An Introduction to ECMs
An Introduction to Error Correction Models
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 equilibrium after after a change in an independent independent variable - X.
Robin Best Oxford Spring School for Quantitative Methods in Social Research 2008
ECMs are useful for for estimating both short term and long term effects effects of one time series on another. •
Thus, they often mesh well with our theories of political and social processes.
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Theoretically-driven 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 The basic structure of an ECM
An Introduction to ECMs
∆Yt = α + β∆X β∆Xt-1 - βECt-1 + εt 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.
As we will see, the versatility of ECMs give them a number of desirable properties. •
Estimates of short and long term effects
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Easy interpretation of short and long term effects
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Applications to both integrated and stationary time series data
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Can be estimated with OLS
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Model theoretical relationships
ECMs can be appropriate whenever (1) we have time time series data and (2) are interested in both short and long term relationships between multiple time series.
Applications of Applications of ECMs in the (Political Science) Literature
Overview of the Course I.
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U.S. Presidential Approval/ U.K. Prime Ministerial Satisfaction
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Policy Mood/Policy Sentiment
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Support for Social Security
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2-step error correction estimators
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Consumer Confidence
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Stata session session #1
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Economic Expectations
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Health Care Cost Containment/ Government Spending /Patronage Spending / Redistribution
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Interest Rates/ Purchasing Power Parity
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Growth in (U.S.) Presidential Staff
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Arms Transfers
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U.S. Judicial Influence
Motivating Motivating ECMs with cointegrated cointegrated data
II.
Integration and cointegration
Motivating ECMs with stationary data •
The single equation ECM
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Interpretation of long and short term effects
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The Autoregressive Distributive Lag (ADL) model
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Equivalence of the ECM and ADL
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Stata session session #2
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ECMs and Cointegrat Cointegration: ion: Stationary vs. Integrated Time Series
Stationary time series data are mean reverting. That is, they have a finite mean and variance that do n ot depend on time.
Yt = α + ρYt-1 + εt Where | p | < 1 and εt is also stationary with a mean of zero and variance σ2
Note that when 0 < | p | < 1 the time series is stationary stationary but contains autocorrelation.
ECMs and Cointegrat ECMs Cointegration: ion: 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 ng
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appear to be on a ‘random walk’
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Have current values that can be expressed as the sum of all previous changes
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The effect of any shock is permanently incorporated incorporated into the series
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Thus, the best predictor of the series at time t is t is the value at time t-1
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Have a (theoretically) infinite variance and no mean.
ECMs and Cointegrat Cointegration: ion: Integrated Time Series
Formally, an integrated series can be expressed as a function of of all past disturbances at any point in time. t
Y t
∑e
ECMs and Cointegrat ECMs Cointegration: ion: Integrated Time Series Order of Integration
Integrated time series data that are stationary after being d ifference I(d ) d times d times are Integrated of order d : I(d
For our purposes, we focus on time series d ata that are I(1). • Data that are stationary after being first-differenced.
I(1) processes are fairly common in time series data
i
i =1
Or Yt = α + ρYt-1 + εt Where p = 1
Or Yt - Yt-1 = ut Where ut = εt And εt is still a stationary process
ECMs and Cointegrat Cointegration: ion: Integrated Time Series
A Drunk’s Random Walk
(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.
0
20
40
60
time
2
ECMs and Cointegrat Cointegration: ion: Integrated Time Series •
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Analyzing time series data in differenced form solves the spurious regression problem, but may “throw the baby out with the bathwater.”
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A model that includes only (lagged) differenced variables assumes the effects of the X variables on Y never last longer than one time period.
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What if our time series share a long run relationship?
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If the time series share an equilibrium r elationship with an errorcorrection mechanism, then the stochastic trends of the time series will be correlated with one another.
Analyzing integrated time series in level form dramatically increases the likelihood of making a Type-II error.
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ECMs and Cointegrat ECMs Cointegration: ion: Integrated Time Series
Problem of spurious associations. R2
High
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
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ECMs and Coin Cointegra tegration tion
ECMs and Coin Cointegra tegration tion
Two time series are cointegrated if cointegrated if
Cointegration
Lets go back to the drunk’s random walk and call the drunk X. The random walk can be expressed as
Both are integrated of the same order. Xt - Xt-1 = u t
There is a linear combination of the two time series that is I(0) - i.e. stationary.
