An Introduction to Error Correction Models

8/2/2019 An Introduction to Error Correction Models

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An Introduction to

Error Correction Models

Robin Best

Oxford Spring School for QuantitativeMethods in Social Research

2008

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 ofone time series on another.

Thus, they often mesh well with our theories of political and socialprocesses.

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.


The basic structure of an ECM

Yt = + Xt-1 - ECt-1 + t

Where EC is the error correction component of the model and measures the speedat which prior deviations from equilibrium are corrected.

Error correction models can be used to estimate the following quantities ofinterest 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.


As we will see, the versatility of ECMs give them a number of desirableproperties.

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.

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

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


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ECMs and Cointegration:Stationary vs. Integrated Time Series

Stationary time series data are mean reverting. That is, they have afinite mean and variance that do not depend on time.

Yt = + Yt-1 + t

Where | p | < 1 and t is also stationary with a mean of zero and variance 2

Note that when 0 < | p | < 1 the time series is stationary but contains

autocorrelation.

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 tis the value at time t-1

Have a (theoretically) infinite variance and no mean.

ECMs and Cointegration:Integrated Time Series

Formally, an integrated series can be expressed as a function of allpast disturbances at any point in time.

Or Yt = + Yt-1 + tWhere p = 1

Or Yt

- Yt-1

= utWhere ut = t

And t is still a stationary process

=

t

i

it eY

1


Order of Integration

Integrated time series data that are stationary after being d ifferencedtimes 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


(Theoretical) Sources of integration

The effect of past shocks is permanently incorporated into the

memory of the series.

The series is a function of other integrated processes.

A Drunks Random Walk

0 20 40 60time


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Analyzing integrated time series in level form dramatically increases thelikelihood of making a Type-II error.

Problem of spurious associations.

High R2

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


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 theeffects 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 error-

correction mechanism, then the stochastic trends of the time series willbe correlated with one another.

Cointegration

ECMs and Cointegration

Two time series are cointegratedif

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 eachother, maintaining an equilibrium relationship.


Lets go back to the drunks random walk and call the drunk X. Therandom walk can be expressed as

Xt - Xt-1 = ut

Where utrepresents the stationary, white-noise shocks.

Another rather trivial example of a random walk is the walk (or jaunt) of adog, which can be expressed as

Yt - Yt-1 = wt Where wtrepresents the stationary, while-noise process of the dogs

steps.

A Dogs Random Walk

0 20 40 60time


But what if the dog belongs to the drunk?

Then the two random walks are likely to have an equilibrium relationship and tobe 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 willcorrect deviations the equilibrium

Xt - Xt-1 = ut + c(Yt-1 - Xt-1)

Yt - Yt-1 = wt + d(Xt-1 - Yt-1)

Where the terms in parentheses are the error correcting mechanisms


4/16

The Drunk and Her Dog

0 20 40 60time

drunk dog


Two I(1) time series (Xt and Yt) are cointegrated if there is some linearcombination that is stationary.

Zt = Yt - Xt

Where Z is the portion of (levels of) Y that are not shared with X: the equilibriumerrors.

We can also rewrite this equation in regression form

Yt = Xt + Zt

Where the cointegrating vector - Zt - can be obtained by regressing Yt on Xt.


Yt = Xt + Zt

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 thedegree 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.


Yt will be a function of the degree to which the two time series were out ofequilibrium in the previous period: Zt-1

Zt-1 = Yt-1 - Xt-1

When Z = 0 the system is in its equilibrium state

Yt will respond negatively to Zt-1.

If Z is negative, then Y is too high and will be adjusted downward in the nextperiod.

If Z is positive, then Y is too low and will be adjusted upward in the next timeperiod.


We might theorize that shocks to X have two effects on Y.

Some portion of shocks to X might immediately affect Y in the next time

period, so that Yt responds to Xt-1.

A shock to Xt will also disturb the equilibrium between Y and X, sending Y

on a long term movement to a value that reproduces the equilibrium stategiven the new value of X.

Thus Yt is a function of both Xt-1 and the degree to which the two

variables were out of equilibrium in the previous time period.

Engle and Granger Two-Step ECM

If two time series are integrated of the same order AND some linearcombination of them is stationary, then the two series are cointegrated.

Cointegrated series share a stochastic component and a long termequilibrium relationship.

Deviations from this equilibrium relationship as a result of shocks will becorrected over time.

We can think of Yt as responding to shocks to X over the short and long

term.


5/16


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 Yt on Zt-1 plus any relevant short term

effects.


