Luckiness in Singapore Condo Prices: Number Superstitions (Part 4)

5 Results and Analysis Baseline linear and quadratic regressions produced unexpected negative estimates for the coefficient of the lucky unit number variable, and indeterminate results for the coefficient of the unlucky unit number variable. To address inter-building variations that might have caused endogeneity problems, building fixed effects were added to the model. Results then showed a statistically significant price premium for the lucky numbers 6 and 8, and an extremely robust price discount for the unlucky number 4. Running fixed effects regressions on different districts shows that both the lucky number premium and unlucky number discount increases with income, suggesting that lucky numbers are, indeed, a luxury good. 5.1 Baseline Regressions: Linear and Quadratic Table 5.1.1 summarizes the results from the linear models 1-4, which regressed price on area, floor, height (relative floor), age at the time of transaction, income, dummy variables top floor (within top 5 floors of a building), freehold (ownership rights held in perpetuity rather than 99 years), and the number of years of ownership rights left in a leasehold building [see Equation 1]. Moving across the table, Model 1 did not account for income, Model 2 used the ranked districts income proxy, Model 3 used the price cutoffs income proxy, and Model 4 used income information from the 2000 census. Table 5.1.2 summarizes the results from the quadratic models 5-8, with the same permutations for the income proxy [see Equation 2]. 26 Table 5.1.1: Linear Regressions luckyfloordum Model 1 Model 2 Model 3 Model 4 Price, No Income Controls Price, Income by Ranked Districts Price, Income by Price Cutoffs Price, Income Information 13,453.3 *** 18,299.5 3,591.7 luckyunitdum -15,228.0 *** -9,634.9 3,544.4 unluckyfloordum unluckyunitdum area 9,832.3 3,976.8 -3,618.5 -7,384.9 4,125.7 3,350.3 3,078.1 887.1 16,866.3 -164,738.4 -18,761.2 transage freeholdyn -18,038.0 *** 13,672.1 *** -103,422.4 -29,933.4 *** -11,610.6 335.3 -80.8 -8,066.3 402.3 329.7 *** 3,156.5 360,318.0 *** *** 2,334.0 *** 103,852.5 *** 205156.1 *** 717172.9 ** -12,976.6 -9,390.6 -7,804.9 *** *** *** 127,370.6 11,482.8 -7,817.3 *** 872.7 *** ** *** 3.0 *** 22,288.6 *** 299.0 *** -218,928.8 *** 9,355.4 *** -30,946.2 *** 4,216.6 *** -13,823.3 *** 388.1 *** -6,157.9 *** 383.7 *** -72,726.1 26,894.6 32,459.9 -207.9 -4,221.7 305.1 367.9 8,540.0 *** 3,864.6 313.0 ** *** *** 2,934.2 *** 3161.4 highincome -125,492.9 -8,561.8 3,688.3 319.5 2892.4 uppermiddle *** 3,390.8 320.5 lowermiddle 12,772.3 *** 3,315.6 7,550.2 28,292.0 392.5 *** 252.5 3,682.8 34,633.3 yearsleft *** 8,102.3 *** 700.1 14,804.8 3,366.9 3.1 258.7 410.4 601,463.2 *** 2.6 4,525.9 buildingage 2,644.4 2,938.1 9,962.1 topfloors -10,387.5 7,738.9 314.5 height *** 2,878.2 3,198.3 *** *** 2,683.5 2,149.3 985.4 ** 16,144.9 3,938.4 3.1 floor *** 2,917.6 26,946.0 *** 3,441.4 *** 3222.3 416,357.7 *** 4,962.4 incomedata 224.4 *** 1.6 incomesqrdata _cons N R-squared -625,256.6 -120,580.1 -857,270.3 35,630.7 *** -592774.3 29279.4 *** 28,040.5 33,739.4 105891 0.5663 105891 0.7141 105891 0.7590 92829 0.6695 *** *** significant at 1% level; ** significant at 5% level; * significant at 10% level. Models are based on Equation 1 Dummy variables for residential districts are absorbed into Model 3; to avoid collinearity, they are not included in the other models, which depend on assigning average transaction price/income values to each district to determine the income variables. 