Measurement error in FRC/NCREIF returns on real estate.

• Author: Shilling, James D.

1. Introduction

   The problem of measurement error in the Frank Russell Company/National Council of Real Estate Investment Fiduciaries (FRC/NCREIF) real estate returns series is quite clearly an important one. This returns series purports to track the performance of real estate investments made by large institutional investors. But, in practice, measurement errors are likely to occur since the returns are based on appraised values, not on market prices. One recent contribution, that of Geltner [3] and Giliberto [4], suggests that the problem of measurement error in the FRC/NCREIF returns series is apt to make it appear as if real estate offers higher returns and lower portfolio risk than stocks or bonds. Geltner [3] and Giliberto [4] propose testing for the existence of measurement errors in the FRC/NCREIF returns series by measuring the extent to which appraised values depart significantly from market prices.

   A more practical test procedure is to treat the problem as an errors in variables problem and specify an inventory demand model for commercial real estate put in place in which the FRC/NCREIF returns series, as a proxy for the "true" return on real estate, shows up as an explanatory variable. The appeal of this estimation procedure lies in the fact that, although it is in the spirit of Geltner [3] and Giliberto [4], it does not require the analyst to postulate how changes in appraised values are related to changes in market values; rather, the idea is to evaluate the measurement errors in the FRC/NCREIF returns series by assuming that the relationship between the inventory demand for commercial real estate put in place and the return on real estate is a deterministic one, and that the only reason why we observe some unaccounted for variation in the dependent variable is that we do not have exact measurements of the inventory demand for commercial real estate put in place or the return on real estate. For suitably defined parameter estimates, the ratio of the asymptotic error variance in the FRC/NCREIF returns series to the true return variance on real estate can then be obtained by separating out the effects of errors associated with the regression model and measurement errors. It then goes without saying that the larger the variance of the measurement error in the FRC/NCREIF returns series relative to the variance of the true returns on real estate, the less reliable the FRC/NCREIF returns series is in assessing the performance of real estate investments relative to stocks and bonds. Conversely, when the proportion of the variance in the FRC/NCREIF returns series to the variance in the true returns on real estate is small, the theoretical returns on real estate and the FRC/NCREIF returns series may be close enough that the difference is of no real consequence in assessing the diversification advantage of real estate. The problem becomes crucial only when the variance of the measurement error in the FRC/NCREIF returns series is of such size that real estate is seen as offering an attractive diversification opportunity for those invested in stocks and bonds when stocks and bonds are, in fact, clearly superior.

2. An Errors in Variables Approach to Estimate the Measurement Error in the FRC/NCREIF Returns Series

   To begin with, consider a standard flexible accelerator-inventory model:

   [I.sub.t] - [I.sub.t-1] = [theta]([I.sub.t] - [I.sub.t-1]), (1) [Mathematical Expression Omitted] (2) where

   I = the value of commercial real estate put in place at the end of period t, GNP = aggregate gross national product, UNEMP = unemployment rate, r = the actual (before-tax) return on real estate, [delta] = the risk-free (before-tax) rate of return,

   and where O

   Estimation of (1) and (2) tends to be problematic since, instead of [I.sub.t], and [r.sub.t], we normally observe [I.sub.t.sup.*] and [r.sub.t.sup.*]. Thus, a typical reduced-form regression equation - ignoring, for the moment, the variables [GNP.sub.t], [UNEMP.sub.t], [delta].sub.t], and [I.sub.t-i] - might then be

   [I.sub.t.sup.*] = [alpha] + [[beta][r.sub.t.sup.*] + [epsilon].sub.t.sup.*]

   (3) where

   [I.sub.t.sup.*] = [I.sub.t] + [v.sub.t] (4)

   [r.sub.t.sup.*] = [r.sub.t] + [xi].sub.t] (5)

   [epsilon].sub.t.sup.*] = [epsilon].sub.t] + [v.sub.t] - [beta] [xi].sub.t]

   (6) and where [epsilon].sub.t] is a disturbance term with mean zero and [sigma].sub.[epsilon].sup.2] in the "true" regression model [I.sub.t] = [alpha] + [beta][r.sub.t] + [epsilon].sub.t], and [v.sub.t] and [xi].sub.t] represent the errors in measuring the t th value of I and of r.(1) Asterisks in this case denote the observable values of the variables.

   The error terms [v.sub.t] and [xi].sub.t] are distributed

   [v.sub.t] N(O, [sigma].sub.v.sup.2], (7) [xi].sub.t] N(O, [sigma].sub.[sigma].sup.2]; (8) where E([.sub.i][v.sub.j]) = O, E([xi].sub.i][xi].sub.j] = O for i # j, and E([v.sub.i][xi].sub.j]) = O, E([v..sub.i][epsilon].sub.j]) = O, and E([xi].sub.i][epsilon].sub.j]) = O. Assumptions (7) and (8) state that each error is a random variable with zero mean and constant variance. The assumptions E([v.sub.i][v.sub.j]) = O, E([xi].sub.i][xi].sub.j]) = O for i # j rule out situations in which the errors are autoregressive and E([v.sub.i][xi].sub.j]) = O, E([v.sub.i][epsilon].sub.j]) = O, and E([xi].sub.i][epsilon].sub.j]) = O state that the errors are unrelated to each other. Of course, the real problem is not that [I.sub.t] is measured with error, but rather it is because [r.sub.t] is measured with error.

   When ordinary least squares (OLS) is applied to equation (3), the OLS estimators of...