Time series regression with Arima noise and missing observations program TRAM

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PAPERS IN

ECONOMICS

EUI Working Paper EC O No. 92/81

Time Series Regression with ArimaNoise

and Missing Observations

Program Tram VICTOR GOMEZ \ ‘ ‘ and AGUSTlN MARAVALL © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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Please note

As from January 1990 the EUI Working Paper Series is divided into six sub-series, each sub-series is numbered

individually (e.g. EU I Working Paper L A W No. 90/1). ©

The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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EUI Working Paper E C O No. 92/81

Tim e Series R egression with A R IM A Noise and M issing O bservations

P rog ram TRAM VICTOR G 6M EZ

and

AGUSTIN MARA V ALL

BA D IA F IE S O L A N A , SAN D O M EN IC O (F I) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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© Victor G6mez and Agustfn Maravall Printed in Italy in June 1992 European University Institute

Badia Fiesolana 1-50016 San Domenico (FI)

Italy © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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Program T

r a m

Victor Gómez

Institute Nacional de Estadfstica E-28046 Madrid

Spain

Tel.: (+34-1-) 58.39.229 Fax: (+34-1-) 58.39.484

Agustfn Maravall

European University Institute Badia Fiesolana

1-50016 S. Domenico di Fiesole (FI) Italy

Tel.: (+39-55-) 50.92.347 Fax: (+39-55-) 50.92.202

Abstract

The present paper describes the program TRAM , which stands for "Tim e Series Regression with ARIMA Noise and Missing Observations". TRAM has been written in Fortran and is available from the authors for MS-DOS computers and mainframes. The program estimates the parameters of a regression model with possibly nonstationary noise and any sequence of missing observations, interpolates the missing values, and obtains forecasts for the series. The program incorporates several additional facilities, such as intervention analysis, and easter and/or trading day corrections. The methodology is described in the paper "Estimation, Prediction and Interpolation for Nonstationary Series with the Kalman Filter", by V. Gomez and A . Maravall, EUI Working Paper ECO No. 92/80.

The first part of this document presents a summary of the program. Part two contains the instructions for the user and a description of the parameters. Finally, part three illustrates the program for six well-known examples, which present different regression and time series models with different combinations of missing observations.

1. Introduction

2 . Instructions for the user 3. Examples Preliminary Version © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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restriction is imposed on the location of the missing observations in the series. Given the vector of observations:

z = (z.,.... ZJ ' '

«>

where 0 < tj < ...< tM, the program fits the regression model

z, =y ,'0 +v,,

(

2 )

where /3 = (/),,...,j3n)’, is a vector of regression coefficients, y't = [yit,...,y nl] denotes n regression variables, and v, follows the general ARIMA process

0(B)5(B)vt = 8(B)at + c , (3)

where B is the backshift operator; 0(B), 5(B), and 8(B) are finite polynomials in B; a, is assumed a n.i.i.d. (o,<rw hite-noise variable, and c is a constant.

The polynomial 0(B) contains the stationary autoregressive roots, 5(B) is the polynomial with the nonstationary autoregressive roots, and 8(B) denotes the (invertible) moving average polynomial. In TRAM, they assume the following multiplicative form:

5(B) = (1 - B)IDR(l - Bs)IDS

0(B) = (1 + 0]B + ...+0IPRB,PR)(l + <h1Bs+ ...+ 0 IPSBs>‘,ps)

8(B) = (l + 8,B+...+8iqrBiqr )(l + 0 ,B S+ .. . + 0 IQSBS*,QS )

As explained in the user instructions, initial estimates of the parameters can be input by the user, set to the default values, or computed by the program.

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a) dummy variables (additive outliers); b) any possible sequence of ones and zeros;

c) l/(l - dB) of any sequence of ones and zeros, where 0 < 5 < 1; d) l/(l - <SsBs) of any sequence of ones and zeros, where 0 < <5S < 1;

e) l/(l - B)(l - Bs) of any sequence of ones and zeros.

As indicated in the user instructions, missing observations can also be treated as additive outliers. In this case, the likelihood is corrected so that it coincides with that of the standard missing-observations case (see example 4 in section 3).

