3 Generalized regression estimation

Jan de Haan and Rens Hendriks

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3.1  A simple GREG method

In this section we will outline an alternative approach to measuring house price change that makes use of appraisal data. The appraisals now serve as auxiliary information in a generalized regression (GREG) framework. Consider the following simple two-variable linear regression model:

p n 0 = α 0 + β 0 a n 0 + ε n 0 ,       ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaakiabg2da9iabeg7aHnaaCaaa leqabaGaaGimaaaakiabgUcaRiabek7aInaaCaaaleqabaGaaGimaa aakiaadggadaqhaaWcbaGaamOBaaqaaiaaicdaaaGccqGHRaWkcqaH 1oqzdaqhaaWcbaGaamOBaaqaaiaaicdaaaGccaGGSaGaaCzcamaabm aabaaeaaaaaaaaa8qacaaIZaGaaiOlaiaaigdaa8aacaGLOaGaayzk aaaaaa@4FE5@

where ε n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyTdu 2aa0baaSqaaiaad6gaaeaacaaIWaaaaaaa@3D18@  is the error term. Unlike hedonic regression models, which postulate a causal relation between the selling price p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C66@  and a set of characteristics relating to the structure and the location of the housing units, this model does not say anything about how house prices are generated; equation (3.1) is merely a descriptive model.

Estimating model (3.1) by least squares regression on the data of sample S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaaGimaaaaaaa@3B56@  yields predicted prices

p ^ n 0 = α ^ 0 + β ^ 0 a n 0 .       ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aajaWaa0baaSqaaiaad6gaaeaacaaIWaaaaOGaeyypa0JafqySdeMb aKaadaahaaWcbeqaaiaaicdaaaGccqGHRaWkcuaHYoGygaqcamaaCa aaleqabaGaaGimaaaakiaadggadaqhaaWcbaGaamOBaaqaaiaaicda aaGccaGGUaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaIZaGaaiOlai aaikdaa8aacaGLOaGaayzkaaaaaa@4BAB@

The regression residuals for n S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBai abgIGiolaadofadaahaaWcbeqaaiaaicdaaaaaaa@3DCD@  are e n 0 = p n 0 p ^ n 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyzam aaDaaaleaacaWGUbaabaGaaGimaaaakiabg2da9iaadchadaqhaaWc baGaamOBaaqaaiaaicdaaaGccqGHsislceWGWbGbaKaadaqhaaWcba GaamOBaaqaaiaaicdaaaGccaGGUaaaaa@44CC@  Assuming random sampling, as before, we can write the Horvitz-Thompson estimator n S 0 p n 0 / n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGtbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai 4laiaad6gadaahaaWcbeqaaiaaicdaaaaaaa@4517@  of the mean value n U 0 p n 0 / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGvbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai 4laiaad6eadaahaaWcbeqaaiaaicdaaaaaaa@44F9@  as

n S 0 p n 0 / n 0 = n S 0 p ^ n 0 / n 0 + n S 0 e n 0 / n 0 = α ^ 0 + β ^ 0 n S 0 a n 0 / n 0 + n S 0 e n 0 / n 0 .       ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabuae aacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWaaaaOGaai4laiaad6ga daahaaWcbeqaaiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaa adbeqaaiaaicdaaaaaleqaniabggHiLdGccqGH9aqpdaaeqbqaaiqa dchagaqcamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaacaWGUbGaey icI4Saam4uamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaa c+cacaWGUbWaaWbaaSqabeaacaaIWaaaaOGaey4kaSYaaabuaeaaca WGLbWaa0baaSqaaiaad6gaaeaacaaIWaaaaOGaai4laiaad6gadaah aaWcbeqaaiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaaadbe qaaiaaicdaaaaaleqaniabggHiLdGccqGH9aqpcuaHXoqygaqcamaa CaaaleqabaGaaGimaaaakiabgUcaRiqbek7aIzaajaWaaWbaaSqabe aacaaIWaaaaOWaaabuaeaacaWGHbWaa0baaSqaaiaad6gaaeaacaaI WaaaaOGaai4laiaad6gadaahaaWcbeqaaiaaicdaaaaabaGaamOBai abgIGiolaadofadaahaaadbeqaaiaaicdaaaaaleqaniabggHiLdGc cqGHRaWkdaaeqbqaaiaadwgadaqhaaWcbaGaamOBaaqaaiaaicdaaa GccaGGVaGaamOBamaaCaaaleqabaGaaGimaaaaaeaacaWGUbGaeyic I4Saam4uamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaac6 cacaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaIZaGaaiOlaiaa iodaa8aacaGLOaGaayzkaaaaaa@8442@

Replacing the sample average of appraisals, n S 0 a n 0 / n 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGHbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGtbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai 4laiaad6gadaahaaWcbeqaaiaaicdaaaGccaGGSaaaaa@45C2@  by its population counterpart n U 0 a n 0 / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGHbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGvbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai 4laiaad6eadaahaaWcbeqaaiaaicdaaaaaaa@44EA@  yields the generalized regression (GREG) estimator:

p ¯ ^ GREG 0 = α ^ 0 + β ^ 0 n U 0 a n 0 / N 0 + n S 0 e n 0 / n 0 = n U 0 p ^ n 0 / N 0 + n S 0 e n 0 / n 0 .       ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aaryaajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeaacaaI WaaaaOGaeyypa0JafqySdeMbaKaadaahaaWcbeqaaiaaicdaaaGccq GHRaWkcuaHYoGygaqcamaaCaaaleqabaGaaGimaaaakmaaqafabaGa amyyamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaacaWGUbGaeyicI4 SaamyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaac+ca caWGobWaaWbaaSqabeaacaaIWaaaaOGaey4kaSYaaabuaeaacaWGLb Waa0baaSqaaiaad6gaaeaacaaIWaaaaOGaai4laiaad6gadaahaaWc beqaaiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaai aaicdaaaaaleqaniabggHiLdGccqGH9aqpdaaeqbqaaiqadchagaqc amaaDaaaleaacaWGUbaabaGaaGimaaaakiaac+cacaWGobWaaWbaaS qabeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWGvbWaaWbaaWqabeaa caaIWaaaaaWcbeqdcqGHris5aOGaey4kaSYaaabuaeaacaWGLbWaa0 baaSqaaiaad6gaaeaacaaIWaaaaOGaai4laiaad6gadaahaaWcbeqa aiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaaiaaic daaaaaleqaniabggHiLdGccaGGUaGaaCzcaiaaxMaadaqadaqaaaba aaaaaaaapeGaaG4maiaac6cacaaI0aaapaGaayjkaiaawMcaaaaa@7D7B@

Model-assisted sampling theory shows that GREG estimators are asymptotically design unbiased (Särndal, et al. 1992), irrespective of the choice of regressors. Unless the sample would be small, the bias can be neglected. It is obvious that the GREG estimator (3.4) will be more efficient MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  in the sense that it has a lower variance MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  than the Horvitz-Thompson estimator (3.3). As a result, the GREG estimator will usually outperform the Horvitz-Thompson estimator in terms of the mean square error (the sum of the variance and the squared bias).

The same procedure can be applied to the comparison period t. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai aac6caaaa@3B43@  After estimating the model

p n t = α t + β t a n 0 + ε n t       ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaamiDaaaakiabg2da9iabeg7aHnaaCaaa leqabaGaamiDaaaakiabgUcaRiabek7aInaaCaaaleqabaGaamiDaa aakiaadggadaqhaaWcbaGaamOBaaqaaiaaicdaaaGccqGHRaWkcqaH 1oqzdaqhaaWcbaGaamOBaaqaaiaadshaaaGccaWLjaGaaCzcamaabm aabaaeaaaaaaaaa8qacaaIZaGaaiOlaiaaiwdaa8aacaGLOaGaayzk aaaaaa@50D7@

through least squares regression on the data of the current period sample S t , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaamiDaaaakiaacYcaaaa@3C4F@  we obtain predicted prices

p ^ n t = α ^ t + β ^ t a n 0 ,       ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aajaWaa0baaSqaaiaad6gaaeaacaWG0baaaOGaeyypa0JafqySdeMb aKaadaahaaWcbeqaaiaadshaaaGccqGHRaWkcuaHYoGygaqcamaaCa aaleqabaGaamiDaaaakiaadggadaqhaaWcbaGaamOBaaqaaiaaicda aaGccaGGSaGaaCzcaiaaxMaadaqadaqaaabaaaaaaaaapeGaaG4mai aac6cacaaI2aaapaGaayjkaiaawMcaaaaa@4D0C@

which lead to the GREG estimator of the mean value of the housing stock in period t: MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai aacQdaaaa@3B4F@

p ¯ ^ GREG t = α ^ t + β ^ t n U t a n 0 / N t + n S t e n t / n t = n U t p ^ n t / N t + n S t e n t / n t ,       ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aaryaajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeaacaWG 0baaaOGaeyypa0JafqySdeMbaKaadaahaaWcbeqaaiaadshaaaGccq GHRaWkcuaHYoGygaqcamaaCaaaleqabaGaamiDaaaakmaaqafabaGa amyyamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaacaWGUbGaeyicI4 SaamyvamaaCaaameqabaGaamiDaaaaaSqab0GaeyyeIuoakiaac+ca caWGobWaaWbaaSqabeaacaWG0baaaOGaey4kaSYaaabuaeaacaWGLb Waa0baaSqaaiaad6gaaeaacaWG0baaaOGaai4laiaad6gadaahaaWc beqaaiaadshaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaai aadshaaaaaleqaniabggHiLdGccqGH9aqpdaaeqbqaaiqadchagaqc amaaDaaaleaacaWGUbaabaGaamiDaaaaaeaacaWGUbGaeyicI4Saam yvamaaCaaameqabaGaamiDaaaaaSqab0GaeyyeIuoakiaac+cacaWG obWaaWbaaSqabeaacaWG0baaaOGaey4kaSYaaabuaeaacaWGLbWaa0 baaSqaaiaad6gaaeaacaWG0baaaOGaai4laiaad6gadaahaaWcbeqa aiaadshaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaaiaads haaaaaleqaniabggHiLdGccaGGSaGaaCzcaiaaxMaadaqadaqaaaba aaaaaaaapeGaaG4maiaac6cacaaI3aaapaGaayjkaiaawMcaaaaa@80EE@

where e n t = p n t p ^ n t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyzam aaDaaaleaacaWGUbaabaGaamiDaaaakiabg2da9iaadchadaqhaaWc baGaamOBaaqaaiaadshaaaGccqGHsislceWGWbGbaKaadaqhaaWcba GaamOBaaqaaiaadshaaaaaaa@44CD@  denote the period t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaa aa@3A91@  regression residuals. For a fixed housing stock we have U t = U 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaCaaaleqabaGaamiDaaaakiabg2da9iaadwfadaahaaWcbeqaaiaa icdaaaGccaGGSaaaaa@3F22@  hence n U t a n 0 / N t = n U 0 a n 0 / N 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGHbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGvbWaaWbaaWqabeaacaWG0baaaaWcbeqdcqGHris5aOGaai 4laiaad6eadaahaaWcbeqaaiaadshaaaGccqGH9aqpdaaeqaqaaiaa dggadaqhaaWcbaGaamOBaaqaaiaaicdaaaaabaGaamOBaiabgIGiol aadwfadaahaaadbeqaaiaaicdaaaaaleqaniabggHiLdGccaGGVaGa amOtamaaCaaaleqabaGaaGimaaaakiaacYcaaaa@5285@  and it follows that

