tag:blogger.com,1999:blog-5359270546348328302024-03-13T05:09:30.826-07:00ecatometricsprimitive yet comprehensible guide to econometricsMhttp://www.blogger.com/profile/14144482883436360085noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-535927054634832830.post-40633440677930231322013-03-04T00:45:00.000-08:002013-03-04T10:17:22.515-08:005. Time lags<br />
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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).<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.</span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#1: Finite Distributed Lags</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">M = constant + H<sub>1</sub> + H<sub>2</sub> + H<sub>3</sub>
+ H<sub>4 </sub>+ H<sub>5<o:p></o:p></sub></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">The instant impact on cat’s mood will equal H<sub>1</sub>.
H<sub>2</sub>, H<sub>3</sub>, H<sub>4</sub>,
H<sub>5</sub> 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. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">The generated model gives coefficients for H<sub>1</sub>
and its 4 lags (H<sub>2</sub>, H<sub>3</sub>, H<sub>4</sub>, H<sub>5</sub>).
Adding them up will tell us the value of long-run multiplier. H<sub>1</sub>
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. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#2: Infinite Distributed Lags</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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:<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><span style="font-size: large;"> </span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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:<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">F coefficient / ( 1 – F_1 coefficient )<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">0.331132 / (1 - 0.01196) = 0.3351<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. <o:p></o:p></span></span></div>
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Mhttp://www.blogger.com/profile/14144482883436360085noreply@blogger.comtag:blogger.com,1999:blog-535927054634832830.post-61177144884014447252013-03-03T22:53:00.000-08:002013-03-04T10:14:59.790-08:004. Time series data<br />
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">It was no coincidence that our exemplary model was a
cross-sectional one. Their creation is much simpler since cats due to their
geographic dispersion don’t influence each others’ behavior. Taking a
time-series into account we have to deal with phenomena like trends or
seasonality. Granny’s cat for example getting older and older becomes harder to
please – that’s a trend. Those distort a
model and hamper comparison of variables. We’ll dodge it (gretl will) but first
we may need to comprehend a couple of things. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">There are two types of time-series data. The one on
the left resembles a typical cross-sectional data graph. It’s called stationary
meaning its variance, covariance and expected value are equal zero the way it
was in cross-sectional model. It means it’s perfectly random so all of the
previous analysis tools will work properly. The trouble begins as we proceed to
the right.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="wykresy.jpg" id="Obraz_x0020_0" o:spid="_x0000_i1029" style="height: 156pt; mso-wrap-style: square; visibility: visible; width: 403.5pt;" type="#_x0000_t75">
<v:imagedata o:title="wykresy" src="file:///C:\Users\M\AppData\Local\Temp\msohtmlclip1\01\clip_image002.jpg">
</v:imagedata></v:shape></span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">That’s a non stationary model. There is a trend
present meaning that we can no longer use any analysis tool we’ve already
learned until we do some arrangements first. The difference is that even before
we check we’ve got a slight idea what the next value of a variable would be.
That’s not random. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">We’ve got a ‘base’ and in each following period of
time we add to it ‘something’. As time goes by this ‘something’ becomes part of
the ‘base’.</span></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"><br /></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">0</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> =
</span><span style="color: red; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">base</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">1</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> =</span><span style="color: red; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> base</span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> + </span><span style="color: #1f497d; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">something <sub>1</sub></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">2</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> =
</span><span style="color: red; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">(base + something <sub>1</sub>)</span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> + </span><span style="color: #1f497d; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">something <sub>2</sub></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">3</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> =
</span><span style="color: red; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">(base + something <sub>1</sub> <sub> </sub>+ something <sub>2</sub>)</span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> + </span><span style="color: #1f497d; font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">something <sub>3 </sub></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="color: #1f497d; mso-themecolor: text2;"><sub><span style="font-size: large;"><br /></span></sub></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">The main idea is to separate the red from the blue. But
again there are two main types of base+something models we can encounter and
each has to be treated accordingly. <o:p></o:p></span></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">#1: Stochastic model<o:p></o:p></span></span></b></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></b></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Stochastic means random. If variables are so random then
what’s the whole point of discussing it in the first place? Well they are
random to some extent. Again, let’s assume there’s a cat sleeping on a couch.
Chances are good that if we close our eyes, count to 10 and check again it’ll
still be asleep. At least we expect it to be so. We know that it’s highly
impossible that within those 10 seconds cat would depart to the moon or take up
cooking class. At best he could have woke up and jumped off the couch. So as
you can see the randomness is quite limited. The limits are set by the last
cat’s activity. If the last thing we saw cat doing before closing our eyes
would be entering a space shuttle at Cape Canveral well then it would be highly
probable that he would actually be departing to the moon. Therefore:</span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">activity </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">1</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">
= activity </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">0</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> + something </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">1</sub></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"><br /></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">1</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> =
y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">0</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> + ε</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">t</sub></div>
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<sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;"><br /></sub></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"><br /></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">what we expect
to be the next activity </span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> </span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">= </span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;"> </span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">present activity</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">E(y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">1</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">)
= y</span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">0</sub></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><sub><span style="font-size: large;"><br /></span></sub></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">So y<sub>1</sub> is related to y<sub>0</sub>
and y<sub>2</sub> but it doesn’t influence y<sub>3</sub>, y<sub>4</sub>, y<sub>5</sub>
etc. This is called first order autoregressive process. The bigger the number of
the following variables determined by y<sub>0</sub> the higher the order
of autoregression. In the previous model we analyzed a mutual regression of two
variables. Now it is the same variable having a regression in terms of itself. <o:p></o:p></span></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">#2
Deterministic model<o:p></o:p></span></span></b></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></b></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Opposing to the limited randomness there’s continuity.
