1 Hedonic Regreions: A Review of Some Unresolved ues Erwin Diewert1, - PDF document

1 Hedonic Regreions: A Review of Some Unresolved ues Erwin Diewert1, Deparent of Eccs, Universy of Brsh Columbia, Vancouver, B.C., Canada, V6T 1Z email: diewert@c. January 2, 2003. 1. Introction Three recent s have revived interest in the tiof heic regreis. The first on is Pakes () who proposed a somewhat ctroversiaview of the topic.2 The second pubis Chapter 4 in Schuze and Mkie (2002), where a rather us aroh to the use of heic regreis was ated due to the fact that many iues had not yet bn completely resolved. A third paper by Heravi and Sver () also raised quess at the usefulness of heic regreis since this paper presented several alternative heiregreion methodologies and obtained dierent empiriresults using the aernativmels.3 Some of the more rtant iues that nd to be resolved before heiregreis n be rnely aed by stastil agencies incle: · Should the depeent variable be transformed or t? · Should separate heiregreis be run for h of the comparison periods or should we use the y variable adjacent yr regreion tique inally suested by Crt (-11) and used by Bert, Grches and Rappaport (1995; ) and many others? · Should regreicoeicients be sign restricted or t? · Should the heiregreis be weighted or weighted? If they sld be weighted, should quany or expeure weights be used?4 · How sld ouers in the regreis be treated? Can influence analysis be used? The present paper takes a systemalook at the ae ess. Single period hedonic regreion iues are areed in ss 2 to 5 whe two yr e y variable regreion iues are areed in ss 6 and 7. Some of the more tecil material relang to sis in an Appendix, which exanes the preres of bateral weighted hedonic regreis. S discues the treaent of ers and 1 The author is indebted to Ernst Berndt and Ace Nakamura for helpful coents. 2 See Huen () for a nice review of the iues raised in Pakes paper. 3 The servaon that dierenvariants of henic regreion tiques n generate qudierent answers empiriy dates bk to Triplett and McDonald (1977; ) at lst. 4 Diewert () rny lked at these weing iues in the context of a simpfied adjnt yr henic regreion mel where the only charteriscs were duy variables. 2 influenal servas and saddrees the iuof whether the signs of heic regreicoeicients sld be restricted. Sconcludes. 2. To Log or Not to Log We sse that price data have been coteK mels or variees of a coy over T+1 periods.5 Thus pkt is the price of mel k in period t for t = T and kÎS(t) where S(t) is the set of mels that are actuay sold in period t. For kÎS(t), denote the number of these type k models sold during period t by qkt.6 We sslso that informaon is avaable on N relevant charteriscs of h model. The amt of charterisc n that mek poesses in period t is deted as zt for t = ,...,T, n = 1,...,N and kÎS(t). Define the N densial vtor of charteriscs for mel k in period t as zkt º [zt,zt,...,zkNt] for t = T and kÎS(t). We shall csider ly nr hedonic regreis in this review. Henthe unweighted near heic regreion for period t has the foowing for7 (1) f(pkt) = b0t + ån=1N fn(zt)bnt + ekt ; t = T; kÎS(t) where ekt is an independeny distrited eor term with mand variance s2, x) is eher the ideny fcon x) º x or the natural logarithm fcon x) º ln x and the funcs of e variable fn are eher the ideny fcon, the logarhm fcon or a duy variable which takes on the value 1 if the characterisc n is present in model k or 0 otherwise. We are restricng the f and fn in this way since the ideny, log and duy variable fcs are by far the most commly used transformafuncs used in hedonic regreis. Rll that the period t charteriscs vtor for mel k was defined as zkt º [zt,zt,...,zkNt]. We define also the period t vtor of the b’s as bt º [b0t,b1t,...,bNt]. Using these defins, we splify the taon on the right hand side of (1) by defining: (2) ht(zkt,bt) º b0t + ån=1N fn(zt)bnt t = T; kÎS(t). The eson we w want to address is: sld the depeent variable pkt) on the left hand side of (1) be pkt or lkt; i.e., should f be the ideny fcon or the log fc?8 We also would ke to know if the ciof ideny or log for the fcon f sld at our cice of ideny or log for the fn that coespond to the cs (i.e., non duy variable) charteriscs. 5 Mels sold in dierent outlets n be regarded as separate variees or not, depending on the context. 6 If a parcular mel k is sold at various priduring period t, then we interpret qkt as the total quantity of mel k that is sold in period t and pkt as the coesnding average price or it value. 7 Note that the nr regreion mel defined by (1) n only provide a first order aroximaon to a general henic fcon. Diewert () made a case for considering sond order aroximaons but in this paper, we will follow cuent prce and consider only nr aroximaons. 8 Grches (a; ) noted that an advantage of the log formulaois that bnt would provide an esmate of the percentage chain price due to a one it chae in zn, provided that fn was the ideny fcon. Court () impcitly noted this advantage of the log formulaon. 3 Suppose that we csf to be the ideny fcon. Suppose further that there is ly one cs charterisc so that N = 1. In this suation, the heiregreion is eenally a regreion of price on pkage size and so if we want to have as a special se, that price per it of useful charterisc is a cstant, then we sld set f1(z1) = z1.9 Under these cs, the mel defined by (1) and p) = p will be csistent with the cstant per it price hythesis if b0t = 0. In the case of N cs charteriscs, a generazation of the cstant per it charterisc price hythesis is the hythesis of cstant returns to sle in the vector of charteriscs, so that if all charteriscs are led, then the resung mel price is led. If period t model is defined by (1) and p) = p, then ht must satisfy the foowing prerty: (3) b0t + ån=1N fn(lzt)bnt = l[b0t + ån=1N fn(zt)bnt] for all l � In order to sasfy (3), we must cse b0t = 0 and the fn to be ideny fcs. Thus if f is csen to be the ideny fcon, then it is natural to choose the fn that coespond to cons charteriscto be ideny fcs as we. Now sse that we csf to be the log fcon. Suppose again that there is ly one cs charterisc so that N = 1. In this suation, again the heiregreion is essenally a regreion of price on pkage size and so if we want to have as a spial se, that price per it of useful charterisc is a cstant, then we need to set f1(z1) = lnz1 and b1t = 1. Under these cs, the mel defined by (1) and p) = lnp will be consistent with the cstant per it price hythesis. In the se of N cs charteriscs, a generazation of the cstant per it price hythesis is the hythesis of cstant returns to sle in the vector of charteriscs. If period t mes defined by (1) and p) = lnp, then ht must satisfy the foowing prerty: (4) b0t + ån=1N fn(lzt)bnt = lnl + b0t + ån=1N fn(zt)bnt for all l � In order to sasfy (4), we must cse the fn(zn) to be log fcs and the bnt must sasfy the foowing nr restric: (5) ån=1N bnt = Thus if f is csen to be the log fcon, then it is natural to choose the fn that coesto cons charteriscto be log fcs as we. 9 We are not aring that this constant returns to sle hythesis must nary hold (usuay, it will not hold); we are just aring that it is useful for the heniregreion mel to be able to model this suation as a special se. The constant returns to sle hythesis is required in some henic mels; e.g., see Muebauer (; ) and Poak’s () “L Charteriscs” mel, which is also used by Triplett (). 10 If we chathe s of msurement for the conous charteriscs, then the nr henic regreion mel will be ated by this chae in the s; i.e., the chain the s for the nth charterisc can be absorbed into the regreion coeicient bn. 11 Note that all of the conous charteriscmust be measured in sve s in this se. 4 An extremely rtant prerty that a heiregreion model sld poess is that the mel be iariant to changes in the s of msurement of the cs charteriscs. Thus sse that we have ly cs charteriscand the period t mes defined by (1) wh f arbrary and the fn(zn) = lnzn. Suppose further that new uns of msurement for the N charteriscs are csen, say Zn, where (6) Zn º zn/cn ; n = N where the cn are svstants. The iariance prerty reires that we can find new regreion coeicientsbnt*, such that the foowing eaon e sasfied ideny: (7) b0t + ån=1N (lnzn)bnt = b0t* + ån=1N (lnZn)bnt* = b0t* + ån=1N (lnzn/cn)bnt* using (6) = b0t* - ån=1N (lncn)bnt* + ån=1N (lnzn)bnt*. Hence to sasfy (7) ideny, we need only set bnt* = bnt for n = N and set b0t* = b0t - ån=1N (lncn)bnt. Thus in parcular, the heiregreion model where f and the fn are all log fcs will satisfy the rtant iariance to changes in the s of msurement of the cs charteriscprerty, provided that the regreion has a cstant term in . We w aress the foowing es: sld the depeent variable pkt) on the left hand side of (1) be pkt or lkt ? If f is the ideny fcon, then using defins (2), equas (1) n be rewren as foows: (8) pkt = ht(zkt,bt) + ekt ; t = T; kÎS(t) where ekt is an independeny distrited eor term with mand variance s2. On the other hand, if f is the logarhm fcon, then equas (1) are eivalent to the foowing eas: (9) pkt = exp[ht(zkt,bt)]exp[ekt] ; t = T; kÎS(t) = exp[ht(zkt,bt)]hkt ; where hkt is an independeny distrited eor term with mand constant variance. Which is more plausible: the mel spified by (8) or the mel spified by (9)? We argue that it is more kely that the eors in (9) are moskedasompared to the eors in (8) since mels wh very large characterisvectors zkt will have high pris pkt and 12 Note that the avrment is independent of the fconal form for f; i.e., if the fn for the conous charteriscs are log funcons, then for any f, the henic regreion must include a constant term to be iariant to chaes in the s of these conous charteriscs. 5 are very kelto have relavely large eor terms. On the other hand, models wh very small amts of charteriscs will have small pris and small mns and the deviaon of a mel price om its mn will be neceary sma. In other words, it is more plausible to aume that the rao of mel price to s mn price is ray distrited wh mand constant variance than to aume that the dierenbetween model price and s mn is ray distrited wmn 0 and constant varian. Hen, om an a priori inof view, we wld favor the logarhmiregreion model (9) (or (1) wh p) º lnp) es nr cterpart (8). The regreion models csidered in this son were weighted models and could be esated wt a wledgof the amts sold for h model in eriod. In the foowing son, we aumthat meany informaon qkt is avaable and we consider w this extra informaon could be use 3. Quantity Weights vsus Expeiture Weights Usuay, discuis of w to use any or expeure weights in a heic regreion are centered around discuis on how to reduce the heteroskedascity of eor terms. In this son, we aempt a somewhat dierent aroh based on the idea that the regreion model sld be representa. In other words, if mel k sold qkt es in period t, then perhaps mel k should be repeated in the period t heic regreion qkt es so that the period t regreion is representave of the sales that tuay oued during the peri To ustrate this id, suppose that in period t, only three mels were sold and there is only e cs charterisc. Let the period t price of the three mels be p1t, p2t and p3t and suppose that the three mels have the amts zt, zt and zt of the single charterisc respecvely. Then the period t weighted regreimodel (1) has ly the foowing 3 observas and 2 unknown parameters, b0t and b1t : (10) f(p1t) = b0t + f1(zt)b1t + e1t ; p2t) = b0t + f1(zt)b1t + e2t ; p3t) = b0t + f1(zt)b1t + e3t . Note that h of the 3 observas gets an equal weight in the period t heic regreion model defined by (). However, if say models 1 and 2 are vasy more popular than model 3, then it es t so be aropriatthat me3 gets the same rtance as mel1 and 2. 13 Ts our representave aroh foows along the nes of The’s (; -) stochasc aroh to index number theory, which is also pursued by Rao (2002). The use of weits that reflhe economic imrtance of mels was roended by Grches (; 8): “But even here, we should use a weited regreion aroh, since we are interested in an esmate of a weited average of the pure price chae, rather than just an unweighted average over all ible mels, no maer how puar or rare.” However, he did not make any expcit weing suggesons. 