The program:

1) estimates by exact maximum likelihood (or unconditional least squares) the parameters in (2) and (3);

2) computes optimal forecasts for the series, together with their MSE; and

3) yields optimal interpolators of the missing observations and their associated MSE.

The methodology followed is described in the paper "Estimation, Prediction, and Interpolation for Nonstationary Series with the Kalman Filter" by V. Gomez and A. Maravall. This paper will be referred to as the "background paper"; references made in this Introduction can be found in the Reference section of the background paper.

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When the differenced series can be used, the algorithm of Morf, Sidhu and Kailath (1974), (as improved by Melard, 1984) is employed.

For the nondifferenced series, it is possible to use the ordinary Kalman filter, as described in the background paper (default option), or its square root version (see Anderson and Moore, 1979). The latter is adequate when numerical difficulties arise; however it is markedly slower and does not permit (at present) to concentrate the regression parameters out of the likelihood. By default, the exact maximum likelihood method is employed, and the unconditional least squares method is available as an option. Nonlinear maximization of the likelihood function and computation of the parameter estimates standard errors is made using Marquardt's method and first numerical derivatives.

As detailed in the background paper, estimation of regression parameters is made by using first the Cholesky decomposition of the error covariance matrix to transform the regression equation (the Kalman filter provides an efficient algorithm to compute the variables in this transformed regression). Then, the resulting least squares problem is solved by orthogonal matrix factorization using the Householder transformation. This procedure yields a numerically stable method to compute GLS estimators of the regression parameters, which avoids matrix inversion.

For forecasting and interpolation, the ordinary Kalman filter or the square root filter options are available. Interpolation of missing values is made by the simplified Fixed Point Smoother, as described in the paper. When concentrating the regression parameters out of the likelihood, mean squared errors of the forecasts and interpolations are obtained following the approach of Kohn and Ansley (1985).

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in this way, all parameters of the ARIMA model can be estimated because the function to minimize does not depend on the free parameters. Moreover, it will be evident which forecasts and interpolations are affected by these arbitrary values because they will strongly deviate from the rest of the estimates (see example 6 in section 3). However, if all unknown parameters are jointly estimated, the program may not flag all free parameters. It may happen, that there is convergence to a valid arbitrary set of solutions (i.e., that some linear combinations of the initial missing observations, including the free parameters, are estimable.) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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2. Instruction for the user

INSTALLATION :

Insert the diskette in drive A or B, and change the default drive (type "A:" or "B:"). When the prompt appears type:

INSTALL

The installation procedure creates a directory "TRAM"; be sure it doesn't already exist.

(If you have a partitioned diskette, you will be asked in which drive the program should be written.)

TO RUN THE PROGRAM :

Prepare the input file following the instructions in the next pages.

Once the input file has been prepared in the SERIES subdirectory, to execute the program simply type:

"TRAM filename"

where filename is the name of the input file, at the directory where the program has been installed (TRAM). The results can be seen by editing or printing the file OUTPUT.

Typing "GRAPH", several graphics can be readily obtained. Moreover, from the subdirectory GRAPH (which is then created), the relevant arrays can be retrived for further use in other econometrics/statistics/graphics package.

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The input starts with the series to be modelled, comprising no more than 250 observations, followed by one set of control parameters for the series model plus a list of instructions for the regression variables.

To specify the set of control parameters for the series model, as well as the instructions for the regression variables, the NAMELIST facility is used, so that only those parameters which are not at their default values (see below) need to be set.

The series is set up as:

Card 1 TITLE (no more than 72 characters) Card 2 NZ NYER NPER NFREQ (free format) Card 3 et seq Z(I): 1=1,NZ (free format),

where NZ is the number of observations, NYER the start y e a r , NPER the start period, and NFREQ the observational frequency in the year. Z(.) is the array of observations. For each missing observation, the code -99999. must be entered.

