The Efficiency Of OLS In The Presence Of Auto- Correlated Disturbances In Regression Models

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1 Jornal of Modern Applied Statistical Methods Volme 5 Isse Article 5--6 The Efficiency Of OLS In The Presence Of Ato- Correlated Distrbances In Regression Models Samir Safi James Madison University Alexander White Texas State University Follow this and additional works at: Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Safi, Samir and White, Alexander (6) "The Efficiency Of OLS In The Presence Of Ato-Correlated Distrbances In Regression Models," Jornal of Modern Applied Statistical Methods: Vol. 5 : Iss., Article. DOI:.37/jmasm/ Available at: This Reglar Article is broght to yo for free and open access by the Open Access Jornals at DigitalCommons@WayneState. It has been accepted for inclsion in Jornal of Modern Applied Statistical Methods by an athorized editor of DigitalCommons@WayneState.

2 Jornal of Modern Applied Statistical Methods Copyright 6 JMASM, Inc. May, 6, Vol. 5, No., /6/$95. The Efficiency Of OLS In The Presence Of Ato-Correlated Distrbances In Regression Models Samir Safi Department of Mathematics and Statistics James Madison University* Alexander White Department of Mathematics and Statistics Texas State University The ordinary least sqares (OLS) estimates in the regression model are efficient when the distrbances have mean zero, constant variance, and are ncorrelated. In problems concerning time series, it is often the case that the distrbances are correlated. Using compter simlations, the robstness of varios estimators are considered, inclding estimated generalized least sqares. It was fond that if the distrbance strctre is atoregressive and the dependent variable is nonstochastic and linear or qadratic, the OLS performs nearly as well as its competitors. For other forms of the dependent variable, rles of thmb are presented to gide practitioners in the choice of estimators. Key words: Atocorrelation, atoregressive, ordinary least sqares, generalized least sqares, efficiency Introdction Let the relationship between an observable random variable y and k explanatory variables X, X,, X k in a T-finite system be specified in the following linear regression model: y = X β + () where y is a ( T ) vector of observations on a response variable, X is a ( T k) design matrix, β is a ( k ) vector of nknown regression parameters, and is a ( T ) random vector of distrbances. For convenience, it is assmed that *This article was accepted while Samir Safi was at James Madison University. He is now an Assistant Professor of Statistics at the Islamic University of Gaza. His research interests are in time series analysis, concerning the comparison of estimators in regression models with atocorrelated distrbances and efficiency of OLS in the presence of atocorrelated distrbances. Alexander White is Associate Professor in Mathematics Edcation. His research interests are in mathematics edcation, statistics and mathematical finance. X is fll colmn rank k < T and its first colmn is 's. The ordinary least sqares (OLS) estimator of β in the regression model () is ( X X) X y ˆ β = () In problems concerning time series, it is often the case that the distrbances are, in fact, correlated. Practitioners are then faced with a decision, se OLS anyway, or try to fit a more complicated distrbance strctre. The problem is difficlt becase the properties of the estimators depend highly on the strctre of the independent variables in the model. For more complicated distrbance strctres, many of the properties are not well nderstood. If the distrbance term has mean zero, i.e. E() =, bt is in fact, atocorrelated, i.e. Cov ( ) = σ, where is a T T positive definite matrix and the variance σ is either known or nknown positive and finite scalar, then the OLS parameter estimates will contine to be nbiased, i.e. E ( βˆ ) = β. Bt it has a different covariance matrix; Cov ( ˆ ) = σ ( X X) X X ( X X). β (3) 7

