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

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ECMs, Causality, and Theory � In the social sciences, our theories (usually) tell us which time series should be on the left side of the equation and which should be on the right. � The Engle and Granger approach assumes endogeneity between the cointegrating time series. Integration Issues Error correction approaches that rely on cointegration of two or more I(1) time series become problematic when we are dealing with data that are not truly (co)integrated. � I(1) processes may be incorrectly included into the cointegrating regression - producing spurious associations - if two other I(1) cointegrated time series are already included (Durr 1992) � This problem increases with sample size. � The low power of unit root tests can lead us to conclude our data are integrated when they are not. More Integration Issues Under these conditions, we are likely to draw faulty inferences from the two-step procedure. We might conclude: � Our data are integrated when they are not. � Our data are cointegrated when they are not. � Our data are not cointegrated, therefore, an ECM is not appropriate Engle and Granger Two-Step Technique: Issues and Limitations � Does not clearly distinguish dependent variables from independent variables. � In the social sciences the Engle and Granger two-step ECM might not be consistent with our theories. � Is appropriate when dealing with cointegrated time series. � Can we clearly distinguish between integrated and stationary processes? More Integration Issues In the social sciences, we are more likely to have data that are � Near integrated (p = 0, but there is memory. p may not = 0 in finite samples.) � Fractionally integrated (0 < p < 1, where when 0 < p < .5 the data are mean-reverting and have finite variance, and when .5 ≤ p < 1 the data are mean-reverting but have infinite variance) � A combined process of both stationary and integrated data � Aggregated data Integration Issues and ECMs � Under these conditions, we are often better off estimating a single equation ECM. � Single equation ECMs solve some of these problems and avoid others. � However, single equation ECMs require weak exogeneity. 10
Single Equation Error Correction Models � Following theory, Single Equation ECMs clearly distinguish between dependent and independent variables. � Single Equation ECMs are appropriate for both cointegrated and longmemoried, but stationary, data. � Cointegration may imply error correction, but does error correction imply cointegration? � Single Equation ECMs estimate a long term effect for each independent variable, allowing us to judge the contribution of each. � Allow for easier interpretation of the effects of the independent variables. Single Equation ECMs � Single Equation Error Correction Models are useful � When our theories dictate the causal relationships of interest � When we have long-memoried/stationary data A basic single equation ECM: ∆Y t = α + β 0 *∆X t - β 1 (Y t-1 - β 2 X t-1 ) + ε t The Single Equation ECM ∆Y t = α + β 0*∆X t - β 1(Y t-1 - β 2X t-1) + βε t The portion of the equation in parentheses is the error correction mechanism. (Y t-1 - β 2 X t-1 ) = 0 when Y and X are in their equilibrium state β 0 estimates the short term effect of an increase in X on Y β 1 estimates the speed of return to equilibrium after a deviation. If the ECM approach is appropriate, then -1 < β 1 < 0 β 2 estimates the long term effect that a one unit increase in X has on Y. This long term effect will be distributed over future time periods according to the rate of error correction - β 1 Single Equation ECMs � Our theories might specify long and short term effects of independent variables on a dependent variable even when our data are stationary. � The concepts of error correction, equilibrium , and long term effects are not unique to cointegrated data. � Furthermore, an ECM may provide a more useful modeling technique for stationary data than alternative approaches. � Our theories may be better represented by a single equation ECM. The Single Equation ECM Basic form of the ECM Engle and Granger two-step ECM The Single Equation ECM ΔY t = α + βΔX t-1 - βEC t-1 + ε t ∆Y t = β 0 ∆X t-1 - β 1 Z t-1 Where Z t = Y t - βX t - α ∆Y t = α + β 0 *∆X t - β 1 (Y t-1 - β 2 X t-1 ) + ε t The Single Equation ECM ∆Y t = α + β 0*∆X t - β 1(Y t-1 - β 2X t-1) + ε t The values for which Y and X are in their long term equilibrium relationship are Y = k0 + k1X α Where k0 = β And β 2 k1 = β 1 1 Where k 1 is the total long term effect of X on Y (a.k.a the long run multiplier) - - distributed over future time periods. Single equation ECMs are particularly useful for allowing us to also estimate k 1 ’s standard error, and therefore statistical significance. 11
Page 1 and 2: An Introduction to Error Correction
Page 3 and 4: ECMs and Cointegration: Integrated
Page 5 and 6: Engle and Granger Two-Step ECM �
Page 7 and 8: Dickey-Fuller Tests Basic Dickey-Fu
Page 9: Engle and Granger Two-Step ECM �
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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