Luckiness in Singapore Condo Prices: Number Superstitions (Part 3)

4 Empirical Strategy This paper attempts both to estimate the implicit equilibrium prices of lucky and unlucky numbers, and to investigate their income elasticity of demand. To answer the first question, a hedonic model is used to decompose an apartment unit into its component attributes so that the prices of unit numbers can be analyzed in isolation. To answer the second question, the model is applied to different segments of the data, broken down by income, to see how the implicit prices change. There were several issues related to the variables included in the regressions. How should the model characterize lucky numbers? How should the model proxy for income? Which is the most appropriate dependent variable in this situation: price, price per square foot or log price? There were also model choices to consider: which model specification best isolates the impact of lucky/unlucky numbers from other factors? Given the lack of income data, how can the model infer lucky numbers’ income elasticity of demand? 4.1 Characterization of Key Variables 4.1.1 Lucky Numbers First, a small but crucial question: what makes a lucky number? The sounds of Chinese words are often similar; a classic example is the syllable “ma,” which, when inflected differently, can mean either “mother,” “to numb,” “horse,” or “to curse.” Numbers are considered lucky or unlucky if they are homophonous with good or 16 bad concepts. The numbers 3, 6, 8 and 9 are lucky numbers—3 sounds like Cantonese for “growth” or “vigor,” 6 sounds like Cantonese for “wealth,” 8 sounds like Cantonese for “prosperity,” and 9 sounds like Cantonese for “long life.” The number 4 is an especially unlucky number because it sounds like the word for death. But categorization becomes difficult with multiple-digit numbers, and more so when referring to condominium units with both floor and unit numbers: which is more important, the first or last digit, the floor number or unit number? Is 48 luckier than 84, or are they both neutral? What about the eighth unit on the fourth floor? This paper follows its predecessors in focusing on the last digit when dealing with numbers greater than 9. While other papers (Chau, Ma and Ho, 2001; Choy, Mak and Ho, 2007) break off the analysis at the floor-number level, this one considers unit numbers to be a potentially important attribute as well. Not only is there greater possible variation in the data with unit numbers, unit numbers also better meet the Woo and Kwok (1993) condition for a product used in the study of superstition-influenced pricing: it is less likely to have any other intrinsic—that is, productive—value. A floor number’s value is associated with many other factors aside from luckiness. Higher floors are more prestigious, generally afford better views, and are farther removed from the noise and bustle of the streets below. Controlling for these other factors by the mere inclusion of a “floor” variable in the model may not be sufficient. Other considerations that contribute to a unit number’s value—such as the fact that units with a certain unit number, faces away from the street—are much more likely to be random (assuming that developers do not take lucky numbers into account when making such decisions.) Initially, according to a priori knowledge of Chinese superstition, unit and floor numbers with the luckiest last digits 6 and 8 are grouped into the “lucky” category, while 4 17 comprises the “unlucky” category. 7 But luckiness is not necessarily a yes/no category— rather, one could imagine a scale that ranges from the very unlucky to the very lucky. Thus the lucky categorization is later empirically tested by running regressions on a full range of dummy variables for unit and floor numbers ending in the digits 1-9. Later regressions also include dummy variables for special unit numbers such as 66 or 88, the sixth unit on the sixth floor, the eighth unit on the eighth floor, the sixth unit on the eighth floor and the eighth unit on the sixth floor—to allow for the fact that number combinations, rather than simply the last digit of a number, may be important. 4.1.2 Income One of the key issues in designing the model was a way to proxy for the income variable. (This was important as a control variable to the paper’s first question, but even more important in answering the paper’s second question about income elasticity.) Since the data do not include information on the buyers’ income levels, income could only be imputed to the district level—a coarser and more imperfect measure than ideal. In the initial linear and squared models, three ways to characterize income were considered. First, the price cutoff method divides the data into four categories by transaction prices: low income, lower middle income, upper middle income and high income are defined by the 25th, 50th and 75th percentiles. This approach takes into account changes in desirability—the same unit that was in a high price percentile back in 1996 may fall into a lower price percentile in 2007 as its building ages and its appeal declines. However, segmenting the data by the dependent variable may yield biased coefficient estimates. The results section discusses in detail the fact that coefficient estimates for area, using this 7 “3” and “9” are not included even though they have positive associations in Chinese numerology because they are “second-class” lucky numbers—that is, “6” and “8” have much stronger associations with luck than do “3” and “9” 18 method, are drastically different from estimates when other model specifications were used; in brief, this indicates that this income definition is problematic. Second, the ranked districts method orders each district by the average transaction price of condominium units in that district, and grouped them into the same four income categories. As with the price cutoff method, this assumes that buyers with higher incomes are, in general, more willing and able to afford properties with higher prices. The resulting rankings roughly correspond with the current perceived prestige of the district in the public mind. This characterization of income also suffers the disadvantage of not having an equal number of observations in each category, but, unlike the price cutoff method, does not rely directly on the dependent variable. 