An Introduction to Error Correction Models Robin Best Oxford Spring School for Quantitative Methods in Social Research An Introduction to ECMs Error Correction Models ECMs are a category of mu ltipl PDF document - DocSlides

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An Introduction to Error Correction Models Robin Best Oxford Spring School for Quantitative Methods in Social Research 2008 An Introduction to ECMs Error Correction Models (ECMs) are a category of mu ltiple time series models that directly estimate the speed at which a dependent variable - Y - returns to equilibrium after a change in an inde pendent variable - X. ECMs are useful for estimating both short term and l ong term effects of one time series on another. Thus, they often mesh well with our theories of pol itical and social processes. Theoretically-driven approach to estimating time se ries 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 = + bD t-1 EC t-1 + Where EC is the error correction component of the m odel and measures the speed at which prior deviations from equilibrium are corr ected. 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. An Introduction to ECMs 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 se ries data and (2) are interested in both short and long term relation ships between multiple time series. Applications of ECMs in the (Political Science) Literature U.S. Presidential Approval/ U.K. Prime Ministerial Satisfaction Policy Mood/Policy Sentiment Support for Social Security Consumer Confidence Economic Expectations Health Care Cost Containment/ Government Spending / Patronage Spending / Redistribution Interest Rates/ Purchasing Power Parity Growth in (U.S.) Presidential Staff Arms Transfers U.S. Judicial Influence Overview of the Course I. Motivating ECMs with cointegrated data Integration and cointegration 2-step error correction estimators Stata session #1 II. Motivating ECMs with stationary data The single equation ECM Interpretation of long and short term effects The Autoregressive Distributive Lag (ADL) model Equivalence of the ECM and ADL Stata session #2
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ECMs and Cointegration: Stationary vs. Integrated Time Series Stationary time series data are mean reverting. Th at is, they have a finite mean and variance that do not depend on time = + t-1 + Where | p | < 1 and t is also stationary with a mean of zero and variance Note that when 0 < | p | < 1 the time series is s tationary but contains autocorrelation. ECMs and Cointegration: Stationary vs. Integrated Time Series Often our time series data are not stationary, but appear to be integrated. Integrated time series data Are not mean-reverting appear to be on a ‘random walk Have current values that can be expressed as the su m of all previous changes The effect of any shock is permanently incorporated into the series Thus, the best predictor of the series at time is the value at time t-1 Have a (theoretically) infinite variance and no mea n. ECMs and Cointegration: Integrated Time Series Formally, an integrated series can be expressed as a function of all past disturbances at any point in time. Or Y = + t-1 + Where p = 1 Or Y - Y t-1 = u Where u = And t is still a stationary process ECMs and Cointegration: Integrated Time Series Order of Integration Integrated time series data that are stationary aft er being difference times are Integrated of order : I( For our purposes, we focus on time series data that are I(1). Data that are stationary after being first-differen ced. I(1) processes are fairly common in time series dat ECMs and Cointegration: Integrated Time Series (Theoretical) Sources of integration The effect of past shocks is permanently incorporat ed into the memory of the series. The series is a function of other integrated proces ses. A Drunk’s Random Walk 20 40 60 time
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ECMs and Cointegration: Integrated Time Series Analyzing integrated time series in level form dram atically increases the likelihood of making a Type-II error. Problem of spurious associations. High R Small standard errors and inflated t-ratios A common solution to these problems is to analyze t he data in differenced form. Look only at short term effects ECMs and Cointegration: Integrated Time Series Analyzing time series data in differenced form solv es the spurious regression problem, but may “throw the baby out wit h the bathwater. A model that includes only (lagged) differenced var iables assumes the effects of the X variables on Y never last longer t han one time period. What if our time series share a long run relationsh ip? If the time series share an equilibrium relationshi p with an error- correction mechanism, then the stochastic trends of the time series will be correlated with one another. Cointegration 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 serie s 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 f ar away from each other, maintaining an equilibrium relationship. ECMs and Cointegration Lets go back to the drunk’s random walk and call th e drunk X. The random walk can be expressed as - X t-1 = Where 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-1 = Where represents the stationary, while-noise process of t he dog’s steps. A Dog’s Random Walk 20 40 60 time ECMs and Cointegration But what if the dog belongs to the drunk? Then the two random walks are likely to have an equ ilibrium 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 walk s will be shared and will correct deviations the equilibrium - X t-1 = u + c(Y t-1 - X t-1 - Y t-1 = w + d(X t-1 - Y t-1 Where the terms in parentheses are the error correc ting mechanisms
