(12) United States Patent
`McManus et ai.
`
`111111
`
`1111111111111111111111111111111111111111111111111111111111111
`US006401070Bl
`US 6,401,070 Bl
`Jun. 4,2002
`
`(10) Patent No.:
`(45) Date of Patent:
`
`(54) SYSTEM AND METHOD FOR PROVIDING
`HOUSE PRICE FORECASTS BASED ON
`REPEAT SALES MODEL
`
`(75)
`
`Inventors: Douglas A. McManus, Bethesda, MD
`(US); Sol T. Mumey, McLean, VA
`(US)
`
`(73) Assignee: Freddie Mac, Vienna, VA (US)
`
`( *) Notice:
`
`Subject to any disclaimer, the term of this
`patent is extended or adjusted under 35
`U.S.c. 154(b) by 0 days.
`
`(21) Appl. No.: 09/115,831
`
`(22) Filed:
`
`Jul. 15, 1998
`
`Related U.S. Application Data
`
`(63) Continuation of application No. 08/730,289, filed on Oct.
`11, 1996.
`(60) Provisional application No. 60/059,327, filed on Sep. 17,
`1997, provisional application No. 60/059,194, filed on Sep.
`17,1997, and provisional application No. 60/059,328, filed
`on Sep. 17, 1997.
`
`Int. CI? ................................................ G06F 17/60
`(51)
`(52) U.S. CI. ............................... 705/1; 705/10; 705/35;
`705/37
`(58) Field of Search ................................ 705/1, 10, 35,
`705/37
`
`(56)
`
`References Cited
`
`U.S. PATENT DOCUMENTS
`5,361,201 A * 11/1994 Jost et al. ..................... 705/35
`5,414,621 A * 5/1995 Hough ........................ 705/10
`5,500,793 A * 3/1996 Deming, Jr. et al. .......... 705/37
`5,680,305 A * 10/1997 Apgar ......................... 705/10
`5,857,174 A * 1/1999 Dugan ........................... 705/1
`
`OlliER PUBLICATIONS
`
`Lew Sichelman; "Determining home's proper selling price",
`National Mortgage News, Sep. 29, 1997. *
`Clapp et al. "Estimating Time Adjustments with Sales Prices
`and Assesed Values", The Appraisal Journal, Jul. 1996. *
`Cocheo, "Appraisals: A Trade under Renovation", ABA
`Banking Journal Feb. 1996.*
`
`Fritz Wayne c., "Real Estate Appraisal Cocepts", Economic
`development Review, Winter 1996. *
`Detweiler et aI., "Computer Assisted Real Estate Appraisal",
`The Appraisal Journal, Jan. 1996.*
`Janavicius John M., "A Formula for Tax Appraisal of
`Mult-Tenant Properties", The Appraisal Journal, Oct.
`1996.*
`Jesse M. Abraham and William S. Schauman, Secondary
`Mortgage MarketslWinter 1990/1991, "Measuring House
`Price Inflation, Sizing Up Alternative Methods," pp. 8-12.
`G.A.F. Seber and c.J. Wild, "Nonlinear Regression,"
`Department of Mathematics and Statistics, University of
`Auckland, New Zealand, John Wiley & Sons (1989), pp.
`481-486.
`Martin J. Bailey, Richard F. Muth, and Hugh O. Nourse,
`American Statisical Association Journal, Dec. 1963, "A
`Regression Method for Real Estate Price Index Construc(cid:173)
`tion," pp. 933-942.
`Peter Chinloy, Man Cho, and Isaac F. Megbolugbe, The
`Journal of Real Estate Finance and Economics, "Appraisals,
`Transaction Incentives, and Smoothing," pp. 89-111 (1997).
`Evaluation and Combination of Forecasts, Chapter 8.
`Econometrics, G.S. Maddala, pp. 314-317 (1977).
`Economic Forecasting: An Introduction, pp. 85-107 (1994).
`Forecasting Economic Time Series, pp. 265-276 (2d ed.,
`1986).
`
`* cited by examiner
`
`Primary Examiner-V. Millin
`Assistant Examiner-Jagdish N Patel
`(74) Attorney, Agent, or Firm-Finnegan, Henderson,
`Farabow, Garrett & Dunner, L.L.P.
`
`(57)
`
`ABSTRACT
`
`Method and system for estimating real estate property values
`based on repeat sales model. The method estimates the price
`index using property value data from refinance transactions,
`as well as from purchase transactions. In so estimating, the
`method compensates for the transaction bias arising from
`using data from refinance transactions, which may exhibit
`incentive and selection biases. The property at issue may be
`estimated based on the so-computed price index. The price
`index and/or the bias component may be estimated using
`nonparametric functions.
`
`102 Claims,S Drawing Sheets
`
`140
`
`I
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`WITH REFINANCE
`NONPARAMETRIC
`
`& PURCHASE
`
`146"\
`
`NONPARAMETRIC
`
`ESTIMATION
`
`L+
`
`144"\
`
`REFINANCE &
`
`PURCHASE
`
`142\
`
`[140
`
`REAL ESTATE
`
`DATABASE
`
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`
`~ 100
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`FIG. 1
`
`I+---
`
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`CURSOR
`
`(116
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`DEVICE
`INPUT
`
`(114
`
`DISPLAY I+---
`
`(112
`
`1-------10"6-----~08------~110-1
`
`

