Non-parametrc Estmates of Technology Usng Generalzed Addtve Models: Input Separablty and Regulaton n Canadan Cable Televson Morteza Haghr Memoral Unversty at Corner Brook Unversty Drve, Corner Brook, NL, A2H 6P9, Canada Tel: 1-709-637-6200 ext 6145 E-mal: mhaghr@mun.ca Stephen M. Law (Correspondng author) Department of Economcs, Mount Allson Unversty 144 Man Street, Sackvlle, NB, E4L 1A7, Canada Tel: 1-506-364-2355 E-mal: slaw@mta.ca James F. Nolan Department of Boresource Polcy, Busness and Economcs Unversty of Saskatchewan 51 Campus Drve, Saskatoon, SK, S7N 5A8, Canada Tel: 1-306-966-8412 E-mal: james.nolan@usask.ca Receved: August 11, 2011 Accepted: September 20, 2011 Publshed: January 1, 2012 do:10.5539/jef.v4n1p36 URL: http://dx.do.org/10.5539/jef.v4n1p36 Abstract We develop a non-parametrc cost functon usng generalzed addtve models and demonstrate how to test for nput separablty. Our emprcal example focuses on Canadan cable televson (CATV) provson. We estmate a new non-parametrc cost functon for ths ndustry usng fnancal and operatng data collected between 1990 and 1996. Ths perod s of partcular mportance from a polcy perspectve because CATV n Canada was a rate- and entry-regulated ndustry pror to 1997. The results show that the nput separablty assumpton holds and the generalzed addtve model yelds relable estmates of CATV technology durng ths tme perod. Keywords: Generalzed addtve models, Cable televson ndustry, Input separablty JEL classfcaton codes: C14, L50, L97 1. Introducton Emprcal estmates of producton, cost, and/or supply and demand functons can be dvded nto two major categores dependng on the purpose of the study. The frst category descrbes general results, whereby a problem s analysed by weakenng certan estmaton restrctons founded on economc axoms. In contrast, there are studes that mpose strong restrctons on functonal form, whch often obtan ad hoc results. Rchmond (2000) observed that the structure of most appled mcroeconomc studes calls for dsaggregated data. Ths mples that practtoners must mpose severe restrctons on ther models to make them consstent wth theoretcal prescrptons. As a consequence, many appled studes are n fact data-specfc, snce any modfcatons to the data can yeld dfferent results. In appled mcroeconomc studes of producton, separablty of nputs s a very commonly used restrcton n estmaton. Sono (1961) [frst publshed n Japanese n Kokumn Keza Zassh, LXXIV (1945), 1-51] and Leontef (1947a; 1947b) were the frst to dscuss the ssue of functonal separablty n producer (and consumer) theory. Smply put, a separable technology s a technology that can be dvded nto several stages. For example, n producton theory we assume that the underlyng technology s nherently weakly separable when we estmate producton functons. In consumer demand theory, the role of separablty s more mportant because t allows 36 ISSN 1916-971X E-ISSN 1916-9728
practtoners to group varous commodtes nto prmary budget categores. Snce emprcal results are often dependent on these crucal assumptons, t s almost always a good decson for practtoners to formally examne whether the assumpton of nput separablty holds. Here, we show an example where ths practce s crucal to develop a non-parametrc econometrc estmaton of technology. Ths partcular applcaton s also an extenson of earler research by Law & Nolan (2002), who sought to measure the mpact of regulaton by the Canadan Rado-televson and Telecommuncatons Commsson (CRTC) on the Canadan basc cable televson ndustry usng both parametrc and non-parametrc (data envelopment analyss) methods. We estmate a novel non-parametrc cost functon for the Canadan cable televson ndustry (CATV) usng a generalzed addtve model (GAM). The structure of Canadan CATV leads us to beleve that ths partcular specfcaton s approprate for cost estmaton. In addton, we explan the need to conduct a formal test for nput separablty wth respect to ths estmaton process. Ths paper has fve sectons. The next secton revews those studes examnng nput separablty usng non-parametrc regressons. Secton three provdes an explanaton of generalzed addtve models. Ths s followed by a dscusson of our chosen non-parametrc technque, known as splne smoothng. The technque s used to estmate a Canadan CATV cost functon constructed usng GAMs. Ths secton also motvates a statstcal test for the separablty of nputs n the model. Secton four presents and dscusses the fndngs of the estmates, ncludng the tests for nput separablty. Fnally, secton fve concludes the study and dscusses areas of future research. 2. Studes on Input Separablty Usng GAMs A number of recent studes dscuss aspects of the estmaton of generalzed addtve models n a context smlar to our research, but n most cases the estmaton processes used are somewhat dfferent than the methods to be used here. For example, Lnton & Nelsen (1995) proposed a method to dscrmnate between the addtve and multplcatve specfcatons of a mean response functon. Snce non-parametrc regressons have problems wth rate of convergence (to the true parameters) whenever ndependent varables number more than two, the authors used a smple kernel estmaton procedure to estmate the model. In turn, they estmated a unvarate predctor n both addtve and multplcatve non-parametrc regressons to examne the addtve structure of the model. Chen et al. (1996) proposed a method to test addtve separablty n a generalzed addtve model based on a Cobb-Douglas producton functon wth fve nputs. Ths model s superor to the Lnton-Nelsen model for two reasons: frst, more than two ndependent varables can be used n the model; and second, they developed an estmator for addtve models wth an explct hat matrx whch dd not rely on an teratve process. Gozalo & Lnton (2001) developed a semparametrc generalsed