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

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The Drunk and Her Dog 0 20 40 60 time drunk dog ECMs and Cointegration Y t = βX t + Z t Here, Z represents the portion of Y (in levels) that is not attributable to X. � In short, Z will capture the error correction relationship by capturing the degree to which Y and X are out of equilibrium. Z will capture any shock to either Y or X. If Y and X are cointegrated, then the relationship between the two will adjust accordingly. ECMs and Cointegration � We might theorize that shocks to X have two effects on ΔY. � Some portion of shocks to X might immediately affect Y in the next time period, so that ΔY t responds to ΔX t-1 . � A shock to X t will also disturb the equilibrium between Y and X, sending Y on a long term movement to a value that reproduces the equilibrium state given the new value of X. � Thus ΔY t is a function of both ΔX t-1 and the degree to which the two variables were out of equilibrium in the previous time period. ECMs and Cointegration Two I(1) time series (X t and Y t ) are cointegrated if there is some linear combination that is stationary. Z t = Y t - βX t Where Z is the portion of (levels of) Y that are not shared with X: the equilibrium errors. We can also rewrite this equation in regression form Y t = βX t + Z t Where the cointegrating vector - Z t - can be obtained by regressing Y t on X t . ECMs and Cointegration ΔY t will be a function of the degree to which the two time series were out of equilibrium in the previous period: Z t-1 Z t-1 = Y t-1 - X t-1 � When Z = 0 the system is in its equilibrium state � Y t will respond negatively to Z t-1 . � If Z is negative, then Y is too high and will be adjusted downward in the next period. � If Z is positive, then Y is too low and will be adjusted upward in the next time period. Engle and Granger Two-Step ECM � If two time series are integrated of the same order AND some linear combination of them is stationary, then the two series are cointegrated. � Cointegrated series share a stochastic component and a long term equilibrium relationship. � Deviations from this equilibrium relationship as a result of shocks will be corrected over time. � We can think of ∆Y t as responding to shocks to X over the short and long term. 4
Engle and Granger Two-Step ECM � Engle and Granger (1987) suggested an appropriate model for Y, based two or more time series that are cointegrated. � First, we can obtain an estimate of Z by regressing Y on X. � Second, we can regress ΔY t on Z t-1 plus any relevant short term effects. Engle and Granger Two-Step ECM � In Step 1, where we estimate the cointegrating regression we can - and should - include all variables we expect to 1) be cointegrated 2) have sustained shocks on the equilibrium. � The variables that have sustained shocks on the equilibrium are usually regarded as exogenous shocks and often take the form of dummy variables. Engle and Granger Two-Step ECM The basic structure of the ECM ΔY t = α + βΔX t-1 - βEC t-1 + ε t In the Engle and Granger Two-Step Method the EC component is derived from cointegrated time series as Z. ∆Y t = β 0 ∆X t-1 - β 1 Z t-1 β 0 captures the short term effects of X in the prior period on Y in the current period. β 1 captures the rate at which the system Y adjusts to the equilibrium state after a shock. In other words, it captures the speed of error correction. Step 1: Engle and Granger Two-Step ECM Y t = α + βX t + Z t The cointegrating vector - Z - is measured by taking the residuals from the regression of Y t on X t Z t = Y t - βX t - α Step 2: Regress changes on Y on lagged changes in X as well as the equilibrium errors represented by Z. ∆Y t = β 0 ∆X t-1 - β 1 Z t-1 Note that all variables in this model are stationary. Engle and Granger Two-Step ECM The cointegrating regression is performed as Y t = α + βX t + Z t Which we can also conceptualize as Z t = Y t - (α +βX t ) If we add a series of j exogenous shocks - represented as w j Then Y t = α + βX t + βW 1t + βW 2t +βW 3t + Z t Z t = Y t - (α +βX t + βW 1t + βW 2t +βW 3t ) Engle and Granger Two-Step ECM Note that the Engle and Granger 2-Step method is really a 4-step method. 1) Determine that all time series are integrated of the same order. 2) Demonstrate that the time series are cointegrated 3) Obtain an estimate of the cointegrating vector - Z - by regressing Y t on X t and taking the residuals. 4) Enter the lagged residuals - Z - into a regression of ∆Y t on ∆X t-1 . 5
Page 1 and 2: An Introduction to Error Correction
Page 3: ECMs and Cointegration: Integrated
Page 7 and 8: Dickey-Fuller Tests Basic Dickey-Fu
Page 9 and 10: Engle and Granger Two-Step ECM �
Page 11 and 12: Single Equation Error Correction Mo
Page 13 and 14: ECMs and the ADL We can see that th
Page 15 and 16: 0 .5 1 1.5 Change in Y 0 2 4 6 Time

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