Introduction to Local Level Model and Kalman Filter

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Smoothing ◮ The filter calculates the mean and variance conditional on Yt; ◮ The Kalman smoother calculates the mean and variance conditional on the full set of observations Yn; ◮ After the filtered estimates are calculated, the smoothing recursion starts at the last observations and runs until the first. ˆµt = E(µt|Yn), Vt = var(µt|Yn), rt = weighted sum of future innovations, Nt = var(rt), Lt = 1 − Kt. Starting with rn = 0, Nn = 0, the smoothing recursions are given by rt−1 = F −1 t vt + Ltrt, Nt−1 = F −1 t + L 2 t Nt, ˆµt = at + Ptrt−1, Vt = Pt − P 2 t Nt−1.
Kalman smoothing for Nile Data: (i) ˆµt; (ii) Vt; (iii) rt and (iv) Nt. 1250 1000 750 500 0.02 0.00 −0.02 (i) 1880 1900 1920 1940 1960 (iii) 1880 1900 1920 1940 1960 4000 3500 3000 2500 0.0001 8e−5 6e−5 (ii) 1880 1900 1920 1940 1960 (iv) 1880 1900 1920 1940 1960
Page 1 and 2: Introduction to Local Level Model a
Page 3 and 4: Dow Jones 15000 10000 5000 0.25 0.0
Page 5 and 6: Classical Decomposition Basic Model
Page 7 and 8: Local Level Model General framework
Page 9 and 10: Simulated LL Data 6 4 2 0 −2 −4
Page 11 and 12: Simulated LL Data 5 σε 2 =0.1 σ
Page 13 and 14: Properties of the LL model ◮ The
Page 15 and 16: Nile Data: decomposition 1250 1000
Page 17 and 18: Nile Data: forecasts 1400 1300 1200
Page 19 and 20: Kalman Filter The unobserved variab
Page 21 and 22: Regression theory The proof of the
Page 23 and 24: Kalman Filter Derived ◮ Our best
Page 25: Steady State Kalman Filter Kalman f
Page 29 and 30: Nile Data with missing observations
Page 31 and 32: Nile Data: forecasting 1250 1000 75
Page 33 and 34: Parameter Estimation by ML The para
Page 35 and 36: Nile Data: diagnostics 2 0 −2 2 0
Page 37 and 38: Three exercises (cont.) 2. Derive a

Introduction to Local Level Model and Kalman Filter

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