method of moments model n whch dscrete covarates were permtted. The authors were nterested n examnng the addtvty hypothess for the predctors snce ths structural assumpton provded a bass upon whch the model could be nterpreted, whle ts estmators converged wth reasonable convergence rates. Gozalo & Lnton were able to fnd the asymptotc dstrbuton of the parameter estmators and they derved a locally asymptotc dstrbuton for the addtvty test statstcs. They concluded that the predctors n the model were addtvely separable. However, the nput separablty statstcal test proposed by Gozalo-Lnton has some drawbacks. Specfcally, t s senstve to the choce of bandwdths and the degree of the asymptotc approxmatons beng used. Fnally, Haghr et al. (2004) examned the techncal effcency of dary farmng usng a generalzed addtve model. They estmated ther model usng a non-parametrc technque known as locally-weghted scatterplot smoothng (LOWESS), and specfed a producton functon to evaluate varaton n ther dependent varable. To examne the addtve separablty of nputs, the authors formulated a resdual devance analyss method and were able to conclude that labour and feed costs were ndeed addtvely separable. In the sprt of the recent research on ths topc, we wll address the queston of nput separablty n the Canadan CATV ndustry usng non-parametrc specfcatons for producton. In ths manner, we seek to advance appled econometrc research on measurement of separablty, as well as extend analyss of regulatory transton and effcency n a major ndustry for whch the network characterstc of producton s mportant. 3. Methodology 3.1 Generalzed Addtve Models (GAMs) The model developed for ths research bulds on earler studes. The method of GAMs, as proposed by Haste & Tbshran (1990) s extenson of generalzed lnear models, whch n turn, are extensons of classcal lnear models. A GAM mantans the addtvty assumpton for the predctors and relaxes the lnearty postulate (Haste & Tbshran, 1990). Suppose there are n observatons on a random response Y, whose mean varaton s measured by Publshed by Canadan Center of Scence and Educaton 37
X x1,..,xn predetermned and/or random varable covarates be wrtten as Y x11 x22... xkk or x. In ths case, multple regresson models can Y m, for =1,, n (1) where m x x,,..., k, 2 E 0, and Var. The key assumpton n equaton [1] s that the relatonshp between the expected value of Y and each of k-covarate elements of x s lnear and addtve (Haste & Tbshran, 1990). From Haste & Tbshran (1986), one way to relax the lnearty assumpton n equaton [1] s to use surface smoothers, whch can be thought of as non-parametrc estmates of the regresson model. A group of canddates for surface smoothers are kernel functons, but wth respect to fndng a local neghbourhood n k dmensons, these have major drawbacks. Perhaps the most mportant shortcomng of kernel functons s the curse of dmensonalty, whch precludes practtoners from ncludng more than two varables n the model (Haste & Tbshran, 1986). Ths problem can be avoded f the method of generalzed addtve models s used. Schmek & Turlach (2000) explan how a GAM s constructed n ths context. Consder equaton [2] E 1 k j j (2) j 1 k Y x x,...,x G f x G f x k n whch f x f jx j, s a constant, the f j s are arbtrary unvarate smooth functons, one for j 1 each predctor, and G (.) s a fxed lnk functon. The dstrbuton of Y follows an exponental famly, n a manner smlar to generalzed lnear models. To avod havng free constants n each of the functons f j, t s assumed that E f jx j 0. Ths requrement, whch s set n the range of 1 n and 1 j k, mples E f x and s necessary for the purpose of dentfcaton (Schmek & Turlach, 2000). Thus GAMs allow the condtonal mean of a response to be dependent on a sum of ndvdual unvarate functons, where each of them contans one predctor of the covarate matrx. By relaxng the lnearty assumpton n a GAM, the predctor s effects n equaton [2] mght be non-lnear, because the functons f are now arbtrary (Schmek & Turlach, 2000). j For ths paper, we employ a non-parametrc technque known as the splne smoothng approach. Ths was developed by Wahba (1990) and we use t to estmate the mean response functon modelled n our generalzed addtve models. Splne smoothng provdes a flexble methodology for fttng data n a nonparametrc regresson, and has been used n a wde varety of applcatons, ncludng analyss of growth data, medcal research, remote sensng experments, and economcs (e.g., Pagan & Ullah, 1999, pp. 91-93). 3.2 Splne Smoothng Suppose the prmary goal of equaton [1] s to estmate m from the sample data. A conventonal econometrc approach for estmatng m and the parameters of the regresson functon s to use the least squares method by mnmzng the resdual sum of squares RSS m y m x 1 assumed functonal form of m x x n 2 ˆ over all observatons n relaton to the. However, the problem wth usng the least squares approach s that there may not be a lnear relatonshp between the response and predctors (Wahba, 1990). One way to avod ths problem s to employ a Taylor-seres expanson, such as m x m x m x xx o xx 0 0 0 0 n whch m, an unknown functon, s at least twce dfferentable and there s a pont x close to some fxed pont x 0. Equaton [3] states that for x close to x 0, m follows a lnear model whose ntercept and slope are, m x m x x and 0 0 0 lnear, whch mples the slope m x 0, respectvely. Ths may occur n two extreme cases. Frst, f m s assumed to be 2 0 2 m x remans nvarant and the resdual term o x x 0 (3) s small, the latter 38 ISSN 1916-971X E-ISSN 1916-9728
