Assignment 2: Pricing Diamonds

Overview of Assignment

The purpose of this exercise is to use the Wald test to determine if the reputation of the certifying agency is significant to the diamond pricing model. Note the reputation of the certifying agency is not a single variable so that we could use the t-test to determine significance but it is a set of dummy variables. Thus, we will need to use the Wald Test.

All of the information contained in Assignment 1 is still valid. Merely extend that analysis to est the significance of the “reputation of the certifying agency.

The Multivariate Analysis from Assignment is the starting place for performing the Wald test. Analyses from Assignment 1 are provided with this assignment.

Table 1. Dummy Variables Used in Regression Analysis

Condition Description / Variable for use
in Regression Analysis / Value if True /

Value if False

Color Indicator "D" / Cd / 1 / 0
Color Indicator "E" / Ce / 1 / 0
Color Indicator "F" / Cf / 1 / 0
Color Indicator "G" / Cg / 1 / 0
Color Indicator "H" / Ch / 1 / 0
Clarity = IF / CL1 / 1 / 0
Clarity = VS1 / CL2 / 1 / 0
Clarity = VVS1 / CL4 / 1 / 0
Clarity = VVS2 / CL5 / 1 / 0
Certification Body "GIA" / CE1 / 1 / 0
Certification Body "IGI" / CE3 / 1 / 0

Note: Conditions omitted are “color I” is the omitted condition for color. “Clarity VS2” is the omitted condition for clarity and Hoge Raad Voor Diamant(HRD) for certification body.

The data provided in the embedded spreadsheet (with dummy variables as shown in Table 1) to perform the regression analysis resulted in the following output.

Figure 1. Unrestricted Regression Model

SUMMARY OUTPUT
Regression Statistics
Multiple R / 0.978840159
R Square / 0.958128058
Adjusted R Square / 0.956424792
Standard Error / 710.3893619
Observations / 308
ANOVA
df / SS / MS / F / Significance F
Regression / 12 / 3406554699 / 283879558.2 / 562.5242149 / 2.8172E-195
Residual / 295 / 148872648.4 / 504653.0455
Total / 307 / 3555427347
Coefficients / Std. Er. / t Stat / P-value / Lower 95% / Upper 95% / Lower 95.0% / Upper 95.0%
Intercept / (4,920.7106) / 247.2030 / (19.9055) / 1.741E-56 / (5,407.2152) / (4,434.2059) / (5,407.2152) / (4,434.2059)
CT / 12,766.3960 / 190.0244 / 67.1829 / 7.622E-181 / 12,392.4208 / 13,140.3712 / 12,392.4208 / 13,140.3712
Cd / 3,313.1024 / 212.7145 / 15.5753 / 2.482E-40 / 2,894.4722 / 3,731.7326 / 2,894.4722 / 3,731.7326
Ce / 1,874.0171 / 158.4446 / 11.8276 / 1.115E-26 / 1,562.1922 / 2,185.8419 / 1,562.1922 / 2,185.8419
Cf / 1,471.4119 / 141.2508 / 10.4170 / 7.630E-22 / 1,193.4250 / 1,749.3987 / 1,193.4250 / 1,749.3987
Cg / 1,136.4302 / 145.7750 / 7.7958 / 1.107E-13 / 849.5396 / 1,423.3208 / 849.5396 / 1,423.3208
Ch / 565.9524 / 146.6172 / 3.8601 / 1.393E-04 / 277.4042 / 854.5006 / 277.4042 / 854.5006
CL1 / 1,792.0109 / 171.1855 / 10.4682 / 5.145E-22 / 1,455.1115 / 2,128.9103 / 1,455.1115 / 2,128.9103
CL2 / 317.4448 / 128.0887 / 2.4783 / 1.376E-02 / 65.3614 / 569.5281 / 65.3614 / 569.5281
CL4 / 1,102.7205 / 144.4522 / 7.6338 / 3.184E-13 / 818.4333 / 1,387.0077 / 818.4333 / 1,387.0077
CL5 / 600.8467 / 130.2751 / 4.6121 / 5.947E-06 / 344.4603 / 857.2330 / 344.4603 / 857.2330
CE1 / (15.2267) / 107.2475 / (0.1420) / 8.872E-01 / (226.2938) / 195.8404 / (226.2938) / 195.8404
CE3 / 126.0357 / 147.3853 / 0.8551 / 3.932E-01 / (164.0240) / 416.0954 / (164.0240) / 416.0954