This is followed by namelist DATEN. The namelist starts with &DATEN (in the second column) and terminates with /. The parameters in namelist DATEN are:

Parameter Meanina Default

LAMDA = 1 No tranformation of data 1

= 0 Take logs of data

IMEAN — _{0 No mean correction} ₀

= 1 Mean correction

IDR = # of non-seasonal differences 1

IDS = # of seasonal differences 1

I PR = _{# of non-seasonal autoregressive} 0

terms

IPS = _{# of seasonal autoregressive terms} 0

IQR = _{# of non-seasonal moving average} 0

terms

IQS = # of seasonal moving average terms 0

IREG = # of regression variables 0

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I EAST 0 No Easter effect adjustment 0 1 Easter effect adjustment

\

IDUR = Duration of Easter affecting period 0

LAG S3 # of autocorrelations and partial autocorrelations printed

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INCON

-0 Exact maximum likelihood estimation 1 Unconditional least squares

0

NBACK = # of observations back from the end of the data that the multistep forecasts are to begin

0

NPRED = # of multistep forecast values to compute

0

INTERP 35

0 No interpolation of unobserved values 1 Interpolation of unobserved values

0

IESTIM « 0 No estimation of unknown parameters 1 Estimation of unknown parameters

1

THR IQR initial estimates of the regular moving average parameters (not input if there are not missing observations and INIC=0)

All -.1

THS IQS initial estimates of the seasonal moving average parameters (not input if there are not missing observations and INIC=0)

All -.1

FIR IPR initial estimates of the regular autoregressive parameters (not input if there are not missing observations and INIC=0)

All -.1

FIS IPS initial estimates of the seasonal autoregressive parameters (not input if there are not missing observations and INIC=0)

All -.1

VA 35 Residual variance to be used for inter- 1 .0

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IFILT IGRBAR IGRRES RG IDENSC INVER INIC

1 Square root filter 2 (No

2 M o r f , Sidhu and Kailath algorithm, missing)

as improved by Melard 3

3 Kalman filter (Missing)

1 Graph of autocorrelations printed 0 0 " " " not printed

1 Graph of model residuals printed 0 0 " " " " not printed

IMEAN + IREG + ITRAD + IEAST initial All 0.1 estimates of the regression parameters,

not including initial missing observa tions (not input if there are not missing observations or IC0NCE=1)

1 Denominator of residual sum of squares 1 is that of Ansley and Newbold = number of non-initial observations minus number of unknown parameters (AR and MA parame ters plus regression parameters, inclu ding initial missing observations) 0 Denominator of residual sum of squares

is equal to the number of non-initial observations

1 Parameters of the MA polynomial res- 0 tricted to remain in the invertible region

0 Parameters of the MA polynomial not restricted to remain in the invertible region

1 Initial estimates of AR and MA parame- 0 ters input

0 Initial estimates of AR and MA parame ters calculated

At present, the program only calcu lates initial estimates of AR and MA parameters if there are no missing ob servations. With missing observations, the program always takes as initial es timates for the AR and MA parameters those input by the the user (or their default values if none are input)

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input if IFILT=1)

= 0 only a2 concentrated out of the like lihood

If IREG in namelist DATEN is greater than zero, then namelist DATEN should be followed by a certain number of namelists REG, to be described below. Each namelist REG starts with &REG (in the second column), terminates with / and contains the set of instructions for the corresponding regression variable/s.

Missing observations can also be treated as additive outliers. That is, each missing observation is assigned a tentative value (now the code -99999. should not be entered) and an additive outlier is specified for each missing observation. In this case, one namelist REG corresponding to all missing observations should be written before the other namelists REG. The determinantal term in the function to be minimized when this approach is used is adjusted so that it coincides with that of the function used in our approach.

The total number of namelists REG is as follows: There must be one namelist REG for all missing observations to be treated as additive outliers (in case there are any), specifying their time indices, and as many namelists REG following as there are regression variables, either input by the user or calculated by the program. It is not possible to treat some missing observations as additive outliers while specifying others with the code -99999. , simultaneously. Only one procedure can be used.