beng an unrealstc assumpton (Wahba, 1990). Although ths case uses too lttle of the nformaton avalable n the data, t does provde a useful summary of the sample observatons and a satsfactory descrpton of the features n the data. Second, m could be assumed to have varyng slope, meanng that at each pont x the slopes of the lnes that connect each par of responses may be dfferent. Unlke the frst case, the latter demands too much nformaton, and t does not provde a useful summary of the data whle falng to delver a satsfactory descrpton of basc trends n the sample observatons. Note that ths falure can occur because of the regresson functon n equaton [3] rather than the random-nose component of the model (Wahba, 1990). To crcumvent ths problem, Wahba (1990) assumes that the rate of change n the slope of a functon m s gven by m. Snce ths slope vares from one pont to another, we can take the ntegral from the set of changes n the slope of the ftted functon. Ths generates equaton [4] x n m m x x1 2 dx (4) whch suggests a new approach that can account for the possblty that as the predctor changes, the slope may change rapdly (Wahba, 1990). Hence, we consder the followng functon m m, 0, RSS (5) that can be mnmsed over all admssble functons provded that they are twce dfferentable. In equaton [5], s called the smoothng parameter (or span degree) ndcatng the level of mportance placed on the structure of the functon gven that the slope of the ftted functon, to some extent, s flexble. By way of example, as approaches nfnty we obtan lnear regressons wth fxed slopes and, as approaches zero, we get lnear regressons wth flexble slopes (Wahba, 1990). Eubank (2000) showed that f m n equaton [4] s greater than or equal to two, then there exsts a unque and computable mnmzer for equaton [5], a technque known as cubc splne smoothng. Cubc splne smoothng estmators are lnear n the sense that one can fnd constants such as g x, 1,..., n for each estmaton pont x such that x g x y Cubc splne smoothng estmators can solve the problem of fttng regressons wth varant slopes, but at a cost. The problem concerns how an approprate smoothng degree s determned, gven a set of sample observatons. Consequently, cubc splne smoothng estmators are nconsstent and senstve to the choce of smoothng degree, meanng that the choce of span degree s data specfc. In order to mtgate ths ssue we employ the cross valdaton (CV) method of Stone (1974), a technque explaned n the next secton. 3.3 Model Specfcaton Consder a mult-varable cost functon, whch s seeks a relatonshp between a response and predctors. Such a model can be wrtten as Y f X, 1,2,..., n, and t 1,2,..., T. or t t f t k t n 1 X t + f j X jt t j=1 where Y t s total cost, f s an unknown functonal form of the cost relatonshp, X t s a multdmensonal seres k of nput prces and outputs wth real values,.e., X t and s a random error term, assumed to be t dstrbuted dentcally and ndependently wth zero mean and constant varance. In equaton [7], f must be estmated throughout a seres of common k regressors employed by each frm n the sample observatons. These are drawn ndependently from a populaton and can be estmated va non-parametrc methods, such as the aforementoned splne smoothng. Moreover, the dentfcaton condton descrbed earler stll holds. The estmaton process s descrbed below for two predctors. For smplcty of exposton, the subscrpts and t are dropped. Assume a smple generalzed addtve model such as that found n equaton [2], that can be wrtten as Furthermore, notce that E Y x,x f x,x f x f 1 2 1 2 1 1 + 2 x2 (8) (6) (7) Publshed by Canadan Center of Scence and Educaton 39
d 1 1 1 2 2 1 2 x 2 2 0 f x gx dx 0 2 2 2 2 f x f 1 1 f ˆ x, 1 x t 2 s some semparametrc estmator fx x By consderng the assumpton E f Note that Chen et al. (1996) show that f x f x, x g x x E f x, X (9), we obtan the followng result (Stone, 1974), (10) 1 can then be estmated by x T fˆ 1 1 x1, x t 2 T ˆ where,. Gven the assumpton that f s a twce-dfferentable 1 2 smooth functon, relable sem-parametrc estmators can be obtaned by usng the backfttng algorthm of Fredman & Stuetzle (1981), a technque that was subsequently modfed by Breman & Fredman (1985), as the teratve smoothng process. The latter algorthm frst estmates f x n equaton [8] and then, whle fxng the ftted 1 1 functon f x, estmates the mean regresson functon on x by smoothng the resdualy ˆ fˆ1 1 1 2 x 1, leadng to the estmaton of f x. Snce the backfttng algorthm s an teratve process, the next step s to mprove the 2 2 estmaton of f x by smoothng the resdual Y ˆ fˆ2 x 2 on x 1, whch, n turn, enhances the estmators 1 1 that are used to smooth the resdual Y ˆ fˆ1 x 1 on x 2 n the second step. Ths procedure contnues untl statstcally relable and effcent estmators are acheved. It s mportant to know that the algorthm needs an ntal estmaton of f x and hence an estmate of f ˆ x, and ths s provded usng the splne smoothng 1 1, 1 x t 2 technque. For those nterested, addtonal nformaton about the backfttng algorthm can be found n Schmek & Turlach (2000). The smoothng process s assocated wth the nconsstency and senstvty of the estmates to the choce of span degree. The CV method, whch provdes an optmal smoothng degree and thus generates consstent estmators, can solve ths problem (Stone, 1974). In partcular, gven equaton [9] and [10], the CV method mnmses (, t) ˆ fˆ 2 t1 Y (11) The estmaton process of fˆ follows two steps. In the frst step, for a fxed ndvdual data generator, 1, 2,...,n t and n every sequence of tme perod t, t 12,,..., T, one par of sample data,.e., the -th and t-th observatons, are put asde and the mean response functon f, defned n equaton [7] s re-estmated based on the n - 1 remanng observatons. In the second step, the algorthm s repeated and contnues to estmate f untl convergence. Others have also used ths algorthm for non-parametrc estmaton. In fact, Knep & Smar (1996, p. 192) refer to ths method as leavng out the observatony, x. t t 3.4 Statstcal Inference We use ths combned approach to