Adaptive Expectations

1

• Adaptive Expectations
• Partial Adjustment Models
• Presented prepared
• by
• Marta Stepien and Cinnie Tijus

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?

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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
• Xet1 - Xet ? (Xt - Xet) 0 lt ? lt 1
• Xet1 ? Xt (1- ?) Xet
• Substitute in for Xet the same equation
• Xet1 ? Xt (1- ?) ? Xt-1 (1- ?) Xe t-1
• Repeat this substitution to get
• Xet1 ? Xt (1- ?) ? Xt-1 (1- ?)2 ? X t-1
  ...
• Thus adaptive expectations assume people weight
  all past values with the weights falling off
  exponentially.

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How are the models built?Adaptive expectations
(2)

• Suppose that Y depends on next periods expected
  X
• Yt ?0 ?1 Xet1 ut
  (1)
• Xet1 ? Xt (1 -?) Xet
  (2)
• or ? Xt Xet1 - (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 ?1Xet1 ut -
• - (1-?)?0 ?1(1- ?) Xet (1- ?) ut-1
• Yt -(1-?)Yt-1 ??0 ?1 (Xet1 - (1- ?) Xet)
  vt

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How are the models built?Adaptive expectations
(3)

• After substitution
• Yt -(1-?)Yt-1 ??0 ?1?Xt 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 lt ? lt 1,
  towards optimal or desired level, Yt
• Yt - Yt-1 ? (Yt - Yt-1)

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How are the models built?Partial Adjustment (2)

• So we do the following transformation
• Yt - Yt-1 ? (Yt - 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
• 
  
• g(1-b1) ab0/(1-b1) bb2/(1-b1)

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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, YtYt-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
• -gt Durbins h (for first order correlation)
• h(1-0.5d)(n/(1-n(var(? ))0.5 -gtStandard Normal
  distribution
• Where dDW, n is the sample size and ?, the
  estimated coefficient on Yt-1.
• H0 No serial correlation. Reject of H0 if
  hgt1.96

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Problems in these models (2)

• -gt 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 a30 using the
  following statistic LMnR2 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
• -gtThe desirable stock of money depends on
  anticipated incomes and rates of return for the
  different past periods
• -gtThe actual stock of money will adjust to the
  desired level via the standard PAM
• -gtThe 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 a ßYtp
• 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 ?(Ctp Ct-1) et , 0lt ? lt 1
• where ? is the partial adjustment coefficient
• Ytp Yt-1p d(Yt Yt-1p) , 0lt d lt 1
• where d is the adaptive expectations
  coefficient
• Estimated equation (in logs)
• Ct ad ßdYt (1 d) Ct-1 error term
• where
• ßd is the elasticity of consumption with respect
  to actual income
• ß is the elasticity of consumption with respect
  to permanent income

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How can we use AE and PAM?(3)

• Data

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
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How can we use AE and PAM?(4)Results
Country ßd d ß
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 ßd d ß
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.