TIME SERIES DATA ANALYSIS USING EVIEWS potx

TIME SERIES DATA ANALYSIS USING EVIEWS I Gusti Ngurah Agung Graduate School Of Management Faculty Of Economics University Of Indonesia Ph.D in Biostatistics and MSc in Mathematical Statistics from University of North Carolina at Chapel Hill TIME SERIES DATA ANALYSIS USING EVIEWS TIME SERIES DATA ANALYSIS USING EVIEWS I Gusti Ngurah Agung Graduate School Of Management Faculty Of Economics University Of Indonesia Ph.D in Biostatistics and MSc in Mathematical Statistics from University of North Carolina at Chapel Hill STATISTICS IN PRACTICE Series Advisory Editors Marian Scott University of Glasgow, UK Stephen Senn University of Glasgow, UK Founding Editor Vic Barnett Nottingham Trent University, UK Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title’s special topic area The books provide statistical support for professionals and research workers across a range of employment fields and research environments Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on The books also provide support to students studying statistical courses applied to the above areas The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs Feedback of views from readers will be most valuable to monitor the success of this aim A complete list of titles in this series appears at the end of the volume Copyright # 2009 John Wiley & Sons (Asia) Pte Ltd, Clementi Loop, #02-01, Singapore 129809 Visit our Home Page on www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy fee to the Copyright Clearance Center Requests for permission should be addressed to the Publisher, John Wiley & Sons (Asia) Pte Ltd, Clementi Loop, #02-01, Singapore 129809, tel: 65-64632400, fax: 65-64646912, email: enquiry@wiley.com Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The Publisher is not associated with any product or vendor mentioned in this book All trademarks referred to in the text of this publication are the property of their respective owners This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Screenshots from EViews reproduced with kind permission from Quantitative Micro Software, 4521 Campus Drive, #336, Irvine, CA 92612-2621, USA Other Wiley Editorial Offices John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Agung, Ign Time series data analysis using EViews / Ign Agung p cm Includes bibliographical references and index ISBN 978-0-470-82367-5 (cloth) Time-series analysis Econometric models I Title QA280.A265 2009 519.5’5–dc22 2008035077 ISBN 978-0-470-82367-5 (HB) Typeset in 10/12pt Times by Thomson Digital, Noida, India Printed and bound in Singapore by Markono Print Media Pte Ltd, Singapore This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Dedicated to my wife Anak Agung Alit Mas, children Ningsih A Chandra, Ratna E Lefort, and Dharma Putra, sons in law Aditiawan Chandra, and Eric Lefort, daughter in law Refiana Andries, and all grand children Indra, Rama, Luana, Leonard, and Natasya Time Series Data Analysis Using EViews 556 This model could be extended to the AR(p) growth model, as a special case of the AR(p) model in (B.48), with the following equation: logYt ị ẳ b0 ỵ b1 t ỵ mt mt ẳ p X ri mti ỵ ôt B:58ị iẳ1 B.6.3 Alternative 3: The AR(p) polynomial model This model is defined as Yt ¼ b þ k X bi Xti þ mt i¼1 p X mt ẳ ri mti ỵ ôt B:59ị iẳ1 B.6.4 Alternative 4: The AR(p) return rate model This model is defined as d logYt ị ẳ b0 ỵ b1 Xt ỵ mt mt ẳ p X ri mti ỵ ôt B:60ị i¼1 B.6.5 Alternative 5: The bounded translog linear (Cobb-Douglas) AR(p) model This model is defined as Yt ÀL ¼ b0 þ b1 logðXt Þ þ mt UÀYt p X mt ẳ ri mti ỵ ôt log B:61ị iẳ1 where U and L are the upper and lower bounds of the expected values of the series Yt B.7 Lagged-variable model A qth lagged endogenous variable model, namely the LV(q) model, with an exogenous variable Xt is defined as q X bj Ytj ỵ dXt ỵ ôt B:62ị Yt ẳ b þ j¼1 Appendix B: Simple Linear Models 557 Based on this model, the error sum of squares function is defined as follows: Qẳ T X ô2 t ẳ tẳ1 T X Yt Àb0 À q X !2 bj YtÀj ÀdXt B:63ị jẳ1 tẳ1 Furthermore, under the assumption that ôt is i.i.d Gaussian or N(0,s2), the following log-likelihood function is obtained: T X ô2 T T t LL ẳ log2pị logs2 ị 2s2 2 tẳ1 q T X T T X LL ẳ log2pị logs2 ị yt Àb0 À bj ytÀj Àdxt 2 2s t¼1 jẳ1 !2 B:64ị To estimate the parameters b0, bj, d and s2, in a mathematical sense, the following normal equation is considered: T X tẳq ỵ T X tẳq þ T X q X yt Àb0 À bj ytj dxt yt b0 yt b0 tẳq ỵ s2 ¼ ! ¼0 ! j¼1 q X bj ytj dxt ytj ẳ 0; ! tẳq ỵ B:65ị bj ytÀj Àdxt xt ¼ j¼1 T X j ¼ 1; ; q j¼1 q X yt Àb0 À q X !2 bj ytÀj Àdxt j¼1 B.8 Lagged-variable autoregressive models B.8.1 The simplest lagged-variable autoregressive model The simplest lagged-variable autoregressive model, namely the LVAR(1,1) model, with an exogenous variable is dened as Yt ẳ b0 ỵ b1 Yt1 ỵ dXt ỵ mt mt ẳ rmt1 ỵ «t ðB:66Þ Under the assumption that «t is i.i.d N(0,s2), then ôt ẳ Yt b0 b1 Yt1 dXt ịrmt1 ẳ ðYt Àb0 Àb1 YtÀ1 ÀdXt ÞÀrðYtÀ1 Àb0 Àb1 YtÀ2 ÀdXtÀ1 Þ ðB:67Þ Time Series Data Analysis Using EViews 558 with the normal density function as follows: ( ) ½ðYt Àb0 Àb1 YtÀ1 ÀdXt ÞÀrðYtÀ1 Àb0 Àb1 YtÀ2 ÀdXtÀ1 ފ2 À1=2 f ôt ị ẳ 2ps2 ị exp 2s2 B:68ị for t ¼ 3, , T, since YtÀ2 is on the right-hand side and the series considered is Yt, for t ¼ 1, , T In order to estimate the parameters b, d, r and s2, either the error sum of squares function or the LL function may be used, as follows The error sum of squares function is given by Q ¼ Qðb0 ; ; bq ; r; dị ẳ T X ô2 t tẳ3 T X ẳ ẵYt b0 b1 Yt1 dXt ịrYt1 b0 b1 Yt2 dXt1 ị2 B:69ị tẳ3 The LL function is given by T X «2  T2 T2 t log2pị logs2 ị 2s2 2 tẳ3 T2 T2 LL ẳ log2pị logs2 ị 2 LL ẳ B:70ị T X ẵyt b0 Àb1 ytÀ1 Àdxt ÞÀrðytÀ1 Àb0 Àb1 ytÀ2 ÀdxtÀ1 ފ2 2s t¼3 Based on this LL function the following normal equations would be derived: T X t¼3 T X t¼3 T X ẵyt b0 b1 yt1 dxt ịryt1 b0 b1 yt2 dxt1 ị ẳ ẵyt b0 b1 yt1 dxt ÞÀrðytÀ1 Àb0 Àb1 ytÀ2 ÀdxtÀ1 ފðytÀ1 ÀrytÀ2 Þ ¼ ½ðyt Àb0 Àb1 ytÀ1 Àdxt ÞÀrðytÀ1 Àb0 Àb1 ytÀ2 ÀdxtÀ1 ịxt rxt1 ị ẳ tẳ3 T X ẵyt b0 Àb1 ytÀ1 Àdxt ÞÀrðytÀ1 Àb0 Àb1 ytÀ2 ÀdxtÀ1 ފ ðytÀ1 b0 b1 yt2 dxt1 ị ẳ T X ½ðyt Àb0 Àb1 ytÀ1 Àdxt ÞÀrðytÀ1 Àb0 Àb1 ytÀ2 ÀdxtÀ1 ފ2 s2 ¼ TÀ2 t¼3 t¼3 ðB:71Þ Appendix B: Simple Linear Models 559 Note that in (B.71) there are five equations with five unknowns or parameters, so that, in general, a unique solution would be expected However, it is very difficult to present an explicit solution for each parameter Therefore, EViews provides an iteration estimation process, as presented in Section A.7 B.8.2 