Stationarity Consider an Ma(1) process without drift It can be shown, regardless of the value of 0, that E(y1)=0 var (y1)=σ(1+) if s COV(D= otherwise
16 Stationarity Consider an MA(1) process without drift: It can be shown, regardless of the value of , that yt = t −1 t−1 − = = = + = − 0 otherwise if s 1 cov( ) var( ) (1 ) ( ) 0 2 1 2 1 2 t t s t t y y y E y 1
Stationarity ◆ For an Ma(2) process E(y1)=0 var(y1)=(1+61+2) o(1-62)ifs=1 coV(J,s)=3-020 if s=2 0 otherwise 17
17 Stationarity For an MA(2) process yt = t −1 t−1 −2 t−2 − = − − = = = + + = − 0 otherwise if s 2 (1 ) if s 1 cov( ) var( ) (1 ) ( ) 0 2 2 2 2 1 2 2 2 1 2 t t s t t y y y E y
Stationarity In general, Ma processes are stationarity regardless of the values of the parameters, but not necessarily Invertible An ma process is said to be invertible if it can be converted into a stationary ar process of infinite order The conditions of invertibility for an MA(k) process is analogous to those of stationarity for an aR(k) process. More on invertibility in tutorial 18
18 Stationarity In general, MA processes are stationarity regardless of the values of the parameters, but not necessarily “invertible”. An MA process is said to be invertible if it can be converted into a stationary AR process of infinite order. The conditions of invertibility for an MA(k) process is analogous to those of stationarity for an AR(k) process. More on invertibility in tutorial
Differencing Often non-stationary series can be made stationary through differencing Examples YI=V-1+E, IS not stationary, but W,=y,-y=E, is stationary 2)y,=1.7y-1-0.7y-2+E, is not stationary, but w,=y-y=0.7w-+& is stationary
19 Differencing Often non-stationary series can be made stationary through differencing. Examples: 0.7 is stationary 2) 1.7 0.7 is not stationary , but is stationary 1) is not stationary , but 1 1 1 2 1 1 t t t t t t t t t t t t t t t t w y y w y y y w y y y y = − = + = − + = − = = + − − − − − −
Differencing Differencing continues until stationarity is achieved Ay,=y-y A2y=△(4y,)=△(y1-y21)=y-2y21+y=2 The differenced series has n-l values after taking the first- difference, n-2 values after taking the second difference, and so on The number of times that the original series must be differenced in order to achieve stationarity is called the order of integration, denoted by d In practice, it is almost never necessary to go beyond secon difference because real data generally involve only first or second level non-stationarity
20 Differencing Differencing continues until stationarity is achieved. The differenced series has n-1 values after taking the firstdifference, n-2 values after taking the second difference, and so on. The number of times that the original series must be differenced in order to achieve stationarity is called the order of integration, denoted by d. In practice, it is almost never necessary to go beyond second difference, because real data generally involve only first or second level non-stationarity. t t t 1 y y y = − − 2 1 1 2 ( ) ( ) 2 t t t t t t t y y y y y y y = = − = − + − − −