6 Suppose that the integers q1t, q2t and q3t are the amts sold in period t of mels 1,2 and 3 respvely. Then one way of cstrucng a heiregreion that weights models ording to their c rtanis to repeat h model servaon ording to the mber of es it sold in the period. This ldto the foowing more representave heiregreion model, where the eor terms have been oed: (11) 11p1t) = 11b0t + 11f1(zt)b1t ; 12p2t) = 12b0t + 12f1(zt)b1t ; 13p3t) = 13b0t + 13f1(zt)b1t where 1k is a vector of es of dension qkt for k = . Now csider the foowing any transformaon of the original weighted hedonic regreimodel (): (12) (q1t)1/2 p1t) = (q1t)1/2 b0t + (q1t)1/2 f1(zt)b1t + e1t* ; (q2t)1/2 p2t) = (q2t)1/2 b0t + (q2t)1/2 f1(zt)b1t + e2t* ; (q3t)1/2 p3t) = (q3t)1/2 b0t + (q3t)1/2 f1(zt)b1t + e3t* . Comparing () and (12), it n be seen that the servas in (12) are eal to the coesing servas in (10), expt that the depeent and independent variables in observaon k of (have been mupliey the sarrt of the any sold of model k in period t for k = 1,2,3 in order to obtain the servas in (12). A sampng amework for (is avaable if we aumthat the transformed residuals ekt* are independeny rmadistrited wmzero and constant varian. Let b0t and b1t denote the least sares esatorfor the parameters b0t and b1t in (11) and let b0t* and b1t* denote the least sares esatorfor the parameters b0t and b1t in (12). Then it is straightforward to show that these two sets of lst sares esatorare the same; i.e., we have: (13) [b0t,b1t] = [b0t*,b1t*]. Thus a srtcut method for taining the least sares esatorfor the wn parameters, b0t and b1t, which our in the “representavmel () is to obtain the lst sares esatorfor the transformed model (). This eivalence between the two models pridea jusfion for using the weighted model () in place of the original me(). The aantagin using the transformed model () ethe “representave” mel () is that we can develop a sampng amework for (t t 14 S, for example, Grne (; -). However, the meril equivalence of the least squares esmates tained by repng muple servaons or by the square rt of the weit transformation was nod long ago as the foowing quotaoindites: “It is evident that an servaoof weit w enters into the equaons exy as if it were w separate servaons h of weit y. The best prl method of ong for the weit is, however, to prepare the equaons of condon by muplying equaothroughout by the square rt of s weit.” E. T. Whaker and G. Rinson (; ). 7 for (), since the (oed) error terms in (11) t be aumed to be distrited independeny of eh other. However, in view of the eivalenbetween the least squares esatorfor mels () and (12), we can now be comfortable that the regreion model () weights servaording to their anative rtanin period t. Henwe definely roend the use of the weighted hedonic regreion model () es weighted counterpart (). However, rather than weighng mels by their quany sold in eriod, it is poible to weight h model ording to the value of s sales in eriod. Thus define the value of sales of mel k in period t to be: (14) vkt º pktqkt ; t = T ; kÎS(t). Now csider again the splweighted hedonic regreion model defined by () above and round off the sales of h of the 3 models to the nearest ar (or pey). Let 1k* be a vector of es of dension vkt for k = 1,2,3. Repng h model in (10) ording to the value of s sales in period t lds to the foowing more representave period t heic regreion model (where the eors have been oed): (15) 11*p1t) = 11*b0t + 11*f1(zt)b1t ; 12*p2t) = 12*b0t + 12*f1(zt)b1t ; 13*p3t) = 13*b0t + 13*f1(zt)b1t . Now csider the foowing value transformaon of the original weighted hedonic regreimodel (): (16) (v1t)1/2 p1t) = (v1t)1/2 b0t + (v1t)1/2 f1(zt)b1t + e1t ; (v2t)1/2 p2t) = (v2t)1/2 b0t + (v2t)1/2 f1(zt)b1t + e2t ; (v3t)1/2 p3t) = (v3t)1/2 b0t + (v3t)1/2 f1(zt)b1t + e3t . Comparing () and (16), it n be seen that the servas in (12) are eal to the coesing servas in (10), expt that the depeent and independent variables in observa of () have been mupliey the sarrt of the value sold of model k in period t for k = 1,2,3 in order to obtain the servas in (16). Again, a sampng amework for (is avaable if we aume that the transformed residuals ekt are iependeny distrited normal ravariables wh mn zero and constant varian Again, it is straightforward to show that the least sares esatorfor the parameters b0t and b1t in (15) and (16) are the same. Thus a srtcut method for taining the lst squares esatorfor the wn parameters, b0t and b1t, which our in the value weights representave mel () is to obtain the least sares esatorfor the transformed model (). This eivalence between the two models pridea jusfion for using the value weighted model () in place of the original me(). As before, the aantagin using the transformed model () ethe value weights representave mel () is that we can develop a sampng amework for (t t 8 for (), since the (oed) error terms in (15) t be aumed to be distrited independeny of h other. It sms to us that the any weighted and value weighted models are clear prements er the original weighted model (). Our rsing here is sar to that used by Fisher (; Chapter in developing bateral iex mber theory, who argued that pris nded to be weighted ording to their anative or value rtance in the two periods being compare In the present ctext, we have a weighng prlem that ilves ly e period so that r weighng prlems are tuay much spler than those csidered by Fisher: we need only cse between quany or value weights! But which system of weighng is better in our present ctexany or value weighng? The prlem with quany weighng is this: it will tend to give too e weight to models that have high pris and too much weight to chmodels that have low amounts of useful charteriscs. Hence it ars to us that value weighng is clrly preferable. Thus we are taking the int of view that the main purpose of the period t hedonic regreion is to enable us to domse the market value of h model sold, pktqkt, into the prct of a period t price for a ay adjusted unit of the heic coy, say Pt, es a cstant utility total any for mel k, Qkt. Hence observaon k in period t sld have the representavweight Qkt in constant utility uns that are comparable across mels. But Qkt is eal to pktqkt/Pt, which in turn is equal to vkt/Pt, which in turn is prral to vkt. Thus weighng by the values vkt sms to be the most aropriate form of weighng. Our cclusions at single period hedonic regreis at this int n be suarized as foows: · Wh respt to taking transformas of the depeent variable in a period t hedonic regreion, taking of logarhms of the mel pris is r preferred transforma 15 ªIt has alrdy bn served that the pursof any index number is to strike a air average’ of the price movements—or movements of other groups of maudes. At first a simple average seemed fair, just buse it trted all terms ake. And, in the absence of any owledge of the relavimrtanof the various coeincluded in the average, the simple average is fair. But it was rly roid that there are enormous dierens in imrtan. Everyone ows that rk is more imrtant than coee and wht than quinine. Ts the quest for fairness led to the intrucon of wei.º ving Fisher (; ). ªBut on what principle shall we weit the terms? Artr Yo’s eand other ees at weing represent, consciously or un consciously, the idea that relative mey values of the various coehould determine their weits. A value is, of course, the pruct of a price per , muplied by the mber of s taken. Such values aord the only coon msure for comparing the streams of coes prud, exchaedor consumed, and aoralmost the only basis of weing which has ever bn seriously prsed.º ving Fisher (; ). 9 · If informaon on the mber of mels sold in eriod is avaable, then weighng h observay the sarrt of the value of mel sales is r prefeemethod of weighng. · If the log transformaon is csen for the depeent variable, then we have a d preference for transforng the cs charteriscby the logarhm transformaon as we. If the cs charteriscare transformed by the logarhmic transformaon, then the regreion must have a cstant term to ensure that the resus of the regreion are iariant to the ciof s for the charteriscs. · If the depeent variable is sply the mel pri, then we have a d preference for t transforming the cs charteriscas we. Wh the ageneral csideraons in we w turn to a discuion of w single period hedonic regreis n be used by stastil agencies in a sampng ctext. 4. The Use of Single Period Hedonic Regreions in a Replament Sampng Context In this son, we csider the use of single period hedonic regreis in the ctext of stastil agency sampng prores where a sampled model that was avaable in period s is t available in a later period t and is repld wa new mel that is avaable in period t. We aumthat s t and that me1 is avaable in period s (wh price p1s and charteriscs vtor z1s) t is t available in period t. We further aume that me1 is replad by mel 2 in period t, wh price p2t and charteriscs vtor z2t. The problem is to somehow adjust the price relavp2t/p1s so that the adjusted price rela n be averaged wther price relaves of the form pkt/pks that coespond to models k that are present in both periods s and t in order to form an overall price relavfor the em level, goinom period s to t. If the em level iex is a chain type iex, then s will be eal to t-1 and if the em level iex is a fixed base type iex, then s will be equal to the base period 0. Rll the faly of single period hedonic regreis defined in sabove by equas (1). If we use defins (2) and aume that the fcon of e variable x) has an inverse fcon f-1, then we may rewre eas (1) as foows: (17) pkt = f-1[ht(zkt,bt) + ekt] ; t = T; kÎS(t). Aume that we have a vtor of esates bt for the period t vtor of parameters bt and define the model k sample resils for period t, ekt, as foows: (18) ekt º pkt) - ht(zkt,bt) ; t = T; kÎS(t). 16 Definons () nd to be mified if weited regreions are run instd of weighted regreions. 10 Thus the sample cterparts to equas () are the foowing eas: (19) pkt = f-1[ht(zkt,bt) + ekt] ; t = 0,1,...,T; kÎS(t). Now sse that the period s heic regreion is avaable to the stastil agency. Thus eaon (19) for period s and model 1 is: (20) p1s = f-1[hs(z1s,bs) + e1s]. Rll that mel 2, the replacement for mel 1 in period t, has the vector of charteriscs z2t. Hensing the period s heic regreion, a comparable price for model 2 in period s is f-1[hs(z2t,bs)], the predicted period s price using the period t hedonic regreion for a mel with the vector of charteriscs z2t. Thus r first esator for an adjusted price relavfor mels 1 and 2 going om period s to t is: (21) r(1) º p2t/f-1[hs(z2t,bs)]. However, there is a prlem with the use of () as an adjusted price relave. The problem will bome aarent if z2t = z1s, so that the two models are in ft identil. In this se, we want r price relavto equal the actual price rao: (22) p2t/p1s = p2t/f-1[hs(z1s,bs) + e1s] using () ¹ p2t/f-1[hs(z1s,bs)] if e1s ¹ 0. Hence if the regreion residual for mel 1 in period s, e1s, is t eao zero, then 1) defined by () will t be an appropriate adjusted price relave. In order to compare ke wh kewe must multiply 1) by an adjusent ftor eal to (23) f-1[hs(z1s,bs)]/p1s = f-1[hs(z1s,bs)]/f-1[hs(z1s,bs) + e1s]. Thus r sond esator r(2) for an adjusted price relavis 1) defined by () es the adjusent ftor defined by (), which adjusts the period s served price for model 1, p1s, onto the period s heic regreion su: (24) r(2) º {p2t/f-1[hs(z2t,bs)]}{f-1[hs(z1s,bs)]/p1s} = 2t/f-1[hs(z2t,bs)]}/{p1s/f-1[hs(z1s,bs)]}. The second expreifor r(2) in (24) is instrucve. We can interpret p2t/f-1[hs(z2t,bs)] as the period t price for mel 2 expreein constant ay uy s, using the period s hedonic regreion as the ay adjusent mhanism. Sarlywe can interpret p1s/f-1[hs(z1s,bs)] as the period s price for mel 1 expreein constant ay uy uns, using the period s heic regreion as the ay adjuster. Thus the price relave defined by () compares the price of mel 2 in period t to the price of mel 1 17 If e1s = 0, then 1) will equal 2). 