14mesmet

14 : APPENDIX

Polynomials andDifierence Equations

A time-series model is one which postulates a relationship amongst a num-ber of temporal sequences or time series. For example, we have the regressionmodel

(1) y(t) = x(t)fl + "(t)

where x(t) is an observable sequence indexed by the time subscript t and "(t) isan unobservable sequence of independently and identically distributed randomvariables.

A more general model, which we shall call the general temporal regres-sion model, is one which postulates a relationship comprising any number ofconsecutive elements of x(t), y(t) and "(t). Thus we have

(2)pXi=0

fiiy(t i) =kXi=0

flix(t i) +qXi=0

i"(t i)

Any of the sums in this expression can be inflnite, but if the model is to beviable, the sequences of coecients ffiig, fflig and fig can depend on only alimited number of parameters.

We are particularly interested in a number of specialisations of the model.The model represented by the equation

(3) y(t) = flx(t i) +qXi=0

i"(t i)

is described as a regression model with a serially correlated disturbance se-quence (t) =

Pi"(t i). If this sum is flnite, then (t) is called a moving

average process. The model represented by

(4)pXi=0

fiiy(t i) = flx(t) + "(t)

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D.S.G. POLLOCK : ECONOMETRIC THEORY

is described as an autoregressive regression model. The equation

(5) y(t) =Xi=0

flix(t i) + "(t)

represents the distributed-lag regression model.The foregoing models are all termed regression models by virtue of the

inclusion of the observable explanatory sequence x(t). When x(t) is deleted, weobtain the simpler unconditional linear stochastic models. Thus the equation

(6) y(t) =qXi=0

i"(t i);

where the sum is flnite, represents a moving average process; whereas

(7)pXi=0

fiiy(t i) = "(t);

again with a flnite sum, represents an autoregressive process. The equation

(8)pXi=0

fiiy(t i) =qXi=0

i"(t i)

represents an autoregressive-moving average process.

The Algebra of the Lag Operator

A sequence x(t) is any function mapping from the set of integers Z =f0;1;2; : : :g to the real line. If the set of integers represents a set of datesseparated by unit intervals, then we say that x(t) is a temporal sequence ortime series.

The set of all time series represents a vector space, and we can deflnevarious linear transformations or operators over the space. For example, wedeflne the lag operator by

(9) Lx(t) = x(t 1):

Now, LfLx(t)g = Lx(t 1) = x(t 2); so it makes sense to deflne L2 byL2x(t) = x(t 2). More generally, Lkx(t) = x(t k) and, likewise, Lkx(t) =x(t+ k). Other operators are the difierence operator r = I L which has theefiect that

(10) rx(t) = x(t) x(t 1);

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D.S.G. POLLOCK : POLYNOMIALS

the forward-difierence operator = L1 I, and the summation operatorS = (I L)1 = fI + L+ L2 + g which has the efiect that

(11) Sx(t) =1Xi=0

x(t i):

In general, we can deflne polynomials of the lag operator of the form p(L) =p0 + p1L+ p2L2 + + pnLn =

PpiL

i having the efiect that

(12)

p(L)x(t) = p0x(t) + p1x(t 1) + + pnx(t n)

=nXi=0

pix(t i):

The advantage which comes from deflning polynomials in the lag operatorstems from the fact that they are isomorphic to the set of ordinary algebraicpolynomials. Thus we can rely upon what we know about ordinary polynomialsto treat problems concerning lag-operator polynomials.

Algebraic Polynomials

Consider the equation fi0 + fi1z + fi2z2 = 0. On dividing the equation byfi2, we can factorise it as (z 1)(z 2) where 1, 2 are the roots of theequation that are given by the formula

(13) =fi1

pfi21 4fi2fi0

2fi2:

If fi21 4fi2fi0, then the roots 1, 2 are real. If fi21 = 4fi2fi0, then 1 = 2.If fi21 4fi2fi0, then the roots are the conjugate complex numbers = fi+ ifl, = fi ifl where i = p1.

Complex Roots

There are three alternative ways of representing the conjugate complexnumbers and :

(14) = fi+ ifl = (cos + i sin ) = ei;

= fi ifl = (cos i sin ) = ei;

where

(15) = p(fi2 + fl2) and = tan1fl

fi

:

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These are called, respectively, the Cartesian form, the trigonometrical form andthe exponential form.

The polar representation is understood by considering the Argand diagram:

*

Re

Im

Figure 1. The Argand Diagram showing a complex

number = fi+ ifl and its conjugate = fi ifl.The exponential form is understood by considering the following series

expansions of cos and i sin about the point = 0:

(16)cos = 1

2

2!+4

4!

