2 Horvitz-Thompson estimators and the SPAR index

Jan de Haan and Rens Hendriks

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The typical aim of survey sampling is to estimate the total or (arithmetic) mean of some variable for a finite population. In a housing context we may want to estimate the total value of the housing stock in, say, period 0. Let U 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaCaaaleqabaGaaGimaaaaaaa@3B58@  denote the housing stock of size N 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtam aaCaaaleqabaGaaGimaaaaaaa@3B51@  and p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C66@  the value of house n(n=1,, N 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBai aacIcacaWGUbGaeyypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamOt amaaCaaaleqabaGaaGimaaaakiaacMcacaGGUaaaaa@438F@  The target to be estimated is

V 0 = n U 0 p n 0 .       ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOvam aaCaaaleqabaGaaGimaaaakiabg2da9maaqafabaGaamiCamaaDaaa leaacaWGUbaabaGaaGimaaaaaeaacaWGUbGaeyicI4SaamyvamaaCa aameqabaGaaGimaaaaaSqab0GaeyyeIuoakiaac6cacaWLjaGaaCzc amaabmaabaaeaaaaaaaaa8qacaaIYaGaaiOlaiaaigdaa8aacaGLOa Gaayzkaaaaaa@4B75@

Suppose we have a sample S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaaGimaaaaaaa@3B56@  consisting of n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBam aaCaaaleqabaGaaGimaaaaaaa@3B71@  houses sold in the base period. If the houses were selected by simple random sampling from the housing stock U 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvam aaCaaaleqabaGaaGimaaaakiaacYcaaaa@3C12@  where each house had the same inclusion probability, then the Horvitz-Thompson estimator

V ^ 0 =( N 0 / n 0 ) n=1 n 0 p n 0       ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOvay aajaWaaWbaaSqabeaacaaIWaaaaOGaeyypa0Jaaiikaiaad6eadaah aaWcbeqaaiaaicdaaaGccaGGVaGaamOBamaaCaaaleqabaGaaGimaa aakiaacMcadaaeWbqaaiaadchadaqhaaWcbaGaamOBaaqaaiaaicda aaaabaGaamOBaiabg2da9iaaigdaaeaacaWGUbWaaWbaaWqabeaaca aIWaaaaaqdcqGHris5aOGaaCzcamaabmaabaaeaaaaaaaaa8qacaaI YaGaaiOlaiaaikdaa8aacaGLOaGaayzkaaaaaa@5050@

is an unbiased estimator of (2.1); see e.g., Cochran (1977).

A natural target MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  though not the only possibility MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacba qcLbyaqaaaaaaaaaWdbiaa=nbiaaa@39DE@  for a house price index would be the value change of a fixed housing stock. Conditioning on the base period stock has two implications: additions to the stock (mostly newly-built houses) should be excluded and the price changes of existing properties should be adjusted for quality changes, i.e., for the impact of depreciation, renovations and extensions. For convenience we assume that such quality changes are negligible. In that case the target price index going from the base period 0 to the comparison period t(>0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai aacIcacqGH+aGpcaaIWaGaaiykaaaa@3DAB@  is defined as

P 0t = n U 0 p n t n U 0 p n 0 ,       ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuam aaCaaaleqabaGaaGimaiaadshaaaGccqGH9aqpdaWcaaqaamaaqafa baGaamiCamaaDaaaleaacaWGUbaabaGaamiDaaaaaeaacaWGUbGaey icI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0GaeyyeIuoaaOqa amaaqafabaGaamiCamaaDaaaleaacaWGUbaabaGaaGimaaaaaeaaca WGUbGaeyicI4SaamyvamaaCaaameqabaGaaGimaaaaaSqab0Gaeyye IuoaaaGccaGGSaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaIYaGaai Olaiaaiodaa8aacaGLOaGaayzkaaaaaa@554A@

with obvious notation. Suppose that we also have a sample S t , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaamiDaaaakiaacYcaaaa@3C4F@  consisting of n t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBam aaCaaaleqabaGaamiDaaaaaaa@3BB0@  houses sold in period t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaa aa@3A91@  and assume that it is an independent random draw from the base period stock. The ratio of the Horvitz-Thompson estimators (the sample means) in both periods

