••
THE FITTING
OF TlME-SERIES MODELS
by
James Durbin
University of North Carolina
and
University of London
This research was supported by the Office of
Naval Research under Contract No. Nonr-855(09)
for research in probability and statistics at
Chapel Hill. Reproduction in whole or in part
for any purpose of the United States Government
is permitted.
Institute of Statistics
Mimeograph Series No. 244
December 1 1959
THE Fl'TTING OF TD'lE-SERIES MODELS 1,2
by
J. Durbin
University of North Carolina
and London School of Economics
1.
Introduction
The purpose of this paper is to reT1ew methods of efficient estima-
tion of the parameters in some of the models commonly employed in timeseries analysis.
The models we shall consider are the following:
The autoregressive model
(1)
ut + (tlut _l +. • • + (tkUt_k =
Regression on fixed XIS and lagged yls
(2)
et
(t
= 1, ••• , n);
Yt + (tlYt-l +. • .+ (tpYt _p - ~lXlt +. • .+ ~qX~ + Et
(t = 1, ••• , n)i
Regression on fixed XIS with autoregressive disturbances
(3)
Yt· ~lXlt ~. • .+ ~qXqt + ut
ll.t
+
~Ut_l +. • .+ (tput _p = ~t
where
(t
= 1, ••• , n);
(t
= 1, ••• ,
The moving-average model
n);
lThis research was supported by the Office of Naval Research
under Contract No. Nonr-855(09) for research in probability and
statistics at Chapel Hill. Reproduction in Whole orin part for any
pUrpose of the ~ted States Government is permitted.
- 2
Invited review paper presented at a meeting of the Econometric
Society at Washing;on, D. C. on December 30, 1959.
•
2
The autoregressive model with moving-average errors
6 £.
q t-q
(t
= 1,.
., n).
~
In all cases [£t~ is assumed to be a sequence of independently
and identically distributed random variables with mean zero, variance
,l
and with finite 'moments
of all orders.
For discussions of effi-
ciency, but only then, the etts will be assumed to be normally distributed.
•
The sequences u , u l' u 2'0 •
0-
-
-
0
and y
J
0
Y l'Y 2'.
-
-
~
• are regarded
as sequences of fixed constants. For model (1) we assume that all
k
k-l
the roots of the equation x + ct X
+." .+"It = 0 !;lave modulus less
l
than one; a similar assumption holds for the eq~ation xP + (tlxP- l +. • .+
h-l
.
h
( )
ct
p = 0 in model (2) and model 3 J the equat~on; x + ~lx + • • • + ~h:o
q
l
in model (4) and the equations x P + 'YlxP- + • • • + 'Y = 0 and x + ~l
p
q
x - l + • • • + ~ = 0 in model (5). These assumptions are more restricq
tive than is necessary for certain parts of the exposition; they have
been made at the outset in this rather sweeping fashion for the sake
of
sir~licity
of presentation later on.
Throughout the paper an es-
'timator will be said to be efficient when it is asymptotically ef-
ficient.
20
the autoregressive model
There is an underlying unity in the methods of estimation to
be discussed in this paper arising from the fact that they all depend
fundamentally on the method of least squares.
model
For the autoregressive
•
3
n
~l
the minimisation of
to estimates
(ut +
~Ut_l
+. • 0+
~Ut_k)2
leads directly
•• J
a which are efficient and easy to compute.
k
The sampling properties of a , ••• , a were first investigated by
k
l
Mann and Wald [8J who showed that they are the same asymptotically
~'o
as those of least-squares estimates of regression coefficients in
multivariate normal systems.
(Actually Mann and Waldls model con-
tained a constant a o and therefore differs slightly from (1) but this
does not substantially affect the conclusions).
It is sometimes preferable to work with the asymptotically equivalent values a "
1
0
0
•
J
a
IC I obtained frol.l the equations
(6)
+
where r
i
~r=o
is the i th sample serial correlation coefficient.
If
desired the rila can be replaced by estimates of the serial covariances.
For all except small vailiues of k the equations (6) are easier to solve
than the least-squares equations owing to their possession of a more
symmetric structure.
It is found that a pivotal reduction of (6)
reduces to the recurrence relations
r a + a a-l.l r s-l + a a-1 0 2 r a- 2+. • .+as-l.s-l
= ... - - - - - - - -+.
- -• -• - - - -+a
- - - - r- a-l.a-1 s-1
(s
= 1,.
•• J It)
•
4
(8)
a
... a
+a a
s-l.r
ss s-l~s-r
sr
(r
= 1,.
