An Introduction to Error Correction Models An Introduction to ECMs ...
An Introduction to Error Correction Models An Introduction to ECMs ...
An Introduction to Error Correction Models An Introduction to ECMs ...
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An Introduction to
Error Correction Models
Robin Best
Oxford Spring School for Quantitative
Methods in Social Research
2008
An Introduction to ECMs
The basic structure of an ECM
ΔY t = α + βΔX t1  βEC t1 + ε t
Where EC is the error correction component of the model and measures the speed
at which prior deviations from equilibrium are corrected.
Error correction models can be used to estimate the following quantities of
interest for all X variables.
� Short term effects of X on Y
� Long term effects of X on Y (long run multiplier)
� The speed at which Y returns to equilibrium after a deviation has occurred.
Applications of ECMs in the
(Political Science) Literature
• U.S. Presidential Approval/ U.K. Prime Ministerial Satisfaction
• Policy Mood/Policy Sentiment
• Support for Social Security
• Consumer Confidence
• Economic Expectations
• Health Care Cost Containment/ Government Spending /Patronage Spending /
Redistribution
• Interest Rates/ Purchasing Power Parity
• Growth in (U.S.) Presidential Staff
• Arms Transfers
• U.S. Judicial Influence
An Introduction to ECMs
Error Correction Models (ECMs) are a category of multiple time series
models that directly estimate the speed at which a dependent variable 
Y  returns to equilibrium after a change in an independent variable  X.
� ECMs are useful for estimating both short term and long term effects of
one time series on another.
• Thus, they often mesh well with our theories of political and social
processes.
• Theoreticallydriven approach to estimating time series models.
� ECMs are useful models when dealing with integrated data, but can
also be used with stationary data.
An Introduction to ECMs
� As we will see, the versatility of ECMs give them a number of desirable
properties.
• Estimates of short and long term effects
• Easy interpretation of short and long term effects
• Applications to both integrated and stationary time series data
• Can be estimated with OLS
• Model theoretical relationships
� ECMs can be appropriate whenever (1) we have time series data and (2)
are interested in both short and long term relationships between multiple
time series.
Overview of the Course
I. Motivating ECMs with cointegrated data
• Integration and cointegration
• 2step error correction estimators
• Stata session #1
II. Motivating ECMs with stationary data
• The single equation ECM
• Interpretation of long and short term effects
• The Autoregressive Distributive Lag (ADL) model
• Equivalence of the ECM and ADL
• Stata session #2
1
ECMs and Cointegration:
Stationary vs. Integrated Time Series
� Stationary time series data are mean reverting. That is, they have a
finite mean and variance that do not depend on time.
Y t = α + ρY t1 + ε t
Where  p  < 1 and ε t is also stationary with a mean of zero and variance σ 2
� Note that when 0 <  p  < 1 the time series is stationary but contains
autocorrelation.
ECMs and Cointegration:
Integrated Time Series
� Formally, an integrated series can be expressed as a function of all
past disturbances at any point in time.
t
t∑
ei
i=
1
Or Y t = α + ρY t1 + ε t
Where p = 1
Or Y t  Y t1 = u t
Where u t = ε t
And ε t is still a stationary process
ECMs and Cointegration:
Integrated Time Series
(Theoretical) Sources of integration
Y
� The effect of past shocks is permanently incorporated into the
memory of the series.
� The series is a function of other integrated processes.
ECMs and Cointegration:
Stationary vs. Integrated Time Series
Often our time series data are not stationary, but appear to be integrated.
Integrated time series data
• Are not meanreverting
• appear to be on a ‘random walk’
• Have current values that can be expressed as the sum of all previous changes
• The effect of any shock is permanently incorporated into the series
• Thus, the best predictor of the series at time t is the value at time t1
• Have a (theoretically) infinite variance and no mean.
ECMs and Cointegration:
Integrated Time Series
Order of Integration
� Integrated time series data that are stationary after being difference
d times are Integrated of order d: I(d)
� For our purposes, we focus on time series data that are I(1).
• Data that are stationary after being firstdifferenced.
� I(1) processes are fairly common in time series data
A Drunk’s Random Walk
0 20 40 60
time
2
ECMs and Cointegration:
Integrated Time Series
• Analyzing integrated time series in level form dramatically increases the
likelihood of making a TypeII error.
� Problem of spurious associations.
� High R 2
� Small standard errors and inflated tratios
• A common solution to these problems is to analyze the data in differenced form.
� Look only at short term effects
ECMs and Cointegration
� Two time series are cointegrated if
� Both are integrated of the same order.
� There is a linear combination of the two time series that is I(0)  i.e. 
stationary.
� Two (or more) series are cointegrated if each has a long run component,
but these components cancel out between the series.
� Share stochastic trends
� Conintegrated data are never expected to drift too far away from each
other, maintaining an equilibrium relationship.
A Dog’s Random Walk
0 20 40 60
time
ECMs and Cointegration:
Integrated Time Series
• Analyzing time series data in differenced form solves the spurious
regression problem, but may “throw the baby out with the bathwater.”
• A model that includes only (lagged) differenced variables assumes the
effects of the X variables on Y never last longer than one time period.
• What if our time series share a long run relationship?
• If the time series share an equilibrium relationship with an errorcorrection
mechanism, then the stochastic trends of the time series will
be correlated with one another.
• Cointegration
ECMs and Cointegration
� Lets go back to the drunk’s random walk and call the drunk X. The
random walk can be expressed as
X t  X t1 = u t
� Where u t represents the stationary, whitenoise shocks.
� Another rather trivial example of a random walk is the walk (or jaunt) of a
dog, which can be expressed as
Y t  Y t1 = w t
� Where w t represents the stationary, whilenoise process of the dog’s
steps.
ECMs and Cointegration
But what if the dog belongs to the drunk?
� Then the two random walks are likely to have an equilibrium relationship and to
be cointegrated (Murray 1994).
� Deviations from this equilibrium relationship will be corrected over time.
� Thus, part of the stochastic processes of both walks will be shared and will
correct deviations the equilibrium
X t  X t1 = u t + c(Y t1  X t1 )
Y t  Y t1 = w t + d(X t1  Y t1 )
Where the terms in parentheses are the error correcting mechanisms
3
The Drunk and Her Dog
0 20 40 60
time
drunk dog
ECMs and Cointegration
Y t = βX t + Z t
Here, Z represents the portion of Y (in levels) that is not attributable to X.
� In short, Z will capture the error correction relationship by capturing the
