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(1)

*

### , Maria Stoica

**

* Petroleum-Gas University of Ploieşti, 39 Bd. Bucureşti, Ploieşti, Romania e-mail: [email protected]

** “Nichita Stănescu” High School, 3 Nalbei, Ploieşti, Romania e-mail: [email protected]

### Abstract

The construction of the statistic or econometric models of the chronological series is also made with the help of stochastic processes. This article presents a modeling problem of stationarity as the property of chronological series using the stochastic equation of a chronological series and the stochastic equation of the mobile average.

Key words:covariant, corelograma , stochastic process JEL Classification:A12

### Introduction

The statistic series presents itself as a multitude of statistic data reflecting yearly, quarterly, monthly results etc. regarding the economic activities which are characterized not by stability but mainly by fluctuations within a specified time period. The analysis of a chronological series can be made by determining some indicators regarding the values variation, by determining the adjusted values using suitable statistic methods and which can not allow obtaining some accurate information regarding the future or anticipated values.

The evolution of a social-economic phenomenon is determined by different factors requiring thus the emphasis of the chronological series nature and of its va value as related to the possible result of the phenomenon or uncontrollable complex process F.

Statistically speaking, process F can be considered a positive real aleatory variable V , the variable being defined by a probability field associated to Fand the chronological series is defined by

vt /tÎT

, T Í

0,¥

### )

, vt represents the value of an economic variable at timet, noted with Vt and defined on the borelian field

### ( W , K , P )

.

The construction of the statistic or econometric series is also made with the help of the stochastic processes. If the space of elementary events, K is a borelian body or σ-algebra over Ω and P is a completely additional probability or a probabilistic measure on the borlian fields of events

### ( W ,K , )

and a totally ordered multitude is

### ( )

T,£ , then we define a stochastic process on

### ( W , K , P )

and the ordered multitude

### ( )

T,£ as the aleatory variables family

(2)

V(t,w) wÎW,tÎT

### }

, where V(t,w):T´W®R has partial functions V(t,·):W®R as a real onedimensional aleatory variable and V(·;w):T ®R which is a real function called achievement or trajectory of the stochastic process andT is called indexes multitude.

The notations for this stochastic process are:

V(t,w)

sau

Vt(w)

tÎT or

Vt(·)

### )

tÎT.

For the fixed wÎW the adequate achievement of the stochastic process is noted by

### ( )

Vt tÎT where vt===notedVt

### ( )

w .

According to the definition of the stochastic process induced in [3] we can consider the stochastic process an aleatory variables family

Vt(·)

### )

tÎT, defined on the same probability borelian field

### ( W , K , P )

.

The existence of the stochastic process is presented in [5], the aleatory variables

Vt(·)

### )

tÎT are discrete or of continuous type and the schievements of the stochastic process

Vt(·)

### )

tÎT can be real continuous functions or not.

If T =

t1,t2,...,tn

### }

is finite then the stochastic process

Vt(·)

### )

tÎT presents itself as an aleatory

vector

### ( ) ( )

n tj j

V

V ( ) 1,

· =

=

· and probabilistically speaking, it is completely determined by the common repartition function:

n

### )

=

t t

t v v v

F , ,...,n 1, 2,...,

2

1 P

Vt

v Vt

vj Vt

vn

## } )

n

j w < w <

<

w

w 1,..., ,..,

1

Î

"v1,v2,...,vn R, "nÎN*,

Which fulfills the compatibility condition:

x x x x x

F

x x x x x

### )

i n

Ft t t i i i n t t t t i i i n

xlimi 1,2,...,n 1,... -1, , +1..., = 1,...,i 1,i 1...,n 1,... -1, , +1..., ," =1,

¥

® - +

The chronological series obtained after the statistic observation is

vt tÎTc

, where

0,¥

### )

,

c Í

T Tc finite multitude, that is the result of a temporal observation of an economic variable.

Stochastically speaking, the taking over of the chronological series is made on the probability borelian field

### ( W , K , P )

associated to the process determining the economic variable.

We define the time series as the stochastic process

Vt(·)

tÎT over

### ( W , K , P )

and Tc ÌT Í R, having the property of being wÎW, so thatVt(w)=vt,"tÎTc.

