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Unformatted text preview: A Guide to Diamond Pricing Using the “4 C’s” Econometric Analysis of Retail Diamond Pricing Michael Anderson
Econ 388 4 April 17, 2007 Abstract Diamonds have become a large part of the American culture Since the late 1920’s
when DeBeers started an aggressive advertising campaign to boost slumping diamond
sales. De Beers established a monopoly on diamond sales through aggressively
purchasing diamond mines throughout the world Because of the monopolistic properties
of the diamond market the pricing of diamonds is different than we would expect in more
competitive markets. Another aspect of diamonds that makes the pricing of diamonds
rather unique is the nature of diamond purchasing. Many Americans will purchase only a
few diamonds in their life. Additionally these diamonds are usually purchased for a
future spouse and the spouse may have signiﬁcant power over what is bought. Sometimes
this can lead the husband to forego information gathering since it may be hard to haggle
over a price when the recipient of the diamond may be oﬁ’ended by overt price sensitivity.
This research paper is going to provide several models in order to attempt to show how
diamonds are priced by typical retailers that operate throughout the United States. This
can operate as a guide for people to estimate prices on diamonds with characteristics
they desire.
Introduction Diamonds form a large part of many thirdworld economies. Because the diamond
industry has such a large impact on some small economies, many studies have been done
on diamond market growth. Economic risk is also evaluated for these small economies in
Africa since any fall in global demand for diamonds can pose serious dangers to the economic wellbeing of countries such as Botswana and Angola. (Africa Monitor: Southern Africa, 2007) One of the more interesting studies done with more lasting impact is on
environmental impact. With the world becoming increasingly green the impact on the
environment from mining diamonds could become a more important aspect of a
company. In part because of attempts by highproﬁle people to make us aware of the
origin of diamonds, society is beginning to become more aware of the costs of procuring
diamonds. Canadian diamonds are being marketed as “clean” as opposed to Africa’s
“blood” diamonds that were made famous by movies like Blood Diamond and songs such
as “Diamonds From Sierra Leone” that demonize the diamond trade and the social
atrocities associated with it.(Barney, 2007) This paper Will provide several models that will be available for use by any
person who would like to get a rough estimate on prices of diamonds. There are many
different retail options to choose from when buying a diamond. The options run from
mall retailers, which are popular but expensive, to single employee wholesale operations,
which are small, may be less reliable but far cheaper. The data used in this analysis was
gathered from a online diamond retailer with over 20,000 diamonds in the desired range
to use in this analysis. The ﬁrst section will outline the models we wish to analyze. The second section is
going to describe the origin of our data and give a brief summary of the variables that we
will be using. The third section is dealing primarily with the analysis of the models
proposed in the next section. Lastly we will go over the conclusions that have been
obtained through our regression analysis. Description of model In order to develop a model that accurately predicts the price of diamonds we
need to understand the characteristics that are quantiﬁable or at least deﬁned by the
diamond retailers. The characteristics shown to retail customer are usually referred to as
the “4 C’s”. These “C’s” are cut, color, clarity, and carat. The only one of these that is
quantiﬁed is carat. Cut quality is based upon comparing the diamond with the
mathematically established ideal cut. This is quantiﬁed but then split up into ﬁve
categories which are not quantiﬁed. The remaining two are split up into categories that
are not based upon a quantitative scale. Color and clarity. are both based upon scales that
are produced by the certiﬁcation associations. Price= [31 + '32 (carat) + [53 Up + at This model implies the price of the diamond has a minimum regardless of the
size, which would be the intercept. After you pay the intercept price you have to pay a
certain price per carat increase in size. In addition I have added a variable that is going to
separate my diamonds into two groups. in this model. Up=1 if the diamond’s price is
above $3000. I set it up this way becausel don’t think that just relying on the size of the
diamond is going to be very predictive. I believe that this model will be predictive over a
short range. After adding Up I will have two small ranges where my residuals are fairly
close to zero. I have two reasons for choosing such a simple model to start my paper
with. First, the most common question asked when talking about diamonds is the size and
it is important to realize how much of the price of diamonds can or cannot be attributed to
the size of diamonds. Next, this is a simple model that we can use as a baseline to compare our other models too. Our second model is: Price= [51+ B2 carat+ B3 (AGSL cert.)+ [34 good + [35 vgood + [56 ideal + B7 sigideal +138 D + [59 E + [310 F + [511 G +B12H + [513 IF + B14VVS1 +ﬁ15‘VV82 +
I316 V81 + 8t This model can be interpreted as the change in price of diamonds due to categorical
change among the 4 CS. It seems somewhat intuitive that the price of diamonds would be
very highly correlated with the “4 C's”. Differences in these four characteristics are
visible to people who take even a casual interest in diamond quality so it seems
reasonable to assume that differences in these would be able to accurately predict the
price of diamonds. I would expect two factors to inﬂuence the correlation with price
these variables have. The ﬁrst one would be visibility. If ﬂaws in one category can be
seen by onlookers you can imagine that the premium to ensure that you don’t have those
ﬂaws would be higher than those ﬂaws which only experts are able to see. Because of
this I would expect color and clarity to have a large impact on price. The other factor V
would be whether the characteristic is natural or not. Clarity and cut are two
characteristics that can illustrate this point. Clarity is an innate part of the diamonds. If
the diamond has internal ﬂaws you are not going to be able to remove them without .
