Sample Skewness and Kurtosis in Gujarati book page 72.
- Thread starter fashepard
- Start date
Hi Frank,
Me neither! See attached XLS. In prepping the notes, 3.46 & 3.48 frustrated me b/c i could not get his answer. But we've since discovered more than a few errors in Gujarati so I gave up... There is an errata, but strangely, even it is incorrect, it says "Table 3-3. Sample var (Y) should be 368,871 instead of 368,872" which is wrong!
Here is my XLS from this:
gujarat sample skew & kurt
Note: I was satisfied that i could reconcile with Excel's SKEW() and KURT(). Note: under Gujarati, I got 0.467 and 1.546.
Also note: re: 3.47 and 3.48, I assume you realize, are not the skew and kurtosis per se: they must be divided by, respectively, the sample standard dev ^3 and ^4. This is called normalizing the moments.
The key takeaway is that sample skew (e.g.) is sample third moment divided by cube of sample standard deviations.
mean: 1st moment
variance: 2nd moment (such that std dev is a normalized 2nd moment)
skew: function of [3rd moment], not 3rd moment per se; i.e., a normalized 3rd moment
kurt: function of [4th moment], not 4th moment per se; i.e., a normalized 4th moment
wikipedia is helpful on this.
So you can see, the general form is
nth moment / standard dev ^ (n)
David
Me neither! See attached XLS. In prepping the notes, 3.46 & 3.48 frustrated me b/c i could not get his answer. But we've since discovered more than a few errors in Gujarati so I gave up... There is an errata, but strangely, even it is incorrect, it says "Table 3-3. Sample var (Y) should be 368,871 instead of 368,872" which is wrong!
Here is my XLS from this:
gujarat sample skew & kurt
Note: I was satisfied that i could reconcile with Excel's SKEW() and KURT(). Note: under Gujarati, I got 0.467 and 1.546.
Also note: re: 3.47 and 3.48, I assume you realize, are not the skew and kurtosis per se: they must be divided by, respectively, the sample standard dev ^3 and ^4. This is called normalizing the moments.
The key takeaway is that sample skew (e.g.) is sample third moment divided by cube of sample standard deviations.
mean: 1st moment
variance: 2nd moment (such that std dev is a normalized 2nd moment)
skew: function of [3rd moment], not 3rd moment per se; i.e., a normalized 3rd moment
kurt: function of [4th moment], not 4th moment per se; i.e., a normalized 4th moment
wikipedia is helpful on this.
So you can see, the general form is
nth moment / standard dev ^ (n)
David
Hi Frank,
The link is to an editgrid, which seems to be working.
The editgrid is pretty universal https://wiki.editgrid.com/display/helpcentre/Supported+Browsers
can you check your browser?
David
The link is to an editgrid, which seems to be working.
The editgrid is pretty universal https://wiki.editgrid.com/display/helpcentre/Supported+Browsers
can you check your browser?
David
Frank
do you refer to p 53 and 54? if so, just ignore the u^3 (notice, i didn't speak to them, u^3 is shorthand for 3rd moment). if you don't mean 53 and p 54, i don't know what you mean. It is as we have above, in the XLS. And in Gujarati 3.39 and 3.40.
if you have a doubt like this, sometimes it may help to look at the XLS:
https://www.editgrid.com/bt/frm_2008/gujarati_3-3
David
do you refer to p 53 and 54? if so, just ignore the u^3 (notice, i didn't speak to them, u^3 is shorthand for 3rd moment). if you don't mean 53 and p 54, i don't know what you mean. It is as we have above, in the XLS. And in Gujarati 3.39 and 3.40.
if you have a doubt like this, sometimes it may help to look at the XLS:
https://www.editgrid.com/bt/frm_2008/gujarati_3-3
David
Hi David,
Gujarati Pg.67 says leptokurtic Kurtosis has slim or long tail. You mentioned in recent AIMs discussions as leptokurtic having fat tails? Which is true? Pls. give one real world example each for leptokurtic and platykurtic.
Thanks for the briefcasts. We should appreciate(you) that it must be very tough to be so detailed on all three parts of Risk subject(i.e. Market, Credit and Rest all).
Also, if possible, can you pls. briefcast on Metalgesellschaft, I mean, audio,video and excel with assumed numbers.
Thanks
Regars
mudaltiru
Gujarati Pg.67 says leptokurtic Kurtosis has slim or long tail. You mentioned in recent AIMs discussions as leptokurtic having fat tails? Which is true? Pls. give one real world example each for leptokurtic and platykurtic.
Thanks for the briefcasts. We should appreciate(you) that it must be very tough to be so detailed on all three parts of Risk subject(i.e. Market, Credit and Rest all).
Also, if possible, can you pls. briefcast on Metalgesellschaft, I mean, audio,video and excel with assumed numbers.
Thanks
Regars
mudaltiru
Hi mudaltiru,
Re kurtosis, you raise an excellent point (a prior discussion thread here)
Traditionally (for as long as I can remember), we say "asset returns tend to be non-normal with fat tails. They exhibit leptokurtosis." You will see the authors (e.g., Jorion) refer to leptokurtosis = fat tails. What really matters for us is that asset returns tend to follow distribution with leptokurtosis; i.e., kurtosis > 3 or excess kurtosis > 0
I currently believe that "fat tails" is an ambiguous term. Gujarati is correct that leptokurtosis = long tails.