This is the cat getting older and older. We know that one day we’ll bury it in
the backyard. It’s not random, that’s something to be sure of. So we look at
the cat, close our eyes, count to ten, and there he is – 10 seconds older. No
matter how many times we do it, the cat is always heading in the same
direction. Let’s assume that the observed variable here would be the cat’s
speed. Obviously as time goes by he’s getting slower, that’s our trend. There
will be of course some random moments the cat will go berserk, running around
the house or quite contrary laying all day by the fire. Those are what we
cannot predict, the noise distorting our observations. Therefore:</span></span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">speed </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">t</sub></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">=</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large;">constant (some
average speed each non-dead cat performs)</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">+</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">time trend </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">t</sub></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">+</span></div>
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<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">something </span><sub style="font-family: 'Trebuchet MS', sans-serif; text-indent: 35.4pt;">t</sub><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; text-indent: 35.4pt;">
(random amok/ laziness)</span></div>
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<br /></div>
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<br /></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">Generally speaking variables are not related to one
another. They’re determined by the time instead. Since the time goes always in
one direction we might get the wrong impression that y<sub>1</sub> depends on y<sub>0</sub>
and so on. That’s why stochastic models and deterministic ones are frequently
confused. Unluckily we have to be sure which type of model we’re dealing with
since there’s a different approach meant for each of them</span><span style="font-family: 'Trebuchet MS', sans-serif;">.</span></span></div>
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<br /></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">#3: Stochastic
model approach<o:p></o:p></span></span></b></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></b></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">While stochastic model is not stationary, its deltas
are. Those would be our ‘something’ separated from the base. Then we could
substitute the variables in model (F,C,H...) with values of their deltas. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Given that the variable shows autoregression of first
order then instead of F<sub>1</sub>, F<sub>2</sub>, F<sub>3</sub>, F<sub>4 </sub>...
we would have <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Δ F<sub>1 </sub>= (F<sub>1 </sub>- F<sub>0</sub>)<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Δ F<sub>2</sub> = (F<sub>2 </sub>- F<sub>1</sub>)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Δ F<sub>3</sub> = (F<sub>3 </sub>- F<sub>2</sub>) ...<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Those are calculated in gretl using Add menu (tool
bar)<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Differences of variables in gretl are marked ‘d’ so
instead of F we’ll have d_F. Those d values are the ones we put in the model
instead of the original ones. We can get back to the procedure of model creation
that we’ve already learned.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">#4:
Deterministic model approach<o:p></o:p></span></span></b></div>
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<b><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></b></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">We know that in that kind of model some part of
variable F is determined by the passing time. What we would like to do is pull
out only pure F values, omitting this time-contaminated part. To do so first
we’ll have to figure out what is the relation between time and F thus create
a model F = time. Time variable can be added
to the data using Add > Time trend.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Once we get the model we save its residuals – the
differences between the estimated values and the real ones, those are our pure
Fs . Select Save > Residuals.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Then once more create a model you had in your mind
replacing F with its residuals’ values. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Of course you can do all of that absentmindedly
clicking where told. This is what you do at the exams but that’s not the case
here. Creating a model of your own requires understanding even if done with blissful ignorance of mathematical
dimension of the issue. Once you comprehend the simple you’ll be able to
explore the rest. <o:p></o:p></span></span></div>
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Mhttp://www.blogger.com/profile/14144482883436360085noreply@blogger.comtag:blogger.com,1999:blog-535927054634832830.post-22091768847649409862013-03-03T08:31:00.000-08:002013-03-04T10:14:13.099-08:003. More testing<br />
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<a href="http://1.bp.blogspot.com/-r7ehmvx_D6c/UTN5fRLK1KI/AAAAAAAAAGI/BCgEVtDfVus/s1600/k31.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="320" src="http://1.bp.blogspot.com/-r7ehmvx_D6c/UTN5fRLK1KI/AAAAAAAAAGI/BCgEVtDfVus/s320/k31.jpg" width="177" /></span></a></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Apart from the indicators automatically spitted out by
gretl we can conduct a couple of additional test. Do not fear, cause it takes
no more than a couple of mouse clicks. Once again the program will calculate everything
for us, we only have to know where to look for the answers. Of course the
question may arise what do we need all this testing for. Well, as mere human
beings, we have a great capacity to err. Each mistake distorts the model,
making it grow further apart from reality. Why building a model in a first
place if we don’t want it to reflect anything?<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#1: Testing for collinearity</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Collinearity
is simply a relationship between two independent variables. Think of a cat
caressed only while he’s asleep. This would cause the values for sleep and
caress to behave alike. Of course some level of collinearity doesn’t do any
harm to the model. In economics it’s perfectly normal that some indicators sway
together in a response to markets’ mood. We just have to make sure that those
similarities in reactions lay within safe limits.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Having
our previous model already calculated we select: Tests > Collinearity (menu
bar).<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Since
everything has been already explained by the program I can only repeat. Check
if any variable has VIF value exceeding 10. Our results seem ok (3.959 being
the highest number in the column). A piece of cake. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#2: RESET (Regression Specification Error Test)</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">This test checks whether the model has a correct
structure. At the beginning we’ve assumed that relation between independent
variables and cat’s mood are linear. This is just the simplest possible form of relation between sets of
values. As you know there are no squares nor logarithms in our formula, but
they could’ve been. RESET won’t tell us what the appropriate formula should be,
it would just inform us that the one we’ve chosen is wrong, and how wrong it
is. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Once again select: Tests > Romney’s Reset<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">The
result of the test is a new model with two new variables yhat^2 and yhat^3.
