forecasting_lecture_10

forecasting_lecture_10 - Lecture Notes 10 1. Stationary f...

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1 Lecture Notes 10 1. Stationary – fancy word 1. Stationary – time series does not depend on time. Does not vary with time Weakly stationary – the mean and variance of a time series does not vary with time Examples “I” means integrative Integration is the branch of calculus that deals with infinite sums, like area under a function. (i) Stationary time series, I(0) – no trend I(0) Time Series -6 -4 -2 0 2 4 6 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 (ii) Non-stationary time series, I(1) – linear trend I(1) Time Series -40 -20 0 20 40 60 80 100 120 140 160 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59
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2 (ii) Non-stationary time series, I(2) – Quadratic trend I(2) Time Series -400 -200 0 200 400 600 800 1000 1200 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 2. MA(q) Process Moving average has q lags q t q t t t t Z Z Z Z X 2 2 1 1 Z t ~ white noise Z t ~ (0, 2 ) Since X t is a function of white noise, then it is stationary. Series does not move in any direction. 3. AR(1) Process t t t Z X X 1 Z t ~ white noise Very important 1 Note – AR(1) can be written as t t t Z X X 1 We will do a trick, here is the AR(1) t t t Z X X 1
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3 If we go back one unit in time, then 1 2 1 t t t Z X X Substitute X t-1 into the X t equation to get:   1 2 2 1 2 t t t t t t t t Z Z X X Z Z X X Let’s go back another unit in time, so 2 3 2 t t t Z X X Substitute this into X t equation, yielding   2 2 1 3 3 1 2 3 2 t t t t t t t t t t Z Z Z X X Z Z Z X X We keep doing this trick until we get the infinite series: 3 3 2 2 1 t t t t t Z Z Z Z X Thus, X t as an AR(1) process can be written as an infinite Moving Average time series. Did you notice the ’s; the further back in time you go, the smaller the ’s are because they are raised to a power. Since ’s are fractions, they get smaller when raised to higher and higher powers. Thus, the AR(1) is a stationary process. You can show an AR(p) is stationary too in the same manner. It just involves more algebra. Note – The Random Walk is not a stationary process t t t Z X X 1 Z t ~ white noise Note 1 Do the same thing, the time series for the last time period is: 1 2 1 t t t Z X X
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4 Substitute X t-1 into X t equation, yielding: 1 2 t t t t Z Z X X The time series for two periods ago is:
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forecasting_lecture_10 - Lecture Notes 10 1. Stationary f...

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