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The Drunk and Her Dog 20 40 60 time drunk dog ECMs and Cointegration Two I(1) time series (X and Y ) are cointegrated if there is some linear combination that is stationary. = Y Where Z is the portion of (levels of) Y that are no t shared with X: the equilibrium errors. We can also rewrite this equation in regression for = + Z Where the cointegrating vector - Z - can be obtained by regressing Y on X ECMs and Cointegration = + Z Here, Z represents the portion of Y (in levels) tha t is not attributable to X. In short, Z will capture the error correction relat ionship 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 an d X are cointegrated, then the relationship between the two will adjust accord ingly. ECMs and Cointegration will be a function of the degree to which the two t ime series were out of equilibrium in the previous period: Z t-1 t-1 = Y t-1 - X t-1 When Z = 0 the system is in its equilibrium state will respond negatively to Z t-1 If Z is negative, then Y is too high and will be ad justed downward in the next period. If Z is positive, then Y is too low and will be adj usted upward in the next time period. ECMs and Cointegration We might theorize that shocks to X have two effects on Y. Some portion of shocks to X might immediately affec t Y in the next time period, so that responds to t-1 A shock to X 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 is a function of both t-1 and the degree to which the two variables were out of equilibrium in the previous t ime 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 ser ies 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 as responding to shocks to X over the short and lon g term.
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Engle and Granger Two-Step ECM Engle and Granger (1987) suggested an appropriate m odel 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 on Z t-1 plus any relevant short term effects. Engle and Granger Two-Step ECM Step 1: = + + Z The cointegrating vector - Z - is measured by taking t he residuals from the regression of Y on X = Y Step 2: Regress changes on Y on lagged changes in X as well as the equilibrium errors represented by Z. = t-1 t-1 Note that all variables in this model are stationar y. Engle and Granger Two-Step ECM In Step 1, where we estimate the cointegrating regre ssion 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 equ ilibrium are usually regarded as exogenous shocks and often take the form of dummy variables. Engle and Granger Two-Step ECM The cointegrating regression is performed as Y = + + Z Which we can also conceptualize as = Y - ( If we add a series of exogenous shocks - represented as w = + 1t + 2t 3t + Z Then = Y - ( 1t + 2t 3t Engle and Granger Two-Step ECM The basic structure of the ECM = + bD t-1 EC t-1 + In the Engle and Granger Two-Step Method the EC com ponent is derived from cointegrated time series as Z. = t-1 t-1 captures the short term effects of X in the prior p eriod on Y in the current period. 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 er ror correction. Engle and Granger Two-Step ECM Note that the Engle and Granger 2-Step method is re ally a 4-step method. 1) Determine that all time series are integrated of the same order. 2) Demonstrate that the time series are cointegrate 3) Obtain an estimate of the cointegrating vector - Z - by regressing on X and taking the residuals. 4) Enter the lagged residuals - Z - into a regression of on t-1
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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 coint egration (we will come back to this later). Integrated time series present a problem for time s eries analysis - at least in terms of long term relationships. When integrated time series variables are also coin tegrated, error correction models provide a nice solution to this p roblem. 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 integrati on and cointegration among time series is theoretically common. Error correction models were imported from economic s. Would we expect data from the social sciences to fo llow similar patterns of integration and cointegration? Cointegration and Error Correction in Political Science Prime Ministerial Statisfaction (U.K.) and Conservat ive Party Support Arms transfers by the U.S. and Soviet Union Economic expectations and U.S. Presidential Approva U.S. Domestic Policy Sentiment and Economic Expecta tions Support for U.S. Social Security and the Stock Mark et The Engle and Granger Two-Step ECM: Putting it into Practice Lets imagine we have two time series - perhaps the d runk 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 response s in Y, as Y adjusts back to the equilibrium state. X and Y: Cointegrated? 10 15 20 25 1960m1 1961m1 1962m1 1963m1 1964m1 1965m1 months Y X Engle and Granger Two-Step ECM First, we need to determine that both X and Y are i ntegrated 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 controversia l, primarily due to their low power. For our purposes, we will focus on Dickey-Fuller (D F) and Augmented Dickey-Fuller tests to examine the (non)stationarity of our time s eries.
