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The properties of some two step estimators of ARMA Models
•
EI~SEVIER
MATHEMATICS AND COMPUTERS I N SIMULATION
Mathematics and Computers in Simulation 43 (1997) 451-456
The properties of some two step estimators of ARMA Models C.R. McKenzie* Osaka School of International Public Policy, Osaka University, Toyonaka, Osaka 560, Japan
Abstract
This paper analyzes the large sample properties of several two step estimators that have recently been suggested for estimating autoregressive moving-average models. As these estimators typically involve generated regressors, the generated regressor literature suggests that, in general, they will be inefficient and their estimated formula standard errors will be inconsistent estimates of the true standard errors. Deriving the covariance matrix of the true disturbances in these models enables consistent estimates of the true standard errors and efficient generalized least squares estimators to be computed.
1. Introduction
Univariate autoregressive moving-average (ARMA) models are commonly used for modelling and forecasting time-series data. Various techniques are available for estimating these models, the computationally expensive maximum likelihood method which is available in SHAZAM and MICROFIT but not PC-GIVE ([1-3]) and computationally simple two step methods (e.g. [4-6]). The two step methods typically rely on estimates of unobserved variables and as a result could be expected to suffer from a generated regressor problem (see [7-9] for a general discussion of this issue). Generated regressors typically cause the error term to become serially correlated and heteroscedastic in small and large samples making ordinary least squares (OLS) an inefficient estimator. As McKenzie and McAleer [9] indicate, there are various ways around this problem; for example, by applying generalized least squares (GLS) using a consistent estimate of the covariance matrix of the true errors, or some other transformation of the equation to eliminate asymptotically the serial correlation and heteroscedasticity in the errors. In this paper, several two step estimators are discussed and their asymptotic properties are derived very simply based on the approach developed in [9]. Section 2 contains a discussion of the model and the two step estimators considered in the paper. The asymptotic properties of these estimators are analyzed in Section 3 while Section 4 contains some concluding comments.
* Corresponding author. Tel.: 81-6-850-5622; fax: 81-6-850-5274; e-mail: [email protected]. 0378-4754/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 0 3 1- 1
C.R. Mc Kenzie /Mathematics and Computers in Simulation 43 (1997) 451-456
452
2. The model and estimators The ARMA (p, q) model for a mean adjusted series is P
q
(1)
Zt= Z~iZt_i+at--ZOjat~ j=l
i=l
where at is a normal white-noise process with zero mean and variance (r2, and t ---- 1 . . . T. It is assumed that the autoregressive and moving average parameters satisfy the conditions for stationarity and invertibility, respectively. Since the moving average component of Eq. (1) is assumed to be invertible, Eq. (1) can be rewritten as an infinite order autoregression, namely, Z, = ) j
(2)
7 ~ i Z t - i + at.
