Method of Moments Estimation in Linear Regression with Errors in both Variables J.W. Gillard and T.C. Iles

Transcription

1 Method of Moment Etimation in Linear Regreion with Error in both Variable by J.W. Gillard and T.C. Ile Cardiff Univerity School of Mathematic Technical Paper October 005 Cardiff Univerity School of Mathematic, Senghennydd Road, Cardiff, CF4 4AG

2 Content 1. Introduction 3. Literature Survey 6 3. Statitical aumption 9 4. Firt and econd order moment equation 1 5. Etimator baed on the firt and econd order moment Etimator making ue of third order moment Etimator making ue of fourth order moment 1 8. Variance and covariance of the etimator 4 9. Dicuion A ytematic approach for fitting line with error in both variable Reference Appendice Figure 43

3 1. Introduction The problem of fitting a traight line to bivariate (x, y) data where the data are cattered about the line i a fundamental one in tatitic. Method of fitting a line are decribed in many tatitic text book, for example Draper and Smith (1998) and Kleinbaum et al (1997). The uual way of fitting a line i to ue the principle of leat quare, finding the line that ha the minimum um of the quare of ditance of the point to the line in the vertical y direction. Thi line i called the regreion line of y on x. In the jutification of the choice of thi line it i aumed that deviation of the obervation from the line are caued by unexplained random variation that i aociated with the variable y. Implicitly it i aumed that the variable x i meaured without error or other variation. Clearly, if it i felt that deviation from the line are due to variation in x alone the appropriate method would be to ue the regreion line of x on y, minimiing the um of quare in the horizontal direction. The random deviation of the obervation from the uppoed underlying linear relationhip are uually called the error. Although the word error i a very common term it i an unfortunate choice of word; the variation may incorporate not jut meaurement error but any other ource of unexplained variation that reult in catter from the line. Some author have uggeted that other term might be ued, diturbance, departure, perturbation, noie and random component being amongt the uggetion. In thi report, however, becaue of the wide ue of the word, the variation from the line will be decribed a error. In many invetigation the catter of the obervation arie becaue of error in both meaurement. Thi problem i known by many name, the commonet being error in variable regreion and meaurement error model. The former name i ued throughout thi report. Caella and Berger (1990) wrote of thi problem, '(it) i o different from imple linear regreion... that it i bet thought of a a completely different topic'. There i a very extenive literature on the ubject, but publihed work i mainly in the form of article in the technical journal, mot of which deal with a particular apect of the problem. Relatively few tandard text book on regreion theory contain comprehenive decription of olution to the problem. A brief literature urvey i given in the next ection. 3

4 We believe that the error in variable regreion problem i potentially of wide practical application in the analyi of experimental data. One of the aim of thi report therefore i to give ome guidance for practitioner in deciding how an error in variable traight line hould be fitted. We give imple formula that a practitioner can ue to etimate the lope and intercept of an optimum line together with variance term that are alo included in the model. Very few previou author have given formula for the tandard error of thee etimator, and we offer ome advice regarding thee. Indeed, a detailed expoition on the variance covariance matrice for mot of the etimator in thi report i included in Gillard and Ile (006). In our approach we make a few aumption a are neceary to obtain etimator that are reliable. We have found that traightforward etimator of the parameter and their aymptotic variance can be found uing the method of moment principle. Thi approach ha the advantage of being imple to follow for reader who are not principally intereted in the methodology itelf. The method of moment technique i decribed in many book of mathematical tatitic, for example Caella and Berger (1990), although here, a elewhere, the treatment i brief. In common with many other mathematical tatitical text, they gave greater attention to the method of maximum likelihood. Bowman and Shenton (1988) wrote that 'the method of moment ha a long hitory, involve an enormou literature, ha been through period of evere turmoil aociated with it ampling propertie compared to other etimation procedure, yet urvive a an effective tool, eaily implemented and of wide generality'. Method of moment etimator can be criticied becaue they are not uniquely defined, o that if the method i ued it i neceary to chooe amongt poible etimator to find one that bet uit the data being analyed. Thi prove to be the cae when the method i ued in error in variable regreion theory. Neverthele the method of moment ha the advantage of implicity, and alo that the only aumption that have to be made are that low order moment of the ditribution decribing the obervation exit. We alo aume here that thee ditribution are mutually uncorrelated. It i relatively eay to work out the theoretical aymptotic variance and covariance of the etimator uing the delta method outlined by Cramer (1946). The information in thi report will enable a practitioner to fit the line and calculate approximate confidence interval for the 4