P ^ 0t = ( N 0 / n t ) n S t p n t ( N 0 / n 0 ) n S 0 p n 0 = n S t p n t / n t n S 0 p n 0 / n 0       ( 2.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaaWbaaSqabeaacaaIWaGaamiDaaaakiabg2da9maalaaabaGa aiikaiaad6eadaahaaWcbeqaaiaaicdaaaGccaGGVaGaamOBamaaCa aaleqabaGaamiDaaaakiaacMcadaaeqbqaaiaadchadaqhaaWcbaGa amOBaaqaaiaadshaaaaabaGaamOBaiabgIGiolaadofadaahaaadbe qaaiaadshaaaaaleqaniabggHiLdaakeaacaGGOaGaamOtamaaCaaa leqabaGaaGimaaaakiaac+cacaWGUbWaaWbaaSqabeaacaaIWaaaaO GaaiykamaaqafabaGaamiCamaaDaaaleaacaWGUbaabaGaaGimaaaa aeaacaWGUbGaeyicI4Saam4uamaaCaaameqabaGaaGimaaaaaSqab0 GaeyyeIuoaaaGccqGH9aqpdaWcaaqaamaaqafabaGaamiCamaaDaaa leaacaWGUbaabaGaamiDaaaakiaac+cacaWGUbWaaWbaaSqabeaaca WG0baaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqabeaacaWG0baa aaWcbeqdcqGHris5aaGcbaWaaabuaeaacaWGWbWaa0baaSqaaiaad6 gaaeaacaaIWaaaaOGaai4laiaad6gadaahaaWcbeqaaiaaicdaaaaa baGaamOBaiabgIGiolaadofadaahaaadbeqaaiaaicdaaaaaleqani abggHiLdaaaOGaaCzcaiaaxMaadaqadaqaaabaaaaaaaaapeGaaGOm aiaac6cacaaI0aaapaGaayjkaiaawMcaaaaa@7A96@

might seem a natural estimator of our target index (2.3). However, if 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@  are independently drawn, the variance of estimator (2.4) can be substantial. Moreover, an estimated ratio such as (2.4) has a bias that depends on the variance of the numerator and the covariance of the numerator and the denominator (Cochran 1977). From an index number perspective the issue at stake is that the mix of properties traded in period t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaa aa@3A91@  differs from that in period 0. That is, we are not comparing like with like.

The standard approach to estimating price indexes relies on the matched model methodology where prices p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C66@  and p n t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaamiDaaaaaaa@3CA5@  are observed for a fixed panel of items. The use of panel data ensures that like is compared with like and will reduce the variance of the ratio estimator because p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C66@  and p n t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaamiDaaaaaaa@3CA5@  are typically positively correlated. However, unless 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@  are extraordinary large, there will only be few matched houses, if any. Hence, while prices p n t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaamiDaaaaaaa@3CA5@  are observed for the houses belonging to S t , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uam aaCaaaleqabaGaamiDaaaakiaacYcaaaa@3C4F@  for most of those houses the base period prices p n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C66@  are 'missing'. What may be available instead are government assessments a n 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyam aaDaaaleaacaWGUbaabaGaaGimaaaakiaac6caaaa@3D13@  We could use these as base period values and construct the following (pseudo) matched-model estimator of house price change:

P ˜ 0t = n S t p n t / n t n S t a n 0 / n t .       ( 2.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aaiaWaaWbaaSqabeaacaaIWaGaamiDaaaakiabg2da9maalaaabaWa aabuaeaacaWGWbWaa0baaSqaaiaad6gaaeaacaWG0baaaaqaaiaad6 gacqGHiiIZcaWGtbWaaWbaaWqabeaacaWG0baaaaWcbeqdcqGHris5 aOGaai4laiaad6gadaahaaWcbeqaaiaadshaaaaakeaadaaeqbqaai aadggadaqhaaWcbaGaamOBaaqaaiaaicdaaaaabaGaamOBaiabgIGi olaadofadaahaaadbeqaaiaadshaaaaaleqaniabggHiLdGccaGGVa GaamOBamaaCaaaleqabaGaamiDaaaaaaGccaGGUaGaaCzcaiaaxMaa daqadaqaaabaaaaaaaaapeGaaGOmaiaac6cacaaI1aaapaGaayjkai aawMcaaaaa@5C16@

A problem associated with estimator (2.5) is that the base period index number will differ from 1 because the appraisals a n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyam aaDaaaleaacaWGUbaabaGaaGimaaaaaaa@3C57@  differ from the selling prices p n 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCam aaDaaaleaacaWGUbaabaGaaGimaaaakiaac6caaaa@3D22@  Rescaling (2.5) by dividing it by its base period value is an obvious solution, yielding