`

`u.s. Patent
`
`Jun. 4, 2002
`
`Sheet 2 of 5
`
`US 6,401,070 Bl
`
`ACCESS REAL ESTATE DATABASE 130
`
`V 210
`
`COMPUTE I(t) AND d(t) BASED ON
`log (P s/Pt) = Is-lt+ds2RsTdtIRtl+ S
`
`v 220
`t----~ OUTPUT l(t)
`AND d(t)
`
`COMPUTE F s BASED ON
`
`V 230
`
`F s= exp(log(Pt)+ls-lt);
`
`OUTPUT Fs
`
`~240
`
`FIG. 2
`
`

`

`u.s. Patent
`
`Jun. 4, 2002
`
`Sheet 3 of 5
`
`US 6,401,070 Bl
`
`r ACCESS REAL ESTATE DATABASE 130
`
`310
`
`__ ----------~------~/'--32~0
`f325
`USE A NON PARAMETRIC FUNCTION I---~~! OUTPUT l(t)J
`TO COMPUTE I(t)
`I
`
`COMPUTE Fs BASED ON
`
`F s=exp( log(Pt) + l(s)-I(t))
`
`1---330
`
`OUTPUT F
`s
`
`v340
`
`FIG. 3
`
`

`

`u.s. Patent
`
`Jun. 4, 2002
`
`Sheet 4 of 5
`
`US 6,401,070 Bl
`
`ACCESS REAL ESTATE DATABASE 130
`
`V 410
`
`USE NONPARAMETRIC FUNCTIONS f../420
`TO COMPUTE
`I(t), 0 1 (t) AND D2(t)
`
`OUTPUT I(t)
`0 1 (t) AND 02(t)
`
`COMPUTE F s BASED ON
`
`F s= exp(log(Pt)+Is-It);
`
`OUTPUT Fs
`
`V 440
`
`FIG. 4
`
`

`

`u.s. Patent
`
`US. Patent
`
`Jun. 4, 2002
`Jun. 4, 2002
`
`Sheet 5 of 5
`Sheet 5 0f 5
`
`US 6,401,070 Bl
`US 6,401,070 B1
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`