3 8 OLS IN THE PRESENCE OF AUTO-CORRELATED DISTURBANCES The most serios implication of atocorrelated distrbances is not the reslting inefficiency of OLS, bt the misleading inference when standard tests are sed. The atocorrelated natre of distrbances is acconted for in the generalized least sqares (GLS) estimator given by: ( X X) X y ~ β = (4) which is nbiased, i.e. ( ) = β covariance matrix ~ ( ) = σ ( X X). E β ~, with Cov β (5) The speriority of GLS over OLS is de to the fact that GLS has a smaller variance. According to the Generalized Gass Markov Theorem, the GLS estimator provides the Best Linear Unbiased Estimator (BLUE) of β. Bt the GLS estimator reqires prior knowledge of the matrix correlation strctre, Σ. The OLS estimator βˆ is simpler from a comptational point of view and does not reqire a prior knowledge of Σ. A common approach for modeling nivariate time series is the atoregressive model. The general finite order atoregressive process of order p or briefly, AR(p), is t = φ t + φ t + + φp t p + ε t, ε t ~ i.i.d. N(, σ ) (6) ε There are nmeros articles describing the efficiency of the OLS coefficient estimator βˆ, which ignore the correlation of the error, relative to the GLS estimator β ~, which takes this correlation into accont. One strand is concerned with conditions on regressors and error correlation strctre, which garantee that OLS is asymptotically as efficient as GLS (e.g. Chipman, 979; Krämer, 98). The efficiency of the OLS estimators in a linear regression containing an atocorrelated error term depends on the strctre of the matrix of observations on the independent variables (e.g. Anderson, 948; 97; Grenander & Rosenblatt, 957). For a linear regression model with first order atocorrelated distrbances, several alternative estimators for the regression coefficients have been discssed in the literatre, and their efficiency properties have been investigated with respect to the OLS and GLS estimators (e.g. Kadiyala, 968; Maeshiro, 976; 979; Ullah et al., 983). The relative efficiency of GLS to OLS in the important cases of atoregressive distrbances of order one, AR(), with atoregressive coefficient ρ and second order, AR(), with atoregressive coefficients ( φ,φ ) for specific choices of the design vector have been investigated. Bilding on work on the economics and time series literatre, the price one mst pay for sing OLS nder sboptimal conditions reqired investigation. Different designs are being explored, nder which relative efficiency of the OLS estimator to that of GLS estimator approaches to one or zero, determining ranges of first-order atoregressive coefficient, ρ, in AR() distrbance and second order of atoregressive coefficients, ( φ,φ ) in AR() for which OLS is efficient and qantifying the effect of the design on the efficiency of the OLS estimator. Frthermore, a simlation stdy has been condcted to examine the sensitivity of estimators to model misspecification. In particlar, how do estimators perform when an AR() process is appropriate and the process is incorrectly assmed to be an AR() or AR(4)? Performance Comparisons In this section, nmerical reslts are presented sing the formlas in (3) and (5). Focs will be placed on two isses; first, the relative efficiency of GLS estimator as compared with the OLS estimator when the strctre of the design vector, X, is nonstochastic. For example, linear, qadratic, and exponential design vectors with an intercept term inclded in the design vector. Secondly, the relative efficiency of the GLS estimator as compared with the OLS for a stochastic design vector. In the example considered here, a standard Normal stochastic design vector of length was generated. The three finite sample sizes sed are 5,, and for

4 SAFI & WHITE 9 selected vales of the atoregressive coefficients. Both AR() and AR() error processes are considered to discss the behavior of OLS as compared to GLS. Performance Comparisons for AR () Process The relative efficiencies of OLS to GLS are discssed when the distrbance term follows an AR() process, t = ρ t +εt, t =,,, T, assming that the atoregressive coefficient, ρ, is known priori. The three finite sample sizes sed are 5,, and for the elected vales of ρ. 9, evalated in steps of.. Table () shows the relative efficiencies of the variances of GLS to OLS for a regression coefficient on linear trend with an intercept term inclded in the design. For estimating an intercept term, the relative efficiency of the OLS estimator as compared to the GLS estimator decreases with increasing vales of ρ. For small and moderate sample sizes, the efficiency of the OLS estimator appears to be nearly as efficient as the GLS estimator for ρ. 7. In addition, for large size sample data, the OLS estimator performs nearly as efficiently as the GLS estimator for the additional vales of ρ = ±. 9. Frther, the efficiency for estimating the slope mimics the efficiency of the intercept, except for large sample size; the efficiency of the OLS estimator appears to be nearly as efficient as the GLS estimator for ρ ±. 9. The efficiency of GLS estimator to the OLS estimator for the qadratic design agrees with the behavior for the linear design vector. In contrast, the gain in efficiency of the GLS estimator for different design vectors sch as exponential and standard Normal, N(,) Table : Relative Efficiency of GLS to OLS for Linear Design ρ Intercept Slope T = 5 T = T = T = 5 T = T =

