Ch. 13 Difference Equations 1 First-Order Difference Equations Suppose we are given a dynamic equation relating the value y takes on at date t to another variables Wt and to the value y took in the previous period: where o is a constant. Equation(1)is a linear first-order difference equation a difference equation is an expression relating a variable yt to its previous values This is a first-order difference equation because only the first lag of the variable (Yt-1)appears in the equation. Note that it expresses Yt as a linear function of t-1 and w Y In Chapter 14 the input variable Wt will be regarded as a random variable, and the implication of (1)for the statistical properties of the output variables Yt will be explored. In preparation for this discussion, it is necessary first to understand the mechanics of the difference equations. For the discussion in Chapter 13, the values for the input variables W1, W2, . will simply be regarded as a sequence of deterministic numbers. Our goal is to answer the following question: If a dynamic system is described by(1), what are the effects on Y of changes in the value of w 1.1 Solving a Difference equations by Recursive Substitu- tion The presumption is that the dynamic equation(1) governs the behavior of Y for all dates t. that is Yt=Y-1+Wt,t∈T We first consider the index set T=10, 1, 2, 3,. By direct substitution Yt = Yt-1+Wt (Yt-2+Wt-1)+W
Ch. 13 Difference Equations 1 First-Order Difference Equations Suppose we are given a dynamic equation relating the value Y takes on at date t to another variables Wt and to the value Y took in the previous period: Yt = φYt−1 + Wt , (1) where φ is a constant. Equation (1) is a linear first-order difference equation. A difference equation is an expression relating a variable Yt to its previous values. This is a first-order difference equation because only the first lag of the variable (Yt−1) appears in the equation. Note that it expresses Yt as a linear function of Yt−1 and Wt . In Chapter 14 the input variable Wt will be regarded as a random variable, and the implication of (1) for the statistical properties of the output variables Yt will be explored. In preparation for this discussion, it is necessary first to understand the mechanics of the difference equations. For the discussion in Chapter 13, the values for the input variables {W1, W2, ...} will simply be regarded as a sequence of deterministic numbers. Our goal is to answer the following question: If a dynamic system is described by (1), what are the effects on Y of changes in the value of W ? 1.1 Solving a Difference equations by Recursive Substitution The presumption is that the dynamic equation (1) governs the behavior of Y for all dates t, that is Yt = φYt−1 + Wt , t ∈ T . We first consider the index set T = {0, 1, 2, 3, ...}. By direct substitution Yt = φYt−1 + Wt = φ(φYt−2 + Wt−1) + Wt = φ 2Yt−1 + φWt−1 + Wt = φ 2 (φYt−2 + Wt−1) + φWt−1 + Wt = ..... 1
Assume the value of Y for date t=-l is known(Y-1 here is an"initial value") we can express(1) by repeated substitution in the form dW0+a-W1+0-2w2+….+ This procedure is known as solving the difference equation(1) by recursive substitution 1.2 Dynamic Multipliers Note that(2)expresses Y as a linear function of the initial value Y-1 and the historical value of W. This makes it very easy to calculate the effect of Wo(say) on Yt. If Wo were to change with Y-1 and Wi, W2, . Wt taken as unaffected (this is the reason that we need the error term to be a white noise sequence in the ARMa model in the subsequent chapters) the effect on Yt would be given by ' --backword Note that the calculation would be exactly the same if the dynamic simulation were started at date t(taking Yi-I as given); then Yi+; can be described as a function of Yt-1 and Wt, Wi+1,., Wi+ Y计+=+1-1+m+-W+1+-2W+2+…+t+-1+Wt+y-(3) The effect of Wt on Yi+i is given by aW.=d--foreword Thus the dynamic multiplier(or also refereed as the impulse-response func tion)(4)depends only on 3, the length of time separating the disturbance to the input variable Wt and the observed value of output Yi+i. the multiplier does not depend on t; that is, it does not depend on the dates of the observations them selves. This is true for any difference equation Different value of in(1)can produce a variety of dynamic responses of If 0<o< l, the multiplier aY(+i/awt in(4) decays geometrically toward zero. If -1 <o<0, the absolute value of the multiplier aYt+i/aw in(4) also decays geometrically toward zero. If o> l, the dynamic multiplier
