HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT?

Transcription

1 11 HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT? An mportant assumpton of the classcal lnear regresson model (Assumpton 4) s that the dsturbances u appearng n the populaton regresson functon are homoscedastc; that s, they all have the same varance. In ths chapter we examne the valdty of ths assumpton and fnd out what happens f ths assumpton s not fulflled. As n Chapter 10, we seek answers to the followng questons: 1. What s the nature of heteroscedastcty?. What are ts consequences? 3. How does one detect t? 4. What are the remedal measures? 11.1 THE NATURE OF HETEROSCEDASTICITY As noted n Chapter 3, one of the mportant assumptons of the classcal lnear regresson model s that the varance of each dsturbance term u, condtonal on the chosen values of the explanatory varables, s some constant number equal to σ. Ths s the assumpton of homoscedastcty, or equal (homo) spread (scedastcty), that s, equal varance. Symbolcally, E ( u ) = σ = 1,,..., n (11.1.1) Dagrammatcally, n the two-varable regresson model homoscedastcty can be shown as n Fgure 3.4, whch, for convenence, s reproduced as 387

2 388 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL Densty Savngs Y β 1 + β X Income X FIGURE 11.1 Homoscedastc dsturbances. Densty Savngs Y β 1 + β X Income X FIGURE 11. Heteroscedastc dsturbances. Fgure As Fgure 11.1 shows, the condtonal varance of Y (whch s equal to that of u ), condtonal upon the gven X, remans the same regardless of the values taken by the varable X. In contrast, consder Fgure 11., whch shows that the condtonal varance of Y ncreases as X ncreases. Here, the varances of Y are not the same. Hence, there s heteroscedastcty. Symbolcally, E ( u ) = σ (11.1.) Notce the subscrpt of σ, whch remnds us that the condtonal varances of u ( = condtonal varances of Y ) are no longer constant. To make the dfference between homoscedastcty and heteroscedastcty clear, assume that n the two-varable model Y = β 1 + β X + u, Y represents savngs and X represents ncome. Fgures 11.1 and 11. show that as ncome ncreases, savngs on the average also ncrease. But n Fgure 11.1

3 CHAPTER ELEVEN: HETEROSCEDASTICITY 389 the varance of savngs remans the same at all levels of ncome, whereas n Fgure 11. t ncreases wth ncome. It seems that n Fgure 11. the hgherncome famles on the average save more than the lower-ncome famles, but there s also more varablty n ther savngs. There are several reasons why the varances of u may be varable, some of whch are as follows Followng the error-learnng models, as people learn, ther errors of behavor become smaller over tme. In ths case, σ s expected to decrease. As an example, consder Fgure 11.3, whch relates the number of typng errors made n a gven tme perod on a test to the hours put n typng practce. As Fgure 11.3 shows, as the number of hours of typng practce ncreases, the average number of typng errors as well as ther varances decreases.. As ncomes grow, people have more dscretonary ncome and hence more scope for choce about the dsposton of ther ncome. Hence, σ s lkely to ncrease wth ncome. Thus n the regresson of savngs on ncome one s lkely to fnd σ ncreasng wth ncome (as n Fgure 11.) because people have more choces about ther savngs behavor. Smlarly, companes wth larger profts are generally expected to show greater varablty n ther dvdend polces than companes wth lower profts. Also, growthorented companes are lkely to show more varablty n ther dvdend payout rato than establshed companes. 3. As data collectng technques mprove, σ s lkely to decrease. Thus, banks that have sophstcated data processng equpment are lkely to Densty Typng errors Y Hours of typng practce β 1 + β X FIGURE 11.3 Illustraton of heteroscedastcty. X 1 See Stefan Valavans, Econometrcs, McGraw-Hll, New York, 1959, p. 48. As Valavans puts t, Income grows, and people now barely dscern dollars whereas prevously they dscerned dmes, bd., p. 48.

4 390 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL commt fewer errors n the monthly or quarterly statements of ther customers than banks wthout such facltes. 4. Heteroscedastcty can also arse as a result of the presence of outlers. An outlyng observaton, or outler, s an observaton that s much dfferent (ether very small or very large) n relaton to the observatons n the sample. More precsely, an outler s an observaton from a dfferent populaton to that generatng the remanng sample observatons. 3 The ncluson or excluson of such an observaton, especally f the sample sze s small, can substantally alter the results of regresson analyss. As an example, consder the scattergram gven n Fgure Based on the data gven n exercse 11., ths fgure plots percent rate of change of stock prces (Y) and consumer prces (X) for the post World War II perod through 1969 for 0 countres. In ths fgure the observaton on Y and X for Chle can be regarded as an outler because the gven Y and X values are much larger than for the rest of the countres. In stuatons such as ths, t would be hard to mantan the assumpton of homoscedastcty. In exercse 11., you are asked to fnd out what happens to the regresson results f the observatons for Chle are dropped from the analyss Chle 10 Stock prces (% change) Consumer prces (% change) FIGURE 11.4 The relatonshp between stock prces and consumer prces. 3 I am ndebted to Mchael McAleer for pontng ths out to me.

5 CHAPTER ELEVEN: HETEROSCEDASTICITY Another source of heteroscedastcty arses from volatng Assumpton 9 of CLRM, namely, that the regresson model s correctly specfed. Although we wll dscuss the topc of specfcaton errors more fully n Chapter 13, very often what looks lke heteroscedastcty may be due to the fact that some mportant varables are omtted from the model. Thus, n the demand functon for a commodty, f we do not nclude the prces of commodtes complementary to or competng wth the commodty n queston (the omtted varable bas), the resduals obtaned from the regresson may gve the dstnct mpresson that the error varance may not be constant. But f the omtted varables are ncluded n the model, that mpresson may dsappear. As a concrete example, recall our study of advertsng mpressons retaned (Y) n relaton to advertsng expendture (X). (See exercse 8.3.) If you regress Y on X only and observe the resduals from ths regresson, you wll see one pattern, but f you regress Y on X and X, you wll see another pattern, whch can be seen clearly from Fgure We have already seen that X belongs n the model. (See exercse 8.3.) 6. Another source of heteroscedastcty s skewness n the dstrbuton of one or more regressors ncluded n the model. Examples are economc varables such as ncome, wealth, and educaton. It s well known that the dstrbuton of ncome and wealth n most socetes s uneven, wth the bulk of the ncome and wealth beng owned by a few at the top. 7. Other sources of heteroscedastcty: As Davd Hendry notes, heteroscedastcty can also arse because of (1) ncorrect data transformaton (e.g., rato or frst dfference transformatons) and () ncorrect functonal form (e.g., lnear versus log lnear models). 4 Note that the problem of heteroscedastcty s lkely to be more common n cross-sectonal than n tme seres data. In cross-sectonal data, one (a) (b) FIGURE 11.5 Resduals from the regresson of (a) mpressons of advertsng expendture and (b) mpresson on Adexp and Adexp. 4 Davd F. Hendry, Dynamc Econometrcs, Oxford Unversty Press, 1995, p. 45.