5 aociated parameter. Significance tet can alo be done. A limitation of the formula i that they are aymptotic reult, o they hould only be ued for moderate or large data et. 5

6 . Literature Survey A mentioned above, the error in variable regreion problem i rarely included in tatitical text. There are two text devoted entirely to the error in variable regreion problem, Fuller (1987) and Cheng and van Ne (1999). Caella and Berger (1990) ha an informative ection on the topic, Sprent (1969) contain chapter on the problem, a do Kendall and Stuart (1979) and Dunn (004). Draper and Smith (1998) on the other hand, in their book on regreion analyi, devoted only 7 out of a total of almot 700 page to error in variable regreion. The problem i more frequently decribed in Econometric text, for example Judge et al (1980). In thee text the method of intrumental variable i often given prominence. Intrumental variable are uncorrelated with the error ditribution, but are highly correlated with the predictor variable. The extra information that thee variable contain enable a method of etimating the parameter of the line to be obtained. Carroll et al (1995) decribed error in variable model for non-linear regreion, and Seber and Wild (1989) included a chapter on thi topic. Probably the earliet work decribing a method that i appropriate for the error in variable problem wa publihed by Adcock (1878). He uggeted that a line be fitted by minimiing the um of quare of ditance between the point and the line in a direction perpendicular to the line, the method that ha come to be known a orthogonal regreion. Kummel (1879) took the idea further, generaliing to a line that ha minimum um of quare of ditance of the obervation from the line in a direction other than perpendicular. Pearon (1901) generalied the error in variable model to that of multiple regreion, where there are two or more different x variable. He alo pointed out that the lope of the orthogonal regreion line i between thoe of the regreion line of y on x and that of x on y. The idea of orthogonal regreion wa included in Deming' book (1943), and orthogonal regreion i ometime referred to a Deming regreion. Another method of etimation that ha been ued in error in variable regreion i the method of moment. Geary (194, 1943, 1948 and 1949) wrote a erie of paper on the method, but uing cumulant rather than moment in the later paper. Drion (1951), in a paper that i infrequently cited, ued the method of moment, and gave 6

7 ome reult concerning the variance of the ample moment ued in the etimator that he uggeted. More recent work uing the moment approach ha been written by Pal (1980), van Montfort et al (1987), van Montfort (1989) and Cragg (1997). Much of thi work centre on a earch for optimal etimator uing etimator baed on higher moment. Dunn (004) gave formula for many of the etimator of the lope that we decribe later in thi report uing a method of moment approach. However, he did not give information about etimator baed on higher moment and it turn out that thee are the only moment baed etimator that can be ued unle there i ome information about the relationhip additional to the (x, y) obervation. Neither did he give information about the variance of the etimator. Another idea, firt decribed by Wald (1940) and taken further by Bartlett (1949), i to group the data, ordered by the true value of the predictor variable, and ue the mean of the group to obtain etimator of the lope. The intercept i then etimated by chooing the line that pae through the centroid ( x,y) of the complete data et. A difficulty of the method, noted by Wald himelf, i that the grouping of the data cannot, a may at firt be thought, be baed on the oberved value without making further aumption. In order to preerve the propertie of the random variable underlying the method it i neceary that the grouping be baed on ome external knowledge of the ordering of the data. In depending on thi extra information, Wald' grouping method i a pecial cae of an intrumental variable method, the intrumental variable in thi cae being the ordering of the true value. Gupta and Amanullah (1970) gave the firt four moment of the Wald etimator and Gibon and Jowett (1957) invetigated optimum way of grouping the obervation. Madanky (1959) reviewed ome apect of grouping method. Lindley (1947) and many ubequent author approached the problem of error in variable regreion from a likelihood perpective. Kendall and Stuart (1979), Chapter 9, reviewed the literature and outlined the likelihood approach. A diadvantage of the likelihood method in the error in variable problem i that it i only tractable if all of the ditribution decribing variation in the data are aumed to be Normal. In thi cae a unique olution i only poible if additional aumption are made concerning the parameter of the model, uually aumption about the error variance. 7