Where u t represents the stationary, white-noise shocks.
Another rather trivial example of a ra ndom walk is the walk (or jaunt) of a dog, which can be expressed as
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
Yt - Yt-1 = w t Where w t represents the stationary, while-noise process of the dog’s steps.
ECMs and Coin Cointegra tegration tion But what if the dog belongs to th e drunk?
Then the two random walks are likely to have an equilibrium relationship and to be cointegrated (Murray 1994). 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
Xt - Xt-1 = ut + c(Yt-1 - Xt-1) Yt - Yt-1 = wt + d(Xt-1 - Yt-1) 0
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Where the terms in parentheses are the error correcting mechanisms
time
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The Drunk and Her Dog
ECMs and Coin Cointegra tegration tion Two I(1) time series (Xt and Yt) are cointegrated if there is some linear combination that is stationary. Zt = Yt - βXt 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
Yt = βXt + Zt 0
20
40
60
time drunk
Where the the cointegr cointegrating ating vector vector - Z t - can be obtained by regressing Yt on Xt.
dog
ECMs and Coin Cointegra tegration tion Yt = βXt + Zt
ECMs and Coin Cointegra tegration tion ∆Yt will be a function of the degre e to which the two time series were out of equilibrium in the previous period: Zt-1
Here, Z represents the portion of Y (in levels) that is not attributable to X.
In short, Z will capture the error correction rel ationship by capturing the degree to which Y and X are out of equilibrium.
Zt-1 = Yt-1 - Xt-1
When Z = 0 the system is in its equilibrium state
Yt will respond negatively to Z t-1.
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 Coin Cointegra tegration tion
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 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 ∆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.
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 coi ntegrated. 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 ∆Yt as responding to shocks to X over the short and long term.
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Engle and Granger Two-Step ECM
Engle and Granger Two-Step ECM Step 1: Yt = α + βXt + Zt
Engle and Granger (1987) suggested an appropriate model for Y, based two or more time series that are cointegrated.
The cointegrating vector - Z - is measured by taking the residuals from the regression of Yt on Xt
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
Zt = Yt - β Xt - α
Step 2: effects.
Regress changes on Y on lagged changes in X as well as the equilibrium errors represented by Z. ∆Yt
= β0∆Xt-1 - β1Zt-1
Note that all variables in this model are stationary.
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
The cointegrating regression is performed as Yt = α + βXt + Zt Which we can also conceptualize as Zt = Yt - (α +βXt)
1) be cointegrated 2) have sustained shocks on the equilibrium.
Engle and Granger Two-Step ECM
If we add a series of j of j exogenou exogenous s shocks - represented represented as w j
The variables that have sustained shocks on the equilibrium are usually regarded as exogenous shocks an d often take the form of dummy variables.
Yt = α + βXt+ βW1t + βW2t +βW3t + Zt Then Zt = Yt - (α +βXt + βW1t + βW2t +βW3t)
Engle and Granger Two-Step ECM
Engle and Granger Two-Step ECM
The basic structure of the ECM
Note that the Engle and Granger 2-Step method is really a 4-step method. ∆Yt = α + β∆X β∆Xt-1 - βECt-1 + εt In the Engle and Granger Two-Step Method the EC component is derived from cointegrated time series as Z.
1) Determine that all time series are integrated of the same order. 2) Demonstrate that the time series are cointegrated
∆Yt
= β0∆ Xt-1 - β1Zt-1
β0 captures the short term effects of X in the prior period on Y in the current period.
3) Obtain an estimate estimate of the cointegrati cointegrating ng vector - Z - by regressing regressing Yt on Xt and taking the residuals.
β1 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. correction.
4) Enter the lagged lagged residuals - Z - into a regression regression of ∆Yt on ∆Xt-1.
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Cointegrat Coint egration ion and Error Correction Correction
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.
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?
Cointegration Cointegratio n and Error Correction in Political Science
The Engle and Granger Two-Step ECM: Putting itit into Practice
Prime Ministerial Statisfaction (U.K.) and Conservative Party Support
Lets imagine we have two time series - perhaps the drunk and her dog but lets call the drunk ‘X’ and the dog ‘Y’.
