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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 t-1 - βEC t-1 + 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.

• Theoretically-driven 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

• 2-step 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 t-1 + 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 t-1 + ε t

Where p = 1

Or Y t - Y t-1 = 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 mean-reverting

• 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 t-1

• 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 first-differenced.

� 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 Type-II error.

� Problem of spurious associations.

� High R 2

� Small standard errors and inflated t-ratios

• 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 t-1 = u t

� Where u t represents the stationary, white-noise 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 t-1 = w t

� Where w t represents the stationary, while-noise 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 t-1 = u t + c(Y t-1 - X t-1 )

Y t - Y t-1 = w t + d(X t-1 - Y t-1 )

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 t-1 .

� 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 t-1 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 t-1

Z t-1 = Y t-1 - X t-1

� When Z = 0 the system is in its equilibrium state

� Y t will respond negatively to Z t-1 .

� 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 Two-Step 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 Two-Step 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 t-1 plus any relevant short term

effects.

Engle and Granger Two-Step 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 Two-Step ECM

The basic structure of the ECM

ΔY t = α + βΔX t-1 - βEC t-1 + ε t

In the Engle and Granger Two-Step Method the EC component is derived from

cointegrated time series as Z.

∆Y t = β 0 ∆X t-1 - β 1 Z t-1

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 Two-Step 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 t-1 - β 1 Z t-1

Note that all variables in this model are stationary.

Engle and Granger Two-Step 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 Two-Step ECM

Note that the Engle and Granger 2-Step method is really a 4-step 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 t-1 .

5

Engle and Granger Two-Step 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 Two-Step

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 Two-Step 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 unit-root process tend to be controversial, primarily due to their low power.

For our purposes, we will focus on Dickey-Fuller (DF) and Augmented Dickey-Fuller

tests to examine the (non)stationarity of our time series.

6

Dickey-Fuller Tests

Basic Dickey-Fuller 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 Dickey-Fuller

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 Dickey-Fuller test for unit root Number of obs = 59

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) 0.690 -3.567 -2.923 -2.596

------------------------------------------------------------------------------

MacKinnon approximate p-value 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

Dickey-Fuller Tests

Basic Dickey-Fuller 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?

Dickey-Fuller test for unit root Number of obs = 63

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -1.852 -3.562 -2.920 -2.595

------------------------------------------------------------------------------

MacKinnon approximate p-value 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

Dickey-Fuller test for unit root Number of obs = 62

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -10.779 -3.563 -2.920 -2.595

------------------------------------------------------------------------------

MacKinnon approximate p-value for Z(t) = 0.0000

t

t

t

t

7

Is Y Integrated?

dfuller Y, regress

Dickey-Fuller test for unit root Number of obs = 63

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -1.323 -3.562 -2.920 -2.595

------------------------------------------------------------------------------

MacKinnon approximate p-value 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

Dickey-Fuller test for unit root Number of obs = 63

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -5.487 -3.562 -2.920 -2.595

------------------------------------------------------------------------------

MacKinnon approximate p-value for Z(t) = 0.0000

Is Y Integrated?

dfuller dif_Y, regress

Dickey-Fuller test for unit root Number of obs = 62

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -9.071 -3.563 -2.920 -2.595

------------------------------------------------------------------------------

MacKinnon approximate p-value 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 R-squared = 0.5987

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 Two-Step 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 t-1 + β 2 *R t-1 +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 Engle-Granger 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 R-squared = 0.1722

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 Two-Step 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 R-squared = 0.1473

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 R-squared = 0.2917

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

two-step 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 Two-Step

Technique: Issues and Limitations

� Does not clearly distinguish dependent variables from independent

variables.

� In the social sciences the Engle and Granger two-step 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

mean-reverting and have finite variance, and when .5 ≤ p < 1 the data are

mean-reverting 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 long-memoried/stationary data

A basic single equation ECM:

∆Y t = α + β 0 *∆X t - β 1 (Y t-1 - β 2 X t-1 ) + ε t

The Single Equation ECM

∆Y t = α + β 0*∆X t - β 1(Y t-1 - β 2X t-1) + βε t

The portion of the equation in parentheses is the error correction mechanism.

(Y t-1 - β 2 X t-1 ) = 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 two-step ECM

The Single Equation ECM

ΔY t = α + βΔX t-1 - βEC t-1 + ε t

∆Y t = β 0 ∆X t-1 - β 1 Z t-1

Where Z t = Y t - βX t - α

∆Y t = α + β 0 *∆X t - β 1 (Y t-1 - β 2 X t-1 ) + ε t

The Single Equation ECM

∆Y t = α + β 0*∆X t - β 1(Y t-1 - β 2X t-1) + 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.

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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 t-1 + 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

� 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 t-1 + β 1X t + β 2X t-1 + 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 t-1 - βX 1t-1 - βX 2t-1 - βX 3t-1 ) + 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 two-step estimator.

Single Equation ECMs

� Single Equation ECMs

� Provide the same information about the rate of error correction as the

Engle and Granger two-step method.

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?

Y t = α + β 0 Y t-1 + β 1 X t + β 2 X t-1 + ε t

∆Y t = α + (β 0 - 1)Y t-1 + β 1 X t + β 2 X t-1 + ε t

∆Y t = α + (β 0 - 1)Y t-1 + β 1 ΔX t + (β 1 + β 2 )X t-1 + ε t

Where φ 0 = β 0 - 1 and φ 1 = β 1 + β 2

∆Y t = α + φ 0 Y t-1 + β 1 ΔX t + φ 1 X t-1 + ε t

We can rewrite this equation in error correction form as

∆Y t = α + β 1 ΔX t - φ 0 (Y t-1 - φ 1 X t-1 ) + ε t

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We can see that the ADL model provides information similar to the ECM.

Y t = α + β 0 Y t-1 + β 1 X t + β 2 X t-1 + ε t

β 0 estimates the proportion of the deviation from equilibrium at t-1 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 t-1 in the ECM)

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

Y t = α + β 0 Y t-1 + β 1 X t + β 2 X t-1 + 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

∆Y t = α + β 0∆X t - β 1(Y t-1 - β 2X t-1) + ε t

Y t = α + β 0Y t-1 + β 1X t + β 2X t-1 + 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

Dickey-Fuller test for unit root Number of obs = 55

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -3.353 -3.573 -2.926 -2.598

------------------------------------------------------------------------------

MacKinnon approximate p-value 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 t-1 - β 2X t-1) + ε t

∆Y t = α + β 0∆X t + β 1Y t-1 + β 2X t-1 + 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 t-1 + .52*X t-1 + ε t

Which we can write in error correction form as

∆Y t = 13.12 + 1.32*∆X t -.42(Y t-1 - 1.22*X t-1 ) + 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 t-1 - 1.22*X t-1 ) + 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 R-squared = 0.5572

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 t-1 - 1.22*X t-1) + 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 t-1 - 1.22*X t-1 ) + 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 t-1 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 t-1 + 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 R-squared = 0.5572

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 R-squared = 0.7374

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 t-1 + β 1 X t + β 2 X t-1 + 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 R-squared = 0.7374

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

------------------------------------------------------------------------------

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Equivalence of the EC and ADL models

The results imply the equation Y t = 13.12 + .58*Y t-1 + 1.32*X t -.81*X t-1 + 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 two-step 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

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 t-1 + 1.32*X t -.81*X t-1 + 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

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