8 Neverthele, maximum likelihood etimator have certain optimal propertie and it i poible to work out the aymptotic variance-covariance matrix of the etimator. Thee were given for a range of aumption by Hood et al (1999). The likelihood approach wa alo ued by Dolby and Lipton (197), Dolby (1976) and Cox (1976) to invetigate the error in variable regreion problem where there are replicate meaured value at the ame true value of the predictor variable. Lindley and el Sayyad (1968) decribed a Bayeian approach to the error in variable regreion problem and concluded that in ome repect the likelihood approach may be mileading. A decription of a Bayeian approach to the problem, with a critical comparion with the likelihood method, i given by Zellner (1980). Golub and van Loan (1980), van Huffel and Vanderwalle (1991) and van Huffel and Lemmerling (00) have developed a theory that they have called total leat quare. Thi method allow the fitting of linear model where there are error in the predictor variable a well a the dependent variable. Thee model include the linear regreion one. The idea i linked with that of adjuted leat quare, that ha been developed by Kukuh et al (003) and Markovky et al (00, 003). Error in variable regreion ha ome imilaritie with factor analyi, a method in multivariate analyi decribed by Lawley and Maxwell (1971) and Johnon and Wichern (199) and elewhere. Factor analyi i one of a family, called latent variable method (Skrondal and Rabe-Heketh, 004), that include the error in variable regreion problem. Dunn and Robert (1999) ued a latent variable approach in an error in variable regreion etting, and more recently extenion combining latent variable and generalied linear model method have been devied (Rabe-Heketh et al, 000, 001). Over the year everal author have written review article on error in variable regreion. Thee include Kendall (1951), Durbin (1954), Madanky (1959), Moran (1971) and Anderon (1984). Rigg et al (1978) performed imulation exercie comparing ome of the lope etimator that have been decribed in the literature. 8

9 3. Statitical Aumption The notation in the literature for the error in variable regreion problem differ from author to author. In thi report we ue a notation that i imilar to that ued by Cheng and van Ne (1999), and that appear to be finding favour with other modern author. It i, unfortunately, different from that ued by Kendall and Stuart (1979), and ubequently adopted by Hood (1998) and Hood et al (1999). We uppoe that there are n individual in the ample with true value (ξ i, η i ) and oberved value (x i, y i ). It i believed that there i a linear relationhip between the two variable ξ and η. η i = α + βξ i (1) However, there i variation in both variable that reult in a deviation of the obervation (x i, y i ) from the true value (ξ i, η i ) reulting in a catter about the traight line. Thi catter i repreented by the addition of random error repreenting the variation of the oberved from the true value. x i = ξ i + i () y i = η i + i = α + βξ i + i (3) The error i and i are aumed to have zero mean and variance that do not change with the uffix i. E[ i ] = 0, Var[ i ] = E[ i ] = 0, Var[ i ] = We aume that higher moment alo exit. E[ 3 i. ] = µ 3, E[ ] = µ 4 E[ 3 i ] = µ 3, E[ 4 i ] = µ 4. 4 i We alo aume that the error are mutually uncorrelated and that the error i are uncorrelated with i. E[ i j ] = 0, E[ i j ] = 0 (i j) E[ i j ] = 0 for all i and j (including i = j). 9

