20.04.2006adaptive expectations & partial adjustment models 1 adaptive expectations &...

20.04.2006 Adaptive expectations & partial adjustment models

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Adaptive Expectations &

Partial Adjustment Models

Presented & prepared

by Marta Stępień and Cinnie Tijus


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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?


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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


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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


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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

0 vt


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How are the models built?Adaptive expectations (1)

Suppose that expectations of future income is formed as follows:

Xet+1 - Xe

t = Xt - Xe

t) 0 < Xe

t+1 = Xt + (1- Xet

Substitute in for Xet the same equation:

Xet+1 = Xt + (1- Xt-1 + (1- Xe t-1]

Repeat this substitution to get:

Xet+1 = Xt + (1- Xt-1 + (1- X t-1 + ...

Thus adaptive expectations assume people weight all past values with the weights falling off exponentially.


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Suppose that Y depends on next period’s expected X:Yt = 0 + 1 Xe

t+1 + ut (1)

Xet+1 = Xt + (1 -Xe

t (2)

or Xt = Xet+1 - (1- Xe

t (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- Xe

t) + vt


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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 )


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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)


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So we do the following transformation:

Yt - Yt-1 = (Y*t - Yt-1) = (Yt* = + Xt + ut –Yt-1)= + Xt - Yt-1+ ut t

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:^ ^ ^ ^ ^ ^ ^ ^


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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 is 1/


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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 distributionWhere 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


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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


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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


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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


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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)


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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 termwhere:

βδ is the elasticity of consumption with respect to actual income;

β is the elasticity of consumption with respect to permanent income;


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How can we use AE and PAM?(3)

In the paper In our model

EU15 EU25

annual data quarterly data

1980 - 2005 1995 - 2005

GDP at constant prices GDP at constant prices

Private consumption expenditure Private consumption expenditure

from the EC from the Eurostat

Data:


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How can we use AE and PAM?(4)Results

Country βδ δ β

Belgium 0.415 0.508 0.817

Germany 0.356 0.427 0.834

Greece 0.582 0.737 0.790

Spain 0.458 0.445 1.029

France 0.282 0.266 1.060

Ireland 0.2 0.255 0.784

Italy -0.026 0.006 -4.333

Netherlands 0.191 0.225 0.849

Austria 0.215 0.283 0.760

Finland 0.03 0.044 0.682

Denmark 0.061 0.086 0.709

Sweden 0.415 0.469 0.885

UK 0.612 0.49 1.249

EU15 0.451 0.446 1.011

Country βδ δ β

Belgium 0.421 0.493 0.854

Germany 0.503 0.547 0.920

Greece 0.194 0.198 0.980

Spain 0.63 0.724 0.870

France 0.543 0.586 0.927

Ireland 0.461 0.675 0.683

Italy 0.685 0.719 0.953

Netherlands 0.676 0.735 0.920

Austria 0.657 0.721 0.911

Finland 0.396 0.402 0.985

Denmark 0.513 0.652 0.787

Sweden 0.493 0.513 0.961

UK 0.582 0.601 0.968

EU15 0.531 0.609 0.872


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How can we use AE and PAM?(5)Results

Czech Rep. 0.216 0.208 1.038

Estonia 0.468 0.442 1.059

Cyprus 0.625 0.576 1.085

Lithuania 0.173 0.13 1.331

Poland 0.043 0.077 0.558

Slovenia 0.619 0.853 0.726

Slovakia 0.131 0.154 0.851

EU25 0.407 0.403 1.010


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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.

20.04.2006adaptive expectations & partial adjustment models 1 adaptive expectations &...

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