0
.,s-l),
USi:ng;, ~l ... -r
a
l as the starting value. The quantities asl '. Ii . ,
are the coefficients of the best-fitting autoregressive model of
ss
order s, while -a
,o • o,-akk are estimates of the partial correlation
22
coefficients between observations 2, •• ~,k time periods apart with
intermediate observations held fixed.
Apart from yielding this in-
formation of incidental interest, the use of (1) and (8) is decidedly
more expeditious than a direct solution of (6). The final coefficients
akl ,. c .,akk are identically equal to the values aI', ••
obtained from (6) ..
3.
"~',
Regression on fixed XIS and lagged yls
The extension of the autoregressive model to include fixed XIS
appears to have been first considered about
1945 by Cowles Commission
Writers in connection with the study of simultaneous regression systems
(see Koopmans et al
l11 )..
In this paper we are concerned only with
the single-equation model
Yt + alYt_l +. • .. +apyt _p - ~lXlt +0 • .+~qXqt + Bt (t = 1,. • .,n)
where [~t} ,.
0' qt are sequences of constants. Putting Yt-i
It
x q+i • t and a i
the form
[x 1
Q
= -~q+i (i ... 1, ••• ,p) the model can be written in
(t ... 1,. ... ,n),
or in an obvious matrix notation,
Y
= X~
+ £',
where X is a nx(p+q)
•
matrix•
The least-squares estimatOrs of
of the vector b
= (X'X)-lX'y.
~l'.
0
.'~Q+P
are the elements
If X were a matrix of constants, least-
squares theory would tell us that the vector of discrepancies b has vector mean zero and variance matrix 02(X1X)-1.
~
For the present
model these results do not hold since some of the elements of X
are random variables.
However we can obtain analogous results by
introducing the matrix t = [E(XtX)] -1 XIX, where E(X'X) denotes the
matriX whose elements are the expected values of the corresponding
elements of XIX.
The matrix t will usually converge stochastically
to the unit matrix as n ~
It is show~ in
2
matrix 0 E(XIX).
00.
[41 that
t(b-~) has vector mean zero and variance
[t is also shown that when the 8
variance matrix is minimal in a certain sense.
( 9)
s2
J.
IS
are normal this
Letting
~
)2
= i;1f. t;l (Yt - blxlt • ...~bq+p..t x'l+P. t
it is shown further that under certain assumptions E(s 2 )
1
= 0 2+O(~)
n
and that bl ,., ... ,b are asymptotically multinormal. The implication
k
of these results is that least-squares theory applies asymptotically
to the model (2).
40
Regression on fixed
XIS
with autoregressive
err~
Of greater relevance in many investigations is the model
(t =
where the uttS are autocorrelated o
1,.
Q
.,n),
It is well known that a simple least-
squares analysis of data from such a model can be seriously misleading
6
owing to the inefficiency of the least-squares estimators of the
~'s
and to the biasedness of their estimates of variance (see for
example the discussion by Cochrane and Orcutt
and Anderson Ll] )0
[3] ,
Watson [loJ
It is true that for certain special cases, in-
cluding regressions on polynomial trends and seasonal constants, the
least-squares coefficients have been shown to be asymptotically efficient (Grenander and Rosenblatt
[6J
and Anderson and Anderson [2]
)~
Nevertheless the least-squares estimator of variance remains biased
and the use of analysis-of-variance methods for testing hypotheses
and setting confidence limits can be expected to give incorrect results.
It is often reasonable to assume that the ut's have the autoregressive structure
(11) ut
+
CLlut _l
+0 • .. +
CLpUt ...p .. f t .
(10) may then be transformed to the form
(12) Yt
+
CL1Yt _l
+. • 0+
CLpYt _p = ~l~t
+ CLl~lXlot_l +.
This now has the same structure as (2) except that relations exist
between the coefficients.
Conceptually the simplest approach to the
n
estimation problem would be to minimise
£t2 , when this is expressed
t=l
in terms of the Y's and a's, with respect to the a I s and ~ IS. How-
L
ever, since some of the coefficients in (12) are quadratic in the
unknowns this procedure results in non-linear estimating equations
which are usually unmanageable for practical use.
•
7
Before going on to discuss general methods we draw attention to
an important special case in whioh simple methods based on leastsquares theory do give satisfactory results.
This arises when each
x. t j (i=l, ••• ,q; j=l, ••• ,p) can be expressed as a linear function
1.0 -
of xlt'o • .,Xqt • Examples are polynomial trends, seasonal constants
and periodic regressions.