11 in period s in constant utility ans. Henthe period s heic regreion may be used to express mel pris in homogeneous ay adjusted uns. Obviously, if the stastil agency has the period t heic regreion avaable to then the analysis n be repeated, wsome mifis. In this se, equa(19) for period t and model 2 is: (25) p2t = f-1[ht(z2t,bt) + e2t]. R that me1 has the vector of charteriscs z1s. Hensing the period t hedonic regreion, a comparable price for mel 1 in period t is f-1[ht(z1s,bt)], the predicted period t price using the period t heic regreion for a mel with the vector of charteriscs z1s. Thus r third esator for an adjusted price relavfor mels 1 and 2 going om period s to t is: (26) r(3) º f-1[ht(z1s,bt)]/p1s. However, again, there is a prlem with the use of (as an adjusted price relave. As above, the prlem bomes aarent if z2t = z1s, so that the two models are in ft idenlIn this se, we want r price relavto equal the actual price rao: (27) p2t/p1s = f-1[ht(z2t,bt) + e2t]/p1s using () ¹ f-1[ht(z2t,bt)]/p1s if e2t ¹ 0. Hence if the regreion residual for mel 2 in period t, e2t, is t eao zero, then 3) defined by () will t be an appropriate adjusted price relave. In order to compare ke wh kewe must multiply 3) by an adjusent ftor eal to (28) p2t/f-1[ht(z2t,bt)] = f-1[ht(z2t,bt) + e2t]/f-1[ht(z2t,bt)]. Thus r frth esator r(4) for an adjusted price relavis 3) defined by () es the adjusent ftor defined by (), which adjusts the period t served price for mel 2, p2t, onto the period t heic regreion su: (29) r(4) º {f-1[ht(z1s,bt)]/p1s}{p2t/f-1[ht(z2t,bt)]} = 2t/f-1[ht(z2t,bt)]}/{p1s/f-1[ht(z1s,bt)]}. The second expreifor r(4) in (29) is again instrucveWe can interpret p2t/f-1[ht(z2t,bt)] as the period t price for mel 2 expreein constant ay uy uns, using the period t heic regreion as the ay adjusent mhanism. Sarly, we can interpret p1s/f-1[ht(z1s,bt)] as the period s price for mel 1 expreein 18 This basic idea can be traced bk to Court () as his henic seson number one. The idea was expcitly laid out in Grches (a; -) (; 6) and Dhrymes (; -). It was implemented in a stastil agency sampng context by Triplett and McDonald (1977; ). 19 Of course, if e2t = 0, then 3) will equal 4). 12 constant ay us, usinthe period t heic regreion as the ay adjuster. Thus the price relavdefined by () compares the price of mel 2 in period t to the price of mel 1 in period s in constant utility ans, using the period t heic regreito do the ay adjusent. If the period s and t heic regreis are th avaable to the stastil agency, then it is best to make use of both of the adjusted price relaves 2) and 4) and generate a final adjusted price relavthat is a syetriverage of the two esates. Thus define our final prefeed adjusted price relav5) as the geometric mean of r(2) and 4): (30) r(5) º [2)r(4)]1/2. We csthe geometric mean in (30) eother sple syetrimeans ke the arhmetic average because the use of the geometric average leads to an adjusted price relave that will satisfy the reversal test. Finay, suppose that period s and t heic regreis are t available to the stastil agency t a base period hedonic regreion is avaable. In this case, the ious adjusted replment price rao is: (31) r(6) º {p2t/f-1[h0(z2t,b0)]}/{p1s/f-1[h0(z1s,b0)]}. Thus the price relavdefined by () compares the price of mel 2 in period t to the price of mel 1 in period s in constant utility anty s, using the period 0 hedonic regreito do the ay adjusent. Obviously, the adjusted price relav5) wld generay be preferable to the price relave defined by 6), since the period 0 hedonic regreion may be t of date if period 0 is distant operis s and t. Sar csideras suest that more reable resus will be tained if the chain principle is used in forng the adjusted price relaves defined by (5); i.e., the gap betwthe eay vad 2) and 4) is kely to be nimizeif period s is csen to be period t-1. 20 Grches (a; ) noted the existence of these two equay vad esmates. Grches (; 7) also sested taking an average of the two esmates and, as an aernative method of averaging or smthi, he sested using adjnt yr regreions, which will be studied in sons 7 and 8 below. 21 See Diewert () for an argument along these nes. 22 Tastes will prably chae over me and the characteriscs main of definon for mels that exist in period 0 may be qudierent from the mains of definon for the mels that exist in peris s and t; i.e., the z region spanned by the period 0 hedonic regreion may be quout of date for the later peris. 23 Our advoy of the chain principle and of averaging equay vad resus smto be consistent with the son advoted by Grches (; 6-7): ªThis aroh lls for relavely rnt and often changing ‘pri' weits. Since such stastics come to us in discrete intervals, we are also fwh the usual Laspeyres-Psche prlem. The oftener we can change such weights [i.e., run a new henic regreion], the less of a prlem it will be. In pr, whe one may want to use the most rncross son to derive the relevant price weits, such esmates may fluctuate too much for comfort as the result of mucollinry and sampng fluctuaons. They should be smthed in some way, eher by chsing wi = (1/2wi(t) + wi(t+1or by using ‘adjnt yr' regreions in esmatithese weits.º 13 In the foowing son, we shall aume that the stastil agency has esated single period hedonic regreis as in this son but in addwe aume that information on quanes sold of h model is avaable. HenPscheLaspeyres and superlave indexes of the type ated by Sver and Heravi () (aand Pakes (2001) n be calculate 5. Single Period Hedonic Regreions in the ScaData Context In this son, we aumthat the stastil agency has th price and quany (or value) data for the sset of the K mels that are avaablin eriod. As in the previous period, we will aume that the stastil agency has run single period hedonic regreis for peris s and t. The heiregreion of period s n be used in order to lculate the foowing Paasche type iing from period s to t: (32) PP(s,t) º åkÎS(t) pktqkt/{åkÎ[S(t)ÇS(s)] pksqkt + åkÎ[S(t)~S(s)] f-1[hs(zkt,bs)]qkt}. The suation in the merator of the right hand side of (is sply the sum of price pkt es any qkt over all of the mels k sold during period t, which is the set of indexes k represented by S(t). The first summation in the denator of the right hand side of (is the prct of the period s mel k priks, over all mels that are present in both periods s and t while the second set of terms uses the period s esated hedonic price of a mel k that is sold in period t (which has charteriscdefined by the vector zkt) t is t sold in period s, f-1[hs(zkt,bs)], es the period t any sold for this mel, qkt. If we make the strg aumps on demandes period s preferens that are sted in Diewert (), then we can interpret f-1[hs(zkt,bs)] as an approxate Hicksian (1940; ) reservaon price for mel k that is sold in period t n period s; i.e., if price is ae this ng pri, then purchasers will t want to buy any uns of it in period s. Thus er aropriate aumps on consumes preferens, the Paasche iex defined by () will be an approxate lower bound to a theorel Psche-Konüs cost of ving iex; see Diewert (). Thus the esated period s heic regreion enables us to lculate a matched model type Paasche iex betw 24 Wh the avaability of quany informaon on the mels sold, value weited hedonic regreions of the type recoended in so4 n be run for h peri. 25 This is Pakes' (; ) Psche complete henihybrid price index. Expt for error terms, it is also equal to one of Sver and Heravi's () Psche type lower ding indexes for a true cost of ving index 26 A stroer but simpler set of aumpons than those of Diewert () are that all period s demanders of the henic coy evaluate the uy of a mel with charteriscs vtor z according to the maude gs(z), where gs(z) is a separable (cardinal) uy fcon. Under these aumpons, the equbrium price of a mel with charteriscs vtor z should have the period s henic price fcon equal to gs(z) mes a constant. If f-1[hs(z,bs)] n aroximate this true period s henic price fcon and if the fit of the period s henic regreion is good so that bs is close to bs, then f-1[hs(zkt,bs)] will be an aroximate Hicksian reservaoprice for mel k that is sold in period t but not in period s. 27 See Diewert (-) for an exsoof the use of Hicksian reservaopris for new and disaring coes in the context of Psche and Laspeyres indexes. 14 periods s and t, where the prices for the mels that were sold in period t period s are fed in using the period s heic regrei In a sar maer, we can use the heiregreion for period t to fill in the ing reservaon pris for mels that were sold in period s t and we can lculate the foowing Laspres type iing from period s to t: (33) PL(s,t) º {åkÎ[S(s)ÇS(t)] pktqks + åkÎ[S(s)~S(t)] f-1[ht(zks,bt)]qks}/{åkÎS(s) pksqks}. The suation in the denator of the right hand side of (is sply the sum of price pks es any qks over all of the mels k sold during period s, which is the set of iexes k represented by S(s). The first summation in the merator of the right hand side of (is the prct of the period t mek prikt, over all mels that are present in both periods s and t while the second set of terms uses the period t estimated hedonic price of a mel k that is sold in period s (which has charteriscs defined by the vector zks) t is t sold in period t, f-1[ht(zks,bt)], es the period s any sold for this mel, qks. Under aropriate aumps on consumes preferens, the Laspeyres iex defined by () will be an approxate er bound to a theorel Laspeyres-Konüs cost of ving iex; see Diewert (). Thus the esated period t heic regreion enables us to lculate a matched model type Laspeyres iex betwn periods s at, where the prices for the mels that were sold in period s t t period t are fed in using the period t heic regreion. If th period s and t heic regreis are avaablto the stastil agency, then since the Paasche and Laspeyres msureof price change between periods s and t are eay vad, it is aropriate to take a syetriverage of these two esators of price change as a ªfinalº esator of price change between the peris. As usual, we cse the geometric mean of PL and PP over other sple syetrimeans ke the arhmetic average because the use of the geometric average leads to an index that will satisfy the e reversal test. Henefine the Fisher (1922) index between periods s and t as: (34) PF(s,t) º [PL(s,t) PP(s,t)]1/2 where PP and PL are defined by () and (33). 28 Expt for error terms, it is equal to one of Sver and Heravi's () Laspeyres type uer ding indexes for a true cost of ving index. 29 If all mels are present in tperis, then the Laspeyres type index defined by () redus to an ordinary Laspeyres index betwperis s and t and the Paasche type index defined by () redus to an ordinary Psche index. It n be seen that the weits for h of these indexes is not representative of th peris and hence each of the indexes () and (33) will be subject to subsution or representavity bias; see Diewert (a) on the concept of representavity bias. Hento eminate this bias, it is nary to take an average of the two indexes defined by () and (33). 30 See Diewert (). 