Consider the model for the time series {y| t e T} y, = x, + v, and x, = Bx, +u, where {v| t e T} and {u| t e T} are independent white noise time series with variances o and o, respectively. a....


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Consider the model for the time series {y| t e T}<br>y, = x, + v, and x, = Bx, +u,<br>where {v| t e T} and {u| t e T} are independent white noise time series with<br>variances o and o, respectively.<br>a. Assuming that E(x,) = 0 and Var(x,) = o , determine a means of<br>calculating:<br>i. î, (1 –1) = E[x, .Y-3, Yr2»Yr1]<br>X,<br>1.<br>iii. £ (T)= E[x, .. Y7-2, Yr-1» Yr] for t<T<br>b. Assuming that ß = 0.9,0 = 2.0,o; = 1.5 and o =15, Suppose that the<br>first T= 15 observations on {y| t e T} are given below:<br>t<br>2<br>3<br>4<br>5<br>6<br>7<br>8<br>0.89<br>0.19<br>1.9<br>2.96<br>2.43<br>0.25<br>0.18<br>-1.03<br>t<br>9<br>10<br>11<br>12<br>13<br>14<br>15<br>-1.79<br>-4.74<br>-2.69<br>-0.78<br>-2.31<br>-0.89<br>-4.21<br>Compute<br>i. î, (1– 1) = E[x, ... y,-3» Yr-2, Yr-1] ! = 1, 2,...,15<br>ii. i, (1) = E[x, ...y,3» Y,2»Y;1»9%] t = 1,2,.,15<br>iii. , (15) = E[x, ..Yr-2, Yr-1» Vis] t = 1,2,.,15<br>= E] x,<br>1» Yi5| t = 1,2,...,15<br>

Extracted text: Consider the model for the time series {y| t e T} y, = x, + v, and x, = Bx, +u, where {v| t e T} and {u| t e T} are independent white noise time series with variances o and o, respectively. a. Assuming that E(x,) = 0 and Var(x,) = o , determine a means of calculating: i. î, (1 –1) = E[x, .Y-3, Yr2»Yr1] X, 1. iii. £ (T)= E[x, .. Y7-2, Yr-1» Yr] for t

Jun 07, 2022
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