Arms transfers by the U.S. and Soviet Union
From a theoretical perspective, we believe changes in X will have both short and long term effects on Y, since we e xpect X and Y to have an
Economic expectations and U.S. Presidential Approval
U.S. Domestic Policy Sentiment and Economic Expectations
Support for U.S. Social Security and the St ock Market
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?
Engle and Granger Two-Step ECM
5 2
First, we need to determine that both X and Y are integrated of the same order. •
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•
5 1
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.
0 1
Tests for unit-root process tend to be controversial, controversial, primarily due to their low power. 5
For our purposes, we will focus on Dickey-Fuller (DF) and Augmented Dickey-Fuller 0
196 0m1
tests to examine the (non)stationarity (non)stationarity of our time series. 1 9 6 1m 1
1962m 1
19 63m 1 months Y
196 4m 1
19 65m1
X
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Dickey-Fuller Tests
Dickey-Fuller Tests ∆ xt = γ xt −1 + ε t
Basic Dickey-Fuller test
∆ xt = γ xt −1 + ε t
Basic Dickey-Fuller test
With a constant (drift)
∆ xt = α t + γ xt −1 + β t + ε t
With a time trend
∆ xt = α t + γ xt −1 + ε t
With a constant (drift)
∆ xt = α t + γ xt −1 + ε t
If X is a random walk process, then γ = γ = 0 The null hypothesis is that X is a random walk
∆ xt = α t + γ xt −1 + β t + ε t
With a time trend
MacKinnon values for statistical significance Note that in small samples the standard error of γ will γ will be large, making it likely that we fail to reject the null when we really should
Augmented Dickey-Fuller
Is X Integrated? dfuller X, regress regress
We can remove any remaining serial correlation in εt by introducing an appropriate number of lagged differences of X in the equation.
Dickey-Fuller Dickey-Fuller test for unit root
k
xt −1 + ∑ β i ∆x1t −i + ε t ∆ xt = γ i =1 k
xt −1 + ∑ β i ∆x1t −i + ε t ∆ xt = α t + γ
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-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Where i = 1, 2, …k Null hypotheses are the same as the DF tests
Is X Integrated?
Is X Integrated? If X is I(1), then the first difference of X should be stationar y.
regress regress
Augmented Augmented Dickey-Fulle Dickey-Fuller r test for unit root
=
-----------------------------------------------------------------------------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
i =1
dfuller X, lags(4) lags(4)
Number of obs
---------- 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
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 ------------------------------------------------------------------------------
dfuller dfuller dif_X dif_X Dickey-Fuller test for unit root
Number 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
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Is Y Integrated?
Is Y Integrated? dfuller dif_Y, dif_Y, regress
dfuller Y, regress regress Dickey-Fuller test for unit root Dickey-Fuller test for unit root
Number Number of obs
=
Number Number of obs
=
62
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 ------------------------------------------------------------------------------
---------- 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] -------------+---------------------------------------------------------------d if if _Y _Y | L1. | -1.159903 .1278662 -9.07 0.000 -1.415674 -.9041329 _cons | .2219184 .3259962 0.68 0.499 -.4301711 .8740078 ------------------------------------------------------------------------------
Cointegration
Cointegra Coin tegrating ting Regre Regression ssion
Both X and Y appe ar to be integrated of the same order: I(1).
If they are cointegrated, then they share stochastic trends.
In the following regression, εt should be stationary and β should be
regress Y X So ur ce | SS df MS - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Model | 1009.22604 1009.22604 1 1009.22604 1009.22604 Residual | 676.523964 62 10.9116768 - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total | 1685.75 63 26.7579365
statistically significant and in the expected direction.