6

6!+

i sin = i i3

3!+i5

5! i

7

7!+

Adding these gives us Eulers equation:

(17)cos + i sin = 1 + i

2

2! i

3

3!+4

4!+

= ei:

Likewise, by subtraction, we get

(18) cos i sin = ei:

It follows readily from (17) and (18) that

(19) cos =ei + ei

2

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and

(20)sin =

i(ei ei)2

=ei ei

2i:

The n-th Order Polynomial

Now consider the general equation of the nth order:

(21) fi0 + fi1z + fi2z2 + + finzn = 0:

On dividing by fin, we can factorise this as

(22) (z 1)(z 2) (z n) = 0;

where some of the roots may be real and others may be complex. The complexroots come in conjugate pairs, so that, if = fi + ifl is a complex root, thenthere is a corresponding root = fiifl such that the product (z)(z) =z2 + 2fiz + (fi2 + fl2) is real and quadratic. When we multiply the n factorstogether, we obtain the expansion

(23) 0 = zn Xi

izn1 +

Xi

Xj

ijzn2 (1)n12 n:

We can compare this with the expression (fi0=fin)+(fi1=fin)z+ +zn = 0. Byequating coecients of the two expressions, we flnd that (fi0=fin) = (1)n

Qi

or, equivalently,

(24) fin = fi0nYi=1

(i)1:

Thus we can express the polynomial in any of the following forms:

(25)

Xfiiz

i = finY

(z i)= fi0

Y(i)1

Y(z i)

= fi0Y

1 zi

:

We should also note that, if is a root of the equationPfiiz

i = 0,then = 1= is a root of the equation

Pfiiz

ni = 0. This follows since

5

D.S.G. POLLOCK : ECONOMETRIC THEORYPfii

ni = nPfiii = 0 implies that

Pfiii =

Pfiii = 0. Confusion

can arise from not knowing which of the two equations one is dealing with.

Rational Functions of Polynomials

If (z) and (z) are polynomial functions of x of degrees n and m respec-tively with n < m, then the ratio (z)=(z) is described as a proper rationalpolynomial. We shall often encounter expressions of the form

y(t) =(L)(L)

x(t):

For this to have a meaningful interpretation in the context of a time-seriesmodel, we normally require that y(t) should be a bounded sequence wheneverx(t) is bounded. The necessary and sucient condition for the boundedness ofy(t), in that case, is that the series expansion of (z)=(z) should be convergentwhenever jzj 1. We can determine whether or not the sequence will convergeby expressing the ratio (z)=(z) as a sum of partial fractions. The basic resultis as follows:

(26) If (z)=(z) = (z)=f1(z)2(z)g is a proper rational fraction, andif 1(z) and 2(z) have no common factor, then the fraction canbe uniquely expressed as

(z)(z)

=1(z)1(z)

+2(z)2(z)

;

where 1(z)=1(z) and 2(z)=2(z) are proper rational fractions.

Imagine that (z) =Q

(1z=i). Then repeated applications of this basicresult enables us to write

(27) (z) =1

1 z=1 +2

1 z=2 + +n

1 z=n :

By adding the terms on the RHS, we flnd an expression with a numerator oforder n 1. By equating the terms of the numerator with the terms of (z),we can flnd the values 1, 2; : : : ; n.

Consider, for example,

(28)

3x1 + x 2x2 =

3x(1 x)(1 + 2x)

=2

1 x +2

1 + 2x

=1(1 + 2x) + 2(1 x)

(1 x)(1 + 2x) :

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Equating the terms of the numerator gives

(29) 3x = (21 2)x+ (1 + 2);

so that 2 = 1 which gives 3 = (21 2) = 31; and thus we have 1 = 1,2 = 1.

The conditions for the convergence of the expansion of (z)=(z) arestraightforward. For the rational function converges if and only if the expansionof each of its partial fractions converges. For the expansion

(30)

(1 z=) = n

1 + z=+ (z=)2 + o

to converge when jzj 1, it is necessary and sucient that jj > 1.

Linear Difierence Equations

An nth-order linear difierence equation is a relationship amongst n + 1consecutive elements of a sequence x(t) of the form

(31) fi0x(t) + fi1x(t) + + finx(t n) = u(t);

where u(t) is some specifled sequence. We can also write this as

(32) fi(L)x(t) = u(t);

where fi(L) = fi0 + fi1L + fi2L2 + + finLn. If we are given n consecutivevalues of x(t), say x1; x2; : : : ; xn, then we can use this relationship to flnd thesucceeding value xn+1. In this way, so long as u(t) is fully specifled, we cangenerate all the succeeding elements of the sequence. Likewise, we can generateall the values of the sequence prior to t = 1; and thus, in efiect, we can deducethe function x(t) from the difierence equation. However, instead of a recursivesolution, we usually seek an analytic expression for x(t).