p ¯ ^ GREG t = α ^ t + β ^ t n U 0 a n 0 / N 0 + n S t e n t / n t = n U 0 p ^ n t / N 0 + n S t e n t / n t .       ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aaryaajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeaacaWG 0baaaOGaeyypa0JafqySdeMbaKaadaahaaWcbeqaaiaadshaaaGccq GHRaWkcuaHYoGygaqcamaaCaaaleqabaGaamiDaaaakmaaqafabaGa amyyamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaacaWGUbGaeyicI4 SaamyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaac+ca caWGobWaaWbaaSqabeaacaaIWaaaaOGaey4kaSYaaabuaeaacaWGLb Waa0baaSqaaiaad6gaaeaacaWG0baaaOGaai4laiaad6gadaahaaWc beqaaiaadshaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaai aadshaaaaaleqaniabggHiLdGccqGH9aqpdaaeqbqaaiqadchagaqc amaaDaaaleaacaWGUbaabaGaamiDaaaaaeaacaWGUbGaeyicI4Saam yvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaac+cacaWG obWaaWbaaSqabeaacaaIWaaaaOGaey4kaSYaaabuaeaacaWGLbWaa0 baaSqaaiaad6gaaeaacaWG0baaaOGaai4laiaad6gadaahaaWcbeqa aiaadshaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaaiaads haaaaaleqaniabggHiLdGccaGGUaGaaCzcamaabmaabaaeaaaaaaaa a8qacaaIZaGaaiOlaiaaiIdaa8aacaGLOaGaayzkaaaaaa@7F53@

The GREG estimator of house price change results simply from taking the ratio of equations (3.8) and (3.4):

P ^ GREG 0t = p ¯ ^ GREG t p ¯ ^ GREG 0 = α ^ t + β ^ t a ¯ 0 + n S t e n t / n t α ^ 0 + β ^ 0 a ¯ 0 + n S 0 e n 0 / n 0 = n U 0 p ^ n t / N 0 + n S t e n t / n t n U 0 p ^ n 0 / N 0 + n S 0 e n 0 / n 0 ,       ( 3.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeaacaaIWaGa amiDaaaakiabg2da9maalaaabaGabmiCayaaryaajaWaa0baaSqaai aabEeacaqGsbGaaeyraiaabEeaaeaacaWG0baaaaGcbaGabmiCayaa ryaajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeaacaaIWa aaaaaakiabg2da9maalaaabaGafqySdeMbaKaadaahaaWcbeqaaiaa dshaaaGccqGHRaWkcuaHYoGygaqcamaaCaaaleqabaGaamiDaaaaki qadggagaqeamaaCaaaleqabaGaaGimaaaakiabgUcaRmaaqafabaGa amyzamaaDaaaleaacaWGUbaabaGaamiDaaaakiaac+cacaWGUbWaaW baaSqabeaacaWG0baaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqa beaacaWG0baaaaWcbeqdcqGHris5aaGcbaGafqySdeMbaKaadaahaa WcbeqaaiaaicdaaaGccqGHRaWkcuaHYoGygaqcamaaCaaaleqabaGa aGimaaaakiqadggagaqeamaaCaaaleqabaGaaGimaaaakiabgUcaRm aaqafabaGaamyzamaaDaaaleaacaWGUbaabaGaaGimaaaakiaac+ca caWGUbWaaWbaaSqabeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWGtb WaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aaaakiabg2da9maa laaabaWaaabuaeaaceWGWbGbaKaadaqhaaWcbaGaamOBaaqaaiaads haaaGccaGGVaGaamOtamaaCaaaleqabaGaaGimaaaaaeaacaWGUbGa eyicI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoaki abgUcaRmaaqafabaGaamyzamaaDaaaleaacaWGUbaabaGaamiDaaaa kiaac+cacaWGUbWaaWbaaSqabeaacaWG0baaaaqaaiaad6gacqGHii IZcaWGtbWaaWbaaWqabeaacaWG0baaaaWcbeqdcqGHris5aaGcbaWa aabuaeaaceWGWbGbaKaadaqhaaWcbaGaamOBaaqaaiaaicdaaaGcca GGVaGaamOtamaaCaaaleqabaGaaGimaaaaaeaacaWGUbGaeyicI4Sa amyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoakiabgUcaRm aaqafabaGaamyzamaaDaaaleaacaWGUbaabaGaaGimaaaakiaac+ca caWGUbWaaWbaaSqabeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWGtb WaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aaaakiaacYcacaWL jaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaIZaGaaiOlaiaaiMdaa8 aacaGLOaGaayzkaaaaaa@AFCB@

where a ¯ 0 = n U 0 a n 0 / N 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyyay aaraWaaWbaaSqabeaacaaIWaaaaOGaeyypa0ZaaabeaeaacaWGHbWa a0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWGvb WaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai4laiaad6ea daahaaWcbeqaaiaaicdaaaGccaaMb8UaaiOlaaaa@4A25@  Some additional small sample bias will be introduced due to the non-linear (ratio) structure. When using Ordinary Least Squares (OLS) regression to estimate the models (3.1) and (3.5), the unweighted sample means of regression residuals in (3.9), n S 0 e n 0 / n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGLbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacqGH iiIZcaWGtbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aOGaai 4laiaad6gadaahaaWcbeqaaiaaicdaaaaaaa@450C@  and n S t e n t / n t , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabeae aacaWGLbWaa0baaSqaaiaad6gaaeaacaWG0baaaaqaaiaad6gacqGH iiIZcaWGtbWaaWbaaWqabeaacaWG0baaaaWcbeqdcqGHris5aOGaai 4laiaad6gadaahaaWcbeqaaiaadshaaaGccaGGSaaaaa@4683@  will be equal to 0 and the GREG index reduces to