6 39 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL usually deals wth members of a populaton at a gven pont n tme, such as ndvdual consumers or ther famles, frms, ndustres, or geographcal subdvsons such as state, country, cty, etc. Moreover, these members may be of dfferent szes, such as small, medum, or large frms or low, medum, or hgh ncome. In tme seres data, on the other hand, the varables tend to be of smlar orders of magntude because one generally collects the data for the same entty over a perod of tme. Examples are GNP, consumpton expendture, savngs, or employment n the Unted States, say, for the perod 1950 to 000. As an llustraton of heteroscedastcty lkely to be encountered n crosssectonal analyss, consder Table Ths table gves data on compensaton per employee n 10 nondurable goods manufacturng ndustres, classfed by the employment sze of the frm or the establshment for the year Also gven n the table are average productvty fgures for nne employment classes. Although the ndustres dffer n ther output composton, Table 11.1 shows clearly that on the average large frms pay more than the small frms. TABLE 11.1 COMPENSATION PER EMPLOYEE ($) IN NONDURABLE MANUFACTURING INDUSTRIES ACCORDING TO EMPLOYMENT SIZE OF ESTABLISHMENT, 1958 Employment sze (average number of employees) Industry Food and kndred products Tobacco products Textle mll products Apparel and related products Paper and alled products Prntng and publshng Chemcals and alled products Petroleum and coal products Rubber and plastc products Leather and leather products Average compensaton Standard devaton Average productvty ,81 11,750 Source: The Census of Manufacturers, U.S. Department of Commerce, 1958 (computed by author).

7 CHAPTER ELEVEN: HETEROSCEDASTICITY Standard devaton Mean compensaton FIGURE 11.6 Standard devaton of compensaton and mean compensaton. As an example, frms employng one to four employees pad on the average about $3396, whereas those employng 1000 to 499 employees on the average pad about $4843. But notce that there s consderable varablty n earnng among varous employment classes as ndcated by the estmated standard devatons of earnngs. Ths can be seen also from Fgure 11.6, whch plots the standard devaton of compensaton and average compensaton n each employment class. As can be seen clearly, on average, the standard devaton of compensaton ncreases wth the average value of compensaton. 11. OLS ESTIMATION IN THE PRESENCE OF HETEROSCEDASTICITY What happens to OLS estmators and ther varances f we ntroduce heteroscedastcty by lettng E(u ) = σ but retan all other assumptons of the classcal model? To answer ths queston, let us revert to the two-varable model: Y = β 1 + β X + u Applyng the usual formula, the OLS estmator of β s ˆβ = x y x = n X Y X Y n X ( X ) (11..1)

8 394 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL but ts varance s now gven by the followng expresson (see Appendx 11A, Secton 11A.1): var ( ˆβ ) = x σ ( x ) (11..) whch s obvously dfferent from the usual varance formula obtaned under the assumpton of homoscedastcty, namely, var ( ˆβ ) = σ x (11..3) Of course, f σ = σ for each, the two formulas wll be dentcal. (Why?) Recall that ˆβ s best lnear unbased estmator (BLUE) f the assumptons of the classcal model, ncludng homoscedastcty, hold. Is t stll BLUE when we drop only the homoscedastcty assumpton and replace t wth the assumpton of heteroscedastcty? It s easy to prove that ˆβ s stll lnear and unbased. As a matter of fact, as shown n Appendx 3A, Secton 3A., to establsh the unbasedness of ˆβ t s not necessary that the dsturbances (u ) be homoscedastc. In fact, the varance of u, homoscedastc or heteroscedastc, plays no part n the determnaton of the unbasedness property. Recall that n Appendx 3A, Secton 3A.7, we showed that ˆβ s a consstent estmator under the assumptons of the classcal lnear regresson model. Although we wll not prove t, t can be shown that ˆβ s a consstent estmator despte heteroscedastcty; that s, as the sample sze ncreases ndefntely, the estmated β converges to ts true value. Furthermore, t can also be shown that under certan condtons (called regularty condtons), ˆβ s asymptotcally normally dstrbuted. Of course, what we have sad about ˆβ also holds true of other parameters of a multple regresson model. Granted that ˆβ s stll lnear unbased and consstent, s t effcent or best ; that s, does t have mnmum varance n the class of unbased estmators? And s that mnmum varance gven by Eq. (11..)? The answer s no to both the questons: ˆβ s no longer best and the mnmum varance s not gven by (11..). Then what s BLUE n the presence of heteroscedastcty? The answer s gven n the followng secton THE METHOD OF GENERALIZED LEAST SQUARES (GLS) Why s the usual OLS estmator of β gven n (11..1) not best, although t s stll unbased? Intutvely, we can see the reason from Table As the table shows, there s consderable varablty n the earnngs between employment classes. If we were to regress per-employee compensaton on the sze of employment, we would lke to make use of the knowledge that there s consderable nterclass varablty n earnngs. Ideally, we would lke to devse