The function x(t; c), expressing the analytic solution, will comprise a setof n constants in c = [c1; c2; : : : ; cn]0 which can only be determined once we aregiven a set of n consecutive values of x(t) which are called initial conditions.The general analytic solution of the equation fi(L)x(t) = u(t) is expressed asx(t; c) = y(t; c) + z(t), where y(t) is the general solution of the homogeneousequation fi(L)y(t) = 0, and z(t) = fi1(L)u(t) is called a particular solution ofthe inhomogeneous equation.

We solve the difierence equation in three steps. First, we flnd the generalsolution of the homogeneous equation. Next, we flnd the particular solutionz(t) which embodies no unknown quantities. Finally, we use the n initial valuesof x to determine the constants c1; c2; : : : ; cn. We shall discuss in detail onlythe solution of the homogeneous equation.

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Solution of the Homogeneous Difierence Equation

If j is a root of the equation fi(z) = fi0 +fi1z+fi2z2 + +finzn = 0 suchthat fi(j) = 0, then yj(t) = (1=j)t is a solution of the equation fi(L)y(t) = 0.We can see this by considering the expression

(33)

fi(L)

1j

t= (fi0 + fi1L+ + finLn)

1j

t= fi0

1j

t+ fi1

1j

t1+ + fin

1j

tn= (fi0 + fi1j + + finnj )

1j

t= fi(j)

1j

t:

Alternatively, consider the factorisation fi(L) = fi0Qi(1 L=j). Within this

product, we have the term 1 L=j and, clearly, since1 L

j

1j

t=

1j

t

1j

t= 0;

we must have fi(L)(1j)t = 0.The general solution, in the case where fi(L) = 0 has distinct real roots, is

given by

(34) y(t; c) = c1

11

t+ c2

12

t+ + cn

1n

t;

where c1; c2; : : : ; cn are the constants which are determined by the initial con-ditions.

In the case where two roots coincide at a value of , the equation fi(L)y(t)= 0 has the solutions y1(t) = (1=)t and y2(t) = t(1=)t. To show this, letus extract the term (1L=)2 from the factorisation fi(L) = fi0

Qi(1L=j)

given under (25). Then, according to the previous argument, we have (1 L=)2(1=)t = 0, but, also, we have

(35)

1 L

2t

1

t=

1 2L

+L2

2

t

1

t= t

1

t 2(t 1)

1

t+ (t 2)

1

t= 0:

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In general, if there are r repeated roots, then (1=)t, t(1=)t, t2(1=)t, : : : ;tr1(1=)t are all solutions to the equation fi(L)y(t) = 0.

The 2nd-order Difierence equation with Complex Roots

Imagine that the 2nd-order equation fi(L)y(t) = fi0y(t) + fi1y(t 1) +fi2y(t2) = 0 is such that fi(z) = 0 has complex roots = 1= and = 1=.Let us write

(36) = + i = (cos! + i sin!) = ei!;

= i = (cos! i sin!) = ei!:

These will appear in a general solution of the difierence equation of the form of

(37) y(t) = ct + c()t:

This is a real-valued sequence; and, since a real term must equal its own con-jugate, we require c and c to be conjugate numbers of the form

(38)c = (cos + i sin ) = ei;

c = (cos i sin ) = ei:

Thus we have

(39)

ct + c()t = ei(ei!)t + ei(ei!)t

= tnei(!t) + ei(!t)

o= 2t cos(!t ):

To analyse the flnal expression, consider flrst the factor cos(!t ). Thisis a displaced cosine wave. The value !, which is a number of radians per unitperiod, is called the angular velocity or the angular frequency of the wave. Thevalue f = !=2 is its frequency in cycles per unit period. The duration of onecycle, also called the period, is r = 2=!.

The term is called the phase displacement of the cosine wave, and itserves to shift the cosine function along the axis of t so that the peak occurs atthe value of t = =! instead of at t = 0.

Next consider the term t wherein = p(2 + 2) is the modulus of thecomplex roots. When has a value of less than unity, it becomes a dampingfactor which serves to attenuate the cosine wave as t increases.

Finally, the factor 2 represents the initial amplitude of the cosine wavewhich is the value that it assumes when t = 0. Since is just the modulus ofthe values c and c, this amplitude reects the initial conditions.

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