nvestgate features of a Canadan cable televson cost functon. Lke most regressons, the estmaton of a cost functon usng GAMs has some drawbacks. From a theoretcal pont of vew, although GAMs relax the lnearty assumpton of generalsed lnear models, they mantan the addtvty structure of these types of models. Ths mples that the use of GAMs also assumes that the predctors used n the model must be addtvely separable. But as some have noted, the addtve separablty of the predctors could be a wrong approxmaton of the real functon (Knep & Smar, 1996, p. 209). In fact, the addtvty assumpton s not as mportant as separablty because the former s a feature of parametrc estmaton. And separablty n producton can be tested through examnaton of the estmated margnal rates of techncal substtuton (MTRS). However, the assumpton of nput separablty needs to be addressed n applcable non-parametrc regressons that use GAMs. Therefore to verfy the general applcablty of the results obtaned from these non-parametrc approaches to the estmaton of producton, we must construct a statstcal test to examne whether the nput separablty assumpton holds. 40 ISSN 1916-971X E-ISSN 1916-9728
Another major ptfall that occurs usng GAMs s a consequence of how these models are estmated, notng that ths problem also occurs wth other non-parametrc technques such as LOWESS and splne smoothng. In fact, fndng the optmal smoothng degree s data-specfc, meanng that the span degree wll necessarly be dfferent from one model to another. The latter s the prmary reason why the CV method s ncorporated nto ths study. Ultmately, gven that we need to make relatvely nnocuous assumptons wth respect to the structure of technology n the ndustry under analyss, we offer that the gans n estmaton and nference from employng a GAM over the alternatves argue for ts use n ths type of appled econometrc analyss. Input separablty n Canadan CATV wll be examned through an analyss of resdual devance. When performng estmates wth GAMs, analyss of devance can be used for statstcal nference (Bowman & Azzaln, 1997). Resdual devances obtaned from our generalzed addtve model estmates are aggregated nto a devance table. The value of the devance s the logarthm of the lkelhood rato (LR) and follows a ch-squared dstrbuton (Haste & Tbshran, 1990). Note as well that the value for resdual devance s dentcal to the value of the resdual sum of squares f only a sngle predctor s used (Schmek & Turlach, 2000). Resdual devance s obtaned by computng two LR statstcal tests (restrcted and unrestrcted). Equaton [12] shows the computed value of devance, ˆ D m ; ˆ= 2l max; m l ; m max and l ; m Tbshran (1990, p. 282) showed that m;ˆ n whch l ;m (12) ndcate the restrcted and unrestrcted LR values, respectvely. Haste & D has asymptotc degrees of freedom equal to the dfferences n the dmensons between the restrcted and unrestrcted models. Thus, a ch-square dstrbuton s a good asymptotc approxmaton for testng the applcablty of generalzed addtve models (Schmek & Turlach, 2000). 4. Emprcal Analyss Gven that GAMs are approprate to the estmaton of producton, we estmate a new non-parametrc cost functon for the Canadan cable televson ndustry coverng the perod between 1990 and 1996. In turn, the GAMs model s estmated usng splne smoothng technques. We specfy the cost functon as a functon of total output,.e., the total number of basc and non-basc cable subscrbers, as well as nput prces, ncludng the prces of labor, captal, and materals. In performng ths estmaton, we are also ndrectly assumng that standard dualty assumptons hold, so that a producton functon can be retreved va estmaton of ths cost functon. One reason for our nterest n nput separablty n the context of Canadan CATV s the mportance of nput choce as a consequence of ndustry regulaton. In partcular, the regulatory rules n ths ndustry at that tme provded for allowable prce ncreases based on the costs of nstallaton of addtonal captal for the provson of basc CATV servce. Ths aspect of the CATV regulatory ssue n Canada s related to concerns about rate of return regulaton because of the potental presence of the Averch-Johnson-Wellsz (AJW) effect (Averch & Johnson, 1962, and Wellsz, 1963). As s well known n the regulatory economcs lterature, the AJW effect dstorts the captal-labour rato for the output level chosen by the regulated frm. Whle t s possble that the regulated frm mght have chosen the same captal-labour rato whle provdng a dfferent level of servce, one nterpretaton of the AJW model s that regulated frms make a smultaneous choce of captal, labour and output that leads to some dstorton or productve neffcency when compared to the choces made by an dentcal unregulated frm. Although Law (1997, 2002) dentfed key features of the regulatory structure establshed by the CRTC for Canadan CATV that led to cost neffcences, and Henderson et al. (1992, p.22) suggested that the CRTC prcng structure for Canadan CATV resulted n sgnfcantly hgher captal expendtures than would be expected, to date there are no studes that conclusvely dentfy the AJW effect n ths canddate ndustry. Furthermore, theory tells us that f aggregated captal and labor nputs are not addtvely separable, then the margnal rate of techncal substtuton (MRTS) between these nputs depends on the amount of each nput chosen by the frm. Ths stuaton ncreases the possblty that regulatory prcng constrants wll dstort nput ratos from those levels that would have been chosen n the absence of regulaton. However f nput separablty n producton s confrmed, then the lkelhood that ths negatve effect of regulaton could have occurred s reduced (see also Law, 2008). We conclude that f the latter stuaton s dentfed emprcally, although regulaton as practced n Canadan CATV at that tme mght have led to other unfortunate consequences for economc effcency, ths regulaton would not lkely have been responsble for any dstorton of nput choce. Estmatng ths partcular non-parametrc cost functon for the Canadan cable televson ndustry allows for testng whether varous captal and labour nputs n ths ndustry were addtvely separable. Necessarly, we are studyng the technology of the regulated frms n the sample. Färe and Logan (1983) noted that to reconstruct the rate-of-return Publshed by Canadan Center of Scence and Educaton 41