i=l
Truncating the autoregression to a finite order, say L, where L < T, and ignoring the error induced by the truncation gives L Zt =
(3)
~_~ 7fiZt i + at. i=1
This equation can be estimated by OLS to obtain estimates of 7ri, #i, and the innovation series where
at,
at
L
Clt = Zt - Z
(4)
frizt-i
i=1
These estimates of the innovation series can then be substituted back into Eq. (1) to give P
q
z, -- Z
Z 0j ,_j + v,
i:1
(5)
j=l
where v, : a t - ~-~)ql O j ( a t _ j - gtt~). OLS applied to Eq. (5) is the second stage of Hannan and Rissanen's [4] three stage procedure for efficiently estimating the parameters of Eq. (1). Alternatively, the second stage of Koreisha and Pukkila's [6] three stage estimation procedure is OLS applied to a slight variation of Eq. (5): P
Y, = ~
q
~iZt-i - Z
i=1
(6)
Oj~lt--J "~ Wt,
j=l
where y, = zt - gzt, and w t = at -- glt rewritten as a moving average of et:
-- ~J
l Oj(at-j
-- (it-j) •
Defining
et
= a,
-
~l t,
then wt can be
q
wt = e,-
~ Ojet_j. j=l
(7)
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451~t56
453
Another slight variation of Eq. (5) is P q xt = ~ ~iZt-i - Z Oj~lt~ -~- bit' i=1 j=l
(8) ,
where xt=zt-f~q_lOjfit-j, ut=at-~~q=lOjat~-~~J=l(Oj-Oj)(at~-ht~), and the t~j are consistent estimates of Oj obtained by, say, applying OLS to Eq. (5). OLS applied to Eq. (8) will be inconsistent since the regressors are correlated with ut. In ut the term y-~q_l(Oj-Oj)(at~-fitS) is asymptotically irrelevant so it is easy to see that applying the moving average version of the CochraneOrcutt method to Eq. (8) using Oj will be asymptotically efficient. This is in fact the third stage of Hannan and Rissanen's [4] three stage procedure. The structure of the errors in Eqs. (5) and (6) suggests there may be efficiency gains by applying GLS to Eqs. (5) and (6). The form of wt in Eq. (7) suggests that wt follows a moving average of order q, so that in this case the moving average equivalent of the Cochrane-Orcutt transformation applied to Eq. (6) seems appropriate. This is in fact the third stage of Koreisha and Pukkila's [6] three stage estimator. In the standard linear regression model, it is well-known that a combination of lagged dependent variables and serially correlated errors generally leads to OLS being inconsistent suggesting a potential problem for OLS applied to Eq. (6). However, as is shown later, this is not the case here.
3. Asymptotic properties Eq. (3) can be written in the more compact matrix form
Z = WH + A
(9)
where Z, Z-i, A and A i are T× 1 vectors with typical jth element zj, zj-i, aj and aj-i, respectively, W = [Z_I ... Z_L] and H ' = [7rl... 7rL]. Estimation of Eq. (9) by OLS gives the following estimators of 7r, A and A-i
11= (W'W) 1W'Z .4
= Z -
W l) =
(I -
10) 11)
W(W'W)-lWt)A
.~ i ~ Z - i - W_il~---- A _ i - W _ i ( W t w ) - l w t A
12)
where I/V i = [ Z _ i _ l . . . Z_i_L] , and A and J~--i are T x 1 vectors with typical jth element hj and aj-i, respectively. Eq. (5) can be rewritten as Z = X/3 + ¢ where X = [Z-1... Z_p, - A - 1 • .. - / I - q ] , / ~ t typical jth element vj.
(13) = [ ~ 1 . . . q~p, 0 1 . . . Oq], and ~ is a vector T × 1 vector with
454
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
For OLS applied to Eq. (13) to be consistent requires that (X'~/T) = 0. Defining e and e_j to be T x 1 vectors with typical elements ei and el-j, respectively, then using Eqs. (10)-(12) ~, c_j and ~ can be rewritten as
(14)
= A - A = W(W'W)-' W'A e_j = A ~ - A ~ q
=A - ~
= W _ j ( W f W ) -1
W'A
(15)
q
Oje_j = A - ~
j=l
Oj(A~ - A~) = SA
(16)
j=l
