The Effect of Nonzero Autocorrelation Coefficients on the Distributions of Durbin-Watson Test Estimator: Three Autoregressive Models

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1 EJ Exper Journal of Economi c s ( 4 ), Th e Au h or. Publi sh ed by Sp rin In v esify. ISS N Econ omics.e xp erjou rn a ls.com The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models Mei-Yu LEE * Yuanpei Universiy, Taiwan This paper invesigaes he effec of he nonzero auocorrelaion coefficiens on he sampling disribuions of he Durbin-Wason es esimaor in hree ime-series models ha have differen variance-covariance marix assumpion, separaely. We show ha he expeced values and variances of he Durbin-Wason es esimaor are slighly differen, bu he skewed and kurosis coefficiens are considerably differen among hree models. The shapes of four coefficiens are similar beween he Durbin-Wason model and our benchmark model, bu are no he same wih he auoregressive model cu by one-lagged period. Second, he large sample case shows ha he hree models have he same expeced values, however, he auoregressive model cu by one-lagged period explores differen shapes of variance, skewed and kurosis coefficiens from he oher wo models. This implies ha he large samples lead o he same expeced values, ( ρ ), whaever he variance-covariance marix of he errors is assumed. Finally, comparing wih he wo sample cases, he shape of each coefficien is almos he same, moreover, he auocorrelaion coefficiens are negaively relaed wih expeced values, are invered-u relaed wih variances, are cubic relaed wih skewed coefficiens, and are U relaed wih kurosis coefficiens. Keywords: Nonzero auocorrelaion coefficien, he d saisic, serial correlaion, auoregressive model, ime series analysis JEL Classificaion: C3, C5, C5. Inroducion Serial correlaion has he mos imporan role in auoregressive models, which is based on he regression analysis. If he daa has serial correlaion, hen he researchers have o pay aenion o i and use he correc variance-covariance marix for esimaion and forecasing. However, Lee (4a) indicaes he reasons of he difference beween he errors and he residuals in regression analysis where X Tˆ ε = is is inernal consrain for he residuals, ha is also affeced by he values of he independen variables. Anoher one imporan facor is degree of freedom which consrains he relaionship of sample size and he number of independen variables. Lee found ha X Tˆ ε = is very imporan when he degree of freedom is no very large in he regression analysis, hus, he auoregressive model will have o pay considerable aenion on he above facors when he researchers use he serial correlaion es esimaor. Lee (4b) also discusses he Z * Corresponding Auhor: Mei-Yu Lee, Yuanpei Universiy, Deparmen of Applied Finance, Taiwan Aricle Hisory: Received 9 Ocober 4 Acceped 5 November 4 Available Online 4 November 4 Cie Reference: Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp

2 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp es ha can be used in he serial correlaion es of he d saisic when he degree of freedom is larger han 5. In ha paper, Lee invesigaed he effecs of he facors, including he variances of he errors, he values of he independen variables, on he disribuions of he d saisic. Therefore, we do no repea he invesigaions in his paper. Due o he inernal consrain and degree of freedom, we use hree models, including Durbin-Wason model (Durbin and Wason, 95, 95), he common auoregressive model, and auoregressive model wih one-lagged period, AR() model, (Savin and Whie, 978), o discuss he effec of nonzero auocorrelaion coefficiens on he disribuions of he Durbin-Wason es esimaor, he d saisic. The reason we choose he d saisic is ha is formula is he combinaion of he residuals and no one researches from he viewpoin of he degrees of freedom. In fac, he Durbin-Wason model has unfixed variance-covariance marix, and he AR() model is resriced in he range from -.5 o.5. We inend o show he differences from he disribuions of he d saisic among hree models, and o compare he coefficiens of he d saisic beween any wo models. This paper complemens and explains if he null hypohesis is he nonzero auocorrelaion coefficiens, H : ρ = ρ, ρ, hen how he disribuions of he d saisic will become and wha are he differences among hree models. We show ha he hree models have he same auocorrelaion coefficiens as he null hypohesis in he robus analysis, bu in he small samples, he hree models have differen disribuions of he d saisic. I is worhy noing ha he imporance of he null hypohesis wih he nonzero auocorrelaion coefficien. When he researchers can know he daa of he exacly auocorrelaion coefficien, hey can accuraely forecas and judge he fuure. Even he criical value able can be buil wihou neglecing he properies of he errors and he values of he independen variables. The srucure of he paper is as follows. Secion describes he hree model seings and he simulaion procedure. Secion 3 explores our simulaion resuls ha have () he paerns of four coefficiens among hree models when he sample is 57 and he number of independen variables is 6, and () he paerns of four coefficiens beween any wo models when he sample is and he number of independen variables is 6. Secion 4 presens he conclusions and discussion of he resuls.. The model Consider a linear regression model wih k regressors and T sample sizes, as Y = X (T ) (T k) (k ) B + e (T ) Each ε is he error marix, e, and saisfied wih hree condiions ha are (i) ε is i.i.d. Normal disribuion. (ii) E(ε ) = and Var(ε ) = σ for all. (iii) E(ε ε -) = and E(ε i ε j) =, i j >, for all and i, j =,,, T. Y = X B + e is consrained by E(e) = and E(X T e) =. Use OLS and ge he esimaor of coefficiens, ˆB = (X T X) - X T Y, due o he consrain of X T e=. Thus he residuals are = (I X(X T X) - X T )ε, which is saisfied wih E( ) = and X T = and he degree of freedom being T-p-. The sum of square residuals will be E( ) = σ (I X(X T X) - X T ) () The condiion (iii) guaranees he errors are independen from each oher. However, he serial correlaion model has broken condiion (iii). In order o es he exensibiliy of he d saisic in he serial correlaion models, he models are ha Model A is he Durbin-Wason model inroduced by Durbin and Wason in 95. Model B is he serial correlaion model in common. This is also he benchmark model. Model C is he auoregressive error procedure wih one lagged period. 86

3 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp Model A The serial correlaive condiion of Model A is ε + = ρε + µ +, where =,,, T-, ρ is populaion auocorrelaion coefficien of ε + and ε, µ + is i.i.d. Normal disribuion wih E(µ +) = and Var(µ +) = σ for all. The specialis propery of Model A is he unfixed variance of he error when increases, ha is, ( ) + j ρ + = ρ ε + µ = µ + + j j= ε, Var E ρ + j= ( ) + j ( ε ) σ ( ρ ) = + + j ( ) ( ε ε ) = ρ ( ρ ) + ( ε, ε ) + = j= j= ρ j= σ, + j ( ρ ) ) + + j ( ρ ) µ ) ( ρ ) j ( ), + j µ j j= where ρ(ε, ε +) is he sample auocorrelaion coefficien. If approaches o infinie, hen Var(ε ) = σ / ( ρ ), E(ε ε +) = ρ Var(ε ) and ρ(ε, ε +) = ρ (see he proofs in Appendix I)... Model B The serial correlaion condiion in Model B is ε + = ρε + µ +, where E(ε ε +) =ρσ, =,,, T-, hus, µ + is i.i.d. Normal disribuion wih E(µ +) = and Var(µ +) = ( ρ ) σ for all. This serial correlaion condiion indicaes he condiional ε + on ε is Normal disribuion wih E(ε + ε ) = ρε and Var(ε + ε ) = ( ρ ) σ. Therefore, he variance-covariance marix is, The special propery of Model B is ( ε ε ) = ρ E, + σ ρ σ ρ, = + σ σ ( ε ε ) = ρ..3. Model C The serial correlaion condiion in Model C is ε + = ρε + µ +, where E(ε ε +) = ρσ, =,,, T-, and E(ε ε +j) =, j >. Thus, he variance-covariance marix is 87