The econometric pattern of the chronological series noted

### ( )

Vt tÎT is defined by the time series

Vt(·)

### )

tÎT and the time series achievement is a family of values

Vt tÎT .

(3)

### The Time Independence of the Chronological Series

The chronological series with an evolution independent of time or stationary is the series which has neither tendency nor circularity, nor seasonality, thus it is not influenced by the number of observations (which can be small or big) and neither by the beginning moment of the observation. The problem of the predictions for this chronological series is simplified, because a similar set of factors constantly acts on the economic variable.

Econometrically speaking, we deal with the issue of moulding the stationarity of a chronological series by defining the stationary notion for the time series which are econometric patterns for the chronological series.

We define time series

### ( )

Vt tÎT strictly stationary if:

n

t h tn h

n

### )

tn

t V t t t V V t t t

V v v v F v v v

F , ,..., , ,...,

2 1 1

2

1,..., 1 = + ,..., +

¹

### }

¹Æ

"t1,..,tn t1+h,..,tn+h and all value sequences

vt ,vt ,...,vtn

### )

2

1 from the domain of

the aleatory variable values Vt.

We mention that the indexes t1,...,tnare not consecutive; if the time series is strictly stationary then the repartition function of an aleatory variable is the same for any value from the indexes multitude and the common repartition depends only on the distance among the elements of the indexes multitude and not on their actual values.

If

### ( )

Vt tÎT is a strictly stationary time series then the average and the variant of the variable Vt are constant, the same for anyt.

We call a time series

### ( )

Vt tÎT weakly stationary or stationary if the following conditions are fulfilled:

a. M(Vt)=c,"tÎT,cÎ R;

b. D2(Vt)<¥,"tÎT;

c. the covariant matrix of

Vt ,Vt ,...,Vtn

### )

2

1 coincides with that of

### (

Vt+h,Vt +h,...,Vtn+h

2

1 for

### { }

¹Æ

"t1,...,tn and finite and all h for which

### {

t1,..,tn,t1+h,..,tn+h

### }

ÌT .

h ‘s feature is to take those values for which the indexes multitude

### {

t1,..,tn,t1+h,..,tn+h

### }

is contained inT we note it with p0.

Observations:

1. the constant c from the definition of the weakly stationary (a) by convention can be considered as being zero.

2. the covariant matrix is a time function only among the observations.

Given condition (c) and observation 2. we can say that a weakly stationary series isstationary in covariant. The selfcovariant function or the covariant function of the time series

### ( )

Vt tÎT stationary is noted and defined as follows:

γ :R® R, g(h)=cov(Vt,Vt+h)=cov(Vt+h,Vt)=M(Vt,Vt+h)

(4)

where M(Vt)=0andh has the feature p0.

Theconsequences of the previous definition are as follows:

a) g(0)=cov(Vt,Vt)=M

Vt2 -

M

Vt

2 =D2

### ( )

Vt , "tÎT; b) g(-h)=g(h),"h with the property p0;

c) the covariant function of a stationary time series is non negatively defined.

The selfcovariant function of a stationary time series depends on the data measure which forms the chronological series and consequently to compare the time series features we must eliminate the measure by introducing a new function called self correlation function or correlation function.

The time series correlation function

### ( )

Vt tÎT is noted and defined as:

r:R® R,

) 0 (

) ) (

( g

= g

r h

h .

The correlation coefficient between two aleatory varaiables defined on the same probability field justifies the notation used for the selfcorrelation function and r(h) £1,"h with the feature p0.

The selfcorrelation function diagram can be achieved using different softwares, such as Mathematics, Statistics etc. and it is called corelogram.

### The Stochastic Equation of the Selfregressive Series

The stochastic equation of a chronological series

### ( )

Vt tÎTT =Z , non homogeneous linear with differences of ordern with the constant coefficients is :

t n t n t

t

t aV a V a V b

V

a0 + 1 -1+ 2 -2+...+ - = , tÎ

n,n+1,...