degree to which Y and X are out of equilibrium.
Z will capture any shock to either Y or X. If Y and X are cointegrated, then
the relationship between the two will adjust accordingly.
ECMs and Cointegration
� We might theorize that shocks to X have two effects on ΔY.
� Some portion of shocks to X might immediately affect Y in the next time
period, so that ΔY t responds to ΔX t1 .
� A shock to X t will also disturb the equilibrium between Y and X, sending Y
on a long term movement to a value that reproduces the equilibrium state
given the new value of X.
� Thus ΔY t is a function of both ΔX t1 and the degree to which the two
variables were out of equilibrium in the previous time period.
ECMs and Cointegration
Two I(1) time series (X t and Y t ) are cointegrated if there is some linear
combination that is stationary.
Z t = Y t  βX t
Where Z is the portion of (levels of) Y that are not shared with X: the equilibrium
errors.
We can also rewrite this equation in regression form
Y t = βX t + Z t
Where the cointegrating vector  Z t  can be obtained by regressing Y t on X t .
ECMs and Cointegration
ΔY t will be a function of the degree to which the two time series were out of
equilibrium in the previous period: Z t1
Z t1 = Y t1  X t1
� When Z = 0 the system is in its equilibrium state
� Y t will respond negatively to Z t1 .
� If Z is negative, then Y is too high and will be adjusted downward in the next
period.
� If Z is positive, then Y is too low and will be adjusted upward in the next time
period.
Engle and Granger TwoStep ECM
� If two time series are integrated of the same order AND some linear
combination of them is stationary, then the two series are cointegrated.
� Cointegrated series share a stochastic component and a long term
equilibrium relationship.
� Deviations from this equilibrium relationship as a result of shocks will be
corrected over time.
� We can think of ∆Y t as responding to shocks to X over the short and long
term.
4
Engle and Granger TwoStep ECM
� Engle and Granger (1987) suggested an appropriate model for Y, based
two or more time series that are cointegrated.
� First, we can obtain an estimate of Z by regressing Y on X.
� Second, we can regress ΔY t on Z t1 plus any relevant short term
effects.
Engle and Granger TwoStep ECM
� In Step 1, where we estimate the cointegrating regression we can 
and should  include all variables we expect to
1) be cointegrated
2) have sustained shocks on the equilibrium.
� The variables that have sustained shocks on the equilibrium are
usually regarded as exogenous shocks and often take the form of
dummy variables.
Engle and Granger TwoStep ECM
The basic structure of the ECM
ΔY t = α + βΔX t1  βEC t1 + ε t
In the Engle and Granger TwoStep Method the EC component is derived from
cointegrated time series as Z.
∆Y t = β 0 ∆X t1  β 1 Z t1
β 0 captures the short term effects of X in the prior period on Y in the current period.
β 1 captures the rate at which the system Y adjusts to the equilibrium state after a
shock. In other words, it captures the speed of error correction.
Step 1:
Engle and Granger TwoStep ECM
Y t = α + βX t + Z t
The cointegrating vector  Z  is measured by taking the residuals from the
regression of Y t on X t
Z t = Y t  βX t  α
Step 2:
Regress changes on Y on lagged changes in X as well as the equilibrium errors
represented by Z.
∆Y t = β 0 ∆X t1  β 1 Z t1
Note that all variables in this model are stationary.
Engle and Granger TwoStep ECM
The cointegrating regression is performed as Y t = α + βX t + Z t
Which we can also conceptualize as
Z t = Y t  (α +βX t )
If we add a series of j exogenous shocks  represented as w j
Then
Y t = α + βX t + βW 1t + βW 2t +βW 3t + Z t
Z t = Y t  (α +βX t + βW 1t + βW 2t +βW 3t )
Engle and Granger TwoStep ECM
Note that the Engle and Granger 2Step method is really a 4step method.
1) Determine that all time series are integrated of the same order.
2) Demonstrate that the time series are cointegrated
3) Obtain an estimate of the cointegrating vector  Z  by regressing
Y t on X t and taking the residuals.
4) Enter the lagged residuals  Z  into a regression of ∆Y t on ∆X t1 .
5
Engle and Granger TwoStep ECM
� Viewed from this perspective, it is easy to see why error correction
models have become so closely associated with cointegration (we will
come back to this later).
� Integrated time series present a problem for time series analysis  at
least in terms of long term relationships.
� When integrated time series variables are also cointegrated, error
correction models provide a nice solution to this problem.
Cointegration and Error
Correction in Political Science
� Prime Ministerial Statisfaction (U.K.) and Conservative Party
Support
� Arms transfers by the U.S. and Soviet Union
� Economic expectations and U.S. Presidential Approval
� U.S. Domestic Policy Sentiment and Economic Expectations
� Support for U.S. Social Security and the Stock Market
X and Y: Cointegrated?
0 5 10 15 20 25
1960m1 1961m1 1962m1 1963m1 1964m1 1965m1
months
Y X
Cointegration and Error Correction
� One of the first instances of error correction was Davidson et. al.’s
(1978) study of consumer expenditure and income in the U.K..
� The Engle and Granger approach to error correction models follows
nicely from the field of economics, where integration and cointegration
among time series is theoretically common.
� Error correction models were imported from economics.
� Would we expect data from the social sciences to follow similar
patterns of integration and cointegration?
The Engle and Granger TwoStep
ECM: Putting it into Practice
� Lets imagine we have two time series  perhaps the drunk and her dog 
but lets call the drunk ‘X’ and the dog ‘Y’.
� From a theoretical perspective, we believe changes in X will have both
short and long term effects on Y, since we expect X and Y to have an
equilibrium relationship.
� We expect changes in X to produce long run responses in Y, as Y
adjusts back to the equilibrium state.
Engle and Granger TwoStep ECM
First, we need to determine that both X and Y are integrated of the same order.
• Which means we first need to demonstrate that both X and Y are, in fact,
integrated processes.
• We should also think about the likely stationary or nonstationary nature of our
time series from a theoretical perspective.
Tests for unitroot process tend to be controversial, primarily due to their low power.
For our purposes, we will focus on DickeyFuller (DF) and Augmented DickeyFuller
tests to examine the (non)stationarity of our time series.
6
DickeyFuller Tests
Basic DickeyFuller test
With a constant (drift)
With a time trend
Δ t = xt−1
x γ + ε
x α γ + ε
Δ t = t + xt−1
x α γ + β + ε
Δ t = t + xt−1
Augmented DickeyFuller
We can remove any remaining serial correlation in ε t by introducing an
appropriate number of lagged differences of X in the equation.
k
Δ t = xt−1
+ ∑
i=
1
x γ β Δx
+ ε
k
Δ t = t + γxt
−1
+ ∑
i=
1
x α β Δx
+ ε
Where i = 1, 2, …k
Null hypotheses are the same as the DF tests
Is X Integrated?
dfuller X, lags(4) regress
i
1t−i
1t
−i
Augmented DickeyFuller test for unit root Number of obs = 59
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 0.690 3.567 2.923 2.596