8 Third, income information by district from Singapore’s 2000 population census is used. As the data section describes, the census groups monthly income data into increments of $1,000 and reports the number of people in each district who fall within each income category. From this breakdown, this paper calculates the average monthly income of each district. However, the census divides Singapore into different districts than does the real estate data, which means that there was no way of imputing income information on 8 of the 25 districts. Consequently, the analysis drops 10541 observations (about 10% of the total), when using this income definition [see Section II of Appendix for table of income proxy methods]. 4.1.3 Dependent Variable 8 The districts in order from lowest to highest average transaction price (as proxy for income) are: 22, 25, 27, 26, 17, 19, 18, 23, 28, 8, 14, 16, 13, 5, 21, 20, 12, 3, 15, 11, 9, 1, 4, 10. There are different numbers of districts in each income group in order to keep the number of observations in each category as equal as possible. In the low income group: districts 22, 25, 27, 26, 17, 19, 18, 23. In the lower-middle income group: districts 28, 8, 14, 16, 13, 5, 21, 20, 21. In the upper-middle income group: 20, 12, 3, 15. In the high income group: 11, 9, 1, 4, 10. 19 This paper uses price rather than log price or price per square foot (PSF) as the main dependent variable. Given the log-normal distribution of transaction price, the log price is also used as the dependent variable, once the model is refined to eliminate endogeneity problems. Regarding PSF, though a common metric in Singapore, the absolute price is, arguably, a more important consideration for a buyer. A buyer can, within limits, choose to buy a larger, lower-PSF unit or a smaller, higher-PSF unit. But given the unit itself, it is ultimately its absolute price that determines if a buyer can or cannot afford to purchase it, and it is from this total amount that a buyer (consciously or subconsciously) must make his decisions on how much he is willing to pay for each attributes, including its unit number. Regressing lucky/unlucky numbers on transaction price itself also produces implicit price estimates that can be compared to prices of numbers in other areas, such as license plate auction markets. Further, similar regressions with either price or PSF as the dependent variable do not produce substantially different coefficient estimates. 4.2 Main Model Specifications: Equilibrium Price Estimation 4.2.1 Basic Linear Models After deciding on the characterizations of lucky/unlucky numbers, price, and income, the hedonic approach is used to break out the impact of lucky unit and floor numbers from the other factors influencing price in order to answer the paper’s first question: how much does a lucky unit number of a condominium unit contribute to its ultimate price? In the model’s most basic form, lucky/unlucky groups are simply added to a linear equation relating the condominium unit’s attributes and the transaction price. Price = β1Area + β2Floor + β3Height + β4TopFloors + β5BuildingAge + β6TransAge + β7FreeholdYN + β8YearsLeft + [IncomeDummies]β9 + β10LuckyUnitDum + β11UnluckyUnitDum 20 + β12LuckyFloorDum + β13UnluckyFloorDum + ε 9 (1) Table 4.1: Variable Descriptions Variable Description Price Transaction price of unit. Area Area in square feet of the unit. Floor Floor number of the unit. Height Ratio of the unit’s floor to the highest floor of the building—a relative floor measure. Top Floors Dummy variable; equals 1 when unit is within five floors of the building’s highest floor. Building Age Age of the building calculated from the date its Temporary Occupation Permit (TOP) is issued. Trans Age Age of the building at time of transaction. FreeholdYN Dummy variable; equals 1 when a building is owned in perpetuity (freehold) and 0 when a building’s ownership rights are leased from the government for 99 years. Years Left Years remaining in a 99-year leasehold agreement; takes value of 0 for freehold buildings. Income Dummies Set of four income dummy variables that each takes the value of 1 for a different income group of the population. Their criteria changes according to three characterizations of income, depending on the model used. Lucky Unit Dum Dummy variable; equals 1 when the unit’s number ends in 3, 6, 8, 9. Unlucky Unit Dum Dummy variable; equals 1 when the unit’s number ends in 4. Lucky Floor Dum Dummy variable; equals 1 when the floor’s number ends in 3, 6, 8 or 9. Unlucky Floor Dum Dummy variable; equals 1 when the floor’s number ends in 4. 4.2.2 Basic Quadratic Models Multiple papers have pointed out that house prices do not necessarily vary in a linear manner with respect to attributes. For example, a condominium unit’s value may drop substantially from brand-new to one year old, analogous to the drop in a car’s value as it is driven off the dealership lot; the change in value between the first and second year, on the other hand, may not be nearly as large. Accounting for nonlinear relationships is especially important if some of the nonlinear effects are being captured in the coefficients for lucky and 9 A set of dummy variables for districts were included in the models where income not accounted for using districts (i.e. for the income characterizations using price cutoffs and the 2000 Census data). District dummy variables were left out in the version of the model with ranked districts to proxy for income, to avoid multicollinearity. 