5 OLS IN THE PRESENCE OF AUTO-CORRELATED DISTURBANCES compared to the OLS estimator is sbstantial for moderate and large vales of ρ. However, for small vales of ρ the OLS appears to be nearly as efficient as GLS. Performance Comparisons for AR () Process The relative efficiencies of OLS to GLS are discssed for linear, qadratic, and exponential design vectors when the distrbance term follows an AR() process, t = φ t +φt +εt, t =,,,T, assming that the atoregressive coefficients φ CPF φ are known priori. The three finite sample sizes sed are 5,, and for the selected 45 pairs of the atoregressive coefficients. These coefficients were chosen according to stationary φ + φ <, φ φ <, and φ < conditions ( ) and so that ( ) ρ = φ φ is positive. This second condition was chosen since this is the case in most econometric stdies. To demonstrate the efficiency of OLS, consider the linear design vector. When the distrbance term follows an AR() process for the linear design with small sample size, OLS performs nearly as efficiently as GLS for estimating the slope for all AR() parametrizations except when φ' s are close to the stationary bondary. As the sample size increases, the difference between the performance of OLS and GLS decreases. Only when φ =. 9, does OLS perform badly regardless of the sample size. The efficiency of GLS to OLS for the qadratic design mimics the behavior for the linear design. Finally, for exponential and standard Normal design vectors, the efficiency of OLS appears to be nearly as efficient as GLS for φ =. and small vales of φ for all sample sizes. Otherwise, OLS performs poorly. Simlation Stdy In this section, the robstness of varios estimators are considered, inclding estimated generalized least sqares (EGLS). These simlations examine the sensitivity of estimators to model misspecification. In particlar, how do estimators perform when an AR() process is appropriate and it is incorrectly assmed that the process is an AR()? The finite sample efficiencies of the OLS estimator relative to for GLS estimators are compared: the GLS based on the correct distrbance model strctres and known AR() coefficients denoted as GLS- AR(); the GLS based on the correct distrbance model strctres, bt with estimated AR() coefficients denoted as EGLS-AR(); the GLS based on AR() incorrect distrbance model strctres with an estimated AR() coefficient denoted as EIGLS-AR(); and the GLS based on AR(4) incorrect distrbance model strctres with estimated AR(4) coefficients denoted as EIGLS-AR(4). This stdy focses only on AR(p) GLS corrections distrbances which are widely sed in econometric stdies. The Simlation Setp Three finite sample sizes (5,, and ) and three nonstochastic design vectors of the independent variable are sed; linear, qadratic, and exponential. A standard Normal stochastic design vector of length is also generated (Assming that the variance of the error term in AR() process σ ε = ). Frther, observations for each of the AR() error distrbances with for pairs of atoregressive coefficients; (.,-.9), (.8,-.9), (.,-.7), and (.,-.) were also generated. Table () shows the vales of atocorrelation coefficients ρ, ρ, distrbance variances, σ, σ [( )( )] = φ ρ and the relative efficiencies for estimating an intercept β, and the slope, β of GLS to OLS for linear design with T=5, denoted RE( β ), and RE( β ). Looking at the table, it may be seen that the choices (., -.9) and (.8, -.9) give the worst performance of OLS as compared to GLS for estimating ( β, β ) of the regression coefficients and the largest vales of σ. However, the choices (., -.7) and (., -.) give the moderate and best performance of OLS as compared to GLS and the smallest vales of σ. Reslts for other sample sizes and designs demonstrate a similar pattern as in Table ().