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Dickey-Fuller Tests Basic Dickey-Fuller test With a constant (drift) With a time trend Dickey-Fuller Tests Basic Dickey-Fuller test With a constant (drift) With a time trend If X is a random walk process, then = 0 The null hypothesis is that X is a random walk MacKinnon values for statistical significance Note that in small samples the standard error of will be large, making it likely that we fail to reject the null when we really should Augmented Dickey-Fuller We can remove any remaining serial correlation in e t by introducing an appropriate number of lagged differences of X in th e equation. Where i = 1, 2, …k Null hypotheses are the same as the DF tests Is X Integrated? dfuller X, regress Dickey-Fuller test for unit root Num ber of obs = 63 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- Z(t) -1.852 -3.562 -2.920 -2.595 --------------------------------------------------- --------------------------- MacKinnon approximate p-value for Z(t) = 0.3548 --------------------------------------------------- --------------------------- D.X | Coef. Std. Err. t P>|t| [9 5% Conf. Interval] -------------+------------------------------------- --------------------------- X | L1. | -.1492285 .0805656 -1.85 0.069 -.3 103293 .0118724 _cons | 1.365817 .7149307 1.91 0.061 - .0637749 2.79541 --------------------------------------------------- --------------------------------------------------- ------------------------------------------ Is X Integrated? dfuller X, lags(4) regress Augmented Dickey-Fuller test for unit root Number of obs = 59 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- 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| [9 5% Conf.Interval] -------------+------------------------------------- --------------------------- X | L1. | .0696672 .1008978 0.69 0.493 -.1 327082 .2720426 LD. | -.5724812 .1738494 -3.29 0.002 -.9 211789 .2237835 L2D. | -.4935811 .1776346 -2.78 0.008 -. 8498709 -.1372912 L3D. | -.2891465 .1677748 -1.72 0.091 -. 6256601 .0473671 L4D. | -.0898266 .1468121 -0.61 0.543 -. 3842943 .2046412 _cons | -.2525666 .839646 -0.30 0.765 - 1.936683 1.43155 --------------------------------------------------- ------------- -------------- Is X Integrated? If X is I(1), then the first difference of X should be stationary. dfuller dif_X Dickey-Fuller test for unit root Number of obs = 62 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- 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? dfuller Y, regress Dickey-Fuller test for unit root Number of obs = 63 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- 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| [9 5% Conf. Interval] -------------+------------------------------------- --------------------------- Y | L1. | -.0854922 .064599 -1.32 0.191 -.2 146659 .0436814 _cons | 1.061271 .7208156 1.47 0.146 - .3800884 2.502631 --------------------------------------------------- --------------------------- Is Y Integrated? dfuller dif_Y, regress Dickey-Fuller test for unit root Number of obs = 62 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- 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] -------------+------------------------------------- --------------------------- dif_Y | L1. | -1.159903 .1278662 -9.07 0.000 -1. 415674 -.9041329 _cons | .2219184 .3259962 0.68 0.499 - .4301711 .8740078 --------------------------------------------------- --------------------------- Cointegration Both X and Y appear to be integrated of the same or der: I(1). If they are cointegrated, then they share stochasti c trends. In the following regression, should be stationary and should be statistically significant and in the expected direc tion. = + Lets see if this is the case Cointegrating Regression regress Y X Source | SS df MS Number of obs = 64 -------------+------------------------------ F( 1, 62) = 92.49 Model | 1009.22604 1 1009.22604 Pro b > F = 0.0000 Residual | 676.523964 62 10.9116768 R-squared = 0.5987 -------------+------------------------------ Adj R-sq uared = 0.5922 Total | 1685.75 63 26.7579365 Roo t MSE = 3.3033 --------------------------------------------------- --------------------------- Y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+------------------------------------- --------------------------- X | 1.206126 .1254135 9.62 0.000 .955 4281 1.456824 _cons | .0108108 1.135884 0.01 0.992 - 2.259789 2.28141 --------------------------------------------------- --------------------------- Cointegrating Regression predict r, resid dfuller r Dickey-Fuller test for unit root Number of obs = 63 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% C ritical Statistic Value Value Va lue --------------------------------------------------- --------------------------- Z(t) -5.487 -3.562 -2.920 -2.595 --------------------------------------------------- --------------------------- MacKinnon approximate p-value for Z(t) = 0.0000 -15 -10 -5 10 Residuals 1960m1 1961m1 1962m1 1963m1 1964m1 1965m1 months
