1. 20.04.2006Adaptive expectations & partial adjustment models 1 Adaptive Expectations & Partial Adjustment Models Presented & prepared by Marta Stępień and Cinnie Tijus

2. 20.04.2006Adaptive expectations & partial adjustment models 2 Outline of the presentation  What are Adaptive Expectations and Partial Adjustments?  How are the models built?  Where are they used?  How can we use AE and PAM?

3. 20.04.2006Adaptive expectations & partial adjustment models 3 What are adaptive expectations and partial adjustments?  In Adaptive Expectations Model: Expected level of Y t in the future (not observable) based on current expectations or on what happened in the past  In Partial Adjustment Model: Desirable or optimal level of Y t which is unobservable. Agents cannot adjust fully to changing conditions

4. 20.04.2006Adaptive expectations & partial adjustment models 4 How are the models built? Introduction (1)  Suppose the effect of a variable X on the dependent variable Y is spread out over several time periods; we get a distributed lag model (finite or infinite): Y t = a 0 +  0 X t +  1 X t-1 +  2 X t-2 +  3 X t-3 +... + u t  we have to constraint the coefficients to follow the pattern; for the geometric lag we assume that the coefficients decline exponentially (Koyck lag):  i =  0 i  so: Y t = a 0 +  0 ( X t + X t-1 + 2 X t-2 +... ) + u t

5. 20.04.2006Adaptive expectations & partial adjustment models 5 How are the models built? Introduction (2)  We use Koyck transformation: Y t = a 0 +  0 ( X t + X t-1 + 2 X t-2 +... ) + u t Y t-1 = a 0 +  0 ( X t-1 + X t-2 + 2 X t-3 +... ) + u t-1 Y t-1 = a 0 +  0 ( X t-1 + 2 X t-2 + 3 X t-3 +... ) + u t-1 Y t - Y t-1 = (1-)a 0 +  0 X t + u t - u t-1  The estimated equation becomes: Y t = (1-)a 0 +  0 X t + Y t-1 + u t - u t-1  0 v t

6. 20.04.2006Adaptive expectations & partial adjustment models 6 How are the models built? Adaptive expectations (1)  Suppose that expectations of future income is formed as follows: X e t+1 - X e t = X t - X e t ) 0 <  X e t+1 =  X t + (1- X e t  Substitute in for X e t the same equation: X e t+1 =  X t + (1-  X t-1 + (1-  X e t-1 ]  Repeat this substitution to get: X e t+1 =  X t + (1-  X t-1 + (1-    X t-1 +... Thus adaptive expectations assume people weight all past values with the weights falling off exponentially.

7. 20.04.2006Adaptive expectations & partial adjustment models 7 How are the models built? Adaptive expectations (2)  Suppose that Y depends on next period’s expected X: Y t =  0 +  1 X e t+1 + u t (1) X e t+1 =  X t + (1 -X e t (2) or X t = X e t+1 - (1- X e t (2a)  Use Koyck transformation for equation (1): (1-Y t-1 =(1- 0 +  1 (1-  X e t + (1-  u t-1 (3) Y t - (1-Y t-1 =  0 +  1 X e t+1 + u t - - [(1- 0 +  1 (1-  X e t + (1-  u t-1 ] Y t -(1-Y t-1 =  0 +  1 (X e t+1 - (1-  X e t ) + v t

8. 20.04.2006Adaptive expectations & partial adjustment models 8 How are the models built? Adaptive expectations (3)  After substitution: Y t -(1-Y t-1 =  0 +  1 X t + v t Y t =  0 +  1  X t + (1-Y t-1 + v t  Estimate: Y t =  0 +  1 X t +  2 Y t-1 + v t  Where: ^ ^ ^ ^  = 1 -  2  1 =  1 /(1 -  2 )