MacKinnon approximate pvalue for Z(t) = 0.9896

D.X  Coef. Std. Err. t P>t [95% Conf.Interval]
+
X 
L1.  .0696672 .1008978 0.69 0.493 .1327082 .2720426
LD.  .5724812 .1738494 3.29 0.002 .9211789 .2237835
L2D.  .4935811 .1776346 2.78 0.008 .8498709 .1372912
L3D.  .2891465 .1677748 1.72 0.091 .6256601 .0473671
L4D.  .0898266 .1468121 0.61 0.543 .3842943 .2046412
_cons  .2525666 .839646 0.30 0.765 1.936683 1.43155

i
t
t
t
t
t
t
DickeyFuller Tests
Basic DickeyFuller test
With a constant (drift)
With a time trend
Δ t = xt
−1
x γ + ε
Δ t = t + γxt−1
x α + ε
x α γ + β + ε
Δ t = t + xt−1
If X is a random walk process, then γ = 0
The null hypothesis is that X is a random walk
MacKinnon values for statistical significance
Note that in small samples the standard error of γ will be large, making it likely that
we fail to reject the null when we really should
dfuller X, regress
Is X Integrated?
DickeyFuller test for unit root Number of obs = 63
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 1.852 3.562 2.920 2.595

MacKinnon approximate pvalue for Z(t) = 0.3548

D.X  Coef. Std. Err. t P>t [95% Conf. Interval]
+
X 
L1.  .1492285 .0805656 1.85 0.069 .3103293 .0118724
_cons  1.365817 .7149307 1.91 0.061 .0637749 2.79541

Is X Integrated?
If X is I(1), then the first difference of X should be stationary.
dfuller dif_X
DickeyFuller test for unit root Number of obs = 62
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 10.779 3.563 2.920 2.595

MacKinnon approximate pvalue for Z(t) = 0.0000
t
t
t
t
7
Is Y Integrated?
dfuller Y, regress
DickeyFuller test for unit root Number of obs = 63
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 1.323 3.562 2.920 2.595

MacKinnon approximate pvalue for Z(t) = 0.6184

D.Y  Coef. Std. Err. t P>t [95% Conf. Interval]
+
Y 
L1.  .0854922 .064599 1.32 0.191 .2146659 .0436814
_cons  1.061271 .7208156 1.47 0.146 .3800884 2.502631

Cointegration
� Both X and Y appear to be integrated of the same order: I(1).
� If they are cointegrated, then they share stochastic trends.
� In the following regression, ε t should be stationary and β should be
statistically significant and in the expected direction.
Lets see if this is the case
predict r, resid
dfuller r
Y t = α t + βX t +ε t
Cointegrating Regression
DickeyFuller test for unit root Number of obs = 63
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 5.487 3.562 2.920 2.595

MacKinnon approximate pvalue for Z(t) = 0.0000
Is Y Integrated?
dfuller dif_Y, regress
DickeyFuller test for unit root Number of obs = 62
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 9.071 3.563 2.920 2.595

MacKinnon approximate pvalue for Z(t) = 0.0000

D.dif_Y  Coef. Std. Err. t P>t [95% Conf. Interval]
+dif_Y

L1.  1.159903 .1278662 9.07 0.000 1.415674 .9041329
_cons  .2219184 .3259962 0.68 0.499 .4301711 .8740078

regress Y X
Cointegrating Regression
Source  SS df MS Number of obs = 64
+ F( 1, 62) = 92.49
Model  1009.22604 1 1009.22604 Prob > F = 0.0000
Residual  676.523964 62 10.9116768 Rsquared = 0.5987
+ Adj Rsquared = 0.5922
Total  1685.75 63 26.7579365 Root MSE = 3.3033

Y  Coef. Std. Err. t P>t [95% Conf. Interval]
+
X  1.206126 .1254135 9.62 0.000 .9554281 1.456824
_cons  .0108108 1.135884 0.01 0.992 2.259789 2.28141

15 10 5 0 5 10
Residuals
1960m1 1961m1 1962m1 1963m1 1964m1 1965m1
months
8
Engle and Granger TwoStep ECM
� Our residuals from the cointegrating regression capture deviations from
the equilibrium of X and Y.
� Therefore, we can estimate both the short and long term effects of X on
Y by including the lagged residuals from the cointegrating regression as
our measure of the error correction mechanism.
ΔY t = α + β 1 *ΔX t1 + β 2 *R t1 +ε t
Granger Causality and ECMs
Granger Causality:
� A variable  X – Granger causes another variable – Y – if Y can be
better predicted by the lagged values of both X and Y than by the lagged
values of Y alone (see Freeman 1983).
� Standard Granger causality tests can result in incorrect inferences about
causality when there is an error correction process.
� The EngleGranger approach to ECMs begins by assuming all variables
in the cointegrating regression are jointly endogeneous.
� Thus, in the previous example we should also estimate a cointegrating
regression of X on Y.
Granger Causality
regress dif_Y l.dif_Y l.dif_X lag_Y lag_X
Source  SS df MS Number of obs = 62
+ F( 4, 57) = 2.97
Model  69.5277246 4 17.3819311 Prob > F = 0.0270
Residual  334.149695 57 5.86227535 Rsquared = 0.1722
+ Adj Rsquared = 0.1141
Total  403.677419 61 6.61766261 Root MSE = 2.4212
dif_Y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+dif_Y