where S = I - $1 and S1 = ~-]~q_, OjW j(W'W) 1W'. Since plim(W'a/T) = 0, it is easily shown that plim(X'c~/T)=O ( j = 1 .ql-In addition, plim(,4 jA/T)=plim(A~_jA/T)=O ( j = 1 . . . q ) , so plim(X~A/T) = 0 ensuring the consistency of OLS applied to Eq. (13). From Eqs. (14)-(16), it is clear that if Wand W j were non-stochastic, then any et (and any vt) is a linear combination of all the at. So that et would not be serially uncorrelated and homoscedastic (cf [6]). If all the elements of W were non-stochastic, then ~2 = Var(~)/~r 2 = SS' so that we can see immediately that ~ in Eq. (13) is heteroscedastic and serially correlated. Here it suffices to note that under the assumption of the model X ' ( / T ____+DN(O, cr2V)), where V =plim(X'f2X/T). Since plim[R'~X/T] ¢ 0 for any non-zero R such that plim[R'X/T] = 0, OLS applied to Eq. (13) will not be efficient (see [ 10], Theorem 6.1.1 and [9], Theorem 1). The form of V also indicates that the formula standard errors produced by an OLS regression will not be consistent estimates of the true standard errors (see [9] Theorem 1). So even though this OLS regression provides consistent estimates of the parameters it cannot be used for conducting hypothesis testing. The application of GLS to Eq. (13) requires the inversion of the T x T matrix, Y2. The complicated nature of f2 suggests that GLS applied to Eq. (13) is not going to lead to a computationally simple procedure. What about the second and third stages of Koreisha and Pukkila's [6] procedure applied to Eq. (6)? Eq. (6) can be rewritten as Z-,4
= X/3 + 7/
(17)
where ~/= ¢ - A = K(W'W)-Iw'A and K = [W - ~q=, OjW_j]. Given that plim(X'¢/T) = 0 has a l r e a d y been shown, the c o n s i s t e n c y of OLS applied to Eq. (17) is assured since plim(X'.4/T) = p l i m ( X ' A / T ) = 0 . If all the e l e m e n t s o f W were n o n - s t o c h a s t i c , then A = Var(7/)/a 2 = K(W~W) 1K' which does not have the form of the covariance matrix of a moving average of order q. Here it suffices to note that under the assumptions of the model X~I/T ._~o N(O, a2M)), where M = plim(X~AX/T) = plim(X~K(W'W) 'KZX/T). The form of A also indicates that OLS applied to Eq. (17) will not be efficient and the OLS formula standard errors will not be consistent estimates of the true standard errors (see [9], Theorem 1). So even though this OLS regression provides consistent estimates of the parameters it also cannot be used for conducting hypothesis testing. The application of GLS to Eq. (17) requires the inversion of the T x T matrix, A. The complicated nature of A suggest that GLS applied to Eq. (17) is not going to lead to a computationally simple procedure. Applying the appropriate moving average transformation to Eq. (17) would still leave ~ as the error and as is obvious from Eq. (14) e is heteroscedastic and serially correlated. Hence, Koreisha and
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
455
Pukkila's [6] GLS estimator based on the incorrect assumption that wt follows a moving average of order q will neither lead to efficient estimates of the parameters nor consistent estimates of the covariance matrix of the parameter estimates. The truncation error that arises in using a finite rather than an infinite autoregression to generate the initial estimates of at has been ignored. To ensure that the truncation error is asymptotically negligible it is necessary to assume that as T tends to infinity L also tends to infinity at an appropriate rate (see, e.g., [11-13]).
4. Conclusion The correct covariance matrix for Koreisha and Pukkila's [6] GLS estimator of a univariate ARMA(p,q) does not have the form of the covariance matrix associated with an MA(q) model. As a result, the computation of the correctly specified GLS estimator is not computationally attractive since it requires the inversion of a T x T matrix. Similar problems arise if the GLS approach is applied to the estimation methods proposed by [5] for vector ARMA models (see [14] for a discussion of this case). Since the estimation method suggested by Hannan and Rissanen [4] for univariate ARMA models is both computationally simple and has desirable asymptotic properties there is little to be gained by using a computationally unattractive GLS estimator.
Acknowledgements The author would like to thank Sergio Koreisha, Michael McAleer and seminar participants at Osaka University for their helpful comments. He also would like to acknowledge the financial support of the Foundation to Promote Research on the Japanese Economy, the University of Western Australia and a travel grant awarded by the Japanese Department of Education.