P ^ GREG,OLS 0t = n U 0 p ^ n t / N 0 n U 0 p ^ n 0 / N 0 = α ^ t + β ^ t a ¯ 0 α ^ 0 + β ^ 0 a ¯ 0 = α ^ t / a ¯ 0 + β ^ t α ^ 0 / a ¯ 0 + β ^ 0 .       ( 3.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeacaqGSaGaae4t aiaabYeacaqGtbaabaGaaGimaiaadshaaaGccqGH9aqpdaWcaaqaam aaqafabaGabmiCayaajaWaa0baaSqaaiaad6gaaeaacaWG0baaaaqa aiaad6gacqGHiiIZcaWGvbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcq GHris5aOGaai4laiaad6eadaahaaWcbeqaaiaaicdaaaaakeaadaae qbqaaiqadchagaqcamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaaca WGUbGaeyicI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0Gaeyye Iuoakiaac+cacaWGobWaaWbaaSqabeaacaaIWaaaaaaakiabg2da9m aalaaabaGafqySdeMbaKaadaahaaWcbeqaaiaadshaaaGccqGHRaWk cuaHYoGygaqcamaaCaaaleqabaGaamiDaaaakiqadggagaqeamaaCa aaleqabaGaaGimaaaaaOqaaiqbeg7aHzaajaWaaWbaaSqabeaacaaI WaaaaOGaey4kaSIafqOSdiMbaKaadaahaaWcbeqaaiaaicdaaaGcce WGHbGbaebadaahaaWcbeqaaiaaicdaaaaaaOGaeyypa0ZaaSaaaeaa cuaHXoqygaqcamaaCaaaleqabaGaamiDaaaakiaac+caceWGHbGbae badaahaaWcbeqaaiaaicdaaaGccqGHRaWkcuaHYoGygaqcamaaCaaa leqabaGaamiDaaaaaOqaaiqbeg7aHzaajaWaaWbaaSqabeaacaaIWa aaaOGaai4laiqadggagaqeamaaCaaaleqabaGaaGimaaaakiabgUca Riqbek7aIzaajaWaaWbaaSqabeaacaaIWaaaaaaakiaac6cacaWLja GaaCzcamaabmaabaaeaaaaaaaaa8qacaaIZaGaaiOlaiaaigdacaaI WaaapaGaayjkaiaawMcaaaaa@86F6@

As the first expression on the right-hand side of (3.10) indicates, the (OLS) GREG approach essentially imputes prices pertaining to the base period and the current period using equations (3.2) and (3.6). The difference with the hedonic double imputation method is twofold: a descriptive model, not a hedonic one, is used to estimate predicted prices MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  so that we cannot speak of unbiased predicted prices MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  and prices are imputed for all houses of the housing stock instead of the sub-set of sampled houses.

3.2  Properties of the GREG index

The (OLS) GREG index has several properties worth mentioning. First, the computation of the GREG index is very simple. Once the population mean of appraisals a ¯ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyyay aaraWaaWbaaSqabeaacaaIWaaaaaaa@3B7C@  and the base period regression coefficients α ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySde MbaKaadaahaaWcbeqaaiaaicdaaaaaaa@3C2D@  and β ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqOSdi MbaKaadaahaaWcbeqaaiaaicdaaaaaaa@3C2F@  have been calculated, all that is needed is running a regression each month of selling prices against appraisals and plugging the coefficients α ^ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySde MbaKaadaahaaWcbeqaaiaadshaaaaaaa@3C6C@  and β ^ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqOSdi MbaKaadaahaaWcbeqaaiaadshaaaaaaa@3C6E@  into (3.10). Note that the GREG index can be written as a pseudo chain index:

P ^ GREG,OLS 0t = α ^ t / a ¯ 0 + β ^ t α ^ 0 / a ¯ 0 + β ^ 0 = τ=1 t α ^ τ / a ¯ 0 + β ^ τ α ^ τ1 / a ¯ 0 + β ^ τ1 .       ( 3.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeacaqGSaGaae4t aiaabYeacaqGtbaabaGaaGimaiaadshaaaGccqGH9aqpdaWcaaqaai qbeg7aHzaajaWaaWbaaSqabeaacaWG0baaaOGaai4laiqadggagaqe amaaCaaaleqabaGaaGimaaaakiabgUcaRiqbek7aIzaajaWaaWbaaS qabeaacaWG0baaaaGcbaGafqySdeMbaKaadaahaaWcbeqaaiaaicda aaGccaGGVaGabmyyayaaraWaaWbaaSqabeaacaaIWaaaaOGaey4kaS IafqOSdiMbaKaadaahaaWcbeqaaiaaicdaaaaaaOGaeyypa0ZaaebC aeaadaWcaaqaaiqbeg7aHzaajaWaaWbaaSqabeaacqaHepaDaaGcca GGVaGabmyyayaaraWaaWbaaSqabeaacaaIWaaaaOGaey4kaSIafqOS diMbaKaadaahaaWcbeqaaiabes8a0baaaOqaaiqbeg7aHzaajaWaaW baaSqabeaacqaHepaDcqGHsislcaaIXaaaaOGaai4laiqadggagaqe amaaCaaaleqabaGaaGimaaaakiabgUcaRiqbek7aIzaajaWaaWbaaS qabeaacqaHepaDcqGHsislcaaIXaaaaaaaaeaacqaHepaDcqGH9aqp caaIXaaabaGaamiDaaqdcqGHpis1aOGaaiOlaiaaxMaacaWLjaWaae WaaeaaqaaaaaaaaaWdbiaaiodacaGGUaGaaGymaiaaigdaa8aacaGL OaGaayzkaaaaaa@7D33@