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Engle and Granger Two-Step ECM Our residuals from the cointegrating regression capt ure 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 cointe grating regression as our measure of the error correction mechanism. = + t-1 + *R t-1 Engle and Granger Two-Step ECM regress dif_Y dlag_X lag_r Source | SS df MS Number of obs = 62 -------------+------------------------------ F( 2, 59) = 5.09 Model | 59.4494524 2 29.7247262 Pro b > F = 0.0091 Residual | 344.227967 59 5.83437232 R-squared = 0.1473 -------------+------------------------------ Adj R-sq uared = 0.1184 Total | 403.677419 61 6.61766261 Roo t MSE = 2.4154 --------------------------------------------------- --------------------------- dif_Y | Coef. Std. Err. t P>|t| [ 95% Conf. Interval] -------------+------------------------------------- --------------------------- dlag_X | -.1161038 .1609359 -0.72 0.473 - .4381358 .2059282 lag_r | -.3160139 .0999927 -3.16 0.002 -. 5160988 -.1159291 _cons | .210471 .3074794 0.68 0.496 - .4047939 .8257358 --------------------------------------------------- --------------------------- The error correction mechanism is negative and sign ificant, suggesting that deviations from equilibrium are corrected at about 32% per month. However, X does not appear to have significant shor t term effects on Y. 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 inco rrect inferences about causality when there is an error correction process . The Engle-Granger approach to ECMs begins by assumin g all variables in the cointegrating regression are jointly endogene ous. Thus, in the previous example we should also estima te a cointegrating regression of X on Y. Granger Causality Granger causality can be ascertained in the ECM fra mework 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 b e negative and significant. The lagged level of the other variable should be in the expected direction and significant. Granger Causality regress dif_Y l.dif_Y l.dif_X lag_Y lag_X Source | SS df MS Number of obs = 62 -------------+------------------------------ F( 4, 57) = 2.97 Model | 69.5277246 4 17.3819311 Pro b > F = 0.0270 Residual | 334.149695 57 5.86227535 R-squared = 0.1722 -------------+------------------------------ Adj R-sq uared = 0.1141 Total | 403.677419 61 6.61766261 Roo t MSE = 2.4212 --------------------------------------------------- --------------------------- dif_Y | Coef. Std. Err. t P>|t| [ 95% Conf. Interval] -------------+------------------------------------- --------------------------- dif_Y | L1. | .0483244 .1399056 0.35 0.731 -.2 318318 .3284806 dif_X | L1. | -.2205689 .1802099 -1.22 0.226 -. 581433 .1402952 lag_Y | -.3557259 .1161894 -3.06 0.003 -. 5883911 -.1230606 lag_X | .5675793 .1899981 2.99 0.004 . 1871146 .948044 _cons | -.928984 .9426534 -0.99 0.329 - 2.816615 .9586468 --------------------------------------------------- --------------------------- Granger Causality regress dif_X l.dif_X l.dif_Y lag_X lag_Y Source | SS df MS Number of obs = 62 -------------+------------------------------ F( 4, 57) = 5.87 Model | 74.2042429 4 18.5510607 Pro b > F = 0.0005 Residual | 180.182854 57 3.1611027 R-squared = 0.2917 -------------+------------------------------ Adj R-sq uared = 0.2420 Total | 254.387097 61 4.17028027 Roo t MSE = 1.7779 --------------------------------------------------- --------------------------- dif_X | Coef. Std. Err. t P>|t| [ 95% Conf. Interval] -------------+------------------------------------- --------------------------- dif_X | L1. | -.0640245 .132332 -0.48 0.630 -.3 290147 .2009657 dif_Y | L1. | .0014809 .1027357 0.01 0.989 -.2 042438 .