regulated producton functon, t s necessary to have knowledge about the rate-of-return constrant as well as to know the rate-of-return regulated cost functon. In fact, we do not attempt to construct an assumed regulatory constrant that captures exactly the nteracton between the frm and the regulator, and f we were to make ths attempt all conclusons would be condtonal on the accuracy of the formulaton of the regulatory constrant. Instead, followng Nelson and Wohar (1983, 1987) among others, we present results and conclusons whch pertan to the operatons of the actual regulated frms rather than those of hypothetcal unregulated (but otherwse equvalent) frms. Gven the latter set of related studes n regulatory economcs, we offer that our chosen approach stll generates useful results about the mpact of regulatory constrants. Ths secton s dvded nto three sub-sectons. Frst, the sources and the general characterstcs of data are dscussed and we provde a summary of the descrptve analyss of the varables used n the model. The second sub-secton presents the estmaton results followed by statstcal nferences drawn from examnng the null hypotheses of separablty amongst nputs n the model. Fnally, the thrd sub-secton dscusses the fndngs from the estmated model and dscusses mplcatons of ths research. 4.1 Data and Varable Descrptons The non-parametrc cost functon we estmate for the Canadan cable televson ndustry uses an unbalanced panel of data drawn from several Canadan Rado-televson and Telecommuncatons Commsson (CRTC) databases. The CRTC databases contan nformaton on cable televson provders who operated n Canada between 1990 and 1996. Cable televson frms were assgned operatng areas by the CTRC, and each of these s referred to as a lcensed servce area (LSA). Each CATV operator was oblged to submt extensve data on operatons, whch was compled by the CRTC. The unbalanced panel contans 1,041 observatons related to 242 unque undertakng dentfcaton (UID) codes. A UID code s allocated to each LSA so that the number of the codes ndcates the number of ndvdual cable operatons n the sample. Durng the sample perod, a new UID code would also be ssued for an LSA whenever the dentty of the operatng CATV frm changed. We consder each of the LSAs as one data-generatng unt of CATV producton over the tme perod of the study. In addton, we do not pool any ndvdual operaton s nformaton n any sample perod from precedng years snce prevous work suggests that the data at each pont n tme (annual) s suffcent to reflect long run decsons n the ndustry (Law & Nolan, 2002, p. 234). Fnally, the rchness of the database allows us to specfy a relatonshp between total cost wth nput prces and the level of output for the entre sample of observatons. Interested readers can fnd comprehensve descrptons of the Canadan CATV ndustry, the CRTC and ts regulatons, and the structure of LSAs n Law (1997) or Law & Nolan (2002). Here we use dependent and ndependent varables smlar to the set used by Law & Nolan (2002), wth the excepton that the data for each ndependent varable are provded separately for basc and non-basc cable subscrbers. Subscrbers to basc servce receve a package of sgnals correspondng to the set of local televson broadcasts; subscrbers to non-basc servce also receve pay or dscretonary servces whch are packages of sgnals assembled by the CATV operator and nclude specalty servces such as sports channels, move channels, or communty programmng. Note that a subscrpton to basc servce s a condton of a subscrpton to non-basc servce. Table 1 presents statstcal summares of the varables. Operatng expenses, ncludng the cost of labor, are computed by takng the dfference between total expenses and those expenses ncurred for programmng. Total cost s obtaned by addng operatng expenses to the user cost of captal, for both basc and non-basc servces. Output s defned as the number of basc and non-basc subscrbers. The prce of labor s computed through a smple rato of total salares over the total number of staff employed to provde servces for basc and non-basc subscrbers wthn each of the LSAs. The rental prce of captal s found by addng deprecaton and fnancng rates. The former rate s derved from the CRTC database, whle the latter rate s obtaned by the summaton of a rsk free nterest rate and a rsk premum specfc to CATV operatons. The rsk free nterest rate s the average of monthly ten-year-plus bond rates for a fnancal year, and the product of the market rsk premum and a systematc rsk factor yelds the rsk premum for captal. Accordng to Law & Nolan (2002, p.247), the values of market rsk premum and the systematc rsk factor n Canada were equal to 0.042 and 0.60, respectvely. The prce of materals s the per unt expendture on ntermedate materal nputs, meanng that cost not allocated to expendtures for labor or for captal s dstrbuted on a per cable klometre and per channel bass for basc and for non-basc servce for all years. Fnally, all varables are computed per unt of output and transformed to logarthms n order to mtgate possble heteroskedastcty n the data. Insert Table 1 Here 42 ISSN 1916-971X E-ISSN 1916-9728
