1a
E  yt   μ
(constant mean)
Eq. 12.1b
var  yt   σ 2
(constant variance)
Eq. 12.1c
cov  yt , yt  s   cov  yt , yt  s   γ s (covariance depends on s, not t )
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 9
12.1
Stationary and
Nonstationary
Variables
Table 12.1 Sample Means of Time Series Shown in Figure 12.1
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 10
12.1
Stationary and
Nonstationary
Variables
Nonstationary series with nonconstant means are
often described as not having the property of
mean reversion
– Stationary series have the property of mean
reversion
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 11
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
The econometric model generating a time-series
variable yt is called a stochastic or random
process
– A sample of observed yt values is called a
particular realization of the stochastic process
• It is one of many possible paths that the
stochastic process could have taken
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 12
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
The autoregressive model of order one, the AR(1)
model, is a useful univariate time series model for
explaining the difference between stationary and
nonstationary series:
yt  yt 1  vt ,
Eq. 12.2a
 1
– The errors vt are independent, with zero mean
2
σ
and constant variance v , and may be normally
distributed
– The errors are sometimes known as ‘‘shocks’’
or ‘‘innovations’’
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 13
12.1
Stationary and
Nonstationary
Variables
FIGURE 12.2 Time-series models
12.1.1
The First-Order
Autoregressive
Model
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 14
12.1
Stationary and
Nonstationary
Variables
FIGURE 12.2 (Continued) Time-series models
12.1.1
The First-Order
Autoregressive
Model
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 15
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
The value ‘‘zero’’ is the constant mean of the
series, and it can be determined by doing some
algebra known as recursive substitution
– Consider the value of y at time t = 1, then its
value at time t = 2 and so on
– These values are:
y1  y0  v1
y2  y1  v2  (y0  v1 )  v2   2 y0  v1  v2
yt  vt  vt 1  2 vt  2  .....  t y0
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 16
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
The mean of yt is:
E  yt   E  vt  vt 1  2vt 2 
0
Real-world data rarely have a zero mean
– We can introduce a nonzero mean μ as:
( yt  )  ( yt 1  )  vt
– Or
Eq. 12.2b
Principles of Econometrics, 4th Edition
yt     yt 1  vt ,
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
 1
Page 17
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
With α = 1 and ρ = 0.7:
E ( yt )     / (1  )  1 / (1  0.7)  3.33
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 18
12.1
Stationary and
Nonstationary
Variables
12.1.1
The First-Order
Autoregressive
Model
An extension to Eq. 12.2a is to consider an AR(1)
model fluctuating around a linear trend: (μ + δt)
– Let the ‘‘de-trended’’ series (yt -μ - δt) behave
like an autoregressive model:
( yt    t )  ( yt 1    (t  1))  vt ,
 1
Or:
Eq. 12.2c
Principles of Econometrics, 4th Edition
yt    yt 1  t  vt
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 19
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
Consider the special case of ρ = 1:
yt  yt 1  vt
Eq. 12.3a
– This model is known as the random walk model
• These time series are called random walks
because they appear to wander slowly
upward or downward with no real pattern
• the values of sample means calculated from
subsamples of observations will be
dependent on the sample period
–This is a characteristic of nonstationary
series
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 20
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
We can understand the ‘‘wandering’’ by recursive
substitution:
y1  y0  v1
2
y2  y1  v2  ( y0  v1 )  v2  y0   vs
s 1
t
yt  yt 1  vt  y0   vs
s 1
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 21
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
The term  s 1 vs is often called the stochastic
trend
– This term arises because a stochastic
component vt is added for each time t, and
because it causes the time series to trend in
unpredictable directions
t
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 22
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
Recognizing that the vt are independent, taking the
expectation and the variance of yt yields, for a
fixed initial value y0:
E ( yt )  y0  E (v1  v2  ...  vt )  y0
var( yt )  var(v1  v2  ...  vt )  tσv2
– The random walk has a mean equal to its initial
value and a variance that increases over time,
eventually becoming infinite
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 23