4 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp E( εε T σ ρ σ ) = ρ σ σ ρ σ ρ σ σ ρ σ σ ρ σ ρ σ σ I should be noed ha he auocorrelaion coefficiens canno be more han.5 and less han -.5, or he model would be flawed..4. The Durbin-Wason es As o ρ =, he hree models become one model where he errors i.i.d. Normal disribuion wih E(ε ) = and Var(ε ) = σ. Thus, he join probabiliy densiy funcion of he errors is where < ε < and =,,,T, and hen hose residuals ha are calculaed from he Original Leas Square (OLS) mehod will be also resriced by he inernal consrain, X T =. The DW es saisic is no noised by σ. Unforunaely, he lack of discussions of he d saisic is no only propery of cenral limied heorem, ha has been discussed by Lee (4b), bu also he effec of nonzero auocorrelaion coefficiens on he disribuions of he d saisic which has differen variance-covariance marices in hree models. As o he hypoheses, H : ρ = ρ and H : ρ ρ, he join probabiliy densiy funcion is, where Σ = E( ) and < ε < and =,,, T. Durbin and Wason (95, 95) build he d saisic for esing he serial correlaion of he daa when he null hypohesis is a zero auocorrelaion coefficien. The d saisic is DW = T ( ˆε + ˆε ) = T ˆε = where ˆε = Y ˆ Y. However, he mahemaical ransformaions of join probabiliy densiy funcions, from he errors o he residuals and from he residuals o he d saisic, are no - relaion and canno find he jocabian funcions, ha is, and f ( ˆε,..., T ˆε ) = f ( ε,...,ε T ) ( ε,...,ε T ) ( ˆε,..., T ˆε ), ( ) ( ˆε,..., ˆε T ) ( DW ) DW = f ˆε,..., ˆε T. 88

5 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp Because he exacly sampling disribuions of he d saisic canno be found, he compuer simulaion is needed whaever he auocorrelaion coefficien is zero or nonzero. 3. Simulaion procedure The compuaions are performed in C++ on Inel Core i7 deskop. In order o conrol he inernal consrain of regression and o derive he probabiliy densiy funcions of he d saisic, we use a new simulaion process based on random number mehod o overcome he problems of probabiliy ransformaion. Thus, he Durbin-Wason es esimaor can be simulaed under null hypohesis, H : ρ = ρ, where ρ. The compuer repeas 6 calculaions per ime o ge 6 values of he Durbin-Wason es. The research mehod is as followed. Sep : Give he inercep and slope value, β = β = β = = β k =, and he daa se of independen variables. Sep : Ge he error daa se of normal disribuion which sample size is T. Here, he error value is independenly. Sep 3: According o he linear regression model seing and compuing he daa se of dependen variable, Y = XB + ε. Sep 4: Calculae he poin-esimaed values of regression coefficien and geing he esimaed values of dependen variable, Yˆ = XBˆ. Sep 5: Calculae he daa se of residual,. Sep 6: Ge he value of he d saisic. Every ime generae 6 values by repeaing Sep o Sep 6. Those values can generae a frequency able and hen calculae he sampling disribuions and coefficiens. Because 6 values per ime is large enough, he sampling disribuions of he d saisic can be viewed as populaion disribuions. The error of coefficiens beween real value and esimaed value is from / o /. When he sampling disribuions of he d saisic are generaed, he compuer calculaes he means, variances, skewedness, and kurosis coefficiens. The skewedness and kurosis coefficiens can ensure more wheher he sampling disribuions of he d saisic are Normal disribuion or no in he hree models. The paper defines he coefficiens of he d saisic as ρ = X is he auocorrelaion coefficien of he errors. E(DW) = X is he mean of he d saisic. Var(DW) = X3 is he variance of he d saisic. σ(dw) = X4 is he sandard deviaion of he d saisic. γ (DW) = X5 is he skewedness of he d saisic. γ (DW) = X6 is he kurosis of he d saisic. 4. Simulaion resuls Firs, he compuer calculaion depends on he values of independen variables (Appendix II), 6 regressors, he variance and auocorrelaion coefficiens of he errors. The sampling disribuions of he d saisic have four coefficiens which are paerned by he auocorrelaion coefficiens of he errors from -.99 o.99 for Model A and Model B, and form -.49 o.49 for Model C, as shown in Table. The small sample case, T = 57, shows he effec of he auocorrelaion coefficien on he coefficiens of he sampling disribuions of he d saisic. Table. The exreme values of he coefficiens of in hree models when he auocorrelaion coefficien is nonzero (T = 57, k =6) X X3 X4 X5 X6 Model A Max Min The sofware of Durbin-Wason es is provided by C.C.C. Ld. The sofware of Durbin-Wason es model (Model B) is available online on he websie: hps:// The radiional Durbin-Wason es model is based on Imhof (96) and Pan (968). 89