The parameters in namelist REG are:

Parameter Meaning Default

IUSER = 0 The program will generate one interven- 0 tion variable if IAUS = -k (k = a posi tive integer): either k sequences of ones or the result of applying a filter to this intervention variable. Possible filters are: l/(l-SB), 0<6<1, 1/(1-5BS), S = NFREQ, 0<«<1, and l/(1-B)(1-BS) , S = NFREQ. If IAUS = k (a positive integer), it means that missing observations are to be treated as additive outliers. In this case, the program will generate k interven tion variables, one for each missing obser vation

= 1 The user will enter a series for this regression variable. After the present namelist REG, the user will write the

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IAUS

DELTA

DELTAS

ID1DS

if IUSER=1. The rest of the series, up to a total length of NZ + NPRED is filled up with zeros (not input if IUSER = 0)

k (k = a positive integer) There are k 0 missing observations to be treated as addi tive outliers. After the present namelist REG, the user must write the k time indices corresponding to these missing observations (free format). The program will generate k intervention variables of length NZ +

NPRED, one for each additive outlier (=missing observation). The k time indi ces are to be followed by the next name- list REG. (no input if IUSER=1)

-k (k = a positive integer) The program will generate one intervention varible of length NZ + NPRED consisting of k se quences of ones. After the present name- list REG, the user will write k pairs of numbers (free format); the j-th pair in dicates the time index where the j-th sequence of ones is to beguin and its length, respectively (j=l,...,k). The k

pairs of numbers are to be followed by the next namelist REG.(no input if IUSER=1)

0 The program will generate no regression variable (no input if IUSER=0)

5 (0<6<1); the filter 1/(1-5B) will be 0 applied to the k sequences of ones genera ted by the program when IAUS = -k (no input if IUSER=1 or IAUS > 0)

Ss (0<5S<1); the filter 1/(1-6SBS), S = 0 NFREQ, will be applied to the k sequences of ones generated by the program when IAUS = -k (no input if IUSER=1 or IAUS > 0)

1 The program will generate l/(1-B)(1-BS), 0 S = NFREQ, of the k sequences of ones ge nerated by the program when IAUS = -k

(no input if IUSER=1 or IAUS > 0)

The regression variables used to make the Trading Day or Easter Effect adjustment are generated by the program in the same way as that described in Hillmer, S.C., Bell, W. R . , and Tiao,

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constraints: IFILT=2 2 + 2*IMRTE < 42 IR + 2 + ICON < 42 IFILT=3 IFILT=1 1 + 2*IMRTE + IMISP < 4 2 MAX (IR + 1, ICON) + ICON < 42 like IFILT=3

In all cases MAX (N, ICON) + N + 1 < 42,

where IMRTE = IMEAN + IREG + 7*ITRAD + IEAST, IMISP = number of initial missing values, ICON - ICONCE*(IMRTE + IMISP), IR = MAX { ID + IP, IQ + 1), ID = IDR + NFREQ*IDS, IP = IPR + NFREQ*IPS, N = number of parameters to be estimated by Marquardt's method.

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b) Since the state space representation we use directly provides [in x(tlt)] the (r—1)- periods-ahead forecast function (where r is the dimension of the state vector), the program computes (NPRED + r -1 ) forecasts. Standard errors, however, are only computed for the first NPRED forecasts.

c) When there is a missing observation among the initial values of the series, the missing observation is estimated by regression as explained in the background paper. In the output, the interpolated value appears as the coefficient of ZJ, under the heading "Estimates of Regression Parameters."

The six examples present different regression and time series models, with different combination of missing observations. They have been taken from the following references:

B ox, G .E.P. and Jenkins, G.M. (1970), Time Series Analysis, San Francisco: Holden- Day.

B o x, G .E.P . and Tiao, G.C. (1975), "Intervention Analysis with Applications to Econom ic and Environmental Problems", Journal of the American Statistical

Association 70,70-79.

Harvey, A.C. and Pierce, R.G. (1984), "Estimating Missing Observations in Economic Time Series", Journal of the American Statistical Association 79, 125-131. Hillmer, S .C ., Bell, W .R. and Tiao, G.C. (1983), "Modelling Considerations in the

Seasonal Adjustment of Economic Time Series", in A. Zellner (ed.), Applied

Time Series Analysis of Economic Data, Washington, D.C.: Bureau of the

Census.

Kohn, R. and Ansley, C.F. (1986), "Estimation, Prediction and Interpolation for AR1MA Models with Missing Data", Journal of the American Statistical Association 81, 751-761.