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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10 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 whic h should be on the right. The Engle and Granger approach assumes endogeneity b etween the cointegrating time series. Engle and Granger Two-Step Technique: Issues and Limitations Does not clearly distinguish dependent variables fr om independent variables. In the social sciences the Engle and Granger two-st ep ECM might not be consistent with our theories. Is appropriate when dealing with cointegrated time s eries. Can we clearly distinguish between integrated and s tationary processes? Integration Issues Error correction approaches that rely on cointegrat ion 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 1 992) This problem increases with sample size. The low power of unit root tests can lead us to con clude our data are integrated when they are not. More Integration Issues In the social sciences, we are more likely to have data that are Near integrated (p = 0, but there is memory. p may not = 0 in finit e 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 integrate d data Aggregated data More Integration Issues Under these conditions, we are likely to draw fault y 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 Integration Issues and ECMs Under these conditions, we are often better off est imating a single equation ECM. Single equation ECMs solve some of these problems an d avoid others. However, single equation ECMs require weak exogeneit y.
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11 Single Equation Error Correction Models Following theory, Single Equation ECMs clearly disti nguish between dependent and independent variables. Single Equation ECMs are appropriate for both cointe grated and long- memoried, but stationary, data. Cointegration may imply error correction, but does e rror 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 t he independent variables. Single Equation ECMs Our theories might specify long and short term effe cts of independent variables on a dependent variable even when our dat a 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 model ing technique for stationary data than alternative approaches. Our theories may be better represented by a single equation ECM. 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-1 t-1 ) + The Single Equation ECM Basic form of the ECM = + bD t-1 EC t-1 + Engle and Granger two-step ECM = t-1 t-1 Where Z = Y The Single Equation ECM = + (Y t-1 t-1 ) + The Single Equation ECM = + (Y t-1 t-1 ) + be The portion of the equation in parentheses is the e rror correction mechanism. (Y t-1 t-1 ) = 0 when Y and X are in their equilibrium state estimates the short term effect of an increase in X on Y 1 estimates the speed of return to equilibrium after a deviation. If the ECM approach is appropriate, then -1 < 1 < 0 2 estimates the long term effect that a one unit incr ease in X has on Y. This long term effect will be distributed over future time pe riods according to the rate of error correction - The Single Equation ECM = + (Y t-1 t-1 ) + The values for which Y and X are in their long term equilibrium relationship are Y = k + k Where And Where k 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 all owing us to also estimate k ’s standard error, and therefore statistical significa nce.
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12 The Single Equation ECM Since the long term effect is a ratio of two coeffi cients, we could calculate its standard error using the variance and covariance ma trix Alternatively, we can use the Bewley transformation to estimate the standard error. This requires estimating the following regression. = + + + Where 1 is the long term effect and is estimated with a sta ndard error Notice the problem: we have on the right side of the equation We can proxy as: = + t-1 + + + And use our predicted values of in the Bewley transformation regression The Single Equation ECM We can easily extend the single equation ECM to inc lude more independent variables = + 1t + 2t 3t (Y t-1 1t-1 2t-1 3t-1 ) + Note that each independent variable is now forced t o make an independent contribution to the long term relations hip, solving one of the problems in the two-step estimator. Single Equation ECMs in the (Political Science) Literature Judicial Influence Health Care Cost Containment Interest Rates Patronage Spending Growth in Presidential Staff Government Spending Consumer Confidence Redistribution Single Equation ECMs Single Equation ECMs Provide the same information about the rate of erro r 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 En gle and Granger two- step method. Illustrate that ECMs are appropriate for both cointe grated and stationary data. How do we know Single Equation ECMs are appropriate with stationary data? ECMs and ADL Models We know Autoregressive Distributive Lag models are appropriate for stationary data (stationary data is, in fact, a req uirement of these models). Forms of single equation ECMs and ADL models are equ ivalent. We can derive a single equation ECM from a general ADL model: = + t-1 + + t-1 + ECMs and the ADL = + t-1 + + t-1 + = + ( 0 - 1)Y t-1 + + t-1 + = + ( 0 - 1)Y t-1 + + ( 1 + )X t-1 + = + t-1 + + t-1 + Where = 0 - 1 and = 1 + We can rewrite this equation in error correction fo rm as = + (Y t-1 t-1 ) +