6 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp Model B Max Min Model C Max Min Table illusraes he maximum and minimum of five coefficiens in hree models. By comparison of Model A and B, wo models have he same minimum of E(DW), maximum of Var(DW) and σ(dw), bu slighly differen maximum of E(DW), minimum of Var(DW) and σ(dw). Moreover, Model B is more posiive-skewed and cenralized han Model A. However, Model C has he mos exreme differences of E(DW), γ (DW) and γ (DW) han Model A and Model B, excep for maximum of Var(DW) and σ(dw). Alhough Table shows he five coefficiens of he d saisic, we sill do no know he effec of auocorrelaion coefficiens on he sampling disribuion of he d saisic. Therefore, Table illusraes he plos of four coefficiens where he verical axis is E(DW), Var(DW), γ (DW) and γ (DW), separaely, and he horizonal axis represens he auocorrelaion coefficiens. Those plos assis us o invesigae wheher he d saisic is Normal disribuion and how he auocorrelaion coefficiens affec he sampling disribuion of he d saisic. As o he whole range of ρ in hree models, E(DW) is negaively and linearly relaed wih ρ. This implies ha de(dw) / dρ <. The plo of E(DW) also passes hrough around.8 (Model A and B) and.7 (Model C) as ρ =. The reason is ha he negaive ρ causes he errors and he residuals o flucuae up and down from o + period, however, he posiive ρ leads o one and fixed direcion for he errors and residuals. Second, Var(DW) is an invered-u shape. This implies ha he higher he ρ is, he lower he Var(DW) is. However, he maximum of Var(DW) occurs a ρ =. in Model A, a ρ =. in Model B and a ρ = -. in Model C. This also shows ha he maximum of Var(DW) is no a ρ =. This is because he differen assumpion of variance-covariance marix. From he view of E(DW) and Var(DW), E(DW) canno be used o derive Var(DW) because he linear relaionship canno represen he U-shape relaionship, especially when he auocorrelaion coefficien is nonzero. Table also illusraes he plos of he skewed and kurosis coefficiens. The plos of skewed coefficien are cubic shape which shows he higher he ρ is, he higher he skewed coefficien is. The skewed coefficiens are posiive when ρ >. Model C has considerable shape of skewed coefficiens by comparison wih Model A and Model B. Alhough he kurosis coefficien is.8978 in Model A,.8975 in Model B, and 899 in Model C when ρ =, he minimum of he kurosis coefficien occurs in ρ =. in hree models. If ρ becomes larger han., he kurosis coefficien increases, in paricular, he higher he negaive ρ is, he higher he kurosis coefficien is in Model A and Model B. The plos show ha he assumpions of Model B lead o he higher kurosis coefficien han Model A when ρ becomes larger and close o higher relaion. We also find ha he kurosis coefficien is larger han 3 when ρ -.37 in hree models, bu occurs when ρ >.39 in Model B and Model C, and when ρ >.38 in Model A. Thus, we can obain he proposiion as follows. Proposiion. () de(dw) / dρ <. When ρ =, E(DW) =. accurae o he second decimal place. () When ρ, dvar(dw) / dρ > and dvar(dw) / dρ < when ρ >. The second-order condiion is d Var(DW) / dρ <. (3) dγ (DW) / dρ >. When ρ =, γ (DW) = -. accurae o he second decimal place. (4) When ρ, dγ (DW) / dρ < and dγ (DW) / dρ > when ρ >. The second-order condiion is d γ (DW) / dρ >. In he small sample case, he assumpions of E(ε ε +) and he variance- covariance marix leads o he differences among hree models when he samples and he number of regressors are he same. This is because Model B has he fixed variance in he assumpion of firs-order auoregressive errors, meanwhile, Model C is based on he only one-lagged period effec on he errors. The four coefficiens in hree models show ha he sampling disribuions of he d saisic are no Normal disribuion in he range of ρ. One reason is from Lee (4a, 4b), oher reason is ha nonzero ρ disurbs he errors, he residuals, is mahemaical combinaion and he variance-covariance marix, hus, he d saisic canno display a Normal disribuion. 9

