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```					MACROECONOMETRICS

LAB 3 – DYNAMIC MODELS

   What if we know that the effect lasts in
time?
   Distributed lags
–   ALMON
–   KOYCK
   STATA not really too complicated here 
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...
Unrestricted lags (no structure)
–   It is always finite!
yt =  + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
   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
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: un=u1+d(n-1)
Arithmetic lag - structure

i
0 = (n+1)
1 = n
Linear
2 = (n-1)                      lag
.                                  structure
.
.
n = 

0        1   2    .   .      .   .     .   n   n+1 i
Arithmetic lag - maths
   X (log of) money supply and Y (log of) GDP, n=12 and  is
estimated to be 0.1
   the effect of a change in x on GDP in the current period is
0=(n+1)=1.3
   the impact of monetary policy one period later has declined to
1=n=1.2
   n periods later, the impact is n= =0.1
   n+1 periods later the impact is zero

 E ( yt )
i =
xt i
Arithmetic lag - estimation
   OLS, only need to estimate one parameter: 
   STEP 1: impose restriction

yt =  + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
   STEP 2: factor out the parameter
yt =  +  [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et
   STEP 3: define z
zt = [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n]
   STEP 4: decide n (no. of lags)          ???
For n = 4:      zt = [ 5xt + 4xt-1 + 3xt-2 + 2xt-3 + xt-4]
Arithmetic lag – pros & cons

–   Only one parameter to be estimated!
   t-statistics ok., better s.e., results more reliable
–   Straightforward interpretation
–   If restriction untrue, estimators biased and inconsistent

   Solution? F-test! (see: end of the notes)
Polynomial lag (ALMON)

   If we want a different shape of IRF...
–   It’s 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)
Polynomial lag - structure

i = 0 + 1i + 2i
2
i             2
E ( yt )
0
. . .
1           3

xt i
= i
.                          4
.

0 1       2    3       4               i
Polynomial lag - maths
   n – the lenght of the lag
   p – degree of the polynomial

i = 0 + 1i + 2i2 +...+ pip, where i=1, . . . , n
   For example a quadratic polynomial

i = 0 + 1i + 2i2 , where p=2       and   n=4

0 =  0                        1 =  0 +  1 +  2
2 = 0 + 21 + 42            3 = 0 + 31 + 92
4 = 0 + 41 + 162
Polynomial lags - estimation
   OLS, only need to estimate p parameters: 0,...,p
   STEP 1: impose restriction

yt =  +0xt + 0 + 1 + 2xt-1+(0 +21 +42)xt-2+(0+31 +92)xt-3+
(0 +41 + 162)xt-4 + et
   STEP 2: factor out the unknown coefficients
yt =  +0 [xt +xt-1+xt-2+xt-3 +xt-4]+1[xt+xt-1+2xt-2+3xt-3 +4xt-4] +
2 [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et

   STEP 3: define z
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]
   STEP 4: do OLS on yt =  + 0 z t0 + 1 z t1 + 2 z t2 + et
Polynomial lag – pros & cons

–   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

–  If the restriction untrue, biased and inconsistent
(see F-test in the end of the notes)
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.
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 )
Geometric lag - structure


0 = 
i   .
Geometrically
1 =       .     declining
weights
2 = 2          .
3 = 3
4 = 4
. .
0 1    2   3   4       i
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 || < 1 and i0
   
   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 => infinite geometric lag
yt =  + xt +  xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et
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: ,
   Have to do some algebra to rewrite the model in form
that can be estimated.
Geometric lag – Koyck transformation

  Original equation:
yt =  + xt +  xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et
 Koyck rule: lag everything once, multiply by and
substract from the original
yt =  + xt +  xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et
 yt-1 =  +  xt-1 + 2 xt-2 + 3 xt-3 + . . .) +  et-1
yt   yt-1 = 1 + xt + (et  et-1)
yt = 1 +  yt-1 + xt + (et  et-1) so yt = 1+ 2 yt-1 + 3xt + t
   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)
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
Geometric lag – pros & cons

–   You only estimate two parameters!
–   We allow neither for heterogenous nor for unsmooth declining
   It has many well specified versions, among which two
have particular importance:
(for both: see next student presentation)
F-tests of restrictions
1.   Estimate the unrestricted model
2.   Estimate the restricted (any lag) model
3.   Calculate the test statistic
( SSER  SSEU ) / df1
F=
SSEU / df 2
4.   Compare with critical value F(df1,df2)
   df1 = number of restrictions
   df2 = number of observations-number of variables in the
unrestricted model (incl. constant)
5.   H0: residuals are ‘the same’, restricted model OK

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 views: 4 posted: 12/24/2011 language: English pages: 23