Simulating Time Series Analysis Using SAS Part III Error Correction Model (ECM) Ismail E. Mohamed, L-3 STRATIS, Reston VA

Transcription

1 Simulating Time Series Analysis Using SAS Part III Error Correction Model (ECM) Ismail E. Mohamed, L-3 STRATIS, Reston VA ABSTRACT The purpose of this series of articles is to present a simplified discussion and SAS programming techniques specifically designed to simulate various steps in time series data analysis. Part I of this series discussed in detail the Augmented Dickey-Fuller (ADF) test of time series variables, a test for a time series stationarity in a time series sample. Part II continued the discussion beyond the ADF and examined the concept of cointegration, a powerful econometric technique for investigating trends in time series and a sound methodology for modeling both the long and short run dynamics in a time series system. Each part presented simple SAS techniques that could be used to perform and accomplish these tests. Part III will conclude this series and present a simple discussion on Error Correction Model (ECM), a powerful mechanism discussed by many authors including Granger (1983), and Banergee et al (1993) that is used to determine time series shortrun deviations from long-run equilibrium. A simple SAS technique to simulate the Error Correction mechanism will also be presented and discussed. INTRODUCTION Empirical research in financial economics and many others sectors is largely based on time series data. Such data representing a phenomenon is collected over long and different periods of time. Many analysts erroneously use the framework of linear regression (OLS) models to model variables of such data and to predict change over time or extrapolate from present conditions to future conditions. Part I of this series cautioned against interpretation of the results of regression models estimated using time series data and suggested a simple framework to assist SAS programmers in understanding, modeling, and carrying out stationarity testing (ADF) using a time series data on a univariate series (Mohamed, 2009) Part II suggested a general framework to determine the path forward with time series analysis after concluding an ADF Test and introduced the concept of cointegration stating that if there exists a stationary linear combination between two non-stationary time series, the two variables combined are said to be cointegrated (Granger (1981, 1983). Part II also introduced Engle and Granger s (1987) two-step procedure to test for cointegration between two variables x and y. Discussion and simple techniques to generate the error term and test it for stationarity were presented and discussed. Part II concluded that residuals error from the two series regression seems to be stationary and suggested that the residuals (error) captures the deviations from the equilibrium of x and y. Part II draws the implication that there must be some adjustment process that would prevent this error in the long term from becoming larger and larger and suggested that if two time series are found to be cointegrated then their residuals (the long-term error) could be used to explain the movement in the short term as suggested by many authors. Part III introduces the concept of Error Correction Model (ECM), a dynamic modeling technique that directly estimates the speed at which a dependent variable returns to equilibrium after a change in an independent variable. ECM has been a long tradition in time series econometrics and has recently been popularized by Engle and Granger. ECM is widely used to estimate the acceleration speed of the short-run deviation to the long-run equilibrium. BASICS AND TERMINOLOGY Terms such as long-run, or long-term, relationship imply a relatively long period of time. Such relationships usually measure the relationship, if any, between any two or more variables over a very long time period. People understand this kind of relationship intuitively in their everyday life. For example, a farmer in the most remote place would easily tell you that his corn production (y) is much larger in years when there is more rainfall during the rainy season (x), and a mother knows that in the colder months (x) her household will incur higher energy bills (y), and so forth. Mathematically a long-term relationship between two variables, say x and y could be written in the following linear form 1

2 (1) The two variables are designated as dependent (y) and independent (x). The α measures a fixed difference between (y) and (x), the intercept, while β measures the change in (y) when there is a change in (x), the slope. Nevertheless, what is indistinct to many in these situations is what appeares and is symbolized as ( ε ) in the above equation. This term (ε ) denotes a gap or a discrepancy from accuracy or correctness. In other words, the ( ε ) term captures all other factors which influence the dependent variable y t other than the independent variable x t. For our purpose we will refer to this term as the long-term error. On the other hand, terms such as short-run or short-term relationship imply a limited time duration. Such relationships usually measure adjustments, if any, between a variable s values x and/or y at relatively short time intervals such as between two adjacent periods of time (for example, today versus yesterday or this month versus the previous month). In other words, the 1 st -differenced value(s) of the variable y (y t y t-1 ) and/or the 1 st -differenced value(s) of the variable x (x t x t-1 ). Next we emphasize Murray s 1994 notion that non-stationary time series are likely to have an equilibrium relationship and are cointegrated. Murray suggested that the cointegrated time series are by no means expected to drift too far away from each other, maintaining an equilibrium relationship and those deviations from this equilibrium relationship will be corrected over time. The principle behind ECM models as suggested by many authors, including Engle and Granger (1987), is that there often exists a long-run equilibrium relationship between two economic variables (for example, supply and demand, consumption and income, wages and prices, and so on). In the short run, however, there may be disequilibrium. Engle and Granger (1987) suggested that with the error correction mechanism, a proportion of the disequilibrium (error) in one period (t 1 ) is corrected in the next period (t 2 ). For instance, the change in prices in one period (t 2 ) may depend on the excess demand in the previous period (t 1 ). The error correction process therefore could be seen as a powerful tool to reconcile short-run and long-run behavior. It relates the change in y to the change in x and the past period s disequilibria, or the error from the previous period ( ). We also point out ideas by Singh, et al. (2005) and other authors that the Error Correction Model is used to estimate the acceleration speed of the short-run deviation to the long-run equilibrium. In light of the above discussion, a short-term model between x and y could be explained by the long-term error ( ) in the form of,,, (2) Equation (2) shows that in ECM the movement of time series in any time period is related to the previous period's gap (error) derived from the long-run equilibrium. In other words, ECM is a mechanism for the dependent variable to adjust, and return, to its existing long-term trend. ECM is all about how the long-term error will explain the movement in the short-run. Now, in the context of the first to papers in this series, we introduce the Engle and Granger (1987) Two-Step ECM: 1. Determine that all time series are integrated of the same order (ADF test- Part I, Mohamed 2009) 2. Demonstrate that the time series are cointegrated (cointegration test Part II, Mohamed 2009) 3. Obtain an estimate of the cointegrating vector by regressing y t on x t and taking the residuals (generate residual series Part II) 4. Enter the lagged residuals into a regression of y t on x t-1 such as Y, X, ε ε, (2) Note that this 2-Step method is really a 4-Step method for the fact that an ECM can only be estimated between variables that are integrated of first order I(1), and cointegrated, hence the requirement to test for the order of integration (via unit root tests) of the variables, and then for 2