This can be helpful in practice, particularly when new appraisal data becomes available. New appraisal data often becomes available to the statistical agency with a considerable time lag, up to more than a year. There are two reasons for using the latest appraisal information. The quality of the appraisals may improve over time, which seems to have been the case in the Netherlands (de Vries et al. 2009). Also, the assumption of a fixed housing stock can be relaxed so that newly-built properties can be incorporated through chaining; the resulting chained GREG index takes the dynamics of the housing stock into account. The same advantages of chaining apply to the SPAR method. Suppose new appraisals, relating to period T(0<Tt), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamivai aacIcacaaIWaGaeyipaWJaamivaiabgsMiJkaadshacaGGPaGaaiil aaaa@41BE@  are available in period t+1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai abgUcaRiaaigdacaGGUaaaaa@3CDF@  The time series can then be updated through chain-linking, i.e., by multiplying P ^ GREG,OLS 0t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeacaqGSaGaae4t aiaabYeacaqGtbaabaGaaGimaiaadshaaaaaaa@42B3@  by the month-to-month change ( α ˜ t+1 / a ¯ T + β ˜ t+1 )/( α ˜ t / a ¯ T + β ˜ t ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikai qbeg7aHzaaiaWaaWbaaSqabeaacaWG0bGaey4kaSIaaGymaaaakiaa c+caceWGHbGbaebadaahaaWcbeqaaiaadsfaaaGccqGHRaWkcuaHYo GygaacamaaCaaaleqabaGaamiDaiabgUcaRiaaigdaaaGccaGGPaGa ai4laiaacIcacuaHXoqygaacamaaCaaaleqabaGaamiDaaaakiaac+ caceWGHbGbaebadaahaaWcbeqaaiaadsfaaaGccqGHRaWkcuaHYoGy gaacamaaCaaaleqabaGaamiDaaaakiaacMcacaGGSaaaaa@53A8@  where the coefficients now pertain to a regression of selling prices on the period T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamivaa aa@3A71@  appraisals.

Second, standard errors of the GREG index can be estimated rather easily using the variance-covariance matrix of the regression coefficients, which is standard output of most statistical packages. An expression for the approximate standard error is derived in the Appendix. The standard error of the GREG index depends on the goodness of fit ( R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikai aadkfadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@3CBA@  of the regression model. It is most likely that R 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOuam aaCaaaleqabaGaaGOmaaaaaaa@3B57@  for the base period regression is higher than that for the current period regressions. This is because we expect to find a strong linear relation between appraisals and sale prices in the appraisal reference period while in later periods this relation will probably be weaker due to differing price trends across different types of houses or regions. The derivation of approximate standard errors for the SPAR index is a bit more complex because there is an additional source of sampling error, namely the sampling variability of the mean appraisals; see de Haan (2007).

The latter point brings us to the third property of the GREG index, namely its dependence on the quality of the appraisal data. For two reasons at least the appraisals may not exactly represent the transaction prices during the base period so that the model fit is not perfect ( R 2 <1). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikai aadkfadaahaaWcbeqaaiaaikdaaaGccqGH8aapcaaIXaGaaiykaiaa c6caaaa@3F2B@  The assessment authorities may not have (real time) access to the actual sale prices and therefore have to make their own judgements based on other information. But even if they knew the selling prices, the authorities may still decide to make adjustments when determining the property values. It can be argued that selling prices do not always properly measure the unknown market values MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  which can be seen as a latent variable MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  and tend to be more volatile. In this respect, Francke (2010) and others have used the term transaction noise.

The way in which the appraisals have been determined will affect the standard error of the GREG index. As long as the quality of the appraisal data is the same for all houses in stock, no bias arises since the appraisals only serve as an auxiliary variable in regressions run on the samples S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaaGimaaaaaaa@3B56@  and S t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaamiDaaaaaaa@3B95@  of properties sold in periods 0 and t(t=1,,T). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai aacIcacaWG0bGaeyypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamiv aiaacMcacaGGUaaaaa@42B0@  However, in general we expect the quality of the appraisals to be higher for properties belonging to the appraisal reference (base) period sample S 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaaGimaaaakiaacYcaaaa@3C10@  although this will most likely differ across valuation methods. In the Netherlands the properties are assessed for tax purposes, both for income tax and local taxes. The municipalities are responsible for the valuations. Several municipalities value the houses which are sold during the reference period (January) by the selling price. Houses which were not sold are sometimes valued by comparing them to similar traded houses. Some municipalities apparently use a form of hedonic regression to value the houses, but the methodology is unfortunately not made publicly available. For more information on the Dutch appraisal system, see de Vries et al. (2009).

So far we have assumed that the quality of the individual houses stays the same over time. This is a strong assumption. Thus, the fourth property MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  and most important drawback MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  of the GREG method is that the resulting price index suffers from quality change bias since explicit quality adjustments are not carried out. The same drawback holds true for the SPAR method and for the standard repeat sales method. In principle, hedonic regression methods can deal with the quality change problem, although it may prove difficult to control for all relevant price determining characteristics, in particular micro location. The SPAR method automatically controls for micro location, provided of course that the appraisals sufficiently account for this, as it is based on the matched-model methodology where the matching is done at the address level.

