MIT OpenCourseWare http://ocw.mit.edu 11.433J / 15.021J Real Estate Economics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Recitation 3 Real Estate Economics: Housing Attributes & Density Sep 23, 2008 Jinhua ZhaoOutline 1 • Simple Richardian model expansion – How to price location Æ how to price capital • Housing Attributes & Density – Housing attributes: structure, neighborhood, regional – Marginal utility and diminishing marginal utility – Expenditure vs. price – Why do we need hedonic model? • Hedonic Regression Analysis: – Regressions: linear, log linear, log log – Logic of applying HRA – Hedonic housing price model: variation within a city (in contrast to price variation among cities) • Land value maximization – First order derivatives – Think on the marginSummary: Price vs. Rent; Land vs. House House 2 Land Rent 1 R(d) = ra *q + c + k(b − d) r(d) = ra + k(b − d) / q Price 3 4 pt (d ) = ra + k(bt − d ) + kbtgPt (d) = raq + c + k(bt − d) + kbtg i iq i(i − g)qi i i i(i − g) Price accounts for future; Rent does not!Expansion of the Ricardian model Assumptions in a stylized city • Monocentric: all opportunities are in the center • Location purely defined by transportation • houses identical: no physical differences except location • q is fixed: 1/q density; no substitution between structure capital and land • households identical: same income, same preference StructureR(d) = ra *q + c + k(b − d) N *q =π*b2*V Boundary condition Expansion of the Ricardian model: relaxation of the assumptions • Identical households Æ different household segments • Identical houses Æ different density • Identical houses Æ different characteristics • Mono-center Æ multi-centerDiminishing marginal utility Marginal Utility (MU) Additional satisfaction obtained from consuming one additional unit of good. We might write MU as ∆U/∆x. Graphically, MU is the slope of the utility function. In mathematical terms: dU/dx Diminishing Marginal Utility Principle that as more of a good is consumed, the consumption of additional amounts will yield smaller addition to utility. The more, the merrier, but less so. Graphs of utility and marginal utilityHedonic prices Hedonic Prices • Implicit prices of attributes of differentiated goods • Derived by observing the joint variation of product prices and bundles of product characteristics • Expenditure vs. price Construction of Hedonic Price Model OLS Regression Model P = f (Z) – P = product price – Z = vector of product attributes: structural and locational Estimated OLS coefficients represent the shadow prices of product attributes (i.e., the value of an additional unit of attribute i, holding all other attributes constant)Elements of a regression equation 6 Linear relationship y =α+β1* x1 +β2* x2 +β3* x3 +ε x1, x2, x3 Independent (explanatory) variables Observed from data y Dependent (response) variables α Constant term Unobserved and to be estimated β , β , β Coefficients of the independent variables 1 2 3 ε Error term Unobserved, Assumptions about itCorrespondence between a linear regress and a linear y =α+β1* x1 +ε α Intercept β Slope What is the best estimate of the location of the line? • How to define the best? Curve fitting • How to identify it?How to identify the best line? To minimize the distance between actual y and the estimated y The line of best fit is the one that minimizes the sum of the squared errors SSE =∑(Yi −Y))2 i SST =∑(Yi − Y )2 i In order to minimize the Sum of Squared Errors SSE =∑(Yi −Y))2 , what is the best α and β iOLS (Ordinary Least Square) method • Derive alpha and beta from the first order condition of minimizing the SSE ) ∑(Xi − X )(Yi −Y ) β= i ∑(Xi − X )(Xi − X ) i α) = Y −β)X SST Measures of goodness of fit Standard Error of the Estimate 21− SSE Coefficient of Determination: R-square: [0, 1]: % of the variance R = in y explained by the variances of the x Standard error of the slope: a measure for the accuracy of betasStatistical property of regression and assumptions of the error term Assumptions • 1. Regarding the shape of the relationship: linear • 2. Regarding the expected value of the error term: 0 • 3. Regarding the variance of the error term: constant • 4. Regarding the relationship between the error terms: independent • 5. That the error term is normally distributed Properties • Unbiased: the mean of the parameter estimates is equal to the true value of the parameter that we are trying to estimate • Efficient / best : the minimum variance of the unbiased parameter estimates Gauss-Markov Theorem • Assumptions 1 and 2 Î unbiased • Assumptions 1, 2 and 3, 4 Î unbiased and efficient: best linear unbiased estimator (BLUE) • Assumptions 1~5 Î useful t-statistic valueRegression: housing price variation among cities (p56) • A simple model to explain the housing price variation among cities • Three key factors: – Size of the city – Growth of the city – Construction cost • Data: – 1990, CMSAs in the US • Variables: – Price: median house price in 1990 (PRICE) – Size of the city: # of households (HH) – Growth of the city: % difference between 1980 and 1990 households (HHGRO) – Construction cost: 1990 Construction Cost Index (COST) • Model: PRICE =α+β1* HH +β2* HHGRO +β3*COST +ε • Expected results: – Size of the city – Growth of the city – Construction costRegression: Housing price variation among cities • Results: PRICE = −298,138 + 0.019* HH +152,156* HHGRO +1622*COST (10.0) (2.4) (2.3) (4.2) R-square=0.76 • Interpretation – Betas • Constant • Size of the city • Growth of the city • Construction cost –t-statistics –R2 • Notes: – Different scale of the variables Æ different scale of the betas – 3 variables but quite a powerful explanations – CMSA as the unit – HHGRO as the growth rate proxyHedonic housing price model: price variation within the city (p69) • Expenditure vs. price / a true measure of price • Factors – Number of bedrooms – Number of bathrooms – Age of structure – Single family attached – Garage – Poor-quality unit – Poor neighborhood – Central city • Hedonic model εββ βββ βββα +++ +++ +++= CENTRALCITYBADAREA POORQUALSFAAGE