#1: Stealing
gretl
Luckily it’s
one of those open source boons. It can be downloaded from http://gretl.sourceforge.net/
#2: Creating a model
Once you download it you need to import
your data. The data file should be saved as excel 97-2003 document. More up to date versions are beyond gretl’s
reach.
If done successfully you’ll be able to
have a glimpse at each variable separately (left mouse double click on
variable’s tag). You can check each just
to make sure that everything went as planned.
After this initial familiarization with
the new environment select:
Model > Ordinary Least Squares, then
arrange variables as shown below and click OK.
In less than a second, the program generates exactly
the same parameters that we’ve already calculated in excel. While it’s not very
motivating, the program did that faster and more accurately producing plenty of
additional indicators.
#3:
Interpreting the results
The
numbers are pure work of fiction but we still can interpret them showing how it
works. We’ll omit some of those more complicated indicators since at this level
of knowledge they could’ve at best litter our minds.
Coefficients (also known as regressors)
We already know that these are
consecutive elements of the equation and we’ve also mastered how to put them in
the right order. Still, we haven’t yet mentioned how exactly should those be
understood. For example:
If F equals 22.15 it means that ceteris paribus[1]
each additional gram of food makes cat (on average) 22.15 units happier.
H coefficient, the negative one, indicates that increasing
human disturbance per 1 unit causes on average 14.7 units decrease in cats mood
(ceteris paribus).
S, the only zero-one variable in the model, can be
interpreted as follows: cat’s mood ceteris
paribus is on average better by 7.25 units if the cat has slept (that is if
S equals 1).
The last question concerning those parameters would be
what does the constant mean. The idea is to put 0 in place of all the remaining
variables meaning that cat didn’t get any food, caress etc. Then the default
mood would be 13. The imaginary graph of the cat’s mood would thus originate
from point (0,13). Sometimes the constant cannot be interpreted because
variables won’t ever be equal to zero in a real world.
Standard errors
Those are estimators of what the variance of coefficients‘
distribution will be. Tells you how much will parameters of different cats sway
in reference to the calculated coefficients. The bigger the standard error the
weaker the model. But as long as it doesn’t exceed half of the coefficient it’s
ok. The mathematical interpretation of the rule goes as follows:
(standard error/coefficient) x 100 < 50
Calculating that for variable C we’ll get:
(0,0235 / 5) x 100 <50
0,47 < 50
This means that the variable’s parameter is good, too
good I would say. Since numbers in the example are quite random I guess the
indicators will keep reaching extremes.
R-squared (R2)
It’s a relation of parameters variance to dependent
variable variance. It tells you how much the change in the cat’s mood is explained
by the change in quantity of food, caress etc. In other words how well the
model we created explains moods alterations.
R2 varies from 0 to 1, 0 being the worst and 1 the best possible
model. There is no fixed number separating the good from the bad. In big
cross-sectional data models based on international statistics R2
should be contained within 0.30 – 0.70 limit, while for those little ones
concerning individual households, companies or cats the acceptable R2
values vary from 0.05 to 0.40. Those limits get stricter in time-series models
where R2 is expected to be greater than 0.70.
Our little cats-based model with its R2
amounting to 0.1010 seems all right.
Adjusted R2
The tricky thing about the R2
indicator is that its value increases as we put more and more variables into
the model and so the temptation arises to come up with as many variables as
possible. The end to this sick fantasy puts adjusted R2. It
calculates exactly the same thing as plain R2 does. The only
difference is that this indicator punishes us for each additional variable
added to the model deducting a little from the original indicator. That’s why
adjusted R2 is always a bit smaller than R2. Though I’ve
never seen it having a negative value...
p-value of
t-ratio (empirical significance value)
In order to interpret that one we have to turn on more
abstract thinking. So there’s a hypothesis, that a variable (let’s say F) lacks
empirical significance. The term empirical significance comes from statistics
and tells whether something is important or not. So one more time:
hypothesis 0
: F is not important to the model
and it’s
alternative hypothesis 1 : F is important to the model
There’s also
given a so called significance level (α) of 0.05. If:
p value >
α then the hypothesis 0 is
true, F is not important to the model
p value < α
then we accept the alternative hypothesis 1, F is important
to the model
It all comes
down to a very simple thing. You look at p-value column. Something bigger than
0.05 – bad, not important.
In our model
only the constant has empirical significance (constants always have it), so as
you can see that’s not a very good model.
The column t-ratio is given so that you can check for
yourself which hypothesis to accept and which to reject. You can do it using
statistical table for student’s t-distribution. Since we already got p-value it
seems just pointless.
Wald test (F
statistics)
It calculates the same thing as t-ratio significance
test. The difference is that the former does it for each variable separately.
Wald test checks whether the variables makes sense altogether. Instead of
student’s t it is based on F distribution. One more time we are given both F
value (0.927766) for our own calculations
and p-value ( 0.459739)which will make things much easier.
Ho says that all variables taken together are
unimportant to the model.
The alternative H1 states the
opposite.
Significance
level is always the same, α=0.05
p value
0.459739 > α 0.05
Therefore we
have to accept Ho and admit that the model is totally wrong. It’s
not like it comes as a surprise.
Information
criterion
Shwarz, Akaike and Hannah-Quinn are all information
criterions. Their values serve for comparison purposes only. If you have two
models and wonder which to choose you pick the one characterized by lower
values of information criterions. Those values themselves tell you nothing
more.
That would be all of the most basic analysis of a
model. Not much but as you can see it can already tell if a model makes sense
or not. This one clearly makes none.
[1] ceteris paribus- ‘with other things the same’, That’s just a formal requirement.
Each time you interpret something you have to add ‘ceteris paribus’ somewhere
in the sentence. I’m never quite sure where to put it, just make sure it’s
there, anywhere.