(12) then reduces to the form (2) with
funotionally independent coefficients which can be legituaately
estimated by least squares. We illustrate the procedure by considering the case of regression on a pure periodic function with firstorder autoregressive disturbances, i.e.
Yt =
~l
cos'At +
~2
siriAt + u " where ut + aUt _l = Et and
t
where A is known.
We have
Yt +ayt-l .. ~l cosAt+a~l cos'A( t-l) + ~ 2
.sin
'A t+aJJ 2 s inA( t-l) '+ t,
" == "(1 cos'At + "(2 sin'At + 8 t,
where
'Yl == ~l
+
a~l
Efficient estimates a,
n
L
(Yt
+
t=l
of
~l' ~2
cos'A ..
a~2
sinA,
and c2 are then obtained by minimising
aYt_l .. cl cos'At .. c 2 SiriAt)2 , whence estimates bl , b2
~
result from the equations
(1 + a cos 'A) bl .. a sin ~2 = 01
a sin 'A b + (1+acos'A)b .. c2 •
2
1
8
a
2
is estimated by s
2
.1
= n-3
2
n
~ (Yt+aYt _1 - cl cOSAt - c2sinAt) •
Fbr simPlicity of exposition our discussion of the general
problem will be based mainly on the two-coefficient model
(13)
Yt
= pXt
+ ~t ' where
(t=l,.
(14)
0
.,n).
It was pointed out by Cochrane and Orcutt (3 ]
that if a were
known we could employ an autoregressive transformation Yt I
= xt
Xt '
Y
t
I
=
AX
t'
= Yt +..
aYt _l
+ axt-l to put the model in the form
t
I
+
et ,
to Which least squares can be applied quite validly.
For the case
of unknown a they suggested that one should insert in the autoregressive
transformation either a value of a guessed on a priori grounds or
a value estimated from the residuals of a fitted least-squares regression, further iterations being carried out if desired.
these
suggestione~~
The first of
is computationally attractive, though inefficient,
while the second, though efficient, is computationally burdensome.
An
approach will now be outlined Which leads to estimates
which are efficient and which are not too onerous to compute.
From (13) and (14) we have
Yt + aYt_l - pXt + r.x t - l + c t '
where y "'" ape If we were to ignore the restriction y =
(15)
a~
and regard
y as a free parameter, (15) would have the form (2) so that the leastn
squares estimators a, b, c obtained by minimising ~ (Yt+aYt_l-bXt-CXt_l)2
t=l
9
would be efficient estimators of a,
mators of a and
~
bution of a, b,
Co
since Yt-l
~,y.
To obtain efficient esti-
we need only therefore consider the joint distri-
= ~Xt_l +
ut _l • Consequently a, band c -
a~
are the
least-squares coefficients of regression of Yt on -ut _l , xt and xt_lo
The corresponding true coefficients are a,
~
and zero in virtue of the
relation
(16)
By a slight extension of the results of the previous section we
know that least-squares regression theory applies asymptotically
to (16).
asymptotically normally distributed with
matrix
(i
CD
~ero
a~
and c -
are
means and variance
A-I, where A is the expected value of the matrix
l: u 2_
t 1
l:Ut_1Xt
I: ut_1X t
l: x t
i: ut -J3't-l
I: XtX _
distribution of a, band c n4
~
Consequently the quantities a - a, b -
of the expression
2
i: XtX _
t 1
t 1
~
% ut_1X t _1
i: x 2_
t 1
•
has a density which is the limit as
10
(11)
lf
2
2
constant x exp [ - 2a 2t(a-a) E(Zut _l ) + (b-~)
+ 2( b-jl )(
ZXt
a-.~) ],xt"t_l + (a-all)2 ],xt:l~] •
On maximising the exponent of (17) with respect to a and
th~t
22
~
we find
their efficient estimates are
1\
a ... a
Z(xt + axt_l)(Yt + aYt_l)
(18)
Z(x + ax _ ) 2
t l
t
It is remarkable that
"~
..
is precisely the same estimator as is obtained
by using a as an estimator of a in an autoregressive transformation.
The same procedure was arrived at earlier by the author
[4 ]
using a
rather different approacho
2
t and ~ ZX~_l' as :Ls legitimate to the order of accuracy considered here, we find that (18)
Ignoring the difference between
~
'l:x.
reduces to
(19)
R...
t'
(1 + ar)b + (a + r)c
2'
1 + 2ar + a
2
where r ... 1::xt X _ / ZX
t
t 1
depently distributed.