L1.  .0483244 .1399056 0.35 0.731 .2318318 .3284806
dif_X 
L1.  .2205689 .1802099 1.22 0.226 .581433 .1402952
lag_Y  .3557259 .1161894 3.06 0.003 .5883911 .1230606
lag_X  .5675793 .1899981 2.99 0.004 .1871146 .948044
_cons  .928984 .9426534 0.99 0.329 2.816615 .9586468

Engle and Granger TwoStep ECM
regress dif_Y dlag_X lag_r
Source  SS df MS Number of obs = 62
+ F( 2, 59) = 5.09
Model  59.4494524 2 29.7247262 Prob > F = 0.0091
Residual  344.227967 59 5.83437232 Rsquared = 0.1473
+ Adj Rsquared = 0.1184
Total  403.677419 61 6.61766261 Root MSE = 2.4154
dif_Y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+dlag_X
 .1161038 .1609359 0.72 0.473 .4381358 .2059282
lag_r  .3160139 .0999927 3.16 0.002 .5160988 .1159291
_cons  .210471 .3074794 0.68 0.496 .4047939 .8257358

The error correction mechanism is negative and significant, suggesting that
deviations from equilibrium are corrected at about 32% per month.
However, X does not appear to have significant short term effects on Y.
Granger Causality
• Granger causality can be ascertained in the ECM framework by
regressing each time series in differenced form on all time series in
both differenced and level form.
• If an EC representation is appropriate, then in at least one of the
regressions:
� The lagged level of the predicted variable should be negative and
significant.
� The lagged level of the other variable should be in the expected
direction and significant.
Granger Causality
regress dif_X l.dif_X l.dif_Y lag_X lag_Y
Source  SS df MS Number of obs = 62
+ F( 4, 57) = 5.87
Model  74.2042429 4 18.5510607 Prob > F = 0.0005
Residual  180.182854 57 3.1611027 Rsquared = 0.2917
+ Adj Rsquared = 0.2420
Total  254.387097 61 4.17028027 Root MSE = 1.7779
dif_X
 Coef. Std. Err. t P>t [95% Conf. Interval]
+dif_X

L1.  .0640245 .132332 0.48 0.630 .3290147 .2009657
dif_Y 
L1.  .0014809 .1027357 0.01 0.989 .2042438 .2072056
lag_X  .4676537 .1395197 3.35 0.001 .7470371 .1882703
lag_Y  .2847586 .0853204 3.34 0.001 .1139075 .4556097
_cons  1.194109 .6922106 1.73 0.090 .1920183 2.580237