27 Table 5.1.2: Quadratic Regressions luckyfloordum luckyunitdum Model 5 Model 6 Model 7 Model 8 Price, No Income Controls Price, Income by Ranked Districts Price, Income by Price Cutoffs Price, Income Information 1,554.5 6,378.6 3,606.4 2,932.8 -12,733.8 *** -8,161.7 3,517.7 unluckyfloordum unluckyunitdum area 13,436.8 *** 6,474.1 transage freeholdyn -4,232.6 -7,548.8 3,328.7 3,047.1 *** 738.7 17,699.3 352,494.7 12,230.2 -14,409.5 *** 20,186.3 *** ** 171,053.6 *** floorsqr heightsqr 334,997.2 buildingagesqr transagesqr yearsleftsqr *** -4,486.5 ** -8,730.2 *** 590.0 *** 29,310.2 *** -22,998.8 23,463.4 *** -5,084.8 *** -6,520.7 4,593.9 *** -4,173.3 486.0 -2,531.3 18,269.8 2,843.6 3,397.0 -199.5 20.5 -342,103.5 -518.1 -128.1 *** -281,667.7 *** -383.8 -217.2 *** -256,273.9 *** 13.8 459.0 33.6 27.5 25.3 32.1 6.4 11.8 -143.2 18.5 17.0 20.4 256.4 *** 22.6 lowermiddle 107,813.0 24.9 -599.9 26.6 *** 2,886.4 uppermiddle 211,241.6 717,593.1 39,440.0 *** *** *** *** 31.1 *** 234.7 *** *** *** 3,059.8 *** 3,154.9 highincome 378.5 *** 19,714.7 32.6 *** *** 24.7 15,595.3 *** 0.1 *** 0.0 19.0 16,973.9 *** *** 0.0 *** *** 591.1 3,085.2 25.1 *** 825.2 395.9 0.1 *** 12.1 1,420.8 *** ** 3,819.2 142,113.8 0.0 *** -9,028.3 *** 3,651.5 491.0 *** 14,721.1 801,486.2 -34.7 -219.7 *** * 3,277.3 7,382.4 0.0 *** 3,364.3 119,204.4 *** 20,846.7 -6,180.4 129,412.0 0.0 -547,830.4 112,859.9 3,662.8 421.3 0.1 *** 15,065.0 -10,044.3 *** 648.5 3,999.6 513.1 *** 16,253.1 6,244.0 678.0 -39,785.4 *** 18,518.3 511.4 *** 304.7 20,097.8 -4,689.6 5,340.5 11.7 697.9 623.1 -1,183,631.0 *** 10.5 3,783.2 areasqr 11,760.1 4,093.9 158,832.0 yearsleft ** 2,961.5 659.8 ** 2,617.9 2,912.8 4,915.4 buildingage -8,924.7 3,182.8 24,686.2 topfloors *** 3,914.4 855.1 height 6,488.3 2,688.1 2,860.1 12.9 floor ** 79,604.4 *** 3,770.7 *** 3,229.6 481,087.1 *** 5,345.5 incomedata -123.5 *** 14.1 incomesqrdata 0.0 *** 0.0 _cons N R-squared 1,311,949.0 *** -542,551.9 *** 228,005.4 * -713,004.9 159,087.0 129,662.7 119,734.7 147,155.9 105891 0.6 105891 0.7 105891 0.8 92829 0.7 *** significant at 1% level; ** significant at 5% level; * significant at 10% level. Models are based on Equation 2. Dummy variables for residential districts are absorbed into Model 3. 28 *** The coefficient estimates for the control variables in both the linear and quadratic models were generally of the expected sign and significant. A marginal square foot of space in a condominium unit costs about $800, which is similar to the data’s average price per square foot of $795.37. This suggests that area’s relationship to price is linear, or at least close to linear. The quadratic regressions bear out this conjecture. Estimates for the linear area term were between $600 and $750, except for the $305 estimated by Model 7, which used the price cutoffs income characterization. Meanwhile, squared area terms were also positive but very small, ranging from 0.045 to 0.11. The passage of one year reduces the unit’s price by about $8,000. Also as anticipated, buyers in the lower middle, upper middle and high income brackets pay progressively higher prices for their units relative to those in the low income bracket. Interpreting the coefficient estimates for floor, height and the topfloors dummy variable is more complex: they should not be considered in isolation because they all address the same underlying issue. Counter-intuitively, according to the linear models, being within the top five floors of a building subtracts between $13,000 to $30,000 from the total price, controlling for other measures. The quadratic regressions estimate a positive marginal effect for the topfloor dummy, so it is likely that the earlier negative top floor coefficients were capturing the nonlinear effects of absolute and relative floor ignored by the linear regressions. The best way to interpret these coefficients together may be to apply the coefficient estimates to an average unit in an average Singaporean condominium: one on the ninth floor of a 20storey building. From Model 6, 11 the unit’s absolute floor contributes $165,510 to the transaction price, while its floor relative to the 20-storeys accounts for $7,697. In total, moving from the eighth floor to the ninth floor of an average building adds $173,207 to a 11 Quadratic regression using ranked districts income characterization. 29 unit’s transaction price. 