~
Note that a and
~
~
are asymptotically in-
The treatment of the general model 0) follows along similar lineso
We shall confine ourselves here to the presentation of a brief summary
L
of the computing routine, referring the reader to 4} for further
theoretical discussion.
For simplicity of exposition let us suppose
11
independent.
(The outstanding case to the contrary is the common
one in which the model contains a constant term, i.e. x
equals
it
unity for some i and all t; this, however, is easily dealt with by
working throughout with deviations from sample means as in ordinary
regression analysis.
MOdifications for other cases are easily worked
out ad ~.~)o
Suppose that the normal equations for the least-squares fitting
Xqot _p' -Yt_l'O •• ,-Yt - p ' the variables being taken in this order,
are denoted by
where A is the (p + q + pq) x (p + q'+ pq) matrix of sums of
l
and products of the variables
L
.L t
,· • .. , xq.. t -p '-Yt- 1 ,.
0
.,
sq~ares
-Yt
.... ,
'~.,:p
where c is the vector of sums of products of these variables with
l
Y ' and where b is the vector of regression coefficients. If the
t
equations are solved by a method such as the abbreviated Doolittle
method note that it is only necessary to carry the back solution
far enough to give the coefficients al ,. • .,a of -Yt 1'· •
p
-
0'
-Yt
-po
Form a new matrix A2 whose first row is obtained by multiplying
the first p+l rows of ~ by 1, a , .... ,ap respectively and adding,
l
whose second row is obtained by taking the second group of p+l roes
of A , multiplying by 1, a , ••• ,ap respectively and adding. Conl
l
tinue in this way until A2 has q rows. Repeat the process on 01
to give a new vector c containing q :elements.
2
Repeat the process on the
-1'
·12
columns of A2 to give a new matrix A with q rows anu columns. Let "3 be
3
the vector obtained by subtracting from c 2 the sum of ~ times the
last column of A2 , a times the second last column, •• • , ap times
2
the ~ column of A2 counting backwards from the last column. Then
the solution
"-
~
of the equation
(20)
is the vector of efficient estimators of
~l'o
•• '~q.
Its estimated
.
· ·1.S s 2A-1 I were
h
varl.ance
matr:1,X
3
5.
The moving-average model
The special problems of fitting moving-average models can be
appreciated from a consideration of the first-order model
(21)
(t • 1, • • • ,n).
A simple estimator of
~
can be obtained by equating the theoretical
value of the first serial correlation, namely ~/(1+~2), to the sample
value r
l
<,
However, the estimator was shown to be inefficient by
Whittle [11,12J who proposed the use of an approximate maximumlikelihood estimator equivalent to the solution to' the equation
~ [~
(1 - 2~rl + 2~2r2
l-~
of
where r. is the i th sample serial
1.
-
0
•
0)1~ ,. 0
co~relation.
Although efficient, this estimator is difficult to calculate
and the method does not easily extend to higher-order models.
The
13
J
author [ 5 has therefore suggested a different method in which a
~th .. order autoregressive model is first fitted to the data, k
being taken to be large.
cients
~,
0
•
0,
It is 'shown in
l?Jthat the fitted coeffi-
ak have an asymptotic distribution with the approximate
density
2 1/2
\" nfk-l
2
2 2}U
constant x (l-~ ) ..
exp
(ai +l +~ai) + ~ ~U
L- '2'li:C
Neglecting the factor
•
(1_~2) -1/2 since this is of small order in
n compared with the remainder and maximising the exponent With respect
to
~
we obtain as our estimator of
~,
k-l
.2:
( 22 )
b .... -
1=0
a i ai +l
""k~--2
(a -cl),
o
Z a.
i-o 1
the efficiency of which can be made as close to unity as desired by
taking k sufficiently largeo
For the gener al model
(t == 1, ...
",n)
the same approach yields estimators b ". " .,b Which are obtained
l
h
as the solution to the linear equations
h
(23)
E At
.' b ... -A
J n 'r
i=l r-Jl
(r
= 1, .... ,h),
k-r
where A .... 2: aiai + ' the a1. f s being as before.
r 1=0
r
6.
The autoregressive model with moving-average errors.
This model"which has the generating equation
has greater theoretical importance than the attention paid to it in
time-series literature would appear to indicate.
Firstly, it is the
general model of which the autoregressive and moving-average models
are special cases.
Secondly, when q=p-l it is the only one of the
three models whose structure is invariant under changes in the timeperiod between successive observations" a fact pointed out by
Quenouille
[9J .
