An Introduction to Error Correction Models An Introduction to ECMs ...

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 


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. 


� 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. 


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 



(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. 


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. 



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



• 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 



• 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 


� 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. 


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 


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. 


� 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. 


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 . 


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 (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. 


� 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. 


The basic structure of the ECM 


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: 


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. 


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 ) 


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


� 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. 


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 


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




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

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

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


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

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 

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


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 

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


t 

t 

t 

t 

7

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 

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

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 





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

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

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


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 

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

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 

-------------+------------------------------ 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 

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

-15 -10 -5 0 5 10 

Residuals 

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

months 

8


� 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 


-------------+------------------------------ F( 4, 57) = 2.97 

Model | 69.5277246 4 17.3819311 Prob > F = 0.0270 


-------------+------------------------------ 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 

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


regress dif_Y dlag_X lag_r 


-------------+------------------------------ 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 


-------------+---------------------------------------------------------------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 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. 


regress dif_X l.dif_X l.dif_Y lag_X lag_Y 


-------------+------------------------------ F( 4, 57) = 5.87 

Model | 74.2042429 4 18.5510607 Prob > F = 0.0005 


-------------+------------------------------ Adj R-squared = 0.2420 

Total | 254.387097 61 4.17028027 Root MSE = 1.7779 

-----------------------------------------------------------------------------dif_X 


-------------+---------------------------------------------------------------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 

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 


� 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 


� 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. 


Basic form of the ECM 

Engle and Granger two-step ECM 



∆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 


∆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. 

11


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 

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 t-1 + β 1X t + β 2X t-1 + ε t 


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 

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

Engle and Granger two-step 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 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 

12


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


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) 

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 



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 t-1 - β 2X t-1) + ε t 

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


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 

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


0 

13


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. 


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 


∆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. 


regress dif_y dif_x lag_y lag_x 


-------------+------------------------------ 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 


-------------+---------------------------------------------------------------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 

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


∆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 


∆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 





� 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 


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 


-------------+------------------------------ 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 


-------------+---------------------------------------------------------------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 

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


� 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 


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 


-------------+------------------------------ 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 


-------------+---------------------------------------------------------------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 


-------------+------------------------------ 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 


-------------+---------------------------------------------------------------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 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). 


� 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 

� 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 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 

16

An Introduction to Error Correction Models An Introduction to ECMs ...

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