P ^ SPAR 0t = n S t p n t / n t n S t a n 0 / n t [ n S 0 p n 0 / n 0 n S 0 a n 0 / n 0 ] 1 = n S t p n t / n t n S 0 p n 0 / n 0 [ n S 0 a n 0 / n 0 n S t a n 0 / n t ].       ( 2.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuay aajaWaa0baaSqaaiaabofacaqGqbGaaeyqaiaabkfaaeaacaaIWaGa amiDaaaakiabg2da9maalaaabaWaaabuaeaacaWGWbWaa0baaSqaai aad6gaaeaacaWG0baaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqa beaacaWG0baaaaWcbeqdcqGHris5aOGaai4laiaad6gadaahaaWcbe qaaiaadshaaaaakeaadaaeqbqaaiaadggadaqhaaWcbaGaamOBaaqa aiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaaiaads haaaaaleqaniabggHiLdGccaGGVaGaamOBamaaCaaaleqabaGaamiD aaaaaaGcdaWadaqaamaalaaabaWaaabuaeaacaWGWbWaa0baaSqaai aad6gaaeaacaaIWaaaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqa beaacaaIWaaaaaWcbeqdcqGHris5aOGaai4laiaad6gadaahaaWcbe qaaiaaicdaaaaakeaadaaeqbqaaiaadggadaqhaaWcbaGaamOBaaqa aiaaicdaaaaabaGaamOBaiabgIGiolaadofadaahaaadbeqaaiaaic daaaaaleqaniabggHiLdGccaGGVaGaamOBamaaCaaaleqabaGaaGim aaaaaaaakiaawUfacaGLDbaadaahaaWcbeqaaiabgkHiTiaaigdaaa GccqGH9aqpdaWcaaqaamaaqafabaGaamiCamaaDaaaleaacaWGUbaa baGaamiDaaaaaeaacaWGUbGaeyicI4Saam4uamaaCaaameqabaGaam iDaaaaaSqab0GaeyyeIuoakiaac+cacaWGUbWaaWbaaSqabeaacaWG 0baaaaGcbaWaaabuaeaacaWGWbWaa0baaSqaaiaad6gaaeaacaaIWa aaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqabeaacaaIWaaaaaWc beqdcqGHris5aOGaai4laiaad6gadaahaaWcbeqaaiaaicdaaaaaaO WaamWaaeaadaWcaaqaamaaqafabaGaamyyamaaDaaaleaacaWGUbaa baGaaGimaaaaaeaacaWGUbGaeyicI4Saam4uamaaCaaameqabaGaaG imaaaaaSqab0GaeyyeIuoakiaac+cacaWGUbWaaWbaaSqabeaacaaI WaaaaaGcbaWaaabuaeaacaWGHbWaa0baaSqaaiaad6gaaeaacaaIWa aaaaqaaiaad6gacqGHiiIZcaWGtbWaaWbaaWqabeaacaWG0baaaaWc beqdcqGHris5aOGaai4laiaad6gadaahaaWcbeqaaiaadshaaaaaaa GccaGLBbGaayzxaaGaaiOlaiaaxMaacaWLjaWaaeWaaeaaqaaaaaaa aaWdbiaaikdacaGGUaGaaGOnaaWdaiaawIcacaGLPaaaaaa@AE1D@

Note that the rescaling factor is stochastic, as it is a ratio of sample means for the base period, and will increase the variance of (2.6) as compared to the estimator given by (2.5), depending on the correlations between the appraisals and the selling prices. Details can be found in de Haan (2007). But we cannot circumvent rescaling since a price index that does not start at the value 1 would be meaningless.

Expression (2.6) is called a Sale Price Appraisal Ratio (SPAR) index. The SPAR method has been applied in the Netherlands since January 2008 to measure the price change of owner-occupied dwellings. As mentioned earlier, we assume that the SPAR index aims at tracking the price change of the housing stock, which is a measure of the change in wealth. In the context of the Harmonized Index of Consumer Prices on the other hand, the house price index should measure the price change of the houses sold during the base period (Makaronidis and Hayes 2006; Eurostat 2010). Under the latter concept there would be no sampling involved if all transactions are recorded and used in the compilation of the index, as is the case in the Netherlands.

The second expression on the right-hand side of (2.6) writes the SPAR index as the product of two factors, the ratio of sample means and a factor between brackets. As the SPAR index is essentially based on the matched model methodology (using base period appraisals instead of sale prices), this factor adjusts the ratio of sample means for changes in the quality mix of the samples that occur between period 0 and period t. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai aac6caaaa@3B43@  A potential problem is that the SPAR index is not a panel-type estimator. A SPAR time series, say for periods t=0,,T, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFfeu0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDai abg2da9iaaicdacaGGSaGaeSOjGSKaaiilaiaadsfacaGGSaaaaa@405B@  might therefore suffer from short-term volatility due to mix changes, especially when the number of sales is low.

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