References [1] K.J. White, S.D. Wong, D. Whistler and S.A. Hayn, SHAZAM Econometric Computer Program: Users Reference Manual version 6.2 (McGraw-Hill, Toronto, 1990). [2] M.H. Pesaran and B. Pesaran, MICROFIT 3.0: An Interactive Econometric Software Package (Oxford University Press, Oxford 1991). [3] J.A. Doornik and D.E Hendry, PcGive 8.0: An Interactive Econometric Modelling System (International Thomson Publishing, London, 1994). [4] E.J. Hannan and J. Rissanen, Recursive estimation of mixed autoregressive-moving average order, Biometrika 69 (1982) 81-94. [5] S. Koreisha and T. Pukkila, Fast linear estimation methods for vector autoregressive moving-average models, J. Time Series Analysis 10 (1989) 1325-1339. [6] S. Koreisha and T. Pukkila, A generalized least-squares approach for estimation of autoregressive moving-average models, J. Time Series Analysis 11 (1990) 139-151. [7] A.R. Pagan, Econometric issues in the analysis of regressions with generated regressors, Intl. Econom. Rev. 25 (1984) 221-247. [8] A.R. Pagan, Two stage and related estimators and their applications, Rev. Econom. Studies 53 (1986) 517-538.
456
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
[9] C.R. McKenzie and M. McAleer, On efficient estimation and correct inference in models with generated regressors: A general approach, forthcoming, Japanese Econom. Rev., 48 (1997). [10] T. Amemiya, Advanced Econometrics (Basil Blackwell, Oxford, 1985). [11] K.N. Berk, Consistent autoregressive spectral estimation, The Annals of Statist. 2 (1974) 489-502. [12] R.J. Bhansali, Linear prediction by autoregressive model fitting in the time domain, The Annals of Statist. 6 (1976) 224231. [13] R. Lewis and G.C. Reinsel, Prediction of multivariate time series by autoregressive model fitting, J. Multivariate Analysis 6 (1985) 393-411. [14] C.R. McKenzie, Fast linear estimation methods for vector autoregressive moving-average models: A comment, Discussion Paper No. 95, Faculty of Econom. Osaka University, (1990).
EI~SEVIER
MATHEMATICS AND COMPUTERS I N SIMULATION
Mathematics and Computers in Simulation 43 (1997) 451-456
The properties of some two step estimators of ARMA Models C.R. McKenzie* Osaka School of International Public Policy, Osaka University, Toyonaka, Osaka 560, Japan
Abstract
This paper analyzes the large sample properties of several two step estimators that have recently been suggested for estimating autoregressive moving-average models. As these estimators typically involve generated regressors, the generated regressor literature suggests that, in general, they will be inefficient and their estimated formula standard errors will be inconsistent estimates of the true standard errors. Deriving the covariance matrix of the true disturbances in these models enables consistent estimates of the true standard errors and efficient generalized least squares estimators to be computed.
1. Introduction
Univariate autoregressive moving-average (ARMA) models are commonly used for modelling and forecasting time-series data. Various techniques are available for estimating these models, the computationally expensive maximum likelihood method which is available in SHAZAM and MICROFIT but not PC-GIVE ([1-3]) and computationally simple two step methods (e.g. [4-6]). The two step methods typically rely on estimates of unobserved variables and as a result could be expected to suffer from a generated regressor problem (see [7-9] for a general discussion of this issue). Generated regressors typically cause the error term to become serially correlated and heteroscedastic in small and large samples making ordinary least squares (OLS) an inefficient estimator. As McKenzie and McAleer [9] indicate, there are various ways around this problem; for example, by applying generalized least squares (GLS) using a consistent estimate of the covariance matrix of the true errors, or some other transformation of the equation to eliminate asymptotically the serial correlation and heteroscedasticity in the errors. In this paper, several two step estimators are discussed and their asymptotic properties are derived very simply based on the approach developed in [9]. Section 2 contains a discussion of the model and the two step estimators considered in the paper. The asymptotic properties of these estimators are analyzed in Section 3 while Section 4 contains some concluding comments.