`US 6,401,070 Bl
`
`1
`SYSTEM AND METHOD FOR PROVIDING
`HOUSE PRICE FORECASTS BASED ON
`REPEAT SALES MODEL
`
`RELATED APPLICATIONS
`
`2
`timeliness of the evaluation sample. However, the conven(cid:173)
`tional repeat sales model treated the data from refinance and
`purchase transactions identically, and the potential gains of
`a larger sample size were outweighed by a transaction type
`5 bias introduced into the forecasts. In short, the absence of an
`adequate control for transaction type could cause the fore(cid:173)
`casts of the repeat transactions model to exhibit material
`bias, thereby reducing their usefulness for business pur-
`poses.
`Another shortcoming with the conventional repeat sales
`method is that the price index used was estimated discretely,
`as a series of numbers which represent constant house price
`levels at particular time intervals (typically quarters). This
`method produces counter-intuitive results: first, house price
`appreciation is constant during the discrete intervals (here
`represented as quarters); and second, there are 'jumps' in the
`appreciation at the end of each quarter, and thus, the path of
`house price appreciation is not continuous.
`
`This application is based on a provisional application, Ser.
`No. 60/059,327, entitled METHOD FOR FORECASTING
`HOUSE PRICE VALUES USING SPLINE TECHNIQUES
`TO ESTIMATE REFINANCE BIAS COMPONENT OF 10
`REPEAT SALES PROCESS, filed Sep. 17, 1997, which is
`hereby incorporated by reference. This application is also
`based on provisional applications, Ser. No. 60/059,194,
`entitled METHOD FOR FORECASTING HOUSE PRICE
`VALUES USING REFINANCE AND PURCHASE 15
`TRANSACTIONS, filed Sep. 17, 1997, and Ser. No. 60/059,
`328, entitled METHOD FOR FORECASTING HOUSE
`PRICE VALUES USING NONPARAMETRIC ESTIMA(cid:173)
`TION TECHNIQUES, filed Sep. 17, 1997, and a continu(cid:173)
`ation of U.S. patent application, Ser. No. 081730,289, 20
`entitled METHOD FOR COMBINING HOUSE PRICE
`FORECASTS, filed Oct. 11, 1996, all of which are hereby
`incorporated by reference.
`
`BACKGROUND OF THE INVENTION
`
`A. Field of the Invention
`The present invention relates generally to estimating
`property values, and more particularly, to providing property
`value estimates based on a repeat sales model.
`B. Description of the Prior Art
`Financial institutions and businesses involved with sales
`of property have long tried to estimate values of property
`accurately. Accurate estimation serves many important pur(cid:173)
`poses. For example, financial institutions use property value 35
`estimates as one of the key factors in approving mortgage
`applications for real estate sales. Relying on the soundness
`of the estimate, financial institutions accept the risk of
`lending large sums of money and typically attach the prop(cid:173)
`erty as security for the transaction. Accordingly, the accu(cid:173)
`racy of estimated value of the real estate entity is critical.
`There are several ways of forecasting house prices. See,
`for example, 1. M. Abraham and W. S. Schauman's article,
`"Measuring House Price Inflation," Secondary Mortgage
`Markets (Winter 1990/91), which is incorporated herein by
`reference, for various different approaches using an overall
`price index of the real estate market at issue, such as the
`median sale price index (essentially a median average of all
`the properties sold in a market) and the hedonic index
`(calculating the house price increases by estimating and
`tracking the average prices of various features of a house,
`such as the square footage and the presence or absence of the
`garage).