General lagged-variable autoregressive model A general lagged-variable autoregressive model, namely the LVAR(p, q) model, with an exogenous variable, is dened as Y t ẳ b0 ỵ mt ẳ p X q X bj Ytj ỵ dXt ỵ mt jẳ1 B:72ị ri mti ỵ ôt iẳ1 Compared to the AR(p) model presented in Section A.7, where the term AR(p) is related to the endogenous variable Yt, in the model (B.72) the term AR(p) is related to the error term or residual mtÀ1, , mtÀp Under the assumption that «t is i.i.d N(0,s2), then the model parameters can be estimated by using either the error sum of squares function or the LL function, as follows The error or residual sum of squares function is Q ¼ ¼ T X ô2 t tẳk ỵ " T X iẳk ỵ q p q X X X yt Àb0 À bj ytÀj Àdxt ÞÀ ri ðytÀi Àb0 À bj ytij dxti ị jẳ1 iẳ1 #2 jẳ1 B:73ị where k ẳ p ỵ q ẳ max{i ỵ j, 8i and j} Then the log-likelihood function is given by LL ẳ T X ô2 Tk Tk t log2pị logs2 ị 2s2 2 tẳkỵ1 Tk Tk log2pị logs2 ị 2 " #2 q p q T X X X X ðyt Àb0 À bj ytÀj Àdxt ÞÀ ri ðytÀi Àb0 À bj ytÀiÀj ÀdxtÀi Þ À 2s tẳkỵ1 jẳ1 iẳ1 jẳ1 LL ẳ B:74ị 560 Time Series Data Analysis Using EViews By using the same technique as presented in Section A.7, it is easy to obtain the estimates of the parameters, as well as testing hypotheses, using EViews B.9 Special notes and comments Considering the application of the basic model in (B.1), namely Yt ¼ b0 ỵ b1Xt ỵ ôt, the following notes and comments are made: (1) This model represents the linear trend of an endogenous variable Yt with respect to an exogenous variable Xt in the population Even though their pattern of relationship could be nonlinear, the true population model will never be known However, the linear trend of Yt with respect to Xt can always be considered Therefore, it could be said that this model can be defined as a true population model with trend of Yt with respect to Xt (2) It is well known that the moment product correlation in the population, namely r ¼ r(Xt, Yt), is a measure of a linear correlation Therefore, this moment product correlation can also be used to present the linear trend of Yt with respect to Xt Refer to the standardized regression of Yt on Xt, which can be presented as ZYt ¼ rZXt ỵ ôt, where ZYt and ZXt are the Z-scores of the variable Yt and Xt respectively Hence, testing the null hypothesis H0: b1 ¼ is exactly the same as testing the null hypothesis H0: r ¼ r(Xt, Yt) ¼ (3) On the other hand, to study their pattern of relationship in more detail, as well as the growth curve of Yt with respect to Xt, there should be a high dependence on the data set that happens to be available In this case, the scatter graph or plot of the bivariate (Xt,Yt) with a regression or kernel fit should be observed, as presented in this book Then personal judgment should be used to define a model or alternative models, as presented in Section 2.6 Refer to Section 2.14 for more detailed comments (4) In order to present the causal effect of an exogenous variable Xt on an endogenous variable Yt, it is suggested that XtÀi should be used for some selected i > 0, instead of Xt, since a cause factor needs to be measured prior or before the impact factor However, in general, researchers have been using XtÀ1 (5) It has been recognized that any time series models should be using either the lag(s) of