Maddala, G.S. (1977), Econometrics, N .Y.: McGraw-Hill Book Co.

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' © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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form

COn » CO-1 £ COj £

y I = £ll + J _ gl2 ^2t + J_gl2 ^31 + n

V^n, = ( l - 0 1B ) ( l - 0 12B,2)at,

where f 2l and |3l are "intervention" variables such that t * Jan 1960

t = Jan 1960

months June - October, beginning in 1966 otherwise

months November - May, beginning in 1966 otherwise

Eight observations were randomly removed and estimated as missing values, and one of them falls among the first 12 values. The missing observations can be identified by the number -99999 in the file. For this model, the three regression variables are constructed by the program. (For their particular meaning, see the paper by Box and Tiao.)

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NUMB ER O F WH ITE NO IS E RESID UAL S

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OBS FOR ECA ST STD ERR OR AC TUAL RES IDU AL F OR EC AS T ( T R . S ER IE S) ( OR IGI NAL S ER IE S) 2 1 5 1 .8 4 4 0 .7 9 0 3 1 .6 0 0 0 -. 2 4 4 0 1 .8 4 sfeON.KiN'-cvji-'Oi-'j'O-^OcOOin'jKifNJNtNJincos*- mmeO'-d-eo*-'OK»'»OK>eor\j(\Jh-roN-oO'ON-(\J'OOrO'd-• - « - mmeO'-d-eo*-'OK»'»OK>eor\j(\Jh-roN-oO'ON-(\J'OOrO'd-• - ( M M f O f ' J K i f O O J O j » - * - * - ' — v t ' t i n i n ' O ' O - O i n K i K i K i o o o fO m r o N r o M M o o - o o " - lO'O fflffl w m»m Wffl mracoara (Mom'te0 !>iMMMsfsfwc0 '0 »-c0 T-r0 '0 «-f O i n O ' - e O C M M S M M ' O f V J s f O ' O ^ f M I O r O O ' N i«oeoN.ro»-orvJO'0«-fO'0'a-oj^-'Oiri'4-ro^;t - r s j r O f n r o p o r o r j^ r - ^ T - s f stinu •o o -o iri ro ro ro

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The estimated model is of the form:

yt = <o0 + <o1x,_1 + <u2zt_1+ n t

( l - 0 , B - 0 2B2)nt = a„

where x denotes a measure of the value of the firm, and z the stock of plant and equipment.

Two of the 20 observations were removed and treated as missing values. The regressors in this case are input by the user.

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EL ECT RIC : GR O S S I N VE ST . ( MA DD A LA , 1 97 7 P.21 4) , 2 R E G R E S S I O N VAR ., 2 M. O VO r* cn cda» rH ■«a* r* CO - H C'' o O il rH CN CO VO II W • • z u O rH ro o s • • CN CD VO o o CO CD CO CTl •z o CO rH • COcdH H in CO 0^ CO •CO •. •> rH H •CO VO rH CN H CD CO in H II rH in rH •II 04 • • CO CO coCD o 05 o • •CD < w CO rH CO o H CD 3 EH CO VO •o VO CD rH rH CO CD O H CO •1 II - CN CO CN rH cn CO CO • • • cn CO_a ii CN rH CD VO cd •H Eh CO CO CN cd rHH* O r " CD in O H CNrH CN 1rHII (H CN CO 05 H CN • • CN a *• • • VOH ro •H rH VO • rHin - II in O VO vo rHCO O Z CN VO CO rHII

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TIME SE RIES REGRESSION MO DELS WITH AR IM A ER RORS AND MISSIN G V A L U E S . - r \ j i ' 0 - < f u ' i ' O N - c o o o * - r \ j r O ' J i n ' O r ' ' - c o o o LU cc C3 O © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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MOD EL PAR AME TE RS I F IL T © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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9 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 'l = 301 O O CO ►— © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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REGULA R A R I NV ER SE ROOTS A R E O ' I''-lo o f\l CM 33 t— cmcm

