SOME CONVERGENCE THEOREMS ON LINEAR MODELS
GENERATING A PAIR OF RELATED TIME SERIES
[1]
Siddamsetty Upendra, Department of Statistics, SVU, Tirupati
[2]
R. Abbaiah, Department of Statistics, SVU, Tirupati
Abstract: The main aim of this paper is to establish some convergence theorems on
Linear Models Generating a Pair of Related Time Series of certain covariance type
functions relating to the model specified. The estimates of residual are obtained on using
the estimators defined under different placements of the roots 𝜌1 and 𝜌2 of P (z). This
work is motivated by similar studies on linear stochastic difference equations for scalar
time series. The pivotal lemmas concerned with the statements and proofs of some
lemmas.
Keywords: Time series definition, Stationary process, non-stationary process, assumptions of
stationary, stochastic models for time series, some of the pivotal lemmas.
The linear stochastic model generating a pair of related time series
𝑋1 = {𝑋1 (𝑡); 𝑡 = 1,2, … . } And 𝑋2 = {𝑋2 (𝑡); 𝑡 = 1 ,2 … . } discusses in this chapter has the
following specifications [1].
X1(t  1)  11X1(t )  12 X 2 (t )  13  U1(t  1) 
    [1]
X 2 (t  1)   21X1(t )   22 X 2 (t )   23  U1(t  1)
on being governed by the following basic and explanatory assumptions.
{𝑈1 (𝑡); 𝑡 = 1, 2, … . } , {𝑈2 (𝑡); 𝑡 = 1 , 2, … }are linear processes
ASSUMPTION 1:
generated by the two independent families of i.i.d random variables {∈1 (𝑡) ; 𝑡 = 1,
2, … } and {∈2 (𝑡) ; 𝑡 = 1, 2, … }say, such that (i) E (∈𝑖 (𝑡)) ; i = 1, 2 is zero and
(ii) 0 < 𝜎𝑖2 = 𝐸 ( ∈2𝑖 (t)) < ∞ ; 𝑖 = 1, 2. To be specific, let for i= 1, 2
t 1
 p (r) (t  r);t  1
(t )   b (r ) (t  r ); b (0)  1
U i (t ) 
i
l.i
i
i
r 0
 l.i
i
r 0
On setting ∈𝑖 (𝑡) = 0 for t ≤ 0, and under the stipulation that
and 𝑃𝑖 (𝑠) ; 𝑠 ≥ 0 are real numbers such that
𝑏𝑖 (𝑟); 𝑟 = 0, …,

0
 pi(s)  
s 0
That is, ∑ 𝑝𝑖 (𝑠) is absolutely summable.
Solving for 𝑋1 (𝑡) and 𝑋2 (𝑡) in terms of 𝑈1 (𝑡) and 𝑈2 (𝑡) from [1], one has [2].
(a) 𝑋1 (𝑡 + 2) − (𝛼11 + 𝛼22 ) 𝑋1 (𝑡 + 1) + (𝛼11 𝛼22 − 𝛼12 𝛼21 ) 𝑋1 (𝑡)
- [(1 − 𝛼22 ) 𝛼13 + 𝛼12 𝛼23 ]
= 𝑈1 (𝑡 + 2) − 𝛼22 𝑈1 (𝑡 + 1) + 𝛼12 𝑈2 (𝑡 + 1)
= 𝐿1 (𝑡 + 2)
(say)
(b) 𝑋2 (𝑡 + 2) − (𝛼11 + 𝛼22 ) 𝑋2 (𝑡 + 1) + (𝛼11 𝛼22 − 𝛼12 𝛼21 ) 𝑋2 (𝑡)
_ [(1 − 𝛼11 ) 𝛼23 + 𝛼21 𝛼13 ]
= 𝑈2 (𝑡 + 2) − 𝛼11 𝑈2 (𝑡 + 1) + 𝛼21 𝑈1 (𝑡 + 1)
= 𝐿2 (𝑡 + 2)
(say)
----- [2]
Equations [2]-(a), (b) lead to the explicit solutions [3].
t 1
X1(t )  1 


 (r ) L1(t  r ) 


    [3]

