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 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 1, the dynamic multiplier 2
increase exponentially over time and if 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 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
Thus, for a pth-order difference equation, the dynamic multiplier is given by fi re fi denotes the(, 1)element of F3 Example The(1, 1)elements of F is 1 and the(1, 1)elements of F2([91, 2,op 1, 1,0,0) is 1+o2. Thus D+1=o aY+2 01+2 in a pth-order syster For larger values of j, an easy way to obtain a numerical value for the dynamic multiplier aYt+i/awt in terms the eigenvalues of the matrix F. Recall that the eigenvalues of a matrix F are those numbers a for which F-A=0 (1 For example, for p=2 the eigenvalues are the solutions to 入0 10 0入 (1-入)2 2-02)-02=0 on For a general pth-order system, the determinant in(12)is a pth-order ploy- minal in A whose p solutions characterize the p eigenvalues of F. This polyno- mial turns out to take a very similar form to(13 The eigenvalues of the matrix F defines in equation(12)are the values of A that satisfy -01-1-02-2- 中p
Thus, for a pth-order difference equation, the dynamic multiplier is given by ∂Yt+j ∂Wt = f j 11, where f j 11 denotes the (1, 1) element of F j . Example: The (1, 1) elements of F 1 is φ1 and the (1, 1) elements of F 2 (= [φ1, φ2, ..., φp][φ1, 1, 0, ..., 0]0 ) is φ 2 1 + φ2. Thus, ∂Yt+1 ∂Wt = φ1; and ∂Yt+2 ∂Wt = φ 2 1 + φ2 in a pth-order system. For larger values of j, an easy way to obtain a numerical value for the dynamic multiplier ∂Yt+j/∂Wt in terms the eigenvalues of the matrix F. Recall that the eigenvalues of a matrix F are those numbers λ for which |F − λIp| = 0. (12) For example, for p = 2 the eigenvalues are the solutions to φ1 φ2 1 0 − λ 0 0 λ = 0 or (φ1 − λ) φ2 1 −λ = λ 2 − φ2λ − φ2 = 0. (13) For a general pth-order system, the determinant in (12) is a pth-order ploynominal in λ whose p solutions characterize the p eigenvalues of F. This polynomial turns out to take a very similar form to (13). Proposition: The eigenvalues of the matrix F defines in equation (12) are the values of λ that satisfy λ p − φ1λ p−1 − φ2λ p−2 − ... − φp−1λ − φp = 0. 6
2.1 General Solution of a pth-order Difference Equation with Distinct Eigenvalues Recall that if the eigenvalues of a(p x p)matrix F are distinct, there exists a nonsingular(p x p) matrix T such that F=TAT-I where T=[x1, x2, .,xp], xi, i= 1, 2, ., p are the eigenvectors of F corresponding to its eigenvalues Ai; and A is a(p x p) matrix such that h100 0A20 00 0 This enables us to characterize the dynamic multiplier (the(1, 1)elements of F2) very easily. In general, we have F=TAT-1×TAT-1×….×TAT TAT-I (15) where A00 0
2.1 General Solution of a pth-order Difference Equation with Distinct Eigenvalues Recall that if the eigenvalues of a (p × p) matrix F are distinct, there exists a nonsingular (p × p) matrix T such that F = TΛT−1 where T = [x1, x2, ..., xp], xi , i = 1, 2, ..., p are the eigenvectors of F corresponding to its eigenvalues λi ; and Λ is a (p × p) matrix such that Λ = λ1 0 0 . . . 0 0 λ2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λp . This enables us to characterize the dynamic multiplier (the (1,1) elements of F j ) very easily. In general, we have F j = TΛT −1 × TΛT−1 × ... × TΛT −1 (14) = TΛjT −1 , (15) where Λ j = λ j 1 0 0 . . . 0 0 λ j 2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λ j p . 7
Let tii denote the row i column j element of T and let to denote the row i column element of T. Equation(15) written out explicitly become A00 0 t11t1 00 000 tpa t21t22 t12 入 from which the(1, 1)element of F3 is given by f1=c1+c2 where and a1+c2+…+cp=t1t1+th21+…+tpt=1 Therefore the dynamic multiplier of a pth-order difference equation fi1=c1+c2+…+cp2 that is the dynamic multiplier is a weighted average of each of the p eigenvalues raised to the jth power The following result provides a closed-form expression for the constant c1, C2,.,p If the eigenvalues(A1, A2, ...,Ap) of the matrix F are distinct, then the magnitude Ci can be written as I=1,k≠(A2-Ak)