6 SAFI & WHITE The regression coefficients β, and β for an intercept and the slope were each chosen to be eqal one. Bresch (98) has shown that β ˆ EGLS β for a fixed design, the distribtion of σ does not depend on the choice for β and σ, and the reslt holds even if the covariance matrix Σ is misspecified. When the design vector is stochastic, the assmption of a fixed design can be constrcted as conditioning pon a given realization of the design, provided that the design is independent of t, Koreisha et al. (). Definition The efficiency of the GLS estimates relative to that of OLS in terms of the mean sqared error of the regression coefficient, ζ ˆ β, j is given by: k ~ ( βij,gls β j ) = i β = (7) j k βˆ β i= ( ) ij,ols where j =,, for for GLS estimates, and k is the nmber of simlations. A ratio less than one indicates that the GLS estimates is more efficient than OLS, and if β is close to one, j j then the OLS estimate is nearly as efficient as GLS estimates. The Simlation Reslts for ζ ˆ β j Tables (3) throgh (6) show the complete simlation reslts of the ratios of the GLS estimators relative to the OLS estimator in terms of the mean sqared error of the regression coefficients, ζ ˆ β and ζ ˆ β in (7), when the serially correlated distrbance follows an AR() process. Each table presents the reslts for the three sample sizes considered, as well as all for selected pairs of AR() parametrizations. Each of the different designs is presented in a separate table. Note that regardless of the sample size, selected design vectors, and AR() parametrizations the efficiency in estimating an intercept, β, and the slope, β, of the regression coefficients is higher for the GLS- AR() estimator than OLS. This reslt emphasizes that GLS is the BLUE. However, OLS performs nearly as efficiently as GLS for all selected sample sizes and designs when Φ = (., -.). This reslt is not srprising since the choice of Φ = (., -.) gives the highest performance of OLS as compared to GLS, in addition, it gives the smallest vales of ρ, ρ, and σ. Table : Atocorrelation Coefficients, Distrbance Variances and the Relative Efficiencies of GLS to OLS for Standardized Linear Design with T = 5 (, φ ) φ ρ ρ σ RE( β ) RE( β ) (., -.9) (.8, -.9) (., -.7) (., -.)

7 OLS IN THE PRESENCE OF AUTO-CORRELATED DISTURBANCES When the order of the distrbance term is nder estimated, i.e. EIGLS-AR(), the GLS estimate performs poorly. In fact, OLS is more efficient for nearly every sitation considered here. For example, when Φ = (.8, -.9) for ζ ˆ, ˆ = qadratic design with T = 5, ( ) (.796) β ζ β.479, as shown in Table (3). This shows that EIGLS-AR() can be mch less efficient than OLS. The poor performance of EIGLS-AR() relative to OLS is most marked when the sample size is relatively estimation is smaller than an appropriate estimated AR strctre. This sggests the small (i.e. T = 5) and the order of the atoregressive process sed in the GLS srprising reslt that OLS may often be better than assming an AR() when the actal process is AR(). However, for the choice of Φ = (., -.) there is little difference between OLS and EIGLS-AR(). For example, for linear design with T=, ( ζ ˆ ˆ β, ζ ) β = (.9998,.9984) as presented in Table (4). Table 3: Efficiency for MSEs of the Regression Coefficients of the GLS Estimators Relative to OLS Estimator for Qadratic Design Size Estimator ( Φ, Φ ) (., -.9) (.8, -.9) (., -.7) (., -.) β 5 GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) β β β

8 SAFI & WHITE 3 Table 4: Efficiency for MSEs of the Regression Coefficients of the GLS Estimators Relative to OLS Estimator for Linear Design ( Φ, Φ ) Size Estimator (., -.9) (.8, -.9) (., -.7) (., -.) β β β β 5 GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) This reslt is expected becase the choice of φ =. indicates that the serially correlated distrbance very nearly AR() since φ is close to zero. To frther demonstrate the efficiency of OLS, consider the qadratic and linear designs. OLS is nearly as efficient or more efficient in estimating ( β, ) than the GLS estimators; EGLS-AR(), and EIGLS-AR(4), for moderate and large sample sizes (i.e. T= and ) with AR() parametrizations Φ = (., -.7) and (., -.) Tables (3) and (4). However, there are examples where OLS performs poorly as well. For the exponential design, OLS is nearly as efficient as EGLS-AR(), and EIGLS-AR(4) for all sample sizes only when Φ = (., -.). Otherwise, OLS performs poorly as shown in Table (5). For example, when T = 5 with Φ = (., -.9), ζ ˆ β = (.35,.8). However, even in this case, the performance of the OLS estimator for estimating the intercept is not bad, ζ ˆ β = (.756,.766). In fact, the performance of OLS is always better for estimating the intercept than the slope. For the standard Normal stochastic design model, OLS fares more poorly. Only for Φ = (., -.) does the efficiency of OLS match GLS as shown in Table (6). However, regardless of the sample size, OLS performs as nearly as efficiently or better than EIGLS-AR() for all selected atoregressive coefficients for estimating β.