9 CHAPTER ELEVEN: HETEROSCEDASTICITY 395 the estmatng scheme n such a manner that observatons comng from populatons wth greater varablty are gven less weght than those comng from populatons wth smaller varablty. Examnng Table 11.1, we would lke to weght observatons comng from employment classes and 0 49 more heavly than those comng from employment classes lke 5 9 and , for the former are more closely clustered around ther mean values than the latter, thereby enablng us to estmate the PRF more accurately. Unfortunately, the usual OLS method does not follow ths strategy and therefore does not make use of the nformaton contaned n the unequal varablty of the dependent varable Y, say, employee compensaton of Table 11.1: It assgns equal weght or mportance to each observaton. But a method of estmaton, known as generalzed least squares (GLS), takes such nformaton nto account explctly and s therefore capable of producng estmators that are BLUE. To see how ths s accomplshed, let us contnue wth the now-famlar two-varable model: Y = β 1 + β X + u (11.3.1) whch for ease of algebrac manpulaton we wrte as Y = β 1 X 0 + β X + u (11.3.) where X 0 = 1 for each. The reader can see that these two formulatons are dentcal. Now assume that the heteroscedastc varances σ are known. Dvde (11.3.) through by σ to obtan ( ) ( ) ( ) Y X0 X u = β 1 + β + (11.3.3) σ σ σ σ whch for ease of exposton we wrte as Y * = β 1 * X* 0 + β* X* + u* (11.3.4) where the starred, or transformed, varables are the orgnal varables dvded by (the known) σ. We use the notaton β1 and β, the parameters of the transformed model, to dstngush them from the usual OLS parameters β 1 and β. What s the purpose of transformng the orgnal model? To see ths, notce the followng feature of the transformed error term u : var (u * ) = E(u* ) = E = 1 σ ( ) u σ E ( u ) = 1 ( ) σ σ = 1 snce σ s known snce E ( u ) = σ (11.3.5)

10 396 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL whch s a constant. That s, the varance of the transformed dsturbance term u * s now homoscedastc. Snce we are stll retanng the other assumptons of the classcal model, the fndng that t s u * that s homoscedastc suggests that f we apply OLS to the transformed model (11.3.3) t wll produce estmators that are BLUE. In short, the estmated β1 and β * are now BLUE and not the OLS estmators ˆβ 1 and ˆβ. Ths procedure of transformng the orgnal varables n such a way that the transformed varables satsfy the assumptons of the classcal model and then applyng OLS to them s known as the method of generalzed least squares (GLS). In short, GLS s OLS on the transformed varables that satsfy the standard least-squares assumptons. The estmators thus obtaned are known as GLS estmators, and t s these estmators that are BLUE. The actual mechancs of estmatng β 1 * and β* are as follows. Frst, we wrte down the SRF of (11.3.3) or Y σ = ˆβ * 1 ( X0 σ ) + ˆβ * ( X σ ) + Now, to obtan the GLS estmators, we mnmze ) (û σ Y * = ˆβ * 1 X* 0 + ˆβ * X* +û* (11.3.6) that s, ( û σ û* = (Y * ˆβ * 1 X* 0 ˆβ * X* ) ) = [( ) Y ˆβ 1 * σ ( X0 σ ) ˆβ * ( X σ )] (11.3.7) The actual mechancs of mnmzng (11.3.7) follow the standard calculus technques and are gven n Appendx 11A, Secton 11A.. As shown there, the GLS estmator of β * s ( )( ) ( )( ) ˆβ * = w w X Y w X w Y ( )( ) ( ) w w X w X (11.3.8) and ts varance s gven by w var ( ˆβ * ) = ( )( ) ( ) w w X w X (11.3.9) where w = 1/σ.

11 CHAPTER ELEVEN: HETEROSCEDASTICITY 397 Dfference between OLS and GLS Recall from Chapter 3 that n OLS we mnmze û = (Y ˆβ 1 ˆβ X ) ( ) but n GLS we mnmze the expresson (11.3.7), whch can also be wrtten as w û = w (Y ˆβ * 1 X 0 ˆβ * X ) ( ) where w = 1/σ [verfy that ( ) and (11.3.7) are dentcal]. Thus, n GLS we mnmze a weghted sum of resdual squares wth w = 1/σ actng as the weghts, but n OLS we mnmze an unweghted or (what amounts to the same thng) equally weghted RSS. As (11.3.7) shows, n GLS the weght assgned to each observaton s nversely proportonal to ts σ, that s, observatons comng from a populaton wth larger σ wll get relatvely smaller weght and those from a populaton wth smaller σ wll get proportonately larger weght n mnmzng the RSS ( ). To see the dfference between OLS and GLS clearly, consder the hypothetcal scattergram gven n Fgure In the (unweghted) OLS, each û assocated wth ponts A, B, and C wll receve the same weght n mnmzng the RSS. Obvously, n ths case the û assocated wth pont C wll domnate the RSS. But n GLS the extreme observaton C wll get relatvely smaller weght than the other two observatons. As noted earler, ths s the rght strategy, for n estmatng the Y C u Y = β1 + β X u{ A B u 0 X FIGURE 11.7 Hypothetcal scattergram.

12 398 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL populaton regresson functon (PRF) more relably we would lke to gve more weght to observatons that are closely clustered around ther (populaton) mean than to those that are wdely scattered about. Snce ( ) mnmzes a weghted RSS, t s approprately known as weghted least squares (WLS), and the estmators thus obtaned and gven n (11.3.8) and (11.3.9) are known as WLS estmators. But WLS s just a specal case of the more general estmatng technque, GLS. In the context of heteroscedastcty, one can treat the two terms WLS and GLS nterchangeably. In later chapters we wll come across other specal cases of GLS. In passng, note that f w = w, a constant for all, ˆβ * s dentcal wth ˆβ and var ( ˆβ * ) s dentcal wth the usual (.e., homoscedastc) var ( ˆβ ) gven n (11..3), whch should not be surprsng. (Why?) (See exercse 11.8.) 11.4 CONSEQUENCES OF USING OLS IN THE PRESENCE OF HETEROSCEDASTICITY As we have seen, both ˆβ * and ˆβ are (lnear) unbased estmators: In repeated samplng, on the average, ˆβ * and ˆβ wll equal the true β ; that s, they are both unbased estmators. But we know that t s ˆβ * that s effcent, that s, has the smallest varance. What happens to our confdence nterval, hypotheses testng, and other procedures f we contnue to use the OLS estmator ˆβ? We dstngush two cases. OLS Estmaton Allowng for Heteroscedastcty Suppose we use ˆβ and use the varance formula gven n (11..), whch takes nto account heteroscedastcty explctly. Usng ths varance, and assumng σ are known, can we establsh confdence ntervals and test hypotheses wth the usual t and F tests? The answer generally s no because t can be shown that var ( ˆβ *) var ( ˆβ ), 5 whch means that confdence ntervals based on the latter wll be unnecessarly larger. As a result, the t and F tests are lkely to gve us naccurate results n that var ( ˆβ ) s overly large and what appears to be a statstcally nsgnfcant coeffcent (because the t value s smaller than what s approprate) may n fact be sgnfcant f the correct confdence ntervals were establshed on the bass of the GLS procedure. OLS Estmaton Dsregardng Heteroscedastcty The stuaton can become serous f we not only use ˆβ but also contnue to use the usual (homoscedastc) varance formula gven n (11..3) even f heteroscedastcty s present or suspected: Note that ths s the more lkely 5 A formal proof can be found n Phoebus J. Dhrymes, Introductory Econometrcs, Sprnger- Verlag, New York, 1978, pp In passng, note that the loss of effcency of ˆβ [.e., by how much var ( ˆβ ) exceeds var ( ˆβ * )] depends on the sample values of the X varables and the value of σ.