9. 20.04.2006Adaptive expectations & partial adjustment models 9 How are the models built? Partial Adjustment (1)  We get this equation to estimate: Y t * =  +  X t + u t Where Y* are the desired inventories, X are the sales  inventories partially adjust, 0 <  < 1, towards optimal or desired level, Y* t : Y t - Y t-1 =  (Y* t - Y t-1 )

10. 20.04.2006Adaptive expectations & partial adjustment models 10 How are the models built? Partial Adjustment (2)  So we do the following transformation: Y t - Y t-1 =  (Y* t - Y t-1 ) =  (Y t * =  +  X t + u t –Y t-1 ) =   +   X t -  Y t-1 +  u t t  We obtain: Y t =  + (1 -  Y t-1 +  X t +  t  Then we have the estimated equation: Y t =  0 +  1 Y t-1 +  2 X t + t  And we can use ordinary least squares regression to get: ^ ^ ^ ^           

11. 20.04.2006Adaptive expectations & partial adjustment models 11 How are the models built? Partial Adjustment (3) Long-run & short-run effects in PAM:  Suppose our model is: Y t * =  0 +  1 X t + e t Y t - Y t-1 = (Y t * - Y t-1 )  We estimate: Y t =  0 + (1- ) Y t-1 +  1 X t + e t An increase in X of 1 unit increases Y in the ST by  1 units  In the LR, Y t =Y t-1, so we get: Y t =  0 +  1 X t + e t the LR effect of X on Y is  1 /

12. 20.04.2006Adaptive expectations & partial adjustment models 12 Problems in these models (1)  If the error term is serially correlated, then the error term is correlated with lagged dependent variable. Y t =  0 +  1 X t +  2 Y t-1 +  t And  t =   t-1 + v t Y t-1 depends in part on  t-1 and hence Y t-1 and  t are correlated.  Tests: -> Durbin’s h (for first order correlation) h=(1-0.5d)(n/(1-n(var( )) 0.5 ->Standard Normal distribution Where d=DW, n is the sample size and, the estimated coefficient on Y t-1. H 0 : No serial correlation. Reject of H 0 if |h|>1.96

13. 20.04.2006Adaptive expectations & partial adjustment models 13 Problems in these models (2) -> Lagrange Multiplier Test a) Estimate the model by OLS and get the residual e t b) Estimate the following equation by OLS e t = a 0 + a 1 X t + a 2 Y t-1 + a 3 e t-1 + u t c) Test the hypothesis that a 3 =0 using the following statistic LM=nR 2 with n, the sample size.  Instrumental Variable Estimation Method: replace the lagged dependent variable with an instrument that is correlated with Y t-1 but not with error

14. 20.04.2006Adaptive expectations & partial adjustment models 14 Where AE & PA models are used? Literature Review (1)  On the Long-Run and Short-Run Demand for Money, Chow G. C.,1966; Maximum Likelihood Estimates of a Partial Adjustment-Adaptive Expectations Model of the Demand for Money, D. L. Thornton, The Review of Economics and Statistics, Vol.64, 1982. to estimate the short-run demand for money: ->The desirable stock of money depends on anticipated incomes and rates of return for the different past periods ->The actual stock of money will adjust to the desired level via the standard PAM ->The expectational variables will adjust via the AEM

15. 20.04.2006Adaptive expectations & partial adjustment models 15 Where AE & PA models are used? Literature Review (2)  How the Bundesbank Conducts Monetary Policy, R. Clarida, M. Gertler, NBER, Working Paper No. 5581, 1996; Monetary Policy and the Term Structure of Interest Rate, B. McCallum, NBER, Working Paper No. 4938.  PAM is used for capturing the type of smoothing of interest-rate;  It is taken as given that the target interest rate is set and it is changed in pursuit of macroeconomic objectives;  The target interest rate tends to adjust slowly and in relatively smooth pattern;  Estimating the European Union Consumption Function under the Permanent Income Hypothesis, Athanasios Manitsaris, International Research Journal of Finance and Economics, 2006