* Corresponding author. Tel.: 81-6-850-5622; fax: 81-6-850-5274; e-mail: [email protected]. 0378-4754/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 0 3 1- 1
C.R. Mc Kenzie /Mathematics and Computers in Simulation 43 (1997) 451-456
452
2. The model and estimators The ARMA (p, q) model for a mean adjusted series is P
q
(1)
Zt= Z~iZt_i+at--ZOjat~ j=l
i=l
where at is a normal white-noise process with zero mean and variance (r2, and t ---- 1 . . . T. It is assumed that the autoregressive and moving average parameters satisfy the conditions for stationarity and invertibility, respectively. Since the moving average component of Eq. (1) is assumed to be invertible, Eq. (1) can be rewritten as an infinite order autoregression, namely, Z, = ) j
(2)
7 ~ i Z t - i + at.
i=l
Truncating the autoregression to a finite order, say L, where L < T, and ignoring the error induced by the truncation gives L Zt =
(3)
~_~ 7fiZt i + at. i=1
This equation can be estimated by OLS to obtain estimates of 7ri, #i, and the innovation series where
at,
at
L
Clt = Zt - Z
(4)
frizt-i
i=1
These estimates of the innovation series can then be substituted back into Eq. (1) to give P
q
z, -- Z
Z 0j ,_j + v,
i:1
(5)
j=l
where v, : a t - ~-~)ql O j ( a t _ j - gtt~). OLS applied to Eq. (5) is the second stage of Hannan and Rissanen's [4] three stage procedure for efficiently estimating the parameters of Eq. (1). Alternatively, the second stage of Koreisha and Pukkila's [6] three stage estimation procedure is OLS applied to a slight variation of Eq. (5): P
Y, = ~
q
~iZt-i - Z
i=1
(6)
Oj~lt--J "~ Wt,
j=l
where y, = zt - gzt, and w t = at -- glt rewritten as a moving average of et:
-- ~J
l Oj(at-j
-- (it-j) •
Defining
et
= a,
-
~l t,
then wt can be
q
wt = e,-
~ Ojet_j. j=l
(7)
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451~t56
453
Another slight variation of Eq. (5) is P q xt = ~ ~iZt-i - Z Oj~lt~ -~- bit' i=1 j=l
(8) ,
where xt=zt-f~q_lOjfit-j, ut=at-~~q=lOjat~-~~J=l(Oj-Oj)(at~-ht~), and the t~j are consistent estimates of Oj obtained by, say, applying OLS to Eq. (5). OLS applied to Eq. (8) will be inconsistent since the regressors are correlated with ut. In ut the term y-~q_l(Oj-Oj)(at~-fitS) is asymptotically irrelevant so it is easy to see that applying the moving average version of the CochraneOrcutt method to Eq. (8) using Oj will be asymptotically efficient. This is in fact the third stage of Hannan and Rissanen's [4] three stage procedure. The structure of the errors in Eqs. (5) and (6) suggests there may be efficiency gains by applying GLS to Eqs. (5) and (6). The form of wt in Eq. (7) suggests that wt follows a moving average of order q, so that in this case the moving average equivalent of the Cochrane-Orcutt transformation applied to Eq. (6) seems appropriate. This is in fact the third stage of Koreisha and Pukkila's [6] three stage estimator. In the standard linear regression model, it is well-known that a combination of lagged dependent variables and serially correlated errors generally leads to OLS being inconsistent suggesting a potential problem for OLS applied to Eq. (6). However, as is shown later, this is not the case here.
3. Asymptotic properties Eq. (3) can be written in the more compact matrix form
Z = WH + A
(9)
where Z, Z-i, A and A i are T× 1 vectors with typical jth element zj, zj-i, aj and aj-i, respectively, W = [Z_I ... Z_L] and H ' = [7rl... 7rL]. Estimation of Eq. (9) by OLS gives the following estimators of 7r, A and A-i
11= (W'W) 1W'Z .4
= Z -
W l) =
(I -
10) 11)
W(W'W)-lWt)A
.~ i ~ Z - i - W_il~---- A _ i - W _ i ( W t w ) - l w t A
12)
where I/V i = [ Z _ i _ l . . . Z_i_L] , and A and J~--i are T x 1 vectors with typical jth element hj and aj-i, respectively. Eq. (5) can be rewritten as Z = X/3 + ¢ where X = [Z-1... Z_p, - A - 1 • .. - / I - q ] , / ~ t typical jth element vj.