### }

,nÎ N * (1) where an ¹0,aiÎR"iÎN *, bt non-zero function of t .

for bt =0 equation (1) is called homogeneous stochastic equation with differences of order n with constant coefficients.

In the stochastic equations the terms are aleatory variables of some stochastic processes and the regressive differences are functions on multitudeZ.

The chronological series

### ( )

Vt tÎTis associated to a real function family corresponding to a stochastic process and consequently to solve the homogeneous and linear equation with finite differences of order n with constant coefficients, we attach the characteristic equation:

0

1 ...

1

0rt +art- + +anrt-n =

a (2)

whose roots are inC.

(5)

The chronological series

### ( )

Vt tÎT is called selfregressive order p if T =Z and there are real coefficients ai, i=0,p,a0 ¹0,ap ¹0, so that a0Vt +a1Vt-1+a2Vt-2+...+apVt-p =et,tÎ Z,

Î

p N *şi

### ( )

et tÎZ ÎZA(0,s2).

We consider AR(p) the multitude of all selfregressive chronological series of order p.

If there is a row of numbers

### ( )

an nÎN and a0 =1so that

Z T = Î

= +

+ +

+aV- a V- a V- e t V

a0 t 1 t 1 2 t 2 ... n t n t, then

### ( )

Vt tÎT is self regressive of indefinite order.

For

### ( )

Vt tÎTÎAR(1)andT =Z the self regressive chronological series presents itself like:

0 , 0

, 0 1

1 1

0V +aV- =e a ¹ a ¹

a t t t şi

### ( )

et tÎZ ÎZA(0,s2) (3) Equation (3) is a linear stochastic equation, non homogeneouus with differences of order 1 and constant coefficients, having attached the homogeneous equation:

0 , 0 ,

0 0 1

1 1

0V +aV- = a ¹ a ¹

a t t (4)

And the characteristic equation in its formal form is:

0 , 0 ,

0 0 1

1

0r+a = a ¹ a ¹ a

With the solution: , 0

0

1 =- ¹

-

= a a

a

r a .

For a <1 şi M2

### ( )

Vt <M <¥,"tÎZ the series

### ( )

Vt tÎZ is a mobile average of indefinite order with =

### å

¥ - -

i

i t i

t a e

V ( ) .

The covariant function of the chronological series is defined as follows:

÷÷= - -

=

ø ö çç

è æ

÷÷ ø ö çç

è

æ -

÷÷ø çç ö

è

æ -

=

×

=

g

### å å å

¥

= - -

¥

= - -

¥

= -

-

h i

i t h i i j

j h t j i

i t i h

t

t V M a e a e a a M e

V M

h 2

0 0

) ( ) ( )

( )

( )

(

### å

¥

=

¥ -

= - - - - =s - - Î

=

h i

h i i n

i

i t h i

i a D e a a h

a) ( ) ( ) ( ) , N

( 2 2 .

Considering that ,

0

1 a

a

a = equation (3) is equivalent to Vt +aVt-1=et,a¹0, from where we get:

h t t h t t h t

t V aV V e V

V × - + -1× - = × - (5)

Using the average operator, relation (5) becomes:

Vt Vt h

aM

Vt Vt h

M

et Vt h

### )

M × - + -1× - = × - From where we obtain:

ïî ïí ì

³

=

= s - g + g

1 , 0

0 ) ,

1 ( ) (

2

h h h

a

h (6)

(6)

From relation (6) we deduce that any h³1,hÎN the covariant functionis solution of the homogeneous stochastic equation Vt +aVt-1=0 and for g(0)¹0 we get

* ,

0 ) 1 ( )

( + r - = ÎN

r h a h h with the initial conditions r(0)=1 and r(1)=-a. The solution for this equation is the correlation function r(h)=

### ( )

-a h,hÎN .

The representation theorem of the self regressive series Be

### ( )

Vt tÎT ÎAR(p) so that:

1) , , *, 0 1, 0

0

¹

= Î

Î

=

= - p

p i

t i t

iV e t p a a

a Z N ;

2) M2

### ( )

Vt <M,"tÎZ,MÎR*;

3) the roots of the characteristic equation attached to the non homogeneous stochastic equation 1) have the module strictly subunitary.