X 2 (t )   2 
 (r ) L2 (t  r )

r 0

r 0
t 1

on identifying [4].
1  (1  11   22  11 22  12 21) 1[(1   22 )13  12 23 ]
    [4]
 2  (1  11   22  11 22  12 21) 1[(1  11) 23   2113 ]
And defining λ(r) =0 for r<0, λ (0) =1, and for r≥1
 (r )  (11   22 ) (r  1)  (1111  12 21) (r  2)  0      [5]
The dynamic stability or otherwise of the solution in [3] depend on the placements
of the roots of the characteristic polynomial associated with the stochastic difference
equations in [2].
ASSUMPTION 2:
The roots 𝑃1 and 𝑃2 of the polynomial
𝑃(𝑧) = 𝑍 2 − (𝛼11 + 𝛼22 ) 𝑍 + (𝛼11 𝛼22 − 𝛼12 𝛼21 ) have either of the following placements.
(i)
(ii)
(iii)
|𝑃𝑖 | < 1 ; 𝑖 = 1, 2
|𝑃1 | > 1 > |𝑃2 |
|𝑃1 | > |𝑃2 | > 1
(Auto regressive)
(partially explosive)
(puyely explosive)
However, under partially explosive or purely explosive situations the following additional
assumption is imposed.
ASSUMPTION 3: whenever the two infinite series ∑ 𝛼𝑟 and ∑ 𝑐𝑟 are absolutely summable it

follows the random variable

 a  (r )  c  (r )
r 1
r 1
r 2
r 1
is continuous at zero.
NOTE :
All random variables represented by infinite series in this thesis are presumed to
be the mean square limits of absolutely mean square convergent series.
The model specified in [1] has wide applications in Econometrics. Our basic assumptions
imply that the identifiability conditions are necessarily satisfied so that estimation and
testing problems associated with the model [1] can be carried out without much theoretical
limitations.
Estimation of the parameters in [1] under autoregressive placements on the roots 𝜌1 and 𝜌2
of P (z) have been extensively discussed in the literature ( Hannan (1970), Anderson (1971), Fuller
(1976)), viewing the model [1] as a first order vector process. A detailed investigation on the models
of the type [1] is due to Venkataraman (1974) wherein he has studied, in depth, the asymptotic
properties of the least squares estimators of the parameters under the both explosive placement of
the roots 𝜌1 and 𝜌2 specified in assumption (ii).
It is interesting to note that generally, the least squares estimator (ˆ1,ˆ2 )  (ˆ11,ˆ12,ˆ13,ˆ21,ˆ22,ˆ23) of
the parameter vector (1,2 )  (11,12,13,21,22,23) occurring in [1] obtained by minimizing the sum
of squares
N 1

N 1
[ X1(t  1)  11X1(t )  12 X 2 (t )  13 ]2 
t 1
 (X (t  1)  
2
2
21X1 (t )   22 X 2 (t )   23 )    [6]
t 1
With respect to (𝛼1 ,𝛼2 ) is asymptotically well behaved under all placements of the roots 𝜌1 and 𝜌2
specified in assumption (ii). Further, under the purely explosive placements of the roots 𝜌1 and 𝜌2 in
assumption (ii), i.e., when 1   2  1 the modified least squares estimator
(~1,~1)  (~11,~12,~13;~21,~22,~23)
of (𝛼1 ,𝛼2 ) defined below is also asymptotically well behaved.
(a) Let (𝛼̃1𝑖 , 𝛼̃2𝑖 ); i=1,2 be the least squares estimators of (𝛼1𝑖 , 𝛼2𝑖 ); i=1,2 respectively by
minimising the sum of squares
N 1
N 1
 (X (t  1) - 
1
2
11X1 (t )  12 X 2 (t )) 
t 1
2
2
21X1 (t )   22 X 2 (t ))
t 1
N 1

(X1(t  1) - ~11X1(t )  ~12 X 2 (t )) 


t 1
    [7 ]
N 1

~
~
~
1
 23  ( N  1)
( X 2 (t  1)   21X1(t )   22 X 2 (t ))

t 1

~13  ( N  1) 1
(b) [7].
 ( X (t  1)  


