DYNAMIC MODELS

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DYNAMIC MODELS

• Week 10
• Prepared by
• Dr. Zerihun Gudeta Alemu

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TYPES OF DYNAMIC MODELS

• A) Distributed-lag Models if a model contains
  both the current and past (lagged) values of the
  explanatory variables.
• Yt ? ?0Xt ?1Xt-1 ?kXt-k ut
• B) Autoregressive Models if a model includes one
  or more past values of the dependent variable
  among its explanatory variables.
• Yt ?0 ?1Xt ?2Yt-1 vt
• C) Causality Relationship between two variables
  can be unidirectional, bidirectional, and neither
  bilateral nor unilateral (i.e. independence).

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A) DISTRIBUTED LAG MODELS

• Finite versus infinite distributed lag models
• Yt ? ?0Xt ?1Xt-1 ?kXk ut Finite
  model
• Yt ? ?0Xt ?1Xt-1 ut Infinite
  Model
• ?0 is short run multiplier ?0 ?1 ?3 .. ?k
  give interim
• Multiplier ? ?I ? if ? exists is long-term
  multiplier.
• Problem of estimation No guideline as to the
  maximum
• length of the lag. A bottom-up approach (adding
  lags until
• insignificant or one of explanatory variables
  sign changes)
• is commonly used. This result in fewer degree of
  freedom,
• multicollinearity problem and data mining

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Koyck Model An example of distributed lag model

• Assume all ?s have the same sign and that the
  lag structure is infinite
• ?k ?0?k , k 0, 1, 2, 0 ? ? ? 1
• ? ? rate of decay 1 - ? ? speed of adjustment.
• i.e. effect of lagged Xs is geometrically damped
  as lag increases
• ?s cant change sign (0 ? ?)
• less weight to distant ?s (? ? 1)
• Sum of ?k up to infinity ?0(1/(1-?))

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Koyck Transformation

• Yt ? ?0Xt ?0?Xt-1 ?0?2Xt-2 ut (i)
• lag (i) by one period and multiply by ?
• ?Yt-1 ?? ?0?Xt-1 ?0?2Xt-2 ?0?3Xt-3
  ?tut-1 (ii)
• subtract (ii) from (i)
• Yt - ?Yt-1 ?(1 - ?) ?0Xt (ut - ?ut-1)
  (iii)
• or
• Yt ?(1 - ?) ?0Xt ?Yt-1 vt
  (iv)

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Koyck Model (Cont)

• Equation (iv) is an AR (autoregressive) model
• Yt-1 is a stochastic regressor (this violates
  CLRM assumption why?)
• vt is autocorrelated if ut is not why?
• Durbin Watson test statistic is not appropriate
  to test for the presence of fist order
  autocorrelation instead Durbin H is used for
  large samples. h statistic is asymptotically
  normally distributed with zero mean and unit
  variance recall P(-1.96 ? h ? 1.96) 0.95The

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Koyck model (Cont)

• Interpretation
• Median Lag
• ? time required for 50 of total ?Y following a
  unit sustained ?X
• e.g. Koyck Med Lag
• so ? ? ? lower speed of adjustment ( longer
  median lag)
• Mean Lag
• ? weighted average of all lags (lag-weighted
  average of time)
• If ? 0, mean lag
• e.g. Koyck Mean lag
• ? ? ? lower speed of adjustment ( longer mean
  lag)
• Koyck transformation is a useful mathematical
  device. In practice it is rationalized using
  appropriate theoretical underpinnings from
  Adaptive Expectation or Partial Adjustment
  Models.

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Adaptive Expectation (AE) Partial Adjustment
Models (PAM)

• Koyck model is devoid of any theoretical
  underpinning.
• AE and PAM Applied in Agricultural Supply
  Response Analysis.
• Assume we want to estimate the responsiveness of
  maize producers to incentive changes.
• If AE theory is used, maize output is expressed
  as a function of expected price.
• If PAM is used, desired level of maize output is
  expressed as a function of actual price.

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B) Autoregressive Models

• The following are requirements
• Stochastic explanatory variables (i.e. Yt-1) must
  be independent of vt .
• Test for serial correlation must be conducted
  using Durbin H test.
• Yt-1 is correlated with vt by construction
  (why?), so OLS yields biased and inconsistent
  estimators.
• To solve the latter problem, instrumental
  variables (Zt,) may be used as a proxy for Yt-1.
  Zt must be highly correlated with Yt-1 but
  uncorrelated with vt.

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C) Granger Causality

• Here we are interested in the investigation of
  the direction of causality between two variables,
  say, price of a certain agricultural product (Pt)
  and demand for the same product (Dt).
• Pt Granger-causes Dt if lagged Pts help predict
  Dt (when demand is regressed on its own past
  values/lags).

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Granger Causality

• Four Possible cases
• P ? D (unidirectional
  causality)
• D ? P (unidirectional
  causality)
• Bilateral causality All sets of
  coefficients are significant
• Independence Not significant

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Granger Causality (cont)

• Steps to test for granger causality
• a) Regress Dt on lagged Dt to obtain restricted
  RSSR. Lag length may be determined based on AIC
  or SIC criteria.
• b) Regress Dt on lagged Dt and lagged Pt to
  obtain unrestricted RSSUR. Lag length may be
  determined based on AIC or SIC criteria.
• c) H0 Pt does not Granger cause Dt
• H0 lagged Pt terms are insignificant
• d) Fm, n-k k is
  the of parameters in UR regression
• e) If Fcalc ? Fcrit then reject H0 ? lagged P
  terms are significant, ? P does Granger-cause GDP
• f) Repeat with P regressed on D to test H0 D
  does not Granger cause P