16. 20.04.2006Adaptive expectations & partial adjustment models 16 How can we use AE and PAM?(1)  The specifications adopted in the paper refer to the combined partial adjustment and adaptive expectation model;  The permanent income hypothesis:  Provided by Milton Friedman in 1957;  People in trying to maintain a rather constant standard of living base their consumption on what they consider their ‘normal’ (permanent) income, althought their actual income may very over time changes in actual income are assumed to be temporary and thus have little effect on consumption; C tp = α + βY tp  PROBLEM: permanent income and consumption expenditure are unobservable; they need to be transformed into observable variables (we use AE and PAM)

17. 20.04.2006Adaptive expectations & partial adjustment models 17 How can we use AE and PAM?(2)  C t – C t-1 = γ(C tp – C t-1 ) + ε t, 0< γ < 1 where γ is the partial adjustment coefficient;  Y tp – Y t-1p = δ(Y t – Y t-1p ), 0< δ < 1 where δ is the adaptive expectations coefficient;  Estimated equation (in logs): C t = αδ + βδY t + (1 – δ) C t-1 + error term where:  βδ is the elasticity of consumption with respect to actual income;  β is the elasticity of consumption with respect to permanent income;

18. 20.04.2006Adaptive expectations & partial adjustment models 18 How can we use AE and PAM?(3) In the paperIn our model EU15EU25 annual dataquarterly data 1980 - 20051995 - 2005 GDP at constant prices Private consumption expenditure from the ECfrom the Eurostat  Data:

19. 20.04.2006Adaptive expectations & partial adjustment models 19 How can we use AE and PAM?(4) Results Country βδδβ Belgium0.4150.5080.817 Germany0.3560.4270.834 Greece0.5820.7370.790 Spain0.4580.4451.029 France0.2820.2661.060 Ireland0.20.2550.784 Italy-0.0260.006-4.333 Netherlands0.1910.2250.849 Austria0.2150.2830.760 Finland0.030.0440.682 Denmark0.0610.0860.709 Sweden0.4150.4690.885 UK0.6120.491.249 EU150.4510.4461.011 Country βδδβ Belgium0.4210.4930.854 Germany0.5030.5470.920 Greece0.1940.1980.980 Spain0.630.7240.870 France0.5430.5860.927 Ireland0.4610.6750.683 Italy0.6850.7190.953 Netherlands0.6760.7350.920 Austria0.6570.7210.911 Finland0.3960.4020.985 Denmark0.5130.6520.787 Sweden0.4930.5130.961 UK0.5820.6010.968 EU150.5310.6090.872

20. 20.04.2006Adaptive expectations & partial adjustment models 20 How can we use AE and PAM?(5) Results Czech Rep.0.2160.2081.038 Estonia0.4680.4421.059 Cyprus0.6250.5761.085 Lithuania0.1730.131.331 Poland0.0430.0770.558 Slovenia0.6190.8530.726 Slovakia0.1310.1540.851 EU250.4070.4031.010

21. 20.04.2006Adaptive expectations & partial adjustment models 21 Sources  On the Long-Run and Short-Run Demand for Money, Chow G. C.,1966;  Maximum Likelihood Estimates of a Partial Adjustment-Adaptive Expectations Model of the Demand for Money, D. L. Thornton, The Review of Economics and Statistics, Vol.64, 1982.  How the Bundesbank Conducts Monetary Policy, R. Clarida, M. Gertler, NBER, Working Paper No. 5581, 1996;  Monetary Policy and the Term Structure of Interest Rate, B. McCallum, NBER, Working Paper No. 4938, 1994;  Estimating the European Union Consumption Function under the Permanent Income Hypothesis, Athanasios Manitsaris, International Research Journal of Finance and Economics, 2006;  The Estimation of Partial Adjustment Models with Rational Expectations, Kennan J., 1979.