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diamond pricing

diamond pricing - A Guide to Diamond Pricing Using the “4...

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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 third-world 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 well-being 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 high-profile 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 upper-limit 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 Chi-squared 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 variance-weighted 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 variance-weighted least squares is not the best method to regress these models I still must use robust standard errors to fix the t-stats of my estimators. Using robust standard errors fixes the t-stats by using the correct standard errors for our t-stats. 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 auto-correlation is not expected to have much impact on the validity of my estimators or t-stats. Assumption four and five hold since auto-correlation doesn’t exist and my depenfiéit variables are all / UKJL’ non-stochastic. 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) *= p-Value<.10 * *=p-va1ue<.05 * * *=p-value<.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 F-stat=24936.50*** F-stat=175.36*** F-stat=4151.68*** In my actual analysis of the data I decided not to use the variance-weighted 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 p-value 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 likelihood-ratio test with the hypothesis'that g00d=very goodiemeraldéO. My chi-square 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 signature-ideal 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 _— esi-ide~vvs1 ———— —_—— _——_ _——_ __—— —___ —_—— —_—_ -——— -_—_ —_—— _——_ _——— ———— -——_ ———— ———— _——_ —_—— -—- 346.1 fideaIVSZ 2705.7*** h - oodvvs1 3146.0*** ___— ___— ___— ___— ___— ___— ___— ___— ___— ___— ___— ___— ___— ___- ___— ___— ___— ___— ___— ___— ___— ___- -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 IIIIIIIIIIIIIMMEIIIIIIIIIIIIHEM EQflEMIIIIIMflMHHflMflMMflIIIIflflBEH IIIIIIIIIIIIEEEIIIIIIIIIIIIEEI lfiiflfiflflflIllfiflflfifllfiflflW&llllflflflfii IIIIIIIIIIIIIflMflfllllllllllllflzfl lmflflflfillIIIIIEEMHEMMEIIIIIQMMEI IIIIIIIIIIIIIEIEIIIIIIIIIIIIIEHI flfififlflflllIllflflflfiiflflfiflllllllflflflfifl IIIIIIIIIIIIIEEEIIIIIIIIIIIEEM! EMHWQIIIllfiflflfiflfliflaflflfllllflflflfil IIIIIIIIIIIIIEEMIIIIIIIIIIIIIEHE IH&M§IIIIIIE&flfiIfl$flEMEIIIlflflmfii IIIIIIIHIIIIIMHEIIIIIIIIIIIEMEI lflflflflflllIIIEEEHIEiHEMEIIflIEEflfiI IIIIIIIIIIIIIMEHIIIIIIIIIIIIIEEI lfifiEQEZIIIlfiEMHHIflMEIIIIIlflflfififl IIIIIIIIIIIIEEEIIIIIIIIIIIIEEE R=02m3 F-stat=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 t-stats 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 that-the 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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