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
Another nonstationary model is obtained by
adding a constant term:
yt    yt 1  vt
Eq. 12.3b
– This model is known as the random walk with
drift
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 24
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
A better understanding is obtained by applying
recursive substitution:
y1    y0  v1
2
y2    y1  v2    (  y0  v1 )  v2  2  y0   vs
s 1
t
yt    yt 1  vt  t   y0   vs
s 1
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 25
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
The term tα a deterministic trend component
– It is called a deterministic trend because a fixed
value α is added for each time t
– The variable y wanders up and down as well as
increases by a fixed amount at each time t
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 26
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
The mean and variance of yt are:
E ( yt )  t   y0  E (v1  v2  v3  ...  vt )  t   y0
var( yt )  var(v1  v2  v3  ...  vt )  tv2
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 27
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
We can extend the random walk model even
further by adding a time trend:
Eq. 12.3c
Principles of Econometrics, 4th Edition
yt    t  yt 1  vt
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 28
12.1
Stationary and
Nonstationary
Variables
12.1.2
Random Walk
Models
The addition of a time-trend variable t strengthens
the trend behavior:
y1      y0  v1
2
y2    2  y1  v2    2  (    y0  v1 )  v2  2  3  y0   vs
s 1
t
 t (t  1) 
yt    t  yt 1  vt  t   
   y0   vs
2
s 1


where we used:
1 2  3 
Principles of Econometrics, 4th Edition
 t  t  t  1 2
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 29
12.2
Spurious Regressions
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 30
12.2
Spurious
Regressions
The main reason why it is important to know
whether a time series is stationary or nonstationary
before one embarks on a regression analysis is that
there is a danger of obtaining apparently
significant regression results from unrelated data
when nonstationary series are used in regression
analysis
– Such regressions are said to be spurious
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 31
12.2
Spurious
Regressions
Consider two independent random walks:
rw1 : yt  yt 1  v1t
rw2 : xt  xt 1  v2t
– These series were generated independently and,
in truth, have no relation to one another
– Yet when plotted we see a positive relationship
between them
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 32
12.2
Spurious
Regressions
FIGURE 12.3 Time series and scatter plot of two random walk variables
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 33
12.2
Spurious
Regressions
FIGURE 12.3 (Continued) Time series and scatter plot of two random
walk variables
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 34
12.2
Spurious
Regressions
A simple regression of series one (rw1) on series
two (rw2) yields:
rw1t  17.818  0.842 rw2t ,
(t )
R 2  0.70
(40.837)
– These results are completely meaningless, or
spurious
• The apparent significance of the relationship
is false
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 35
12.2
Spurious
Regressions
When nonstationary time series are used in a
regression model, the results may spuriously
indicate a significant relationship when there is
none
– In these cases the least squares estimator and
least squares predictor do not have their usual
properties, and t-statistics are not reliable
– Since many macroeconomic time series are
nonstationary, it is particularly important to
take care when estimating regressions with
macroeconomic variables
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 36
12.3
Unit Root Tests for Stationarity
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 37
12.3
Unit Root Tests for
Stationarity
There are many tests for determining whether a
series is stationary or nonstationary
– The most popular is the Dickey–Fuller test
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 38
12.3
Unit Root Tests for
Stationarity
12.3.1
Dickey-Fuller Test 1
(No constant and No
Trend)
The AR(1) process yt = ρyt-1 + vt is stationary when
|ρ| < 1
– But, when ρ = 1, it becomes the nonstationary
random walk process
– We want to test whether ρ is equal to one or
significantly less than one
• Tests for this purpose are known as unit root
tests for stationarity
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 39
12.3
Unit Root Tests for
Stationarity
12.3.1
Dickey-Fuller Test 1
(No constant and No
Trend)
Consider again the AR(1) model:
yt  yt 1  vt
Eq. 12.4
– We can test for nonstationarity by testing the
null hypothesis that ρ = 1 against the alternative
that |ρ| < 1
• Or simply ρ < 1