Nu mb er of obs F( 1, 6 2) 2) Prob > F R-squared A dj dj R -s -s qu qu ar ar ed ed Root MSE
= = = = = =
64 9 2. 2. 49 49 0.0000 0.5987 0 .5 .5 92 92 2 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 ------------------------------------------------------------------------------
Yt = αt + βXt +εt Lets see if this is the case
0 1
Cointegratin Cointe grating g Regre Regression ssion 5 predict r, resid
d f ul ul le le r r Dickey-Fuller 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 ------------------------------------------------------------------------------
s 0 l a u d i s e R 5 -
0 1 -
MacKinnon approximate p-value for Z(t) = 0.0000 5 1 -
1960m1
1 9 6 1m 1
1 962m1
1 9 6 3m 1 months
1 9 6 4m 1
1965m 1
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Engle and Granger Two-Step ECM
Engle and Granger Two-Step ECM regress regress dif_Y dif_Y dlag_X dlag_X lag_r lag_r
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.
∆Yt = α + β 1*∆Xt-1 + β 2*Rt-1 +εt
So ur ce | SS df MS - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Model | 59.4494524 59.4494524 2 29.7247262 29.7247262 Residual | 344.227967 59 5.83437232 - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total | 403.677419 61 6.61766261
Nu mb er of obs = 62 F( 2, 5 9) 9) = 5 .0 .0 9 Prob > F = 0.0091 R-squared = 0.1473 A dj dj R -s -s qu qu ar ar ed ed = 0 .1 .1 18 18 4 Root MSE = 2.4154
-----------------------------------------------------------------------------dif_Y | Coef. Std. Err. t P>|t| [95% Conf. Interval] 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 and ECMs Granger Causality:
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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.
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If an EC representation is appropriate, then i n at least one of the regressions:
A variable variable - X – Granger causes another variable variable – Y – if Y can be better predicted by the lagged val ues 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
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 regress dif_Y dif_Y l.dif_ l.dif_Y Y l.dif_ l.dif_X X lag_Y lag_X So ur ce | SS df MS - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Model | 69.5277246 4 17.3819311 Residual | 334.149695 57 5.86227535 - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total | 403.677419 61 6.61766261
r eg eg r es s d if if _X _X Nu mb mber of obs = 62 F( 4, 5 7) 7) = 2 .9 .9 7 Prob > F = 0.0270 R-squared = 0.1722 A dj dj R -s -s qu qu ar ar ed ed = 0 .1 .1 14 14 1 Root MSE = 2.4212
-----------------------------------------------------------------------------dif_Y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------d if if _Y _Y | L1. | .0483244 .1399056 0.35 0.731 -.2318318 .3284806 d if if _X _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 ------------------------------------------------------------------------------
l .d .d if if _X _X l .d .d i f_ f_ Y l ag ag _X _X
l ag ag _Y _Y
So ur ce | SS df MS - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Model | 74.2042429 74.2042429 4 18.5510607 18.5510607 Residual | 180.182854 57 3.1611027 - --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total | 254.387097 61 4.17028027
Nu mb er o f ob s = F( 4, 5 7) 7) Prob > F R-squared A dj dj R -s -s qu qu ar ar ed ed Root MSE
= = = = =
62 5 .8 .8 7 0.0005 0.2917 0 .2 .2 42 42 0 1.7779
-----------------------------------------------------------------------------dif_X | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------d if if _X _X | L1. | -.0640245 .132332 -0.48 0.630 -.3290147 .2009657 d if if _Y _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 ------------------------------------------------------------------------------
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Engle and Granger Two-Step T wo-Step Technique: Issues and Limitations
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 cointegrating time series.
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?
Integration Issues Error correction approaches that rely on cointegration of two or more I(1) I(1) time series become problematic when we are dealing with data that are not truly (co)integrated.
In the social sciences, we are more likely to have data that are
I(1) processes may be incorrectly included into the cointegrating regression regression - producing producing spurious associations associations - if two other I(1) cointegrated cointegrated time series are already included included (Durr 1992)
More Integration Issues
Near integrated (p 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
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.
Aggregated data
Integration Issues and ECMs
Under these conditions, we are often better off estimating a single equation ECM.
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
Single equation ECMs solve some of these problems and avoid others.
However, single equation ECMsrequire weak exogeneity.
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Single Equation Error Correction Models
Following theory, Single Equation ECMs clearly distinguish between dependent and independent variables.
Equation ECMs are appropriate appropriate for both cointegrated and long Single Equation
Single Equation ECMs
Our theories might specify long and shor t 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 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.
memoried, but stationary, data.
Cointegration may imply error correction, but does error correction imply cointegration?
effect for each independent Single Equation ECMs estimate a long term effect variable, allowing us to judge the contribution of each.