13 CHAPTER ELEVEN: HETEROSCEDASTICITY 399 case of the two we dscuss here, because runnng a standard OLS regresson package and gnorng (or beng gnorant of) heteroscedastcty wll yeld varance of ˆβ as gven n (11..3). Frst of all, var ( ˆβ ) gven n (11..3) s a based estmator of var ( ˆβ ) gven n (11..), that s, on the average t overestmates or underestmates the latter, and n general we cannot tell whether the bas s postve (overestmaton) or negatve (underestmaton) because t depends on the nature of the relatonshp between σ and the values taken by the explanatory varable X, as can be seen clearly from (11..) (see exercse 11.9). The bas arses from the fact that ˆσ, the conventonal estmator of σ, namely, û /(n ) s no longer an unbased estmator of the latter when heteroscedastcty s present (see Appendx 11A.3). As a result, we can no longer rely on the conventonally computed confdence ntervals and the conventonally employed t and F tests. 6 In short, f we persst n usng the usual testng procedures despte heteroscedastcty, whatever conclusons we draw or nferences we make may be very msleadng. To throw more lght on ths topc, we refer to a Monte Carlo study conducted by Davdson and MacKnnon. 7 They consder the followng smple model, whch n our notaton s Y = β 1 + β X + u (11.4.1) They assume that β 1 = 1, β = 1, and u N(0, X α ). As the last expresson shows, they assume that the error varance s heteroscedastc and s related to the value of the regressor X wth power α. If, for example, α = 1, the error varance s proportonal to the value of X; f α =, the error varance s proportonal to the square of the value of X, and so on. In Secton 11.6 we wll consder the logc behnd such a procedure. Based on 0,000 replcatons and allowng for varous values for α, they obtan the standard errors of the two regresson coeffcents usng OLS [see Eq. (11..3)], OLS allowng for heteroscedastcty [see Eq. (11..)], and GLS [see Eq. (11.3.9)]. We quote ther results for selected values of α: Standard error of ˆβ 1 Standard error of ˆβ Value of α OLS OLS het GLS OLS OLS het GLS Note: OLS het means OLS allowng for heteroscedastcty. 6 From (5.3.6) we know that the 100(1 α)% confdence nterval for β s [ ˆβ ± t α/ se ( ˆβ )]. But f se ( ˆβ ) cannot be estmated unbasedly, what trust can we put n the conventonally computed confdence nterval? 7 Russell Davdson and James G. MacKnnon, Estmaton and Inference n Econometrcs, Oxford Unversty Press, New York, 1993, pp

14 400 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL The most strkng feature of these results s that OLS, wth or wthout correcton for heteroscedastcty, consstently overestmates the true standard error obtaned by the (correct) GLS procedure, especally for large values of α, thus establshng the superorty of GLS. These results also show that f we do not use GLS and rely on OLS allowng for or not allowng for heteroscedastcty the pcture s mxed. The usual OLS standard errors are ether too large (for the ntercept) or generally too small (for the slope coeffcent) n relaton to those obtaned by OLS allowng for heteroscedastcty. The message s clear: In the presence of heteroscedastcty, use GLS. However, for reasons explaned later n the chapter, n practce t s not always easy to apply GLS. Also, as we dscuss later, unless heteroscedastcty s very severe, one may not abandon OLS n favor of GLS or WLS. From the precedng dscusson t s clear that heteroscedastcty s potentally a serous problem and the researcher needs to know whether t s present n a gven stuaton. If ts presence s detected, then one can take correctve acton, such as usng the weghted least-squares regresson or some other technque. Before we turn to examnng the varous correctve procedures, however, we must frst fnd out whether heteroscedastcty s present or lkely to be present n a gven case. Ths topc s dscussed n the followng secton. A Techncal Note Although we have stated that, n cases of heteroscedastcty, t s the GLS, not the OLS, that s BLUE, there are examples where OLS can be BLUE, despte heteroscedastcty. 8 But such examples are nfrequent n practce DETECTION OF HETEROSCEDASTICITY As wth multcollnearty, the mportant practcal queston s: How does one know that heteroscedastcty s present n a specfc stuaton? Agan, as n the case of multcollnearty, there are no hard-and-fast rules for detectng heteroscedastcty, only a few rules of thumb. But ths stuaton s nevtable because σ can be known only f we have the entre Y populaton correspondng to the chosen X s, such as the populaton shown n Table.1 or Table But such data are an excepton rather than the rule n most 8 The reason for ths s that the Gauss Markov theorem provdes the suffcent (but not necessary) condton for OLS to be effcent. The necessary and suffcent condton for OLS to be BLUE s gven by Kruskal s Theorem. But ths topc s beyond the scope of ths book. I am ndebted to Mchael McAleer for brngng ths to my attenton. For further detals, see Denzl G. Febg, Mchael McAleer, and Robert Bartels, Propertes of Ordnary Least Squares Estmators n Regresson Models wth Nonsphercal Dsturbances, Journal of Econometrcs, vol. 54, No. 1 3, Oct. Dec., 199, pp For the mathematcally nclned student, I dscuss ths topc further n App. C, usng matrx algebra.