31 An argument due originay to Kos () n be used to prove that a theorel cost of ving index es betwn the Paasche and Laspeyres indexes; see also Diewert (). However, this arment will only go through for the case where all of the characteriscs are of the conous type. 15 It is of some interest to compute PP, PL and PF defined by ()-(34) ae for the case where there are ly two models: me1, which is avaable in period s t period t, and model 2, which is avaable in period t t period s; i.e., we are revisng the sampng mel that was stied in sabove. Under these cs, PP defined by () splifies to the foowing exprei: (35) PP(s,t) º p2tq2t/f-1[hs(z2t,bs)]q2t = p2t/f-1[hs(z2t,bs)] = 1) where 1) was defined in sy (). SarlyPL defined by () splifies to the foowing exprei: (36) PL(s,t) º f-1[ht(z1s,bt)]q1s/p1sq1s = f-1[ht(z1s,bt)]/p1s = r(3) where 3) was defined in sy (). Rll that r preferred replment price raos tained in s were 2) and 4) rather than 1) and 3)Hence the resus obtained in this son sm to be sghtly incsistent with the resus tained in s This sght incsistency n be resolved if we make strg aumps at the preferens of rchasers of the heioes. Suppose a purchasers of the hedonic coy evaluate the relavuy of eh model in period s ording to the rdinal utility fcon gs(z) so that the relavvalue to purchasers of a mel with charteriscs vtor z1 versus a mel with charteriscs vtor z2 is gs(z1)/gs(z2). Then in equbriumthe period s relave price of the two models sld also be gs(z1)/gs(z2). Thus the period s price of a mel with charteriscs vtor z sld be proporal to gs(z). Finaysuppose that the period s metriy esated hedonic fcon, f-1[hs(z,bs)], n provide an adequate aroxation to the theorel hedonic fcon, rsgs(z), where rs is a sve cstant. Under these strg aumps, the total market utility for period s that is prided by rchases of the hedonic coes is eal to: (37) Qs º åkÎS(s) rsgs(zks)qks » åkÎS(s) f-1[hs(zks,bs)]qks where we have aroxated the uy to purchasers of mel k in period s, rsgs(zks), by the period s heic regreion esatealue, f-1[hs(zks,bs)]. Thus Qs e interpreted as the aregatany of all of the mels rchased in period s, where h model has bn quay adjusted into constant utility s using the period s heic aregator fcon, gs(z). In what follows, we will neglhe aroxation eor betwn nes 1 and 2 of (so that we idenfy the period s aregate any purchased of the heioy, Qs(s), using the period s heic regreion to do the ay adjusent, as foows: 32 We say sy inconsistent buse usuay the heniregreion served eors e1s and e2t will be small and hence the dierens betwn 1) and 2) and 3) and 4) will also be sma. 16 (38) Qs(s) º åkÎS(s) f-1[hs(zks,bs)]qks. For h period t, we can define the value of all mels rchased as: (39) Vt º åkÎS(t) pktqkt ; t = T. For later referen, we also define the period t peure srof mek as foows: (40) skt º pktqkt / åiÎS(t) pitqit ; t = T; kÎS(t). Coesing to the period s any aregate defined by (), we can define an aggregate period s price level, Ps(s), by dividing Qs(s) into the period s value aregate, Vs: (41) Ps(s) º Vs/Qs(s) = Vs/åkÎS(s) f-1[hs(zks,bs)]qks using () = 1/[åkÎS(s) {f-1[hs(zks,bs)]/pks}pksqks/Vs] = 1/[åkÎS(s) {f-1[hs(zks,bs)]/pks}sks] using () for t = s = [åkÎS(s) sks{pks/f-1[hs(zks,bs)]}-1]-1. Thus the aregatperiod s price level using the period s heic regreion, Ps(s), is equal to a period s share weighted harmonic mean of the period s tual mepris, pks, relave to the coesing predicted period s mel pris using the period s heic regreion, f-1[hs(zks,bs)]. Since pks = f-1[hs(zks,bs)ks] where eks is the regreion residual for mel k in period s and these resials are typicay close to 0 and randoy distrited around 0, it n be seen that er rmal cs, Ps(s) defined by () will be close to 1. Now let us use the period s heic regreion to form a cstant utility any aregate for the mels sold in period t. Thus mel k in period t, using the esated hedonic valuaon funcf period s, will have the cstant utility value f-1[hs(zkt,bs)]. Hen, the period t aregate any rchased of the heioy, Qt(s), using the period s heic regreion to do the ay adjusent into constant utility s, be defined as foows: (42) Qt(s) º åkÎS(t) f-1[hs(zkt,bs)]qkt. 33 It n be seen that the expreion on the rit hand side of () is a type of Psche price index, where the price and quany data of period s, pks and qks for kÎS(s), t as the comparison period data and the henic regreion period s predicted pris, f-1[hs(zks,bs)] for kÎS(s)t as base period pris. 34 Our algebra here aumes that weighted hedonic regreions have been run. If a value weited henic regreion has bn run for period s, then the equaon pks = f-1[hs(zks,bs)+eks] must be replaced bpks = f-1[hs(zks,bs)+ (vks)-(1/2) eks] where the eks are the residuals for the transformed period s henic regreion. 17 Coesing to the period t any aregate defined by (), we can define an aggregate period t price level using the preferences of period s to do the ay adjusentPt(s), by dividing Qt(s) into the period t value aregate, Vt: (43) Pt(s) º Vt/Qt(s) = Vt/åkÎS(t) f-1[hs(zkt,bs)]qkt using () = 1/[åkÎS(t) {f-1[hs(zkt,bs)]/pkt}pktqkt/Vt] = 1/[åkÎS(t) {f-1[hs(zkt,bs)]/pkt}skt] using defins () = [åkÎS(t) skt{pkt/f-1[hs(zkt,bs)]}-1]-1. Thus the aregatperiod t price level using the period s heic regreion, Pt(s), is equal to a period t share weighted harmonic mean of the period t tuaeprices, pkt, relave to the coesing predicted period s mel pris using the period s heic regreif-1[hs(zkt,bs)]. Having defined the period s price level Ps(s) by () and the period t price level Pt(s) by (43) using the heiregreion of period s to do the cstant utility aadjusent, we can take the rao of these two price levels to form a Paasche type price iing from period s to t, using the heiregreion of period s, as foows: (44) Pst(s) º Pt(s)/Ps(s) = [åkÎS(t) skt{pkt/f-1[hs(zkt,bs)]}-1]-1/[åkÎS(s) sks{pks/f-1[hs(zks,bs)]}-1]-1. The aPaasche type iex n be compared wr rePsche type iex defined by (): (45) PP(s,t) º åkÎS(t) pktqkt/{åkÎ[S(t)ÇS(s)] pksqkt + åkÎ[S(t)~S(s)] f-1[hs(zkt,bs)]qkt} = Vt/{åkÎ[S(t)ÇS(s)] pksqkt + åkÎ[S(t)~S(s)] f-1[hs(zkt,bs)]qkt} using definon (39) = 1/{åkÎ[S(t)ÇS(s)] [pks/pkt]pktqkt + åkÎ[S(t)~S(s)] -1[hs(zkt,bs)]/pkt)pktqkt}/Vt = 1/{åkÎ[S(t)ÇS(s)] [pks/pkt]skt + åkÎ[S(t)~S(s)] -1[hs(zkt,bs)]/pkt)skt} using () = {åkÎ[S(t)ÇS(s)] skt [pkt/pks]-1 + åkÎ[S(t)~S(s)] skt [pkt/f-1[hs(zkt,bs)-1}-1 = {åkÎ[S(t)ÇS(s)] skt (pkt/f-1[hs(zkt,bs)ks])-1 + åkÎ[S(t)~S(s)] skt [pkt/f-1[hs(zkt,bs)-1}-1 since pks = f-1[hs(zkt,bs)ks] for kÎS(t)ÇS(s) » {åkÎ[S(t)ÇS(s)] skt (pkt/f-1[hs(zkt,bs)])-1 + åkÎ[S(t)~S(s)] skt (pkt/f-1[hs(zkt,bs)])-1}-1 neglng the regreion residuals eks for kÎS(t)ÇS(s) = {åkÎS(t) skt(pkt/f-1[hs(zkt,bs)])-1}-1 = Pt(s) using (). Thus r old Psche type iex PP(s,t) is aroxately eal to the merator of r new Psche type iex Pst(s). However, as we mened before, the denator of Pst(s), Ps(s), will be aroxately eal to 1, and henr new Psche type iex will be aroxately eal to our old Psche type iex; i.e., we have 35 It n be seen that the expreion on the rit hand side of () is a Paasche price index, where the price and quany data of period t, pkt and qkt for kÎS(t), t as the comparison period data and the henic regreion period s predicted pris, f-1[hs(zkt,bs)] for kÎS(t)t as base period pris. 18 (46) Pst(s) » PP(s,t). Now csider new Psche type iex for the case where there are ly two models: model 1, which is avaable in period s t period t, and model 2, which is avaable in period t period s so that we are revisng the sampng mel that was stied in sabove. Under these cs, Pst(s) defined by () splifies to 2) defined in son 4 by (). Hence r new Psche type iex is pey csistent wthe heiy adjusted sampng price rao 2) defined rer in s Obviously, the analysis n be repeated expt that the heiregreion for period t is used to do the ay adjusent rather than the period s heic regreion. Thus, we w sse that a purchasers of the heioy evaluate the relave uy of h model in period t ording to the cardinal utility fcon gt(z). Then in equbrium, the period t price of a mel with charteriscs vtor z sld be proporal to gt(z). Suppose that the period t metriy esated hedonic funcon, f-1[ht(z,bt)], rovide an adequate aroxation to the period t theorel hedonic fcon, rtgt(z), where rt is a svstant. Under these strg aumps, the total market utility for period t that is prided by rchases of the hedonic coes is eal to: (47) Qt º åkÎS(t) rtgt(zkt)qkt » åkÎS(t) f-1[ht(zkt,bt)]qkt where we have aroxated the uy to purchasers of mel k in period t, rtgt(zkt), by the period t heic regreion esatealue, f-1[ht(zkt,bt)]. Thus Qt e interpreted as the aregatany of all of the mels rchased in period t, where each model has bn quay adjusted into constant utility s using the period t heic aregator funcon, gt(z). In what follows, we will again neglt the aroxation eor betwn nes 1 and 2 of () so that we idenfy the period t aregate any rchased of the hedonic coy, Qt(t), usinthe period t heic regreion to do the ay adjusentas foows: (48) Qt(t) º åkÎS(t) f-1[ht(zkt,bt)]qkt. Coesing to the period t any aggregate defined by (), we can define an aggregate period t price level, Pt(t), by dividing Qt(t) into the period t value aregate, Vt: (49) Pt(t) º Vt/Qt(t) = Vt/åkÎS(t) f-1[ht(zkt,bt)]qkt using () = 1/[åkÎS(t) {f-1[ht(zkt,bt)]/pkt}pktqkt/Vt] = 1/[åkÎS(t) {f-1[ht(zkt,bt)]/pkt}skt] using defins () = [åkÎS(t) skt{pkt/f-1[ht(zkt,bt)]}-1]-1. 19 Thus the aregatperiod t price level using the period t heic regreion, Pt(t), is equal to a period t share weighted harmonic mean of the period t tuaepris, pkt, relave to the coesing predicted period t mepris using the period t hedonic regreion, f-1[ht(zkt,bt)]. Since pkt = f-1[ht(zkt,bt)kt] where ekt is the regreion residual for mel k in period t and these resials are typicay close to 0 and randoy distributed around 0, it n be seen that er rmal cs, Pt(t) defined by () will be close to 1. Now use the period t heic regreion to form a cstant utility any aregate for the mels sold in period s. Thus mel k in period s, using the esated hedonic valuaon funcf period t, will have the cstant utility value f-1[ht(zks,bt)]. Hen, the period s regate y purchased of the heioy, Qs(t), using the period t heic regreion to do the ay adjusent into constant utility s, e defined as foows: (50) Qs(t) º åkÎS(s) f-1[ht(zks,bt)]qks. Coesing to the period s any aregate defined by (), we can define an aggregate period s price level using the preferences of period t to do the ay adjusentPs(t), by dividing Qs(t) into the period s value aregate, Vs: (51) Ps(t) º Vs/Qs(t) = Vs/åkÎS(s) f-1[ht(zks,bt)]qks using () = 1/[åkÎS(s) {f-1[ht(zks,bt)]/pks}pksqks/Vs] = 1/[åkÎS(s) {f-1[ht(zks,bt)]/pks}sks] using defins () = [åkÎS(s) sks{f-1[ht(zks,bt)]/pks}]-1. Thus the aregatperiod s price level using the period t heic regreion, Ps(t), is equal to the reciprocal of a period s share weighted arhmetic mean of the predicted period s mel pris in period t using the period t heic regreion, f-1[ht(zks,bt)], relave to the period s tual mepris, pks. Having defined the period s price level Ps(t) by () and the coesing period t price level Pt(t) by () using the heiregreion of period t to do the cstant utility quay adjusent, we can take the rao of these two price levels to form a Laspres type price iing from period s to t, using the hedonic regreion of period t, as foows: (52) Pst(t) º Pt(t)/Ps(t) 36 It n be seen that the expreion on the rit hand side of () is a type of Psche price index, where the price and quany data of period t, pkt and qkt for kÎS(t), t as the comparison period data and the henic regreion period t predicted pris, f-1[ht(zkt,bt)] for kÎS(t)t as base period pris. 37 It n be seen that the expreion on the rit hand side of () is the reciprocal of a kind of Laspeyres price index, where the price and quany data of period s, pks and qks for kÎS(s)t as the base period price and quany data and the heniregreion period t predicted pris, f-1[ht(zks,bt)] for kÎS(s), t as comparison period pris. 20 = [åkÎS(t) skt{pkt/f-1[ht(zkt,bt)]}-1]-1/[åkÎS(s) sks{f-1[ht(zks,bt)]/pks}]-1 = [åkÎS(s) sks{f-1[ht(zks,bt)]/pks}åkÎS(t) skt{pkt/f-1[ht(zkt,bt)]}-1]-1 » åkÎS(s) sks{f-1[ht(zks,bt)]/pks} where the last line afoows om the aumpon that [åkÎS(t) skt{pkt/f-1[ht(zkt,bt)]}-1]-1 will be aroxately eal to 1. The aLaspeyres type iex n be compared wr reLaspeyres type iex defined by (): (53) PL(s,t) º {åkÎ[S(s)ÇS(t)] pktqks + åkÎ[S(s)~S(t)] f-1[ht(zks,bt)]qks}/{åkÎS(s) pksqks} = {åkÎ[S(s)ÇS(t)] pktqks + åkÎ[S(s)~S(t)] f-1[ht(zks,bt)]qks}/Vs using () for t = s = {åkÎ[S(s)ÇS(t)] [pkt/pks]pksqks + åkÎ[S(s)~S(t)] -1[ht(zks,bt)]/pks)pksqks}/Vs = åkÎ[S(s)ÇS(t)] [pkt/pks]sks + åkÎ[S(s)~S(t)] -1[ht(zks,bt)]/pks)sks using () = åkÎ[S(s)ÇS(t)] [f-1[ht(zkt,bt)kt]/pks]sks + åkÎ[S(s)~S(t)] -1[ht(zks,bt)]/pks)sks since pkt = f-1[ht(zkt,bt)kt] for kÎS(s)ÇS(t) » åkÎ[S(s)ÇS(t)] [f-1[ht(zkt,bt)]/pks]sks + åkÎ[S(s)~S(t)] -1[ht(zks,bt)]/pks)sks neglng the regreion residuals ekt for kÎS(s)ÇS(t) = 1/Ps(t) using definon (51) » Pst(t) since Pt(t) is aroxately eal to 1. Thus r old Laspeyres type iex PL(s,t) is aroxately eal to our new Laspeyres type iex Pst(t). Consider new Laspeyres type index for the case where there are ly two models: model 1, which is avaable in period s t period t, and model 2, which is avaable in period t period s so that we are revisng the sampng mel that was stied in sabove. Under these cs, Pst(t) defined by () splifies to 4) defined in sy (). Hence r new Laspeyres type iex, Pst(t), is pey consistent with the heiy adjusted sampng price rao 4) defined rer in s As usual, if heic regreis are avaablfor th periods s and t, then the two indexes Pst(s) and Pst(t), defined by () and (52) respvely, should be averaged geometriy to form a final Fisher type esatof price change going om period s to t. We w turn our aention to bateral heic regreis (i.e., hedonic regreis that involve the data of two periods instd of ly e peri) that also make use of a e duy variable. 