9
ECMs, Causality, and Theory
� In the social sciences, our theories (usually) tell us which time series
should be on the left side of the equation and which should be on the
right.
� The Engle and Granger approach assumes endogeneity between the
cointegrating time series.
Integration Issues
Error correction approaches that rely on cointegration of two or more I(1)
time series become problematic when we are dealing with data that are
not truly (co)integrated.
� I(1) processes may be incorrectly included into the cointegrating
regression  producing spurious associations  if two other I(1)
cointegrated time series are already included (Durr 1992)
� This problem increases with sample size.
� The low power of unit root tests can lead us to conclude our data are
integrated when they are not.
More Integration Issues
Under these conditions, we are likely to draw faulty inferences from the
twostep procedure.
We might conclude:
� Our data are integrated when they are not.
� Our data are cointegrated when they are not.
� Our data are not cointegrated, therefore, an ECM is not appropriate
Engle and Granger TwoStep
Technique: Issues and Limitations
� Does not clearly distinguish dependent variables from independent
variables.
� In the social sciences the Engle and Granger twostep ECM might not be
consistent with our theories.
� Is appropriate when dealing with cointegrated time series.
� Can we clearly distinguish between integrated and stationary processes?
More Integration Issues
In the social sciences, we are more likely to have data that are
� Near integrated (p = 0, but there is memory. p may not = 0 in finite
samples.)
� Fractionally integrated (0 < p < 1, where when 0 < p < .5 the data are
meanreverting and have finite variance, and when .5 ≤ p < 1 the data are
meanreverting but have infinite variance)
� A combined process of both stationary and integrated data
� Aggregated data
Integration Issues and ECMs
� Under these conditions, we are often better off estimating a single
equation ECM.
� Single equation ECMs solve some of these problems and avoid others.
� However, single equation ECMs require weak exogeneity.
10
Single Equation
Error Correction Models
� Following theory, Single Equation ECMs clearly distinguish between
dependent and independent variables.
� Single Equation ECMs are appropriate for both cointegrated and longmemoried,
but stationary, data.
� Cointegration may imply error correction, but does error correction imply
cointegration?
� Single Equation ECMs estimate a long term effect for each independent
variable, allowing us to judge the contribution of each.
� Allow for easier interpretation of the effects of the independent variables.
Single Equation ECMs
� Single Equation Error Correction Models are useful
� When our theories dictate the causal relationships of interest
� When we have longmemoried/stationary data
A basic single equation ECM:
∆Y t = α + β 0 *∆X t  β 1 (Y t1  β 2 X t1 ) + ε t
The Single Equation ECM
∆Y t = α + β 0*∆X t  β 1(Y t1  β 2X t1) + βε t
The portion of the equation in parentheses is the error correction mechanism.
(Y t1  β 2 X t1 ) = 0 when Y and X are in their equilibrium state
β 0 estimates the short term effect of an increase in X on Y
β 1 estimates the speed of return to equilibrium after a deviation.
If the ECM approach is appropriate, then 1 < β 1 < 0
β 2 estimates the long term effect that a one unit increase in X has on Y. This long
term effect will be distributed over future time periods according to the rate of
error correction  β 1
Single Equation ECMs
� Our theories might specify long and short term effects of independent
variables on a dependent variable even when our data are stationary.
� The concepts of error correction, equilibrium , and long term effects are
not unique to cointegrated data.
� Furthermore, an ECM may provide a more useful modeling technique for
stationary data than alternative approaches.
� Our theories may be better represented by a single equation ECM.
The Single Equation ECM
Basic form of the ECM
Engle and Granger twostep ECM
The Single Equation ECM
ΔY t = α + βΔX t1  βEC t1 + ε t
∆Y t = β 0 ∆X t1  β 1 Z t1
Where Z t = Y t  βX t  α
∆Y t = α + β 0 *∆X t  β 1 (Y t1  β 2 X t1 ) + ε t
The Single Equation ECM
∆Y t = α + β 0*∆X t  β 1(Y t1  β 2X t1) + ε t
The values for which Y and X are in their long term equilibrium relationship are
Y = k0 + k1X α
Where k0
=
β
And β 2
k1
=
β
1
1
Where k 1 is the total long term effect of X on Y (a.k.a the long run multiplier)  