The limit theorems that are established in this chapter depend on these estimators and their
limit distribution properties. To this end, some of the well known results on these estimators
are recorded for easy reference.
PROPOSITION (I): Let (i) 1  1 (ii) Ui (t)  i (t ) for i=1, 2. Then under basic assumptions the
following statements holds.
( N 1/ 2 (ˆ11  11), N 1/ 2 (ˆ12  12 ), N 1/ 2 (ˆ13  13 ), N 1/ 2 (ˆ 21   21), N 1/ 2 (ˆ 22   22 ), N 1/ 2 (ˆ 23   23 ))
Converges in
L
law ( 
) as N   , to a normal vector 1  (1(11), 1(12), 1(13), 1(21), 1(22), 1(23)) , say with mean
zero and a well specified non-singular covariance matrix.
Towards stating the asymptotic properties of the estimators under explosive situations it is
necessary to introduce, for i=1, 2.[8].

Gi 
r 1

Hi 


r

1 Li ( r ), when  1  1 

r
2 Li ( r ), when  2
r 1
PROPOSITION (II): Let (i)
1  1   2
, (ii)

    [8]

 1


Ui (t )  i (t ) , (iii) P(G i=0)=0 for i=1, 2. Then under basic
assumptions, the following statements hold.
( N 1/ 2 (ˆ11  11), N 1/ 2 (ˆ12  12 ), N 1/ 2 (ˆ13  13 ), N 1/ 2 (ˆ 21   21), N 1/ 2 (ˆ 22   22 ), N 1/ 2 (ˆ 23   23 ))
L
Converges in law ( 
) as N   , to a normal vector
 2  ( 2 (11),  2 (12),  2 (13),  2 (21),  2 (22),  2 (23)) say, with mean zero, such that
(i)
M02 (11)  2 (12)  0
(ii)
M02 (21)  2 (22)  0
M0
Being defined as
PROPORSITION (III): Let (i)
G1 / G2 = 12 /( 1  11)  ( 1   22 ) /  21
 1   2  1 , (ii) P(Gi  0)  0
for i=1, 2, (iii)
P(Hi  0)  0
for i=1, 2.
Then under the basic assumptions following statements hold.
L
(a) (2N (ˆ11  11), 2N (ˆ12  12), 2N (ˆ21  21), 2N (ˆ22  22)) Converges in law ( 
) as N   , to a
normal vector
3  (3(11),3(12),2 (21),3(22)) ; say such that
(b)

M03 (11)  3 (12)  0

M 03 (21)  3 (22)  0
( N 1/ 2 (ˆ13  13 ), N 1/ 2 (ˆ 23   23 ))
 4  ( 4 (13),  4 (23))
(c)
say, with mean zero and a non-singular covariance matrix.
(2N (~11  11), 2N (~12  12 ), 2N (~21  21), 2N (~22  22))
5  (5 (11),5 (12),5 (21),5 (22))
(d)
L
Converges in law ( 
) as N   , to a normal vector
(i)
M05 (11)  5 (12)  0
(ii)
M 05 (21)  5 (22)  0
( N 1/ 2 (~13  13 ), N 1/ 2 (~23   23 ))
6  (6 (13),6 (23))
L
Converges in law ( 
) as
N  ,
Say, such that
L
Converges in law ( 
) as N   , to a normal vector
Say, with mean zero and a non-singular covariance matrix.
The main aim of this chapter is to establish some pivotal lemmas certain covariance type
functions relating to the model [1]. The estimates of residual are obtained on using the
estimators defined in [6], and [7] under different placements of the roots 𝜌1 and 𝜌2 of
P (z). This work is motivated by similar studies on linear stochastic difference equations
for scalar time series. The preliminary ground work that is necessary to establish the
basic results of this chapter is presented in the next section. Section, pivotal lemmas
concerned with the statements and proofs of two pivotal lemmas.
SOME OF THE PIVOTAL LEMMAS
The following lemma, which is elementary and classical, providing the essential
ingredient for the validity of propositions (i),(ii) and (iii) finds repeated reference
throughout this chapter.
LEMMA (1): Under the basic assumption on i (t) ; i=1, 2.
N