These are squares and cubes of the cat’s mood values. In a correctly
constructed model those should be valid, I mean their p values of t-ratio
should be less than 0.05 (α ). They are not. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Another
approach would be checking the F statistics as we did it in Wald test. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">H<sub>o</sub> :
variables are not valid<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">H<sub>1</sub> : the opposite<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">p value 0.227 > α 0.05 so we stick to H<sub>o</sub>
what only confirms our previous assumptions. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#3: Omitted variable test</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">This one is the most comprehensible one. We indicate a
variable we don’t like and the program checks if we are better off without it.
If you like to you can repeat the test for each variable, but at the higher
level of initiation you would already be able to tell which one stinks. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Select: Tests > Omit variable > Wald omit test,
and choose a variable you don’t like the most.<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="omit variable.jpg" id="Obraz_x0020_2" o:spid="_x0000_i1027" style="height: 4in; mso-wrap-style: square; visibility: visible; width: 347.25pt;" type="#_x0000_t75">
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">I’ve chosen the H. One more time we are given the Wald
test statistics (F). Every time it works
the same way. Two hypothesis, H<sub>0</sub>: the variable is useless, H<sub>1</sub>:
it’s not. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">p-value 0.457033 > α 0.05 , we throw out the H<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#4: Davidson Mc-Kinnon test</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Time
for dessert. It’s a bit more complicated meaning it requires more clicking. We
have to get back to creation of a model. Then instead of the one we already
have construct two models with divergent variables sets. This is:<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">M = F + C<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">M = H + S<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">or any
other combination you can think of. Just remember that both models have to
include the same number of variables. I wonder what happens when the original
model has an odd number of those.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">After
creating the first model we select Save > Fitted value<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">What
we’ll get is a new variable yhat1 that has been automatically added to the list
of variables in the first gretl’s window. Repeat the procedure for the second
model (Save> Fitted...) and therefore you’ll obtain another variable yhat2. <o:p></o:p></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="david mckinnon.jpg" id="Obraz_x0020_3" o:spid="_x0000_i1026" style="height: 273.75pt; mso-wrap-style: square; visibility: visible; width: 359.25pt;" type="#_x0000_t75">
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">Another
name for fitted value is predicted value what is a bit more clear. So yhat1 is
a column of cats’ moods predicted by model based on F and C. The same with
yhat2. Now once again we create two separate models. <o:p></o:p></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;">1: </span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">M = F + C + yhat2</span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;">2: </span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">M= H + S + yhat1</span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">In each
model we have to check whether yhat variable is valid or not. Hope you remember
that it’s the p-value of t-ratio that decides. In this exemplary model yhat2
has a p-value of 0.1946. It’s greater than α meaning it lacks statistical
significance. It means that this little model F+C is complete without it, it
works. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">In the
Davidson McKinnon test we can accept or reject both models. There’s no need to
choose between the winner and the loser.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US;"><span style="font-size: large;">I guess
we are done testing for now. Though it’s one of the most important part of the
modeling process. After creation of a hypothetic model there’s always a list of
possible defects waiting to be ticked off. <u>Never</u> skip it . A defective
model can do more harm than good.<o:p></o:p></span></span></div>
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Mhttp://www.blogger.com/profile/14144482883436360085noreply@blogger.comtag:blogger.com,1999:blog-535927054634832830.post-83442036313730797102013-03-03T08:07:00.000-08:002013-03-04T10:13:44.070-08:002. Technology serving science - introduction to econometric software<br />
<div class="separator" style="clear: both; text-align: center;">
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#1: Stealing
gretl</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">Luckily it’s
one of those open source boons. It can be downloaded from </span><span style="font-family: "Trebuchet MS","sans-serif";"><span lang="EN-US"><a href="http://gretl.sourceforge.net/">http://gretl.sourceforge.net/</a></span></span><span style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> <span lang="EN-US"><o:p></o:p></span></span></span></div>
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<span style="font-family: "Trebuchet MS","sans-serif";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-family: "Trebuchet MS","sans-serif";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#2: Creating a model </b> </span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"> <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. </span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-size: large;"><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="jpeg2.jpg" id="Obraz_x0020_6" o:spid="_x0000_i1029" style="height: 253.5pt; mso-wrap-style: square; visibility: visible; width: 328.5pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">After this initial familiarization with
the new environment select: <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Model > Ordinary Least Squares, then
arrange variables as shown below and click OK.<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="kot2.jpg" id="Obraz_x0020_8" o:spid="_x0000_i1028" style="height: 271.5pt; mso-wrap-style: square; visibility: visible; width: 336.75pt;" type="#_x0000_t75">
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<span style="font-size: large;"><span lang="EN-US"></span></span><br />
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<span style="font-size: large;"><span lang="EN-US"><span style="font-family: 'Trebuchet MS', sans-serif; line-height: 17px;"><br /></span></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="zfabrykowane wyniki gretl.jpg" id="Obraz_x0020_2" o:spid="_x0000_i1027" style="height: 255pt; mso-wrap-style: square; visibility: visible; width: 359.25pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><b>#3:
Interpreting the results</b><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 150%; text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">Coefficients (also known as regressors)</span> </span></span></div>
<div class="MsoNormal" style="line-height: 150%; text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"> <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 150%; text-align: justify;">
<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">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:</span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-bidi-font-family: "Courier New"; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">If F equals 22.15 it means that<i> <span style="color: red;">ceteris paribus<a href="file:///D:/ecatometrics%20cd/kotII.docx#_ftn1" name="_ftnref1" title=""><span class="MsoFootnoteReference"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><b><span lang="EN-US" style="color: red; line-height: 115%;">[1]</span></b></span><!--[endif]--></span></a></span></i>
each additional gram of food makes cat (on average) 22.15 units happier. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">H coefficient, the negative one, indicates that increasing
human disturbance per 1 unit causes on average 14.7 units decrease in cats mood
(<i>ceteris paribus</i>).<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">S, the only zero-one variable in the model, can be
interpreted as follows: cat’s mood <i>ceteris
paribus </i>is on average better by 7.25 units if the cat has slept (that is if
S equals 1).<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">Standard errors</span><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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:<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;">(standard error/coefficient) x 100 < 50</span></div>
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<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:f><m:fPr><span
style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-fareast-font-family:"Hand Of Sean";mso-ansi-language:
EN-US'><m:r>Standard</m:r></span></i><i style='mso-bidi-font-style:normal'><span
lang=EN-US style='font-family:"Cambria Math","serif";mso-fareast-font-family:
"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";mso-ansi-language:EN-US'><m:r>
</m:r></span></i><i style='mso-bidi-font-style:normal'><span lang=EN-US
style='font-family:"Cambria Math","serif";mso-fareast-font-family:"Hand Of Sean";
mso-ansi-language:EN-US'><m:r>error</m:r></span></i></m:num><m:den><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-fareast-font-family:"Hand Of Sean";mso-ansi-language:
EN-US'><m:r>Coefficient</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:
normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US'><m:r> </m:r></span></i><i style='mso-bidi-font-style:
normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-ansi-language:EN-US'><m:r>x</m:r></span></i><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US'><m:r> 100 <50</m:r></span></i></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-size: large;"><span style="font-family: Calibri, sans-serif; line-height: 115%;"><v:shape id="_x0000_i1025" style="height: 29.25pt; width: 136.5pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:\Users\M\AppData\Local\Temp\msohtmlclip1\01\clip_image006.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Calculating that for variable C we’ll get:<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">(0,0235 / 5) x 100 <50</span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">0,47 < 50</span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> <br />
</span><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:f><m:fPr><span
style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:
"Trebuchet MS";mso-ansi-language:EN-US'><m:r>0.0235</m:r></span></i></m:num><m:den><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:
"Trebuchet MS";mso-ansi-language:EN-US'><m:r>5</m:r></span></i></m:den></m:f><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US'><m:r> </m:r></span></i><i style='mso-bidi-font-style:
normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-ansi-language:EN-US'><m:r>x</m:r></span></i><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US'><m:r> 100 <50</m:r></span></i></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-family: Calibri, sans-serif; line-height: 115%;"><v:shape id="_x0000_i1025" style="height: 27pt; width: 93pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:\Users\M\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<!--[if gte msEquation 12]><m:oMathPara><m:oMath><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-fareast-font-family:"Hand Of Sean";mso-hansi-font-family:"Trebuchet MS";
mso-ansi-language:EN-US'><m:r>0,47<50</m:r></span></i></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-size: large;"><span style="font-family: Calibri, sans-serif; line-height: 115%;"><v:shape id="_x0000_i1025" style="height: 15pt; width: 47.25pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:\Users\M\AppData\Local\Temp\msohtmlclip1\01\clip_image008.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">R-squared (R<sup>2</sup>)</span><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">R<sup>2</sup> 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 R<sup>2</sup>
should be contained within 0.30 – 0.70 limit, while for those little ones
concerning individual households, companies or cats the acceptable R<sup>2</sup>
values vary from 0.05 to 0.40. Those limits get stricter in time-series models
where R<sup>2</sup> is expected to be greater than 0.70. <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Our little cats-based model with its R<sup>2</sup>
amounting to 0.1010 seems all right.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="background-color: #cccccc; font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">Adjusted </span><span style="font-family: 'Trebuchet MS', sans-serif;">R</span><sup style="font-family: 'Trebuchet MS', sans-serif;">2</sup></span></div>
<div class="MsoNormal" style="text-align: justify;">
<sup style="font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: large;"><br /></span></sup></div>
<span style="font-size: large;">
</span><br />
<div class="MsoNormal" style="text-align: justify;">
<span class="apple-converted-space"><span lang="EN-US" style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: 'Trebuchet MS', sans-serif;"><span style="font-size: large;">The tricky thing about the R<sup>2</sup>
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 R<sup>2</sup>. It
calculates exactly the same thing as plain R<sup>2</sup> 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 R<sup>2</sup> is always a bit smaller than R<sup>2</sup>. Though I’ve
never seen it having a negative value...<o:p></o:p></span></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">p-value of