compromising the diamond itself. Therefore the difference in price between a v52 and a
vvsl diamond would be greater than the difference in price between a fair cut diamond
and a very good cut diamond. The cut is made by men and so if it was shown to have a
large impact on the retail value you would simply cut the diamonds in a manner so that
more and more of the diamonds have better cuts. In this way man can directly inﬂuence the price of the diamonds independent of nature. My third model is: Price=‘[51+ [32 ln(carat)+ B3( AGSL cert.)+ [54 good + [35 vgood + [is ideal + [37 sigideal + 68 D +3913 + [510 F + [311 G + B12 H + [31311T + B14 VVSl + [515 VVSZ + B16
vsl + B17 UP + at The most signiﬁcant difference from model two is the variable UP included in the second
model. This UP variable is slightly different than the UP variable in the ﬁrst model. Here
I described UP=1 if the price of the diamond was above $4000. I propose that this will
have the same effect as in our ﬁrst model. I hope it will in effect give us two areas where
we have more predictive capabilities. Because all cut, color and clarity variables are
binary variables we have left out one variable from each category from our model to
avoid what some call “The Dummy Variable Trap.” The intercept will include the
variables we left out as the default price so all these variable are still implicitly analyzed.
We expect all these variables to have signiﬁcant impact on the price. The variables left
out of our functional form are the poorest grade variables in each category. Since the
model was set up this way we expect that each variable will have a positive coefﬁcient as.
each is an improvement in quality over the base. The intercept in this model is a diamond
with the following characteristics: AGSL certiﬁcation, fair cut, I color, and v32 clarity.
The last model to be discussed in this paper is going to include many interaction
variables, which will hopefully betable to tease out the more intricate relationships
between the 4 C’s and price. I. am choosing a model with all combinations of interaction
variables between cut, color, and clarity because it makes sense that marginal price effects are greater for a clear diamond improving clarity than a colored diamond improving its clarity. In dating girls you can see the same sort of relationship. There are V4 ﬂ
SM many smart girls and many beautiful girls, but the marginal premium on smart, pretty 6’ ‘h l) \t oi A L
V6” girls is higher the premium on just pretty or smart girls alone. I chose to include all
possible interaction variables between color, clarity, and cut. I could have chosen to use
interaction variables between cut type, color, clarity, and cut; but this would have given
the model 540 interaction variables which I believe is too many variables for a pricing
model. With the approximately 180 interaction terms I currently have you could still
make a pricing guide out of this and have it be useful and pragmatic for prospective
diamond consumers.
Description of data The data was gathered from the extensive resources of the online retailer Blue
Nile. Their inventory is extremely extensive so some upper boundaries were set on data
to be collected. I used price as the upper boundary set at $10,000. This upper boundary is
not completely arbitrary but was set here as the upperlimit of what a somewhat afﬂuent
college student would spend on an engagement ring. I also set other bounds on my data
gathering. This was to weed out portions of data that would not serve the general purpose
that this paper is supposed to serve. I didn’t include any diamonds whose cut was poor or
worse, I also excluded all diamonds with color I or worse. These bounds were set so we
could analyze the price of diamonds primarily over the ranges most people seem to want
their diamonds in. The data is available to anyone on the website by going to the search diamonds section and inputting in the variables which you are interested in inspecting. Diamond Data m_———
[m Ideal
__——_
2325746
m
_——— 22582 .1012311 3016411128600
22582 .1932955 .3948915 4364.99 22582 .3590913 .4797447 8108.99
22582 .2883713 .4530148 6512.00
22582 .8835798 .3207352 19952.99
22582 .1164202 .3207352 629.00 The
units used on
this data may be
a little
unconventional
since most of
my variables are
not quantiﬁed.
