Private Real Estate Investment: Data Analysis and Decision Making_7

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Private Real Estate Investment: Data Analysis and Decision Making_7 133 Uncertainty: Risk in Real Estate Price 300000 280000 260000 240000 220000 200000 180000 Square Feet 1600 1700 1800 1900 2000 FIGURE 6-5 Plot of house data. $132 per square foot. The result is that the empirical test of our theory about houses being worth $132 per square foot shows that it is less than perfect. (Most applications are.) Univariate regression is the process of finding the ‘‘conditional mean’’ in that it helps you predict the mean of one quantity conditioned on knowing some other, independent quantity. In this case the fact that we know is the size of the house. But is that fact fully determinative? Table 6-7 shows the residuals, defined as the difference between the observed price and the predicted price. This difference, in a statistical sense, is a measure of the error between what our claim is (that houses sell for $132 per square foot) and what really happens. This sort of thing, while statistically valid, is only partially helpful in guiding an individual property owner to the value of his property. Compare the output in Table 6-7 with the regression output of the circle example in Table 6-5. The R-square for the house price regression is only 66%, residuals are non-zero, standard errors are positive, and confidence intervals are positive. All of this states the obvious: relying on price per square foot as an indicator of value is less than perfect. From this we conclude that in a more complex world determinism rarely exists, and the rule is uncertainty. Diameter completely determines the circumference of all circles. Size only partially determines the price at which a house sells. The positive standard errors of the residuals are measures of how much our theory is wrong for particular houses. The 34% ‘‘unexplained’’ part encompasses the non-size characteristics that determine value. The effects of these characteristics are embedded in the error terms. This suggests that our $132 theory of house prices based on the single variable (size) and a linear relationship is simplistic, something any home buyer or seller knows.10 10 Mathematicians will object with the casual use of ‘‘linear,’’ a term that has a specific and precise mathematical meaning. There is a thread of linear, however tenuous it may be, that links binary outcomes, the normal distribution and linear regression. We do take liberties here and sometimes indulge in a metaphorical use of linear as ‘‘unduly simplistic.’’ 134 Private Real Estate Investment TABLE 6-7 Regression of House Prices on Size (Square Foot) SUMMARY OUTPUT Regression statistics Multiple R 0.813090489 R square 0.661116143 Adjusted R square 0.618755661 Standard error 21450.49633 Observations 10 ANOVA df SS MS F Significance F Regression 1 7181109658 7181109658 15.6069079 0.004232704 Residual 8 3680990342 460123792.8 Total 9 10862100000 Coefficients Standard error t Stat P-value Lower 95% Upper 95% Intercept À79095.58434 78357.96111 À1.009413507 0.342328491 À259789.4835 101598.3149 SF 178.1603607 45.09751892 3.950557923 0.004232704 74.16522831 282.155493 RESIDUAL OUTPUT Observed price Predicted price Residuals 195000 188144.9567 6855.043317 210000 232685.0469 À22685.04685 225000 205960.9928 19039.00725 240000 223777.0288 16222.97118 275000 259409.101 15590.89905 285000 254955.0919 30044.90806 190000 214869.0108 À24869.01079 239000 243374.6685 À4374.668494 249000 272771.128 À23771.12801 185000 197052.9747 À12052.97472 Despite less than perfect results, we press on. Data analysis does not always lead us directly where we want to go. The process involves many partial glimpses of the truth and the rare epiphany. Letting the numbers talk to us enlarges our understanding about how a process works. The constant reconciliation of objective outcomes based on the numbers with our subjective reasoning based on field experience is a big part of the value of the data analysis exercise. 135 Uncertainty: Risk in Real Estate DETERMINISM AND REAL ESTATE INVESTMENT The above examples include the case (a) where a definite, constant linear relationship (the diameter and circumference of a circle) surely exists between two variables or (b) where a suspected linear relationship (house size and price) may exist between two variables. We now complicate this by examining relationships between variables in an investment context. Recall the general caution of Chapter 3. The capitalization rate rule of thumb claims that value is a function of income and that the functional relationship is I 1 V¼ ¼I ...