12 The average unit is not on among the highest floor of its building, so it does not enjoy the additional $6,244 top floor premium. Estimates for the lucky and unlucky floor dummy variables were mixed. While coefficients for the lucky floor dummy were positive and mostly significant at the 5% level, the results showed that unlucky floors also commanded a price premium relative to the neutral base case, and these estimates were also mostly significant at the 5% level. Using the ranked districts income definition, the linear model estimated that lucky floors added approximately $18,300 (0.9%) to the total price, while the quadratic model estimated a price premium of around $6,400 (0.3%). Unlucky floors added $2,149 (0.06%) to transaction price relative to neutral floors in the linear model, and $6,474 (0.09%) in the quadratic model. It is inadvisable to be too credulous about these estimations, however, because it is likely that the coefficient estimates capture effects other than luckiness.13 (It is for this reason that this paper chooses to focus both on unit numbers.) Though multiple attempts were made to isolate the lucky and unlucky factor from the floor variables, not all of them can be controlled for. As discussed above, the model included a top floor variable for being within the top 5 floors of a building. The models do not account for a top floor premium at all when buildings are shorter than five floors—since all floors would meet the dummy variable’s condition. Space being a precious commodity in Singapore, buildings are more likely to be short if they are aimed at a “luxury” demographic. The fourth floors of four-storey buildings would probably command a top-floor premium above the first, second and third floors, and would also be systematically correlated with higher-priced units since they are located in expensive condominiums. The models do not control for either of these effects, which may explain the positive estimates for the unlucky floor coefficients. 12 (9 · 20186 + 92 · -200) + [(9/20 -8/20) · 171,054 + (9/20-8/20)2 · -342,104] = 173,207. Later, fixed effects coefficient estimates for the lucky floor dummy are all statistically insignificant, which supports this hypothesis. 13 30 The estimates for the lucky and unlucky unit dummy variables were also discouraging. The lucky unit dummy’s coefficient was estimated to be negative by all eight of the baseline models. 14 Meanwhile, the unlucky unit dummy’s results were inconclusive— it was the only variable of interest which had both positive and negative coefficient estimates, and only the Models 11 and 12 produced significant results. Clearly, the results from the baseline models seemed to disqualify the original theory that lucky floor and unit numbers command a price premium and unlucky floor and unit numbers suffer a price discount. But it seems curious that behavior so well-established in one market (like auctions for lucky-numbered license plates) would be completely absent in another market. So it is worth considering whether unobserved heterogeneity in these linear and quadratic models hinders estimation of the true price of lucky and unlucky numbers. 5.2 Fixed Effects Regressions Two possibilities for the baseline regressions’ puzzling results are the existence of omitted variables and poorly-defined lucky/unlucky unit and floor number groups. As the theory section mentioned, Bartik (1974) pointed to the endogeneity problem as one of the principle issues in hedonic regressions. A closer look at Table 1 in the data section hints at a “luxury” effect that trends with unit numbers: the higher the income group, the fewer units per floor. Only 0.5% of observations in the low income group had fewer than five units on each floor, compared to more than 10% of the observations in the high income group. Clearly, while the lucky numbers tend to be higher than 5 and unlucky number 4 is lower than 5, it is the lower unit numbers that are systematically correlated with higher-priced units in the data. This omitted variable might be biasing the results for the unlucky unit dummy upward, and results for the lucky unit dummy downward. 