For my dummy
variables there
are no units to
use. Price is measured in nominal dollars, and carat is measured in carats. Carats are measurable in milligrams Where lcarat = 200 milligrams. The accuracy of the regressions done tocalculate the relationship between these variable is highly dependent on the assumptions we make about our data. The assumptions that are most important to test are the ﬁve basic assumptions made about the Classical Normal Linear Regression Model. Using the sktest on Stata we get a probability of 0.000 that our residuals have skewness of zero and kurtosis of 3 individually. This gives us good evidence that the errors are not distributed normally. Assumption two, (errors sum to zero) holds because
Least Squares method regresses so that this holds true. Heteroskedasticity runs rampant through this data. When measured by the white
test performed on my third model the Chisquared value= 5411.386, so we can deﬁnitely
reject our null hypothesis that our data is homoskedastic. This is not unexpected since as
you get higher quality and bigger diamonds you would be able have greater variations in
price due to the interaction effects between bigger diamonds and higher quality
characteristics. With heteroskedasticity having such a strong inﬂuence on our data I will
consider using varianceweighted least squares to predict the price. Using the variance
weighted least squares method allows me to get estimators with the smallest variances.
Even if I think that varianceweighted least squares is not the best method to regress these
models I still must use robust standard errors to ﬁx the tstats of my estimators. Using
robust standard errors ﬁxes the tstats by using the correct standard errors for our tstats.
This may be preferred over vwls because the robust standard errors only affect our t—stats
not the estimators themselves. The data used in this project was cross—sectional so autocorrelation is not
expected to have much impact on the validity of my estimators or tstats. Assumption
four and ﬁve hold since autocorrelation doesn’t exist and my depenﬁéit variables are all /
UKJL’ nonstochastic. lncarat 3 c’s
(50.7341 (54.95811)
(15.46875)
(41.36648) (41.464) Emerald n/a 64.51 1 19 24.43138
(57.28618) (25.4261) D n/a 943.6286*** 984.3508***
(72.26282) (34.29037)
(64.98108) (30.28893) F n/a 653.6381*** 561.4954***
(62.95018) (28.82789) G n/a 514.2412***
(61.60482) (28.61051)
(70.73583) (31.39736) n/a
(123.0841) (54.09016)
(106.8407) (47.53892)
(101.8489) (45.50346) ' (103.9431) (46.65883)
n/a Agsl Constant *d a, ‘13) *= pValue<.10
* *=pva1ue<.05
* * *=pvalue<.01 Results ' (58.16311) (24.43991)
(47.13308) (20.61395)
(54.16268) (23.63501) (42.50969) ' (18.24912) (63.67606) (22.53938)
(20.94332) 19.46469)
2890.262*** 2677.71*** 561.792***
(39.63719) (128.4634) (51.26683)
R =0.7552 R =0.0733 R _= 0.8230 Fstat=24936.50*** Fstat=175.36*** Fstat=4151.68*** In my actual analysis of the data I decided not to use the varianceweighted least
squares regression techniques. When using vwls my coefﬁcients were not different
enough from OLS to show that vwls was necessary. Robust standard errors showed
results that were held well with the basics of our model. The ﬁrst model gave me some
unexpected results. Without the variable UP carat had a strong positive effect on price,
but after adding in UP the correlation between carat and price became strongly negative
with a pvalue less than 0.001. Adding the variable UP didn’t make the model very
predictive. The R2 value is very deceptive. The reason the R2 value is so high is because
UP is based on price so it would be strange if it didn’t increase my R2 value by a large
amount. Regressing price on just carat Without UP we get an R2 value slightly lower than
that for my second model. Looking at the R2 values for these models we can see that if we just regress our
price on the 4 CS we do not have an extremely strong predicting model. With these ﬁrst
three models our main problem is that we have deleted many variables that would help to
explain the price. These models do however give us a great foundation for trying to
model a pricing guide that will more accurately predict the price of diamonds. In order to
see if there are some cumulatively negligible variables in my second model I ran a
likelihoodratio test with the hypothesis'that g00d=very goodiemeraldéO. My chisquare
value=l but with three degrees of freedom I failed to reject this hypothesis. Since the