distributed over future time periods.
Single equation ECMs are particularly useful for allowing us to also estimate k 1 ’s
standard error, and therefore statistical significance.
11
The Single Equation ECM
Since the long term effect is a ratio of two coefficients, we could calculate its
standard error using the variance and covariance matrix
Alternatively, we can use the Bewley transformation to estimate the standard error.
This requires estimating the following regression.
Y t = α+ δ 0 ∆Y t + δ 1 X t  δ 2 ∆X t + μ t
Where δ 1 is the long term effect and is estimated with a standard error
Notice the problem: we have ∆Y t on the right side of the equation
We can proxy ∆Y t as:
∆Y t = α + βY t1 + βX t + β∆X t + ε t
And use our predicted values of ∆Y t in the Bewley transformation regression
Single Equation ECMs in the
(Political Science) Literature
� Judicial Influence
� Health Care Cost Containment
� Interest Rates
� Patronage Spending
� Growth in Presidential Staff
� Government Spending
� Consumer Confidence
� Redistribution
ECMs and ADL Models
� We know Autoregressive Distributive Lag models are appropriate for
stationary data (stationary data is, in fact, a requirement of these
models).
� Forms of single equation ECMs and ADL models are equivalent.
� We can derive a single equation ECM from a general ADL model:
Y t = α + β 0Y t1 + β 1X t + β 2X t1 + ε t
The Single Equation ECM
We can easily extend the single equation ECM to include more
independent variables
∆Y t = α + β∆X 1t + β∆X 2t + β∆X 3t  β(Y t1  βX 1t1  βX 2t1  βX 3t1 ) + ε t
Note that each independent variable is now forced to make an
independent contribution to the long term relationship, solving one of
the problems in the twostep estimator.
Single Equation ECMs
� Single Equation ECMs
� Provide the same information about the rate of error correction as the
Engle and Granger twostep method.
� Provide more information about the long term effect of each independent
variable  including its standard error  than the Engle and Granger twostep
method.
� Illustrate that ECMs are appropriate for both cointegrated and stationary
data.
� How do we know Single Equation ECMs are appropriate with
stationary data?
ECMs and the ADL
Y t = α + β 0 Y t1 + β 1 X t + β 2 X t1 + ε t
∆Y t = α + (β 0  1)Y t1 + β 1 X t + β 2 X t1 + ε t
∆Y t = α + (β 0  1)Y t1 + β 1 ΔX t + (β 1 + β 2 )X t1 + ε t
Where φ 0 = β 0  1 and φ 1 = β 1 + β 2
∆Y t = α + φ 0 Y t1 + β 1 ΔX t + φ 1 X t1 + ε t
We can rewrite this equation in error correction form as
∆Y t = α + β 1 ΔX t  φ 0 (Y t1  φ 1 X t1 ) + ε t
12
ECMs and the ADL
We can see that the ADL model provides information similar to the ECM.
Y t = α + β 0 Y t1 + β 1 X t + β 2 X t1 + ε t
β 0 estimates the proportion of the deviation from equilibrium at t1 that is maintained
at time t. β 0  1 tells us the speed of return.
β 1 estimates the short term effect of X on Y
β 1 + β 2 estimates the long term effect of a unit change in X on Y (the coefficient on
X t1 in the ECM)
ECMs and ADL Models
What does this mean?
� ECMs are isophormic to ADL models
� We can use them with stationary data
� Certain forms of ADL models are  in a general sense  error correction
models. They can be used to estimate:
� The speed of return to equilibrium after a deviation has occurred.
� Long term equilibrium relationships between variables.
� Long and short term effects of independent variables on the dependent
variable.
Single Equation ECM
� Lets imagine our theory about the relationship between X and Y states:
� X causes Y.
� X should have both a short term and a long term effect on Y.
� We don’t have reason to suspect cointegration from a theoretical
standpoint.
� But we believe X and Y share a long term equilibrium relationship
ECMs and the ADL
Y t = α + β 0 Y t1 + β 1 X t + β 2 X t1 + ε t
And the total long term effect/long run multiplier  k 1  is therefore:
β 2 + β1
k1
=
1 − β
Y and X will be in their long term equilibrium state when Y = k 0 + k 1 X
where
0
α
k0
=
1− β
The EC and ADL Models: Notation
Lets use the following notation for the single equation ECM and the ADL
ECM
ADL
∆Y t = α + β 0∆X t  β 1(Y t1  β 2X t1) + ε t
Y t = α + β 0Y t1 + β 1X t + β 2X t1 + ε t
Single Equation ECM
We determine that our Y variable is stationary (with 95% confidence), ruling out an
ECM based on cointegration
dfuller y, regress
DickeyFuller test for unit root Number of obs = 55
 Interpolated DickeyFuller 
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value