14 Except for the models which did not account for income at all: Models 1 (-15228) and 5 (-12734). 31 Building fixed effects provided the solution for this endogeneity problem. The method section provides a detailed discussion of the assumptions of a fixed effects model and compares it against the case of the Singapore condominium market. Since the first aim of this paper is to estimate the implicit price/discount of luckiness and unluckiness, accounting for all the unobservable inter-building variations in the data enables the model to isolate the price premium or discount that a lucky or unlucky unit or floor number had relative to other units which are identical in all respects except its unit number. 15 Another possibility is that the a priori grouping of lucky numbers (6 and 8) and unlucky numbers (4) was not appropriate. Making the model more flexible by running regressions on the full range of unit and floor dummies tests this assumption. Three groups of regressions were run using the fixed effects model. Group A regressions relaxed the original assumption about which digits fall within the “lucky” category. The model included a full set of dummy variables for the last digits of unit and floor numbers (with UnitDum9 and FloorDum0 as the base cases.) Note that this paper focuses on the estimated prices of unit numbers from this point on. Not only were the estimated floor coefficients insignificant, but, as this paper has mentioned several times, choosing a unit number has fewer attendant issues than choosing which floor to live on, and thus the unit dummies offer a much purer way to estimate luckiness’ impact. Next, in Group B, dummy variables for special cases of unit numbers and unit and floor number combinations were included to extend the analysis beyond the last digit of each number. Finally, in Group C, the fixed effects regressions were run with lucky unit number groups redefined to include only those unit numbers that had consistently positive and significant coefficients in the Group A models. Tables 5.2.1, 5.2.2 and 5.2.3 summarize the results. 15 There are 671 unique condominium buildings in the data set. 32 Table 5.2.1: Full Range of Unit/Floor Dummies unitdum1 Model 9 Model 10 Model 11 Model 12 Price, No Income Controls Price, Income by Ranked Districts Price, Income by Price Cutoffs* Price, Income Information 2,290.0 5,723.5 2,546.7 2,546.6 2,406.1 2,710.6 3,148.1 3,148.8 3,613.9 2,409.9 2,558.4 2,558.2 2,417.0 2,724.6 unitdum3 -2,917.9 -2,916.6 -625.5 -2,677.9 unitdum4 -13,043.5 unitdum2 4,955.7 ** 2,610.0 unitdum5 unitdum6 unitdum7 unitdum8 4,982.4 ** 2,609.8 *** -13,007.9 2,465.9 *** -12,895.6 2,770.7 *** -13,435.6 2,646.0 2,645.8 2,499.8 2,815.4 2,556.0 2,525.2 3,003.3 2,534.0 2,715.6 2,715.5 2,565.9 4,596.2 * 4,598.8 * 6,052.5 6,789.1 2,755.8 2,604.1 2,923.4 1,721.1 1,733.8 1,000.5 2,013.9 2,879.4 2,879.3 2,720.4 3,061.8 4,321.5 9,443.6 2,961.7 2,798.3 3,149.4 2,961.9 lowermiddleincome uppermiddleincome highincome ** 7,322.8 ** 4,722.5 62,816.1 106,929.9 2,042.2 41,767.2 123,865.7 126,278.2 2,723.8 -116,909.9 391,622.7 123,561.2 3,900.4 *** *** *** incomedata -157.2 694.6 incomedatasqr 0.0 0.1 _cons N R-squared -8,518.8 8,137.0 254,257.3 *** 444,864.8 7,266.1 67,273.3 37,127.6 1,796,381.0 105891 0.4881 105891 0.3865 105891 0.6539 92829 0.4663 *** significant at 1% level; ** significant at 5% level; * significant at 10% level. Models are based on Equation 3. Floor dummies are excluded from this table because unit dummies are the main focus of analysis, given that they provide a purer way to capture the impact of luckiness and unluckiness on price. 