Let tij denote the row i column j element of T and let t ij denote the row i column j element of T−1 . Equation (15) written out explicitly become F j = t11 t12 . . . . t1p t21 t22 . . . . t2p . . . . . . . . . . . . . . . . . . . . . tp1 tp2 . . . . tpp λ j 1 0 0 . . . 0 0 λ j 2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λ j p t 11 t 12 . . . . t 1p t 21 t 22 . . . . t 2p . . . . . . . . . . . . . . . . . . . . . t p1 t p2 . . . . t pp = t11λ j 1 t12λ j 2 . . . . t1pλ j p t21λ j 1 t22λ j 2 . . . . t2pλ j p . . . . . . . . . . . . . . . . . . . . . tp1λ j 1 tp2λ j 2 . . . . tppλ j p t 11 t 12 . . . . t 1p t 21 t 22 . . . . t 2p . . . . . . . . . . . . . . . . . . . . . t p1 t p2 . . . . t pp from which the (1,1) element of F j is given by f j 11 = c1λ j 1 + c2λ j 2 + ... + cpλ j p where ci = t1it i1 and c1 + c2 + ... + cp = t11t 11 + t12t 21 + ... + t1pt p1 = 1. Therefore the dynamic multiplier of a pth-order difference equation is: ∂Yt+j ∂Wt = f j 11 = c1λ j 1 + c2λ j 2 + ... + cpλ j p , that is the dynamic multiplier is a weighted average of each of the p eigenvalues raised to the jth power. The following result provides a closed-form expression for the constant c1, c2, ..., cp. Proposition 2: If the eigenvalues (λ1, λ2, ..., λp) of the matrix F are distinct, then the magnitude ci can be written as ci = λ p−1 Q i p k=1, k6=i (λi − λk) . 8
Example In then case p we have 入1 A1-A2A1-A2 入2-A1A2-A1 2.1.1 Real roots Suppose first that all the eigenvalues of F are real and all these real eigen- values are less than one in absolute value, then the system is stable, and its dynamics are represented as a weighted average of decaying exponentially or decaying exponentially oscillating in sign Consider the following second-order difference equation Yt=0.6t-1+0.2Y-2+Wt The eigenvalues are the solutions the polynomial 入-0.6入-0.2=0 which are 0.6+√0.6)2 4(0.2)=0 入2 0.6-y(06)2-4(0.2 The dynamic multiplier for this system aY aWt =c1(0.84)2+ is geometrically decaying and is plotted as a function of j in panel(a)of Hamil- ton,p.15. Note that as j becomes larger, the pattern is dominated by the larger eigenvalues(A1), approximating a simple geometric decay at rate(A1) If the eigenvalue are all real but at least one is greater than one in absolute value, the system is explosive. If A1 denotes the eigenvalue that is
Example: In then case p = 2, we have c1 = λ 2−1 1 λ1 − λ2 = λ1 λ1 − λ2 , c2 = λ 2−1 2 λ2 − λ1 = λ2 λ2 − λ1 . 2.1.1 Real Roots Suppose first that all the eigenvalues of F are real and all these real eigenvalues are less than one in absolute value, then the system is stable, and its dynamics are represented as a weighted average of decaying exponentially or decaying exponentially oscillating in sign. Example: Consider the following second-order difference equation: Yt = 0.6Yt−1 + 0.2Yt−2 + Wt . The eigenvalues are the solutions the polynomial λ 2 − 0.6λ − 0.2 = 0 which are λ1 = 0.6 + p (0.6)2 − 4(0.2) 2 = 0.84 λ2 = 0.6 − p (0.6)2 − 4(0.2) 2 = −0.24. The dynamic multiplier for this system, ∂Yt+j ∂Wt = c1λ j 1 + c2λ j 2 = c1(0.84)j + c2(−0.24)j is geometrically decaying and is plotted as a function of j in panel (a) of Hamilton, p.15. Note that as j becomes larger, the pattern is dominated by the larger eigenvalues (λ1), approximating a simple geometric decay at rate (λ1). If the eigenvalue are all real but at least one is greater than one in absolute value, the system is explosive. If λ1 denotes the eigenvalue that is 9
argest in absolute value, the dynamic multiplier is eventually dominated by an exponential function of that eigenvalues lim tty oo aW 2.1.2 Complex Roots It is possible that the eigenvalue of F are complex(Since F is not symmetric. For a symmetric matrix, its eigenvalues are all real). Whenever this is the case, they appear as complex conjugates. For example if p= 2 and 1+ 402<0, then the solutions A1 and A2 are complex conjugates. Suppose that A1 and A2 are complex conjugates, written as A1=a+bi 入2=a-bi By rewritten the definition of the sine and the cosine function we have Rcos(e) and b=Rsin(⊙) where for a given angle 0 and R are defined in terms of a and b by cos(e) R b Therefore we have 入1 8+isin 0 入 By Eular relations(see for example, Chiang, AC.(1984), p. 520)we further h A1=R(cos 0 +isin 0=Rete R[ -isin=Re-
largest in absolute value, the dynamic multiplier is eventually dominated by an exponential function of that eigenvalues: lim j→∞ ∂Yt+j ∂Wt · 1 λ j 1 = c1. 2.1.2 Complex Roots It is possible that the eigenvalue of F are complex (Since F is not symmetric. For a symmetric matrix, its eigenvalues are all real). Whenever this is the case, they appear as complex conjugates. For example if p = 2 and φ 2 1 + 4φ2 < 0, then the solutions λ1 and λ2 are complex conjugates. Suppose that λ1 and λ2 are complex conjugates, written as λ1 = a + bi λ2 = a − bi By rewritten the definition of the sine and the cosine function we have a = R cos(θ) and b = R sin(θ), where for a given angle θ and R are defined in terms of a and b by R = √ a 2 + b 2 cos(θ) = a R sin(θ) = b R . Therefore, we have λ1 = R[cos θ + isin θ] λ2 = R[cos θ − isin θ]. By Eular relations(see for example, Chiang, A.C. (1984), p. 520) we further have λ1 = R[cos θ + isin θ] = Re iθ λ2 = R[cos θ − isin θ] = Re −iθ , 10