The Professor Proposes

The Professor ProposesECO404 | April 2nd, 2014Alejandro Bilbao, Alex Trivanovic, Lisa Guo

1AgendaOverview and Issue StatementAnalysisRecommendation

2LISA2Problem Statement3

What is the fair price of the ring?Budget: $2,000 - $4,000OverviewAnalysisRecommendationLISA

3

Characteristics of Diamonds4CaratColourCutClarityPolishSymmetryCertificationWholesalerOverviewAnalysisRecommendationLISA

4Characteristic Measurements5OverviewAnalysisRecommendationCaratColourCutClarityPolishSymmetryCertificationWholesaler1 carat = 0.2 kgColourless | D E F | G H I | J K | L M N | O P Q R S | T U V W X Y Z | YellowPoor | Fair | Good | Very Good | Excellent | IdealGIA | AGS | EGL | IGI1 | 2 | 3Flawless | FL | IF | VVS 1 2 | VS 1 2 | SI 1 3 | I 1 -3 | IncludedLISAExplain the scale for each characteristicTalk about the ones that were supposedly important we are going to find out with further analysis which are actually significantBriefly about predictions about signs5

The Engagement Ring6

CharacteristicDiamond RatingCarat Weight0.9CutVery GoodColorJ (Faint Yellow)ClarityS12 (Few inclusions at 10x)PolishGoodSymmetryVery GoodCertificationGIAOverviewAnalysisRecommendation$3,100LISA

6Analysis: Overview7OverviewAnalysisRecommendationALEKS7Analysis: Comparable Rings8OverviewAnalysisRecommendationPoint Estimate = $ 2,899.39 Max Estimate = $ 3,069.40 Min Estimate = $ 2,638.69 Comparable Rings:8Estimated Price:CharacteristicThe DiamondComparable SetCarat Weight0.90.88 0.92CutVery GoodGood ExcellentColorJ (Faint Yellow)G- N (Nearly Colorless Very Light Yellow)ClarityS12 (Few inclusions at 10x)SI3 SI1PolishGoodFair Very GoodSymmetryVery GoodGood ExcellentCertificationGIAGIAALEKSComparison method explain methodology in reference to chart regarding comparable set8Methodology: Regressions9OverviewAnalysisRecommendationCaratColourCutClarityPolishSymmetryCertificationWholesalerPrice?Linear RegressionMultiple RegressionSpline Regression (Carat)Non-Linear RegressionLogarithmic (Carat)Quadratic (Carat)ALEKSNow were going to analyze with regression how each of these actually contribute to price holding all other factors constantPoint out the difference between continuous and discrete variablesExplain two different types of regressions 9Analysis: Continuous Variables10OverviewAnalysisRecommendationNumber of Observations: 440AverageRangeStd. DevPrice$1716.74$160 - $3145$1175.689Carat0.669250.09 1.580.3798

Price DistributionCarat Distribution

ALEKSPreliminary diagnosis of price and carat and where our diamond currently lies 10Analysis: Discrete Variables11OverviewAnalysisRecommendation

ALEKSDistribution of discrete dimensions by pie chart and where our diamond currently lies 11Analysis: Wholesaler 12OverviewAnalysisRecommendationWholesaler321Number20018060Average Carat Size0.271.060.84Average Price46826623043Range of ColorD - K (Colorless - Faint yellow)D - L (Colorless - Very light yellow)D - J (Colorless - Faint yellow)Range of CutFair - IdealFair - IdealFair - IdealRange of ClarityI1 - VVS2I1 - VS2SI1 - VS2Range of PolishGood - ExcellentFair - ExcellentGood - IdealRange of SymmetryGood - ExcellentFair - ExcellentGood - IdealCertificationGIA, IGIEGL, DOW, GIAAGS, GIAALEKS12The Price and Carat Relationship13OverviewAnalysisRecommendation

Scatterplot of Price vs. Carat

Fitted Lines of Price vs. CaratALEPreliminary comments about it appears to be clustered, doesnt seem to be linear

13Creating the Best Model14OverviewAnalysisRecommendationFind the best price and carat relationshipFind the optimal division of discrete variablesOptimal Model for Predicting PriceLinearSplineQuadraticLogarithmicColourCutClarityPolishSymmetryCertificationWholesalerUse entire scale? Dummy for a portion of the scale?Which regression gives the best results? ALEHow are we going to create the best model? First we will examine our continuous variable carat. With all the other variables being discrete, this one will determine whether we are working with a linear or non linear regression we have several options with linear regressions, spline regressions acting like two distinct linear regressions, quadratic and logarithmic In terms of the discrete variables, we are interested in whether they differ by every point on the scale or just sections. That is to say, for example, does the difference between a good cut and a very good cut necessarily make a difference to price? Or is it that diamonds who have poor to good cuts are relatively similar while it matters if it becomes a VERY good cut?Lastly, we will judge on fit as well as qualitative assessment on which model is best to use