15 CHAPTER ELEVEN: HETEROSCEDASTICITY 401 economc nvestgatons. In ths respect the econometrcan dffers from scentsts n felds such as agrculture and bology, where researchers have a good deal of control over ther subjects. More often than not, n economc studes there s only one sample Y value correspondng to a partcular value of X. And there s no way one can know σ from just one Y observaton. Therefore, n most cases nvolvng econometrc nvestgatons, heteroscedastcty may be a matter of ntuton, educated guesswork, pror emprcal experence, or sheer speculaton. Wth the precedng caveat n mnd, let us examne some of the nformal and formal methods of detectng heteroscedastcty. As the followng dscusson wll reveal, most of these methods are based on the examnaton of the OLS resduals û snce they are the ones we observe, and not the dsturbances u. One hopes that they are good estmates of u, a hope that may be fulflled f the sample sze s farly large. Informal Methods Nature of the Problem Very often the nature of the problem under consderaton suggests whether heteroscedastcty s lkely to be encountered. For example, followng the poneerng work of Pras and Houthakker on famly budget studes, where they found that the resdual varance around the regresson of consumpton on ncome ncreased wth ncome, one now generally assumes that n smlar surveys one can expect unequal varances among the dsturbances. 9 As a matter of fact, n cross-sectonal data nvolvng heterogeneous unts, heteroscedastcty may be the rule rather than the excepton. Thus, n a cross-sectonal analyss nvolvng the nvestment expendture n relaton to sales, rate of nterest, etc., heteroscedastcty s generally expected f small-, medum-, and large-sze frms are sampled together. As a matter of fact, we have already come across examples of ths. In Chapter we dscussed the relatonshp between mean, or average, hourly wages n relaton to years of schoolng n the Unted States. In that chapter we also dscussed the relatonshp between expendture on food and total expendture for 55 famles n Inda (see exercse 11.16). Graphcal Method If there s no a pror or emprcal nformaton about the nature of heteroscedastcty, n practce one can do the regresson analyss on the assumpton that there s no heteroscedastcty and then do a postmortem examnaton of the resdual squared û to see f they exhbt any systematc pattern. Although û are not the same thng as u, they can be 9 S. J. Pras and H. S. Houthakker, The Analyss of Famly Budgets, Cambrdge Unversty Press, New York, 1955.

16 40 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL u u u Y Y Y (a) (b) (c) u u 0 Y 0 (d) (e) Y FIGURE 11.8 Hypothetcal patterns of estmated squared resduals. used as proxes especally f the sample sze s suffcently large. 10 An examnaton of the û may reveal patterns such as those shown n Fgure In Fgure 11.8, û are plotted aganst Ŷ, the estmated Y from the regresson lne, the dea beng to fnd out whether the estmated mean value of Y s systematcally related to the squared resdual. In Fgure 11.8a we see that there s no systematc pattern between the two varables, suggestng that perhaps no heteroscedastcty s present n the data. Fgure 11.8b to e, however, exhbts defnte patterns. For nstance, Fgure 11.8c suggests a lnear relatonshp, whereas Fgure 11.8d and e ndcates a quadratc relatonshp between û and Ŷ. Usng such knowledge, albet nformal, one may transform the data n such a manner that the transformed data do not exhbt heteroscedastcty. In Secton 11.6 we shall examne several such transformatons. Instead of plottng û aganst Ŷ, one may plot them aganst one of the explanatory varables, especally f plottng û aganst Ŷ results n the pattern shown n Fgure 11.8a. Such a plot, whch s shown n Fgure 11.9, may reveal patterns smlar to those gven n Fgure (In the case of the two-varable model, plottng û aganst Ŷ s equvalent to plottng t aganst 10 For the relatonshp between û and u, see E. Malnvaud, Statstcal Methods of Econometrcs, North Holland Publshng Company, Amsterdam, 1970, pp

17 CHAPTER ELEVEN: HETEROSCEDASTICITY 403 u u u X X X (a) (b) (c) u u 0 X 0 (d) FIGURE 11.9 Scattergram of estmated squared resduals aganst X. (e) X X, and therefore Fgure 11.9 s smlar to Fgure But ths s not the stuaton when we consder a model nvolvng two or more X varables; n ths nstance, û may be plotted aganst any X varable ncluded n the model.) A pattern such as that shown n Fgure 11.9c, for nstance, suggests that the varance of the dsturbance term s lnearly related to the X varable. Thus, f n the regresson of savngs on ncome one fnds a pattern such as that shown n Fgure 11.9c, t suggests that the heteroscedastc varance may be proportonal to the value of the ncome varable. Ths knowledge may help us n transformng our data n such a manner that n the regresson on the transformed data the varance of the dsturbance s homoscedastc. We shall return to ths topc n the next secton. Formal Methods Park Test 11 Park formalzes the graphcal method by suggestng that σ s some functon of the explanatory varable X. The functonal form he 11 R. E. Park, Estmaton wth Heteroscedastc Error Terms, Econometrca, vol. 34, no. 4, October 1966, p The Park test s a specal case of the general test proposed by A. C. Harvey n Estmatng Regresson Models wth Multplcatve Heteroscedastcty, Econometrca, vol. 44, no. 3, 1976, pp

18 404 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL suggested was or σ = σ X β ev ln σ = ln σ + β ln X + v (11.5.1) where v s the stochastc dsturbance term. Snce σ s generally not known, Park suggests usng û as a proxy and runnng the followng regresson: ln û = ln σ + β ln X + v (11.5.) = α + β ln X + v If β turns out to be statstcally sgnfcant, t would suggest that heteroscedastcty s present n the data. If t turns out to be nsgnfcant, we may accept the assumpton of homoscedastcty. The Park test s thus a twostage procedure. In the frst stage we run the OLS regresson dsregardng the heteroscedastcty queston. We obtan û from ths regresson, and then n the second stage we run the regresson (11.5.). Although emprcally appealng, the Park test has some problems. Goldfeld and Quandt have argued that the error term v enterng nto (11.5.) may not satsfy the OLS assumptons and may tself be heteroscedastc. 1 Nonetheless, as a strctly exploratory method, one may use the Park test. EXAMPLE 11.1 RELATIONSHIP BETWEEN COMPENSATION AND PRODUCTIVITY To llustrate the Park approach, we use the data gven n Table 11.1 to run the followng regresson: Y = β 1 + β X + u where Y = average compensaton n thousands of dollars, X = average productvty n thousands of dollars, and = th employment sze of the establshment. The results of the regresson were as follows: Ŷ = X se = ( ) (0.0998) (11.5.3) t = (.175) (.333) R = The results reveal that the estmated slope coeffcent s sgnfcant at the 5 percent level on the bass of a one-tal t test. The equaton shows that as labor productvty ncreases by, say, a dollar, labor compensaton on the average ncreases by about 3 cents. The resduals obtaned from regresson (11.5.3) were regressed on X as suggested n Eq. (11.5.), gvng the followng results: ln û = ln X se = (38.319) (4.16) (11.5.4) t = (0.934) ( 0.667) R = Obvously, there s no statstcally sgnfcant relatonshp between the two varables. Followng the Park test, one may conclude that there s no heteroscedastcty n the error varance Stephen M. Goldfeld and Rchard E. Quandt, Nonlnear Methods n Econometrcs, North Holland Publshng Company, Amsterdam, 197, pp The partcular functonal form chosen by Park s only suggestve. A dfferent functonal form may reveal sgnfcant relatonshps. For example, one may use û nstead of ln û as the dependent varable.