4.2 Estmaton Results & Statstcal Inference Usng the methodologes approprate to the research ssues as descrbed n the prevous sectons, we estmate a total cost functon for the Canadan CATV ndustry usng the 1990-1996 data. We construct the functon as n equaton [7], estmated wth a non-parametrc regresson usng the splne-smoothng approach. The regresson was run usng the R software package (see, Ihaka and Gentleman, 1996), whch s a more recent mplementaton of S-Plus (Verson 4.0). Table 2 reports these estmates of the prmary cost functon usng GAMs ths s our restrcted model. The adjusted R-squared for ths model s 98.6 per cent. All the coeffcents are statstcally sgnfcant at the 95 per cent confdence level, wth the excepton of the prce of labor shared n basc subscrbers, whch s statstcally dfferent from zero at the 0.069 level of sgnfcance (see Table 2). Insert Table 2 Here Next, usng the statstcal test descrbed n equaton [12], we check whether the non-parametrc cost functon estmated satsfes the mplct assumpton of addtve separablty of the predctors. Specfcally, we examne whether changes n one of the factor prces could affect the total cost through the output varable. From a polcy perspectve, regulators mght be nterested to know whether changes n the prce of labor shared by non-basc servces have any mpact on the total cost of provdng servces for basc subscrbers. Or, f the prce of captal shared n basc servces changes, does ths have any effect on the total cost through the component of output that s basc subscrbers? To answer these types of polcy questons along wth the econometrc ssue of nput separablty, we need to estmate an unrestrcted verson of the CATV cost functon n a smlar fashon to the restrcted model. To buld our unrestrcted verson of the CATV cost functon, a new explanatory varable must be produced usng the product of the par of nputs under analyss. By way of example, consder the steps requred to test the null hypothess of whether changes n the prce of labor shared by non-basc subscrbers could affect the total cost of provdng servces for the basc subscrbers. Frst, the new explanatory varable s obtaned from the product of basc subscrbers and the prce of labor shared n non-basc subscrbers. The dea behnd the ntroducton of the nteractve predctor s that f the number of basc CATV subscrbers nfluences the labor shared n non-basc subscrbers, t would affect the total cost of operatons. In other words, f the workloads requred by the non-basc subscrbers affect the amount of labor needed by basc subscrbers, then ths would have an effect on the total cost of operatons. Hence the presence of ths new varable would necessarly add more nformaton to the model, ndcatng that the orgnal par of explanatory varables was not addtvely separable. Fnally, usng the LR statstc as descrbed n Secton 3.4, the null hypothess of addtve separablty between predctors n the prmal non-parametrc regresson (the restrcted model) can be examned aganst the alternatve hypothess (the unrestrcted model). Table 3 presents the results of statstcal nference obtaned from examnng the null hypotheses. In total, we developed and estmated sx dfferent unrestrcted models of the cost functon and tested each one of them aganst the restrcted cost functon. The unrestrcted models have one factor n common whch dstngushes them from the restrcted model, and that s the presence of the nteractve predctor generated to examne the specfc null hypothess beng tested. For the sake of brevty, none of the coeffcent estmates of the unrestrcted models are shown here. Instead, the estmated values of the resdual devance used n the calculaton of the devance value, and, n turn, the LR statstc test are reported. The computed values of devance for all sx models range from zero to 2.018, whch are all less than the crtcal value of the ch-squared dstrbuton (3.84) at the 0.05 level of sgnfcance wth the approprate (one) degree of freedom. We conclude the tests show that the nteractve explanatory varable ntroduced n each of the sx unrestrcted models s not sgnfcantly dfferent from zero, meanng that the new predctor varables add no new nformaton to the ntal non-parametrc (restrcted) cost functon. Hence, we conclude that the addtve separablty assumpton between predctors for cost estmaton n Canadan CATV s vald. Insert Table 3 Here By desgn, our fndngs have both theoretcal and polcy relevance. From a theoretcal perspectve, nput separablty can be examned by studyng how the margnal rate of techncal substtuton s affected by the changes n another nput avalable n a thrd dmenson. By falng to reject the null hypothess on the ncluson of approprate nteracton varables, we conclude that any changes n the total number of basc subscrbers, or n the amount of labor requred by non-basc subscrbers, mght alter the span degree n the space of these nputs ndvdually but do not affect the span degree of basc subscrbers and labor shared n non-basc subscrbers together. In turn, the test shows that the margnal rate of techncal substtuton between these two partcular predctors does not respond to any changes mparted by the new varable, that s, the nteractve ndependent varable. From a polcy perspectve, the ablty to generate such results should be of nterest to all regulators, not just those n Canadan CATV. For example, dscoverng that a change n demand for labor for non-basc CATV servces attrbutable to changes n the Publshed by Canadan Center of Scence and Educaton 43