Step 1:

Yt = + Xt + Zt

The cointegrating vector - Z - is measured by taking the residuals from theregression of Yt on Xt

Zt = Yt - Xt -

Step 2:

Regress changes on Y on lagged changes in X as well as the equilibrium errorsrepresented by Z.

Yt = 0Xt-1 - 1Zt-1

Note that all variables in this model are stationary.


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 areusually regarded as exogenous shocks and often take the form of

dummy variables.


The cointegrating regression is performed as Yt = + Xt + Zt

Which we can also conceptualize as

Zt = Yt - ( +Xt)

If we add a series ofjexogenous shocks - represented as wj

Yt = + Xt+ W1t + W2t +W3t + Zt

Then

Zt = Yt - ( +Xt + W1t + W2t +W3t)


The basic structure of the ECM

Yt = + Xt-1 - ECt-1 + t

In the Engle and Granger Two-Step Method the EC component is derived fromcointegrated time series as Z.

Yt = 0Xt-1 - 1Zt-1

0 captures the short term effects of X in the prior period on Y in the current period.

1 captures the rate at which the system Y adjusts to the equilibrium state after ashock. In other words, it captures the speed of error correction.


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 regressingYt on Xt and taking the residuals.

4) Enter the lagged residuals - Z - into a regression of Yt on Xt-1.


6/16


Viewed from this perspective, it is easy to see why error correctionmodels have become so closely associated with cointegration (we will

come back to this later).

Integrated time series present a problem for time series analysis - atleast in terms of long term relationships.

When integrated time series variables are also cointegrated, errorcorrection models provide a nice solution to this problem.

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 cointegrationamong 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?

Cointegration and Error

Correction in Political Science

Prime Ministerial Statisfaction (U.K.) and Conservative PartySupport

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

The Engle and Granger Two-StepECM: 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.

X and Y: Cointegrated?

0

5

10

15

20

25

1960m1 1961m1 1962m1 1963m1 1964m1 1965m1months

Y X


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 ourtime 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.


7/16

Dickey-Fuller Tests

Basic Dickey-Fuller test

With a constant (drift)

With a time trend

ttttxx ++= 1

tttxx += 1

tttttxx +++= 1

Dickey-Fuller Tests

Basic Dickey-Fuller test

With a constant (drift)

With a time trend

If X is a random walk process, then = 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 will be large, making it likely thatwe fail to reject the null when we really should

ttttxx ++= 1

tttxx += 1

tttttxx +++= 1

Augmented Dickey-Fuller

We can remove any remaining serial correlation in t by introducing anappropriate number of lagged differences of X in the equation.

Where i = 1, 2, kNull hypotheses are the same as the DF tests

t

k

i

ititttxxx +++=

=

1

11

t

k

i

itittxxx ++=

=

1

11

Is X Integrated?

dfuller X, 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) -1.852 -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?

dfuller X, lags(4) regress

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


Test 1% Critical 5% Critical 10% CriticalStatistic Value Value Value

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

Z(t) 0.690 -3.567 -2.923 -2.596

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


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

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

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

Is X Integrated?

If X is I(1), then the first difference of X should be stationary.

dfuller dif_X





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

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

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



8/16

Is Y Integrated?

dfuller Y, regress





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

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

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


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

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

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

Is Y Integrated?dfuller dif_Y, regress





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

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

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


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

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

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

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, t should be stationary and should be

statistically significant and in the expected direction.

Yt = t + Xt +t

Lets see if this is the case

Cointegrating Regression

regress Y X

So ur ce | SS d f MS Nu mb er 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

-------------+------------------------------ Adj R-squared = 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

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

Cointegrating Regression

predict r, resid

dfuller r





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

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

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


-15

-10

-5

0

5

10

Residuals

1960m1 1961m1 1962m1 1963m1 1964m1 1965m1months


9/16


Our residuals from the cointegrating regression capture deviations fromthe 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 asour measure of the error correction mechanism.

Yt = + 1*Xt-1 + 2*Rt-1 +t

Engle and Granger Two-Step ECMregress dif_Y dlag_X lag_r

So ur ce | SS d f MS Nu mb er of obs = 62

-------------+------------------------------ F( 2, 59) = 5.09

Model | 59.4494524 2 29.7247262 Prob > F = 0.0091


-------------+------------------------------ Adj R-squared = 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 thatdeviations from equilibrium are corrected at about 32% per month.

However, X does not appear to have significant short term effects on Y.

Granger Causality and ECMs

Granger Causality:

A variable - X Granger causes another variable Y if Y can bebetter predicted by the lagged values of both X and Y than by the laggedvalues of Y alone (see Freeman 1983).