the endogenous variable or the autoregressive errors, or both (6) Finally, whatever model is used, it is suggested that an additional residual analysis should be done in order to find out the limitation of the final model(s) Appendix C: General linear models C.1 General linear model with i.i.d Gaussian disturbances As an extension of the basic model presented in Appendix B, a (univariate) general linear model (GLM) is presented as yt ẳ b0 ỵ b1 x1t ỵ ỵ bk1 xk1ịt ỵ mt ẳ k1 X bi Xit ỵ mt C:1ị iẳ0 for t ẳ 1, , T, which can be presented in matrix form as y ẳ X Tx1ị b ỵ m Txkị kx1ị ðTx1Þ ðC:2Þ where 03 y1 X1 y2 X 02 7 6.7 6.7 7 y ¼ X ¼ with ðTxkÞ ðTx1Þ 6.7 4.5 yt X 0t 3 2 x0t ¼ m1 b0 x1t b1 m2 7 6 X ¼6 b ¼6 m ¼6 1xðkÀ1Þ ðkx1Þ ðTx1Þ xðkÀ1Þt bkÀ1 mt C:3ị Note that for k ẳ 2, the model is in the form presented in (B.1) Furthermore, also note that the independent variables xit, i ¼ 0, 1, 2, , (k À 1), could be any set of exogenous variables, such as pure exogenous variables and their lags, the time t-variable, as well as selected two-way or higher interactions of the independent variables By selecting a set of relevant independent variables from Time Series Data Analysis Using EViews IGN Agung Ó 2009 John Wiley & Sons (Asia) Pte Ltd Time Series Data Analysis Using EViews 562 all possible types of those variables, it is expected that the error term mt is i.i.d distributed C.1.1 The OLS estimates Under the basic assumptions A1 to A5 presented in Appendix B, namely the multivariate X is deterministic and the error term is an i.i.d Gaussian disturbance, the following OLS estimates are obtained: The unbiased estimator of the vector parameter b: y ¼ Xb ! X y ¼ X Xb ! ðX XÞÀ1 X y ¼ ðX XÞÀ1 ðX XÞb ðC:4Þ If the matrix X0 X is nonsingular then the estimator is ^ b ẳ b ẳ X Xị1 X y C:5ị b ẳ X Xị1 X Xb ỵ mị ẳ b ỵ X Xị1 X m C:6ị kx1ị or with its expected value Ebị ẳ b ðC:7Þ which indicates that b is an unbiased estimator of b The unbiased estimator of the population variance s2: The estimate of the error term vector can be written as m ẳ u ẳ yXX Xị1 X y ẳ ẵIT XX Xị1 X y ẳ Mx y ^ ðTx1Þ ðC:8Þ Therefore, the sum of squared errors (SSE) and the mean of squared errors (MSE) can be written as: P (C.9) SSE ¼ u0 u ¼ yt X 0t bị2 MSE ẳ s2 ẳ where X 0t b indicates Pk1 iẳ0 SSE Tk C:10ị xit bi Furthermore, EMSEị ẳ Es2ị ẳ s2 C:11ị which indicates that the MSE is an unbiased estimator for the population variance Appendix C: General Linear Models 563 The uncentered and centered R-squared, namely R2 and R2 respectively: u c SSE R2 ẳ PT u tẳ1 yt C:12ị SSỀT2 y R2 ¼ P T c ÀT2 y tẳ1 yt C:13ị The variancecovariance matrix of b: Eẵb bịb bị0 ẳ s2 X Xị1 ðC:14Þ The normal distribution of b: b $ Nðb; s2 ðX XÞÀ1 Þ ðC:15Þ C.1.2 Maximum likelihood estimates Under the assumption that the error term mt ¼ Yt ÀX 0t b is i.i.d Gaussian, the following density function is obtained (compare with the density function in (B.3)): 1=2 f mt ị ẳ 2ps ị " #   m2 ðyt À X 0t bÞ2 À1=2 t exp ẳ 2ps ị exp 2s 2s2 C:16ị P where X 0t b ẳ k1 bi xit Therefore, the log likelihood function considered for i¼0 estimation purposes is T T T X ðyt ÀX 0t bị2 LL ẳ ln2pị lns2 ị 2 2s tẳ1 C:17ị The necessary conditions to obtain the maximum value of LL are as follows: T qðLLÞ X ¼ ðYt À X 0t bÞ ¼ qb0 s tẳ1 T qLLị