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Centre of Full Employment and Equity Short-run models and Error Correction Mechanisms Professor Bill Mitchell Director, Centre of Full Employment and Equity Department of Economics University of Newcastle Australia Slide 2 Centre of Full Employment and Equity Objectives • To introduce the concept of a short-run model in economics. • To show how short- and long-run models interact. • To explain the concept of an Error Correction Mechanism (ECM). • To show how ECM and cointegration work together. Slide 3 Centre of Full Employment and Equity Long-run model review • Economic theory is essentially static and mostly considers equilibrium relationships. • Equilibrium (long-run) relations are normally in terms of levels. • The problem is that with non-stationary variables we are prone to finding spurious relationships if we run regressions in levels. Slide 4 Centre of Full Employment and Equity Figure 1 Z1, Z2 and Z4 • The Z variables were simulated using random walk functions with r = 1: y t r y t 1 u t • Any relation between them is spurious and because they contain stochastic trends. 30 20 10 0 -10 -20 -30 60 65 70 75 Z1 80 85 Z2 90 95 00 Z4 Slide 5 Centre of Full Employment and Equity So this equation exhibits “good” econometric results but is in fact spurious and tells us nothing at all. The “good” is qualified b/c the DW statistic is the clue. Slide 6 Centre of Full Employment and Equity The clue is in the residuals… 30 20 10 8 0 4 -10 0 -4 -8 -12 60 65 70 75 R e s id u a l 80 85 Ac tu a l 90 95 00 F itte d Slide 7 Centre of Full Employment and Equity The long-run model quandary • So how do we proceed? • In the 1970s, the approach was to take differences? Slide 8 Centre of Full Employment and Equity Taking differences removes trends 3 1 y t y t y t 1 2 1 0 -1 -2 -3 60 65 70 75 80 85 90 95 00 90 95 00 DZ1 3 2 1 0 -1 -2 -3 60 65 70 75 80 85 DZ2 Slide 9 Centre of Full Employment and Equity Taking differences… • Do we still have a relationship? • To test it we would run Z1t = b0 + b1Z2t + b2Z4t + et Slide 10 Centre of Full Employment and Equity Levels and differences (3) y t a 0 a1 x t a 2 z t e t yt (4) y t 1 a1 x t x t 1 a 2 z t z t 1 e t e t 1 y t a1 x t a 2 z t u t u t et Problems ? Question 1: What are the problems of estimating economic relationships in difference form like Equation (4), given that it can overcome the problem of non-stationarity in the levels of the variables concerned? Slide 11 Centre of Full Employment and Equity Error Correction Approach • This approach attempts to use differenced data to model the short-run adjustments but also take into account and estimate long-run information. • Consider this long-run model: y t 0 1 xt Slide 12 Centre of Full Employment and Equity Equilibrium and disequilibrium (6) y t 0 1 xt (6 a ) y t xt 0 (6 b ) y t xt 0 Question 2: What are the properties of Equation (6)? Does it tell you about the path of adjustment for y if x changes? Question 3: What are some of the reasons why equilibrium may not hold in every period? In a forecasting environment why would it be necessary to know about the nature of disequilibrium adjustment paths? Slide 13 Centre of Full Employment and Equity Equilibrium and disequilibrium (6) y t xt (6 a ) y t xt 0 (6 b ) y t xt 0 When Equation 6(b) holds we cannot observe the relationship in Equation (6). But we can observe the short-run, dynamic relationship that would reduce to Equation (6) whenever equilibrium occurs. So we need to learn a bit about the short-run models. Slide 14 Centre of Full Employment and Equity Short-run model • Short-run models are also called adjustment functions or dynamic models or lagged models. • A typical (simplified) version is the first-order model: y t a 0 1 y t 1 b 1 x t b 2 x t 1 u t • The order is selected to “soak” up the serial correlation (the “missing dynamics”) Slide 15 Centre of Full