Assume the value of Y for date t = −1 is known (Y−1 here is an ”initial value”), we can express (1) by repeated substitution in the form Yt = φ t+1Y−1 + φ tW0 + φ t−1W1 + φ t−2W2 + ... + φWt−1 + Wt . (2) This procedure is known as solving the difference equation (1) by recursive substitution. 1.2 Dynamic Multipliers Note that (2) expresses Y as a linear function of the initial value Y−1 and the historical value of W. This makes it very easy to calculate the effect of W0 (say) on Yt . If W0 were to change with Y−1 and W1, W2, ..., Wt taken as unaffected, (this is the reason that we need the error term to be a white noise sequence in the ARMA model in the subsequent chapters) the effect on Yt would be given by ∂Yt ∂W0 = φ t − −backword. Note that the calculation would be exactly the same if the dynamic simulation were started at date t (taking Yt−1 as given); then Yt+j can be described as a function of Yt−1 and Wt , Wt+1, ..., Wt+j : Yt+j = φ j+1Yt−1 + φ jwt + φ j−1Wt+1 + φ j−2Wt+2 + ... + φWt+j−1 + Wt+j . (3) The effect of Wt on Yt+j is given by ∂Yt+j ∂Wt = φ j − −foreword. (4) Thus the dynamic multiplier (or also refereed as the impulse-response function) (4) depends only on j, the length of time separating the disturbance to the input variable Wt and the observed value of output Yt+j . the multiplier does not depend on t; that is, it does not depend on the dates of the observations themselves. This is true for any difference equation. Different value of φ in (1) can produce a variety of dynamic responses of Y to W. If 0 < φ < 1, the multiplier ∂Yt+j/∂Wt in (4) decays geometrically toward zero. If −1 < φ < 0, the absolute value of the multiplier ∂Yt+j/∂Wt in (4) also decays geometrically toward zero. If φ > 1, the dynamic multiplier 2
increase exponentially over time and if < -l, the multiplier exhibit explosive oscillations Thus, if o< 1, the system is stable; the consequence of a given change in t will eventually die out. If o >l, the system is explosive. An interesting possibility is the borderline case, lo|= 1. In this case, the solution(3)becomes t+=Yt-1+Wt+Wi+1+W++2+.+Wi+i-1+Wi+j Here the output variables y is the sum of the historical input w. A one-unit increase in w will cause a permanent one-unit increase in y ow,=1 for]=0,1 --unit root 2 pth-Order Difference Equations Let us now generalize the dynamic system (1) by allowing the value of y at date t to depend on p of its own lags along with the current value of the input variable Y=1Yt-1+2Yt-1+…+nYt-p+Wt,t∈T Equation(5)is a linear pth-order difference equation It is often convenient to rewrite the pth-order difference equation(5) in the calar Yt as a first-order difference equation in a vector $t. Define the (p x 1) vector st by
increase exponentially over time and if φ < −1, the multiplier exhibit explosive oscillations. Thus, if |φ| < 1, the system is stable; the consequence of a given change in Wt will eventually die out. If |φ| > 1, the system is explosive. An interesting possibility is the borderline case, |φ| = 1. In this case, the solution (3) becomes Yt+j = Yt−1 + Wt + Wt+1 + Wt+2 + ... + Wt+j−1 + Wt+j . Here the output variables Y is the sum of the historical input W. A one-unit increase in W will cause a permanent one-unit increase in Y : ∂Yt+j ∂Wt = 1 for j = 0, 1, .... − −unit root. 2 pth-Order Difference Equations Let us now generalize the dynamic system (1) by allowing the value of Y at date t to depend on p of its own lags along with the current value of the input variable Wt : Yt = φ1Yt−1 + φ2Yt−1 + .... + φpYt−p + Wt , t ∈ T . (5) Equation (5) is a linear pth-order difference equation. It is often convenient to rewrite the pth-order difference equation (5) in the scalar Yt as a first-order difference equation in a vector ξt . Define the (p × 1) vector ξt by ξt ≡ Yt Yt−1 Yt−2 . . . Yt−p+1 , 3