`One such technique is referred to as the repeat sales
`approach. The data used in the repeat sales model comprise
`successive selling prices and the sale dates for the same
`property. In essence, this approach finds the average rate of
`property appreciation in each period that gives the best
`statistical fit to all the overlapping holding periods. By using
`the same house for both prices, the repeat sales model
`eliminates the bias in price changes that are not due to the
`true house price change, but due to external factors such as
`consumer trends for bigger houses.
`This basic repeat sales model can be improved by the use
`of data from refinance transactions, in addition to data from
`purchase transactions, in forming repeat sales forecasts,
`thereby increasing the size of the estimation sample and the
`
`SUMMARY OF THE INVENTION
`In accordance with the purpose of the invention, as
`embodied and broadly described herein, the invention com(cid:173)
`prises: accessing, for a plurality of properties in a database,
`a set of property value data corresponding to each of the
`25 plurality of properties, wherein each property has two or
`more property value data, each property value data derived
`from a refinance or a purchase transaction; and determining,
`based on the set of property value data for the plurality of
`properties, a time-varying price index corresponding to the
`30 overall change over time of the values of the plurality of
`properties, wherein the price index takes into account a
`transaction type bias between the property value data
`derived from refinance transactions and the property value
`data derived from purchase transactions.
`In another aspect, the invention comprises: means for
`accessing, for a plurality of properties in a database, a set of
`property value data corresponding to each of the plurality of
`properties, wherein each property has two or more property
`value data, each property value data derived from a refinance
`40 or a purchase transaction; and means for determining, based
`on the set of property value data for the plurality of
`properties, a time-varying price index corresponding to the
`overall change over time of the values of the plurality of
`properties, wherein the price index takes into account a
`45 transaction type bias between the property value data
`derived from refinance transactions and the property value
`data derived from purchase transactions.
`In a further aspect of the invention the invention com(cid:173)
`prises an article of manufacture capable of configuring a
`50 data processor to estimate the value of a real estate property,
`the article comprising program code to cause the data
`processor to perform the steps of: accessing, for a plurality
`of properties in a database, a set of property value data
`corresponding to each of the plurality of properties, wherein
`55 each property has two or more property value data, each
`property value data derived from a refinance or a purchase
`transaction; and determining, based on the set of property
`value data for the plurality of properties, a time-varying
`price index corresponding to the overall change over time of
`60 the values of the plurality of properties, wherein the price
`index takes into account a transaction type bias between the
`property value data derived from refinance transactions and
`the property value data derived from purchase transactions.
`It is to be understood that both the foregoing general
`65 description and the following detailed description are exem(cid:173)
`plary and explanatory only and are not restrictive of the
`invention, as claimed.
`
`