6. Unweighted Batal Hedonic Regreions with Time as a DuVariable 38 The last line on the rit hand side of () is the heniindex that is advoted by Pakes (; ). Pakes assumes that s = t-1. 21 We w csider the foowing heic regreion model, which uzes the data of periods s and (54) f(pks) = b0 + ån=1N fn(zs)bn + eks ; kÎS(s); (55) f(pkt) = gst + b0 + ån=1N fn(zt)bn + ekt ; kÎS(t); where eks and ekt are iependeny distrited eor terms wh mand variance s2, x) is eher the ideny fcon x) º x or the natural logarithm fcon x) º ln x and the fcs of e variable fn are eher the ideny fcon, the logarhm fcon or a duy variable which takes on the value 1 if the characterisn is present in model k or 0 otherwise. Note that the b regreicoeicients in (54) are cstrained to be the same as the coesing b coeicients in (55). Note also that eas () have aed a e y variable, gst, and this coeicient will summarize the erall price change in the varis melgoing om period s to t. Before proceeding further, we briefly discuss some of the aantages and disadvantages of the y variable mel defined by () and (55) versus ring separate single period regreis of the type defined by (1) for peris s and t and then using these separate regreis to form two separate esates of ay adjusted pris which would be averaged in some way in order to form an overall msure of price change betwn periods s and t. The main advantage of the laer method is that it is more flexible; i.e., changes in tastes betwn periods n rdy be accoated. However, this method has the disaantagthat two disnct estimates of period s to t price change will be generated by the method (one using the regreion for period s and the other using the regreion for period s) and it is somewhat arbitrary w these two esates are to be averaged to form a single esatof price change. The main advantages of the y variable method are that it cserves degrof freedom and is less sjt to mucollinry prlems and there is no ambiguy at the measure of erall price change between periods s and t. 39 This two period me duy variable henic regreion (and s extension to many peris) was first considered expcitly by Court (-) as his henic seson number two. Court () chose to transform the prices by the log transformaoon empiril grodsªPris were included in the form of their logarhms, since preminary analysis indited that this gave more nearly nr and hier simple coelaons.Court () then used adjnt period me duy henic regreions as s in a loer chain of comparisons extending from 1920 to 1939 for US automobes: ªThe net regreions on me shown ave are in et price nk relaves for rs of constant spificaons. By joining these together, a conous index is sured.º If the two periods being compared are consecuve peris, Grches (; 7) coined the term ªadjnt yr regreionº to describe this duy variable henic regreion mel. 40 This advantage was noted by Grches (; 8): ªThe mduy aroh es have the advantage, if the comparaby prlem n be solved, of aowing us to iore the ever present prlem of mucollinry among the various dimensions.º 41 Grches (; 7) has the foowing very nice suary jusfion for the use of the mduy variable meth: ªThe jusfion for this [meth] is very simple and app: we aow as best we can for all of the major dierens in spificaons by `holding them constant' through regreiotiques. That part of the average price chawhich is not oted for by any of the included spifions will be reflected in the coeicient of the mduy and represents our best estimate of the `explained-by-spificaon-chae average price chae. 22 We have csidered only the case of two periods since this is the case of most interest to stastil agencies who must pride measures of price change between two periods. However, the bateral medefined by () and (55) n encompass th the fixed base suation (where s will eahe base period 0) or the chained suatiwhere s will eal t-1. It is also of interest to consider the two period se because in this suation, we can draw on many of the ideas that have been introduinto bateral iex mber theory, which also dls wh the prlem of msuring price change between two periods. We first csider the case where f is the ideny transformaon. Let us esate the unknown parameters in (54) and (55) by lst sares regression and denote the esates for the bn by bn for n = N and the esatfor gst by cst. Denote the least sares residuals for eas () and (55) wh f defined to be the ideny transformaon by eks and ekt respvelyThen we have the foowing eas, which relate the mel pris in the two periods to their predicted values and the sample resials: (56) pks = b0 + ån=1N fn(zs)bn + eks ; kÎS(s); (57) pkt = cst + b0 + ån=1N fn(zt)bn + ekt ; kÎS(t). Now csider a hypothel situation where the mels sold during peris s and t are exy the same so that there are say K coon models pertaining to the two periods. Suppose further that the mel pris in period t are all exy l es grter than the coesponding mel pris in period s, where l is a svstant. Under these conds, it sms rsable to ask that the regreion predicted values for the period t models be exacy eal to l es the regreion predicted values for the same mels in period s; i.e., we want the foowing eas to be sasfied: (58) cst + b0 + ån=1N fn(zt)bn = l[ b0 + ån=1N fn(zs)bn] ; k = K. In general, if K � N+2 and l ¹ 1, it n be seen that eas () t be solved for any coeicients cst, b0, b1,...,bN. Henr cclusion is that the nr e y hedonic regreion model defined by () and (57) is t a very good one, since it will not give us the ªrightº answer in a splsuation where all mel pris are proporal for the two periods. Of crse, this mogeney prlem with the nr duy variable regreion model n be solved if we replace eas () by the foowing eas: (59) pkt = cst[b0 + ån=1N fn(zt)bn] + ekt ; kÎS(t). In equas (), the y variable, cst, now ars in a muplivfashion. Thus, the prlem with the esating eas () and (59) is that we no longer have a nr regreion model; nr esation tiques wld have to be used. 42 Diewert () also aren theorel grods that duy variable heniregreion mels that used transformeris as dependent variables did not have good properes. 23 Since nr regreion models are more diicult to esate and may suer from reproduciby prlems, we will turn our aention to the second set of bateral heic regreion models, where f is the log transformaon. In this se, the cterparts to equas () and (57) are the foowing eas: (60) ln pks = b0 + ån=1N fn(zs)bn + eks ; kÎS(s); (61) ln pkt = cst + b0 + ån=1N fn(zt)bn + ekt ; kÎS(t). Exponenating th sides of (and (61) lds to the foowing eas that will be sasfiey the data and the least sares esatorfor (and (61): (62) pks = exp[b0 + ån=1N fn(zs)bn]exp[eks] kÎS(s); (63) pkt = exp[cst]exp[b0 + ån=1N fn(zt)bn]exp[ekt] ; kÎS(t). Again consider a hythel situation where the mels sold during peris s and t are exy the same so that there are K coon models pertaining to the two periods. Again suppose that the mel pris in period t are all exy l es grter than the coesing mel pris in period s, where l is a svstant. Again we ask that the regreion predicted values for the period t mels be exacy eal to l es the regreion predicted values for the same mels in period s; i.e., we want the foowing equas to be sasfied: (64) exp[cst]exp[b0 + ån=1N fn(zt)bn] = l{exp[b0 + ån=1N fn(zs)bn]} ; k = K. It n be seen that if we csst = ln l, then we can sasfy eas (). Hence we conclude om a test aroh perspve) that it is preferable to run nr bateral duy variable heiregreis using the log transformaon for the depeent variable rather than lving the mel pris transformed. Thus, we have again reinford the case for using the log transformaon on the dependent variable in hedonic regreimodels. The bateral log heic regreion model is defined by () and (55) where f is the log transformaon. It n be seen that in this se, the theorel iex of price change going om period s to t is exp[gst] and the sample esator of this laon msure is: (65) P(s,t) º exp[cst] where cst is the least sares esator for the shift parameter gst. Note that we t the shift parameter in equas () rather than in equas (). The cice of base period should not matter so let us csider the foowing bateral log regreion model which puts the shift parameter gts in the period s earather than in the period t eas: (66) ln pks = gts + b0* + ån=1N fn(zs)bn* + eks ; kÎS(s); (67) ln pkt = b0* + ån=1N fn(zt)bn* + ekt ; kÎS(t). 24 Denote the least sares esatefor bn* by bn* for n = N and the esatfor gts by cts. For the regreion model defined by () and (67), it n be seen that the theorel iex of price change going om period t to s is exp[gts] and the sample esator of this laon msure is: (68) P(t,s) º exp[cts]. The eson now is: w es P(s,t) defined by () relate to P(t,s) defined by ()? Idy, we wld ke these two esators of price change to sasfy the foowing me ral test: (69) P(t,s) =1/P(s,t). If we compare the original log nr regreion model defined by () and (55) (wh f being the log transforma) wh the new mel defined by () and (67), it n be seen that the right hand side exoges variables are idenl exphat gts apprs in the first set of eas in (66) and (67) whe gst apprs in the second set of eas in (54) and (55). The transsof the column in the X matrix that coess to gts in (66) and (67) is eal to [11T,02T] where 11 is a column vtor of es of dension equal to the mber of mels in the set S(s) and 02 is a column vtor of zeros of dension equal to the mber of mels in the set S(t). The transsof the column in the X matrix that coess to gst in (54) and (55) is eal to [01T,12T] where 01 is a column vtor of zeros of densiequal to the mber of mels in the set S(s) and 12 is a column vtor of es of dension equal to the mber of mels in the set S(t). However, note that th models have the cstant term b0 (or b0*) in every eaon and the transsof the column in the X matrix that coess to this cstant term is eal to [11T,12T] in both models. It n be seen that the sspspaed by the X columns coesing to b0 and gst in (54) and (55) is eal to the sspspaed by the X columns coesing to b0* and gts in (66) and (67) and the two sets of parameters are related by the foowing eas: (70) [01T,12T] gst + [11T,12T] b0 = [11T,02T] gts + [11T,12T] b0*. Equas () are eivalent to the foowing 2 equas in the fr variables gst, b0, gts and b0*: (71) 0 gst + 1 b0 = 1 gts + 1 b0* ; 1 gst + 1 b0 = 0 gts + 1 b0*. Thus given gst and b0, the coesing gts and b0* e tained using eas () as: (72) gts = -gst ; b0* = gst + b0. 25 Equas (72) also hold for the least sares esatorfor the two hedonic regreion models. In parcularwe have: (73) cts = -cst. Hen, exponenating th sides of (gives us exp[cts] = 1/exp[cts] and this eaon is equivalent to (69) using definions () and (65). Thus we have swn that the esator of price change P(s,t) defined by () (which coess to the least sares esators of the inal log heic regreion model defined by () and (55) wh p) º ln p) is equal to the reciprol of the esator of price change P(t,s) defined by () (which coess to the second log heic regreion model defined by () and (67) so that the two bateral y variable heiregreis sasfy the reversal test (). The resus in this son strongly srt the use of the logarhms of mel pris as the depeent variables in an unweighted bateral heic regreion model with a e duy variable. In the foowing son, we will sty the preres of weighted bateral heic regreion models. 