3.3  Alternative GREG estimators

Statistics Netherlands not only computes house price indexes for the whole country but also for segments of the housing market, according to type of house (family dwellings and apartments) and region (provinces and large cities), mainly because of user needs. Another motivation behind stratifying the sample can be to mitigate the effect of sample selection bias. This type of bias may arise if the set of houses sold in a particular period is not a random selection from the housing stock. The nationwide index should then be indirectly computed as a weighted average of the stratum indexes instead of directly from all observations.

Suppose the total housing stock U 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaCaaaleqabaGaaGimaaaaaaa@3B58@  is sub-divided into K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4saa aa@3A68@  non-overlapping strata U k 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaDaaaleaacaWGRbaabaGaaGimaaaaaaa@3C48@  of size N k 0 ( k=1 K N k 0 = N 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtam aaDaaaleaacaWGRbaabaGaaGimaaaakmaabmaabaWaaabmaeaacaWG obWaa0baaSqaaiaadUgaaeaacaaIWaaaaOGaeyypa0JaamOtamaaCa aaleqabaGaaGimaaaaaeaacaWGRbGaeyypa0JaaGymaaqaaiaadUea a0GaeyyeIuoaaOGaayjkaiaawMcaaiaac6caaaa@497C@  The target price index (2.3) can now be rewritten as

P 0t = n U 0 p n t n U 0 p n 0 = k=1 K n U k 0 p n t k=1 K n U k 0 p n 0 = k=1 K s k 0 P k 0t ,       ( 3.12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuam aaCaaaleqabaGaaGimaiaadshaaaGccqGH9aqpdaWcaaqaamaaqafa baGaamiCamaaDaaaleaacaWGUbaabaGaamiDaaaaaeaacaWGUbGaey icI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoaaOqa amaaqafabaGaamiCamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaaca WGUbGaeyicI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0Gaeyye IuoaaaGccqGH9aqpdaWcaaqaamaaqahabaWaaabuaeaacaWGWbWaa0 baaSqaaiaad6gaaeaacaWG0baaaaqaaiaad6gacqGHiiIZcaWGvbWa a0baaWqaaiaadUgaaeaacaaIWaaaaaWcbeqdcqGHris5aaWcbaGaam 4Aaiabg2da9iaaigdaaeaacaWGlbaaniabggHiLdaakeaadaaeWbqa amaaqafabaGaamiCamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaaca WGUbGaeyicI4SaamyvamaaDaaameaacaWGRbaabaGaaGimaaaaaSqa b0GaeyyeIuoaaSqaaiaadUgacqGH9aqpcaaIXaaabaGaam4saaqdcq GHris5aaaakiabg2da9maaqahabaGaam4CamaaDaaaleaacaWGRbaa baGaaGimaaaakiaadcfadaqhaaWcbaGaam4AaaqaaiaaicdacaWG0b aaaaqaaiaadUgacqGH9aqpcaaIXaaabaGaam4saaqdcqGHris5aOGa aiilaiaaxMaacaWLjaWaaeWaaeaaqaaaaaaaaaWdbiaaiodacaGGUa GaaGymaiaaikdaa8aacaGLOaGaayzkaaaaaa@8512@

where P k 0t = n U k 0 p n t / n U k 0 p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuam aaDaaaleaacaWGRbaabaGaaGimaiaadshaaaGccqGH9aqpdaWcgaqa amaaqababaGaamiCamaaDaaaleaacaWGUbaabaGaamiDaaaaaeaaca WGUbGaeyicI4SaamyvamaaDaaameaacaWGRbaabaGaaGimaaaaaSqa b0GaeyyeIuoaaOqaamaaqababaGaamiCamaaDaaaleaacaWGUbaaba GaaGimaaaaaeaacaWGUbGaeyicI4SaamyvamaaDaaameaacaWGRbaa baGaaGimaaaaaSqab0GaeyyeIuoaaaaaaa@5261@  is the target price index for stratum U k 0 (k=1,,K). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaDaaaleaacaWGRbaabaGaaGimaaaakiaacIcacaWGRbGaeyypa0Ja aGymaiaacYcacqWIMaYscaGGSaGaam4saiaacMcacaGGUaaaaa@4460@  The base period stock value shares s k 0 = n U k 0 p n 0 / n U 0 p n 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Cam aaDaaaleaacaWGRbaabaGaaGimaaaakiabg2da9maalyaabaWaaabe aeaacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWaaaaaqaaiaad6gacq GHiiIZcaWGvbWaa0baaWqaaiaadUgaaeaacaaIWaaaaaWcbeqdcqGH ris5aaGcbaWaaabeaeaacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWa aaaaqaaiaad6gacqGHiiIZcaWGvbWaaWbaaWqabeaacaaIWaaaaaWc beqdcqGHris5aaaakiaacYcaaaa@5116@  which serve as weights for the stratum indexes, are unknown and have to be estimated. Assuming the variables that define the strata are known for all n U 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBai abgIGiolaadwfadaahaaWcbeqaaiaaicdaaaGccaGGSaaaaa@3E89@  a natural choice for the weights would be the appraisal shares s ^ k 0 = n U k 0 a n 0 / n U 0 a n 0 =( N k 0 / N 0 )( a ¯ k 0 / a ¯ 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4Cay aajaWaa0baaSqaaiaadUgaaeaacaaIWaaaaOGaeyypa0ZaaSGbaeaa daaeqaqaaiaadggadaqhaaWcbaGaamOBaaqaaiaaicdaaaaabaGaam OBaiabgIGiolaadwfadaqhaaadbaGaam4Aaaqaaiaaicdaaaaaleqa niabggHiLdaakeaadaaeqaqaaiaadggadaqhaaWcbaGaamOBaaqaai aaicdaaaaabaGaamOBaiabgIGiolaadwfadaahaaadbeqaaiaaicda aaaaleqaniabggHiLdaaaOGaeyypa0ZaaeWaaeaadaWcgaqaaiaad6 eadaqhaaWcbaGaam4AaaqaaiaaicdaaaaakeaacaWGobWaaWbaaSqa beaacaaIWaaaaaaaaOGaayjkaiaawMcaamaabmaabaWaaSGbaeaace WGHbGbaebadaqhaaWcbaGaam4AaaqaaiaaicdaaaaakeaaceWGHbGb aebadaahaaWcbeqaaiaaicdaaaaaaaGccaGLOaGaayzkaaGaaiOlaa aa@5E94@  Obviously, the stratum-defining housing variables should be included in the appraisal data set. In the Netherlands address and type of dwelling are included. This allows a sub-division of the population into cross classifications of location and type of dwelling. Appraisals may not always be accurate estimates of the 'true' market values of the individual properties but at the stratum level we expect the accuracy of the average appraisals to be sufficient for the computation of the weights.