intercept for the second model represents a princess cut diamond with color I and Vs2
clarity we can see that in this model it is fairly certain that improving the cut to very good
or changing to an emerald cut will have almost no impact on the price of the diamond. My third model strengthens the hypothesis that cut has very little impact on the price. Only the signatureideal cut is signiﬁcant to even the 10% level. We also see that the correlation between clarity and price becomes much stronger in my third model. OLS OLS
Variables Estimators Estimators 256.6*** didealst 642.9
_ 32.46 _ 728.9
I 4671.5*** dsiideallf —I_—
—_——
__——
___
—_—_
__—
——— ——_—
 ——__
m
—__— 2942.8***
— 372.1 — 511.6
EE 499.4*** 3735.9*** _———
M
————
_———
—__—
————
——_—
m—
—_—_ dfairlf 1928.6***
— 799.0  421.5
1463.9 eidealif 1594.4*** — 1374.6 _ 327.6
~2474.1*** “928*
_ 944.0 — 650.4 dfairvs1 465.6 eideaIVVSZ 1538.1** —__— doodlf
_—__
——_— — 984.3 _—
esiide~vvs1
————
—_——
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—_——
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_—_
—_—— _——_
_———
————
——_
———— ————
_——_
—_——
— 346.1 fideaIVSZ 2705.7*** h  oodvvs1 3146.0*** ___—
___—
___— ___—
___—
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___— ___—
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___ ___—
___—
___—
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___— ___—
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___ 443.4 6684.3***
_ 508.8 _ 823.8 1642.4" 7595.1***
— 675.3 — 474.8 . idealvs1 876.2** io oodst 2786.8*** IIIIIIIIIIIIIMEEIIIIIIIIIIIIEEE
EHQMEIIIlllﬂﬂﬁﬂlmﬂﬂﬂlllllﬂﬂﬂﬁi
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R=02m3 Fstat=large*** The fourth model discussed in this paper is the most comprehensive regression
that is possible by only using the 4 CS. The results of this regression are very different
that I expected. I thought that with so many interaction variables we would see very
accurate predictions. This reasoning is based on the differences between including
irrelevant variables and deleting relevant variables. By including most combinations of
variables using only the 4 CS I have attempted to minimize the chance that relevant
variables are being left out. Most of the interaction variables used for this regression
have tstats that make them signiﬁcant to the l% level. This leads me to believe that most
of the interaction variables are all impacting the price signiﬁcantly. Out of the four models discussed in this paper the third model seems to be the
make the most accurate predictions. When we compare how the third and the fourth models predict the last 5% after regressing the ﬁrst 95 %, the third model has a much smaller standard deviation and the maximum and minimum error are also several
thousand dollars smaller for the third model. Including the variable UP in the third seems .
to be the main factor in producing the results that the third model produced. Because the
variable UP produces two small ranges where the price is predicted more accurately we can see thatthe third model is going to have less on the lower and upper extremes like the other models had.
Variable Obs Mean Std. Dev. Min Max
4th 1583 1151.698 2899.863 4890.607 11181.57
3rd 1582 85.48646 1056.899 2398.972 3815.778
Conclusions Having formulated four different models in order to try to show how diamonds
are priced using the 4 C’s I have come to the conclusion that the price of diamonds is not 9) determined by its “C's only. The purpose of this project is to make a guide or “Blue
book” for people that wanted to purchase a diamond and avoid being taken by
opportunistic salesmen. In order to provide a meaningful guide I proposed to use
informatiOn that would be easy for potential diamond consumers to obtain. From my
experience of helping friends and family search for diamonds I found that the information
available when presented with a diamond are the 4 C’s, so I took those characteristics and
started postulating models. One of the changes that I made to my models that probably
wouldn’t be very practical for consumers to use was putting the natural log of carat in my
models. This had good and bad effects. On one hand ln(carat) seems to more accurately
predict price when it is in the natural log form. This is because you can imagine that the price rises exponentially with carat and...
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 Winter '08
 Mcdonald,J
 Econometrics

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