In conclusion:

1’) the solution of the stochastic equation 1) is:

### å

¥

= × -

=

0 i

i t i

t b e

V where the coefficients

### ( )

bi iÎN verify the next system of homogeneous equations with finite differences:

, 1,...

### }

, 0

1 ...

1 + + = " Î +

+ab- a b- i p p

bi i p i p

Having the initial conditions :

### å

= - = " = - +

= i

j

j i j

i a b i p

b b

1

0 1, 0, 1, 1;

2’) the chronological series is stationary in covariance.

Proof. (1’) following the ideas from Introduction to statistical time series by Fuller A.W., we assume that

### ( )

Vt tÎTÎAR(p) is defined on the borelian field

W,K,P

### )

and the homogeneous stochastic equation attached is:

* ,

, ÎZ ÎN

=

### å

×

= a V- t p

p i

i t i 0

0 (7)

And the corresponding characteristic equation is:

1

0 0

0

= Î

=

### å

×

=

- p a

r a

p i

i t

i , N*, (8)

With the roots ri,i=0,pşi ri <1,"i=0,p. From relation (7) for a0=1we get:

### å

= -

-

= p

i i t i

t aV

V

1

and we consider the system:

ïï î ïï í ì

=

= -

=

- - - -

- -

= -

### å

) ( )

( 1 1

1 1

1

p t p t

t t

p i

i t i t

V V

V V

V a V

(7)

having the matrix form Yt =AYt-1+et,tÎZ , as Yt-1=AYt-2+et-1 it results

t t t

t A Y A

Y = 2 -2+ e +e , so:

* ,

, ÎZ ÎN

e +

=

### å

= -

- A t n

Y A Y

n i

i t i n t n t

0

(9)

where

( ) ÷÷

÷÷

÷

ø ö

çç çç ç

è æ

= e

÷÷

÷÷

÷÷

ø ö

çç çç çç

è

æ- - - - -

=

÷÷

÷÷ ø ö çç

çç è æ

=

-

- -

-

0 0

0 1

0 0 0

0 0

0 1 0

0 0

0 0 1

1 3

2 1

1

1 , ...

...

...

...

...

...

...

...

...

...

...

,

t t p p

p t

t t t

a e a

a a a

A V

V V

Y .

For the matrices from relation (9) the following notations are used:

a k N i j

p

### }

Ak = ijk , Î *, , Î1,2,...,

### ( )

a b i p

Ai = 11i = i, =0, and equation (9) becomes:

### å å

=

-

= -

-

- +

= p

j

n i

i t i j n t n

j

t a V be

V

1

1 0

1 (10)

The characteristic polynominal of matrix A is

= l - l

- l -

- -

- - l - -

= l -

-

1 0

0 0

0 0

1 0

0 0

0 1

1 3

2 1

...

...

...

...

...

...

...

...

...

... p p

p

a a

a a a

I A

=

l - l

-

- -

- - l - l

- l -

× l - -

=

L L L L L

L L

L L L L L

L L

0 0

0 1

0 0

0 1

0

0 2 3

1

ap

a a a

l - - -

+ l -

× - - l -

× l - -

= - -

L K K K

L L

0

0 1

3 2 2

1 1

p p

p

a a

a

a =

1

1 0

0

= l

-

=

=

### å

=

- a

a

p i

i p i

p ,

) (

K .

Cayley-Hamilton theorem for matrix A is:

p p p p

p p

p a A a A a A a I O

A + 1 -1+ 2 -2+K+ -1 + = , From where we get by multiplying with Ai-p:

(8)

(A +a1A +a2A +K+ap-1A +apA =Op,i³p (11) According to relation (11) we deduce that bi verifies the conditions from (1’), and a1kj verifies the equation:

p k O a a a

a a

a a a

a1kj + 1 1kj-1+ 2 1kj-2+K+ p-1 1kj-p+1+ p 1kj-p= p," ³ for jÎ

1,2,K,p

### }

,n³p (12) From relation (12) we get:

p

### }

j a a a

a a

a a a

a1kj =- 1 1kj-1- 2 1kj-2-K- p-1 1kj-p+1- p 1kj-p," Î1,2,K, therefore there is c>0 and lÎ

M,1

where max

,det

0

1

0 l -l =

= ££ i i p

i A I

M so that

p

### }

j c

a1kj < lk," Î 1,2,K, and any k ³ p.