7. Weighted Batal Hedonic Regreions with Time as a DuVariable Given the resus in the previs son, we csider ly weighted bateral heic regreis that use the log of mel pris as the depeent variable, before weighng the eas. We also draw on the resus in sand consider ly value weighng. Thus we w csider the foowing value weited hedonic regreion model, which uzes the data of peris s and (74) (vks)1/2 ln pks = (vks)1/2[b0 + ån=1N fn(zs)bn] + eks ; kÎS(s); (75) (vkt)1/2 ln pkt = (vkt)1/2[gst + b0 + ån=1N fn(zt)bn] + ekt ; kÎS(t); where the mel sales values for period t, vkt, were defined by () and eks and ekt are independeny distrited eor terms wh mand variance s2. The weighted model defined by () and (75) is the bateral cterparo our single equaon weighted hedonic regreion model that was stied in s above. However, in the present bilateral ctext, we w encter a prlem that was absent in the single eaon context. The prlem is this: if there is high inflagoing om period s to t, then the period t mesales values vkt e very much bier than the coesing period s mel sales values vks due to this general inflation. Hen, the aumpon of moskedasc resials betwn equas () and (75) is kely to be sasfieHenit is nary to pick new weights that will eliminate this prlem. Our tentave inal suested soluto the aprlem used by general inflation betwn the two periods is to use the mel expeure shares, sks and skt defined rer by () as the weights in (74) and (75) in place of the mel expeures, vks and vkt. 26 Thus we recoend the use of the foowing peure srweited hedonic regreimodel, which uzes the data of peris s and (76) (sks)1/2 ln pks = (sks)1/2[b0 + ån=1N fn(zs)bn] + eks ; kÎS(s); (77) (skt)1/2 ln pkt = (skt)1/2[gst + b0 + ån=1N fn(zt)bn] + ekt ; kÎS(t); where eks and ekt are iependeny distrited eor terms wh mand variance s2. Denote the least sares esatefor bn by bn for n = N and the esatfor gst by cst. For the regreion model defined by () and (77), it n be seen that the theorel index of price change going om period t to s is exp[gst] and the sample esator of this populamsure is: (78) P1(s,t) º exp[cst]. It n be swn that P1(s,t) defined by () in this son has the same desirable prerty that P(s,t) defined by () in the previs son had: namely, if the mels are idenl in the two periods (and the mel expeure shares are idenl for the two periods) and the mel pris in period t are all exy l es grter than the coesing model pris in period s, then P1(s,t) is exy eal to l. The restricon that the expeurshares be idenl in the two periods in the idenl model se is a bit rstic. Moreover, in the idenl mels se, it wld be nice if P1(s,t) defined by () turned out to equal the Törist price iex, since this iex is a prefeed one om the viewints of th the stochasnd c arohes to index mber theory. Hence in place of the mel defined by () and (77), when a model is present in both periods, let us use the are sales share for that mel, (1/2sks+skt), as the weight for that mel in both periods. In this revised weighng scheme, the old period s ea() are replaced by the foowing two sets of equas: (79) (sks)1/2 ln pks = (sks)1/2[b0 + ån=1N fn(zs)bn] + eks ; kÎ[S(s)~S(t)]; (80) [(1/2sks+skt)]1/2 ln pks = [(1/2sks+skt)]1/2 [b0 + ån=1N fn(zs)bn] + eks ; kÎS(s)ÇS(t). Thus if a mel k is present in period s t present in period t, then we use the sare root of the period s sales share for that mel, (sks)1/2, as the weight, which mns this model is incled in equas (). On the other hand, if mel k is present in both periods, , then we use the sarrt of the arhmetiverage of the period s and t sales shares for that mel, [(1/2sks+skt)]1/2, as the weight, which mns this mel is 43 Diewert () considered a mel similar to (76) and (77) expt that all of the explanatory variables were duy variables and showed that weing by the square rts of expendure shares led to a very rsonable index number formula to msure the price chabetween the two periods. Ts the mel defined by () and (77) is consistent with his resus. 44 See Prson 1 in the Aendix. 45 See Diewert (a). 27 included in equas (). Sarlythe old period s ea() are replaced by the foowing two sets of eas: (81) (skt)1/2 ln pkt = (skt)1/2[gst + b0 + ån=1N fn(zt)bn] + ekt ; kÎ[S(t)~S(s)]; (82) [(1/2)(sks+skt)]1/2 ln pkt = [(1/2sks+skt)]1/2 [gst + b0 + ån=1N fn(zt)bn] + ekt ; kÎS(s)ÇS(t). Thus if a mel k is present in period t present in period s, then we use the sare root of the period t sales share for that mel, (skt)1/2, as the weight, which mns this model is incled in equas (). On the other hand, if mel k is present in both periods, then we use the sarrt of the arhmetiverage of the period s and t sales shares for that mel, [(1/2sks+skt)]1/2, as the weight, which mns this mel is included in equas (). As usual, we aumthat eks and ekt are iependeny distributed eor terms wh mand variance s2. Denote the least sares esatefor bn by bn for n = N and the esatfor gst by cst. For the regreion model defined by ()-(82), it n be seen that the theorel index of price change going om period t to s is exp[gst] and the sample esator of this populamsure is: (83) P2(s,t) º exp[cst]. It n be swn that P2(s,t) defined by () has the foowing desirable prerty: if the models are idenl in the two periods, then P2(s,t) is eal to the Törist price iex betwn the two periods. Hence it ears tt the weited hedonic regreion model dened by ()-(82) is a “natural” weited hedonic regreion model tprovides a generazation of the Törnqvist price iex to cor the case where the mels are t matched. If there are no models in cofor the two periods er csideraon, then the mel defined by ()-(82) res to our remel defined by ()-(77). As in the previs son, it is somewhat arbitrary whether we t the e y variable in the period t eas or whether we t it in the period s eas. If we t the y in the period s eaas the parameter gts and obtain a weighted lst squares estate cts for this laon parameter, the theorel iex of price change going om period t to s is exp[gts] and the sample esator of this laon msure is: (84) P*(t,s) º exp[cts]. As in the previs son, we wld ke P*(t,s) to equal the reciprocal of P(s,t). It turns out that this prerty is true for the weighted hedonic regreis defined by () and (77) and (79)-(82) in this son as well as for the weighted ones defined in the previous s; see Prson 4 in the Appendix. Hence it es t matter whether we t the y variable in period s or t: r msure of erall price change 46 This foowfrom Coroary 5.2 in the Aendix. 28 betwn the two periods will be iariant to this cice for the two weighted hedonic regreis csidered in this s Using the resus in the Aendix, we can also show that P1(s,t) and P2(s,t) th sasfy the ideny test (A6), the mogeney tests (A4) and (A5) and the reversal test (A7) as we have already iited. Thus th of these heiprice iexes have some good a priori preres. Which bateral weighted hedonic iex is best? From the viewint of representavity, P1(s,t) sms best: the mels present in eriod are weighted by expeure shares that pertain to that period. However, the loss of representavity for P2(s,t) is prably t large in most as and P2(s,t) has the aantagof being csistent with the use of a Törist price iex in the matched models se. Thus at this stage of resrch, we lean towards the use of P2(s,t). We turn now to a discuion of the treaent of regreion ouers. 8. Outs and Influence Analysis In the ctext of tradal sampling tiques used by stastil agencies, usuay provision is made for the deleon of ers in the samples of pris coted. This raises the iue as to whether ers sld also be deleted in the heiregreion context. In the weighted context, the deleon of sample er servas sld be pered. Since influence analysis is just an extension of eanalysis (an influenal observaon is e which gry influens the esated regreicoeicients)the delef influenal servas sld also be pere However, in the weighted context, the suais somewhat dierenfor two rss. Aung that we have complete market information on the prices and quanes sold of all mels being csidered for the two periods er csideraon, then we are in the same suation as we wld be if we were alying tradal bilateral iex mber theory. In this laer ctext, (aung reable data), tradal iex mber theory does t suesdring t pris and quanes that look a bit sual. Thus the dropping of ers or influenal servas in the ctext of ring a heic 47 Moreover, in the matched models se, P2(s,t) has the advantage of being independent of the henic regreion coeicients b0,b1bNwhers P1(s,t) is not. 48 In the weighted bateral henic regreion context, we need only delete servaons that influence the esmatof gst since the other parameters are not of grt siifince in this context. For an exson of the various aroheto influence analysis, see Chapter 4 in Chaerjee and Hadi (). 49 This is not quite true since there is some eraturon msuricore inflaon that sests the deleon of ouers. However, the goal of this erature is usuay to obtain beer esmates of trend or future inflaon. Tradonal index numbers that do not drop observaons are still ptable as msureof past inflaon. 29 regreion could lto resus that wld not be comparable to the resus of say a maxum erlap superlave iex. Our sond rsfor ang a bit of uon in dropping ers or influenal observas in the weighted hedonic regreion context is that in this weighted context, an individual servaon does t have eal weight! Csidethe case of a gely popular mel that ts for aost all of the sales in a given period. Dropping this weighted observawould eeny ld to a big change in the weighted regreion but on representavity grs, we wld not want to drop this servaon. Hence tradal er and influence analysis wld have to be somew adapted to dl with this suation. Until this new metlogy is avaable, I wld urge a caus aroh to the dring of servas in the ctext of weighted hedonic regreis. 9. Do the Signs of Hedonic Regreion Coeicients Ma? A rnt paper by Pakes () has sulated a certain amount of ctroversy in the erature on hedonic regreis. Hue(2002) has prided a nice suary of the more ctroversial parts of Pakes' analysis and Huen labels these ctroversial Proposis as Pakes I, Pakes II and Pakes . We will follow Huen's interpretaon of Pakes below. Huen (2002; ) states the Pas Prson I as foows: the heifcon is eal to a prrs' marginal cost fcon plus a market wer fcon that depes on the elascities of demand for charteriscs. Huen's (; ) Pas Prson is the foowing coroary to ProposI: the price of a prct n go down when it ires more of a given charteriscIn other words, the sign of a heioeicient for a characterisn go in the ªwrgº dir! Huen's (; ) Pas Prson is that the two single period hedonic regreis pertaining to any two periods being compared may t br any close relaship to ther (this foowom the first Prson that implies that changing market wer betwn the two periods ght ld to que dierent heic 50 Pakes (; ) disishes betwn the compeve case and the more normal market wer se as foows: ªThat is, in the marginal cost pricing equbrium the henifcon is the marginal cost fcon. However in the Bertrand equbrium the henifcon is the sum of the marginal cost fcon and a fcon that suaris the relaonship betwn markups and charteriscs.º 51 ªHenic regreions have been used in resrcfor some mnd they are often found to have coeicients which are `stable' eher over me or ross markets, and which clash wthe naive intuon that the `marginal williess to pay for a characterisc equaled s marginal cost of prucon'. I he this discussion has made it amply clr that these mels n be ry misleing. The derivaves of a heniprice fcon should not be interpreted as eher wess to pay derivaves or cost derivaves; rather they are formed from a complex equbrium pro.º Ariel Pakes (; ). 30 regreis) t ly e of the two regreis is reired to undertake a ay adjusent. We discuss h of these Prss below. I agree wh Pakes that the deternation of mel pris in any e period is deterned by the interacon of demaes preferens, the costs of mel sers and the degree of market wer of suppers. However, I disagree wh Pakes' ctenon that the hedonic fcon must be eal to marginal cost fcon plus a markup funcIt sms to me that Pakes is viewing the heifcon through the eyes of prrs when what is reired is a csumer view. After a, the rposof the heixercise is to find how demande (and not suppers) of the prct value aernativmels in a given period. Thus for the present rpose, it is the preferences of csumers that should be decisive, and not the teclogy and market wer of prrs. The suation is sar to ordinary general ebrium theory where ebrium price and quany for h coy is deterned by the interacon of csumepreferens and produs technology sets and market wer. However, there is a big branch of aed metrics that igres this complex interon and sply uses informaon on the pris that csumers f, the anes that they demand and perhaps demograic informa in order to esate systems of csumedemand funcs. Then these esated demand funcs are used to form estimates of csumer uy fcs and these fcs are often used in appewelfare eccs. What prrs are ing is enrely ielevant to these exercises in appemetrics wh the excepon of the pris that they are oering to sell at. In other words, we do not nd informa produr marginal costs and markups in order to esate csumer preferens: all we nd are seng pris. I beeve that the suation is sar in the ctext of esating 52 The heniindexes that Pakes (; ) considers are all of the type defined by the last line of () ave where the chain principle is used so that s = t-1. 