5-16 The stationary Condition for an ar model ·平稳性使AR模型具有一些很好的性质。如前期误差项对当前 值的影响随时间递减。 The condition for stationarity of a general ar(p) model is that the roots of特征方程 0 all lie outside the unit circle Example 1: Is y,=v-1+ u, stationary The characteristic root is l, so it is a unit root process(so non- stationary) Example 2: p241 A stationary AR(p) model is required for it to have an ma(oo) representation
5-16 • 平稳性使AR模型具有一些很好的性质。如前期误差项对当前 值的影响随时间递减。 • The condition for stationarity of a general AR(p) model is that the roots of 特征方程 all lie outside the unit circle. • Example 1: Is yt = yt-1 + ut stationary? The characteristic root is 1, so it is a unit root process (so nonstationary) • Example 2: p241 • A stationary AR(p) model is required for it to have an MA() representation. The Stationary Condition for an AR Model 1 1 2 0 2 − z − z − − p z = p
5-17 Wold's Decomposition Theorem States that any stationary series can be decomposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part, which will be an ma(oo) For the ar(p) model, (ly, ignoring the intercept, the Wold decomposition is y,=V(l) where v(L)=(1-1L-2D2-…中CP) 1+v1L+v2D2+v3b+ 可以证明,算子多项式RD的集合与代数多项式R()的集 合是同结构的,因此可以对算子L做加、减、乘和比率运算
5-17 • States that any stationary series can be decomposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part, which will be an MA(). • For the AR(p) model, , ignoring the intercept, the Wold decomposition is where, 可以证明，算子多项式R(L)的集合与代数多项式R(z)的集 合是同结构的，因此可以对算子L做加、减、乘和比率运算。 Wold’s Decomposition Theorem t ut (L)y = t L ut y =( ) = + + + + = − − − − 3 3 2 1 2 2 1 1 2 1 ( ) (1 ) L L L L L L L p p
5-18 The moments of an Autoregressive Process The moments of an autoregressive process are as follows The mean is given by E(y2)= 1-1-φ The autocovariances and autocorrelation functions can be obtained by solving what are known as the Yule-Wa alker equations 十z 192 十... +2+…+p-2力 If the ar model is stationary, the autocorrelation function will decay exponentially to zero
5-18 • The moments of an autoregressive process are as follows. The mean is given by* • The autocovariances and autocorrelation functions can be obtained by solving what are known as the Yule-Walker equations: * • If the AR model is stationary, the autocorrelation function will decay exponentially to zero. The Moments of an Autoregressive Process p p p p p p p p = + + + = + + + = + + + − − − − ... ... ... 1 1 2 2 2 1 1 2 2 1 1 1 2 1 p t E y − − − − = 1 1 2 ( )
5-19 Sample ar problem Consider the following simple ar(i)model y1=+1y () Calculate the(unconditional)mean of yr For the remainder of the question, set F0 for simplicity. (ii Calculate the(unconditional) variance ofy (iii) Derive the autocorrelation function fory r
5-19 • Consider the following simple AR(1) model (i) Calculate the (unconditional) mean of yt . For the remainder of the question, set =0 for simplicity. (ii) Calculate the (unconditional) variance of yt . (iii) Derive the autocorrelation function for yt . Sample AR Problem t t ut y = +1 y −1 +
5-20 Solution (i)E()=E(+vn1)=H+E(Uv=1 But also Vi-1=u+oiy1-2+u-1 E(y=H+中1(+nE(U=2) +n+q2E(v12) +n1H+12(+n1E(23 +n1H+12+n3E(3 An infinite number of such substitutions would give E()=(1+n+n12+…)+n So long as the model is stationary, i.e. <l, then o=0 SoE(U)=p(1+1+12+.)
5-20 (i)E(yt )= E(+1 yt-1 ) = +1E(yt-1 ) But also E(yt )= +1 ( +1E(yt-2 )) = +1 +1 2 E(yt-2 )) = +1 +1 2 ( +1E(yt-3 )) = +1 +1 2 +1 3E(yt-3 ) An infinite number of such substitutions would give E(yt )= (1+1+1 2+...) + 1 y0 So long as the model is stationary, i.e. , then 1 = 0. So E(yt )= (1+1+1 2+...) = Solution t−1 = + 1 t−2 +ut−1 y y 1 1 − 1 1