 1/ 2

1
i (t )  j (t  r ), r  1,2,  ; i  1,2 
 N ( i j )


t 1


N


1
/
2

1
 N ( i )

i (t )  j (t  k ), i  1,2


t 1




Converges in law,as N   , to a normal vector
0 (i, j, r), r  1,...,T , i  j  1,2;0 (i),i  1,2 with mean zero and unit covariance matrix.
NOTE: The validity of the statement is a direct consequence of the standard result due to
Diananda (1953) as applied to the vector process i (t ) j (t  r ), r  1,...,T , i (t  k ),i  j  1,2 .
The following lemma which is referred to in this, time and again, plays the role of the
necessary probabilistic tool, towards establishing the limit theorems related to time
series under study.
LEMMA (2): Let (UiN ), (ViN (n)) and (WiN (n)) ; i=1, -----, T, n=1, 2, ------- be a sequences of
random variables such that
(a) For N≥N0 (n) and n≥n0 (say)
For i =1, 2, ---, T.
UiN  Vi N (n)  wiN (n)
(b) For fixed n(≥n0)
iN (n)  xi ; i  1,...,T )  p.d . fFn,T ( x 1 ...x T )
lim p(V
And
N 
n,T ( x1...xT )  F.T ( x 1 ...x T )
lim p.d. fF
for all
n 
(X 1... x T) lying in a set D (T) (say) dense in the T-dimensional Euclidean space RT (say);
and
sup p  wiN (n)     0 For any   0 and for i =1, ---, T.
(c)
lim lim
n 
N 
Then it follows that
iN (n)  xi ; i  1,...,T )  F.T ( x 1 ...x T )
lim p(V
at every continuity point F.T
N 
NOTE: The proof of this useful convergence is found in Venkataraman (1968).
LEMMA (3): Under the basic assumption on i (t ) for i =1, 2 the following statement holds:
( N 1/ 2
N k
Ui(t  k );i  1,2)
t 1
Converges in law, as N   , to 1, 2  , say, where

i  (


 b (s))Z
pi (r ))(
r 0
i
i
for i =1, 2.
s 0
And Z1 and Z2 are independent normal variables with means zero and variance equal to
12 and  2 2 respectively.
PROOF: For n  1 and N>n+k+2 it is possible to represent that
N 1/ 2
N k

U i (t  k ) N 1/ 2
t 1

n

pi (r )
r 0
N
  (t)  w
bi ( s )
s 0
i
iN (n)
--- [9]
t 1
Where wiN (n) ; i =1, 2 satisfy the requirement (c) in Lemma (2). The validity of the
statement in the lemma follows from [9] on invoking Lemmas (1) and (2).

Let
Q(  ) 
  (r) (r   )    [10]
r 0
LEMMA (4): Let |𝑃𝑖 | < 1 ; 𝑖 = 1, 2 and Ui (t)  i (t ) ; i=1, 2. Then under the basic
assumptions, the following statements holds for i=1, 2, as N   .
(a) For fixed integrals p, q and
N 1
(i)
h( N )
N h
 X (t  q)  
p
i
i
t 1
N 1
(ii)
N h
 X (t  q) X (t  q)  
p
1
2
2
2
2 2
1  1 [Q( q  p )   22 (Q( q  p  1)  Q(q  p  1))   22Q(q  p )]   2 (12Q(q  p ))
1
t 1
N 1
(iii)
N h
 X (t  q) X (t  q)   
p
1
2
2
1 2  1 [12Q(q  p  1)   21 22 (Q(q  p )]   2 (12Q(q  p  1))  1112Q(q  p ))
2
t 1
N 1
(iv)
N h
 X (t  q) X (t  q)  
p
2
2
t 1
2
2 2
2
2
2  1 [ 21Q(q  p )]   2 [Q( q  p ))  11(Q(q  p  1)  Q(q  p  1))   11Q( q  p )]
(b)
(i)
For any p≥0 and for any o<h≤N
N h
 X (t  p)  A
N 1/ 2 E
i
0
t 1
(ii)
For any p, q  0 and for any o≤h<N
N 1E
N h
 X (t  p) X (t  h  q)  A
i
j
0
t 1
For any fixed q≥0, for any p≥0 and o≤h<N
(iii)
N h
N 1/ 2 E
U (t  p) X (t  h  q)  A
i
j
0
t 1
(iv)
For any integer p and q and o≤h(<N)
N h
N 1/ 2 E
U (t  p) X (t  q)  A
i
j
0
With p>q
t 1
Q(  )
being defined in [10].
NOTE: In this A0 is being used as a generic notation for any positive memorising constant.
PROOF: The validity of the statements in the lemma follows directly on a substitution
evaluation based on [3] and assumption (I), on remembering that |𝜌𝑖 |<1.
It follows from assumption (II) and (V) that
 (r )  ( 1  2 )1 1r 1  ( 2  1)1 2r 1; 1  2      [11]
An alternative representation for𝑋𝑖 (𝑡); i =1, 2 can now be deduced from [3] and [11] under
the case:  1  1   2 as
X i (t )  ( 1   2 ) 1 1r 1Gi  (  2  1) 1  2r 1Li (t  r )  ( 1   2 ) 1


