MACROECONOMETRICS

1
MACROECONOMETRICS

• LAB 3 DYNAMIC MODELS

2
ROADMAP

• What if we know that the effect lasts in time?
• Distributed lags
• ALMON
• KOYCK
• ADAPTIVE EXPECTATIONS
• PARTIAL ADJUSTMENT
• STATA not really too complicated here ?

3
How to do lags?

• Infinite?
• how many lags do we take?
• how to know?
• Unrestricted?
• do we impose any structure on the lags?
• this structure might be untrue?
• but there is also cost to unrestricted approach...

4
Unrestricted lags (no structure)

• It is always finite!
• N lags and no structure in parameters
• OLS works
• BUT
• n observations lost
• high multicollinearity
• imprecise, large s.e., low t, lots of d.f. Lost
• STRUCTURE COULD HELP

yt ? ?0 xt ?1 xt-1 ?2 xt-2 . . .
??n xt-n et
5
Arithmetic lag

• The effect of X eventually zero
• Linearly!
• The coefficients not independent of each other
• effect of each lag less than previous
• exactly like arithmetic series unu1d(n-1)

6
Arithmetic lag - structure
?i
?0 (n1)?
?1 n?
Linear lag
structure
?2 (n-1)?
?n ?
0 1 2 . . .
. . n n1
i
7
Arithmetic lag - maths

• X (log of) money supply and Y (log of) GDP, n12
  and g is estimated to be 0.1
• the effect of a change in x on GDP in the current
  period is b0(n1)g1.3
• the impact of monetary policy one period later
  has declined to b1ng1.2
• n periods later, the impact is bn g0.1
• n1 periods later the impact is zero

8
Arithmetic lag - estimation

• OLS, only need to estimate one parameter g
• STEP 1 impose restriction
• STEP 2 factor out the parameter
• STEP 3 define z
• STEP 4 decide n (no. of lags)

yt ? ?0 xt ?1 xt-1 ?2 xt-2 . . .
??n xt-n et
yt ? ? (n1)xt nxt-1 (n-1)xt-2 . .
. xt-n et
zt (n1)xt nxt-1 (n-1)xt-2 . . . xt-n
???
For n 4 zt 5xt 4xt-1 3xt-2
2xt-3 xt-4
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Arithmetic lag pros cons

• Advantages
• Only one parameter to be estimated!
• t-statistics ok., better s.e., results more
  reliable
• Straightforward interpretation
• Disadvantages
• If restriction untrue, estimators biased and
  inconsistent
• Solution? F-test! (see end of the notes)

10
Polynomial lag (ALMON)

• If we want a different shape of IRF...
• Its just a different shape
• Still finite the effect eventually goes to zero
• (by DEFINITION and not by nature!)
• The coefficients still related to each other BUT
  not a uniform pattern (decline)

11
Polynomial lag - structure
?i
?2
?1
?3
?0
?4
0 1 2 3 4
i
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Polynomial lag - maths

• n the lenght of the lag
• p degree of the polynomial
• For example a quadratic polynomial

?0 ?0 ?1 ?0 ?1 ?2 ?2 ?0
2?1 4?2 ?3 ?0 3?1 9?2 ?4
?0 4?1 16?2
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Polynomial lags - estimation

• OLS, only need to estimate p parameters g0,...,
  gp
• STEP 1 impose restriction
• STEP 2 factor out the unknown coefficients
• STEP 3 define z
• STEP 4 do OLS on yt ? ?0 z t0 ?1 z t1
  ?2 z t2 et

yt ? ?0?xt ??0 ?1 ?2?xt-1(?0 2?1
4?2)xt-2(?03?1 9?2)xt-3 (?0 4?1
16?2)xt-4 et
yt ? ?0 xt xt-1xt-2xt-3
xt-4?1xtxt-12xt-23xt-3 4xt-4
?2 xt xt-1 4xt-2 9xt-3 16xt-4 et
z t0 xt xt-1 xt-2 xt-3 xt-4
z t1 xt xt-1 2xt-2 3xt-3 4xt- 4
z t2 xt xt-1 4xt-2 9xt-3 16xt- 4
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Polynomial lag pros cons