Using the relation (10) and the average operator we deduce that:

2 0

2

1 1 1 2

0

¾

¾ ®

¾

× l

×

×

÷÷£

÷ ø ö çç

ç è æ

÷÷ ø ö çç

è

= æ

÷÷ ø ö çç

è æ

÷÷ø çç ö

è

æ - ®¥

= - -

-

= -

p n n

j

j n t n

j n

i i t i

t be M a V c p M

V M

21

1 1 1

0

M p c V

M a e

b V

M n

p j

j n t n

j n

i i t i

t ÷÷= × £ × ×l

ø ö ççè

æ -

### å å

= - -

-

= -

so according to the dominated convergency theorem of Lebesque we get

### å

¥

= × -

=

0 i

i t i

t b e

V what

was supposed to be demonstrated.

3’) As bi = a11i <c×li,"i³ pit results that

### å

¥

=0 i

bi is absolutely convergent, so the chronological series

### ( )

Vt tÎTÎAR(p) is stationary in covariance according to Beppa Levi theorem.

Consequence. If the chronological series

### ( )

Vt tÎT verifies the representation theorem of the self regressive series, then the covariance function attached verifies the following relations:

0 + 1g

1 + + g

=s2

g a K ap p

+ 1g

-1

+ + g

-

### )

=0, " ÎN *

g h a h K ap h p h .

Justification. According to relation 1) from the theorem we can write that

= - - - = p

i

i t h t i t h

t e aV V

V

0

so:

ïî ïí ì

³

=

= s - g

=

÷÷= ø çç ö

è

= æ

×

=

= - -

= - -

- 0, 1

0

2,

0 0

0 h

i h h a V

V M a V

V a M e V M

p

i i p

i

i t h t i p

i

i t h t i t

h

t .

(9)

### The Stochastic Equation of the Mobile Average

The chronological series

### ( )

Vt tÎT is a mobile average of order n, if T =Z and

t p

i i t

ie V

a =

### å

= - 0

(13) where nÎN*,aiÎR,"iÎ

0,1,K,n

### }

,a0 =1, an ¹0 and

et tÎZ ÎZA

0,s2 .

We note MA

### ( )

n,nÎN * the multitude of all mobile averages of ordern.

If there is an absolutely convergent numerical series

### å

¥

=0 i

aiwith a0 =1so that relation (13) take place, then the series

### ( )

Vt tÎT is a mobile average of indefinite order.

If

### ( )

Vt tÎT is a finite mobile average then yhis is called inverted, if it can be represented as a self regressive series of indefinite order.

A mobile average of indefinite order

Vt tÎT with

¥

= × -

=

0 i

i t i

t a e

V and

### å

¥

=0 i

aian absolutely covergent series, according to Beppo-Levi theorem it is stationary.

The chronological series

### ( )

Vt tÎT ÎMA(n),T =Z with:

### å

= -

= n

i i t i

t ae

V

0

where nÎN*,a0 =1,an ¹0 and aiÎR,"i=0,1 and

et tÎZ ÎZA

### ( )

0,s2 (14) is inverted if the roots of the characteristic equation corresponding to (14) are strictly lower than 1.

From relation (14) under these circumstances we get: =

### å

¥ × ÎZ

=b V- t e

j

j t j

t ,

0

, where the coefficients

### ( )

bj jÎN verify the next system of homogeneous equations with finite differences:

K

### }

K 0, , 1,

1

1 + + = " Î +

+ab- a b- i n n

bi i p i p

having the following initial conditions:

ïï ï î ïï ï í ì

- - -

-

= - -

= -

=

=

- -

-

-1 1 2 2 3 1

2 1 1 2

1 1

0 1

n n

n

n ab a b a

b

a b a b

a b b

K K

K

### Conclusions

Most economic processes are unstationary, thus the analysis of these processes is made with the appropriate methods. The express use of stationary methods is made in special occasions,

(10)

therefore it is necessary to study the modeling of the stationarity of a chronological series, so that the methods used in analyzing and forecasting will be more efficient.