Standard Granger causality tests can result in incorrect inferences aboutcausality when there is an error correction process.

The Engle-Granger approach to ECMs begins by assuming all variablesin the cointegrating regression are jointly endogeneous.

Thus, in the previous example we should also estimate a cointegratingregression of X on Y.

Granger Causality

Granger causality can be ascertained in the ECM framework byregressing 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 Causalityregress dif_Y l.dif_Y l.dif_X lag_Y lag_X

So ur ce | SS d f MS Nu mber 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-------------+------------------------------ Adj R-squared = 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

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

Granger Causalityregress dif_X l.dif_X l.dif_Y lag_X lag_Y

So ur ce | S S df MS Nu mb er o f ob s = 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-------------+------------------------------ Adj R-squared = 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

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


10/16

ECMs, Causality, and Theory

In the social sciences, our theories (usually) tell us which time seriesshould be on the left side of the equation and which should be on theright.

The Engle and Granger approach assumes endogeneity between thecointegrating time series.

Engle and Granger Two-StepTechnique: Issues and Limitations

Does not clearly distinguish dependent variables from independentvariables.

In the social sciences the Engle and Granger two-step ECM might not beconsistent with our theories.

Is appropriate when dealing with cointegrated time series.

Can we clearly distinguish between integrated and stationary processes?

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 arenot truly (co)integrated.

I(1) processes may be incorrectly included into the cointegratingregression - 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 areintegrated when they are not.

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 finitesamples.)

Fractionally integrated (0 < p < 1, where when 0 < p < .5 the data aremean-reverting and have finite variance, and when .5 p < 1 the data aremean-reverting but have infinite variance)

A combined process of both stationary and integrated data

Aggregated data

More Integration Issues

Under these conditions, we are likely to draw faulty inferences from thetwo-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

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 ECMsrequire weak exogeneity.


11/16

Single EquationError Correction Models

Following theory, Single Equation ECMs clearly distinguish betweendependent and independent variables.

Single Equation ECMs are appropriate for both cointegrated and long-memoried, but stationary, data.

Cointegration may imply error correction, but does error correction implycointegration?

Single Equation ECMs estimate a long term effect for each independentvariable, allowing us to judge the contribution of each.

Allow for easier interpretation of the effects of the independent variables.

Single Equation ECMs

Our theories might specify long and shor t term effects of independentvariables 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.


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:

Yt = + 0*Xt - 1(Yt-1 - 2Xt-1) + t

The Single Equation ECM

Basic form of the ECM

Yt = + Xt-1 - ECt-1 + t

Engle and Granger two-step ECM

Yt = 0Xt-1 - 1Zt-1

Where Zt = Yt - Xt -


Yt = + 0*Xt - 1(Yt-1 - 2Xt-1) + t


Yt = + 0*Xt - 1(Yt-1 - 2Xt-1) + t

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

(Yt-1 - 2Xt-1) = 0 when Y and X are in their equilibrium state

0 estimates the short term effect of an increase in X on Y

1 estimates the speed of return to equilibrium after a deviation.

If the ECM approach is appropriate, then -1 < 1 < 0

2 estimates the long term effect that a one unit increase in X has on Y. This longterm effect will be distributed over future time periods according to the rate oferror correction - 1


Yt = + 0*Xt - 1(Yt-1 - 2Xt-1) + t

The values for which Y and X are in their long term equilibrium relationship are

Y = k0 + k1XWhere

And

Where k1 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 k1sstandard error, and therefore statistical significance.

1

2

1

=k

1

0

=k


12/16


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.

Yt = + 0Yt + 1Xt - 2Xt + t

Where 1 is the long term effect and is estimated with a standard error

Notice the problem: we have Yt on the right side of the equation

We can proxy Yt as:

Yt = + Yt-1 + Xt + Xt + t

And use our predicted values of Yt in the Bewley transformation regression


We can easily extend the single equation ECM to include more

independent variables

Yt = + X1t + X2t + X3t - (Yt-1 - X1t-1 - X2t-1 - X3t-1) + t

Note that each independent variable is now forced to make anindependent contribution to the long term relationship, solving one ofthe problems in the two-step estimator.

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



Provide the same information about the rate of error correction as theEngle and Granger two-step method.

Provide more information about the long term effect of each independentvariable - including its standard error - than the Engle and Granger two-step method.

Illustrate that ECMs are appropriate for both cointegrated and stationarydata.

How do we know Single Equation ECMs are appropriate with

stationary data?