X ẳ Yt X 0t bịxit ẳ 0; qbi s tẳ1 i ẳ 1; 2; ; ðkÀ1Þ T qðLLÞ T X ẳ ỵ Yt X 0t bị2 ¼ qs 2s 2s t¼1 ðC:18Þ Time Series Data Analysis Using EViews 564 As a result, the following normal equations are obtained: T X ðyt À X 0t bị ẳ tẳ1 T X xit yt X 0t bị ẳ 0; tẳ1 s2 ẳ for i ẳ 0; 1; ; ðkÀ1Þ ðC:19Þ T 1X yt X 0t bị2 T tẳ1 It is well known that the first two sets of equations can also be obtained by using the OLS estimation method Therefore, in a mathematical sense, the same estimates of the vector parameter b can be obtained by using either one of the estimation methods As a result, based on the last equation, T kÀ1 X 1X yt À bi xit s2 ¼ ^ T tẳ1 iẳ0 !2 C:20ị C.1.3 Students t-statistic Corresponding to the multivariate distribution of the vector b ¼ [b0, b1, , bkÀ1] ^ as N(b, s2(X0 X)À1) in (C.15), each of its components bi ¼ bi is normally 2 distributed as Nðbi ; sii Þ, where sii is the element in row i and column i of [s2(X0 X)1] By using s2 bi ị ẳ s2 , Student’s t-statistic can be presented as ^ ii bi Àbi is distributed as tðTÀkÞ sðbi Þ ðC:21Þ C.1.4 The Wald form of the OLS F-test C.1.4.1 Testing the Hypothesis H0 : Cb ẳ c and H1 : Otherwise C:22ị where C is a constant (m  k) matrix representing the particular linear combinations of the model parameter b and c is an (m  1) vector of defined values that are believed or judged to be the true values of the corresponding linear combinations From (C.15) it is found that, under H0, Cb $ Nðc; s2 ðX XÞÀ1 C0 Þ (C.23) Furthermore, under H0, the chi-squared test is found to be Cbcị0 ẵs2 X Xị1 C ŠÀ1 ðCb À cÞ $ x2 ðmÞ ðC:24Þ Appendix C: General Linear Models 565 By replacing s2 with its estimate s2 ¼ SSE/(T À k), the Wald form of the OLS F-test is obtained: Cbcị0 ẵs2 X Xị1 C0 Cbcị $ Fm; Tkị m C:25ị Cbcị0 ẵX XÞÀ1 C ŠÀ1 ðCbÀcÞ $ Fðm; TÀkÞ ms2 ðC:26Þ or The hypothesis (C.22) can be represented as H0 : Restricted model H1 : Unrestricted model ðC:27Þ Then the Wald form of the OLS F-test can be written as F¼ ðSSER ÀSSEU Þ=m $ Fðm; TÀkÞ SSEU =ðTÀkÞ ðC:28Þ where SSER indicates the sum of squared errors of the restricted model (i.e if the null hypothesis Cb ¼ c is true) and SSEU indicates the sum of squared errors of the unrestricted or full model Furthermore, it is well known that the numerator and denominator of the F-test are the chi-squared tests as follows: x ẳ SSER SSEU ị=m $ s2 x2 mị x2 ẳ SSEU $ s2 x2 ðTÀkÞ ðTÀkÞ ðC:29Þ ðC:30Þ C.2 AR(1) general linear model Corresponding to the basic model in (C.1), the AR(1) model, without lag of the endogenous variable, should be considered as follows: yt ẳ Xb ỵ mt mt ẳ rmt1 ỵ ôt C:31ị with the assumptions that m ẳ [m1, m2, , mT] $ N(0, s2V), where V is a known (T  T) positive definite matrix and |r| < Compared to the AR(1) model in (A.18) in Appendix A, this model is in fact a linear model with first-order autoregressive errors However, the same terminology is used here, namely the AR(1) model Note that the AR(1) model in (B.19) and (C.31) have different characteristics, and similarly for the AR(p) models presented in Appendix A and the AR(p) model presented in the following section Time Series Data Analysis Using EViews 566 C.2.1 Properties of mt Under the assumption that «t is i.i.d N(0, s2), the residual mt has exactly the same properties as presented in