19 CHAPTER ELEVEN: HETEROSCEDASTICITY 405 Glejser Test 14 The Glejser test s smlar n sprt to the Park test. After obtanng the resduals û from the OLS regresson, Glejser suggests regressng the absolute values of û on the X varable that s thought to be closely assocated wth σ. In hs experments, Glejser used the followng functonal forms: û = β 1 + β X + v û = β 1 + β X + v û = β 1 + β 1 X + v û = β 1 + β 1 X + v û = β 1 + β X + v û = β 1 + β X + v where v s the error term. Agan as an emprcal or practcal matter, one may use the Glejser approach. But Goldfeld and Quandt pont out that the error term v has some problems n that ts expected value s nonzero, t s serally correlated (see Chapter 1), and roncally t s heteroscedastc. 15 An addtonal dffculty wth the Glejser method s that models such as û = β 1 + β X + v and û = β 1 + β X + v are nonlnear n the parameters and therefore cannot be estmated wth the usual OLS procedure. Glejser has found that for large samples the frst four of the precedng models gve generally satsfactory results n detectng heteroscedastcty. As a practcal matter, therefore, the Glejser technque may be used for large samples and may be used n the small samples strctly as a qualtatve devce to learn somethng about heteroscedastcty. 14 H. Glejser, A New Test for Heteroscedastcty, Journal of the Amercan Statstcal Assocaton, vol. 64, 1969, pp For detals, see Goldfeld and Quandt, op. ct., Chap. 3.

20 406 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL EXAMPLE 11. RELATIONSHIP BETWEEN COMPENSATION AND PRODUCTIVITY: THE GLEJSER TEST Contnung wth Example 11.1, the absolute value of the resduals obtaned from regresson (11.5.3) were regressed on average productvty (X ), gvng the followng results: û = X se = ( ) (0.0675) r = (11.5.5) t = (0.643) ( 0.301) As you can see from ths regresson, there s no relatonshp between the absolute value of the resduals and the regressor, average productvty. Ths renforces the concluson based on the Park test. Spearman s Rank Correlaton Test. Spearman s rank correlaton coeffcent as [ d ] r s = 1 6 n(n 1) In exercse 3.8 we defned the (11.5.6) where d = dfference n the ranks assgned to two dfferent characterstcs of the th ndvdual or phenomenon and n = number of ndvduals or phenomena ranked. The precedng rank correlaton coeffcent can be used to detect heteroscedastcty as follows: Assume Y = β 0 + β 1 X + u. Step 1. Ft the regresson to the data on Y and X and obtan the resduals û. Step. Ignorng the sgn of û, that s, takng ther absolute value û, rank both û and X (or Ŷ ) accordng to an ascendng or descendng order and compute the Spearman s rank correlaton coeffcent gven prevously. Step 3. Assumng that the populaton rank correlaton coeffcent ρ s s zero and n > 8, the sgnfcance of the sample r s can be tested by the t test as follows 16 : t = r s n 1 r s (11.5.7) wth df = n. 16 See G. Udny Yule and M. G. Kendall, An Introducton to the Theory of Statstcs, Charles Grffn & Company, London, 1953, p. 455.

21 CHAPTER ELEVEN: HETEROSCEDASTICITY 407 If the computed t value exceeds the crtcal t value, we may accept the hypothess of heteroscedastcty; otherwse we may reject t. If the regresson model nvolves more than one X varable, r s can be computed between û and each of the X varables separately and can be tested for statstcal sgnfcance by the t test gven n Eq. (11.5.7). EXAMPLE 11.3 ILLUSTRATION OF THE RANK CORRELATION TEST To llustrate the rank correlaton test, consder the data gven n Table 11.. The data pertan to the average annual return (E, %) and the standard devaton of annual return (σ, %) of 10 mutual funds. The captal market lne (CML) of portfolo theory postulates a lnear relatonshp between expected return (E ) and rsk (as measured by the standard devaton, σ ) of a portfolo as follows: E = β + β σ Usng the data n Table 11., the precedng model was estmated and the resduals from ths model were computed. Snce the data relate to 10 mutual funds of dfferng szes and nvestment goals, a pror one mght expect heteroscedastcty. To test ths hypothess, we apply the rank correlaton test. The necessary calculatons are gven n Table 11.. Applyng formula (11.5.6), we obtan 110 r s = (100 1) (11.5.8) = Applyng the t test gven n (11.5.7), we obtan t = (0.3333)( 8) = (11.5.9) For 8 df ths t value s not sgnfcant even at the 10% level of sgnfcance; the p value s Thus, there s no evdence of a systematc relatonshp between the explanatory varable and the absolute values of the resduals, whch mght suggest that there s no heteroscedastcty. TABLE 11. RANK CORRELATION TEST OF HETEROSCEDASTICITY E, σ, d, average standard dfference annual devaton û between Name of return, of annual resduals, Rank Rank two mutual fund % return, % Eˆ * (E Eˆ ) of û of σ rankngs d Boston Fund Delaware Fund Equty Fund Fundamental Investors Investors Mutual Looms-Sales Mutual Fund Massachusetts Investors Trust New England Fund Putnam Fund of Boston Wellngton Fund Total *Obtaned from the regresson: Ê = σ. Absolute value of the resduals. Note: The rankng s n ascendng order of values.

