Pushing a cat off a couch has an immediate negative
influence on its mood. For a moment the cat’s a bit angry and as long as it
doesn’t show any vindictive tendencies no further consequences follow. A single
increase in human disturbance (H) leads to sharp and short-term drop in the
mood (M).
Cutting off cat’s tail however may produce a pretty
different outcome. While one knife slash is still perceived as a single
increase in H, the cat may not recover from it for quite a time. Reflecting
this situation in our model requires the application of time lag. Apart from
the butchery the concept also proves to be quite useful in economics.
#1: Finite Distributed Lags
Since the stump example seems to trigger one’s
imagination we’ll stick to it. Let’s
assume that we check up on cat’s mood once a week and that the unlucky day fell
on the first one. We also know that the
cat is rather unforgiving so it’ll take 4-5 weeks for things to go back to
normal. To make things simpler we’ll get rid of all the other variables.
M = constant + H1 + H2 + H3
+ H4 + H5
The instant impact on cat’s mood will equal H1.
H2, H3, H4,
H5 represent the second, third, fourth and fifth week respectively. The
overall effect of cutting off the tail is a sum of all the Hs in the equation.
In other words the damage to the cat’s tail will affect it’s mood for 5 weeks
after the massacre. We’ll be able to calculate its scope as soon as gretl unfolds
all the H values.
In accordance with the previous chapter we transform
Ms into its first differences. (select M, Add> First differences...). Then, creating a model
we add 4 lags of H as independent variables. There’s the ‘lags...’ button at
the bottom of the ‘specify model’ window. They should appear in the ‘Regressors’
column.
The generated model gives coefficients for H1
and its 4 lags (H2, H3, H4, H5).
Adding them up will tell us the value of long-run multiplier. H1
indicates the impact on the cat’s mood at the very week of the accident.
According to the model pasted below the cat actually enjoyed having its body
part sliced off, at least initially. That somehow explains my strong preference
towards dogs.
#2: Infinite Distributed Lags
No more blood, we’re back to mundane everyday life.
Once a day the cat tucks into his munchies. What can’t be eaten gets
immediately hidden behind the couch where the cat stores its supplies. Let’s
say it’s always ¼ of the bowl. As soon as the cat senses first symptoms of the returning
hunger it feasts on the half of what’s hidden behind the sofa excluding today’s
crop. Supposing that M depends on F exclusively then:
M = [ 0.75 Fri F] + [ 0.25 x 0,5 Thu F] + [ 0.25 x 0,25 Wed F] + + [ 0.25 x 0,125 Tue F]... etc.
The equation could continue ad infinitum making itself
unsolvable. The idea is that a single serving provides the cat with an infinite
stimulus. Calculating the overall food intake and its impact on the cat’s mood
requires simplification of the formula. Since this simplified one is not so
simple we’ll use gretl to show how it works only this time we’ll base the model
on days instead.
Again we extract the M’s differences (d_M) evading its
nonstationarity and we add one lag to the F. Surprising how those prior 4 weeks
took more lags than the infinity. This is the promised simplification.
The value of F coefficient is the immediate result of
F serving on the cat’s mood. The
long-run multiplier is given by the formula:
F coefficient / ( 1 – F_1 coefficient )
0.331132 / (1 - 0.01196) = 0.3351
Since the difference between F coefficient and long
run multiplier is so tiny we can conclude that cat derives nearly no additional
profit from its stock. It’s the stupidity of either the model, the data or the
cat. The problem with econometrics is that you can never be sure which one is
to blame.