33 *** 2,886.2 ** 2,756.0 7,317.5 ** ** *** Table 5.2.2: Full Range of Unit/Floor Dummies, with Special Number Combinations unitdum1 unitdum2 unitdum3 unitdum4 unitdum5 unitdum6 unitdum7 unitdum8 DummyUnit66 DummyUnit88 Model 13 Model 14 Model 15 Model 16 Price, No Income Controls Price, Income by Ranked Districts Price, Income by Price Cutoffs* Price, Income Information 5,139.6 2,508.2 5,575.8 2,559.2 2,559.1 2,418.0 2,723.4 3,334.9 3,335.3 3,836.1 2,252.9 2,570.7 2,570.6 2,428.8 2,737.2 -2,733.5 -2,732.4 -406.4 -2,830.5 2,622.0 2,621.9 2,477.4 2,783.2 -12,487.2 ** *** DummyUnit99 DummyUnit44 -12,447.9 ** *** -12,382.7 *** -12,830.0 2,718.5 2,718.3 2,568.4 2,891.9 2,724.6 2,693.7 3,209.2 2,378.0 2,727.1 2,727.0 2,576.8 2,898.0 4,772.8 * 4,785.4 * 6,023.2 ** 7,085.2 2,828.7 2,828.5 2,672.8 2,998.2 1,864.3 1,876.9 1,187.3 1,834.1 2,889.9 2,889.7 2,730.3 3,072.5 6,800.1 3,714.3 8,973.0 3,035.9 3,035.7 2,868.4 3,224.4 -18,653.8 -18,658.6 -8,651.3 -33,748.6 20,551.5 20,550.1 19,415.9 22,219.3 79,086.9 ** * 44,141.3 DummyUnit68/86 5,166.1 -50,179.8 6,808.1 79,084.7 ** * 44,138.3 53,277.9 -28,809.1 -55,737.6 19,900.3 23,031.1 85,906.6 85,910.3 66,826.9 -46,096.6 61,713.9 61,709.7 58,306.8 101,398.0 -12,512.3 -12,537.3 -7,273.4 -18,241.4 13,645.6 13,644.7 12,891.2 14,298.5 highincome ** 4,847.7 62,830.0 106,927.4 2,042.3 41,798.2 123,860.8 126,275.3 2,724.1 -116,862.8 391,570.1 123,558.6 3,900.7 *** ** *** -33,031.2 21,062.1 uppermiddleincome -50,188.8 ** 41,700.5 21,063.6 lowermiddleincome ** 84,026.9 ** ** *** *** *** incomedata -156.8 694.6 incomedatasqr 0.0 0.1 _cons -8,806.4 7,797.2 253,973.8 *** 444,047.4 7,271.0 67,272.2 37,128.6 N 105891 105891 105891 92829 R-squared 0.4881 0.3866 0.6539 0.4663 1,796,353.0 *** significant at 1% level; ** significant at 5% level; * significant at 10% level. Models are based on Equation 3. Floor dummies are excluded from this table because unit dummies are the main focus of analysis, given that they provide a purer way to capture the impact of luckiness and unluckiness on price. The special floor/unit combinations (e.g. 4th unit on 4th floor, 8th unit on 8th floor, etc) are not shown because results were all insignificant. 34 35 0.00116 0.0 105892 0.2401 67,319.4 105892 0.3808 12.6 *** 0.07515 21,188.5 123,780.8 -0.23480 *** 0.07680 -123,831.9 126,502.8 0.06504 -0.09556 31,978.3 107,118.5 0.00121 -0.04203 513.6 1,986.8 -0.00845 *** 1,910.1 -14,744.8 *** -0.00288 * 0.00104 -4,196.5 ** 0.00325 ** 1,707.2 0.00107 4,155.6 ** -0.00047 526.2 1,764.9 LogPrice, Income by Ranked Districts Price, Income by Ranked Districts 102790 0.3426 64,173.4 -8,974.7 120,503.0 -121,749.2 123,154.1 44,903.2 104,281.9 1,634.2 1,978.3 -12,679.6 *** 1,892.0 -4,004.3 ** 1,663.0 4,344.1 *** 1,751.4 316.4 On Buildings with >5 units per floor Model 19C 102790 0.2184 0.0 12.6 *** 0.07469 -0.23304 *** 0.07634 -0.08728 0.06464 -0.04030 0.00123 -0.00824 *** 0.00117 -0.00262 ** 0.00103 0.00323 *** 0.00109 105892 0.5202 6,619.2 -137.5 1,986.9 -14,776.9 *** 1,910.2 -4,217.9 ** 1,707.3 4,154.2 ** 1,765.0 564.3 Price, No Income Controls -0.00055 Model 20 Model 19D Log, On Buildings with >5 units per floor 105892 0.8115 37,058.6 264,936.1 *** 3,902.2 393,681.3 *** 2,726.0 124,729.0 *** 2,044.3 63,239.2 *** 1,876.3 -14,589.9 *** 1,804.2 -1,007.1 1,612.4 3,746.5 ** 1,666.7 105892 0.3236 1799388.0 486,872.1 0.06410 0.01382 695.8 -170.6 2,106.2 -15,316.3 *** 2,024.4 -936.9 1,808.0 6,249.5 *** 1,867.5 605.7 Price, Real Income Information 1,306.5 Model 22 Model 21 Price, Income by Price Cutoffs (Districts Controlled) *** significant at the 1% level; ** significant at the 5% level; * significant at the 10% level. Models are based on Equation 3. Models 19B and 19D using log price as the dependent variable. Models 19C and 19D are run on a subset of the data, in which all condominium buildings have more than 5 units per floor. N R-squared _cons incomedatasqr incomedata highincome uppermiddleincom lowermiddleincom unluckyunitdum unluckyfloordum luckyunitdum2 luckyfloordum2 Model 19B Model 19A Table 5.2.3: Fixed Effects Regressions on Lucky/Unlucky Floor and Unit Number Groups First, a quick overview of the control variable results: as with the baseline regressions using the price cutoffs income proxy, 16 the estimations for the control variables in the fixed effects regressions were mostly unsurprising, with the expected sign and significant. The single exception was the