t-ratio (empirical significance value)</span><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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:<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">hypothesis <sub>0
</sub>: F <u>is not</u> important to the model<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">and it’s
alternative hypothesis<sub> 1</sub> : F <u>is</u> important to the model<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">There’s also
given a so called significance level (α) of 0.05. If:<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">p value >
α then the hypothesis <sub>0</sub> is
true, F is not important to the model<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-size: large;"><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">p value < </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean";">α</span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";">
then we accept the alternative hypothesis <sub>1</sub>, F is important
to the model<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">It all comes
down to a very simple thing. You look at p-value column. Something bigger than
0.05 – bad, not important. <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">Wald test (F
statistics)</span><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">H<sub>o</sub> says that all variables taken together are
unimportant to the model.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">The alternative H<sub>1</sub> states the
opposite.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Significance
level is always the same, α=0.05<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">p value
0.459739 > α 0.05<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">Therefore we
have to accept H<sub>o</sub> and admit that the model is totally wrong. It’s
not like it comes as a surprise.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><span style="background-color: #cccccc;">Information
criterion</span><o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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. <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;"> <o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><span style="font-size: large;">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.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"> </span><span style="font-family: "Trebuchet MS","sans-serif"; mso-fareast-font-family: "Hand Of Sean"; mso-fareast-language: PL; mso-no-proof: yes;"><v:shape alt="misio i pszczolka.jpg" id="Obraz_x0020_10" o:spid="_x0000_i1025" style="height: 270pt; mso-wrap-style: square; visibility: visible; width: 5in;" type="#_x0000_t75">
<v:imagedata o:title="misio i pszczolka" src="file:///C:\Users\M\AppData\Local\Temp\msohtmlclip1\01\clip_image010.jpg">
</v:imagedata></v:shape></span><span lang="EN-US" style="font-family: "Trebuchet MS","sans-serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><o:p></o:p></span></div>
<div>
<div style="text-align: justify;">
<br /></div>
<!--[if !supportFootnotes]-->
<br />
<hr size="1" style="text-align: justify;" width="33%" />
<!--[endif]-->
<br />
<div id="ftn1">
<div class="MsoFootnoteText" style="text-align: justify;">
<a href="file:///D:/ecatometrics%20cd/kotII.docx#_ftnref1" name="_ftn1" title=""><span class="MsoFootnoteReference"><span style="color: red;"><span style="font-family: Trebuchet MS, sans-serif;"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span style="font-size: 10pt; line-height: 115%;">[1]</span></span><!--[endif]--></span></span></span></a><span style="color: red;"><span style="font-family: Trebuchet MS, sans-serif;"> <i><span lang="EN-US">ceteris paribus</span></i></span><span lang="EN-US"><span style="font-family: Trebuchet MS, sans-serif;">- ‘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.</span><span style="font-family: Cambria, serif;"><o:p></o:p></span></span></span></div>
</div>
</div>
Mhttp://www.blogger.com/profile/14144482883436360085noreply@blogger.comtag:blogger.com,1999:blog-535927054634832830.post-22765646915591618592013-03-03T01:00:00.000-08:002013-03-04T10:13:23.143-08:001. The idea behind an econometric model<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
</span><br />
<div class="MsoNormal" style="line-height: 150%;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";">1#
: <b>Encountering a research problem</b><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
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<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";">2#:
<b>Defining a research problem.</b></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
</span><br />
<div style="text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large; line-height: 150%;">The cat is in a
bad mood. The cat poses a threat to the furniture. We like the furniture. We
want the cat to be in a good mood. We have to provide the cat with mood
enhancers and prevent The Annoying. We do not know to what extent various
stimuli influence the cat. We are in need of a simulation – an econometric
model.</span></span></div>
</div>
<div class="MsoNormal" style="line-height: 150%; margin-left: 35.4pt; text-align: justify;">
<div style="text-align: left;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-ascii-theme-font: major-latin; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Hand Of Sean"; mso-hansi-theme-font: major-latin;"><br /></span></div>
</div>
<div class="MsoNormal" style="line-height: 150%; margin-left: 35.4pt; text-align: justify;">
<div style="text-align: left;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-ascii-theme-font: major-latin; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Hand Of Sean"; mso-hansi-theme-font: major-latin;"><br /></span></div>
</div>
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<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";">3#:
<b>Choosing variables</b></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">There are two
types of variables:</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">a. Dependent
variable – the one we are about to exam (cat’s mood) tagged M</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">b. Independent
variables (regressors) - all of the hypothetic mood enhancers and reducers:</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">-food (given in
grams) tagged F</span></span><br />
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">-caress (given
in minutes) tagged C</span></span><br />
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">-human
disturbance (given in annoyance units) tagged H</span></span><br />
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">-sleep (has
slept – good; hasn’t – bad) tagged S</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">While the first
3 variables are quantitative ones (their value is represented in units), the
variable S is a qualitative one. It means that its value can be either 1 (if
the cat has slept) or 0 if he hasn’t.</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">In order to
make it more comprehensible we put all of the above mentioned variables in an
equation with pluses preceding enhancers and minuses in front of reducers.