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13 ECMs and the ADL We can see that the ADL model provides information similar to the ECM. = + t-1 + + t-1 + estimates the proportion of the deviation from equi librium at t-1 that is maintained at time . 0 - 1 tells us the speed of return. estimates the short term effect of X on Y + 2 estimates the long term effect of a unit change in X on Y (the coefficient on t-1 in the ECM) ECMs and the ADL = + t-1 + + t-1 + And the total long term effect/long run multiplier - k - is therefore: Y and X will be in their long term equilibrium stat e when Y = k + k where 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 deviatio n has occurred. Long term equilibrium relationships between variabl es. Long and short term effects of independent variable s on the dependent variable. The EC and ADL Models: Notation Lets use the following notation for the single equa tion ECM and the ADL ECM = + (Y t-1 t-1 ) + ADL = + t-1 + + t-1 + Single Equation ECM Lets imagine our theory about the relationship betw een X and Y states: X causes Y. X should have both a short term and a long term eff ect on Y. We don’t have reason to suspect cointegration from a theoretical standpoint. But we believe X and Y share a long term equilibriu m relationship Single Equation ECM We determine that our Y variable is stationary (wit h 95% confidence), ruling out an ECM based on cointegration dfuller y, regress Dickey-Fuller test for unit root Number of obs = 55 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Va lue --------------------------------------------------- --------------------------- Z(t) -3.353 -3.573 -2.926 -2.598 --------------------------------------------------- --------------------------- MacKinnon approximate p-value for Z(t) = 0.0127
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14 Single Equation ECM We then estimate the single equation ECM = + (Y t-1 t-1 ) + As = + + t-1 + t-1 + If our error correction approach is correct, then 1 should be -1 < 1 < 0 and significant. Single Equation ECM regress dif_y dif_x lag_y lag_x Source | SS df MS Number of obs = 55 -------------+------------------------------ F( 3, 51) = 21.40 Model | 238.216589 3 79.4055296 Pro b > F = 0.0000 Residual | 189.278033 51 3.71133398 R-squared = 0.5572 -------------+------------------------------ Adj R-sq uared = 0.5312 Total | 427.494622 54 7.91656707 Roo t MSE = 1.9265 --------------------------------------------------- --------------------------- dif_y | Coef. Std. Err. t P>|t| [ 95% Conf. Interval] -------------+------------------------------------- --------------------------- dif_x | 1.324821 .200003 6.62 0.000 . 9232986 1.726344 lag_y | -.4248235 .1146587 -3.71 0.001 -. 6550105 -.1946365 lag_x | .5182186 .1971867 2.63 0.011 . 1223498 .9140873 _cons | 13.12112 4.201755 3.12 0.003 4.685745 21.55649 --------------------------------------------------- --------------------------- Single Equation ECM The results indicate the following equation = 13.12 + 1.32* -.42*Y t-1 + .52*X t-1 + Which we can write in error correction form as = 13.12 + 1.32* -.42(Y t-1 - 1.22*X t-1 ) + Where 1.22 is our calculation of the long run multi plier Single Equation ECM = 13.12 + 1.32* -.42(Y t-1 - 1.22*X t-1 ) + Y and X are in their long term equilibrium state wh en Y = 30.89 + 1.22X So that when X = 1 Y = 32.11 Single Equation ECM = + 1.32* -.42(Y t-1 - 1.22*X t-1 ) + Changes in X have both an immediate and long term e ffect 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 equi librium, causing Y to be too low. Y will then increase to correct this disequilibrium , with 42% of the (remaining) deviation corrected in each subsequent time period. Single Equation ECM = + 1.32* -.42(Y t-1 - 1.22*X t-1 ) + A one unit increase in X immediately produces a 1.3 2 unit increase in Y. Increases in X also disrupt the the long term equil ibrium 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 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 effec t on Y