Section B.4.3 C.2.2 Estimation method By presenting the model in (C.31) as yt ẳ Xt b ỵ rmt1 ỵ ôt C:32ị then under the assumption that the error term of this model, namely ôt, is i.i.d Gaussian, ô ẳ [ô1, ô2, , «T] $ N(0, s2I) Furthermore, based on the model in (C.32), the error term is as follows: ôt ẳ yt Xt bị ryt1 Xt1 bÞ ðC:33Þ By using the same process as in Appendix B, the following log-likelihood function is obtained: LL ¼ À TÀp TÀp lnð2pÞÀ lnðs2 Þ 2 ðC:34Þ T X ẵyt X t bịryt1 X t1 bị2 ¼ À 2s t¼2 P where X 0t b ¼ kÀ1 bi xit Then the approach is to maximize this function nui¼0 merically with respect to b0, b1, s2 and r In fact, corresponding to the model in (C.31), the following normal equation is considered for the estimation process, for |r| < 1: " T X ! !# kÀ1 kÀ1 X X yt Àb0 À bi xit Àr yt1 b0 bi xit1ị ẳ0 tẳ2 ! !# k1 kÀ1 X X yt Àb0 À bi xit Àr ytÀ1 b0 bi xit1ị xit rxit1ị ịẳ0 iẳ1 tẳ2 iẳ1 i¼1 " T X i¼1 for i¼1;2; ;kÀ1: " ! !# ! T kÀ1 kÀ1 kÀ1 X X X X yt Àb0 À bi xit Àr ytÀ1 Àb0 À bi xit1ị yt1 b0 bi xit1ị ẳ0 tẳ2 s2 ẳ T X TÀ1 t¼2 " i¼1 yt Àb0 À kÀ1 X i¼1 ! i¼1 bi xit Àr ytÀ1 Àb0 k1 X !#2 iẳ1 bi xit1ị iẳ1 C:35ị Appendix C: General Linear Models 567 Alternatively, instead of using the numerical iteration method, the following regression may be used: ðyt ryt1 ị ẳ b0 ỵ k1 X bi xit rxit1ị ị ỵ ôt C:36ị iẳ1 for various values of r, such as 0.05, 0.10, , 0.95 Then a model could be chosen having the smallest sum of squared errors or other measures of fit, as presented in Section 11.3 C.3 AR(p) general linear model As an extension of the AR(1) model in (C.31) or the model in (2.8), this is an AR (p) GLM, without lag of the endogenous variable, as follows: yt ¼ mt ¼ Xb ỵ mt p X ri mti ỵ ôt C:37ị iẳ1 where ri are the ith autocorrelation or serial correlation parameter such that |ri| < and ôt, t ẳ 1, 2, , T, are i.i.d Gaussian or N(0, s2) In order to estimate the parameters, the following LL function should be considered: TÀp TÀp lnð2pÞÀ lnðs2 Þ 2 " #2 p T X X 0 ðyt ÀX t bÞÀ ri ðytÀi ÀX tÀi bÞ 2s tẳp ỵ iẳ1 LL ẳ where X 0t b ẳ Pk1 iẳ0 C:38ị bi xit (compare this to the LL function in (C.17) C.4 General lagged-variable autoregressive model As an extension of the LVAR(p, q) model in (C.31) with an exogenous variable, a general lagged-variable autoregressive model, namely the LVAR(p, q) model with multivariate exogenous variables, is dened as Yt ẳ b0 ỵ mt ẳ p X iẳ1 q X bi Yti ỵ iẳ1 ri mti ỵ ôt k X di Xit ỵ mt iẳ1 ðC:39Þ Time Series Data Analysis Using EViews 568 Note that the AR(p) model in (C.39) is in factP model with autoregressive a errors, which is indicated by the error terms P ẳ p ri mti ỵ ôt , compared to mt i¼1 the AR(p) model in Appendix A, yt ¼ b0 ỵ p bi yti ỵ ôt , with respect to the i¼1 endogenous variable yt In order to estimate the parameters, the following LL function should be considered: TÀpÀq TÀpÀq lnð2pÞÀ lnðs2 Þ 2 ! 