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rô (vj ro — o> ro (M ( M > > - (M M IA ( M O 3 3 (VJ (VJ o oo K l i a CO (VJ (VJ CO i a S f (VJ O co co i a O (VJ 2 2 CO ■— UJ CO «3 IA O IA CO (VJ KJ 00 CO IA « - (VJ UJ CO §2 Lu O ° g » -< V J K 1 s f l A > O r > -C O C > © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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PARTIAL AUT OCORRELATIONS V A L U E S -.1 0 7 9 1 .0 8 5 6 3 -.1 6 8 2 7 .0 7 5 2 2 .0 2 6 5 5 .0 4 7 6 0 -.2 5 2 8 4 -.2 2 0 4 4 -.0 4 2 8 9 .0 2 5 9 2 -.0 4 5 2 9 -.0 2 6 2 2 o O o o - ' î + CM CM O CO î+ +’ r o cm 1 î + â k î 0 2 5 9 - . 2 5 8 2 + !

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NUMB ER O F WHITE NOI SE RESID UAL S (\i its 52 s f i n in 'O o i n oo i n «— m m 21C o ' nO rvj o co m os r - o ac >o >» w i n s j * - co c c8 rn j -a- ro © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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the last 6 years. Since the estimation problem is identical when the missing values are placed at the beginning, the example illustrates an important possible application of the program: Interpolation of data for frequencies higher than the observed one.

The model has no regressors, and is given by

v v u log y , = (1 - 0,B)(l - 012B12 )at, the so-called Airline Model of Box-Jenkins (1970).