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 40
12.3
Unit Root Tests for
Stationarity
12.3.1
Dickey-Fuller Test 1
(No constant and No
Trend)
A more convenient form is:
yt  yt 1  yt 1  yt 1  vt
yt     1 yt 1  vt
Eq. 12.5a
  yt 1  vt
– The hypotheses are:
H0 :   1  H0 :   0
H1 :   1 
Principles of Econometrics, 4th Edition
H1 :   0
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 41
12.3
Unit Root Tests for
Stationarity
12.3.2
Dickey-Fuller Test 2
(With Constant but
No Trend)
The second Dickey–Fuller test includes a constant
term in the test equation:
yt     yt 1  vt
Eq. 12.5b
– The null and alternative hypotheses are the
same as before
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 42
12.3
Unit Root Tests for
Stationarity
12.3.3
Dickey-Fuller Test 3
(With Constant and
With Trend)
The third Dickey–Fuller test includes a constant
and a trend in the test equation:
yt    yt 1  t  vt
Eq. 12.5c
– The null and alternative hypotheses are
H0: γ = 0 and H1:γ < 0 (same as before)
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 43
12.3
Unit Root Tests for
Stationarity
12.3.4
The Dickey-Fuller
Critical Values
To test the hypothesis in all three cases, we simply
estimate the test equation by least squares and
examine the t-statistic for the hypothesis that
γ=0
– Unfortunately this t-statistic no longer has the
t-distribution
– Instead, we use the statistic often called a τ
(tau) statistic
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 44
12.3
Unit Root Tests for
Stationarity
Table 12.2 Critical Values for the Dickey–Fuller Test
12.3.4
The Dickey-Fuller
Critical Values
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 45
12.3
Unit Root Tests for
Stationarity
12.3.4
The Dickey-Fuller
Critical Values
To carry out a one-tail test of significance, if τc is
the critical value obtained from Table 12.2, we
reject the null hypothesis of nonstationarity if
τ ≤ τc
– If τ > τc then we do not reject the null
hypothesis that the series is nonstationary
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 46
12.3
Unit Root Tests for
Stationarity
12.3.4
The Dickey-Fuller
Critical Values
An important extension of the Dickey–Fuller test
allows for the possibility that the error term is
autocorrelated
– Consider the model:
m
yt     yt 1   as yt s  vt
Eq. 12.6
where
s 1
yt 1   yt 1  yt 2  , yt 2   yt 2  yt 3  ,
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 47
12.3
Unit Root Tests for
Stationarity
12.3.4
The Dickey-Fuller
Critical Values
The unit root tests based on Eq. 12.6 and its
variants (intercept excluded or trend included) are
referred to as augmented Dickey–Fuller tests
– When γ = 0, in addition to saying that the series
is nonstationary, we also say the series has a
unit root
– In practice, we always use the augmented
Dickey–Fuller test
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 48
12.3
Unit Root Tests for
Stationarity
Table 12.3 AR processes and the Dickey-Fuller Tests
12.3.5
The Dickey-Fuller
Testing Procedures
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 49
12.3
Unit Root Tests for
Stationarity
12.3.5
The Dickey-Fuller
Testing Procedures
The Dickey-Fuller testing procedure:
– First plot the time series of the variable and select a
suitable Dickey-Fuller test based on a visual
inspection of the plot
• If the series appears to be wandering or
fluctuating around a sample average of zero, use
test equation (12.5a)
• If the series appears to be wandering or
fluctuating around a sample average which is
nonzero, use test equation (12.5b)
• If the series appears to be wandering or
fluctuating around a linear trend, use test
equation (12.5c)
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 50
12.3
Unit Root Tests for
Stationarity
12.3.5
The Dickey-Fuller
Testing Procedures
The Dickey-Fuller testing procedure (Continued):
– Second, proceed with one of the unit root tests
described in Sections 12.3.1 to 12.3.3
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 51
12.3
Unit Root Tests for
Stationarity
12.3.6
The Dickey-Fuller
Tests: An Example
As an example, consider the two interest rate
series:
– The federal funds rate (Ft)
– The three-year bond rate (Bt)
Following procedures described in Sections 12.3
and 12.4, we find that the inclusion of one lagged
difference term is sufficient to eliminate
autocorrelation in the residuals in both cases
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 52
12.3
Unit Root Tests for
Stationarity
12.3.6
The Dickey-Fuller
Tests: An Example
The results from estimating the resulting equations
are:
Ft  0.173  0.045 Ft 1  0.561Ft 1
(tau )
(  2.505)
Bt  0.237  0.056 Bt 1  0.237 Bt 1
(tau )
(  2.703)
– The 5% critical value for tau (τc) is -2.86
– Since -2.505 > -2.86, we do not reject the null
hypothesis of non-stationarity.