3 cointegration prior to running the ECM. The term ( ε ) in equation (2) is the error correction component of the model and measures the speed at which prior deviations from equilibrium are corrected. In this sense ECM can be seen as a smart way of combining the long run, a cointegrating relationship between the levels variables and the short run relationship between the first differences of the variables. In equation (1) if y and x are I(1) variables, then both y and x are stationary and the error term ( ) is stationary if the variables y and x are cointegrated. Therefore when all the variables in the equation are stationary a valid OLS estimation is possible. Also, since all the variables in the estimated equation are stationary; there is no problem with spurious correlation. STRUCTURE OF AN ECM SYSTEM (EQUATION),,, Some Change in Change in Adjustment Error Value of y Value of x (Long-run error) (regression) At any given at that period from the prior period Period (Lagged value) From the above illustration we can see that the lagged residuals of the long-run model ( ε ) and the 1 st -differences of the x series are used to estimate the error correction model to determine the short-run deviations from equilibrium. We draw the reader s attention here that there is no intercept (constant) in this equation. Given the fact that many time series are not cointegrated, analysts are unable to take advantages of the error correction model's ability to capture both long and short-term dynamics in a single statistical model. Nonetheless, others used analytical results to demonstrate that ECM is appropriate for stationary data as well (DeBoef 2001). SAS TECHNIQUES SAS, EVIEWS and STATA are few of a long list of software that data analysts utilize to setup and estimate an error correction model. For our purpose we will introduce a simple SAS approach that uses SAS data steps and SAS PROC REG to mimic a setup for an error correction model. To demonstrate we will now introduce this example of a time series dataset (REG_SERIES). YEAR QTR x y x and y are two time series variables (Partial output) 3

4 1.) Determine that all time series are integrated of the same order: Test whether the assumed time series are I(1) which is a necessary condition for the further testing procedure. To do that we will employ the very standard Augmented Dickey-Fuller test (ADF) SAS technique presented in Part I. 2.) Demonstrate that the time series are cointegrated: Estimate the long-run relationship y t = x t and Save regression residuals series ( ε ) of the regression y residuals - ( ) - the long-term error. To do this we will employ technique presented and discussed in details in Part II (here we will show SAS code and data structure only) PROC REG DATA= REG_SERIES; MODEL y = x; OUTPUT OUT = RESIDS R = ; RUN; QUIT; Estimated Residual series resulted from fitting the x and y regression in step 1 (partial output) YEAR QTR x y ) Obtain an estimate of the cointegrating vector: Test whether the residuals series ( ε ) is stationary using again the standard ADF. (The procedure is the same as in the step 1), i.e., apply stationarity testing on the residuals series ( ). If we are able to reject the null hypothesis about Non-stationarity, we can conclude that x and y are both cointegrated of the orders CI(1,1) (step 3 was discussed in details in Part II). If we conclude cointegration we will then precede with estimation of the error-correction model (ECM) such as in equation (2) by adding three additional variables to our dataset: DATA ECM; SET RESIDS; = DIF(X); * x first differenced (adjustment in x /period); = DIF(Y); * y first differenced (adjustment in y/period); RUN; = LAG( ); *Previous (lagged) value of the error term; 4