Statistical techniques such as GREG estimation are typically applied to estimate totals or means for small domains for which the number of observations is so small that the standard errors using traditional (Horvitz-Thompson) estimators MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  in our case the ratio of sample means MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  would become unacceptably high. It should be mentioned that, even with the GREG method, the stratification scheme should not be too detailed since that might unduly raise the variance of the stratum indexes and hence of the aggregate index. More importantly perhaps, small sample bias will increase and may become non-negligible with very small samples.

OLS regressions of selling prices on appraisals should now be run in every time period for each stratum in order to compute the aggregate GREG index. The stratified (OLS) GREG index is

P ^ StrGREG 0t = k=1 K s ^ k 0 P ^ k,GREG,OLS 0t = k=1 K s ^ k 0 ( α ^ k t / a ¯ k 0 + β ^ k t α ^ k 0 / a ¯ k 0 + β ^ k 0 ) ;       ( 3.13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabofacaqG0bGaaeOCaiaabEeacaqGsbGaaeyr aiaabEeaaeaacaaIWaGaamiDaaaakiabg2da9maaqahabaGabm4Cay aajaWaa0baaSqaaiaadUgaaeaacaaIWaaaaOGabmiuayaajaWaa0ba aSqaaiaadUgacaGGSaGaae4raiaabkfacaqGfbGaae4raiaabYcaca qGpbGaaeitaiaabofaaeaacaaIWaGaamiDaaaaaeaacaWGRbGaeyyp a0JaaGymaaqaaiaadUeaa0GaeyyeIuoakiabg2da9maaqahabaGabm 4CayaajaWaa0baaSqaaiaadUgaaeaacaaIWaaaaOWaaeWaaeaadaWc aaqaaiqbeg7aHzaajaWaa0baaSqaaiaadUgaaeaacaWG0baaaOGaai 4laiqadggagaqeamaaDaaaleaacaWGRbaabaGaaGimaaaakiabgUca Riqbek7aIzaajaWaa0baaSqaaiaadUgaaeaacaWG0baaaaGcbaGafq ySdeMbaKaadaqhaaWcbaGaam4AaaqaaiaaicdaaaGccaGGVaGabmyy ayaaraWaa0baaSqaaiaadUgaaeaacaaIWaaaaOGaey4kaSIafqOSdi MbaKaadaqhaaWcbaGaam4AaaqaaiaaicdaaaaaaaGccaGLOaGaayzk aaaaleaacaWGRbGaeyypa0JaaGymaaqaaiaadUeaa0GaeyyeIuoaki aacUdacaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaIZaGaaiOl aiaaigdacaaIZaaapaGaayjkaiaawMcaaaaa@8067@

Differences in the slope coefficients β ^ k s (s=0,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqOSdi MbaKaadaqhaaWcbaGaam4AaaqaaiaadohaaaGccaGGOaGaam4Caiab g2da9iaaicdacaGGSaGaamiDaiaacMcaaaa@4321@  across the strata could be the result of sampling error or reflect a real phenomenon. The latter can be of particular importance for periods t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaa aa@3A91@  which are very distant from period 0 as different housing market segments tend to show varying price trends. Whether any differences in the slope coefficients reflect a real phenomenon could be tested.

An alternative model, to be estimated on the entire data set, is one with a single intercept term, but where the βs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOSdi gcbaGaa8xgGiaabohaaaa@3CF1@  are allowed to differ across the strata. Let D n,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiram aaBaaaleaacaWGUbGaaiilaiaadUgaaeqaaaaa@3D1F@  be a dummy variable that has the value 1 if property n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBaa aa@3A8B@  belongs to stratum k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaa aa@3A88@  and 0 otherwise. In period s(s=0,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Cai aacIcacaWGZbGaeyypa0JaaGimaiaacYcacaWG0bGaaiykaaaa@4049@  the model

p n s = α s + k=1 K β k s D n,k a n 0 + ε n s       ( 3.14 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaam4Caaaakiabg2da9iabeg7aHnaaCaaa leqabaGaam4CaaaakiabgUcaRmaaqahabaGaeqOSdi2aa0baaSqaai aadUgaaeaacaWGZbaaaOGaamiramaaBaaaleaacaWGUbGaaiilaiaa dUgaaeqaaOGaamyyamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaaca WGRbGaeyypa0JaaGymaaqaaiaadUeaa0GaeyyeIuoakiabgUcaRiab ew7aLnaaDaaaleaacaWGUbaabaGaam4CaaaakiaaxMaadaqadaqaaa baaaaaaaaapeGaaG4maiaac6cacaaIXaGaaGinaaWdaiaawIcacaGL Paaaaaa@5B25@