9 4 OLS IN THE PRESENCE OF AUTO-CORRELATED DISTURBANCES Table 5: Efficiency for MSEs of the Regression Coefficients of the GLS Estimators Relative to OLS Estimator for Exponential Design ( Φ, Φ ) (., -.9) (.8, -.9) (., -.7) (., -.) Size Estimator β β β β 5 GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4)

10 SAFI & WHITE 5 Table 6: Efficiency for MSEs of the Regression Coefficients of the GLS Estimators Relative to OLS Estimator Standard Normal Stochastic Design ( Φ, Φ ) (., -.9) (.8, -.9) (., -.7) (., -.) Size Estimator β β β β 5 GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) GLS-AR() EGLS-AR() EIGLS-AR() EIGLS-AR(4) Discssion In investigating the simlation reslts in the previos section, the following significant reslts were observed. First and foremost, it was noticed that regardless of the sample size for all design strctres and selected atoregressive coefficients, the efficiency in estimating an intercept, β, and the slope, β, of the regression model is higher for the GLS estimator based on the correct distrbance model strctres and known AR() coefficients. This reslt is expected since GLS is BLUE, bt becase GLS reqires a priori knowledge of Σ, this is not a viable option. In addition, the relative efficiency of OLS is better than EIGLS-AR() in estimating ( β, β ) for all sample sizes and nonstochastic design vectors. The relative efficiency of OLS to be sperior to that of EIGLS in estimating the slope when T=5 with AR() parametrization (.8, -.9) was also observed. This choice of (.8, -.9) gives the highest first-order atoregressive coefficient ( ρ =.4 ) and largest variance of the error process ( σ = ) among the other choices of AR() parametrizations. This explains the poor relative performance of OLS to GLS for this choice of parameter.

11 6 OLS IN THE PRESENCE OF AUTO-CORRELATED DISTURBANCES However, from Table (3) throgh Table (6), it may be seen that the performance of EIGLS- AR() is even worse. This appears to occr becase AR() parametrization (.8, -.9) prodces large vales of ρ, ρ in absolte vale ( ρ =. 563 ) and distrbance variance comparing to the other parameter choices. This means sing OLS is better than assming another incorrect error process. The third general conclsion from the simlation stdy is that regardless of the sample size, all of the estimators perform eqally well with AR() parametrization (., -.). This reslt is not srprising becase the choice of (., -.) gives the smallest variance of the process ( σ =.446), which is sfficiently close to the variance of standard OLS. Forth, for all stochastic and nonstochastic design vectors, the differences in the relative efficiency of OLS and all GLS estimators in estimating β with a few expected exceptions are negligible. In fact, this is so even when the variance of the process is large, in other words, when AR() parametrizations are (., -.9) and (.8, -.9). Similar to reslts for section, when the design vector is linear or qadratic, the relative efficiency of OLS is nearly as good as the EGLS-AR() and EIGLS-AR(4) estimators for moderate and large sample sizes for estimating β with small variance of the distrbances. It is observed that the differences in the relative efficiencies of GLS-AR(), EGLS- AR(), and EIGLS-AR(4) in estimating ( β, β ) are insignificant. Hence, when confronted with an error with nknown order p, it appears that sing AR(4) is the best bet. Finally, OLS may often be more preferable than assming an AR() process when the actal process is AR(). In other words, it is sometimes better to ignore the atocorrelation of the distrbance term and se the OLS estimation rather than to incorrectly assme the process is an AR(). Ftre Research Perhaps, even more important than the efficiency of the different estimation methods in these models, is the effect on forecasting performance. Koreisha et al. (4) investigated the impact that EIGLS correction may have on forecast performance. They developed a new procedre for generating forecasts for regression models with ato-correlated distrbances based on OLS and a finite AR process. They fond that for predictive prposes there is not mch gained in trying to identify the actal order and form of the ato-correlated distrbances or sing more complicated estimation methods sch as GLS or MLE procedres, which often reqire inversion of large matrices. It is necessary to extend Koreisha et al. (4) reslts for different design vectors of the independent variables inclding both stochastic and nonstochastic designs instead of sing one independent variable generated by an AR() process as in their investigation. A second important consideration is the estimation of the standard errors of the estimators. In practice, if one were sing a statistical package to compte the OLS estimators the variance estimate prodced wold be based on σ ( ) X X, which may be biased for the tre variance ( ) ( X σ X X X X X ). For GLS estimation ( Σ known), on the other hand, the variance estimate is nbiased for the tre variance of the GLS estimator. It is nclear, however, how the variance estimators for EGLS estimation behave. The impact that the variance estimators may have on inference based on the OLS estimator is crrently being investigated. Finally, the long range goal is the creation of gidelines or rles of thmb which will aid the practitioner when deciding which regression estimation procedre to se. Conclsion This article has investigated an important statistical problem concerning estimation of the regression coefficients in the presence of atocorrelated distrbances. In particlar, the comparison of efficiency of the ordinary least sqares (OLS) estimation to alternative procedres sch as generalized least sqares (GLS) and estimated GLS (EGLS) estimators in the presence of atocorrelated distrbances was