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 53
12.3
Unit Root Tests for
Stationarity
12.3.7
Order of Integration
Recall that if yt follows a random walk, then γ = 0
and the first difference of yt becomes:
yt  yt  yt 1  vt
– Series like yt, which can be made stationary by
taking the first difference, are said to be
integrated of order one, and denoted as I(1)
• Stationary series are said to be integrated of
order zero, I(0)
– In general, the order of integration of a series is
the minimum number of times it must be
differenced to make it stationary
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 54
12.3
Unit Root Tests for
Stationarity
12.3.7
Order of Integration
The results of the Dickey–Fuller test for a random
walk applied to the first differences are:
  F t   0.447  F t 1
(tau )
(  5.487)
  B t   0.701 B t 1
(tau )
Principles of Econometrics, 4th Edition
(  7.662)
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 55
12.3
Unit Root Tests for
Stationarity
12.3.7
Order of Integration
Based on the large negative value of the tau
statistic (-5.487 < -1.94), we reject the null
hypothesis that ΔFt is nonstationary and accept the
alternative that it is stationary
– We similarly conclude that ΔBt is stationary
(-7.662 < -1.94)
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 56
12.4
Cointegration
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 57
12.4
Cointegration
As a general rule, nonstationary time-series
variables should not be used in regression models
to avoid the problem of spurious regression
– There is an exception to this rule
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 58
12.4
Cointegration
There is an important case when
et = yt - β1 - β2xt is a stationary I(0) process
– In this case yt and xt are said to be cointegrated
• Cointegration implies that yt and xt share
similar stochastic trends, and, since the
difference et is stationary, they never diverge
too far from each other
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 59
12.4
Cointegration
The test for stationarity of the residuals is based
on the test equation:
eˆt  γeˆt 1  vt
Eq. 12.7
– The regression has no constant term because
the mean of the regression residuals is zero.
– We are basing this test upon estimated values of
the residuals
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 60
12.4
Cointegration
Principles of Econometrics, 4th Edition
Table 12.4 Critical Values for the Cointegration Test
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 61
12.4
Cointegration
There are three sets of critical values
– Which set we use depends on whether the
residuals are derived from:
Eq. 12.8a
Equation 1: eˆt  yt  bxt
Eq. 12.8b
Equation 2 : eˆt  yt  b2 xt  b1
Eq. 12.8c
Equation 3: eˆt  yt  b2 xt  b1  ˆ t
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 62
12.4
Cointegration
12.4.1
An Example of a
Cointegration Test
Eq. 12.9
Consider the estimated model:
Bˆt  1.140  0.914 Ft , R 2  0.881
(t ) (6.548) (29.421)
– The unit root test for stationarity in the
estimated residuals is:
eˆt  0.225eˆt 1  0.254eˆt 1
(tau ) (4.196)
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 63
12.4
Cointegration
12.4.1
An Example of a
Cointegration Test
The null and alternative hypotheses in the test for
cointegration are:
H 0 : the series are not cointegrated  residuals are nonstationary
H1 : the series are cointegrated  residuals are stationary
– Similar to the one-tail unit root tests, we reject
the null hypothesis of no cointegration if τ ≤ τc,
and we do not reject the null hypothesis that the
series are not cointegrated if τ > τc
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 64
12.4
Cointegration
12.4.2
The Error Correction
Model
Consider a general model that contains lags of y
and x
– Namely, the autoregressive distributed lag
(ARDL) model, except the variables are
nonstationary:
yt  δ  θ1 yt 1  δ0 xt  δ1 xt 1  vt
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 65
12.4
Cointegration
12.4.2
The Error Correction
Model
If y and x are cointegrated, it means that there is a
long-run relationship between them
– To derive this exact relationship, we set
yt = yt-1 = y, xt = xt-1 = x and vt = 0
– Imposing this concept in the ARDL, we obtain:
y 1  θ1   δ   δ0  δ1  x
• This can be rewritten in the form:
y  β1  β 2 x
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 66
12.4
Cointegration
12.4.2
The Error Correction
Model
Add the term -yt-1 to both sides of the equation:
yt  yt 1  δ   θ1 1 yt 1  δ0 xt  δ1 xt 1  vt
– Add the term – δ0xt-1+ δ0xt-1:
yt  δ   θ1 1 yt 1  δ0  xt  xt 1    δ0  δ1  xt 1  vt
– Manipulating this we get:
 δ

δ0  δ1 

yt   θ1  1 
 yt 1 
xt 1   δ0 xt  vt
  θ  1

θ

1


1
1


Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 67
12.4
Cointegration
12.4.2
The Error Correction
Model
Or:
yt  α  yt 1  β1  β2 xt 1   δ0xt  vt
Eq. 12.10
– This is called an error correction equation
– This is a very popular model because:
• It allows for an underlying or fundamental
link between variables (the long-run
relationship)
• It allows for short-run adjustments (i.e.