10 Some author have treed the importance of a concept known a equation error. Further detail are given by Fuller (1987) and Carroll and Ruppert (1996). Equation error introduce an extra term on the right hand ide of equation (3). y i = η i + ω i + i = α + βξ i +ω i + i Dunn (004) decribed the additional equation error term ω i a '(a) new random component (that) i not necearily a meaurement error but i a part of y that i not related to the contruct or characteritic being meaured'. It i not intended to model a mitake in the choice of equation to decribe the underlying relationhip between ξ and η. Auming that the equation error term have a variance ω that doe not change with i and that they are uncorrelated with the other random variable in the model the practical effect of the incluion of the extra term i to increae the apparent variance of y by the addition of ω. We do not conider in thi report method for ue where there may be expected to be erial correlation amongt the obervation. Sprent (1969) included a ection on thi topic and Karni and Weiman (1974) ued a method of moment approach, making ue of the firt difference of the obervation, auming that a non zero autocorrelation i preent in the erie of obervation. In much of the literature on error in variable regreion a ditinction i drawn between the cae where the ξ i are aumed to be fixed, albeit unknown, quantitie and the cae where ξ i are aumed to be a random ample from a population. The former i known a the functional and the latter the tructural model. Caella and Berger (1990) decribed the theoretical difference in thee two type of model. Uing the approach adopted in thi report it i not neceary to make the ditinction. All that i aumed i that the ξ are mutually uncorrelated, are uncorrelated with the error and that the low order moment exit. Neither the problem of etimation of each individual ξ i in the functional model nor the problem of predicting y i invetigated in thi report. Whether the ξ are aumed to be fixed or a random ample we find only etimator for the low order moment. 10

11 The aumption that we make about the variable ξ are a follow. E[ξ i ] = µ, Var[ξ i ] =. In ome of the work that i decribed later the exitence of higher moment of ξ i alo aumed. E[(ξ i - µ) 3 ] = µ ξ3, E[(ξ i - µ) 4 ] = µ ξ4 The variable ξ i are aumed to be mutually uncorrelated and uncorrelated with the error term and. E[(ξ i - µ)(ξ j - µ)] = 0 (i j) E[(ξ i - µ) j ] = 0 and E[(ξ i - µ) j ] = 0 for all i and j. In order to etimate variance and covariance it i neceary later in thi report to aume the exitence of moment of ξ of order higher than the fourth. The rth moment r i denoted byµ = E[( ξ µ )]. ξr i 11

12 4. Firt and Second Order Moment Equation The firt order ample moment are denoted by x = and y =. n n The econd order moment are notated by (x x)(y y) i i xy =. n x i (x x) i xx =, n y i (y y) i yy = and n No mall ample correction for bia i made, for example by uing (n - 1) a a divior for the variance rather than n. Thi i becaue the reult on variance and covariance that we give later on in the report are reliable only for moderately large ample ize, generally 50 or more, where the adjutment for bia i negligible. Moreover, the algebra needed for the mall ample adjutment complicate the formula omewhat. The moment equation in the error in variable etting are given in the equation below. A tilde i placed over the ymbol for a parameter to denote the method of moment etimator. We have ued thi ymbol in preference to the circumflex, often ued for etimator, to ditinguih between method of moment and maximum likelihood etimator. Firt order moment: x =µ (4) y =α+βµ (5) Second order moment: xx yy xy = + (6) =β + (7) (8) =β It can readily be een from equation (6), (7) and (8) that there i a hyperbolic relationhip between method of moment etimator and of the error variance. Thi wa called the Frich hyperbola by van Montfort (1989). 1