Allow for easier interpretation of the effects of the independent variables.
Single Equation ECMs
The Single Equation ECM Basic form of the ECM
Single Equation Error Correction Models are useful
When our theories dictate the causal relationships of interest
When we have long-memoried/stationary long-memoried/stationary data
∆Yt = α + β∆X β∆Xt-1 - βECt-1 + εt Engle and Granger two-step ECM ∆ Yt
A basic single equation ECM: ∆Yt
= β0∆Xt-1 - β1Zt-1
Where Zt = Yt - βXt - α
= α + β0*∆Xt - β1(Yt-1 - β2Xt-1) + εt The Single Equation ECM ∆Yt
The Single Equation ECM ∆Yt
The Single Equation ECM
= α + β0*∆Xt - β1(Yt-1 - β2Xt-1) + βεt
The portion of the equation in parentheses is the error correction mechanism. (Yt-1 - β2Xt-1) = 0 when Y and X are in their equilibrium state
= α + β0*∆Xt - β1(Yt-1 - β2Xt-1) + εt
∆Yt
= α + β0*∆Xt - β1(Yt-1 - β2Xt-1) + εt
The values for which Y and X are in their long term equilibrium relationship are Y = k0 + k1X α Where k 0 = β 1
β0 estimates the short term effect of an increase in X on Y
And β1 estimates the speed of return to equilibrium after a deviation.
If the ECM approach approach is appropriate, then -1 < β1 < 0 β2 estimates the long term 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 - β1
k 1
=
β 2 β 1
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 k 1’s standard error, and therefore statistical significance.
11
The Single Equation ECM Since the long term effect is a ratio of two coefficients, we could calculate its standard error using the variance and covariance matrix Alternatively, we can use the Bewley transformation transformation to estimate the standard error. This requires estimating the following regression.
The Single Equation ECM We can easily extend the single equation ECM to include more independent variables
∆Yt
= α + β∆X1t + β ∆X2t + β ∆X3t - β(Yt-1 - β X1t-1 - β X2t-1 - βX3t-1) + εt
Yt = α+ δ0∆Yt + δ1Xt - δ2∆Xt + µt Where δ1 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
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.
We can proxy ∆Yt as: ∆Yt =
α + β Yt-1 + βXt + β∆Xt + εt
And use our predicted values of
∆Yt
in the Bewley transformation regression regression
Single Equation Single Equation ECMs 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 Mode Models ls
Single Equation ECMs 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 twotwostep method.
Illustrate that ECMs ECMs are appropriate for both cointegrated cointegrated and stationary data.
How do we know Single Equation ECMs are appropriate with stationary data?
ECMs ECM s and the the ADL ADL Yt = α + β0Yt-1 + β1Xt + β2Xt-1 + εt
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.
∆ Yt
∆Yt
= α + (β (β0 - 1 )Y )Yt-1+ β1Xt + β2Xt-1 + εt
= α + (β (β0 - 1 )Y )Yt-1+ β1∆Xt + (β (β1 + β2)Xt-1 + εt
We can derive a single equation ECM from a general ADL model: ∆Yt
Yt = α + β0Yt-1 + β1Xt + β2Xt-1 + εt
= α + φ0Yt-1 + β1∆Xt + φ1Xt-1 + εt
Where φ0 = β0 - 1 and and φ1 = β1 + β2 We can rewrite this equation in error correction form as ∆Yt
= α + β1∆Xt - φ0(Yt-1 - φ1Xt-1) + εt
12
ECMs ECM s and the the ADL ADL
ECMs ECM s an and d the ADL
Yt = α + β0Yt-1 + β1Xt + β2Xt-1 + εt
We can see that the ADL model provides information similar to the ECM.