5 1 st - differenced values of x and y and value of the lagged Error series make the data ready for step 4 (partial output) YEAR QTR x y ε x y ε Enter the lagged residuals ( into a regression of y t on x t-1 such as in equation (2),,, PROC REG DATA= ECM; MODEL = RUN; QUIT; /NOINT; Again notice PROC REG statement uses of the SAS option NOINT which simply tells SAS there is no intercept (constant) in this regression. SAS will fit the model without the intercept term which is clearly a violation of the classical assumptions for regression models (in many regression analyses the intercept should be included). If it is not, we will then force the mean of y to be zero when x is zero (the line is forced through the origin). A regression without a constant implies that the regression line should run through the origin, i.e., the point where both y and x equal zero. Another problem with leaving out the intercept is that the R-Square statistic will not be defined in the standard way, which means R-Square can't be used in the usual way. Nevertheless, as suggested by Hahn (1977) with time series data the use of the (no-intercept) models are permissible for residuals because the residuals have mean of zero (Please refer to the SAS output from the above PROC REG and the coefficients that fitted the regression page 7). 5

6 DISCUSSION As pointed by Singh and Singh in their 2005 study of wheat and maize market the particular importance of the coefficient of the error correction term ( ), that indicates the speed at which the series returns to equilibrium. For values of that are negative (positive) and less than (equal to) zero, the series converge to (diverge from) the long-run equilibrium. In other words the coefficient, referred to as the speed-of-adjustment factor, measures the short-run deviation from the long run equilibrium. As the coefficient s values near zero, the paths are slow to adjust back to the long-run equilibrium. As they near or go over one, short-run deviation follows rapid paths to the long-run equilibrium. Dependent variables rapid acceleration paths have relatively low -. In contrast, dependent variables with slow paths tend to have high. From the SAS output below it is clear that the regression equation is Y, X, ε (Notice that there is no intercept in this equation). In other words, we can write this as: Short term Adjustments in y = Short term Adjustments in x previous period gap (Error) The REG Procedure Model: MODEL1 Dependent Variable: D1Y Number of Observations Read 72 Number of Observations Used 71 Number of Observations with Missing Values 1 NOTE: No intercept in model. R-Square is redefined. Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Uncorrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t < Residuals estimated from the `long-run relationship' is used in the Error Correction Model (ECM) 6

7 FINAL WORD Part III concludes this series of articles. The three parts were intended for SAS programmers and data analysts to collectively illustrate a simplified framework for time series analyses. The framework in this series shows basic outlines of time series analysis namely the Augmented Dickey Fuller test, cointegration, and error correction models (other steps involved in time series analysis were not covered in any of the 3 parts). In all, the three parts were an effort intended to delineate in a simplified approach the basic steps to time series analysis. They were also meant to show how intuitive simple SAS DATA steps together with SAS procedures, such as PROC REG, could be used to approach solutions for such complex data structures. Future work may include additional steps such as seasonal adjustments in time series and the addition of visual enhancements such as graphs and charts of the time series variables and error series. Also outputs generated by other SAS procedures such as PROC ARIMA and PROC TIMESERIES could be presented and compared with outputs generated using these simple techniques. REFERENCES 1. Banergee, A. Dolado, J., Galbraith, J.W. and Hendry, D.F. (1993), Co-integration, Error Correction, and the Econometric Analysis of Time Series, Oxford University Press, Oxford 2. DeBoef, Suzanna. 2001,Testing for Cointegrating Relationships with Near-integrated Data. Political Analysis 9: Engle, R.F. and C.W.J. Granger (1987), Cointegration and Error-Correction: 4. Representation, Estimation, and Testing, Econometrica 55 (March), pp Granger, Clive W.J., 1981, Some properties of time series data and their use in econometric model specification, Journal of Econometrics, 16, Granger, Clive W.J., 1983, Co-integrated variables and error-correcting models, University of California, San Diego, Department of Economics Working Paper: Hahn, Gerald J. (1977). Fitting Regression Models with No Intercept Term, Journal of Quality Technology, Vol. 9, No. 2, April 1977, pp Mohamed, Ismail E. (2009) Time Series Analysis Using SAS-Part II: Cointegration Examining Time Series Long-run Relationships, Poster presentation, SAS Global Forum, Washington, DC, March 22-25, Mohamed, Ismail E. (2009) Simulating Time Series Testing Using SAS - Part I: The Augmented Dickey-Fuller (ADF) Test, Poster presentation, SAS Global Forum, Washington, DC, March 22-25, N.P. Singh, Ranjit Kumar, R.P. Singh and P.K. Jain (2005).Is Futures Market Mitigating Price Risk: An Exploration of Wheat and Maize Market. Agricultural Economics Research Review Vol. 18 (Conference No.) 2005 pp ACKNOWLEDGEMENTS My sincere thanks to everyone I have had the pleasure of exchanging time Series analysis related ideas with in recent years. Special thanks to Ian Keith, PMP, Senior Program Analyst and Jay Schultz, Ph.D., Senior Economist, both with the Federal Housing Finance Agency (FHFA), for their constructive suggestions which added a great deal to this paper. My sincere appreciation goes to Eric Wolf, VP of L3 STRATIS for his continuous encouragement and support. TRADEMARKS SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. CONTACT INFORMATION Name: Ismail E. Mohamed, Ph.D, Task Lead/Software Engineer 5 Enterprise: L3 STRATIS/ U.S. Department of Housing & Urban Development Address: 451 7th Street, SW, Room 8212, City, State ZIP: Washington, DC Work Phone: (202) ismail.mohamed@l-3com.com, Ismail.e.mohamed@hud.gov 7