13 ( )( ) = ( ) (9) xx yy xy Thi i a ueful equation in that it relate pair of etimator of and that atify equation (6), (7) and (8). The potential application of the Frich hyperbola are dicued further in Section 9. One of the difficultie with the error in variable regreion problem i apparent from an examination of equation (4) - (8). There i an identifiability problem if thee equation alone are ued to find etimator. There are five moment equation of firt or econd order but there are ix unknown parameter. It i therefore not poible to olve the equation to find unique olution without making additional aumption. One poibility i to ue higher moment, and thi i decribed later in the report. Another poibility i to ue additional information in the form of an intrumental variable. A third poibility, and the one that i invetigated firt, i to aume that there i ome prior knowledge of the parameter that enable a retriction to be impoed. Thi then allow the five equation to be olved. There i a comparion with thi identifiability problem and the maximum likelihood approach. In thi approach, the only tractable aumption i that the ditribution of i, i and ξ i, are all Normal. Thi in turn lead to the bivariate random variable (x, y) having a bivariate Normal ditribution. Thi ditribution ha five parameter, and the maximum likelihood etimator for thee parameter are identical to the method of moment etimator baed on the five firt and econd moment equation. In thi cae therefore it i not poible to find olution to the likelihood equation without making an additional aumption, reticting the parameter pace. The retriction that we decribe in Section 5 below are one that have been ued by previou author uing the likelihood method. The likelihood function for any other ditribution than the Normal i complicated and the method i difficult to apply. However the method of moment approach uing higher moment and without auming a retiction in the parameter pace, can be ued without making the aumption of Normality. 13

14 5. Etimator Baed on the Firt and Second Moment So that etimating equation tand out from other numbered equation, they are marked by an aterik. Equation (1) give the etimator for µ directly µ=x (10)* The etimator for all the remaining parameter are eaily expreed in term of the etimator β of the lope. Equation (4) and (5) can be ued to give an equation for the intercept α in term of β. α= y β x (11)* Thu the fitted line in the (x, y) plane pae through the centroid ( x, y) of the data, a feature that i hared by the imple linear regreion equation. Equation (8) yield an equation for, with β alway having the ame ign a xy. = β xy (1)* If the error variance i unknown, it i etimated from equation (6). = (13)* xx Finally if i unknown, it i etimated from equation (7) and the etimator for β. = (14)* yy β Since variance are never negative there are retriction on permiible parameter value, depending on the value taken by the ample econd moment. Thee 14

15 condition are often called admiibility condition. The traightforward condition, enabling non negative variance etimate to be obtained are given below. xx > yy > Alone, thee condition are not ufficient to enure that the variance etimator are non negative. The error in variable lope etimator mut lie between the y on x and x on y lope etimator xy xx and yy xy repectively. Other admiibility condition, relevant in pecial cae, are given in Table 1. Admiibility condition are dicued in detail by Kendall and Stuart (1979), Hood (1998), Hood et al (1999) and Dunn (004). We now turn to the quetion of the etimation of the lope. There i no ingle etimator for the lope that can be ued in all cae in error in variable regreion. Each of the retriction aumed on the parameter pace to to get around the identifiability problem dicued above i aociated with it own etimator of the lope. In order to ue an etimator baed on the firt and econd order moment alone it i neceary for the practitioner to decide on the bai of knowledge of the invetigation being undertaken which retriction i likely to uit the purpoe bet. Table 1 ummarie the implet etimator of the lope parameter β derived by auming a retriction on the parameter. With one exception thee etimator have been decribed previouly; mot were given by Kendall and Stuart (1979), Hood et al (1999) and, in a method of moment context, by Dunn (1989). 15