 r 1
Li (t  r )  i
1
r 1
Further if
*
is a real root of P (Z) such that |  * |>1, and if

J i    * Li (r ); i  1,2      [13]
r 1
r
; i=1, 2; t≥1 ----- [12]
It is known that, on suitable rearrangement of terms
(  *  11 ) J1  12 J 2  0 
    [14]
(  *   22 ) J 2   21J1  0
Next, the discussions in the sequel are based on the properties of the function
 (t )  G 2 X 1(t )  G1X 2(t ); t  1
    [15]
 (t )  0; t  0

Thus settled, the following is an important result for the case:
1   2  1
LEMMA (5): Let (i)
,(ii)
Ui (t )  i (t )
1  1   2
(iii)
.
F (Gi  0)  0 ,
i=1, 2.
Then under the following statements hold for i=1, 2, as N   .
(a) (i) For fixed integers p, q and 0  h( N )
1 2 N
N h
 X (t  p) X (t  q) (   )
p
i
j
1
2
2
Gi G j 1p  q  2h  4 ( 12  1) 1
For i, j=1, 2.
t 1
(ii)For fixed integers p and 0≤ h (<N)
1 N
N h
 X (t  p) is bounded in probability.
i
t 1
(iii)
For fixed non-negative integers p and 0≤ h (<N)
N 1
N h
 (t  p) (G   G  )
p
2 1
1 2
t 1
(iv)
For fixed non-negative integers p, q and 0≤ h (<N)
N 1
N h
 (t  p) (t  q) (G   G  )
p
2 1
1 2
2
 (G2212  G12 22 )  2( p  q ) (1   22 ) 1
t 1
(b) For any integer p and 0≤ h (<N), each of the following terms has absolute expectation
bounded by A0
(i)
1 N
N h
 X (t  p)
i
t 1
(ii)
N 1 / 2
N h
 (t  p)
t 1
(c) For any integral p, q≥0 and 0≤ h (<N) the following terms has absolute expectation
bounded by A0
(i)
N 1
N h
  (t  p) (t  q)
t 1
(ii)
1 N
N h
 (t  p)X (t  h  p)
i
t 1
(iii)
1 N
N h
 (t  h  p)X (t  p)
i
t 1
(iv)
1 2 N
N h
 X (t  p) X (t  h  q)
i
t 1
i
(v)
1 N
N h
U (t  p)X
i
j (t  h  p )
t 1
(vi)
N 1 / 2
N h
U (t  p) (t  h  q)
i
t 1
PROOF: It follows from [14] that when 1  1
( 1  11 )G1  12G2  0 
    [16]
( 1   22 )G2   21G1  0
This information together with (1.2.4) implies that
t 1
t 1
r 0
r 0
 (t )  (G2 1  G1 2 )  G2   2rU i (t  r )  G1   2rU 2 (t  r )    [17]
The validity of the statements (a),(b) and (c) of the lemma follows directly on a substitution
evaluation of the relevant terms based on [12] and [17], together with an apple to lemma(1) on
remembering the basic assumptions.
A parallel result for the case: 1  2  1 is given below;
LEMMA (6): Let (i) 1  2  1 (ii) p(Gi  0)  0 (iii) p( H i  0)  0 , i=1, 2. Then under basic
assumptions the following statements hold, as N   .
(a) (i) For fixed integers p, q and 0≤ h (<N)
N h
p
1 2 N  X i (t  p) X j (t  q) 