(13) = [ ~ 1 . . . q~p, 0 1 . . . Oq], and ~ is a vector T × 1 vector with
454
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
For OLS applied to Eq. (13) to be consistent requires that (X'~/T) = 0. Defining e and e_j to be T x 1 vectors with typical elements ei and el-j, respectively, then using Eqs. (10)-(12) ~, c_j and ~ can be rewritten as
(14)
= A - A = W(W'W)-' W'A e_j = A ~ - A ~ q
=A - ~
= W _ j ( W f W ) -1
W'A
(15)
q
Oje_j = A - ~
j=l
Oj(A~ - A~) = SA
(16)
j=l
where S = I - $1 and S1 = ~-]~q_, OjW j(W'W) 1W'. Since plim(W'a/T) = 0, it is easily shown that plim(X'c~/T)=O ( j = 1 .ql-In addition, plim(,4 jA/T)=plim(A~_jA/T)=O ( j = 1 . . . q ) , so plim(X~A/T) = 0 ensuring the consistency of OLS applied to Eq. (13). From Eqs. (14)-(16), it is clear that if Wand W j were non-stochastic, then any et (and any vt) is a linear combination of all the at. So that et would not be serially uncorrelated and homoscedastic (cf [6]). If all the elements of W were non-stochastic, then ~2 = Var(~)/~r 2 = SS' so that we can see immediately that ~ in Eq. (13) is heteroscedastic and serially correlated. Here it suffices to note that under the assumption of the model X ' ( / T ____+DN(O, cr2V)), where V =plim(X'f2X/T). Since plim[R'~X/T] ¢ 0 for any non-zero R such that plim[R'X/T] = 0, OLS applied to Eq. (13) will not be efficient (see [ 10], Theorem 6.1.1 and [9], Theorem 1). The form of V also indicates that the formula standard errors produced by an OLS regression will not be consistent estimates of the true standard errors (see [9] Theorem 1). So even though this OLS regression provides consistent estimates of the parameters it cannot be used for conducting hypothesis testing. The application of GLS to Eq. (13) requires the inversion of the T x T matrix, Y2. The complicated nature of f2 suggests that GLS applied to Eq. (13) is not going to lead to a computationally simple procedure. What about the second and third stages of Koreisha and Pukkila's [6] procedure applied to Eq. (6)? Eq. (6) can be rewritten as Z-,4
= X/3 + 7/
(17)
where ~/= ¢ - A = K(W'W)-Iw'A and K = [W - ~q=, OjW_j]. Given that plim(X'¢/T) = 0 has a l r e a d y been shown, the c o n s i s t e n c y of OLS applied to Eq. (17) is assured since plim(X'.4/T) = p l i m ( X ' A / T ) = 0 . If all the e l e m e n t s o f W were n o n - s t o c h a s t i c , then A = Var(7/)/a 2 = K(W~W) 1K' which does not have the form of the covariance matrix of a moving average of order q. Here it suffices to note that under the assumptions of the model X~I/T ._~o N(O, a2M)), where M = plim(X~AX/T) = plim(X~K(W'W) 'KZX/T). The form of A also indicates that OLS applied to Eq. (17) will not be efficient and the OLS formula standard errors will not be consistent estimates of the true standard errors (see [9], Theorem 1). So even though this OLS regression provides consistent estimates of the parameters it also cannot be used for conducting hypothesis testing. The application of GLS to Eq. (17) requires the inversion of the T x T matrix, A. The complicated nature of A suggest that GLS applied to Eq. (17) is not going to lead to a computationally simple procedure. Applying the appropriate moving average transformation to Eq. (17) would still leave ~ as the error and as is obvious from Eq. (14) e is heteroscedastic and serially correlated. Hence, Koreisha and
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
455
Pukkila's [6] GLS estimator based on the incorrect assumption that wt follows a moving average of order q will neither lead to efficient estimates of the parameters nor consistent estimates of the covariance matrix of the parameter estimates. The truncation error that arises in using a finite rather than an infinite autoregression to generate the initial estimates of at has been ignored. To ensure that the truncation error is asymptotically negligible it is necessary to assume that as T tends to infinity L also tends to infinity at an appropriate rate (see, e.g., [11-13]).