53 This son sms consistent with the son of Grches (; ), who arethat it is the user value or uy of a mel that is the ªritharacterisfor government statisticians to aempt to msure: ªMost onomists would agree that they would ke the `price' index to be a `price-of-vi' or `uy' inditor. Many government statisticians in charge of prucing tual price indexes will reply that the ct hieve this and that therefore they should not even try, but should conntrate instead on some more `jve' index of `transonpris and/or aow only for those `quay' chaes which are based on `prucon' costs. The fact that `truth' ot be achieved doesn't mn that one shouldn't strive to do so, though I sympathize wh the son that it is beer to msure well something define than to do a very r job oa more interesng but also more nebulous conpt. NevertheleI would deny the contenon that `transon' s or `prucon' costs are much more definvoncepts. In general, they too make sense whout some al to uy consideraons.º Grches (; -) went on to definely rejt the prucon cost viewinof adjusng for quay chaes: ªNor are `prucon costs' an adequate idto quay chaes whout a check of their uy impons. ... Nor should we iore `cose' chaes if we can msure them. If the consumer is in ft buying `horsewer', and if a design chae makes it ible to dever more horsewer from the same size and `cost' eine, then the price of horsewer to the consumer has faen and he is beeo.º 54 Huen (; ) also coents on the simarity of henic regreion esmatiowthe esmation of coenonal suly and demand fcons: ªIndd, the Pakes II result has prdent in convenonal pri-quany analysis. When the price of a good is regressed os quany, it is well own that the derlying suly and demand curves generay ot be idenfied separately, and that the regreion coeicients will be stablnd n sy have the `wro' si.º This is true as far as it goes but what 31 a heiprice fcon for a given period. For welfare rposes, we need to aume that the heimel price fcon is prral to a (separable om everything else) heic uy fcon that gives the uy that demaers will get as a fcon of model charteriscs. We then make the heroic aumpon that the actual pris of models that were sold during the period under csideraon are prral to this aumeedonic uy fc It is clr that there are a mber of prlems wh the aumps: · The separaby aumpon is very rstic. · Dierent demaers may have very dierent heic uy fcs and so over e as charterisc costs and markups change, dierent claes of csumers may be id to enter the market and thus the esated hedonic uy fcon may be stabler e. · The aumpon that the market is in equbrium is also susptarcularly in the ctext of new prcts, where it takes e for demaers to discover the new prcts. · Another csequence of the ebrium aumps that we have made is that all models which are sold in a given period are eay desirable; i.e., they all yield equal utility per ar spent. But in prsome mels are vasy more popular than others and our suested approes t diry take this ft into t. In spe of the aprlems, I beeve that the csumer valuaon approh is more appropriate in the ctext of making ay adjusents for CPI rposes than the produr valuaon approroposed by Pakes. Note that there do not ar to be any welfare plis whatsoever in making henic price adjusents using the ameworf Pakes. We turn now to Pakes Prson ; namely, that the price of a prct n go down when it ires more of a given charteriscIn other words, the signs of heic regreion coeicients not have to be sign restricted. In evaluang this Prson, it is again nary to kin the rposfor ring the heiregreion, which is to provide uy valuas for ibly hythel mels that are sold in one enables consumer demand analysis to ªsudº in this suation is that we do not estimate a sildemand fcon but rather esmate a system of demand fcons, where an exogenous idenfying variable is ªincom. If we were using the same price and quany data to esmate a system of prur suly fcons, there would be dierent exogenous variables aring in the prur system such as pal and labor used by the prurs. 55 However the use of expendure or share weited hedonic regreions n parally overcome this deft of the theory. 56 This son sms to be consistent with the foowing remarks by Grches (; 8): ªThe me duy aroh es have the advantage, if the comparaby prlem n be solved, of aowing us to iore the ever present prlem of mucollinry among the various dimensions. Using , we may not re that in one year the coeicient of weit is high and horsepower is low whe in another ythese coeicients reverse themselves, as long as the two coeicients taken together hold the joint et of weit and horsewer constant.º 32 period but not in the other period. For this welfare eccs type rpose, I beeve that it makes sense to se a priori sign restrics on the regreion coeicients. Hence if we beevthat demaers of a mel will, on average, get a higher uy om a model that has more of an a priori desirable feature, then we sld make sure that r esated hedonic regreion does t ctradichese a priori beliefs. Thus we are taking the int of view here that we sr theory on the data and do the metric esation which fs the data best, consistent with our prior beefs. An aernative (t pey defensible int of view) is that we use the data to discover whether a priori theories are csistent with the data. However, it seems to me that this theory tesng int of view is t what statistiagencies are interested in when they do quay adjusent: they want to make ay adjusents that are csistent wh the c's a priori view that more of a desirable characterisc sld incrse the price of the prct (or at lsdrease ). Finay, we discuss Pakes Prson ; namely that ly one of the two single period hedonic regreis that pertain to the two periods er csideraon nds to be used in order to undertake a ay adjusent. Obviously, this Prson is true! However, as we have argued in the earer ss of this paper, if two esates are avaable, then it always is beer to use an average of the two esates rather than just e of them. This is parcularly true for Pakes' preferred index defined by the last line of (since this iex is kely to suer from ssution or representavity bias. Hence it will usuay be best to match up a Laspeyres type esator ke the esator preferred by Pakes wh a coesing Psche type esator (if th are avaable). 10. Conclusion The theory of heic regreis has left a great dof lway en to the empirical invesgator wh respt to the detas of plementation of the mels. Our strategy in this review of the iues has bn to use some of the ideas that are present in the test approh to index mber theory in an aempt to naow wn some of these somewhat arbrary cis. The problem with arbrary cis is that the end resus may t be invariant to these cis and hence if heic regreion tiques are used by stastil agencies, the resung esates of price change may t be reprcible. We have made a start on naowinwn some of these cis t much work remains to be e. 57 But only for charteriscs where we are fairly rtain that more is beer! 58 These a priori beliefs n be imsed on the regreion by repling coeicients by squares of coeicients and then using nonnr regreion esmation tiques. 59 Again there is an analogy wh tradonal consumer demand analysis: if we are interested in welfare analysis, we will imse the curvature condons imped by onomic theory on the econometric esmation method as is ne by Diewert and Wales () whers if we are interested in tesng tradonal demand theory, then we would not imse these curvature condons. 60 Again, this son sms to be consistent with that of Grches (; 7). 61 This n be most clrly sn in the matched model context where the index defined by the last line of () will be aroximately equal to the ordinary Laspeyres index betwperis s and t. 62 The work of Heravi and Sver (shows that the use of different henic regreion tiques can ld to que dierent estimates of price chae. 33 In order to naow wn the range of tcomes that n result ohe use of heic regreitiques, we make the foowing suggess: · It sms preferable to use the log of the mel price as the depeent variable rather than the mel price self. · If expeure weights are avaable, use them to weight the servas as suested in this son for a y variable heiregreion and as suestein sExpendure weights are preferable to quany weights. · In the ctext of ring single period hedonic regreis, it sms preferable to run separate regreis for th periods and use a syetriverage of the resus om th regreis in the final msure of price change between the two periods. · It is preferable to use heiregreion tiques in the ctext of the chain principle where the prices of period t and t-1 are compared since this will tend nimize the spread betwesates of price change er lger peris that are tained using aernative heiteciques. · It sms preferable to sign restrict regreion coeicients in ordance wh a priori theory. One iuthat we did not discuss ae is whether heic regreions sld include brand duy variables as iependent variables. The argument against ing this is that brand duy variables sld be sels if we have entered all of the rtant charteriscs of the prct into the regreion and hen, by including brand duy variables, we will just use up valuable degrees of frm and incrse mucollinry. The argument for entering brand duy variables is that they pture in an eicient manner rtain charteriscs of the prct that wld be diicult to spify otherwise. At the present stage of resrch in this ar, I wld be incned to aow the use of bras as adible y variables. We ccludby ng that the caus ae towards the use of heic regreis expreed by Scze and Mkie () es the foowing coents made by Bn in his discussion of Crt's () piring paper on hedonic regreis: ªMr. Court's interesng work should be caied much further, as he sests. We should, however, not be disainted if neher pubc agencies nor trade aociaons at the cy of pubshing pris, values and index numbers based on the relavely tricky resus that one is sure to get by alying the device of muple coelaon. The only group who would sponsor such a procedure would be the non-existent Naonal Aociaon of Experts in Muple Coelaon, the demand for whose services would be enormously incrsed.º Louis H. Bn (; ). 63 These diicult to spify charteriscs mit include reability, avaability of the pruct in the local marketplace and the degree of consumer owledge aut the pruct; i.e., some prurs will chse to hvy adverse their pructs whe others will not and the et of this adversing may be to crte a brand premium. 34 Hopefuy, in the next few yrs, as users form a csensus on what the ªbestº prores are, then the use of heic regreis by stastil agencies will bome much more widespread and roune. Aendix: Propties of Batal Weighted Hedonic Regreions We csider some of the mathemal preres of a sght generalization of the share weighted bateral heic regreion model defined by () and (77) in sThe generazation is that we do not restrict the weights to sum up to 1 in eriod. Thus, we replace the period s share weights sks in (76) and the period t share weights skt in (77) by the svweights wks and wkt respvely, where these weights do not nary sum to 1 in eriod. We aumthat these weight fcs are wn funcs of the price and quany data pertaining to periods s and t; i.e., we have for some fcs, gks and gkt: (A1) wks = gks(ps,pt,qs,qt) for kÎS(s) ; wkt = gkt(ps,pt,qs,qt) for kÎS(t) where ps and pt are price vectors of the mel pris for peris s at respvely and qs and qt are the coesing period s and t any vtors of the mels sold in periods s and t. In the Prss below, we will place further restrics on the weighng fcs gks and gkt as they are neede The weighted least sares esatorfor gst, b0, b1,...,bN for this new mel are the solus cst*, b0*, b1*,..., bN* to the foowing adrac weighted lst sares nimizatiroble (A2) n b's and c {åkÎS(s) wks[ln pks - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - cst - b0 - ån=1N fn(zt)bn]2}. The bateral price i P that summarizes the erall change in pris going om period s to t is defined as the exenal of the cst soluto (A2); i.e., we have: (A3) P(ps,pt,qs,qt) º exp[cst*]. We wld ke to show that the heic iex mber formula defined by (A3) has some of the preres that bilateral iex mber formulae defined over matched models usuay have. Thus we are aempting to extend the test aroh to index mber theory to weighted bateral heic regreis. In parcularwe wld ke to estabsthe foowing preres for P: 64 The heniregreion Manual being prepared by Jk Triplett () should help form this consensus. 