area coefficient estimated by Model 11 in Table 3A, which is approximately half the value of the estimations from the other models. This supports the theory that the price cutoffs income proxy is less reliable than the others. On the other hand, the income coefficients for Model 10, which used the ranked districts income proxy, were completely insignificant, while the income coefficients for Model 11 were significant and in the expected direction. This is not necessarily an affirmation of the reliability of the price cutoffs income proxy—on the contrary, it is a symptom of its unreliability. The income groups in Model 11 are defined by specifying buckets of the dependent variable—the low income group includes all observations with transaction prices below the 25th percentile, the lower middle income group includes all observations with prices from the 25th to 50th percentile, and so on. In effect, the variables are designed to have a significant and positive relationship with the dependent variable. Thus the real disappointment is the fact that none of the other models produced statistically significant estimates for their income variables—a result of the lack of income data. Three items of interest were immediately apparent from the estimated coefficients of the variables of interest. First, Group A regressions using the full range of unit and floor dummies reveal that unit numbers ending 6 and 8 did command statistically significant price premiums, while unit numbers ending in 4 did suffer a large and statistically significant discount. Second, Group B’s results showed, surprisingly, that the impact from the singledigit unit dummies were much more significant than the impact from special unit numbers and unit/floor combinations. Finally, Group C’s results, which redefined the lucky unit 16 Model 7 (quadratic version) and to a lesser extent Model 3 (linear version). 36 number group to include only 6 and 8, agreed with Group A’s conclusion that the discount for unit number 4 was much larger in magnitude and much more robust than the price premiums for unit numbers 6 and 8. From Group A’s results, the strongest results of all the unit dummies were by far for UnitDum4, whose large discount of about $13,000 was significant to the 1% level for all the models, large and extremely robust no matter which version of the income proxy was used and even with the inclusion of the dummy variables for special number combinations. As for the lucky numbers, 6 and 8 did qualify. Interestingly, many of the coefficient estimates for the other unit numbers were also positive. This indicates that 9 is lower on the desirability scale than Chinese superstition would suggest. 17 According to the most reliable income proxy (using ranked districts, Model 10), a unit number ending in 6 commands approximately a $4,600 premium over UnitDum9, significant at the 10% level. A unit number ending in 8 commands a $7300 premium, over UnitDum9, significant at the 5% level. This is still only half of the discount caused by UnitDum4. Figure 5.1 compares the unit dummies’ estimated implicit prices when special number variables are included. (A fuller examination of this asymmetry follows in the discussion of the Group C regressions’ results.) 17 Perhaps another effect is obscuring or canceling out the impact of 9’s luckiness. An analysis by Kendall and Smith (1939) shows that, when people are exposed to rapid flashes of numbers equally distributed between 0 and 9, they are more likely to report seeing an even number and exhibit “a bias against the numbers 1, 3 and 9.” (Brown, Chua and Mitchell, 2002) 37 Figure 5.1 Estimated Premiums by Unit Number - Base Case UnitDum9 10000 5000 0 -5000 -10000 -15000 UnitDum1 UnitDum2 UnitDum3 UnitDum4 UnitDum5 UnitDum6 UnitDum7 UnitDum8 Average Model 16 (Ranked Districts) From the regressions that included special number combinations (Group B), the coefficients for the single lucky digit variables (unit dummies 4, 6 and 8 and floor dummies 4, 6 and 8) were much more significant than the dummy variables for the special number combinations. To recap, the special number combinations included variables for special unit numbers and for special floor/unit combinations. 