32 q k X X yt Àb0 À bi ytÀi À di xit 7 T i¼1 i¼1 X 6 !7 À p q k X X 2s tẳp ỵ q ỵ 16 X À r y Àb À b y di xiðtÀ1Þ tÀi j tÀiÀj À i LL ¼ iẳ1 jẳ1 iẳ1 C:40ị C.5 General models with Gaussian errors C.5.1 Gaussian errors with a known variance covariance matrix Corresponding to the general linear model in (C.1), namely y ẳ Xb ỵ m C:41ị the following assumptions are made: A1 X is stochastic A2 Conditional on the full matrix X, the error vector m is N(0, s2V) A3 V is a known positive definite matrix Recall from (C.6) that b bị ẳ X Xị1 X m C:42ị Under the assumption A2, the conditional expectation is Eẵb bịjX ẳ X Xị1 X Emị ẳ ðC:43Þ and by the law of iterated expectation (Hamilton, 1994, p 217), Eb bị ẳ Ex fEẵb bịjX ẳ C:44ị Appendix C: General Linear Models 569 The variance of the vector b conditional on X is given by Eẵb bịb bị0 jX ẳ EẵfX Xị1 X mm0 XX Xị1 gjX ẳ s2 ðX XÞÀ1 X VXðX XÞÀ1 ðC:45Þ As a result, the vector b conditional on X is multivariate normally distributed with E(b) ¼ b and Var(b) ¼ s2(X0 X)À1X0 VX(X0 X)À1, which can be presented as bjX $ Nðb; s2 ðX XÞÀ1 X VXðX XÞÀ1 Þ ðC:46Þ C.5.2 Generalized least squares with a known covariance matrix Under the assumptions A1 to A3 above, namely m|X $ N(0, s2V), where V is a known symmetric and positive (T  T) matrix, there exists a nonsingular (T  T) matrix G such that V À1 ¼ G0 G ðC:47Þ Then the model (C.39) should be transformed to Gy ẳ GXịb ỵ Gm C:48ị with Gm|X $ N(0, s2IT) Under this condition, the estimator of b is as follows: ðGXÞ0 Gy X ðG0 GÞy X V y b ẳ ẳ ẳ ẳ GXị0 GXịbG ¼ ðGXÞ0 ðGXÞb X ðG0 GÞXÞb X V À1 Xb ðX V À1 XÞÀ1 X V À1 y ðC:49Þ which is known as the generalized least squares (GLS) estimator, with Covbị ẳ s2 X V À1 XÞÀ1 ðC:50Þ Furthermore, similar to the estimator of the vector b in (C.46), the conditional distribution of the vector estimator in (C.49) is bjX $ Nðb; s2 ðX V À1 XÞÀ1 Þ ðC:51Þ Similarly, the sum of squared errors has a conditional chi-squared distribution, s2 ẳ y0 ẵV À1 ÀV À1 XðX V À1 XÞÀ1 X V À1 Šy $ s2 :x2 ðTÀkÞ ðC:52Þ Time Series Data Analysis Using EViews 570 Then under the null hypothesis Cb ¼ c in (C.19), the Wald form of the F-test is given by ½CbÀcŠ0 ½s2 CðX V À1 Xị1 C0 ẵCbc $ Fm; Tkị m C:53ị C.5.3 GLS and ML estimations Under the assumption that m|X $ N(0, s2V), then, based on the model in (C.41), yjX $ NðXb; s2 VÞ ðC:54Þ The log-likelihood function of y conditioned on X is given by  LL ¼      ÀT 1 logð2pÞÀ logjs2 VjÀ ðy À XbÞ0 ðs2 VÞÀ1 ðy À XbÞ 2  LL ¼    ÀT 1 logð2pÞÀ logjs2 VjÀ ðy À XbÞ0 V À1 ðy Xbị 2 2s C:55ị Since V1 ẳ G0 G, then     ÀT 1 logð2pÞÀ logjs2 VjÀ ðy À XbÞ0 G0 Gðy À XbÞ LL ¼ 2 2s  LL ¼    ÀT 1 logð2pÞÀ logjs2 VjÀ ðGy À GXbÞ0 ðGy À GXbÞ 2 2s ðC:56Þ This equation shows that the log likelihood function is maximized with respect to b by an OLS regression of Gy on GXb (refer to the model in (C.48) Hence the GLS estimate is also the maximum likelihood estimate C.5.4 The variance of the error is proportional to the square of one of the explanatory variables Under the assumption that Varmt ị ẳ x2 s2 , then 1t s2 V ẳ s2 Diagẵx2 ; x2 ; ; x2 Š 11 12 1T ðC:57Þ ... Statistics from University of North Carolina at Chapel Hill TIME SERIES DATA ANALYSIS USING EVIEWS TIME SERIES DATA ANALYSIS USING EVIEWS I Gusti Ngurah Agung Graduate School Of Management Faculty... will give the options in Figure 1.2 Time Series Data Analysis Using EViews IGN Agung Ó 2009 John Wiley & Sons (Asia) Pte Ltd Time Series Data Analysis Using EViews Figure 1.1 Figure 1.2 The toolbar... types of data sets, such as cross-section, time series, cross-section over time and panel data This book introduces and discusses time series data analysis, and represents the first book of a series