4. Conclusion The correct covariance matrix for Koreisha and Pukkila's [6] GLS estimator of a univariate ARMA(p,q) does not have the form of the covariance matrix associated with an MA(q) model. As a result, the computation of the correctly specified GLS estimator is not computationally attractive since it requires the inversion of a T x T matrix. Similar problems arise if the GLS approach is applied to the estimation methods proposed by [5] for vector ARMA models (see [14] for a discussion of this case). Since the estimation method suggested by Hannan and Rissanen [4] for univariate ARMA models is both computationally simple and has desirable asymptotic properties there is little to be gained by using a computationally unattractive GLS estimator.
Acknowledgements The author would like to thank Sergio Koreisha, Michael McAleer and seminar participants at Osaka University for their helpful comments. He also would like to acknowledge the financial support of the Foundation to Promote Research on the Japanese Economy, the University of Western Australia and a travel grant awarded by the Japanese Department of Education.
References [1] K.J. White, S.D. Wong, D. Whistler and S.A. Hayn, SHAZAM Econometric Computer Program: Users Reference Manual version 6.2 (McGraw-Hill, Toronto, 1990). [2] M.H. Pesaran and B. Pesaran, MICROFIT 3.0: An Interactive Econometric Software Package (Oxford University Press, Oxford 1991). [3] J.A. Doornik and D.E Hendry, PcGive 8.0: An Interactive Econometric Modelling System (International Thomson Publishing, London, 1994). [4] E.J. Hannan and J. Rissanen, Recursive estimation of mixed autoregressive-moving average order, Biometrika 69 (1982) 81-94. [5] S. Koreisha and T. Pukkila, Fast linear estimation methods for vector autoregressive moving-average models, J. Time Series Analysis 10 (1989) 1325-1339. [6] S. Koreisha and T. Pukkila, A generalized least-squares approach for estimation of autoregressive moving-average models, J. Time Series Analysis 11 (1990) 139-151. [7] A.R. Pagan, Econometric issues in the analysis of regressions with generated regressors, Intl. Econom. Rev. 25 (1984) 221-247. [8] A.R. Pagan, Two stage and related estimators and their applications, Rev. Econom. Studies 53 (1986) 517-538.
456
C.R. McKenzie/Mathematics and Computers in Simulation 43 (1997) 451-456
[9] C.R. McKenzie and M. McAleer, On efficient estimation and correct inference in models with generated regressors: A general approach, forthcoming, Japanese Econom. Rev., 48 (1997). [10] T. Amemiya, Advanced Econometrics (Basil Blackwell, Oxford, 1985). [11] K.N. Berk, Consistent autoregressive spectral estimation, The Annals of Statist. 2 (1974) 489-502. [12] R.J. Bhansali, Linear prediction by autoregressive model fitting in the time domain, The Annals of Statist. 6 (1976) 224231. [13] R. Lewis and G.C. Reinsel, Prediction of multivariate time series by autoregressive model fitting, J. Multivariate Analysis 6 (1985) 393-411. [14] C.R. McKenzie, Fast linear estimation methods for vector autoregressive moving-average models: A comment, Discussion Paper No. 95, Faculty of Econom. Osaka University, (1990).