Thirdly" equi-spaced observations from a continuous
stochastic process generated by a linear stochastic differential
equation, cr having a rational spectral density, conform to a
discrete model (24) with q = p-l.
Consequently a solution to the
problem of efficient fitting of (24) also gives as a by-product the
solution to the problems of fitting stochastic differential equation
models and of estimating rational spectral densities from discrete
data~
Yet in spite of the theoretical importance of the model
J
only Quenouille [9 appears to have considered the fitting problem;
however Quenouille did not attempt to discuss efficient methods of
estimation.
Two methods of fitting will now be described.
non-iterative but is not fully efficient.
The first is
The second is an iterative
method in which the autoregressive and moving-average parameters are
estimated alternately.... It is hoped to investigate the performance
1,5
of both methods by means of sampling experiments and to publish
the results
later~
Let us begin by considering the first-order model
(2,5)
Let al,o
(t=l,. • .,n)
D
D
o,a denote the coefficients of a fitted autoregressive
k
model of large order k and let e t denote the residual ut +
+. . . .+ akut_k<l
~Ut_l
In the first method of estimation we replace ~t-l
in (2,5) by e _ and estimate y and 0 by the values of c and of d
t l
which minimise E (u + cU _ - de _ )2
t l
t l
t
co
This leads to the equations
the solution of which is asymptotically equivalent to the expressions
(26)
+. " .+ \:r +
k l
3
c = - alr + a r +. ... +
akr k
2 2
l
(27)
d ... c
~r2
+ a2r
r
+ a r
,
+.
.+ akr +
l 2
k l
"
+ I + alr +• • • + akr
k
l
1
The method is readily extended to cover higher-order systems and can
be used to give starting values for the second method, which we now
describe.
First let us consider the estimation of 5 for a given value of Y.
Suppose that the true values of the first k autoregressive coefficients
16
are
~1'~
before.
the fitted values being denoted by al , ••• ,~ as
Using Mann and Wald's results
we know that for large k
•
~k'
[8J
the asymptotic distribution of a , ••• ,a is normal with density
k
l
(28)
constant x exp [- ..!;,.
~
2a~ 1,j=l
(a - el.) (a ... el j ) E (ut . u .)
1
1
J
-1 t -J~
The quadratic form in the exponent can be represented operationally as
where
Gdenotes the operation of taking an expectation over variation
of the u's, the a's being regarded as fixed constants.
Suppose that the true value of
transformation from a1 ,0
A ,.
.,A were made,
l
k
I>
y
were known and the following
~ ,ak and ell"" • • ,elk to
11'. • ."Ik
and
0
l
=V1
+y
a2
= f/2
+ yf/l
a
ell ... A.1 + Y
~ ~2 = A. 2 + "fA1
.
~ =(
k + "ft k-1
SUbstituting in (29) we have
. •J
(30) Q =
~~'i-il)(Ut_1+rUt_2)+· ~ ~+(tk-l-~-1)(Ut-k+1+YUt-k)
+(fk-Ak)
U
t _k} 2 •
0
11
Now ut + AUt _l = et + 6 e tal • Consequently, on putting Zt
we see that Zt satisfies the moving..average process Zt
Moreover ut _
k
2
:&
Zt_k .. YZ _k_ + Y Zt_k_2 +.
t
l
~.
•
:&
0=
ut + 'YUt _l
€t + 6 S t-l •
Consequently
gives
(31)
•
From the fact that the true autoregressive coefficients are
generatea by the relation 1+alz+a2z2+•• 0 = (l+yz )(1+0z)-1, it
. 1
follows that for large k, A. i • (.6) (i = 1,. e o,k) as accurately
as desired.
Since A can be made arbitrarily small the error committed
k
by taking A + = (_6)r A in (30) in place of ~+r = (_y)r A (r=1,2, ••• )
k r
k
k
can be made arbitrarily small.
Thus to a high degree of accuracy (31)
holds with Zt_l' Zt_2,e
"oorrespgndtng to a moving-average
model for which A , A ,D
2
l
0
0
••
are the true autoregressive coefficients.
Comparing (31) with (29) we see that 11 , 12, • • • behave like
autoregressive coefficients fitted to data corresponding to the model
(t=l, ... GIn) •
Consequently it follows from (22) that the efficient estimator of
18
5 is
(fo
(32)
II;
1)
Q
In practice it will probably suffice to terminate the smmnations in
this expression at about i .. k.
