8.5 Let v equal L'Ev yt/T, V-1 equal 1-1 y/T, and de fine as in (8.19) (8.21), except with 2 = ye - U-1. (a) Show that {( ) (-1 U-)} = 0. (8.32) (b) Why might we expect F, (8.26), to have expectation near 0? V 8.19-8.21 21 = y - y (8.19) for all values of t. We will ignore the difference between definitions (8.14) and (8.19) in what follows, see Problem 8.4. Suppose that b is any guess for the true value of 8 in (8.15). Define the residual squared error for this guess to be RSE(6) -- bzt-1) (8.20) Using (8.15), and the fact that Ep(e) = 0, it is easy to show that RSE(6) has expectation E(RSE(0)) = (1-B)*E(EU 22-1)+(V - U + 1)var f(e). This is minimized when bequals the true value 8. We are led to believe that RSE() should achieve its minimum somewhere near the true value of 8. Given the time series data, we can calculate RSE(b) as a function of b, and choose the minimizing value to be our estimate of 6. RSE() = min RSE(). (8.21) U 8.26 strap disturbance terms are a random sample from F, - (, 3,...,48). (8.26) 8.5 Let v equal L'Ev yt/T, V-1 equal 1-1 y/T, and de fine as in (8.19) (8.21), except with 2 = ye - U-1. (a) Show that {( ) (-1 U-)} = 0. (8.32) (b) Why might we expect F, (8.26), to have expectation near 0? V 8.19-8.21 21 = y - y (8.19) for all values of t. We will ignore the difference between definitions (8.14) and (8.19) in what follows, see Problem 8.4. Suppose that b is any guess for the true value of 8 in (8.15). Define the residual squared error for this guess to be RSE(6) -- bzt-1) (8.20) Using (8.15), and the fact that Ep(e) = 0, it is easy to show that RSE(6) has expectation E(RSE(0)) = (1-B)*E(EU 22-1)+(V - U + 1)var f(e). This is minimized when bequals the true value 8. We are led to believe that RSE() should achieve its minimum somewhere near the true value of 8. Given the time series data, we can calculate RSE(b) as a function of b, and choose the minimizing value to be our estimate of 6. RSE() = min RSE(). (8.21) U 8.26 strap disturbance terms are a random sample from F, - (, 3,...,48). (8.26)