`

`3
`BRIEF DESCRIPTION OF THE DRAWINGS
`
`US 6,401,070 Bl
`
`15
`
`4
`coupled to bus 102 for storing static information and instruc(cid:173)
`tions for processor 104. A storage device 110, such as a
`magnetic disk or optical disk, is provided and coupled to bus
`102 for storing information and instructions.
`Computer system 100 may be coupled via bus 102 to a
`display 112, such as a cathode ray tube (CRT), for displaying
`information to a computer user. An input device 114, includ(cid:173)
`ing alphanumeric and other keys, is coupled to bus 102 for
`communicating information and command selections to
`10 processor 104. Another type of user input device is cursor
`control 116, such as a mouse, a trackball or cursor direction
`keys for communicating direction information and com(cid:173)
`mand selections to processor 104 and for controlling cursor
`movement on display 112. This input device typically has
`two degrees of freedom in two axes, a first axis (e.g., x) and
`a second axis (e.g., y), that allows the device to specify
`positions in a plane.
`Computer system 100 accesses data from real estate
`database 130 and executes one or more sequences of one or
`20 more instructions contained in main memory 106. Both the
`data from real estate database 130 and the instructions may
`be read into main memory 106 from another computer(cid:173)
`readable medium, such as storage device 110. As show in
`FIG. 1, the instructions comprise the repeat sales model
`25 programs used in the systems and methods consistent with
`implementations of the present invention: refinance and
`purchase transactions 142; nonparametric estimation 144;
`and nonparametric with refinance and purchase transactions
`146, each of which is described in detail below. Execution
`30 of the sequences of instructions contained in main memory
`106 causes processor 104 to perform the process steps
`described herein. In an alternative implementation, hard(cid:173)
`wired circuitry may be used in place of or in combination
`with real estate database and/or software instructions to
`35 implement the invention. Thus implementations of the
`invention are not limited to any specific combination of
`hardware circuitry and software.
`The term "computer-readable medium" as used herein
`refers to any media that participates in providing instructions
`40 to processor 104 for execution. Such a medium may take
`many forms, including but not limited to, non-volatile
`media, volatile media, and transmission media. Non-volatile
`media includes, for example, optical or magnetic disks, such
`as storage device 110. Volatile media includes dynamic
`45 memory, such as main memory 106. Transmission media
`includes coaxial cables, copper wire and fiber optics, includ(cid:173)
`ing the wires that comprise bus 102. Transmission media can
`also take the form of acoustic or light waves, such as those
`generated during radio-wave and infra-red data communi(cid:173)
`cations.
`Common forms of computer-readable media include, for
`example, a floppy disk, a flexible disk, hard disk, magnetic
`tape, or any other magnetic medium, a CD-ROM, any other
`optical medium, punch cards, papertape, any other physical
`medium with patterns of holes, a RAM, PROM, and
`EPROM, a FLASH-EPROM, any other memory chip or
`cartridge, a carrier wave as described hereinafter, or any
`other medium from which a computer can read.
`FIG. 2 is a flow chart of the steps used to implement the
`60 repeat sales forecast model, accounting for the differences
`between refinance and purchase transactions. As discussed
`above, if the differences between refinance and purchase
`transactions are not adequately accounted for by the model,
`then these potential gains can be outweighed by a transac-
`65 tion type bias introduced into the forecasts.
`There are qualitative differences between house price data
`derived from purchase transactions and from refinance trans-
`
`The accompanying drawings, which are incorporated in
`and constitute a part of this specification, illustrate embodi(cid:173)
`ments of the invention and together with the description, 5
`serve to explain the principles of the invention.
`In the figures:
`FIG. 1 is an overview of a property value estimation
`system consistent with an implementation of the present
`invention;
`FIG. 2 is a flow chart of the process for the property value
`estimation consistent with an implementation of the present
`invention;
`FIG. 3 is a flow chart of the process for the property value
`estimation consistent with another implementation of the
`present invention;
`FIG. 4 a flow chart of the process for the property value
`estimation consistent with another implementation of the
`present invention; and
`FIG. 5 is a graph showing a comparison between a
`conventional estimation and an estimation consistent with
`one implementation of the present invention.
`
`DETAILED DESCRIPTION
`
`Reference will now be made in detail to the systems and
`methods consistent with implementations of the present
`invention, examples of which are illustrated in the accom(cid:173)
`panying drawings. Where appropriate, the same reference
`numerals refer to the same or similar elements. The
`appended claims define the scope of the invention, and the
`following description does not limit that scope.
`The systems and methods consistent with implementa(cid:173)
`tions of the present invention obviate the limitations and
`disadvantages of traditional repeat sale method for forecast(cid:173)
`ing house price values. The systems and methods estimate
`real estate property values based on repeat sales model. The
`price index is estimated using property value data from
`refinance transactions, as well as from purchase transactions.
`In so estimating, the systems and methods provide for ways,
`such as using dummy variables, to compensate for the
`transaction bias arising from using data from refinance
`transactions, which may exhibit incentive and selection
`biases. The property at issue may be estimated based on the
`so-computed price index. The price index and/or the bias
`component may be estimated using nonparametric tech(cid:173)
`niques.
`Additional advantages of the invention will be set forth in
`part in the description which follows, and in part will be
`obvious from the description, or may be learned by practice 50
`of the invention. The advantages of the invention will be
`realized and attained by means of the elements and combi(cid:173)
`nations particularly pointed out in the appended claims.
`FIG. 1 illustrates the system architecture for a computer
`system with which systems consistent with the present 55
`invention may be implemented. Computer system 100
`includes a bus 102 or other communication mechanism for
`communicating information, and a processor 104 coupled
`with bus 102 for processing information. Computer system
`100 also includes a main memory, such as a random access
`memory (RAM) 106 or other dynamic storage device,
`coupled to bus 102 for storing information and instructions
`to be executed by processor 104. RAM 106 also may be used
`to store temporary variables or other intermediate informa(cid:173)
`tion during execution of instructions to be executed by
`processor 104. Computer system 100 further includes a read
`only memory (ROM) 108 or other static storage device
`
`