7 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp X Table. The coefficiens in hree models when he auocorrelaion coefficien is nonzero (T = 57, k =6) Model A Model B Model C X3 X5 X6 Due o he relaionship of each coefficien and ρ in Table, we can esimae each coefficien of he d saisic by he auocorrelaion coefficiens, ha is, regress each coefficien on ρ by curve-linear regression mehod, which is based on he Taylor expansion funcion, in Table 3. The horizonal axis is he values of ρ and he verical axis is he values of each coefficien. I is noed ha he hree models indicae ha he d saisic is asymmeric a and has significan difference in γ (DW) and γ (DW), in paricular, when he null hypohesis is H : ρ = ρ. Table 3. The esimaion of each coefficien in hree models when he auocorrelaion coefficien is nonzero Symbol Model A Model B Model C X X3 9

8 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp X5 X6 The Appendix III illusraes he residual plos of each coefficien afer esimaion. The residual plos show ha he model seing leads o he differen effec of ρ on he coefficiens of he d saisic, even he shapes of each coefficien are as similar as possible. Moreover, Model A and Model B have similar coefficiens, bu he residual plos are considerably differen wih each coefficien. The special variancecovariance marix assumpion leads o he residual plo of Model C differen from ohers. Due o he considerable pahs of coefficiens as a change of auocorrelaion coefficiens, he d saisic is sill sensiive and is used for hypohesis esing in hree models when he null hypohesis is H : ρ = ρ. 4.. Robus analysis When he samples are large enough, he hree models have he same sampling auocorrelaion coefficien, ρ(ε, ε +) = ρ, however, have differen values of E(ε ε -), ha is, E(ε,ε + ) = ρσ ρ ρσ, Model A, Model B, C. Thus, he zero auocorrelaion coefficien leads o ρ(ε, ε +) = ρ(ε, ε +) = in he hree models. If he null hypohesis is nonzero auocorrelaion coefficien, H : ρ = ρ, and T is infinie, hen E(DW) = ( ρ ). () The expeced values of he d saisic is a consan value away from means ha k has no impac on he robus means of he d saisic whaever he auocorrelaion coefficien is. Furhermore, E(DW) is negaively and linear relaed wih ρ as shown in (). However, E(DW) insufficienly represens he informaion of he sampling disribuions of he d saisic when T is large enough. The second o fourh rows of Table 4 illusrae he effec of he higher momens on he sampling disribuions of he d saisic. The second row shows ha Model A and Model B have he same shape of variance, bu are differen from Model C. moreover, Var(DW) is affeced considerably by he posiive auocorrelaion coefficien in Model A and by he negaive auocorrelaion coefficien in Model B. We also find ha he higher he ρ is, he larger he difference beween Model A and Model C (Model B and Model C) is. The hird row illusraes ha he higher posiive ρ leads o ha () he skewed coefficiens of Model A are larger han ha of Model B and () here is a larger difference from he γ (DW) of Model A (or Model B) minus he γ (DW) of Model C. However, he larger negaive ρ induces in he larger difference from γ (DW) of Model C minus γ (DW) of Model A (or Model B). We also can find ha dγ (DW ) dρ. 9

9 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp The fourh row illusraes ha he smalles values of γ (DW) occurs in ρ =. When he ρ becomes larger, γ (DW) increases in he hree models, in paricular, he paern of γ (DW) has a kinked poin a ρ =.. The relaionship beween γ (DW) and ρ is dγ (DW ) dρ and >,if ρ < <,if ρ >, dγ (DW ) dρ <. Table 4. The comparison of he changes beween hree models coefficiens and auocorrelaion coefficien (T = and k =6) Model A vs Model B Model A vs Model C Model B vs Model C E(DW) Var(DW) skewness kurosis Comparing wih Table and 4, he sampling disribuions of he d saisic have he following properies. Proposiion. A small and large sample cases, () E(DW) passes hrough when null hypohesis is ρ = in hree models. 93