amount of labor needed for basc CATV subscrbers has no sgnfcant effect on the total cost of operatons leads to a better understandng of the relatonshp between nputs n ths ndustry (.e., that they are separable). Ultmately, understandng whch producton nputs are separable n a statstcal sense can better nform the choce among regulatory regmes for that ndustry (for nstance, a choce between rate of return vs. prce cap regulaton) that are more or less lkely to nfluence nput choces by the regulated frm. Fnally to properly assess the usefulness of the suggested methods, one must reman mndful of the followng theoretcal ssues: () not all technologes are separable, and () the concept of separablty s most easly descrbed n the context of contnuously dfferentable technologes (see Chambers, 1997, p. 42). We note that n the precedng sectons we assumed that the mean response varable would be smooth enough to be twce dfferentable, and ths s obvously a crucal assumpton for performng any non-parametrc regresson analyss. 4.3 Dscusson Our fndngs ndcate that Canadan CATV durng ths perod was characterzed by separable nputs. At frst glance, ths result mght seem surprsng, gven the nterconnected nature of CATV provson at that tme. However, consderng the combnaton of the underlyng technology of CATV servce hghly captal-ntensve networks and the nature of the prevalng CRTC regulatory structure, upon further reflecton our separablty fndng s perhaps not so startlng after all. It s mportant to note as well that these general results concur wth prevous research about the nature of the provson of CATV servces n Canada at that tme. Ultmately, a better understandng about the prevalence of nput separablty, especally n network-based utltes, wll serve to assst polcy makers concerned about the nature of producton processes n these often regulated ndustres, many of whch share the captal-ntensve characterstcs of cable televson systems. 5. Summary and Conclusons The purpose of ths research was to develop an example of the use of generalzed addtve models (GAMs) and show that the choce of non-parametrc estmaton technque need not mply an nablty to test structural characterstcs of the estmated model. We show that f the research goal s to estmate a cost functon wthn a GAM non-parametrc framework, one must proceed by evaluatng the addtve separablty condton for the nputs. In ths study, we have shown that labour and captal were addtvely separable n the producton of cable televson servces n Canada through the early to md 1990 s. Whle ths result that may seem counterntutve at frst gven the network nature of CATV provson, other factors (partcularly captal ntensty) seem to have exerted more nfluence on technologcal decsons n Canadan CATV at that tme than the partcular structure of CATV regulaton. Ths study suggests several avenues for research, especally n other regulated ndustres. It would be nterestng to test not only the exstence of separablty, but also substtutablty of regulated nputs. Testng whether nputs are used more as complements or substtutes would have partcular polcy relevance for many ndustres, ncludng CATV. Ths s the case, for example, because rate of return regulaton can take dfferent forms f returns are earned on an nput (captal) that s complementary to, rather than a substtute for, other nputs. Another area of future research would test separablty across output categores. Appled to our partcular example, we note that over the sample perod, the CRTC mantaned a rate system that ndcated the formula by whch CATV operators would be able to obtan rate ncreases on the bass of ncreases n the amount of captal they nstalled for the provson of basc servce. CATV frms were permtted to desgnate assets or declare what fractons of captal expendtures were used for basc servce. In fact, our use of the CRTC data assumes that there was no bas ntroduced by the ncentve to overstate the fracton of captal dedcated to the producton of basc servce. It would also be nterestng to test whether basc and non-basc assets n ths ndustry are ndeed separable. Such results would certanly have mplcatons for other regulated ndustres n whch artfcal reportng requrements create dstnct but non-ntutve categores for producton, ether for nputs or for outputs. References Averch, H., & L. Johnson. (1962). Behavour of the frm under regulatory constrant. Amercan Economc Revew, 52(5): 115-121. http://lnks.jstor.org/sc?sc=0002-8282%28196212%2952%3a5%3c1052%3abotfur%3e2.0.co%3b2-g Bowman, A. D., & A. Azzaln. (1997). Appled smoothng technques for data analyss: The kernel approach wth s-plus llustraton, Oxford: Oxford Unversty Press. Breman, L., & J. H. Fredman. (1985). Estmatng optmal transformatons for multple regresson and correlaton. Journal of Amercan Statstcs Assocaton, 80(391): 580-619. http://dx.do.org/10.2307/2288473 Chambers, R. G. (1997). Appled producton analyss: The dual approach, Cambrdge: Cambrdge Unversty Press. 44 ISSN 1916-971X E-ISSN 1916-9728
Chen, R., W. Hardle, O.B. Lnton, & E. Sevarance-Lossn. (1996). Nonparametrc estmaton of addtve separable regresson models: statstcal theory and computatonal aspects of smoothng, New York: Physca Verlag. Eubank, R. L. (2000). Splne regresson, n M. G. Schmek (ed.) Smoothng and regresson: approaches, computaton, and applcaton, New York: John-Wley and Sons. Färe, R., & J. Logan. (1983). The rate-of-return regulated frm: Cost and producton dualty. Bell Journal of Economcs, 14(2): 405-414. http://www.jstor.org/stable/3003642 Fredman, J. H., & W. Stuetzle. (1981). Projecton pursut regresson. Journal of Amercan Statstcs Assocaton, 76(376): 817-823. http://dx.do.org/10.2307/2287576 Gozalo, P. L., & O. B. Lnton. (2001). Testng addtvty n generalzed nonparametrc regresson models wth estmated parameters. Journal of Econometrcs, 104(1): 1-48. http://dx.do.org/10.1016/s0304-4076(01)00049-5 Haghr, M., J. F. Nolan, & K. C. Tran. (2004). Assessng the mpact of economc lberalsaton across countres: A comparson of dary ndustry effcency n Canada and the Unted States. Appled Economcs, 36(11): 1233-1243. http://dx.do.org/10.1080/0003684042000247406 Henderson, J., A. Elek, & C. Gbson. (1992). A fnancal and regulatory analyss of the Canadan cable televson ndustry