But (IMO) the most accurate is: Leptokurtosis = HEAVY TAILS (as fat/skinny can be interpreted wrongly; wrong is a vertical perspective). One my favorite authors for his precision is Kevin Dowd, and he uses "heavy tails."
A good example of leptokurtosis is student's t (the distribution we use to test the sample mean when the population variance is unknown). Compare the normal at 99% confidence to the student's t:
Normal @ 99% = NORMSINV(99%) = 2.33 standard deviations
Student's t @ 99% = TINV(1%, 20 degrees of freedom; just for example) = 2.85 standard deviations
The excess kurtosis of the student's t = 6/(20-4) = 0.375 or kurtosis = 3.375.
The student's t always gives leptokurtosis and, here is the point: the 1% significance deviation is 2.85 versus 2.33. The issue is not fat versus skinny but rather, the student's t implies gives greater deviation for the same significance. You can see the tail is skinnier but at the same time, it has more density (is heavier) in the extremes.
Thanks for your kind feedback. I am really glad you appreciate the service at BT
Also, I love the name "briefcasts." That is a wonderful term!!
Yes, I will add Metal. to the briefcast schedule.
David
Re kurtosis, you raise an excellent point (a prior discussion thread here)
Traditionally (for as long as I can remember), we say "asset returns tend to be non-normal with fat tails. They exhibit leptokurtosis." You will see the authors (e.g., Jorion) refer to leptokurtosis = fat tails. What really matters for us is that asset returns tend to follow distribution with leptokurtosis; i.e., kurtosis > 3 or excess kurtosis > 0
I currently believe that "fat tails" is an ambiguous term. Gujarati is correct that leptokurtosis = long tails.
But (IMO) the most accurate is: Leptokurtosis = HEAVY TAILS (as fat/skinny can be interpreted wrongly; wrong is a vertical perspective). One my favorite authors for his precision is Kevin Dowd, and he uses "heavy tails."
A good example of leptokurtosis is student's t (the distribution we use to test the sample mean when the population variance is unknown). Compare the normal at 99% confidence to the student's t:
Normal @ 99% = NORMSINV(99%) = 2.33 standard deviations
Student's t @ 99% = TINV(1%, 20 degrees of freedom; just for example) = 2.85 standard deviations
The excess kurtosis of the student's t = 6/(20-4) = 0.375 or kurtosis = 3.375.
The student's t always gives leptokurtosis and, here is the point: the 1% significance deviation is 2.85 versus 2.33. The issue is not fat versus skinny but rather, the student's t implies gives greater deviation for the same significance. You can see the tail is skinnier but at the same time, it has more density (is heavier) in the extremes.
Thanks for your kind feedback. I am really glad you appreciate the service at BT
Also, I love the name "briefcasts." That is a wonderful term!!
Yes, I will add Metal. to the briefcast schedule.
David
sudeepdoon
New Member
HI guys,
I am a bit confused with the calculations thats done in the Excel.... Could someone explain what calculations are being done in D, E and F 13 and not understood
and I was also not able to understand what formulas are being used in the cells E18, and F21.
By theory I have understood Skewness and Kurtosis.. but these calculations are made that a bit messed up ...
Thanks,
Sudeep
I am a bit confused with the calculations thats done in the Excel.... Could someone explain what calculations are being done in D, E and F 13 and not understood
and I was also not able to understand what formulas are being used in the cells E18, and F21.
By theory I have understood Skewness and Kurtosis.. but these calculations are made that a bit messed up ...
Thanks,
Sudeep
Hi Sudeep,
I added those rows only to reconcile (what i perceive to be) a difference between Gujarati's sample skew/kurtosis and the the sample skew/kurtosis returned by Excel. You will notice they are nearly the same. The Gujarati skew/kurtosis (rows 17 & 20) should match your intuition and the reading. I did not mean to confuse with the additional rows (I just like the Excel functions to match, whenever possible). The alternative sample skew/kurtosis are *slightly* larger owing to the adjustment. I really don't think the details are critical, the broader point is: there can be more than one formula for an estimators. Is Gjuarati or Excel right or wrong? Neither. Just like sample variance can divide by (n) or (n-1): estimators are "recipes" and there can be more than one (though often one has the most desirable properties). In short, recommend you ignore my additional rows that merely serve to reconcile b/c Excel gives a different answer (which is no less right).
David
I added those rows only to reconcile (what i perceive to be) a difference between Gujarati's sample skew/kurtosis and the the sample skew/kurtosis returned by Excel. You will notice they are nearly the same. The Gujarati skew/kurtosis (rows 17 & 20) should match your intuition and the reading. I did not mean to confuse with the additional rows (I just like the Excel functions to match, whenever possible). The alternative sample skew/kurtosis are *slightly* larger owing to the adjustment. I really don't think the details are critical, the broader point is: there can be more than one formula for an estimators. Is Gjuarati or Excel right or wrong? Neither. Just like sample variance can divide by (n) or (n-1): estimators are "recipes" and there can be more than one (though often one has the most desirable properties). In short, recommend you ignore my additional rows that merely serve to reconcile b/c Excel gives a different answer (which is no less right).
David
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