16 Table 1: Etimator of the lope parameter b baed on firt and econd moment Retriction Intercept α known Etimator y α β 1 = x Admiibility Condition x 0 Variance Variance known known Reliability ratio κ= known + Variance ratio λ= known β = xy xx β = 3 yy xy xy β 4 = κ xx { } 1/ ( λ ) + ( λ ) + 4 λ( ) yy xx yy xx xy β 5 = xy xx > ( ) xy yy > xx yy > ( ) xy xx > yy None None λ ν= known ( ν 1) + ign( ){( ν 1) ( ) + 4ν } 1/ β xy xy xy xx yy β = 6 ν xx xx 0 Both variance and known. yy β = ign( ) 7 xy xx 1/ xx yy > > There i an ambiguity in the ign to be ued in the equation for β 6 and β 7. Thi i reolved by auming that the lope etimator alway ha the ame ign a xy, a mentioned above to enure that equation (11)* give a non negative etimate of the variance. A dicuion of thee etimator i given in Section 9. 16

17 It may eem that the retriction leading to the etimator β 6 i not one that would often be made on the bai of a priori evidence. The reaon for the incluion of thi etimator, which eem not to have been previouly uggeted, i that it i a generaliation of an etimator that ha been widely recommended, the geometric mean etimator. Thi i the geometric mean of the lope of the regreion of y on x and the reciprocal of the regreion of x on y. Section 9 contain further dicuion. Aymptotic variance concerning thi etimator will not be included in thi report. The aumption that both error variance and are known i omewhat different from the other cae. By auming that two parameter are known there are only four remaining unknown parameter, but five firt and econd moment equation that could be ued to etimate them. One poibility of obtaining a olution i to ue only four of the five equation (4) to (8) incluive, or a imple combination of thee. If equation (6) i excluded, the etimator for the lope β i β 3, but then the aumed value of will almot certainly not agree exactly with the value that would be obtained from equation (1)*. If equation (7) i excluded, the etimator for the lope i β, but then it i mot unlikely that the aumed value of will agree exactly with the value obtained from equation (13)*. If equation (6) and (7) are combined, uing the known ratio λ=, the etimator β 5 i obtained, and then neither of equation (1)* and (13)* will be atified by the a priori value aumed for and. Another poibility that lead to a imple etimator for the lope β i to exclude equation (8), and it i thi that lead to the etimator β 7 in Table 1. 17

18 6. Etimate Making Ue of the Third Moment The third order moment are written a follow. xxx xxy xyy (xi x) = n = 3 (xi x) (yi y) n (xi x)(yi y) = n (y y) 3 i yyy =. n The four third moment equation take a imple form. Some detail on the derivation of thee expreion i given in Appendix 1. (15) xxx =µ ξ3 +µ 3 =βµ (16) xxy ξ3 =βµ (17) xyy ξ3 (18) 3 yyy =βµ ξ3+µ 3 Together with the firt and econd moment equation, equation (4) - (8) incluive, there are now nine equation in nine unknown parameter. The additional parameter introduced here are the third moment µ ξ3, µ 3 and µ 3. There are therefore unique etimator for all nine parameter. However, it i unlikely in practice that there i a much interet in thee third moment a there i in the firt and econd moment, more epecially, the lope and intercept of the line. Thu a impler way of proceeding i probably of more general value. The implet way of making ue of thee equation i to make a ingle further aumption, namely that µ ξ3 i non zero. There i a practical requirement aociated with thi aumption, and thi i that the ample third moment hould be ignificantly different from 0. It i thi requirement that ha probably led to the ue of third moment etimator receiving relatively little attention in recent literature. It i not 18