( 1   2 )  2Gi G j 1p  q  2 h  4 ( 12  1) 1 ; i  j  1,2.
t 1
(ii)For fixed integers p and 0≤ h (<N)
N h
p
1 N  X i (t  p) 

( 1   2 ) 1 Gi 1p  h  2 ( 1  1) 1 ; i  1,2.
t 1
(iii)
For fixed non-negative integers p and 0< h (<N)
N h
p
 2 N   (t  p) 

( 1   2 ) 1 (G2 H1  G1 H 2 )  2p  h  2 (  2  1) 1
t 1
(iv)
For fixed non-negative integers p, q and 0≤ h (<N)
N h
p
 2 2 N   (t  p) (t  q) 

( 1   2 )  2 (G2 H1  G1 H 2 ) 2  2p  q  2 h  4 (  22  1) 1
t 1
(v)
For fixed integers p and q ≥ 0 and 0≤ h (<N)
N h
p
1 N  2 N  X i (t  p) (t  q) 

( 1   2 ) 1 (  2  1 ) 1 Gi (G1 H 2  G2 H1 ) 1p  h  2  2p  h  2 ( 1  2  1) 1
t 1
(b) For any integer p and 0≤ h (<N), each of the following terms has absolute expectation bounded
by A0
N h
(i)
1 N  X i (t  p )
t 1
N h
(ii)
 2 N   (t  p)
t 1
(c) For any integral p, q>0 and 0≤ h (<N) the following terms have absolute expectations by A0
N h
(i)
1 2 N  X i (t  p) X j (t  h  q); i  1,2.
t 1
N h
(ii)
1 N  2 N  X i (t  p) (t  h  q); i  1,2.
t 1
N h
(iii)
 2 2 N   (t  p) (t  h  q)
t 1
N h
(iv)
1 N  U i (t  p)X j (t  h  q)
t 1
N h
(v)
 2 N U i (t  p) (t  h  q)
t 1
PROOF: It follows from [3] and [11] that for t  1


r 1
r 1
X i (t )  ( 1   2 ) 1 1t 1Gi  (  2  1 ) 1  2t 1 H i  ( 1   2 ) 1  1r 1 Li (t  r )  (  2  1 ) 1   2r 1 Li (t  r )   i ; i  1,2.    [18]
Further from [8], [15], [16] and [18], it can be checked on routine manipulations that


r 1
r 1
 (t )  (  2  1 ) 1  2t 1 (G2 H1  G1 H 2 )  (G2 1  G1 2 )  [G2   2r  2 (t  r )  G1   2r  1 (t  r )]    [19]
and the basic assumptions that the expectation within the square brackets on the R.H.S. of
[19] has absolute expectation which is uniformly expressions based on [18] and [19]
together with an appeal to lemma(2) and to standard probabilistic results established the
validity of the statements (a), (b) and (c) of the lemma.
From [6] it can be seen that
(ˆ11  11 )  ˆ 11 / ˆ 10 ;ˆ 21   21  ˆ 21 / ˆ 10
(ˆ   )  ˆ / ˆ ;ˆ    ˆ / ˆ


12
12
12
10
22
22
22
10     [ 20]

(ˆ13  13 )  ˆ 13 / ˆ 10 ;ˆ 23   23  ˆ 23 / ˆ 10 
Where
 X (t ) 
2
1
(i)
ˆ 10   X 1 (t ) X 2 (t ) 
 X 1 (t ) 
 X 1 (t ) X 2 (t )   X 1 (t ) 
 X 22 (t ) 
 X 2 (t ) 
 X 2 (t ) 
N 1
N h
(ii)
 f (t )   f (t ).
(iii)
ˆ 1i ; i  1,2,3 is obtained from ̂10 on replacing its ‘ith’ column by the column vector
t 1
(iv)
 U1 (t 1) X1 (t ) ,  U1 (t 1) X 2 (t ) ,  U1 (t 1) 
ˆ 2i ; i  1,2,3 is obtained from ˆ 1i ; i  1,2,3 respectively,
On replacing U1 (t  1) by U 2 (t  1) ----- [21]
NOTE: let Rij(f) denote the operation on a determinant of adding f times jth row to its ith row
and Cij(f), a similar operation on columns. Successive operations of Rij(.) and Cij(.) are read
from right to left to indicate order of application.