to the year 2001, Toronto: KPMG Peat, Marwck, Stevenson and Kellog. Haste, T. J., & R. J. Tbshran. (1986). Generalzed Addtve Models. Statstcal Scence, 11(3): 297-318. http://dx.do.org/10.1214/ss/1177013604 Haste, T. J., & R. J. Tbshran. (1990). Generalzed addtve models, U.K.: Chapman & Hall/CRC. Ihaka, R., & R. Gentleman. (1996). R: A language for data analyss and graphcs. Journal of Computatonal and Graphcal Statstcs, 5(3): 299-314. http://dx.do.org/10.2307/1390807 Knep, A., & L. Smar. (1996). A general framework for fronter estmaton wth panel data. Journal of Productvty Analyss, 7(2-3): 187-212. http://dx.do.org/10.1007/bf00157041 Law, S. M. (1997). Economc polcy nteractons: Intellectual property rghts and competton polcy; exclusve lcensng and rate regulaton, PhD Dssertaton, Unversty of Toronto. Law, S. M. (2002). The problem of market sze for Canadan cable televson regulaton. Appled Economcs, 34(1): 87-99. http://dx.do.org/10.1080/00036840010025092 Law, S.M. (2008). Assessng evdence for the Averch-Johnson-Wellsz effect for regulated utltes, ACEA Papers and Proceedngs, Atlantc Canada Economcs Assocaton. http://www.unb.ca/econ/acea/papers_proceedngs.html Law, S. M., & J. F. Nolan. (2002). Measurng the mpact of regulaton: A study of Canadan basc cable televson. Revew of Industral Organzaton, 21(3): 231-249. http://dx.do.org/10.1023/a:1020445804113 Leontef, W. W. (1947a). A note on the nterrelaton of subsets of ndependent varables of a contnuous functon wth contnuous frst dervatves. Bulletn of the Amercan Mathematcal Socety, 53 (4): 343-350. http://dx.do.org/10.1090/s0002-9904-1947-08796-6 Leontef, W. W. (1947b). Introducton to a theory of the nternal structure of functonal relatonshps. Econometrca, 15(4): 361-373. http://dx.do.org/10.2307/1905335 Lnton, O. B., & J. P. Nelsen. (1995). A kernel method of estmatng structured nonparametrc regresson based on margnal ntegraton. Bometrka, 82(1): 93-100. http://dx.do.org/10.1093/bomet/82.1.93 Nelson, R.A., & M.E. Wohar. (1983). Regulaton, scale economes, and productvty n steam-electrc generaton. Internatonal Economc Revew, 24(1): 57-79. http://dx.do.org/10.2307/2526115 Nelson, R.A., & M.E. Wohar. (1987). A reply to regulaton, scale and productvty: A comment, Internatonal Economc Revew, 28(2): 535-539. http://dx.do.org/10.2307/2526742 Pagan, A., & A. Ullah. (1999). Nonparametrc econometrcs, Cambrdge: Cambrdge Unversty Press. Rchmond, J. (2000). Separablty and specfcaton tests, Dscusson Paper No.527, Unversty of Essex. Schmek, M. G., & B. A. Turlach. (2000). Addtve and generalzed addtve models, n M.G. Schmek (ed.) Smoothng and regresson: approaches, computaton, and applcaton, New York: John-Wley and Sons. Sono, M. (1961). The effect of prce changes on the demand and supply of separable goods. Internatonal Economc Revew, 2(3): 239-271. http://www.jstor.org/stable/2525430 Publshed by Canadan Center of Scence and Educaton 45
Stone, M. (1974). Cross-valdatory choce and assessment of statstcal predctons. Journal of the Royal Statstcal Socety, Seres B (Methodologcal) 36(2): 111-147. http://www.jstor.org.lbproxy.mta.ca/stable/2984809 Wahba, G. (1990). Splne models for observatonal data, Phladelpha: CBMS-NSF Regonal Conference Seres SIAM (Socety for Industral and Appled Mathematcs). Wellsz, S.H. (1963). Regulaton of natural gas ppelne companes: An economc analyss. Journal of Poltcal Economy, 71(1): 30-43. http://dx.do.org/10.1086/258732 Table 1. Statstcal descrpton of the varables Varable Mean S. D. Mn. Max. Total Costs ($ 000) 6,975.3 16,533.7 30.6 222,821.3 Prce of Labor Shared n Basc Subscrbers ($) 40,388.3 16,275.1 487.6 163,872.6 Prce of Labor Shared n Non-Basc Subscrbers ($) 30,790.8 18,458.6 90.3 177,500.0 Prce of Captal Shared n Basc Subscrbers 0.3211 0.2604 0.1267 7.6618 Prce of Captal Shared n Non-Basc Subscrbers 0.6045 1.3354 0.1202 40.8860 Prce of Materal Shared n Basc Subscrbers ($/km 000 per channel) 181.4 128.9 15.1 1,872.6 Prce of Materal Shared n Non-Basc Subscrbers ($/km 000 per channel) 161.3 143.6 10.2 1,655.9 Number of Basc Subscrbers 28,004.0 58,747.0 91.0 586,606.0 Source: Sample data. Number of Non-Basc Subscrbers 19,581.0 51,744.0 14.0 573,420.0 46 ISSN 1916-971X E-ISSN 1916-9728
Table 2. Estmaton results from GAM (restrcted model) Estmated Estmated Non-parametrc Predctors DF Rank F-value p-value Intercept 1 Prce of labor shared n basc subscrbers 1.000 1.000 3.302 0.0695 Prce of labor shared n non-basc subscrbers 1.747 9.000 2.2530 0.0171 Prce of captal shared n basc subscrbers 8.330 9.000 7.2560 0.0000 Prce of captal shared n non-basc subscrbers 7.806 9.000 11.1850 0.0000 Prce of materal shared n basc subscrbers 8.068 9.000 14.6930 0.0000 Prce of materal shared n non-basc subscrbers 7.106 9.000 11.0810 0.0000 Number of basc subscrbers 8.195 9.000 1720.933 0.0000 Number of non-basc subscrbers 1.000 1.000 98.394 0.0000 Source: Sample data. Resdual Devance 31.2258 Adjusted R-squared 0.986 Resdual Degrees of Freedom 996.7469 Number of observatons 1,041 Table 3. Estmaton results from GAM (unrestrcted models) Resdual Resdual Computed Value New Predctor Devance DF of Devance Model 1: Basc subscrbers & Prce of labor shared n non-basc subscrbers 31.0132 993.9400 0.4252 Model 2: Basc subscrbers & Prce of captal shared n non-basc subscrbers 30.0574 994.2724 0.3368 Model 3: Basc subscrbers & Prce of materal shared n non-basc subscrbers 30.2222 985.9330 2.0072 Model 4: Non-basc subscrbers & Prce of labor shared n basc subscrbers 30.2170 984.7400 2.0180 Model 5: Non-basc subscrbers & Prce of captal shared n basc subscrbers 31.2307 996.8204 0.0000 Model 6: Non-basc subscrbers & Prce of materal shared n basc subscrbers 31.2295 996.8045 0.0000 Source: Sample data. Publshed by Canadan Center of Scence and Educaton 47