19 alway the cae that the oberved value of x and y are ufficiently kewed to allow thee equation to be ued with any degree of confidence. Moreover ample ize needed to identify third order moment with a practically ueful degree of preciion are omewhat larger than i the cae for firt and econd order moment. However, if the aumption can be jutified from the data then a traightforward etimator for the lope parameter i obtained without auming anything known a priori about the value taken by any of the parameter. Thi etimator i obtained by dividing equation (17) by equation (16). xyy β 8 = (19)* xxy The value for β obtained from thi equation can then be ubtituted in equation (11)* - (14)* to etimate the intercept α and all three variance moment µ ξ3 can be etimated from equation (16)., and. The third µ = xxy ξ3 β 8 (0)* Etimator for µ 3 and µ 3 may be obtained from equation (15) and (18) repectfully. Other imple way of etimating the lope are obtained if the additional aumption µ 3 = 0 and µ 3 = 0 are made. Thee would be appropriate aumption to make if the ditribution of the error term and are ymmetric. Note, however, that thi doe not imply that the ditribution of ξ i ymmetric. The obervation have to be kewed to allow the ue of etimator baed on the third moment. With thee aumption the lope β could be etimated by dividing equation (16) by (15) or by dividing equation (18) and (17). β= β= xxy xxx yyy xyy 19

20 We do not invetigate thee etimator further in thi report, ince we feel that etimator that make fewet aumption are likely to be of the mot practical value. 0

21 7. Etimate Making Ue of the Fourth Moment The fourth order moment are written a xxxx xxxy xxyy xyyy yyyy (xi x) = n = 4 3 (xi x) (yi y) n (xi x) (y y) = n i (xi x)(yi y) = n (yi y) = n 4 3 By uing a imilar approach to the one adopted in deriving the third moment etimating equation, the fourth moment equation can be derived. =µ + 6 +µ (1) xxxx ξ4 4 =βµ + 3β () xxxy ξ4 (3) xxyy =βµ ξ4 +β ++ =βµ + 3β (4) 3 xyyy ξ4 =βµ + 6β+µ (5) 4 yyyy ξ4 4 Together with the firt and econd moment equation thee form a et of ten equation, but there are only nine unknown parameter. The fourth moment equation have introduced three additional parameter µ ξ4 µ 4 and µ 4, but four new equation. One of the equation i therefore not needed. The eaiet practical way of etimating the parameter i to ue equation () and (4), together with equation (6), (7) and (8). Equation () i multiplied by β and ubtracted from equation (4). 1

22 β = 3 β ( β ) xxxy xyyy Equation (6) i multiplied by β and ubtracted from equation (7). β =β xx yy Thu, making ue alo of equation (8) an etimating equation i obtained for the lope parameter β. 1/ 3 xyyy xy yy β 9 = (6)* xxxy 3xyxx There may be a practical difficulty aociated with the ue of equation (6)* if the random variable ξ i Normally ditributed. In thi cae the fourth moment i equal to 3 time the quare of the variance. A random variable for which thi property doe not hold i aid to be kurtotic. A cale invariant meaure of kurtoi i given by the following expreion µ γ = (7) If the ditribution of ξ ha zero meaure of kurtoi the average value of the five ample moment ued in equation (6)* are a follow. 3 4 E[ xyyy ] = 3β + 3β 4 E[ xxxy] = 3β + 3β E[ xx] yy = + E[ ] =β + E[ ]=β xy Then it can be een that the average value of the numerator of equation (6)* i approximately equal to zero, a i the average value of the denominator. Thu there i

23 an additional aumption that ha to be made for thi equation to be reliable a an etimator, and that i that equation (7) doe not hold, µ ξ4 mut be different from 3 4. In practical term, both the numerator and the denominator of the right hand ide of equation (6)* mut be ignificantly different from zero. If a reliable etimate of the lope β can be obtained from equation (6)*, equation (10)* - (13)* enable the intercept α and the variance, and to be etimated. The fourth moment µ ξ4 of ξ can then be etimated from equation (), and the fourth moment µ 4 and µ 4 of the error term and can be etimated from equation (0) and (4) repectively, though etimate of thee higher moment of the error term are le likely to be of practical value. Although β 9 ha a compact cloed form, it variance i rather cumberome. Indeed, the variance of β 9 depend on the ixth central moment of ξ. Since it i impractical to etimate thi moment with any degree of accuracy, there will be no dicuion of the aymptotic variance of thi etimator. 3