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? Unrestricted? 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 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: un=u1+d(n-1)

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

7. Arithmetic lag - maths
X (log of) money supply and Y (log of) GDP, n=12 and g is estimated to be 0.1
the effect of a change in x on GDP in the current period is b0=(n+1)g=1.3
the impact of monetary policy one period later has declined to b1=ng=1.2
n periods later, the impact is bn= g=0.1
n+1 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 n xt-n + et
yt =  +  [(n+1)xt + nxt-1 + (n-1)xt xt-n] + et
zt = [(n+1)xt + nxt-1 + (n-1)xt xt-n]
???
For n = 4: zt = [ 5xt + 4xt-1 + 3xt-2 + 2xt-3 + xt-4]

9. 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...
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)

11. Polynomial lag - structure
i = 0 + 1i + 2i 2 i . 2 1 3 0 4 i

12. Polynomial lag - maths
n – the lenght of the lag
p – degree of the polynomial
For example a quadratic polynomial
i = 0 + 1i + 2i pip, where i=1, , n
i = 0 + 1i + 2i2 , where p=2 and n=4
0 = 0 1 = 0 + 1 + 2
2 =   2 3 =   2
4 =  1 + 162

13. 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+(0+31 +92)xt-3+
(0 +4 2)xt-4 + et
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 xt-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 xt- 4]

14. 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
1 = 
2 = 2
3 = 3
4 = 4
0 = 
Geometrically
declining
weights

18. Geometric lag - maths Infinite distributed lag model
yt =  + 0 xt + 1 xt-1 + 2 xt et
yt =  + i xt-i + et
Geometric lag structure
i = i where || < 1 and i
Infinite unstructured geometric lag model
yt =  + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt et
AND:0=, 1=, 2=2, 3=3 ...
Substitute i = i => infinite geometric lag
yt =  + xt +  xt-1 +  xt-2 +  xt ) + et

19. 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 ) + 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 ) + et
 yt-1 =  +  xt-1 +  xt-2 +  xt ) +  et-1
yt   yt-1 =  + xt + (et  et-1)
yt =  +  yt-1 + xt + (et  et-1)
so yt = +  yt-1 + xt + t

21. 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