14Comparison of Models15OverviewAnalysisRecommendationLinearSplineQuadraticLogarithmicConstant303.33-144.86467.101816.59Carat1510.941812.96Carat: 1071.99lnCarat: 285.66

Carat-Sq: 232.150Discrete VariablesFull Scale SignificantColourClarityWholesaler(-)(+)(-) for 3, (+) for 2Partial Scale SignificantCertificateCutGIA/AGS vs. EGL/DOWPoor to Very Good vs. ExcellentNo SignificancePolishWholesalerALEHighlighted cells are insignificantExplain:Colour more yellow, lower priceClarity the higher on the clarity scale (closer to flawless), the higher the price Wholesaler wholesaler 3 lowers price while wholesaler 2 increase priceCertificate it is only significant when differentiating between above 3 or below 3 either GIA and AGS or EGL and DOWCut it is only significant when it is the difference between something better than excellent and ideal 15Comparison of Models16OverviewAnalysisRecommendationLinearSplineQuadraticLogarithmicR20.98180.70570.98200.9784MSE161.75206.25161.52176.75AdvantagesSimpleCaptures carat gapBest fitCaptures non-linearityCaptures non-linearityAll variables significantDisadvantagesDoes not capture non-linearityWholesaler is insignificantWorst fitInsignificant constantDoes not capture non-linearityReduces sample sizeCarat-squared coefficient insignificantWholesaler is insignificant Lower R2 than linearALE16Example Model: Logarithmic17OverviewAnalysisRecommendationVariableCoefficientlnCarat285.6625Scale For Colour2-102.96743-242.15034-554.7601Scale For Clarity3426.37744602.86495867.37556930.5837952.490981003.0279950.4456101054.666Wholesaler2208.78813-2193.272New Scale For Symmetry3188.61934192.07825182.9909Dummies and ConstantDummy cut46.78337dummy_bestlab239.6189_Constant1816.598ALE17The ResultsModelPredicted PriceLower 95% CIUpper 95% CILower 95% PIUpper 95% PILogarithmic (Wholesaler) $2,831.08 $2,754.77 $2,907.38 $2,474.99 $3,187.16 Quadratic (Wholesaler) $2,861.20 $2,793.53 $2,928.86 $2,536.10 $3,186.29 Linear (Wholesaler) $2,857.23 $2,789.67 $2,924.80 $2,531.68 $3,182.79 Logarithmic (No Wholesaler) $2,384.86 $2,282.44 $2,487.27 $1,691.66 $3,078.05 Quadratic (No Wholesaler) $2,615.68 $2,552.07 $2,679.29 $2,190.49 $3,040.87 Linear (No Wholesaler) $2,584.29 $2,515.65 $2,652.94 $2,122.67 $3,045.91 18OverviewAnalysisRecommendation

$3,100ALE

18Youre paying too much!19OverviewAnalysisRecommendation$2,000$4,000$3,100

$2,899 (Comparable)$2,861 (Quadratic)$2,857 (Linear)$2,831 (ln)