Thus:</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">M = F + C – H + S</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span></span>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;">To please the academics we have to put greek epsilon letter at the end of the equation. It stands for 'error term' and represents all the things that we could have possibly messed up in the formula.</span></span><br />
<br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;">
</span>
<br />
<div style="text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">Moreover at the
front of the right hand part of the equation we put constant. We do not yet
know what it’s going to be but for sure at the end of the modeling process it will
be there. Therefore the formula should look as follows:</span></span></div>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">M = const + F
+ C – H + S + epsilon</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">It indicates
that the value of M depends on how big the rest of the variables are. This
relationship is called regression – it’s the most basic concept behind the
whole econometrics science.</span></span></div>
<div class="MsoNormal" style="line-height: 150%; margin-left: 35.4pt; text-indent: .6pt;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="MsoNormal">
<div style="line-height: 150%;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";">4#:
<b>Gathering data</b></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div style="text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large; line-height: 150%;">For gathering
data we have to resort to internet sources or (what is sometimes impossible)
write down our own observations. Disregarding the chosen method there is a
fixed minimum number of observations (n)
the model will require to work properly. It depends on number of independent
variables (k) included and it’s given by the formula:</span></span></div>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<span style="line-height: 150%;">n > 31 + k </span></span><br />
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">In this
particular case n has to be larger that 31 + 4. 31 is a number given in
textbooks so we cannot question that one. Just trust that it will work and don’t
ask.</span></span></div>
<div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">There are 3
types of data we can work at and they are as follows:</span></span></div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span></span></div>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 36px;">Time-series data - The Cat over time (hourly, daily, weekly...)</span></span></div>
<div style="line-height: 150%;">
<br /></div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;">
<span style="line-height: 150%; text-indent: 35.4pt;">Cross-sectional
data – a group of cats in the same time (The Cat, neighbor’s cat, granny’s cat...)</span></span></div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">Panel data (time-series
and cross-sectional data combined) – a group of cats over time</span></span></div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;"><br /></span>
<span style="line-height: 150%;">Once we obtain
data we put it into an excel table.</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large; line-height: 150%;"><br /></span></div>
<div style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;">
<span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="color: #999999; line-height: 150%; text-indent: 35.4pt;">Table #1: Time-series
data</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<a href="http://1.bp.blogspot.com/-V5yeKJ1ugoQ/UTMLBTbqYlI/AAAAAAAAAE4/hskDcSHbm8c/s1600/t6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="84" src="http://1.bp.blogspot.com/-V5yeKJ1ugoQ/UTMLBTbqYlI/AAAAAAAAAE4/hskDcSHbm8c/s320/t6.jpg" width="320" /></a></span></div>
</div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="color: #999999; font-family: Trebuchet MS, sans-serif; font-size: large;">Table #2: Cross-sectional
data</span></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<a href="http://1.bp.blogspot.com/-VuhjW16lxPs/UTMK7NyfgJI/AAAAAAAAAEw/T3kamUx5txk/s1600/t5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><img border="0" height="84" src="http://1.bp.blogspot.com/-VuhjW16lxPs/UTMK7NyfgJI/AAAAAAAAAEw/T3kamUx5txk/s320/t5.jpg" width="320" /></span></a></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="text-indent: 35.4pt;">Make </span><span style="line-height: 150%; text-indent: 35.4pt;">sure that the number of
observations (number of rows) exceeds 35. If you don’t remember why go back to
the beginning of ‘gathering data’ section.</span></span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;">Having everything written down we can finally get to the model itself. </span></span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<br /></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-fareast-font-family: "Hand Of Sean";"><br /></span></div>
<div class="MsoNormal" style="line-height: 150%;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div class="MsoNormal">
<div style="line-height: 150%;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";">5#
<b>Creating the model – Last Squares Method (LS)</b></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%; text-indent: 32.2pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div>
<div style="line-height: 150%;">
<span style="line-height: 150%; text-indent: 32.2pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">There are two
ways to bring our model into existence.</span></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%; text-indent: 32.2pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; line-height: 24px; text-indent: 32.2pt;">a</span><span style="font-family: 'Trebuchet MS', sans-serif; font-size: large; line-height: 150%; text-indent: 32.2pt;">. Excel (feasible)</span><br />
<div style="line-height: 150%;">
<span style="line-height: 150%; text-indent: 32.2pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">b. Gretl (easy)</span></span></div>
</div>
<div style="line-height: 150%;">
<br /></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">Since there are
no cats included in this part the sooner we get through a. the better. So the
procedure goes as follows:</span></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="background-color: #f3f3f3; font-family: Trebuchet MS, sans-serif; font-size: large;">A. Copying the data</span></span></div>
<div style="line-height: 150%;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div style="line-height: 150%; text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">We copy our