Employment and Equity Properties of short-run model (8) y t a 0 1 y t 1 b 1 x t b 2 x t 1 u t Question 4: What are the properties of Equation (8)? Tell a story in words about the process through which the longrun relationship is re-established if x was to change in a particular period? Slide 16 Centre of Full Employment and Equity Properties of short-run model (6) y t xt (8) y t a 0 1 y t 1 b 1 x t b 2 x t 1 u t Question 4: Parameter b1 measures the immediate impact of a change in x on y. It is not the long-run impact that would occur from one equilibrium to another though. Why not? What is the difference between b1 and 1? Can you find an expression that links b1 and 1? Solve the steady-state properties of (8). Slide 17 Centre of Full Employment and Equity Solving for the steady-state… y t xt (6) (8) y t a 0 1 y t 1 b 1 x t b 2 x t 1 u t y t y t 1 y * x t x t 1 x y (1 1 ) a 0 b 1 b 2 x * y * a0 (1 1 ) b1 b 2 (1 1 ) x * * * Slide 18 Centre of Full Employment and Equity Questions 6 to 9 … • Question 6: Assume that 1 = 0.9, b1 = 0.6 and b2 = 0.3. Starting from an equilibrium position, how long does it take for y to return to that equilibrium, if x increases by a unit and remains at that new level? • Question 7: What is the change in y in the first period after the shock? What is the change in y in the second period? What is the total change? • Question 8: How is the shift in equilibrium dependent on the value taken by the AR parameter? • Spreadsheet demonstration. Slide 19 Centre of Full Employment and Equity Error Correction models • The basic dynamic model may also suffer from non-stationarity problems. • We have seen the differencing is unsatisfactory. • Error Correction Mechanism (ECM) models begin with the basic short-run model. • After re-parameterising, the ECM form has both dynamic and steady-state information in the one equation. Slide 20 Centre of Full Employment and Equity ECM questions Question 9: See if you can perform the re-parameterisation to get the ECM model which combines differences and levels. It is shown below as Equation (10). Slide 21 ECM form and the steady-state (8) y t a 0 1 y t 1 b 1 x t b 2 x t 1 u t y t y t 1 a 0 y t 1 1 y t 1 b 1 x t b 1 x t 1 b 1 x t 1 b 2 x t 1 u t b1 b 2 a0 y t b 1 x t (1 1 ) y t 1 x t 1 (1 1 ) (1 1 ) (10) y t b 1 x t (1 1 ) y t 1 0 0 x t 1 Question 10: Would you say that Equation (8) and Equation (10) are equivalent? What are the advantages of Equation (10) relative to Equation (8)? Centre of Full Employment and Equity ECM form and the steady-state (10) b1 b 2 a0 y t b 1 x t (1 1 ) y t 1 x t 1 (1 1 ) (1 1 ) (10) y t b 1 x t (1 1 ) y t 1 0 0 x t 1 y* a0 (1 1 ) b1 b 2 (1 1 ) x * Question 11: Provide an interpretation of the expression in square brackets in Equation (10). Have you already encountered an expression like this earlier in this lecture? Slide 23 Centre of Full Employment and Equity ECM form and the steady-state (10) b1 b 2 a0 y t b 1 x t (1 1 ) y t 1 x t 1 (1 1 ) (1 1 ) (6 ) y t 1 0 1 x t 1 e t 1 e t 1 y t 1 0 1 x t 1 You can see that the term in square brackets is equivalent to the expression for disequilibrium in the steady-state. Slide 24 Centre of Full Employment and Equity ECM form and the steady-state (10) b1 b 2 a0 y t b 1 x t (1 1 ) y t 1 x t 1 (1 1 ) (1 1 ) (6 ) y t 1 0 1 x t 1 e t 1 e t 1 y t 1 0 1 x t 1 Question 12: Is the term in square brackets stationary given that it is in terms of levels? Under what conditions will it be stationary? Slide 25 Centre of Full Employment and Equity Cointegration and ECM model (6) y t 0 1 x t et (10) b1 b 2 a0 y t b 1 x t (1 1 ) y t 1 x t 1 (1 1 ) (1 1 ) (10 ) y t b 1 x e t 1 u t Two-step procedure for estimating the model: • Test for cointegration in Equation (6). • If null accepted then the residuals would be stationary. • Estimate (10) with residuals from CI Equation (6) as the ECM term. Slide 26 Centre of Full Employment and Equity End of Talk Slide 27