Estimated Regression equation:

Price=-4920.710+12766.396XCT+3313.1028XCd+1874.017XCe+1471.412XCf+1136.430XCg+565.953 XCh+1792.010XCL1+317.445XCL2+1102.7205XCL4+600.8465XCL5-15.2265XCE1+126.035XCE3

Hypotheses for WALD Test

H0: B11= B12 = 0 (Dummies for certification agencies are jointly insignificant).

HA: H0 is not true

Figure 1 is the unrestricted regression model. The equation form for the unrestricted model is: Price = B0 + B1CT + B2Cd + B3Ce+ B4Cf+ B5Cg+ B6 Ch + B7 CL1+ B8CL2 + B9CL4+ B10CL5+ B11CE1+ B12CE3

Figure 2 is the restricted model (run without the dummy variables for certification agencies. The equation form for the restricted model is: Equation Form: Price = B0 + B1CT + B2Cd + B3Ce+ B4Cf+ B5Cg+ B6 Ch + B7 CL1+ B8CL2 + B9CL4+ B10CL5

Figure 2. Restricted Regression Model

SUMMARY OUTPUT
Regression Statistics
Multiple R / 0.978751
R Square / 0.957954
Adjusted R Square / 0.956538
Standard Error / 709.4622
Observations / 308
ANOVA
df / SS / MS / F / Significance F
Regression / 10 / 3405936386 / 3.41E+08 / 676.6718 / 7.6634E-198
Residual / 297 / 149,490,961 / 503336.6
Total / 307 / 3555427347
Coefficients / Standard Error / t Stat / P-value / Lower 95% / Upper 95% / Lower 95.0% / Upper 95.0%
Intercept / -4859.78 / 202.2269994 / -24.0313 / 1.28E-71 / -5257.755393 / -4461.8 / -5257.755393 / -4461.8
CT / 12683.75 / 164.2469486 / 77.22367 / 1.2E-198 / 12360.51628 / 13006.99 / 12360.51628 / 13006.99
Cd / 3315.891 / 212.41414 / 15.6105 / 1.58E-40 / 2897.863382 / 3733.918 / 2897.863382 / 3733.918
Ce / 1868.423 / 157.8764441 / 11.83472 / 9.84E-27 / 1557.72479 / 2179.121 / 1557.72479 / 2179.121
Cf / 1472.022 / 140.7465153 / 10.45867 / 5.29E-22 / 1195.035102 / 1749.009 / 1195.035102 / 1749.009
Cg / 1137.095 / 145.0629242 / 7.838631 / 8.21E-14 / 851.6133839 / 1422.576 / 851.6133839 / 1422.576
Ch / 552.7331 / 145.0120876 / 3.811635 / 0.000168 / 267.3518282 / 838.1144 / 267.3518282 / 838.1144
CL1 / 1860.72 / 158.8784056 / 11.7116 / 2.66E-26 / 1548.050243 / 2173.39 / 1548.050243 / 2173.39
CL2 / 311.9098 / 127.7400543 / 2.441754 / 0.015199 / 60.51958855 / 563.3001 / 60.51958855 / 563.3001
CL4 / 1126.811 / 140.0654037 / 8.044894 / 2.08E-14 / 851.165053 / 1402.458 / 851.165053 / 1402.458

CL5 / 625.338 / 127.6997861 / 4.896938 / 1.6E-06 / 374.026972 / 876.6489 / 374.026972 / 876.6489


Critical F = 3.00 (with 2 degrees of freedom for the numerator and 295 (used infinity) degrees of freedom for the denominator.

Conclusion: accept H0, certifying agency is not significant (.612612 < 3.00).