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Adaptive Expectations & Partial Adjustment Models Presented & prepared by Marta St ę pie ń and Cinnie Tijus

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## Adaptive Expectations & Partial Adjustment Models Presented & prepared by Marta St ę pie ń and Cinnie Tijus

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**Adaptive Expectations**& Partial Adjustment Models Presented & prepared by Marta Stępień and Cinnie Tijus Adaptive expectations & partial adjustment models**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? Adaptive expectations & partial adjustment models**What are adaptive expectations and partial adjustments?**• In Adaptive Expectations Model: Expected level of Yt 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 Yt which is unobservable. Agents cannot adjust fully to changing conditions Adaptive expectations & partial adjustment models**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): Yt = a0 + 0Xt + 1 Xt-1+ 2Xt-2+ 3Xt-3+ ... + ut • 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: Yt = a0 + 0( Xt + Xt-1 + 2Xt-2 + ... ) + ut Adaptive expectations & partial adjustment models**How are the models built?Introduction (2)**• We use Koyck transformation: Yt = a0 + 0( Xt + Xt-1 + 2Xt-2 + ... ) + ut Yt-1 = a0 + 0( Xt-1 + Xt-2 + 2Xt-3 + ... ) + ut-1 Yt-1 = a0 + 0( Xt-1+2Xt-2 + 3Xt-3 + ... ) + ut-1 Yt - Yt-1 = (1-)a0 + 0 Xt + ut - ut-1 • The estimated equation becomes: Yt = (1-)a0 + 0 Xt + Yt-1 + ut - ut-1 0vt Adaptive expectations & partial adjustment models**How are the models built?Adaptive expectations (1)**• Suppose that expectations of future income is formed as follows: Xet+1 - Xet = (Xt - Xet) 0 < < 1 Xet+1 = Xt + (1- ) Xet • Substitute in for Xet the same equation: Xet+1 = Xt +(1- ) [ Xt-1 + (1- ) Xet-1] • Repeat this substitution to get: Xet+1 = Xt +(1- ) Xt-1 + (1- )2 X t-1 + ... Thus adaptive expectations assume people weight all past values with the weights falling off exponentially. Adaptive expectations & partial adjustment models**How are the models built?Adaptive expectations (2)**• Suppose that Y depends on next period’s expected X: Yt = 0 + 1 Xet+1 + ut (1) Xet+1 = Xt+ (1 -) Xet (2) or Xt= Xet+1 - (1- ) Xet(2a) • Use Koyck transformation for equation (1): (1-)Yt-1 =(1-)0 + 1(1- ) Xet + (1- ) ut-1 (3) Yt -(1-)Yt-1 =0+1Xet+1 + ut - -[(1-)0 + 1(1- ) Xet + (1- ) ut-1] Yt -(1-)Yt-1 = 0 + 1 (Xet+1 - (1- ) Xet) + vt Adaptive expectations & partial adjustment models**How are the models built?Adaptive expectations (3)**• After substitution: Yt -(1-)Yt-1 = 0 + 1Xt + vt Yt = 0 + 1 Xt + (1-)Yt-1 + vt • Estimate: Yt = 0 + 1 Xt + 2 Yt-1 + vt • Where: ^ ^ ^ ^ = 1 - 2 1 = 1 /(1 - 2 ) Adaptive expectations & partial adjustment models**How are the models built?Partial Adjustment (1)**• We get this equation to estimate: Yt* = + Xt + ut Where Y* are the desired inventories, X are the sales • inventories partially adjust , 0 < < 1, towards optimal or desired level, Y*t : Yt - Yt-1 = (Y*t - Yt-1) Adaptive expectations & partial adjustment models**How are the models built?Partial Adjustment (2)**• So we do the following transformation: Yt - Yt-1 = (Y*t - Yt-1) = (Yt* = + Xt + ut –Yt-1) = + Xt- Yt-1+ utt • We obtain: Yt = + (1- Yt-1 + Xt + t • Then we have the estimated equation: Yt = 0 + 1Yt-1+ 2Xt + t • And we can use ordinary least squares regression to get: ^ ^ ^ ^ ^ ^ ^ ^ g=(1-b1) a=b0/(1-b1) b=b2/(1-b1) Adaptive expectations & partial adjustment models**How are the models built?Partial Adjustment (3)**Long-run & short-run effects in PAM: • Suppose our model is: Yt* = 0 + 1 Xt + et Yt - Yt-1 = (Yt* - Yt-1) • We estimate: Yt = 0 + (1- ) Yt-1 + 1 Xt + et An increase in X of 1 unit increases Y in the ST by 1 units • In the LR, Yt=Yt-1, so we get: Yt = 0 + 1 Xt + et the LR effect of X on Y is1/ Adaptive expectations & partial adjustment models**Problems in these models(1)**• If the error term is serially correlated, then the error term is correlated with lagged dependent variable. Yt = 0 + 1 Xt + 2 Yt-1 + t And t = t-1 + vt Yt-1 depends in part on t-1 and hence Yt-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 Yt-1. H0: No serial correlation. Reject of H0 if |h|>1.96 Adaptive expectations & partial adjustment models**Problems in these models(2)**-> Lagrange Multiplier Test a) Estimate the model by OLS and get the residual et b) Estimate the following equation by OLS et = a0 + a1Xt + a2 Yt-1 + a3 et-1 + ut c) Test the hypothesis that a3=0 using the following statistic LM=nR2 with n, the sample size. • Instrumental Variable Estimation Method: replace the lagged dependent variable with an instrument that is correlated with Yt-1 but not with error Adaptive expectations & partial adjustment models**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 Adaptive expectations & partial adjustment models**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 Adaptive expectations & partial adjustment models**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; Ctp = α + βYtp • PROBLEM: permanent income and consumption expenditure are unobservable; they need to be transformed into observable variables (we use AE and PAM) Adaptive expectations & partial adjustment models**How can we use AE and PAM?(2)**• Ct – Ct-1 = γ(Ctp – Ct-1) + εt , 0< γ < 1 where γ is the partial adjustment coefficient; • Ytp – Yt-1p = δ(Yt – Yt-1p) , 0< δ < 1 where δ is the adaptive expectations coefficient; • Estimated equation (in logs): Ct = αδ + βδYt + (1 – δ) Ct-1 + error term where: • βδis the elasticity of consumption with respect to actual income; • βis the elasticity of consumption with respect to permanent income; Adaptive expectations & partial adjustment models**Data:**How can we use AE and PAM?(3) Adaptive expectations & partial adjustment models**How can we use AE and PAM?(4)Results**Adaptive expectations & partial adjustment models**How can we use AE and PAM?(5)Results**Adaptive expectations & partial adjustment models**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. Adaptive expectations & partial adjustment models