22 408 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL Goldfeld-Quandt Test. 17 Ths popular method s applcable f one assumes that the heteroscedastc varance, σ, s postvely related to one of the explanatory varables n the regresson model. For smplcty, consder the usual two-varable model: Y = β 1 + β X + u Suppose σ s postvely related to X as σ = σ X ( ) where σ s a constant. 18 Assumpton ( ) postulates that σ s proportonal to the square of the X varable. Such an assumpton has been found qute useful by Pras and Houthakker n ther study of famly budgets. (See Secton 11.6.) If ( ) s approprate, t would mean σ would be larger, the larger the values of X. If that turns out to be the case, heteroscedastcty s most lkely to be present n the model. To test ths explctly, Goldfeld and Quandt suggest the followng steps: Step 1. Order or rank the observatons accordng to the values of X, begnnng wth the lowest X value. Step. Omt c central observatons, where c s specfed a pror, and dvde the remanng (n c) observatons nto two groups each of (n c) observatons. Step 3. Ft separate OLS regressons to the frst (n c) observatons and the last (n c) observatons, and obtan the respectve resdual sums of squares RSS 1 and RSS, RSS 1 representng the RSS from the regresson correspondng to the smaller X values (the small varance group) and RSS that from the larger X values (the large varance group). These RSS each have ( ) (n c) n c k k or df where k s the number of parameters to be estmated, ncludng the ntercept. (Why?) For the two-varable case k s of course. Step 4. Compute the rato λ = RSS /df ( ) RSS 1 /df If u are assumed to be normally dstrbuted (whch we usually do), and f the assumpton of homoscedastcty s vald, then t can be shown that λ of ( ) follows the F dstrbuton wth numerator and denomnator df each of (n c k)/. 17 Goldfeld and Quandt, op. ct., Chap Ths s only one plausble assumpton. Actually, what s requred s that σ be monotoncally related to X.

23 CHAPTER ELEVEN: HETEROSCEDASTICITY 409 If n an applcaton the computed λ ( = F) s greater than the crtcal F at the chosen level of sgnfcance, we can reject the hypothess of homoscedastcty, that s, we can say that heteroscedastcty s very lkely. Before llustratng the test, a word about omttng the c central observatons s n order. These observatons are omtted to sharpen or accentuate the dfference between the small varance group (.e., RSS 1 ) and the large varance group (.e., RSS ). But the ablty of the Goldfeld Quandt test to do ths successfully depends on how c s chosen. 19 For the two-varable model the Monte Carlo experments done by Goldfeld and Quandt suggest that c s about 8 f the sample sze s about 30, and t s about 16 f the sample sze s about 60. But Judge et al. note that c = 4 f n = 30 and c = 10 f n s about 60 have been found satsfactory n practce. 0 Before movng on, t may be noted that n case there s more than one X varable n the model, the rankng of observatons, the frst step n the test, can be done accordng to any one of them. Thus n the model: Y = β 1 + β X + β 3 X 3 + β 4 X 4 + u, we can rank-order the data accordng to any one of these X s. If a pror we are not sure whch X varable s approprate, we can conduct the test on each of the X varables, or va a Park test, n turn, on each X. EXAMPLE 11.4 THE GOLDFELD QUANDT TEST To llustrate the Goldfeld Quandt test, we present n Table 11.3 data on consumpton expendture n relaton to ncome for a cross secton of 30 famles. Suppose we postulate that consumpton expendture s lnearly related to ncome but that heteroscedastcty s present n the data. We further postulate that the nature of heteroscedastcty s as gven n ( ). The necessary reorderng of the data for the applcaton of the test s also presented n Table Droppng the mddle 4 observatons, the OLS regressons based on the frst 13 and the last 13 observatons and ther assocated resdual sums of squares are as shown next (standard errors n the parentheses). Regresson based on the frst 13 observatons: Ŷ = X (8.7049) (0.0744) r = RSS 1 = df = 11 Regresson based on the last 13 observatons: Ŷ = X (30.641) (0.1319) r = RSS = df = 11 (Contnued) 19 Techncally, the power of the test depends on how c s chosen. In statstcs, the power of a test s measured by the probablty of rejectng the null hypothess when t s false [.e., by 1 Prob (type II error)]. Here the null hypothess s that the varances of the two groups are the same,.e., homoscedastcty. For further dscusson, see M. M. Al and C. Gaccotto, A Study of Several New and Exstng Tests for Heteroscedastcty n the General Lnear Model, Journal of Econometrcs, vol. 6, 1984, pp George G. Judge, R. Carter Hll, Wllam E. Grffths, Helmut Lütkepohl, and Tsoung-Chao Lee, Introducton to the Theory and Practce of Econometrcs, John Wley & Sons, New York, 198, p. 4.

24 410 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL EXAMPLE 11.4 (Contnued) From these results we obtan λ = RSS /df RSS 1 /df = / /11 λ = 4.07 The crtcal F value for 11 numerator and 11 denomnator df at the 5 percent level s.8. Snce the estmated F ( = λ) value exceeds the crtcal value, we may conclude that there s heteroscedastcty n the error varance. However, f the level of sgnfcance s fxed at 1 percent, we may not reject the assumpton of homoscedastcty. (Why?) Note that the p value of the observed λ s TABLE 11.3 HYPOTHETICAL DATA ON CONSUMPTION EXPENDITURE Y($) AND INCOME X($) TO ILLUSTRATE THE GOLDFELD QUANDT TEST Data ranked by X values Y X Y X Mddle 4 observatons