Too high!ALE

19Recommendation20OverviewAnalysisRecommendationCharacteristicOriginal RingRing ARing BPrice$3,100$3,006$3,064Carat Weight0.90.90.9CutVery GoodVery GoodGoodColorJ (Faint Yellow)J (Faint Yellow)J (Faint Yellow)ClarityS12 (Few inclusions at 10x)VS2 (Very Slightly Included)VS2 (Very Slightly Included)PolishGoodVery GoodExcellentSymmetryVery GoodVery GoodExcellentCertificationGIAGIAAGSLISA20Thank-you,Questions?AppendixLink to spreadsheet for comparability22Appendix: Comparable Diamonds23Comparable Diamonds Carat Dummy_Colour Colour Scale for Clarity Clarity Scale for Cut Cut Dummy_Best LAB Certification Scale for Polish PolishScale for Symmetry Symmetry Price Wholesaler 0.913 J 6 SI1 4 V 1GIA4V4V302310.92 I 6 SI1 4 V 1GIA5X4V302810.912 H 5 SI2 3 G 1GIA4V3G303810.92 H 5 SI2 4 V 1GIA4V3G304310.92 I 5 SI2 3 G 1GIA4V4V306910.92 I 5 SI2 4 V 1GIA4V4V306910.92 I 6 SI1 3 G 1GIA4V3G308110.923 J 6 SI1 4 V 1GIA4V4V30911Recommended Diamonds Carat Dummy_Colour Colour Scale for Clarity Clarity Scale for Cut Cut Dummy_Best LAB Certification Scale for Polish PolishScale for Symmetry Symmetry Price Wholesaler 0.93 J 7 VS2 4 V 1GIA4V4V300610.93 J 7 VS2 3 G 2AGS5X5X30641Appendix: ResultsCarat Graphs24Appendix: Final Quadratic Regression25Final Quadratic Regression SourceSSdfMSNumber of obs=441F( 20, 420)=1142.07Model5977238392029886191.9Prob > F=0Residual10990755.242026168.4649R-squared=0.9819Adj R-squared=0.9811Total6087145944401383442.26Root MSE=161.77priceCoef.Std. Err.tP>|t|[95% Conf.Interval]carat1071.998319.20413.360.001444.56181699.435caratsqr232.1507153.18111.520.13-68.94634533.2477scaleforcolour2-124.692722.64757-5.510-169.2094-80.175963-304.521430.06927-10.130-363.6264-245.41644-661.704155.57112-11.910-770.9362-552.4719scaleforclarity3558.113641.7855413.360475.9788640.24854776.468455.598313.970667.1829885.75451058.52151.432920.580957.42351159.61961136.53457.3088219.8301023.8861249.18271158.61762.1550318.6401036.4431280.79181223.32967.6451718.0801090.3631356.29491156.87891.4466212.650977.12811336.628101295.136131.67949.8401036.3031553.968wholesaler263.4686952.785311.20.23-40.28761167.2253-1760.534103.3932-17.030-1963.767-1557.302newscaleforsymmetry3191.422139.627024.830113.5301269.31414222.771741.240565.40141.7081303.83535219.632546.283474.750128.6564310.6086dummycut40.1457516.595492.420.0167.52518172.76632dummy_bestlab264.479629.812458.870205.8794323.0798_cons467.1027172.74482.70.007127.5507806.6547Appendix: Final Linear Regression26Final Linear RegressionSourceSSdfMSNumber of obs=440F( 19, 420)=1198.52Model5958163751931358756.6Prob > F=0Residual10989146.442026164.6343R-squared=0.9819Adj R-squared=0.9811Total6068055214391382244.92Root MSE=161.75price.CoefStd. Err.tP>|t|[95% Conf.Interval]carat1510.937137.201711.0101241.251780.625scaleforcolour2-132.071922.27041-5.930-175.8473-88.296593-314.991330.1648-10.440-374.2841-255.69854-662.120555.57713-11.910-771.3645-552.8765scaleforclarity3538.752239.4278813.660461.2517616.25284750.274452.1222614.390647.8214852.727451039.66149.1878221.140942.97641136.34661123.75755.977320.0801013.7261233.78871148.79961.2856518.7401028.3341269.26481217.56267.1618818.1301085.5471349.57791141.6790.4620112.620963.85521319.485101283.091130.98189.801025.631540.553wholesaler256.3935451.975161.090.279-45.77031158.55743-1657.75381.22857-20.410-1817.418-1498.087newscaleforsymmetry3189.970339.609674.80112.1124267.82814210.668940.591785.190130.8805290.45735207.083845.489114.550117.6691296.4985dummycut44.1900416.434692.690.00711.8855476.49454dummy_bestlab247.277726.946549.180194.3108300.2446_cons303.33141.43982.140.03325.31188581.3481Appendix: Final Logarithmic Regression27Final Logarithmic RegressionSourceSSdfMSNumber of obs=441F( 19, 421)=998.15Model5954952101931341853.2Prob > F=0Residual1321938442131399.962R-squared=0.9783Adj R-squared=0.9773Total6087145944401383442.26Root MSE=177.2priceCoef.Std. Err.tP>|t|[95% Conf.Interval]lncarat285.662550.722635.630185.9614385.3637scaleforcolour2-102.967424.40753-4.220-150.9432-54.991553-242.150331.89968-7.590-304.8527-179.44784-554.760159.5737-9.310-671.8591-437.6611scaleforclarity3426.377442.671599.990342.5015510.25334602.864956.9721910.580490.8795714.85035867.375551.4701116.850766.2051968.54596930.58357.9738616.050816.62871044.5377952.490963.6560814.960827.36761077.61481003.02769.4211514.450866.5721139.4829950.445697.474859.750758.84761142.044101054.666141.4457.460776.63921332.692wholesaler2208.788153.141363.930104.3326313.24353-2193.27262.87429-34.880-2316.859-2069.685newscaleforsymmetry3188.619343.392284.350103.3268273.91184192.078244.758924.290104.0994280.0575182.990950.252393.64084.21407281.7678dummycut46.7833718.108012.580.0111.1982.37674dummy_bestlab239.618931.987127.490176.7446302.4933_cons1816.59885.3902621.2701648.7531984.442