Z(t) 3.353 3.573 2.926 2.598

MacKinnon approximate pvalue for Z(t) = 0.0127
0
13
Single Equation ECM
We then estimate the single equation ECM
As
∆Y t = α + β 0∆X t  β 1(Y t1  β 2X t1) + ε t
∆Y t = α + β 0∆X t + β 1Y t1 + β 2X t1 + ε t
If our error correction approach is correct, then β 1 should be 1 < β 1 < 0 and
significant.
Single Equation ECM
The results indicate the following equation
∆Y t = 13.12 + 1.32*∆X t .42*Y t1 + .52*X t1 + ε t
Which we can write in error correction form as
∆Y t = 13.12 + 1.32*∆X t .42(Y t1  1.22*X t1 ) + ε t
Where 1.22 is our calculation of the long run multiplier
Single Equation ECM
∆Y t = α + 1.32*∆X t .42(Y t1  1.22*X t1 ) + ε t
Changes in X have both an immediate and long term effect on Y
When the portion of the equation in parentheses = 0, X and Y are in their
equilibrium state.
Increases in X will cause deviations from this equilibrium, causing Y to be too low.
Y will then increase to correct this disequilibrium, with 42% of the (remaining)
deviation corrected in each subsequent time period.
Single Equation ECM
regress dif_y dif_x lag_y lag_x
Source  SS df MS Number of obs = 55
+ F( 3, 51) = 21.40
Model  238.216589 3 79.4055296 Prob > F = 0.0000
Residual  189.278033 51 3.71133398 Rsquared = 0.5572
+ Adj Rsquared = 0.5312
Total  427.494622 54 7.91656707 Root MSE = 1.9265
dif_y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+dif_x
 1.324821 .200003 6.62 0.000 .9232986 1.726344
lag_y  .4248235 .1146587 3.71 0.001 .6550105 .1946365
lag_x  .5182186 .1971867 2.63 0.011 .1223498 .9140873
_cons  13.12112 4.201755 3.12 0.003 4.685745 21.55649