12 SAFI & WHITE 7 discssed. Both stochastic and non-stochastic design vectors were sed with different sample sizes. It was fond that regardless of the sample size, design vector, and order of the ato-correlated distrbances, the relative efficiency of the OLS estimator generally increases with decreasing vales of the distrbance variances. In particlar, if the distrbance strctre is a first or second order atoregressive and the dependent variable is nonstochastic and linear or qadratic, OLS performs nearly as well as its competitors for small vales of the distrbance variances. The gain in efficiency of the GLS estimator for different design vectors sch as exponential and standard Normal compared to the OLS estimator is sbstantial for moderate and large vales of the atoregressive coefficient in the case of an AR() process and large vales of the distrbance variance in the presence of an AR() process. However, for small vales of the atoregressive coefficient and distrbance variance the OLS estimator appears to be nearly as efficient as the GLS estimator. It was also fond that if the error strctre is atoregressive, and the dependent variable is nonstochastic and linear or qadratic, the OLS estimator performs nearly as well as its competitors. When faced with an nknown error strctre, however, AR(4) may be the best choice. References Anderson, T. W. (948). On the theory of testing serial correlation. Skandinavisk Aktarietid skrift, 3, Anderson, T. W. (97). The Statistical analysis of time series. New York:, N.Y: Wiley. Bresch, T. (98). Usefl invariance reslts for generalized regression models. Jornal of Econometrics, 3, Chipman, J. S. (979). Efficiency of least sqares estimation of linear trend when residals are atocorrelated. Econometrica, 47, 5-8. Chodhry, A., Hbata, R. & Lois, R. (999). Understanding time-series regression estimators. The American Statistician, 53, Grenander, U. & Rosenblatt, M. (957). Statistical analysis of stationary time series. New York, N.Y.: Wiley. Jdge, G. G., Griffiths, W. E., Hill, R. C.. Ltkepohl, H., & Lee, T. C. (985). The theory and practice of econometrics. New York, N.Y.: Wiley & Sons Inc. Kadiyala, K. R. (968). A transformation sed to circmvent the problem of atocorrelation. Econometrica, 36, Koreisha, S. G. and Fang, Y. (). Generalized least sqares with misspecified serial correlation strctres. Jornal of the Royal Statistical Society, 63, Series B, Koreisha, S. G. and Fang, Y. (4). Forecasting with serially correlated regression models. Jornal of Statistical Comptations and Simlation, 74, Kramer, W. (98). Finite sample efficiency of ordinary least sqares in the linear regression model with atocorrelated errors. Jornal of the American Statistical Association, 75, 5-9. Maeshiro, A. (976). Atoregressive transformation, trended independent variables and atocorrelated distrbances terms. Review of Economics and Statistics, 58, Maeshiro, A. (979). On the retention of the first observations in serial correlation adjstment of regression models. International Economic Review,, Ullah, A., Srivastava, V. K., Magee, L., & Srivastava, A. (983). Estimation of linear regression model with atocorrelated distrbances. Jornal of Time Series Analysis, 4, 7-35.