Yt = α + β0Yt-1 + β1Xt + β 2Xt-1 + εt
And the total long term effect/long effect/long run multiplier multiplier - k 1 - is therefore: therefore: k 1
=
β0 estimates the proportion of the deviation from equilibrium at t-1 that is maintained at time t . β0 - 1 tells us the the speed of of return. return. β1 estimates the short term effect of X on Y
β 2 + β 1 1 − β 0
Y and X will be in their long term equilibrium um state when Y = k 0 + k1X where
k 0
=
α β 0
1−
β1 + β2 estimates the long term effect of a unit change in X on Y (the coefficient on Xt-1 in the ECM)
ECMs and ADL Mode Models ls
The EC and ADL Models: Notation
What does this mean? Lets use the following notation for the single equation ECM and the ADL
isophormic to ADL models models ECMs are isophormic
We can use them with stationary data
ECM ∆Yt
Certain forms of ADL models models are - in a general sense - error correction correction Certain models. They can be used to 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.
ADL
Yt = α + β0Yt-1 + β1Xt + β2Xt-1 + εt
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 an d a long term effect on Y.
We don’t have reason to suspect cointegration from a theoretical standpoint.
= α + β0∆Xt - β1(Yt-1 - β2Xt-1) + εt
Single Equation ECM We determine that our Y variable is stationary (with 95% confidence), ruling out an ECM based on cointegration dfuller y, regress regress Dickey-Fuller Dickey-Fuller test for unit root
Number of obs
=
55
---------- Interpolated Dickey-Fuller --------Test Statistic
But we believe X and Y share a long term equilibrium relationship
Value
5% Critical Value
10% Critical Value
-----------------------------------------------------------------------------Z(t)
1% Critical
-3.353
-3.573
-2.926
-2.598
-----------------------------------------------------------------------------MacKinnon approximate p-value for Z(t) = 0.0127
13
Single Equation ECM
Single Equation ECM regress regress dif_y dif_y dif_x dif_x lag_y lag_y lag_x
We then estimate the single equation ECM
S ou rc e |
SS
df
MS
N um be ber of obs =
- --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- -
= α + β0∆Xt - β1(Yt-1 - β2Xt-1) + εt
∆Yt
F(
3,
55
5 1) 1) =
2 1. 1. 40 40 0.0000
Model |
238.216589
3
79.4055296
Prob > F
=
Residual |
189.278033
51
3.71133398
R-squared
=
- --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total |
427.494622
54
0.5572
A dj dj R -s -s qu qu ar ar ed ed =
7.91656707
Root MSE
=
0 .5 .5 31 31 2 1.9265
------------------------------------------------------------------------------
As
dif_y |
∆Yt
= α + β0∆Xt + β1Yt-1 + β2Xt-1 + εt
If our error correction approach is correct, then β1 should be -1 < β1 < 0 and significant.
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
Single Equation ECM
The results indicate the following equation ∆Yt
Which we can write in error correction form as ∆ Yt
∆Yt
= 13.12 + 1.32* ∆Xt -.42*Yt-1 + .52*Xt-1 + εt
= 13.12 + 1.32* ∆Xt -.42(Yt-1 - 1.22*X 1.22*Xt-1) + εt
Where 1.22 is our calculation of the long run multiplier
= 13.12 + 1.32* ∆Xt -.42(Yt-1 - 1.22*X 1.22*Xt-1) + εt
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
Single Equation ECM ∆ Yt
∆Yt
= α + 1.32*∆Xt -.42(Yt-1 - 1.22*X 1.22*Xt-1) + εt
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.
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 then increase to correct this disequilibrium, with 42% of the (remaining) deviation corrected in each subsequent time period.
= α + 1.32*∆Xt -.42(Yt-1 - 1.22*X 1.22*Xt-1) + εt
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
5 . 1
5 . 2
1
2
Y n i
e g n a h C
Y
5 . 1
5 .
0
1
0
2
4
Time Period
6
0
Single Equation ECM
2
4
Time Period
6
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 on regression.