Single Equation ECM
∆Y t = 13.12 + 1.32*∆X t .42(Y t1  1.22*X t1) + ε t
Y and X are in their long term equilibrium state when
Y = 30.89 + 1.22X
So that when X = 1
Y = 32.11
Single Equation ECM
∆Y t = α + 1.32*∆X t .42(Y t1  1.22*X t1 ) + ε t
A one unit increase in X immediately produces a 1.32 unit increase in Y.
Increases in X also disrupt the the long term equilibrium relationship between these
two variables, causing Y to be too low.
Y will respond by increasing a total of 1.22 points, spread over future time periods at
a rate of 42% per time period.
� Y will increase .52 points at t
� Then another .3 points at t+1
� Then another .2 points at t+2
� Then another .1 points at t+3
� Then another .05 points at t+4
� Then another .03 points at t+5
� Until the change in X at t1 has virtually no effect on Y
14
0 .5 1 1.5
Change in Y
0 2 4 6
Time Period
Single Equation ECM
We can determine the standard error and confidence level of the total long term
effect of X on Y through the Bewley transformation regression.
First, we can obtain an estimate of ΔY by estimating ∆Y t = α + βY t1 + βX t + β∆X t + ε t
regress dif_y lag_y x dif_x
Source  SS df MS Number of obs = 55
+ F( 3, 51) = 21.40
Model  238.216589 3 79.4055296 Prob > F = 0.0000
Residual  189.278033 51 3.71133398 Rsquared = 0.5572
+ Adj Rsquared = 0.5312
Total  427.494622 54 7.91656707 Root MSE = 1.9265
dif_y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+lag_y
 .4248235 .1146587 3.71 0.001 .6550105 .1946365
x  .5182186 .1971867 2.63 0.011 .1223498 .9140873
dif_x  .8066027 .2278972 3.54 0.001 .34908 1.264125
_cons  13.12112 4.201755 3.12 0.003 4.685745 21.55649

Single Equation ECM
� We can see our estimate of the long term effect of X on Y has a
standard error of .12 and is statistically significant.
� Can we gain similar estimates of the short and long term effects of X
on Y from the ADL model?
1 1.5 2 2.5
Y
0 2 4 6
Time Period
Single Equation ECM
And take the predicted values of ∆Y t to estimate Y t = α+ δ 0 ∆Y t + δ 1 X t  δ 2 ∆X t + μ t
predict deltaYhat
regress y deltaYhat x dif_x
Source  SS df MS Number of obs = 55
+ F( 3, 51) = 47.74
Model  531.551099 3 177.1837 Prob > F = 0.0000
Residual  189.278039 51 3.7113341 Rsquared = 0.7374
+ Adj Rsquared = 0.7220
Total  720.829138 54 13.3486877 Root MSE = 1.9265
y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+deltaYhat
 1.353919 .2698973 5.02 0.000 1.89576 .8120773
x  1.219844 .1245296 9.80 0.000 .9698408 1.469848
dif_x  1.898677 .3963791 4.79 0.000 1.102913 2.694442
_cons  30.88605 2.68463 11.50 0.000 25.49643 36.27567

Equivalence of the EC and ADL models
First, lets estimate Y t = α + β 0 Y t1 + β 1 X t + β 2 X t1 + ε t
regress y lag_y x lag_x
Source  SS df MS Number of obs = 55
+ F( 3, 51) = 47.74
Model  531.551105 3 177.183702 Prob > F = 0.0000
Residual  189.278033 51 3.71133398 Rsquared = 0.7374
+ Adj Rsquared = 0.7220
Total  720.829138 54 13.3486877 Root MSE = 1.9265
y
 Coef. Std. Err. t P>t [95% Conf. Interval]
+lag_y
 .5751765 .1146587 5.02 0.000 .3449895 .8053635
x  1.324821 .200003 6.62 0.000 .9232986 1.726344
lag_x  .8066027 .2278972 3.54 0.001 1.264125 .34908
_cons  13.12112 4.201755 3.12 0.003 4.685745 21.55649

15
Equivalence of the EC and ADL models
The results imply the equation Y t = 13.12 + .58*Y t1 + 1.32*X t .81*X t1 + ε t
Our estimate of the contemporaneous effects of X on Y 1.32 units: the same as in
the ECM.
The long term effect of X on Y at t+1 can be calculated as:
1.32  .81 = .52 which is equivalent to the .52 estimate in the ECM
Deviations from equilibrium are maintained at a rate of 58% per time period, which
implies that deviations from equilibrium are corrected at a rate of 42% per time
period (.58  1).
Error Correction Models
� A Flexible Modeling approach
� Stationary and Integrated Data
� Long and Short Term Effects
� Engle and Granger twostep ECM versus Single Equation ECM
� Importance of Theory
� Integrated or Stationary Data? Single Equation ECMs avoid this debate.
� Single equation ECMs don’t require cointegration and ease interpretation of
causal relationships.
� Single equation ECMs and ADL models
� Equivalence: ADL models can provide the same information about short
and long term effects.
� Standard error for the long term effects of independent variables is
relatively easy to obtain in the single equation ECM
Equivalence of the EC and ADL Models
Y t = 13.12 + .58*Y t1 + 1.32*X t .81*X t1 + ε t
The total long term effect/long run multiplier can be calculated as
(1.32  .81)/(.58  1) = 1.22 which is equivalent to the ECM estimate.
Note, however, that we do not have a standard error for the long run
multiplier.
Y and X will be in their long term equilibrium state when
Y = 30.89 + 1.22X
16