65 Throout this Aendix, we aumthat the X matrix that coesnds to the nr regreion mel defined by (A2) has full column rank so that the soluon to (A2) exists and is ique. 66 The test aroh to index number theory was largely develed by Walsh () (), Fisher (() and Eicorn and Voeer (). For more recent contributions, see Diewert () (), Balk () and von Auer (). 35 (A4) Homogeney of degree in period t pris; i.e., P(ps,lpt,qs,qt) = lP(ps,pt,qs,qt) for all l � (A5) Homogeney of degree mis e in period s pris; i.e., P(lps,pt,qs,qt) = l-1P(ps,pt,qs,qt) for all l � (A6) Ideny; i.e., if the mels in the two periods are idenl and the seng pris are equal so that ps = pt º p and, in addon, the same anes of h model are sold in the two periods so that qs = qt º q, then the resung price iex P() = (A7) Time reveal; i.e., P*(pt,ps,qt,qs) = 1/ P(ps,pt,qs,qt). The aprerty says that if we interchange the order of data and msure the overall change in pris going bkwards om period t to s, then the resung iex P*(pt,ps,qt,qs) is eal to the reciprocal of the original iex P(ps,pt,qs,qt), which msured the degree of erall price change going om period s to t. In order to formay define the price iex P*, let cts*, b0, b1,..., bN be the soluon to the foowing adrac weighted lst sares nimization problem, which coess to reversing the ordering of the two periods: (A8) n b's and c {åkÎS(s) wks[ln pks - cts - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - b0 - ån=1N fn(zt)bn]2}. The bateral price i P* that summarizes the erall change in pris going om period s to t is defined as the exenal of the cts soluto (A8); i.e., we have: (A9) P*(pt,ps,qt,qs) º exp[cts*]. In the remaier of this Aendix, we shall find conds which ensure that the tests (A4)-(A7) are sasfie Proposition 1: Sse that: (i) all mels are idenl in the two periods so that S(s) = S(t) and zs = zt º z for n = N and k = K; (ii) the mel pris in period s are equal to the coesing mel pris in period t so that pks = pkt º pk for k = K; (i) the mel anes sold in period s are eal to the coesing sales in period t so that qks = qkt º qk for k = K; (iv) the mel weights are eal ross the two periods for h model so that wks = wkt º wk for k = K. Under these hypotheses, the ideny test (A6) is sasfie Proof: Uer the ahytheses, the least sares nimization problem (A2) bomes: (A10) n b's and c {åkÎS(s) wk[ln pk - b0 - ån=1N fn(z)bn]2 + åkÎS(t) wk[ln pk - cst - b0 - ån=1N fn(z)bn]2}. 36 From the general preres of nimization problems, it n be seen that the foowing inequay is vad: (A11) n b's and c {åkÎS(s) wk[ln pk - b0 - ån=1N fn(z)bn]2 + åkÎS(t) wk[ln pk - cst - b0 - ån=1N fn(z)bn]2} ³ n b's {åkÎS(s) wk[ln pk - b0 - ån=1N fn(z)bn]2} + n b's and c åkÎS(t) wk[ln pk - cst - b0 - ån=1N fn(z)bn]2}. Let b0*, b1*,..., bN* solve the first minimizati problem on the right hand side of (A). Now look at the second nimizatiroblem on the right hand side of (A). Obviously the parameters cst and b0 t be separately idenfied so one of them n be set eao zero; we csto set cst = 0. But after seng cst = 0, we see that the second nimization problem is idenl to the first minimization problem on the right hand side of (A), and hence cst* = 0 and b0*, b1*,..., bN* solve the second nimizatiroblem. However, cst* = 0 and b0*, b1*,..., bN* are feasible for the nimization problem on the left hand side of (Aand since the jvfcon evaluated at this fsible soluon aains a lower bound, we ccludthat cst* = 0 and b0*, b1*,..., bN* solves (A). But cst* = 0 plies P() º exp[cst*] = exp[0] = 1, which is the desired result (A6). Q.E.D. Proposition 2: Sse that the weight fcs defined by (A1) are mogeneous of degree zero in the coments of the period t price vector pt, so that for a l � 0, gks(ps,lpt,qs,qt) = gks(ps,pt,qs,qt) for kÎS(s) and gkt(ps,lpt,qs,qt) = gkt(ps,pt,qs,qt) for kÎS(t). Then the heic price iex P(ps,pt,qs,qt) defined by (A3) will satisfy the mogeney of degree prerty (A4). Proof: Let cst*, b0*, b1*,..., bN* solve the inal minimization problem (A2) before we muply the period t price vector by l � 0. Now csider a new weighted lst sares nimization problem where pt has bn reply lpt. Under hytheses, the weights will be changed by this change in the period t pris and so the new nimizatiroblem will be: (A12) n b's and c {åkÎS(s) wks[ln pks - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt + ln l - cst - b0 - ån=1N fn(zt)bn]2}. (A13) = n b's and c {åkÎS(s) wks[ln pks - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - cst¢ - b0 - ån=1N fn(zt)bn]2} where the new cst variable is defined as foows: (A14) cst¢ º cst - ln l. 67 We aumthat the period t quantity vtor qt remains the same if the period t pris chae from pt to lpt. 37 Denote the soluon to (A13) as cst¢, b0, b1,..., bN. However, it n be seen that the soluon to (A13) is exy the same as the soluon to the inal prlem, (A2). Hence cst¢ = cst*, and the cst soluto (A12), which we detby cst**, sasfies (A): (A15) cst* = cst¢ = cst - ln l or (A16) cst = cst* + ln l. Hen (A17) P(ps,lpt,qs,qt) º exp[cst] = exp[cst* + ln l] using (A) = l exp[cst*] = l P(ps,pt,qs,qt) using definon (A3) which estabshes the desired result (A4). Q.E.D. Proposition 3: Sse that the weight fcs defined by (A1) are mogeneous of degree zero in the coments of the period s price vector ps, so that for all l � 0, gks(lps,pt,qs,qt) = gks(ps,pt,qs,qt) for kÎS(s) and gkt(lps,pt,qs,qt) = gkt(ps,pt,qs,qt) for kÎS(t). Then the heic price iex P(ps,pt,qs,qt) defined by (A3) will satisfy the mogeney of degree s e prerty (A5). Proof: Let cst*, b0*, b1*,..., bN* solve the inal minimization problem (A2) before we muply the period s price vector by l � 0. Now csider a new weighted lst sares nimizaon problem where ps has bn reply lps. Under r hytheses, the weights will be changed by this change in the period s priand so the new nimizatiroblem will be: (A18) n b's and c {åkÎS(s) wks[ln pks + ln l - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - cst - b0 - ån=1N fn(zt)bn]2}. (A19) = n b's and c {åkÎS(s) wks[ln pks - b0¢ - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - cst¢ - b0¢ - ån=1N fn(zt)bn]2} where the new b0 and cst variables are defined as foows: (A20) b0¢ º b0 - ln l ; cst¢ º cst + ln l. Denote the soluon to (A19) as cst¢, b0¢, b1,..., bN. However, it n be seen that the soluon to (A19) is exy the same as the soluon to the inal prlem, (A2). Hence cst¢ = cst* and b0¢ = b0*. Thus the cst soluto (A18), which we detby cst, sasfies the foowing eas, where we have ssuted into equas (A): 68 We aume that the period s quany vtor qs remains the same if the period s prichae from ps to lps. 38 (A21) b0* º b0 - ln l ; cst* º cst + ln l. Using the second equain (A21), we have: (A22) cst = cst* - ln l. Hen (A23) P(lps,pt,qs,qt) º exp[cst] = exp[cst* - ln l] using (A) = l-1 exp[cst*] = l-1 P(ps,pt,qs,qt) using definon (A3) which estabshes the desired result (A5). Q.E.D. Note that in both Proposs 2 and 3, it is t nary that the weights wks and wkt sum to one for h period s and t. Proposition 4: The bateral heic price iex which msures price change going om period s to t, P(ps,pt,qs,qt) defined by (A3), and the bateral heic price iex which measures price change going om period t to s, P*(pt,ps,qt,qs) defined by (A9), sasfy the reversal test (A7). Proof: As usual, denote the soluon to (A2) as cst*, b0*, b1*,..., bN*. The nimization problem, which coess to reversing the ordering of the two periods, is (A) below and it has the soluon cts*, b0, b1,..., bN: (A24) n b's and c {åkÎS(s) wks[ln pks - cts - b0 - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - b0 - ån=1N fn(zt)bn]2} (A25) = n b's and c {åkÎS(s) wks[ln pks - b0¢ - ån=1N fn(zs)bn]2 + åkÎS(t) wkt[ln pkt - cts¢ - b0¢ - ån=1N fn(zt)bn]2} where we have defined the new variables b0¢ and cts¢ in terms of the old variables b0 and cts as foows: (A26) b0¢ º b0 + cts ; cst¢ º - cst . Denote the soluon to (A25) as cts*¢, b0¢, b1¢,..., bN¢. However, it n be seen that the soluon to (A25) is exy the same as the soluon to the inal prlem, (A2). Hence cts*¢ = cst* and b0¢ = b0*. Thus the cts soluto (A24), which we deted by cts*, sasfies the foowing eas, where we have ssuted into equas (A): (A27) b0* º b0 + cts* ; cst* º - cts*. 39 Using definon (A9), we have: (A28) P*(pt,ps,qt,qs) º exp[cts*] = exp[-cst*] using (A) = 1/exp[cst*] = 1/P(ps,pt,qs,qt) using definon (A3) which estabshes the desired result (A7). Q.E.D. Proposition 5: Let cst*, b0*, b1*,..., bN* denote the soluon to the weighted lst sares problem (A2). Then cst*, which is the logarhm of the bateral heic price iex P(ps,pt,qs,qt) defined by (A3), sasfies the foowing ea: (A29) [åkÎS(t) wkt] cst* = åkÎS(t) wkt ln pkt - åkÎS(s) wks ln pks - [åkÎS(t) wkt] b0* + [åkÎS(s) wks] b0* - åkÎS(t) wkt ån=1N fn(zt)bn* + åkÎS(s) wks ån=1N fn(zs)bn* = åkÎS(t) wkt [ln pkt - b0* - ån=1N fn(zt)bn*] - åkÎS(s) wks [ln pks - b0* - ån=1N fn(zt)bn*]. Proof: The soluon cst*, b0*, b1*,..., bN* to the nimization problem (A2) n be tained by alying lst sares to the foowing nr regreion mode (A30) (wks)1/2 ln pks = (wks)1/2[b0* + ån=1N fn(zs)bn*] + eks ; kÎS(s); (wkt)1/2 ln pkt = (wkt)1/2[cst* + b0* + ån=1N fn(zt)bn*] + ekt ; kÎS(t). We have inserted the mal lssares esators, cst*, b0*, b1*,..., bN*, into equas (A30) so that we can use these eas to define the least sares resialeks and ekt for the period s and t servas. It is well wn that the column vtor of these residuals is ortgonal to the columns of the X matrix, which coesto the exogenous variables on the right hand side of eas (A). These ortgonay relas aed to the columns that coespond to the cstant term b0 and the e duy variable cst give us the foowing 2 equas: (A31) 0 = åkÎS(s) wks ln pks + åkÎS(t) wkt ln pkt - [åkÎS(t) wkt] cst* - [åkÎS(s) wks] b0* - [åkÎS(t) wkt] b0* - åkÎS(s) wks ån=1N fn(zs)bn* - åkÎS(t) wkt ån=1N fn(zt)bn* ; (A32) 0 = åkÎS(t) wkt ln pkt - åkÎS(t) wkt cst* - åkÎS(t) wkt b0* - åkÎS(t) wkt ån=1N fn(zt)bn*. Equa(A32) n be rewren as: (A33) åkÎS(t) wkt ln pkt = [åkÎS(t) wkt]cst* + [åkÎS(t) wkt]b0* + åkÎS(t) wkt ån=1N fn(zt)bn*. Subtrng (A) from (A) lds to the foowing ea: (A34) åkÎS(s) wks ln pks = [åkÎS(s) wks]b0* + åkÎS(s) wks ån=1N fn(zs)bn*. 69 The two equaons in (A) are generaons of a simar formula derived by Triplett and McDonald (; ) in the weighted context. This weighted formula was also used by Triplett (). The tique of prused in this Prson was used in so4 of Diewert (). 40 Finaysubtrng (A) from (A) lds to (A29) after a bit of rangement. Q.E.D. Coroary 5.1: If the mels are idenl ring the two periods and the weights are also idenl ross perifor the same mel, then the heiprice iex P(ps,pt,qs,qt) defined by (A3) is eal to a weighted geometric mean of the mel price relaves, where the weights are prral to the coon model weights, wks = wkt º wk. Proof: Uer the stated hypotheses, the last 4 sets of terms on the right hand side of (A29) sum to zero and hence the logarhm of P(ps,pt,qs,qt) is eal to: (A35) cst* = åk=1K wk ln [pkt /pks]/[åj=1K wj] which estabshes the desired resuQ.E.D. Coroary 5.2: If the mels are idenl ring the two periods and the weight for model k is csen to be the arhmetiverage of the expeurshares on the mel for the two periods, (1/2)sks + (1/2)skt, then the heic price iex P(ps,pt,qs,qt) defined by (A3) is eal to the Törist () price iex. Proof: Aly (A) wh wk º (1/2)sks + (1/2)skt. Q.E.D. 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