• Advantages
• Fewer parameters to be estimated than in the
  unrestricted lag structure
• More precise than unrestricted
• If the polynomial restriction likely to be true
• More flexible than arithmetic DL
• Disadvantages
• If the restriction untrue, biased and
  inconsistent
• (see F-test in the end of the notes)

15
Arithmetic vs. Polynomial vs. ???

• Conclusion no. 1
• Data should decide about the assumed pattern of
  impulse-response function
• Conclusion no. 2
• We still do not know, how many lags!
• Conclusion no. 3
• We still have a finite no. of lags.

16
Geometric lag (KOYCK)

• Distributed lag is infinite ? infinite lag length
  (no time limits)
• BUT cannot estimate an infinite number of
  parameters!
• ? Restrict the lag coefficients to follow a
  pattern
• For the geometric lag the pattern is one of
  continuous decline at decreasing rate
• (we are still stuck with the problem of
  imposing fading out instead of observing it
  gladly, it is not really painful, as most
  processes behave like that anyway ?)

17
Geometric lag - structure
?i
Geometrically declining weights
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Geometric lag - maths

• Infinite distributed lag model
• yt ? ?0 xt ?1 xt-1 ?2 xt-2 .
  . . et
• yt ? ???i xt-i et
• Geometric lag structure
• ?i ???i?? where ? lt 1 and ??i?????
• Infinite unstructured geometric lag model
• yt ? ?0 xt ?1 xt-1 ?2 xt-2 ?3
  xt-3 . . . et
• AND??0?,? ?1??, ??2??2, ?3??3 ...
• Substitute ?i ???i gt infinite geometric
  lag
• yt ? ??xt ? xt-1 ?? xt-2 ??
  xt-3 . . .) et

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Geometric lag - estimation

• Cannot estimate using OLS
• yt-1 is dependent on et-1 ? cannot alow that
  (need to instrument)
• Apply Koyck transformation
• Then use 2SLS
• Only need to estimate two parameters f,b
• Have to do some algebra to rewrite the model in
  form that can be estimated.

20
Geometric lag Koyck transformation

• Original equation
• yt ? ??xt ? xt-1 ?? xt-2 ??
  xt-3 . . .) et
• Koyck rule lag everything once, multiply by ?
  and substract from the original
• yt-1 is dependent on et-1 so yt-1 is correlated
  with vt-1
• OLS will be consistent (it cannot distinguish
  between change in yt caused by yt-1 that caused
  by vt)

yt ? ??xt ? xt-1 ?? xt-2 ?? xt-3
. . .) et
? yt-1 ??? ??? xt-1 ?? xt-2 ?? xt-3
. . .) ? et-1
yt ? ? yt-1 ??????? ?xt (et ??? et-1)
yt ??????? ? yt-1 ?xt (et ??? et-1)
so yt ??? ?? yt-1 ??xt ?t
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Geometric lag - estimation

• Regress yt-1 on xt-1 and calculate the fitted
  value
• Use the fitted value in place of yt-1 in the
  Koyck regression and that is it!
• Why does this work?
• from the first stage fitted value is not
  correlated with et-1 but yt-1 is so fitted value
  is uncorrelated with vt (et -et-1 )
• 2SLS will produce consistent estimates of the
  Geometric Lag Model

22
Geometric lag pros cons

• Advantages
• You only estimate two parameters!
• Disadvantages
• We allow neither for heterogenous nor for
  unsmooth declining
• It has many well specified versions, among which
  two have particular importance
• ADAPTIVE EXPECTATIONS
• PARTIAL ADJUSTMENT MODEL
• (for both see next student presentation)

23
F-tests of restrictions

• Estimate the unrestricted model
• Estimate the restricted (any lag) model
• Calculate the test statistic
• Compare with critical value F(df1,df2)
• df1 number of restrictions
• df2 number of observations-number of variables
  in the unrestricted model (incl. constant)
• H0 residuals are the same, restricted model OK