data table (let it be table #2) excluding the dependent variable (M). Then we
add a column representing constant. It has to be put in the front and consists
of 1s only. I hope that it goes without
saying that the remaining columns should be filled with numbers. This is just
for the simplicity’s sake that they are not.</span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<a href="http://3.bp.blogspot.com/-OEvEkBLd5EQ/UTMKzHYgw5I/AAAAAAAAAEo/Su1FPEwB3nQ/s1600/t4.jpg" imageanchor="1" style="line-height: 150%; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="85" src="http://3.bp.blogspot.com/-OEvEkBLd5EQ/UTMKzHYgw5I/AAAAAAAAAEo/Su1FPEwB3nQ/s320/t4.jpg" width="320" /></a></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div style="line-height: 150%; text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
<div style="line-height: 150%; text-align: justify;">
<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">The table forms
a matrix with number of rows equal to number of observations (n) and number of
columns equal to number of independent variables (k) + 1. Therefore what we
obtain is a n-by-(k + 1) matrix.</span></span></div>
<div style="line-height: 150%; text-align: justify;">
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
<div style="text-align: justify;">
<span style="background-color: #f3f3f3; font-family: Trebuchet MS, sans-serif; font-size: large;">B. Transpositioning the matrix</span><br />
<br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="background-color: white;">Next step would be transpositioning the matrix meaning that we would have to reverse its dimensions. The columns will become rows and the other way round. In order to do so we copy the matrix and paste it using 'paste special' command in Excel (right mouse click).</span></span></div>
<div style="line-height: 150%; text-align: justify;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-ascii-theme-font: major-latin; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Hand Of Sean"; mso-hansi-theme-font: major-latin;"><br /></span></div>
<div style="line-height: 150%; text-align: justify;">
<br /></div>
<div style="line-height: 150%; text-align: justify;">
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-ascii-theme-font: major-latin; mso-fareast-font-family: "Hand Of Sean"; mso-hansi-theme-font: major-latin;"><br /></span></div>
<div style="line-height: 150%; text-align: justify;">
<span lang="EN-US" style="color: #999999; font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-ascii-theme-font: major-latin; mso-fareast-font-family: "Hand Of Sean"; mso-hansi-theme-font: major-latin;">Martix A n - by - (k + 1)</span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<a href="http://3.bp.blogspot.com/-s2ZHoVgwPv0/UTMKtawPPGI/AAAAAAAAAEg/ysu1O8m0r7o/s1600/t3.jpg" imageanchor="1" style="line-height: 150%; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="108" src="http://3.bp.blogspot.com/-s2ZHoVgwPv0/UTMKtawPPGI/AAAAAAAAAEg/ysu1O8m0r7o/s320/t3.jpg" width="320" /></a></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"> </span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<span style="color: #999999;">Martix <span style="line-height: 150%; text-indent: -24px;">A</span><sup style="line-height: 150%; text-indent: -24px;">T</sup><span style="line-height: 150%;"> (k + 1) - by - n</span></span></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<a href="http://2.bp.blogspot.com/-wKi1NAwWNVI/UTMKl8BELHI/AAAAAAAAAEY/8gUh5S65Jt8/s1600/t2.jpg" imageanchor="1" style="line-height: 150%; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="108" src="http://2.bp.blogspot.com/-wKi1NAwWNVI/UTMKl8BELHI/AAAAAAAAAEY/8gUh5S65Jt8/s320/t2.jpg" width="320" /></a></span><br />
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span lang="EN-US" style="text-indent: -18pt;"><span style="background-color: #f3f3f3; line-height: normal;">C. </span></span><span lang="EN-US" style="line-height: 150%; text-indent: -18pt;"><span style="background-color: #f3f3f3;">Multiplying* A<sup>T </sup> and A</span> *(order
matters)</span></span></div>
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<span style="line-height: 150%;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large; line-height: 150%;">One more time excel provides a ready solution. There
is included a special formula for multiplying matrices. The only catch is that
you have to know the size of a newly created matrix before calculating it in
order to select a space for a result of multiplication. So if we multiply two matrices, r- being a
number of rows, and c – a number of columns, the outcome will be:</span></span></div>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
<span style="line-height: 150%;"><span style="line-height: 150%; text-indent: 35.4pt;">r – by - c x r2
– by - c2 = r
– by – c2</span></span></span><br />
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-indent: 35.4pt;"><br /></span>
<span style="line-height: 150%; text-indent: 35.4pt;">(k+1)-by-n x n-by-(k+1) = (k+1)-by-(k+1)</span></span></div>
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<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></span></div>
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<span style="line-height: 150%; text-indent: 35.4pt;"><span style="font-family: Trebuchet MS, sans-serif; font-size: large; line-height: 150%; text-indent: 35.4pt;">Therefore before multiplying we need to select a
rectangle high as multiplicand and wide as multiplier. When using MMULT formula
you also have to remember that when finishing the operation you press ctrl +
shift + enter instead of clicking OK. As the result we should obtain new matrix
with the element (1,1) equal to number of observations (n).</span></span></div>
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
</span><br />
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span lang="EN-US" style="line-height: 150%; text-align: justify; text-indent: -18pt;"><span style="line-height: normal;"><br /></span></span>
<span lang="EN-US" style="line-height: 150%; text-align: justify; text-indent: -18pt;"><span style="line-height: normal;"> <span style="background-color: #f3f3f3;">D. </span></span></span><span lang="EN-US" style="background-color: #f3f3f3; line-height: 150%; text-align: justify; text-indent: -18pt;">Reversing
A<sup>T</sup>A</span></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;"><br /></span>
<span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;">That shouldn’t pose any difficulty either. The
procedure is the same as in multiplication. We select area, find proper
formula, indicate what we would like to inverse and press ctrl + shift + enter.
