By Ernst R. Berndt
This econometrics textual content is helping the reader to use econometric concepts to a number of empirical difficulties, utilizing vintage and modern facts units supplied on a diskette. every one bankruptcy starts with a dialogue of financial thought underlying an program. It then summarizes crucial empirical findings, and contains the reader in a gently designed set of workouts concerning replication and extension of general empirical findings.
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Pt Lt Lt+1 . . Ln−1 Ft Ft+1 Ft+2 Fn = at + Pt rt−1 , where rt−1 = vt vt+1 vt+2 vt+3 + Lt + Lt Lt+1 + Lt Lt+1 Lt+2 + ... + Ft Ft+1 Ft+2 Ft+3 vn + Lt Lt+1 . . 34) is a weighted sum of innovations after t − 1. The value of this at time t is rt = vt+1 vt+2 vt+3 + Lt+1 + Lt+1 Lt+2 + ··· Ft+1 Ft+2 Ft+3 vn + Lt+1 Lt+2 . . Ln−1 . 35) 21 State smoothing Obviously, rn = 0 since no observations are available after time n. 36) with rn = 0, for t = n, n − 1, . . , 1. The smoothed state can therefore be calculated by the backwards recursion rt−1 = Ft−1 vt + Lt rt , α ˆ t = at + Pt rt−1 , t = n, .
These plots are satisfactory and they suggest that the assumptions underlying the local level model are valid for the Nile data. 84. 5 –2 –2 –1 0 1 2 0 5 10 Fig. 7 Diagnostic plots for standardised prediction errors: (i) standardised residual; (ii) histogram plus estimated density; (iii) ordered residuals; (iv) correlogram. 1 1880 1900 1920 1940 1960 –4 –3 –2 –1 0 1 2 3 Fig. 8 Diagnostic plots for auxiliary residuals: (i) observation residual u∗t ; (ii) histogram and estimated density for u∗t ; (iii) state residual rt∗ ; (iv) histogram and estimated density for rt∗ .
11. For a Bayesian treatment a similar argument can be used to show that the posterior mean and variance of the forecast of yn+j is obtained by treating yn+1 , . . , yn+j as missing values, for j = 1, . . , J. 1 Illustration The Nile data set is now extended by 30 missing observations allowing the computation of forecasts for the observations y101 , . . , y130 . Only the Kalman ﬁlter (i) 50000 1250 40000 1000 30000 750 20000 10000 500 1900 1200 (ii) 1950 2000 (iii) 1900 1950 2000 1900 1950 2000 (iv) 60000 1100 50000 1000 40000 900 30000 800 1900 1950 2000 Fig.