10 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp () E(DW) is negaively and linearly relaed wih ρ in hree models (3) Var(DW) is invered-u relaed wih ρ in hree models. (4) Model A and B have he same shape of Var(DW), γ (DW) and γ (DW), ha are differen from Model C. Table 4 also shows ha he sampling disribuion of he d saisic in hree models is a Normal disribuion wih E(DW) = due o γ (DW) = and γ (DW) = 3. This is an evidence of Durbin and Wason (95, 95) when he samples are large enough. The second propery is ha he higher he posiive ρ is, he more posiive-skewed he sampling disribuions of he d saisic are. The small sample case in Table 3 explains ha he differen variance-covariance marix assumpion affec he effec of he auocorrelaion coefficien on he sampling disribuions of he d saisic, ha have differen expeced values and oher coefficiens as shown in Table, and 3. When he samples become large, Table 4 shows he same expeced value, variance and skewed coefficiens among hree models even hrough he kurosis coefficien is a lile bi differen among he hree models. Therefore, he large samples can eliminae he variance-covariance marix assumpion and lead he hree models o become one model. 5. Conclusions The paper runs compuer simulaion of serial correlaion es for an example of Durbin-Wason es esimaor when he errors have nonzero auocorrelaion coefficien in firs-order auoregressive model. We ry o compare hree models o show he effec of nonzero auocorrelaion coefficiens on he sampling disribuions of he d saisic. The resuls can be divided wih hree pars. The firs resul is from he viewpoin of he sample size. We find ha whaever he sample size is, he expeced values, variances, skewed and kurosis coefficiens have he same paerns of he auocorrelaion coefficiens in hree models, separaely, bu par of values are no he same. We also find ha he assumpions of variance-covariance marix can be eliminaed by he increasing samples, herefore, he sampling disribuions of he d saisic have he same expeced values in hree models whaever he auocorrelaion coefficiens are. This resul implies ha in he long run, hree ime series models have he same expeced values, ( ρ ), ha are differen from he small sample case due o he expeced values of Model C. The second resul is from he view of he null hypohesis wih zero auocorrelaion coefficien. We show ha he higher he posiive auocorrelaion coefficien is, he lower expeced values and variances of he d saisic are, bu he higher he skewed and kurosis coefficiens are. There are reversed resuls in he siuaion of he negaive auocorrelaion coefficiens. The hird resul is from he perspecive of whole paerns of each coefficien. We shows ha he auocorrelaion coefficiens are negaively and linearly relaed wih expeced values, invered-u relaed wih variances, cubic relaed wih skewed coefficiens and U-quadraic relaed wih kurosis coefficiens when he auocorrelaion coefficien is from he minimum o he maximum in hree models whaever he samples are. The hree resuls can supplemen he lieraures abou he serial correlaion es for an example of he sampling disribuions of he d saisic. 6. References Durbin, J. and Wason, G.S., 95. Tesing for Serial Correlaion in Leas Squares Regression. I. Biomerika, 37, pp Durbin, J. and Wason, G.S., 95. Tesing for Serial Correlaion in Leas Squares Regression. II. Biomerika, 38, pp Imhof, J.P., 96. Durbin-Wason Tes for Serial Correlaion wih Exreme Sample Sizes or Many Regressors. Biomerika, 48, pp Lee, M.Y., 4a. Compuer Simulaes he Effec of Inernal Resricion on Residuals in Linear Regression Model wih Firs-order Auoregressive Procedures, Journal of Saisical and Economeric Mehods, 3, pp. -. Lee, M.Y., 4b. On Durbin-Wason Tes Saisic Based on Z Tes in Large Samples, working paper. Pan, J.J., 968. Disribuions of he noncircular serial correlaion coefficiens. American Mahemaical Sociey and Insiue of Mahemaical Saisics Seleced Translaions in Prob- abiliy and Saisics, 7, pp Savin, N.E. and Whie, K.J., 978. Esimaion and Tesing for Funcional Form and Auocorrelaion: A Simulaneous Approach. Journal of Economerics, 8, pp