21 unlucky dummies. Thus squared terms were added for area, floor, height, building age, transaction age, and years left in a lease. Price = β1Area + β2Floor + β3Height + β4TopFloors + β5BuildingAge + β6TransAge +β7AreaSqr + β8FloorSqr + β9HeightSqr + β10BuildingAgeSqr + β11TransAgeSqr + β12FreeholdYN + β13YearsLeft + β14YearsLeftSqr + [IncomeDummies]β15 + β16LuckyUnitDum + β17UnluckyUnitDum + β18LuckyFloorDum + β19UnluckyFloorDum + ε 4.2.3 (2) Fixed Effects Models Neither the linear nor the quadratic model addresses an important endogeneity problem. Since luxury condominiums—whose units cost more—generally have fewer numbers of units per floor, the estimated coefficients for the lower-number unit dummy variables captured this “luxury” phenomenon as well as the luckiness premium that they were intended to reflect. As the data section discussed, it is impossible to control for every single inter-building variation such as this, raising the specter not only of omitted variables, but also of heterogeneity. Fortunately, the data set was large and comprehensive enough to use building fixed effects to account for inter-building variations that were unobservable given the data’s limitations. The first aim of this paper is to estimate implicit prices of lucky and unlucky unit and floor numbers relative to neutral cases holding all other factors constant. It is a question, in other words, of intra-building variation: does a unit numbered eight cost more than a unit numbered four in the same building? By essentially including a full set of dummy variables for each condominium building in the data set, the model eliminated the possible differences between buildings from the coefficient estimates of the remaining variables, including lucky/unlucky unit and floor numbers. 22 It is important to note the assumptions made by the fixed effects model and evaluate them against the case of the Singapore condominium market. The model implicitly assumes that there exists a difference between two fourth-floor units numbered four purely by virtue of one being in Building A and one being in Building B, even if they are identical in all other respects. In addition, the model assumes that this inherent difference between Building A and Building B does not itself change over time. For the Singapore real estate market, this is an acceptable assumption, though not perfect. Buildings may change relative to one another if one of them decides to drastically revamp its lobby or enlarge its swimming pool, or if a developer breaks ground on a new construction site right next door to the other. However, Singapore’s regulations are strict and detailed enough to ensure that, in general, residential buildings’ rates of wear and depreciation do not change much relative to each other. For example, buildings are required by law to paint their exteriors and perform basic maintenance every five years. The fixed effects regressions were run using all three characterizations of income— by ranked districts, by price cutoffs, and with the actual income data, and also on data restricted to buildings with more than five units per floor (to allow for at least one lucky and one unlucky unit number per floor in the sample). The fixed effects model addresses inter-building variation, but intra-building variation remains a factor in the analysis. It must be acknowledged that developers are possibly aware that people prefer lucky-numbered units and so choose to bundle those lucky unit numbers with other features (e.g. penthouses, windows that face the morning sun, or a slightly better layout of the apartment unit) in order to take advantage of any interaction effects and command even higher prices for those units. 23 With the inter-building omitted variable problem addressed, the fixed effects model was used to analyze the full set of unit and floor dummies (from 0 to 9) to see which digits were actually lucky. Price = β1Area + β2Floor + β3Height + β4TopFloors + β5BuildingAge + β6TransAge +β7AreaSqr + β8FloorSqr + β9HeightSqr + β10BuildingAgeSqr + β11TransAgeSqr + β12FreeholdYN + β13YearsLeft + β14YearsLeftSqr + [IncomeDummies]β15 + [BuildingFixedEffectsDummies] β18 + [UnitDummies]β16 + [FloorDummies]β17 + ε (3) Later, to account for special number combinations, like 44, 66, 68, 86 and 88, a set of dummy variables for these numbers were included in the regression. As the results section will discuss, these estimates reveal that the numbers 6 and 8 do indeed command a price premium. For the sake of concision, however, the analysis continued to use buckets for “lucky,” “unlucky” and “neutral” numbers. 4.2.3 Interaction Effects After considering the effects of lucky and unlucky numbers separately, interactions between lucky unit numbers and other factors are also studied, including: the age of the building at transaction time, the fact that a building was freehold, the fact that the unit was one of the highest five floors of its building. 4.3 Secondary Model Specifications: Income Elasticity With the implicit prices and discounts of lucky and unlucky numbers estimated from the fixed effects regressions, the paper turns to its second question: if luckiness is a luxury, normal or inferior good. In effect, this is a question of a lucky number’s income elasticity of demand. As in the previous question, the data would ideally have included the income of 24 every buyer. Since it does not, and since the ranked-districts method proved to be the best proxy for income, 10 separate fixed effects regressions were run on each district to estimate the implicit prices of lucky numbers (6 and 8) and the unlucky number 4. Assuming a linear relationship—that is, the increase in the increase in demand for lucky numbers changes proportionally with income—a trend line was estimated. To allow for the fact that people paying higher prices for their apartments might be willing to pay a higher absolute price (but smaller percentage of the total amount) for a lucky number, the process was repeated with the percentages contributions of the lucky and unlucky numbers on the condominium unit’s transaction price. While this is obviously a less-than-perfect measure of income elasticity, it still allows the analysis to reveal the likely trends in the price impact of lucky and unlucky numbers as income changes. 10 Since the price cutoffs method, as the results section discusses, gives coefficient estimates of control variables that are out of line with the other methods, and the use of the 2000 Census income data entails eliminating 8 districts from the analysis for which income information could not be calculated. 25