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EXA MPL E (H A R V E Y -P IE R S E , JA S A 8 4 ) M O N T H L Y IN T E R P O L A T IO N , A IR L IN E M O D E L 1 4 4 1 9 4 9 1 1 2 1 1 2 1 1 8 1 3 2 1 2 9 1 2 1 1 3 5 1 4 8 1 4 8 1 3 6 1 1 9 1 0 4 1 1 8 1 1 5 1 2 6 1 4 1 1 3 5 1 2 5 1 4 9 1 7 0 1 7 0 1 5 8 1 3 3 1 1 4 1 4 0 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 1 .'03 1 03 1 03 ,1 03 1 031 rH 03 03 03 03 03 03 II 03 03 03 03 03 03 s 03 03 03 03 03 03 H 03 03 03 03 03 03 H 03 03 03 03 03 03 cn 1 1 1 1 1 | w H VO rH 03 «■ VO 03 o CM 03 03 03 03 03 03 O H rH CM CM 03 03 03 03 03 03 II 03 03 03 03 03 03 VO CM O CO 03 03 03 03 03 03 () CO o 03 03 03 03 03 03 o H H rH CM 1 1 1 1 1 1 ss H CM H rH 03 x VO 03 rH CM 03 03 03 03 03 03 rH rH CM CM 03 03 03 03 03 03 CM 03 03 03 03 03 03 II 03 r* 03 03 03 03 03 03 03 o CO O CO m 03 03 03 03 03 03 < rH CM CM CM 1 1 1 1 1 1 <n CM CM CO rH 03 03 03 03 03 03 03 03 II rH CM CM CM 03 03 03 03 03 03 C/J 03 03 03 03 03 03 a \ 03 O CM 03 CO 03 VO 03 VO 03 03 in 03 CM H X 03 CO vo O 03 03 o 03 CO 03 CO 03 o 03 co » CM rH CM CM CO 1CM 1 CO 1CO 1 CO 1 ■o- 1 rH H II II CO CO CO ■O' a t''- rH vo 03 03 03 03 03 03 03 03 03 03 030 3 a w rH CM CM CM 03 03 03 03 03 03 03 03 03 03 03 03 H - a 03 03 03 03 03 03 03 03 03 03 03 03 x H P4 CM CD 03 03 03 03 03 03 03 03 03 03 03 03 03 rH II s co CM CO 03 03 03 03 03 03 03 03 03 03 03 03 II « ~ rH H CM CM 1 11 1 1 1 1 11 | 11 w < CO u CQ II co rH in C" H « H vo CO CO CM 03 03 03 03 03 03 03 03 03 03 03 03 x O ^ rH rH CM CM 03 03 03 03 03 03 03 03 03 03 03 03 rH H H 03 03 03 03 03 03 03 03 03 03 03 03 II - P>H CO CO VO in 03 03 03 03 03 03 03 03 03 03 03 03 K H H 03 CO CO 03 03 03 03 03 03 03 03 03 03 03 03 Q II -rH -rH CM CM 11 1 1 1 11 1 1 11 1H d) O « II O O VO CO 53 W < in CO 03 CO 03 03 03 03 03 03 03 03 03 03 03 03 W H U rH rH rH rH 03 03 03 03 03 03 03 03 03 03 03 03 H 53 5 03 03 03 03 03 03 03 03 03 03 03 03 < H 2 in rH VO 03 03 03 03 03 03 03 03 03 03 03 03 Q t-H 03 o 03 03 03 03 03 03 03 03 03 03 03 03 ^5 rH rH rH CM 1 1 1 1 1 1 1 1 1 1 1 1 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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TIME SE RIES RE GRE SSION MO DELS W I T H ARI MA ERRORS AND MISSING V AL UE S. S 2 j rvjm roro -o -o -o -o -o -o -o -o -o -o -o -o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o raoO'O-jwoc_sam-cooc — •-'-rvjojpoOC SS inoeocoro'4-oo-ooo'o ' J O O - O O - O O - O t— .— »— r v j r M r v j O O ' O ' O O O §• O- ^ B- S-o S-o S-o S-o S-o S-o S-o S-o S-o S-o S-o S-o o o o o o o o o o o o o «— m ( M P O O ' ' i - 0 > 0 > 0 0 > 0 0 ( N J N S C O I M I O O - O O O O C _r_\_j₍_\_j_{c * ‘ ‘} O in ro *— in N- C <m ro 'O oo m r\i c CM OJ C >■ Ui «— » oo«-ruM^in'OMOOO ^£ininininininininin<) MI SS IN G OB SERVATION N UMB ER © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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NUMB ER O F WHITE NO I SE RESIDUALS WHI TE NO I SE RE SIDUALS 1 CM O O C f \ J « - s f O C M O O O O O O o o o o o o o o o o o U <N NKI r £ ~ CO > - —• -<r < - ' O i n i \ J O K O ' i O " - o i n O ' - c O i n c o i M - O f ' J ' - ' 0 ' - f f l -j--CM>^0'--jS(\Jcoeo(OK^'joNNOM'Oioro NjiA'j^fin'0'0iri'js}>tNjvtiAimn'0'C'0'0in'tim/\ o o o o o o o o c - C\J O CO CM O' T ■> in -a- in ro «■ « - K I K H O - J i n i n s t C M r - C M C M ’’'O'd'd'O'O'O'O'O'O'O'd'd'O'O'O'O'O'O-O'O'O'O'd'O'O o-*(Miovfin'OMOOO i n i n i n i n i n i n i n i n i n i n ' O C M M M S O O O ' O ' O O ' O ' co g i n ' O S c o o o T - i M f O s n o ' O S c o o o ^ i M M ^ i n ' O N C o O ' ' j s j - j ' j ' j i A i n i n i c i i A i n i c i i n i n i n ' O ' O ' O ' O ' O ' O ' O ' O ' O ' O © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute

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estimating the missing observations.

Example (4a) illustrates the standard approach: the initial missing observation is concentrated out of the likelihood, and estimated with regression. The other missing observations are obtained via the fixed point smoother.

Example (4b) treats all missing observations as additive outliers and estimates them with regression. Arbitrary (reasonable) numbers are plugged in the series holes, and then the following model is fit:

5

y. = X ® A +n.'

i=l

VV12nt = ( l - 0 ,B ) ( l - 0 12BI2)at

where d]t,...d 5t are the dummy variables associated with the additive outliers. The missing observation estimates are obtained as the fitted values, once the outlier effect (ft),) has been removed. When using additive outliers to estimate missing values, the likelihood function is modified by a determinantal term, so that it coincides with that of the standard missing observations case. Computation of the likelihood, however, is made easier, since the algorithm of Morf, Sidhu and Kailath can now be applied.

Comparing (4a) and (4b), it is seen that the forecasts and interpolators obtained with the two approaches are virtually indistinguishable.

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