11 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp Appendix I The model of Durbin and Wason (95) is shown in secion. Based on ε + = ρ ε + µ + and ε =, he firs error is ε = µ and hen subsiue ino ε + = ρ ε + µ + and obain he second error, ε = ρ ε + µ, E ε ( ) σ. ( ) =,Var ( ε ) = + ρ is Thus, E ( ε ε ) = ρ σ and he sampling auocorrelaion coefficien of he firs and second errors ρ ( ε,ε ) = ρ ( + ρ ). Following he same calculaed sep, we can derive ha ε 3 = ρ ε + µ 3 = ρ µ + ρ µ + µ 3, ( ) σ, E ε ε 3 ( + ρ ) ( + ρ ) ( + ρ + ρ 4 ), E ( ε 3 ) =,Var ( ε ) = + ρ + ρ 4 ρ ( ε,ε 3 ) = ε 4 = ρ ε 3 + µ 4 = ρ 3 µ + ρ µ + ρ µ 3 + µ 4, E ( ε 3 ) =,Var ( ε ) = + ρ + ρ 4 + ρ 6 ρ ( ε,ε 3 ) = ( ) = ρ + ρ ( ) σ, E ε 3 ε 4 ( + ρ + ρ 4 ) ( + ρ + ρ 4 ) ( + ρ + ρ 4 + ρ 6 ), Thus, he +h error is ha + ( ) ε + = ρ ε + µ + = ρ + j µ j, =,,,...,T, j= E ( ε + ) =,Var ( ε + ) = ρ ( ) = ρ ρ E ε ε + ρ ( ε,ε + ) = If becomes infinie, hen + j=( ( ) + j ) j=( ( ) + j ) σ, σ, ρ ( ρ ) + j ( ρ ) + j + µ j=( j ) ρ j=( ) ( ) + j µ j=( j ) ( ) σ, ( ) σ, ( ) = ρ + ρ + ρ 4 95

12 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp Var( ε ) = E ε ε + σ ( ρ ), ( ) = ρ σ ρ ( ε,ε + ) = ρ Appendix II ( ρ ), The values of independen variables are as follows. X X X3 X4 X5 X6 : , , , , , , : , , , , , , 3 : , , , , , , 4 : , , , , , , 5 : , , , , , , 6 : , , , , , , 7 : , , , , , , 8 : , , , , , , 9 : , , , , , , : , , , , , , : , , , , 4.549,.64698, : , , , , , , 3 : , , , , , , 4 : ,.34366, , , , , 5 : , , , , ,.5364, 6 : , , , , , , 7 : , , , , , , 8 : , , , , , , 9 : , , , , , , : , , , , , , : , , , , , , : , , , , , , 3 : , , , , , , 4 : , , , , , , 5 :.79634, , , , , , 6 : , , , , , , 7 : , , , , , , 8 :.98896, , , , , , 9 : ,.69676, , , , , 3 : , , , , , , 3 : , , , , , , 3 : , , , , , , 33 : , , , , , , 34 : , , , , , , 35 : , , , , , , 36 : 9.656, , , , , , 37 : , ,.37344, , , , 38 : , , , , , , 39 : , , , , , , 4 : , , , , , , 4 : , , , , , , 4 : , , , , , , 43 : , , , , , , 44 : , , , , , , 45 : , , , , , , 46 : , 3.989, , , , , 47 : , , , , , , 48 : , , , , , , 49 : , , , , , , 5 : , , , , , , 5 : , , , , , , 5 : , ,.87865, , , , 53 : , , , , , , 54 : , , , , , , 55 : ,.8968, , , , , 56 : , , , , ,.64477, 57 : , , , , , , 96

13 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp indepeden sample correlaion coefficien r(x,x)= r(x,x3)= r(x,x4)=.5389 r(x,x5)= r(x,x6)= r(x,x3)= r(x,x4)= r(x,x5)= r(x,x6)= r(x3,x4)= r(x3,x5)= r(x3,x6)= r(x4,x5)= r(x4,x6)= r(x5,x6)= Appendix III Table 3 shows he esimaed line where we regress each coefficien of he d saisic on he auocorrelaion coefficiens in he auoregressive models wih T = 57, k = 6 and hree variance-covariance marices. Appendix III shows he esimaed funcion of each coefficien and he corresponding residual plo. Esima ed funcio n of X X Table A-III. The esimaed funcion and residual plo of each coefficien in hree models Model A Model B Model C X= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^6+ X= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^6+ X= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^6+ Esima ed funcio n of X3 X3= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^9+ X3= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^+ X3= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^+ 97

14 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp X3 Esima ed funcio n of X4 X4= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^9+ X4= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^9+ X4= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^9+ X4 Esima ed funcio n of X5 X5= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^8+ X5= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^8+ X5= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^+ X5 Esima ed funcio n of X6 X6= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^5+ X6= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^5+ X6= *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^ *X^5+ 98

15 Lee, M-Y., 4. The Effec of Nonzero Auocorrelaion Coefficiens on he Disribuions of Durbin-Wason Tes Esimaor: Three Auoregressive Models. Exper Journal of Economics, (3), pp X6 Creaive Commons Aribuion 4. Inernaional License. CC BY 99