The difference is that the size of a newly created matrix (A</span><sup style="line-height: 150%; text-align: justify; text-indent: 0.6pt;">T</sup><span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;">A)</span><sup style="line-height: 150%; text-align: justify; text-indent: 0.6pt;">-1</sup><span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;"> will be the same
as the original one what makes it far more easier comparing to what we’ve
already done. </span></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;"><br /></span>
<span style="line-height: 150%; text-align: justify; text-indent: 0.6pt;">There are only two more steps and we haven’t
encountered anything tough yet. Taking into account that what we’re about to do
now are two more multiplications we shouldn’t be afraid, should we?</span></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%; text-align: justify; text-indent: -18pt;"><br /></span>
<span style="background-color: #f3f3f3;"><span style="line-height: 150%; text-align: justify; text-indent: -18pt;">E. Multiplying A</span><sup style="line-height: 150%; text-align: justify; text-indent: -18pt;">T</sup><span style="line-height: 150%; text-align: justify; text-indent: -18pt;"> </span><span style="line-height: 150%; text-align: justify; text-indent: -18pt;">matrix and Mood vector</span></span></span></div>
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<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: large;">One more time we will use matrix A<sup>T</sup> calculated in the second step, only this time the other element of the operation would be a vector. First we will have to dig out the column representing mood values (M), then again figure out what will be the size of the calculated matrix. If defining a vector as a specific matrix with number of columns equal to 1 then:</span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br />
(k+1)-by-n x n-by-1 = (k+1)-by-1</span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><br /></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span lang="EN-US">So the result of the multiplication should be a vector (A<sup>T</sup>M) with number of rows corresponding to the number of independent variables + 1. Once again we repeat the procedure already described in the third step. This is: select cells, enter formula, indicate elements, ctrl+shift+enter, </span><em><span lang="EN-US" style="background-color: white;">voilà.</span></em></span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span lang="EN-US" style="text-indent: -18pt;"><span style="line-height: 24px;"><br /></span></span>
<span style="background-color: #f3f3f3;"><span lang="EN-US" style="text-indent: -18pt;"><span style="line-height: 24px;">F. </span></span><span lang="EN-US" style="line-height: 24px; text-indent: -18pt;">Multiplying (A<sup>T</sup>A)<sup>-1</sup> matrix and A<sup>T</sup>M vector</span></span></span></div>
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<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: large;">Another matrix-vector operation. Let’s just estimate what will be the size of the calculated vector (looking at the previous step we already know that it’s going to be a vector).</span><br />
<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="text-indent: 0.6pt;"><br /></span>
<span style="text-indent: 0.6pt;">(k+1)-by-(k+1)</span><span style="text-indent: 0.6pt;"> </span><span style="text-indent: 0.6pt;">x</span><span style="text-indent: 0.6pt;"> (k+1)-by-1 = (k+1)-by-1 </span></span></div>
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<span style="font-family: Trebuchet MS, sans-serif; font-size: large;"> What we got from this final equation are parameters of our model. This is what we were looking for all this time. They are given in a form of a column of numbers. How to put them into equation like the one we’ve written down at the beginning? That’s the easy part. Each consecutive cell corresponds to subsequent letters of the formula.<span style="text-indent: 0.6pt;"> </span></span><br />
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<a href="http://3.bp.blogspot.com/-BXIfS0lFscI/UTMKd6huTtI/AAAAAAAAAEQ/wy-CjvlDfEY/s1600/t1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center; text-indent: -18pt;"><img border="0" src="http://3.bp.blogspot.com/-BXIfS0lFscI/UTMKd6huTtI/AAAAAAAAAEQ/wy-CjvlDfEY/s1600/t1.jpg" /></a></span><br />
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<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";"> </span><br />
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";"> M = const + F + C – H + S + epsilon</span><br />
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";"><br /></span>
<span lang="EN-US" style="font-family: Trebuchet MS, sans-serif; font-size: medium; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";"> M = 13 + 22,5F + 5C - 14,7H + 7,25S + epsilon</span><br />
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<span lang="EN-US"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;">This is our
model. Having it we can substitute values for independent variables in order to
calculate what the cat’s mood will be.
Therefore we can seize power over cats mind implementing him a precise
quantity of stimuli.</span></span></div>
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<span lang="EN-US"><span style="font-family: Trebuchet MS, sans-serif; font-size: large;"><span style="line-height: 150%;">The good news is – we don’t have to use this toilsome
procedure, there is a program that will calculate virtually all of that for us.
What’s left is to learn how to thrust the data into that software. Then we’ll
get to know how to interpret the results and how to tell a good model from a
bad one.</span> <o:p></o:p></span></span></div>
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