18 None of the floor/unit combinations produced significant results. Of the special unit numbers, only unit number 88 had positive coefficients, and they were mostly significant only to the 10% level. Dummy unit variable 68/86, surprisingly, had a negative impact on transaction price, significant at the 5% level. Given the strong results for UnitDum4 in Group A’s regressions, it was particularly unexpected that dummy unit variable 44 produced insignificant estimates. These inconclusive results are largely due to sample size: there are only 103 observations of unit number 66, 98 observations of unit number 68, only 23 observations of unit number 88, 12 observations of 18 The complete list is: unit numbers 44, 66, 88, and 99; floor/unit combinations #04-04, #06-06, #06-08, #08-06, #08-08, #09-09 38 unit number 99. Most surprisingly, there are 240 observations of unit number 44. Evidently, concerns other than luckiness trump developers’ unit-numbering decisions. Figure 5.2 Sizing Effects from Model 19A 14,744.8 15,000.0 13,330.9 10,000.0 4,155.6 5,000.0 755.5 0.0 area transage luckyunit unluckyunit The fixed effects regressions on lucky/unlucky number groups (Group C) agreed with Group A that the magnitude of the impact of the unlucky unit number is much greater than that of the lucky unit numbers. Figure 5.2 compares the price impact of lucky and unlucky numbers against the control variables area and age. In order to avoid living in a condominium unit with an unlucky number, buyers had to forego a discount of more than $14,000, about 0.8% of the unit’s total price or about 19 extra square feet in their home; in order to enjoy the benefits of a lucky number, on the other hand, buyers were willing to pay less than $4200 (about 0.3% of the unit’s total price). This is perhaps an indication of a risk-averse population—people may prefer to insure themselves against disaster than to pay for a good that could potentially bring them bonus utility. More generally, this asymmetry is an excellent example of how differences in the perception of two objectively similar situations, depending on which situation is taken as the base case, can result in drastically different decisions, similar to the way people tend to save tax refunds (since they perceive that money 39 to have been theirs in the first place) and spend tax rebates (since they perceive the money as an addition to their normal income.) The behavior of a person in search of any-numbered unit and happens to come across the decision to pay extra for a lucky unit number or less for an unlucky unit number may be different from one who begins looking at unlucky unit numbers and must actively decide to pay more for a neutral number, or even more for a lucky number. Of course, even with fixed effects models, there might still be omitted variables driving the regression results. As mentioned in the empirical strategies section, developers could be bundling lucky unit numbers with unobservable, intra-building features that the model cannot control for—like a better layout within the unit—because the combination of superior features and lucky unit numbers may enable them to charge even more than higherquality features or lucky unit numbers separately. To investigate this, regressions were also run with interactions between the lucky number and unlucky number unit dummy variables and age, the fact that a building was freehold, and the fact that the unit was within the top five floors of its building. Results (not shown here) were inconclusive for the pure lucky and unlucky unit dummies, but, interestingly, the high income group interactions with the lucky and unlucky unit dummy interactions were significant (and the right sign) at a 1% level, and the upper middle income group’s interactions with the unlucky unit dummy were also significant at a 1% level. This hints at the question that the next section investigates—that lucky numbers are indeed a luxury good. In fact, it seems as though superstition in terms of avoiding unluckiness may also be stronger in higher-income groups—an interesting phenomenon, given that it cannot be motivated by conspicuous consumption. Perhaps the phenomenon reflects the desire of high income buyers to satisfy a minimum requirement (the equivalent to getting a “C” on a paper): even if they do not spend on lucky unit numbers, they at least avoid the stigma of living in an unlucky-number apartment. 