is estimated by OLS regression on the data of the sample S s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaam4CaaaakiaacYcaaaa@3C4E@  yielding predicted prices p ˜ n s = α ˜ s + β ˜ k s a n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiCay aaiaWaa0baaSqaaiaad6gaaeaacaWGZbaaaOGaeyypa0JafqySdeMb aGaadaahaaWcbeqaaiaadohaaaGccqGHRaWkcuaHYoGygaacamaaDa aaleaacaWGRbaabaGaam4CaaaakiaadggadaqhaaWcbaGaamOBaaqa aiaaicdaaaaaaa@4811@  for n U k 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBai abgIGiolaadwfadaqhaaWcbaGaam4AaaqaaiaaicdaaaGccaGGUaaa aa@3F7B@  The residuals again sum to zero and the new (unstratified) OLS GREG index becomes

P ˜ GREG,OLS 0t = n U 0 p ˜ n t / N 0 n U 0 p ˜ n 0 / N 0 = k=1 K n U k 0 p ˜ n t / N 0 k=1 K n U k 0 p ˜ n 0 / N 0 = α ˜ t + k=1 K ( N k 0 N 0 ) β ˜ k t a ¯ k 0 α ˜ 0 + k=1 K ( N k 0 N 0 ) β ˜ k 0 a ¯ k 0 .       ( 3.15 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aaiaWaa0baaSqaaiaabEeacaqGsbGaaeyraiaabEeacaqGSaGaae4t aiaabYeacaqGtbaabaGaaGimaiaadshaaaGccqGH9aqpdaWcaaqaam aaqafabaGabmiCayaaiaWaa0baaSqaaiaad6gaaeaacaWG0baaaOGa ai4laiaad6eadaahaaWcbeqaaiaaicdaaaaabaGaamOBaiabgIGiol aadwfadaahaaadbeqaaiaaicdaaaaaleqaniabggHiLdaakeaadaae qbqaaiqadchagaacamaaDaaaleaacaWGUbaabaGaaGimaaaakiaac+ cacaWGobWaaWbaaSqabeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWG vbWaaWbaaWqabeaacaaIWaaaaaWcbeqdcqGHris5aaaakiabg2da9m aalaaabaWaaabCaeaadaaeqbqaaiqadchagaacamaaDaaaleaacaWG UbaabaGaamiDaaaakiaac+cacaWGobWaaWbaaSqabeaacaaIWaaaaa qaaiaad6gacqGHiiIZcaWGvbWaa0baaWqaaiaadUgaaeaacaaIWaaa aaWcbeqdcqGHris5aaWcbaGaam4Aaiabg2da9iaaigdaaeaacaWGlb aaniabggHiLdaakeaadaaeWbqaamaaqafabaGabmiCayaaiaWaa0ba aSqaaiaad6gaaeaacaaIWaaaaOGaai4laiaad6eadaahaaWcbeqaai aaicdaaaaabaGaamOBaiabgIGiolaadwfadaqhaaadbaGaam4Aaaqa aiaaicdaaaaaleqaniabggHiLdaaleaacaWGRbGaeyypa0JaaGymaa qaaiaadUeaa0GaeyyeIuoaaaGccqGH9aqpdaWcaaqaaiqbeg7aHzaa iaWaaWbaaSqabeaacaWG0baaaOGaey4kaSYaaabCaeaadaqadaqaam aalaaabaGaamOtamaaDaaaleaacaWGRbaabaGaaGimaaaaaOqaaiaa d6eadaahaaWcbeqaaiaaicdaaaaaaaGccaGLOaGaayzkaaGafqOSdi MbaGaadaqhaaWcbaGaam4AaaqaaiaadshaaaGcceWGHbGbaebadaqh aaWcbaGaam4AaaqaaiaaicdaaaaabaGaam4Aaiabg2da9iaaigdaae aacaWGlbaaniabggHiLdaakeaacuaHXoqygaacamaaCaaaleqabaGa aGimaaaakiabgUcaRmaaqahabaWaaeWaaeaadaWcaaqaaiaad6eada qhaaWcbaGaam4AaaqaaiaaicdaaaaakeaacaWGobWaaWbaaSqabeaa caaIWaaaaaaaaOGaayjkaiaawMcaaiqbek7aIzaaiaWaa0baaSqaai aadUgaaeaacaaIWaaaaOGabmyyayaaraWaa0baaSqaaiaadUgaaeaa caaIWaaaaaqaaiaadUgacqGH9aqpcaaIXaaabaGaam4saaqdcqGHri s5aaaakiaac6cacaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaI ZaGaaiOlaiaaigdacaaI1aaapaGaayjkaiaawMcaaaaa@B557@

Model (3.14) is more flexible than the original model given by equations (3.1) and (3.5), and could be useful if the proportionality between sale prices and appraisals fails. Estimator (3.15) reduces to the original GREG index (3.10) if the β ˜ k s s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqOSdi MbaGaadaqhaaWcbaGaam4AaaqaaiaadohaaaacbaGccaWFzaIaae4C aaaa@3F1F@  are all equal. In practice this will not happen, and (3.15) and (3.10) will give different answers. A common justification for the use of GREG estimators is that, being asymptotically unbiased, they are relatively robust to model choice. So we would expect the impact of the alternative model specification (3.15) to be moderate. On the other hand, it is well recognized in the literature that model dependence can be an issue under specific circumstances, notably when dealing with highly variable and outlier-prone populations. For example, Hedlin, Falvey, Chambers and Kokic (2001) stress the importance of a careful model specification search while Beaumont and Alavi (2004) focus on the treatment of outliers. It would therefore be worthwhile examining the effect of this alternative model specification.

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