25 CHAPTER ELEVEN: HETEROSCEDASTICITY 411 Breusch Pagan Godfrey Test. 1 The success of the Goldfeld Quandt test depends not only on the value of c (the number of central observatons to be omtted) but also on dentfyng the correct X varable wth whch to order the observatons. Ths lmtaton of ths test can be avoded f we consder the Breusch Pagan Godfrey (BPG) test. To llustrate ths test, consder the k-varable lnear regresson model Y = β 1 + β X + +β k X k + u (11.5.1) Assume that the error varance σ s descrbed as σ = f (α 1 + α Z + +α m Z m ) ( ) that s, σ s some functon of the nonstochastc varables Z s; some or all of the X s can serve as Z s. Specfcally, assume that σ = α 1 + α Z + +α m Z m ( ) that s, σ s a lnear functon of the Z s. If α = α 3 = = α m = 0, σ = α 1, whch s a constant. Therefore, to test whether σ s homoscedastc, one can test the hypothess that α = α 3 = =α m = 0. Ths s the basc dea behnd the Breusch Pagan test. The actual test procedure s as follows. Step 1. Estmate (11.5.1) by OLS and obtan the resduals û 1, û,..., û n. Step. Obtan σ = û /n. Recall from Chapter 4 that ths s the maxmum lkelhood (ML) estmator of σ. [Note: The OLS estmator s û /(n k).] Step 3. Construct varables p defned as p =û / σ whch s smply each resdual squared dvded by σ. Step 4. Regress p thus constructed on the Z s as p = α 1 + α Z + +α m Z m + v ( ) where v s the resdual term of ths regresson. Step 5. Obtan the ESS (explaned sum of squares) from ( ) and defne = 1 (ESS) ( ) Assumng u are normally dstrbuted, one can show that f there s homoscedastcty and f the sample sze n ncreases ndefntely, then χ asy m 1 ( ) 1 T. Breusch and A. Pagan, A Smple Test for Heteroscedastcty and Random Coeffcent Varaton, Econometrca, vol. 47, 1979, pp See also L. Godfrey, Testng for Multplcatve Heteroscedastcty, Journal of Econometrcs, vol. 8, 1978, pp Because of smlarty, these tests are known as Breusch Pagan Godfrey tests of heteroscedastcty.

26 41 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL that s, follows the ch-square dstrbuton wth (m 1) degrees of freedom. (Note: asy means asymptotcally.) Therefore, f n an applcaton the computed ( = χ ) exceeds the crtcal χ value at the chosen level of sgnfcance, one can reject the hypothess of homoscedastcty; otherwse one does not reject t. The reader may wonder why BPG chose 1 ESS as the test statstc. The reasonng s slghtly nvolved and s left for the references. EXAMPLE 11.5 THE BREUSCH PAGAN GODFREY (BPG) TEST As an example, let us revst the data (Table 11.3) that were used to llustrate the Goldfeld Quandt heteroscedastcty test. Regressng Y on X, we obtan the followng: Step 1. Ŷ = X se = (5.314) (0.086) RSS = R = ( ) Step. σ = û /30 = /30 = Step 3. Dvde the squared resduals û obtaned from regresson ( ) by to construct the varable p. Step 4. Assumng that p are lnearly related to X ( = Z ) as per ( ), we obtan the regresson ˆp = X Step 5. se = (0.759) (0.0041) ESS = R = 0.18 ( ) = 1 (ESS) = (11.5.0) Under the assumptons of the BPG test n (11.5.0) asymptotcally follows the chsquare dstrbuton wth 1 df. [Note: There s only one regressor n ( ).] Now from the ch-square table we fnd that for 1 df the 5 percent crtcal ch-square value s and the 1 percent crtcal χ value s Thus, the observed ch-square value of s sgnfcant at the 5 percent but not the 1 percent level of sgnfcance. Therefore, we reach the same concluson as the Goldfeld Quandt test. But keep n mnd that, strctly speakng, the BPG test s an asymptotc, or large-sample, test and n the present example 30 observatons may not consttute a large sample. It should also be ponted out that n small samples the test s senstve to the assumpton that the dsturbances u are normally dstrbuted. Of course, we can test the normalty assumpton by the tests dscussed n Chapter 5. 3 See Adran C. Darnell, A Dctonary of Econometrcs, Edward Elgar, Cheltenham, U.K., 1994, pp On ths, see R. Koenker, A Note on Studentzng a Test for Heteroscedastcty, Journal of Econometrcs, vol. 17, 1981, pp

27 CHAPTER ELEVEN: HETEROSCEDASTICITY 413 Whte s General Heteroscedastcty Test. Unlke the Goldfeld Quandt test, whch requres reorderng the observatons wth respect to the X varable that supposedly caused heteroscedastcty, or the BPG test, whch s senstve to the normalty assumpton, the general test of heteroscedastcty proposed by Whte does not rely on the normalty assumpton and s easy to mplement. 4 As an llustraton of the basc dea, consder the followng three-varable regresson model (the generalzaton to the k-varable model s straghtforward): Y = β 1 + β X + β 3 X 3 + u (11.5.1) The Whte test proceeds as follows: Step 1. Gven the data, we estmate (11.5.1) and obtan the resduals, û. Step. We then run the followng (auxlary) regresson: û = α 1 + α X + α 3 X 3 + α 4 X + α 5 X 3 + α 6X X 3 + v (11.5.) 5 That s, the squared resduals from the orgnal regresson are regressed on the orgnal X varables or regressors, ther squared values, and the cross product(s) of the regressors. Hgher powers of regressors can also be ntroduced. Note that there s a constant term n ths equaton even though the orgnal regresson may or may not contan t. Obtan the R from ths (auxlary) regresson. Step 3. Under the null hypothess that there s no heteroscedastcty, t can be shown that sample sze (n) tmes the R obtaned from the auxlary regresson asymptotcally follows the ch-square dstrbuton wth df equal to the number of regressors (excludng the constant term) n the auxlary regresson. That s, n R asy χ df (11.5.3) where df s as defned prevously. In our example, there are 5 df snce there are 5 regressors n the auxlary regresson. Step 4. If the ch-square value obtaned n (11.5.3) exceeds the crtcal ch-square value at the chosen level of sgnfcance, the concluson s that there s heteroscedastcty. If t does not exceed the crtcal ch-square value, there s no heteroscedastcty, whch s to say that n the auxlary regresson (11.5.1), α = α 3 = α 4 = α 5 = α 6 = 0 (see footnote 5). 4 H. Whte, A Heteroscedastcty Consstent Covarance Matrx Estmator and a Drect Test of Heteroscedastcty, Econometrca, vol. 48, 1980, pp Impled n ths procedure s the assumpton that the error varance of u,σ, s functonally related to the regressors, ther squares, and ther cross products. If all the partal slope coeffcents n ths regresson are smultaneously equal to zero, then the error varance s the homoscedastc constant equal to α 1.