40 5.3 Price Variation with Income Now that a price premium has been established for lucky unit numbers and a discount for unlucky unit numbers, attention can be turned to establishing how those prices vary with income. Of course, any conclusions drawn from this analysis must remain tentative, given the lack of individual-level income information. This paper divides Singapore into its 25 residential districts and runs the fixed effects regression on each district [see Equation 3]. 19 Log price is used as the dependent variable to approximate percentage changes. Earlier in this paper, each district was assigned a rank according to its average transaction price in order to create the ranked-districts income categories. The estimated prices are charted below in Figure 5.3, in ascending order of income, to investigate any correlation. Regression details can be found in the Appendix. Figure 5.3 Lucky Unit Premium v. Income Groups (Ranked Districts) Log(Price) 0.04 0.02 0 -0.02 -0.04 22 25 27 26 17 19 18 23 28 8 14 16 13 5 21 20 12 3 15 11 9 1 4 10 1 4 10 Unlucky Unit Discount v. Income Groups (Ranked Districts) Log(Price) 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 22 25 27 26 17 19 18 23 28 8 14 16 13 5 21 20 12 3 15 11 9 19 District 2 is excluded because it only has 23 observations in total 41 One difficulty is that once the data set is decomposed into district subsets, there are too few observations for significant results. Since the implicit prices and discounts estimated here cannot be relied upon, the analysis returns to the coarser breakdown of the data set, grouping the data into low income, lower middle income, upper middle income and high income groups, to allow for larger subsets of data. The ranked districts income proxy was the most reliable and thus used here. Figure 5.4 below illustrates the results (details of the regressions can be found in the Appendix.) Figure 5.4 The majority of the estimations Lucky Unit Number Premium v. Income Groups (Ranked Districts) Log(Price) broken out by income groups are 0.016 significant to the 1% level, except for the 0.012 0.008 upper middle income group’s coefficient 0.004 for the lucky unit dummy, which was 0.000 low income lower middle income upper middle income high income significant to the 10% level. The Unlucky Unit Number Discount v. Income Groups (Ranked Districts) insignificant results were the lower middle Log(Price) 0.010 income group’s estimate for the unlucky 0.000 -0.010 unit dummy and the low income group’s -0.020 for the lucky unit dummy. -0.030 low income lower middle income upper middle income high income These results imply that lucky numbers are a luxury good (though, of course, since these percentage changes cannot be truly compared to percentage changes in income, the conclusion is necessarily questionable.) Interestingly, it is the lowest and highest income groups which discount unlucky unit numbers most, though the low income group’s implicit price for lucky numbers, even as a 42 percentage of total price rather than as an absolute value, was the smallest and statistically insignificant. As the earlier estimates of implicit prices indicate, the unlucky number effect is much stronger than the lucky number effect. Here, it seems to be much stronger in the lowincome group than expected—perhaps a hint that risk aversion is even stronger among lowincome demographics than high-income demographics. 43