the(P×p) matrix F by 00 010 00 F and the(p x 1) vector vt by W 0 0 0 Consider the following first-order vector difference equation St=FEt_1+ qp-1中 t-1 100 Y 010 00 0 This is a system of p equations. The first equation in this system is identical to equation(5). The remaining p-1 equations is simply the identit t-3 j=1,2,…,p-1 Thus, the first-order vector system(6)is simply an alternative representation of the pth-order scalar system(5). The advantage of rewriting the pth-order system in(5) in the form of a first-order system(6)is that first-order systems are often easier to work with than pth-order systems
the (p × p) matrix F by F ≡ φ1 φ2 φ3 . . φp−1 φp 1 0 0 . . 0 0 0 1 0 . . 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . 1 0 , and the (p × 1) vector vt by vt ≡ Wt 0 0 . . . 0 . Consider the following first-order vector difference equation: ξt = Fξt−1 + vt , (6) or Yt Yt−1 Yt−2 . . . Yt−p+1 = φ1 φ2 φ3 . . φp−1 φp 1 0 0 . . 0 0 0 1 0 . . 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . 1 0 Yt−1 Yt−2 Yt−3 . . . Yt−p + Wt 0 0 . . . 0 . This is a system of p equations. The first equation in this system is identical to equation (5). The remaining p − 1 equations is simply the identity Yt−j = Yt−j , j = 1, 2, ..., p − 1. Thus, the first-order vector system (6) is simply an alternative representation of the pth-order scalar system (5). The advantage of rewriting the pth-order system in (5) in the form of a first-order system (6) is that first-order systems are often easier to work with than pth-order systems. 4
A dynamic multiplier for(5) can be found in exactly the same way as was done for the first-order scalar system of section 1. If we knew the value of $_ then proceeding recursively in this fashion as in the scalar first order difference equation produce a generalization of(2) :=F+E1+F'v+Fv1+F-2v2+….+F Writing this out in terms of the definition of st and vt wt Yt 0 0 0 Yt =F"/Y3 F F1/0 +…+F 0 Consider the first equation of this system, which characterize the value of y Let fi denote the(1, 1)elements of Ft, fi2 denote the(1, 2)elements of Ft, and so on. Then the first equation of( 8)states that Y=什Y-1+1Y-2+…+fYp+1W0+fW1+…+几W-1+W1.(9) This describe the value of y at date t as a linear function of p initial value of Y(Y-1,Y-2, .,Y-p) and the history of the input variables W since date 0 (Wo, Wi,,Wt). Note that whereas only one initial value for Y was needed in the case of a first-order difference equation, p initial values for Y are needed in the case of a pth-order difference equation The obvious on of (3)is t+;=F-1+Fv2+Fv+1+F-v+2+…+Fvt+y-1 from which fi2 yt Y-p+fi,wt+fir
A dynamic multiplier for (5) can be found in exactly the same way as was done for the first-order scalar system of section 1. If we knew the value of ξ−1 , then proceeding recursively in this fashion as in the scalar first order difference equation produce a generalization of (2): ξt = F t+1ξ−1 + F tv0 + F t−1v1 + F t−2v2 + .... + Fvt−1 + vt . (7) Writing this out in terms of the definition of ξt and vt , Yt Yt−1 Yt−2 . . . Yt−p+1 = F t+1 Y−1 Y−2 Y−3 . . . Y−p + F t W0 0 0 . . . 0 + F t−1 W1 0 0 . . . 0 + .... + F 1 Wt−1 0 0 . . . 0 + Wt 0 0 . . . 0 . (8) Consider the first equation of this system, which characterize the value of Yt . Let f t 11 denote the (1, 1) elements of Ft , f t 12 denote the (1, 2) elements of Ft , and so on. Then the first equation of (8) states that Yt = f t+1 11 Y−1 + f t+1 12 Y−2 + ... + f t+1 1p Y−p + f t 11W0 + f t−1 11 W1 + .... + f 1 11Wt−1 + Wt . (9) This describe the value of Y at date t as a linear function of p initial value of Y (Y−1, Y−2, ..., Y−p) and the history of the input variables W since date 0 (W0, W1, ..., Wt). Note that whereas only one initial value for Y was needed in the case of a first-order difference equation, p initial values for Y are needed in the case of a pth-order difference equation. The obvious generalization of (3) is ξt+j = F j+1ξt−1 + F jvt + F j−1vt+1 + F j−2vt+2 + .... + Fvt+j−1 + vt+j (10) from which Yt+j = f j+1 11 Yt−1 + f j+1 12 Yt−2 + ... + f j+1 1p Yt−p + f j 11Wt + f j−1 11 Wt+1 + ... + f 1 11Wt+j−1 + Wt+j .(11) 5