`

`US 6,401,070 Bl
`
`5
`actions. Purchase transactions typically involve arms-length
`arrangements in which the incentives of the parties will tend
`to result in an unbiased sales price, and the information of
`the three parties (buyer, seller and appraiser) will tend to
`result in greater accuracy in ascertaining the value of the
`property. Refinance transactions, on the other hand, have
`valuation based solely on an appraisal and consequently are
`subject to several sources of bias. Incentive biases in
`appraisals arise because appraisers are motivated to arrive at
`valuations that can make the refinance transaction success(cid:173)
`ful. Selection biases arise because, particularly in a down
`market, the properties that are eligible for refinance are more
`likely to be those that have appreciated relative to the market
`as a whole.
`An implementation of the present invention takes into 15
`account the bias introduced by the data from the refinance
`transactions.
`When the differences between transactions (that is,
`between purchase and refinance transactions) are not
`accounted for, the repeat sales model can be expressed as an
`equation of the following form:
`
`5
`
`6
`ters 4 and 5, John Wiley & Sons (1964), describes such a
`technique. The resulting I(t) and d(t) may be outputted on
`display 112 or to storage device 110 of FIG. 1 (step 225).
`In one implementation consistent with the present
`invention, the estimation sample used to compute I(t) and
`d(t) comprise three types of data: purchase to purchase,
`purchase to refinance, and refinance to purchase. In other
`words, data from refinance to refinance transactions are not
`used in the preferred embodiment. An arms-length transac-
`10 tion is preferably used for at least one value.
`Once I(t) and d(t) are computed, the house price forecasts
`can be made using a formula that depends on the nature of
`the type of an earlier transaction. If it is known that at time
`t, the property in question sold for Pt then the house price
`forecast for time s, F" is given by (step 230):
`
`If it is known that at time t, the property in question was
`refinanced and was appraised for P" then the house price
`20 forecast for time s is given by (step 230):
`
`(5) F,~exp(log(P,)+I,-I,-d,,).
`
`where Pt is the first transaction price for a given property, Ps
`is the second transaction price for the same property, It is the
`log index value at time t, Is is the log index value at time s,
`and v is the error term representing the deviation in price of
`the particular property from the model. Point forecasts at
`time s based on a transaction value at time t from this
`equation are made using the following equation:
`
`Finally, there are cases in which the selection effects of a
`25 refinance transaction need to be included. For example, in
`evaluating a pool of refinance transactions funded in a given
`quarter, this pool would be subject to a refinance selection
`effect. In such a case, the forecast of the house value for a
`refinance transaction at time s using a previous sales price
`30 transaction from time t, Pt is given by (step 230):
`
`(6) F,~exp(log(P,)+I,-I,+d'2)
`
`and using a previous refinance transaction from time t, P"
`the forecast is (step 230):
`
`However, this approach does not adequately account for the 35
`differences in the transaction type in estimation and evalu(cid:173)
`ation. Consequently, the estimates of the index L, will
`exhibit bias, which will in turn be reflected in biased
`forecasts.
`In systems and methods consistent with one implemen- 40
`tation of the present invention, the usual set of repeat sales
`regressors is augmented by a set of dummy variables. The
`dummy variables are created from interacting the date of
`transaction with the transaction type (refinance vs. purchase)
`for both transaction dates. Specifically, the model takes the 45
`form:
`
`Where P t is the first transaction price, Psis the second
`transaction price, It is the log index value at time t, Rtl is one 50
`if the first transaction is a refinance and zero otherwise, Rs2
`is one if the second transaction is a refinance and zero
`otherwise, dt1 is coefficient representing the first transaction
`refinance bias at time t, ds2 is coefficient representing the
`second transaction refinance bias at time s, and S is the error
`term. In essence, the refinance bias terms measure the
`difference in appreciation between purchase and refinance
`transactions at the two dates. The dt1 coefficients can be
`thought of as measuring the incentive bias and the ds2
`coefficients as measuring the combined selection and incen(cid:173)
`tive bias. Thus, this implementation of the present invention
`allows for time varying differences between refinance and
`purchase transactions, thereby improving forecast accuracy.
`In steps 210 and 220 of FIG. 2, I(t) (hence Is and It) and
`d(t) (hence ds2 and dt1 ) may be calculated based on equation
`(3) above, using the standard regression technique. For
`example, Arthur S. Goldberger, "Economic Theory," Chap-
`
`The resulting Fs may be outputted on display 112 or to
`storage device 110 of FIG. 1 (step 240).
`In sum, this implementation of the present invention
`introduces a set of time varying transactions effects into
`repeat sales estimation procedures. This serves to control for
`transactions bias while offering the advantage of a larger
`sample size.
`FIG. 3 shows a flow chart illustrating another implemen(cid:173)
`tation of the present invention. This implementation of the
`present invention utilizes nonparametric estimation tech(cid:173)
`niques to obviate the limitations and disadvantages of tra(cid:173)
`ditional repeat sales method for forecasting house price
`values.
`The repeat sales method explicitly models the house price
`level over time. Prior to this invention, the price index used
`for forecasts has been estimated discretely, as a series of
`numbers which represent constant house price levels at
`particular time intervals (typically quarters). See FIG. 5,
`55 wherein the conventional, discrete index is shown as a
`dotted line. There are several disadvantages of the conven(cid:173)
`tional approach. First, a temporal aggregation bias is intro(cid:173)
`duced by treating house price inflation as constant within a
`time interval. Second, this method takes as fixed the tradeoff
`60 between the variability of the estimates and the bias in the
`fineness of time intervals used. For example, it might be
`better to use wider intervals over time periods with relatively
`few observations and conversely, tighter intervals over time
`periods with a large number of observations. This imp le-
`65 mentation of the present invention mitigates both of these
`limitations by using techniques from nonparametric func(cid:173)
`tional estimation.
`
`