Note that we need not take explicit
account of the fact that the "constantlt in (28) depends on the co"
variance determinant of a ,. • ., a , which in turn depends on the
k
l
unknown parameter 5, since the determinant is of small order in n
compared with the exponent of (28)0
Let us now consider the converse problem, i.e. the estimation
of y given the true value of 5.
Define w 'o • o'Wn by the relation
1
where w is either defined arbitrarily O? taken equal to u -5u 1+52u 20 • ••
o
0
-
The wts then satisfy the autoregressive model
Consequently y is efficiently estimated by the expression
where E' denotes smmnation over the range of passible values of t
..
19
divided by the number of terms summed.
Let
s
r
... zr u u
Pr ...
t t+r
Zl U W
t t+r
It is easy to verify that to a good approximation we have the relations
(34)
p
r
= sr
- 8p
r-l
(r - -k+l , -k+2, •• olk)
(r = k-l, k-2, •• 0,0)
(5)
Applying (4) recursively taking P-k
= S_k'
•
and then (35) recursively
taking qk ... Pk' we obtain ql and qo from which we obtain the estimate
of y as
(36)
By applying (32) and (36) alternately we obtain an iterative
method of estimating y and 8.
a starting
(26) or (27) can be used to provide
point~
The treatment of the general model
follows similar lines.
are transformed to
The fitted a\l.toregressive constants a ,,, •• ,a
k
l
i l , •• • ,fk
by the relations
20
'1. .. 11
+ "fl
= i2
+ "fl
a2
.
fit s
and further
fit s
+ "f2
are obtainable from the expression
t
The
11
r +
y'rr..1 +. • .+
f/.
"fp r-p
•
0
behave approximately like autoregressive coefficients fitted
to data generated by the moving-average model
•• ,6
q
can therefore be estimated from equations (23) taking
ro
.1: (JiRi+r
(flo = 1)..
In practice the sUIlIDlation can probably be
J.=O
truncated at about i ... k.
To estimate "f ,.
l
e
."f for given 6
p
1
,0 ... ,6q we
use in place
of (34) and (35) the expressions
(37)
(38)
• • + 6 p.
q 1"... q
q
+ 6 q
r . l r+
.. s
(r
r
= -k+q,o
•• ,k)
(r .. k..q, ••• ,0)
1 +0 ... + 6 q
... p
q r+q
r
Estimates c ' •• o,c are then obtained from the equations
l
p
..
•
21
References
[11
Anderson, R. L.: lIThe Problem of Autocorrelation in Regression
Analysis, It Journal of the American Statistical Association,
Vola 49 (1954)" 113 - 1290
[2]
[3 J
Anderson, R. L.. and Anderson, T. W.:
"Distribution of the
Circular Serial Correlation 60efficient for Residuals from
Fitted Fourier Series, It Annals of Mathematical Statistics,
Vol. 21 (1950), 59 - 81..
Cochrane, D. and Orcutt, G. Ho
'lApplication of Least-Squares
Regression to Relationships Containing Autocorrelated Error
Terms," Journal of the American Statistical Association,
Vol. 44 (1949), 32 - 61 0
Durbin, J. 2 "Estimation of Parameters in Time-Series Regression
Models, II Journal of the Hoyal statistical Society, Series B,
(1960). In the press.
[;J
[6J
,:
ItEfficient Estimation of Parameters in l"1oving-Average
Models," Biometrika, Vol. 46 (1959)g In the press.
Grenander, U. and Rosenblatt, M.:
Statistical Analysis of Stationary
Time Series. New York: John Wiley and Sons, 1957.
Koopmans, To (Editor):
Statistical Inference in Dynamic Economic
Models. New York: John Wiley and Sons, 1950"
L8 J Mann, H. B. and Weld, A.:
llOn the Statistical Treatment of Linear
Stochastic Difference Equations, II Econometrica, Vol. II
(1943), 173 - 220.
[9J
Quenouille, M. H.:
Time-Intervals,"
[lOJ
Watson" G. S. =
Metrika" Volo
1 (1958)" 21 - 21.
"Serial Correlation in Regression Analysis I, It
Biometrika" Vol.
[l~
"Discrete Autoregressive Schemes ·with Varying
42 (1955), 321 - 341.
Whittle, P.:
Hypothesis Testing in Time Series Analysis.
AlmgVist and Wiksells, 1951.
, 2
---~kiv for
Uppsala:
"Estimation and Information in Stationary Time Series,1l
Matematik, Vol.
2 (1953), 423 - 434.