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15 .5 1.5 Change in Y Time Period 1.5 2.5 Time Period 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 r egression. First, we can obtain an estimate of Y by estimating = + t-1 + + + regress dif_y lag_y x dif_x Source | SS df MS Number of obs = 55 -------------+------------------------------ F( 3, 51) = 21.40 Model | 238.216589 3 79.4055296 Pro b > F = 0.0000 Residual | 189.278033 51 3.71133398 R-squared = 0.5572 -------------+------------------------------ Adj R-sq uared = 0.5312 Total | 427.494622 54 7.91656707 Roo t MSE = 1.9265 --------------------------------------------------- --------------------------- dif_y | Coef. Std. Err. t P>|t| [ 95% Conf. Interval] -------------+------------------------------------- --------------------------- lag_y | -.4248235 .1146587 -3.71 0.001 -. 6550105 -.1946365 x | .5182186 .1971867 2.63 0.011 .122 3498 .9140873 dif_x | .8066027 .2278972 3.54 0.001 .34908 1.264125 _cons | 13.12112 4.201755 3.12 0.003 4.685745 21.55649 --------------------------------------------------- --------------------------- Single Equation ECM And take the predicted values of to estimate Y = + + + predict deltaYhat regress y deltaYhat x dif_x Source | SS df MS Number of obs = 55 -------------+------------------------------ F( 3, 51) = 47.74 Model | 531.551099 3 177.1837 Pro b > F = 0.0000 Residual | 189.278039 51 3.7113341 R-squared = 0.7374 -------------+------------------------------ Adj R-sq uared = 0.7220 Total | 720.829138 54 13.3486877 Roo t MSE = 1.9265 --------------------------------------------------- --------------------------- y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+------------------------------------- --------------------------- deltaYhat | -1.353919 .2698973 -5.02 0.000 -1.89576 -.8120773 x | 1.219844 .1245296 9.80 0.000 .969 8408 1.469848 dif_x | 1.898677 .3963791 4.79 0.000 1 .102913 2.694442 _cons | 30.88605 2.68463 11.50 0.000 25.49643 36.27567 --------------------------------------------------- --------------------------- Single Equation ECM We can see our estimate of the long term effect of X on Y has a standard error of .12 and is statistically signific ant. Can we gain similar estimates of the short and long term effects of X on Y from the ADL model? Equivalence of the EC and ADL models First, lets estimate = + t-1 + + t-1 + regress y lag_y x lag_x Source | SS df MS Number of obs = 55 -------------+------------------------------ F( 3, 51) = 47.74 Model | 531.551105 3 177.183702 Pro b > F = 0.0000 Residual | 189.278033 51 3.71133398 R-squared = 0.7374 -------------+------------------------------ Adj R-sq uared = 0.7220 Total | 720.829138 54 13.3486877 Roo t MSE = 1.9265 --------------------------------------------------- --------------------------- y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+------------------------------------- --------------------------- lag_y | .5751765 .1146587 5.02 0.000 . 3449895 .8053635 x | 1.324821 .200003 6.62 0.000 .923 2986 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 --------------------------------------------------- ---------------------------
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16 Equivalence of the EC and ADL models The results imply the equation = 13.12 + .58*Y t-1 + 1.32*X -.81*X t-1 + 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 calcul ated as: 1.32 - .81 = .52 which is equivalent to the .52 esti mate in the ECM Deviations from equilibrium are maintained at a rat e of 58% per time period, which implies that deviations from equilibrium are correc ted at a rate of 42% per time period (.58 - 1). Equivalence of the EC and ADL Models = 13.12 + .58*Y t-1 + 1.32*X -.81*X t-1 + The total long term effect/long run multiplier can be calculated as (1.32 - .81)/(.58 - 1) = 1.22 which is equivalent to t he 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 stat e when = 30.89 + 1.22X Error Correction Models A Flexible Modeling approach Stationary and Integrated Data Long and Short Term Effects Engle and Granger two-step ECM versus Single Equati on 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 infor mation about short and long term effects. Standard error for the long term effects of indepen dent variables is relatively easy to obtain in the single equation EC