And take the predicted values of ∆Yt to estimate Yt = α+ δ0∆Yt + δ1Xt - δ2∆Xt + µt predict deltaYhat regress y deltaYhat x dif_x
First, we can obtain an estimate of ∆Y by estimating ∆Yt = α + βYt-1 + βXt + β∆Xt + εt S ourc e |
SS
df
MS
Nu mbe r o f obs =
- --- -- --- --- -- --- ++- --- --- --- --- --- --- --- -- --- --- -- --- --- -- -
F(
2 1. 1. 40 40
238.216589
3
79.4055296
Prob > F
=
0.0000
Residual |
189.278033
51
3.71133398
R-squared
=
0.5572
A dj dj R -s qu qu ar ar ed =
0 .5 31 31 2
Root MSE
1.9265
- --- -- --- --- -- --- ++- --- --- --- --- --- --- --- -- --- --- -- --- --- -- 427.494622
54
7.91656707
3,
Model |
55
5 1) =
Model |
Total |
S ou rc e |
SS
=
Residual |
531.551099
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lag_y |
-.4248235
x | dif_x |
_cons |
0.001
-.6550105
-.1946365
.5182186
.1971867
2.63
0.011
.1223498
.9140873
.8066027
.2278972
3.54
0.001
.34908
1.264125
13.12112
MS
.1146587
4.201755
-3.71
3.12
0.003
4.685745
3
N um be r o f o bs =
189.278039
51
F(
177.1837 3.7113341
- --- --- --- --- --- --- +- --- --- -- --- --- -- --- --- --- --- --- --- --- --- Total |
720.829138
54
13.3486877
3,
55
5 1) 1) =
4 7. 7. 74
Prob > F
=
0.0000
R-squared
=
0.7374
A dj dj R -s -s qu qu ar ar ed ed =
0 .7 .7 22 22 0
Root MSE
1.9265
=
-----------------------------------------------------------------------------y |
-----------------------------------------------------------------------------dif_y |
df
- --- --- --- --- --- --- +- --- --- -- --- --- -- --- --- --- --- --- --- --- --- -
regress dif_y lag_y x dif_x
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------deltaYhat | x |
-1.353919 1.219844
.2698973 .1245296
-5.02 9.80
0.000 0.000
-1.89576 .9698408
-.8120773 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
------------------------------------------------------------------------------
21.55649
------------------------------------------------------------------------------
Single Equation ECM
Equivalence of the EC and ADL models First, lets estimate Yt = α + β0Yt-1 + β1Xt + β2Xt-1 + εt
We can see our estimate of the long term effect of X o n Y has a standard error of .12 and is statistically significant.
regress y lag_y x lag_x Source |
SS
df
MS
Number of obs =
- --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Model | Residual |
531.551105
3
189.278033
51
F(
177.183702 3.71133398
Can we gain similar estimates of the short and long term effects of X
720.829138
54
13.3486877
55
5 1) 1) =
4 7. 7. 74 74
=
0.0000
R-squared
- --- --- --- --- --- --- ++- --- --- --- --- --- --- --- --- --- --- --- --- --- --- Total |
3,
Prob > F
=
0.7374
A dj dj R -s -s qu qu ar ar ed ed =
0 .7 .7 22 22 0
Root MSE
1.9265
=
------------------------------------------------------------------------------
on Y from the ADL model?
y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lag_y |
.5751765
.1146587
5.02
0.000
.3449895
.8053635
x |
1.324821
.200003
6.62
0.000
.9232986
1.726344
lag_x |
-.8066027
.2278972
-3.54
0.001
-1.264125
-.34908
_cons |
13.12112
4.201755
3.12
0.003
4.685745
21.55649
------------------------------------------------------------------------------
15
Equivalence of the EC and ADL models The results imply the equation
Yt = 13.12 13.12 + .58*Y .58*Yt-1 + 1.32*Xt -.81*Xt-1 + εt
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:
Equivalence of the EC and ADL Models Yt = 13.12 + .58*Y t-1 + 1.32*Xt -.81*Xt-1 + εt The total long term effect/long effect/long run multiplier can be calculated as (1.32 - .81)/(.58 .81)/(.58 - 1) = 1.22 which which is equivalent equivalent to the ECM ECM estimate. estimate.
1.32 - .81 = .52 which is equivalent to the .52 estimate mate in the ECM
Note, however, that we do not have a standard error for the long run multiplier.
Deviations from equilibrium equilibrium are maintained at a rate of 58% per time period, which implies that deviations from equilibrium equilibrium are corrected at a rate of 42% per time period period (.58 - 1).
Y = 30.89 + 1.22X
Y and X will be in their long term equilibrium state when
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 Equation ECMs avoid this debate. Single equation ECMs ECMs don’t require cointegration cointegration and ease interpretation of causal relationships. relationships.
Single equation 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
16