### References

1. C i u c u , G., T u d o r , C., Probabilităţi şi procese stochastice (I,II), Editura Academiei R.S.R,1978,1979.

2. D u f o u r , J.-M., “Logique et tests d'hypothèses: réflexions sur les problèmes mal posés en économétrie”,CIREQ 15/2001.

3. F u l l e r , A.W.,Introduction to statistical time series, John Wiley & Sons ,New York , 1996.

4. I o s i f e s c u , M., M i h o c , Gh., T e o d o r e s c u R., Teoria probabilităţilor şi statisitacă matematică, Editura Tehnică, Bucureşti, 1966.

5. I o s i f e s c u , M., G r i g o r e s c u , Ş., O p r işa n , Gh., P o p e s c u , Gh., Elemente de modelare stochastică, Editura Tehnică, Bucureşti, 1984.

6. I o s i f e s c u , M., Tău t u , P., Stochastic Processes and Applications in Biology and Medicine (I.

Theory; II .Models), Editura Academiei & Springer Verlag, Bucureşti & Berlin, 1973.

7. P r e d a , V., Băl cău , C., “On Maxentropic Reconstruction of Countable Markov Chains and Matrix Scaling Problems”,Studies in Applied Mathematics, 2003.

8. S c h u s s , Z.,Theory and Applications of Stochastic Differtial Equations, John Wiley & Sons, New York , 1980.

### Rezumat

Construcţia modelelor statistice sau econometrice ale seriilor cronologice se face şi cu ajutorul proceselor stochastice. În acest articol se prezintă o problemă de modelare a proprietăţii de staţionaritate a unei serii cronologice folosind ecuaţia stochastică a unei serii cronologice şi ecuaţia stochastică a mediei mobile.

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Abstract: The Canadian Immigration and Refugee Protection Act provides that one of the objectives of immigration is “to see that families are reunited in Canada.” The Act

Using the Fourier series as a projection in the Galerkin method, we approach the solution of the Cauchy singular integral equation1. Numerical examples are developped to show

, Convergence of the family of the deformed Euler-Halley iterations under the H¨ older condition of the second derivative, Journal of Computational and Applied Mathematics,

Keywords: trickster discourse, meaning, blasphemy, social change, transgression of social norms.. The Myth of the trickster and its

babilities of some random variables ixed level by random processes is of.

Funhtion tltLrch [(ctt¿nl¡rüch ];yi.e¿lriclt Gltntnasitt.to trtul llealsclt'ulc... This formula has a lot

In this paper we consider a method to approximate the solution of the following stochastic complex Ginzburg-Landau evolution equation in dimension one perturbed...

The evolution to globalization has been facilitated and amplified by a series of factors: capitals movements arising from the need of covering the external

We then go on to examine a number of prototype techniques proposed for engineering agent systems, including methodologies for agent-oriented analysis and design, formal

De¸si ˆın ambele cazuri de mai sus (S ¸si S ′ ) algoritmul Perceptron g˘ ase¸ste un separator liniar pentru datele de intrare, acest fapt nu este garantat ˆın gazul general,

Un locuitor al oglinzii (An Inhabitant of the Mirror), prose, 1994; Fascinaţia ficţiunii sau despre retorica elipsei (On the Fascination of Fiction and the Rhetoric of Ellipsis),

•  A TS groups a sequence of events of one character or a stable group of characters over a period of. story Kme, which is uninterruptedly told in a span

The study of the series reduces to the study of particular sequences of real numbers, and determining the sum o a series is equivalent to computing the limit of a certain

The purpose of the regulation on volunteering actions is to structure a coherent approach in order to recognise the period of professional experience as well as the

The number of vacancies for the doctoral field of Medicine, Dental Medicine and Pharmacy for the academic year 2022/2023, financed from the state budget, are distributed to

Adrian Iftene, Faculty of Computer Science, “Alexandru Ioan Cuza” University of Iași Elena Irimia, Research Institute for Artificial Intelligence “Mihai Drăgănescu”, Romanian

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