`

`US 6,401,070 Bl
`
`8
`The model is estimated in first differences as given as:
`
`(9) log(PiP,)~bo(s-t)+b, {max[O, s-k,l-max[O, t-k,]}+ . ..
`+bn{max[O, s-knl-max[O, t-kn]}+u
`
`7
`For the purposes of explaining this implementation of the
`present invention, consider FIG. 5. The conventional repeat
`sales method discretely estimates house price appreciation
`for each time interval. As can be seen, the conventional
`method produces several counter-intuitive results: first, 5
`Thus, note that the vector of coefficient b is identified,
`house price appreciation is constant during the discrete
`however the parameter a is not. This is analogous to the need
`intervals (here represented as quarters); and second, there
`to specify a base year to achieve identification of the index
`are 'jumps' in the appreciation at the end of each quarter, and
`function in standard repeat sales estimation.
`thus, the path of house price appreciation is not continuous.
`Similar to the implementation of refinance and purchase
`These limitations of the conventional method can be 10 described above, in steps 310 and 320 of FIG. 3, bo ... bn
`overcome by the introduction of a nonparametric estimator
`may be calculated based on equation (9) above, using the
`of the repeat sales index function, such as using a spline
`standard regression technique. For example, Arthur S.
`function. Systems consistent with the present invention use
`Goldberger, "Economic Theory," Chapters 4 and 5, John
`nonparametric estimation methods to consistently estimate a
`Wiley & Sons (1964), describes such a technique. The
`function whose shape is unknown. There are many non(cid:173)
`15 resulting let) or bo ... bn may be outputted on display 112
`parametric estimators that can be employed to estimate the
`or to storage device 110 of FIG. 1 (step 325).
`repeat sales house price index function I(t). While the
`Point forecasts at time s based on a transaction value at
`current implementation uses one such nonparametric
`time t from this equation are made using the following
`estimators-a linear spline-systems consistent with the
`equation (step 330):
`present invention may also use all nonparametric estimators
`of the index function in repeat sales estimation. The linear
`spline approximates the index function using the following
`formula:
`
`20
`
`(10) F,~exp(log(P,)+I(s)-I(t))
`
`or equivalently,
`
`(8) I(t)~a+bot+b, max[O, t-k,l+b 2 max[O, t-k2 l+ . .. +bnmax[O,
`t-knl
`
`25
`
`(11) F,~exp(log(P,)+bo(s-t)+b, {max[O, s-k,l-max[O, t-k,]}+ ...
`+bn{max[O, s-knl-max[O, t-kn]})
`
`, k,,) and coefficients (a,
`with knot points at (k1 ,
`.
`.
`.
`bo, ... , bn ). The implemented nonparametric functional
`estimator is referred to as a linear regression spline, which
`estimates the unknown index function I(t). As the number of
`knots increases (n---;.oo) and {kJni~l becomes dense in the
`domain of the function, the approximating class of Is has the
`property that over compact domains, minb Ills(t)-I(t) 11---;.0.
`And thus under suitable technical conditions, if the number
`of knot points is allowed to increase with sample size, I( t)
`can be consistently estimated. For additional information on
`spline estimation, see C. de Boor, "A Practical Guide to
`Splines," Springer-Verlag (1978) and T. J. Hastie and R. J.
`Tibshirani, "Generalized Additive Model," Chapman and
`Hall (1990), which are hereby incorporated by reference.
`In particular, a spline function can yield a path of the form
`depicted by the solid curve in FIG. 5. The points at which the
`curve changes slope are called "knot points." The method of
`the present invention allows the placement of the knots to be
`determined by the data. To enhance the performance of the
`spline, endogenous knots selection is used. Thus, in esti(cid:173)
`mating the model, a grid of potential knot points is specified.
`Knot points from the grid are used if their use increases the
`fit of the model beyond a set threshold. As a result, knots will
`typically be finely distributed in time intervals where there
`is either a large sample or strong nonlinear changes in the
`index. Conversely, knots typically will be coarsely distrib(cid:173)
`uted in time intervals where there is either a small sample or
`the index is well approximated by a linear form.
`This strategy both greatly diminishes the bias associated
`with the constant house prices within a time interval in the
`conventional approach and the placement of the knots
`allows knots to be used only when they are needed.
`In addition to the improvements due to mitigating prob- 60
`lems with temporal aggregation, this invention allows the
`index to be estimated at a lower level of geographic aggre(cid:173)
`gation. The smaller the region used in estimation the better
`the model can capture local movements in house prices.
`Because the invention allows more efficient use of data, it 65
`permits more geographically disaggregated indexes to be
`estimated, improving forecast accuracy.
`
`The resulting Fs may be outputted on display 112 or to
`storage device 110 of FIG. 1 (step 340).
`The following references discuss nonparametric estima-
`30 tors including splines and are incorporated herein by refer(cid:173)
`ence: (1) C. de Boor, "A Practical Guide to Splines,"
`Springer-Verlag (1978); (