ECMs and ADL Models

We know Autoregressive Distributive Lag models are appropriate forstationary 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:

Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

ECMs and the ADL

Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

Yt = + (0 - 1)Yt-1+ 1Xt + 2Xt-1 + t

Yt = + (0 - 1)Yt-1+ 1Xt + (1 + 2)Xt-1 + t

Yt = + 0Yt-1 + 1Xt + 1Xt-1 + t

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

We can rewrite this equation in error correction form as

Yt = + 1Xt - 0(Yt-1 - 1Xt-1) + t


13/16

ECMs and the ADL

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

Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

0 estimates the proportion of the deviation from equilibrium at t-1 that is maintainedat time t. 0 - 1 tells us the speed of return.

1 estimates the short term effect of X on Y

1 + 2 estimates the long term effect of a unit change in X on Y (the coefficient onXt-1 in the ECM)

ECMs and the ADL

Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

And the total long term effect/long run multiplier - k1 - is therefore:

Y and X will be in their long term equilibrium state when Y = k0 + k1X

where

0

12

11

+=k

0

01

=k

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 correctionmodels. 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 dependentvariable.

The EC and ADL Models: Notation

Lets use the following notation for the single equation ECM and the ADL

ECM

Yt = + 0Xt - 1(Yt-1 - 2Xt-1) + t

ADL

Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

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 dont have reason to suspect cointegration from a theoreticalstandpoint.

But we believe X and Y share a long term equilibrium relationship

Single Equation ECM

We determine that our Y variable is stationary (with 95% confidence), ruling out an

ECM based on cointegration

dfuller y, regress





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

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

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



14/16

Single Equation ECM

We then estimate the single equation ECM

Yt = + 0Xt - 1(Yt-1 - 2Xt-1) + t

As

Yt = + 0Xt + 1Yt-1 + 2Xt-1 + t

If our error correction approach is correct, then 1 should be -1 < 1 < 0 and

significant.

Single Equation ECMregress dif_y dif_x lag_y lag_x

S ou rc e | SS df M S N um ber of obs = 5 5

-------------+------------------------------ F( 3, 51) = 21.40

Model | 238.216589 3 79.4055296 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 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

The results indicate the following equation

Yt = 13.12 + 1.32*Xt -.42*Yt-1 + .52*Xt-1 + t

Which we can write in error correction form as

Yt = 13.12 + 1.32*Xt -.42(Yt-1 - 1.22*Xt-1) + t

Where 1.22 is our calculation of the long run multiplier

Single Equation ECM

Yt = 13.12 + 1.32*Xt -.42(Yt-1 - 1.22*Xt-1) + 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

Yt = + 1.32*Xt -.42(Yt-1 - 1.22*Xt-1) + 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 theirequilibrium 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

Yt = + 1.32*Xt -.42(Yt-1 - 1.22*Xt-1) + 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 thesetwo variables, causing Y to be too low.

Y will respond by increasing a total of 1.22 points, spread over future time periods ata rate of 42% per time period. Y will increase .52 points at t

Then another .3 points at t+1





Until the change in X at t-1 has virtually no effect on Y


15/16

0

.5

1

1.5

ChangeinY

0 2 4 6Time Period

1

1.5

2

2.5

Y

0 2 4 6Time Period

Single Equation ECM

We can determine the standard error and confidence level of the total long termeffect of X on Y through the Bewley transformation regression.

First, we can obtain an estimate of Y by estimating Yt = + Yt-1 + Xt + Xt + 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


-------------+------------------------------ Adj R-squared = 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

And take the predicted values of Yt to estimate Yt = + 0Yt + 1Xt - 2Xt + 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


-------------+------------------------------ Adj R-squared = 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

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

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?

Equivalence of the EC and ADL models

First, lets estimate Yt = + 0Yt-1 + 1Xt + 2Xt-1 + t

regress y lag_y x lag_x

Source | SS df M S Number of obs = 5 5

-------------+------------------------------ F( 3, 51) = 47.74

Model | 531.551105 3 177.183702 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 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

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


16/16

Equivalence of the EC and ADL models

The results imply the equation Yt = 13.12 + .58*Yt-1 + 1.32*Xt -.81*Xt-1 + 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).

Equivalence of the EC and ADL Models

Yt= 13.12 + .58*Y

t-1+ 1.32*X

t-.81*X

t-1+

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

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 dont require cointegration and ease interpretation ofcausal relationships.

Single equation ECMs and ADL models

Equivalence: ADL models can provide the same information about shortand long term effects.

Standard error for the long term effects of independent variables isrelatively easy to obtain in the single equation ECM

Documents

An Introduction to Error Correction Models