changes) between variables, including
adjustments to achieve the cointegrating
relationship
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 68
12.4
Cointegration
12.4.2
The Error Correction
Model
For the bond and federal funds rates example, we
have:
Bˆt  0.142  Bt 1  1.429  0.777 Ft 1   0.842Ft  0.327Ft 1
t 
 2.857 
 9.387   3.855
– The estimated residuals are
eˆt 1   Bt 1 1.429  0.777Ft 1 
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 69
12.4
Cointegration
The result from applying the ADF test for
stationarity is:
eˆt  0.169eˆt 1  0.180eˆt 1
t   3.929
– Comparing the calculated value (-3.929) with
the critical value, we reject the null hypothesis
and conclude that (B, F) are cointegrated
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 70
12.5
Regression with No-Cointegration
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 71
12.5
Regression When
There is No
Cointegration
How we convert nonstationary series to stationary
series, and the kind of model we estimate, depend
on whether the variables are difference stationary
or trend stationary
– In the former case, we convert the
nonstationary series to its stationary counterpart
by taking first differences
– In the latter case, we convert the nonstationary
series to its stationary counterpart by detrending
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 72
12.5
Regression When
There is No
Cointegration
12.5.1
First Difference
Stationary
Consider the random walk model:
yt  yt 1  vt
– This can be rendered stationary by taking the
first difference:
yt  yt  yt 1  vt
• The variable yt is said to be a first difference
stationary series
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 73
12.5
Regression When
There is No
Cointegration
12.5.1
First Difference
Stationary
A suitable regression involving only stationary
variables is:
yt  yt 1  0 xt  1xt 1  et
Eq. 12.11a
– Now consider a series yt that behaves like a
random walk with drift:
yt    yt 1  vt
with first difference:
yt    vt
• The variable yt is also said to be a first
difference stationary series, even though it
is stationary around a constant term
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 74
12.5
Regression When
There is No
Cointegration
12.5.1
First Difference
Stationary
Suppose that y and x are I(1) and not cointegrated
– An example of a suitable regression equation is:
Eq. 12.11b
Principles of Econometrics, 4th Edition
yt    yt 1  0 xt  1xt 1  et
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 75
12.5
Regression When
There is No
Cointegration
12.5.2
Trend Stationary
Consider a model with a constant term, a trend
term, and a stationary error term:
yt    t  vt
– The variable yt is said to be trend stationary
because it can be made stationary by removing
the effect of the deterministic (constant and
trend) components:
yt    t  vt
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 76
12.5
Regression When
There is No
Cointegration
12.5.2
Trend Stationary
If y and x are two trend-stationary variables, a
possible autoregressive distributed lag model is:
Eq. 12.12
Principles of Econometrics, 4th Edition
yt   yt1 0 xt 1xt1  et
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 77
12.5
Regression When
There is No
Cointegration
12.5.2
Trend Stationary
As an alternative to using the de-trended data for
estimation, a constant term and a trend term can be
included directly in the equation:
yt     t   yt 1  0 xt  1 xt 1  et
where:
  1 (1  1 )   2 (0  1 )  11  1 2
  1 (1  1 )  2 (0  1 )
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 78
12.5
Regression When
There is No
Cointegration
12.5.3
Summary
If variables are stationary, or I(1) and cointegrated,
we can estimate a regression relationship between
the levels of those variables without fear of
encountering a spurious regression
If the variables are I(1) and not cointegrated, we
need to estimate a relationship in first differences,
with or without the constant term
If they are trend stationary, we can either de-trend
the series first and then perform regression
analysis with the stationary (de-trended) variables
or, alternatively, estimate a regression relationship
that includes a trend variable
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 79
12.5
Regression When
There is No
Cointegration
FIGURE 12.4 Regression with time-series data: nonstationary variables
12.5.3
Summary
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 80
Key Words
Principles of Econometrics, 4th Edition
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Page 81
Keywords
autoregressive
process
cointegration
Dickey–Fuller
tests
difference
stationary
mean reversion
nonstationary
Principles of Econometrics, 4th Edition
order of
integration
random walk
process
random walk
with drift
spurious
regressions
stationary
Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
stochastic
process
stochastic trend
tau statistic
trend stationary
unit root tests
Page 82