1 00:00:00,250 --> 00:00:04,210 In this lecture, we will talk about backward shift operator. 2 00:00:04,210 --> 00:00:05,750 Objective is the following. 3 00:00:05,750 --> 00:00:10,690 We're going to define and utilize backward shift operator. 4 00:00:10,690 --> 00:00:12,046 So, let's start. 5 00:00:12,046 --> 00:00:15,260 We have a stochastic process, X1, X2, X3, and so forth. 6 00:00:16,930 --> 00:00:19,850 Each one of these guys are random variables. 7 00:00:19,850 --> 00:00:23,748 And we define backward shift operator as a follow up. 8 00:00:23,748 --> 00:00:27,800 B is our backward shift operator, and we'll take Xt, and 9 00:00:27,800 --> 00:00:32,390 we'll take it back one step, which will take it back one lambda. 10 00:00:32,390 --> 00:00:35,622 So, BXt will go to Xt-1. 11 00:00:35,622 --> 00:00:40,938 In other words, if you have B squared Xt, B squared is defined 12 00:00:40,938 --> 00:00:46,155 B applying backward shift operator twice, so we have BBXt, 13 00:00:46,155 --> 00:00:52,519 BXt is going to be Xt-1, and if take one step back you going to go to Xt-2. 14 00:00:52,519 --> 00:00:59,686 In other words B to the k Xt, Im sorry, B to the k Xt going to the Xt-k. 15 00:00:59,686 --> 00:01:05,450 With the Xt, I would take it back k steps and let's use it. 16 00:01:05,450 --> 00:01:07,850 So, this is the definition of the backward shift operator, so 17 00:01:07,850 --> 00:01:09,690 how can we utilize this? 18 00:01:09,690 --> 00:01:11,260 We're going to do the following. 19 00:01:11,260 --> 00:01:15,990 Think of the random walk right this is one of our examples we have talked about it. 20 00:01:15,990 --> 00:01:17,810 Basically you start from the previous step, 21 00:01:17,810 --> 00:01:21,130 and you add some random disturbance to it right. 22 00:01:21,130 --> 00:01:26,087 Zt is our random disturbance, or you know bayesian, or white noise, 23 00:01:26,087 --> 00:01:31,043 and then we go to the Xt, in other words we can write actually Xt as sum of 24 00:01:31,043 --> 00:01:35,764 the equations of the whole past innovations, all disturbances. 25 00:01:35,764 --> 00:01:40,750 Xt-1 can be expressed as BXt, that's backwards shift operator. 26 00:01:40,750 --> 00:01:44,940 And we take the Xt to the left-hand side, and pull out Xt, so 27 00:01:44,940 --> 00:01:51,080 that we get another operator 1-B is going to be our polynomial operator. 28 00:01:51,080 --> 00:01:56,115 1 is basic identity 1Xt, Xt, or 1-B will give you 29 00:01:56,115 --> 00:02:00,854 (1-B)Xt will give us Xt-Xt-1. 30 00:02:00,854 --> 00:02:04,310 We define (1- B) by phi B. 31 00:02:04,310 --> 00:02:06,890 And then phi (B)Xt becomes Zt. 32 00:02:06,890 --> 00:02:11,830 So, by using back work shift operator we rewrite random walk in this format. 33 00:02:13,820 --> 00:02:14,980 This is a MA(2) process. 34 00:02:14,980 --> 00:02:18,270 One example of MA(2) process is it goes back two steps. 35 00:02:18,270 --> 00:02:21,956 You see this is, the basically, sum weighted sum, 36 00:02:21,956 --> 00:02:27,150 of the three innovations, 37 00:02:27,150 --> 00:02:31,260 right, the present random noise, the previous random noise, and 38 00:02:31,260 --> 00:02:37,640 then the random noise in lack two, and they all add it up to give the Xt stack. 39 00:02:37,640 --> 00:02:39,178 Today, that's Xt. 40 00:02:39,178 --> 00:02:42,488 So, how come you neutralize backward chief operator here. 41 00:02:42,488 --> 00:02:49,109 Instead of Zt-1, we can write BZt, and instead of Zt-2 we can write B squared Zt. 42 00:02:49,109 --> 00:02:51,090 Now, everybody has Zt in it. 43 00:02:51,090 --> 00:02:55,438 We can kind of pull out Zt, and all polynomial operator, 44 00:02:55,438 --> 00:02:59,069 which we call beta B is going to be 1+0.2B. 45 00:02:59,069 --> 00:03:01,710 Plus 0.04B squared. 46 00:03:01,710 --> 00:03:06,832 Now, we write this MA(2) process in this format, Xt = Beta(B)Zt, 47 00:03:06,832 --> 00:03:10,562 where Beta(B) is actually a polynomial operator. 48 00:03:12,682 --> 00:03:18,490 We can utilize backwards shift operator in AR(2) processes. 49 00:03:18,490 --> 00:03:23,370 So, this is our Xt, a regressed on two previous 50 00:03:23,370 --> 00:03:28,420 lags with some random disturbance, innovation, or white noise at that step. 51 00:03:28,420 --> 00:03:31,657 And instead of Xt minus 1, we can write BXt, 52 00:03:31,657 --> 00:03:36,769 Xt minus 2 to write B square Xt take these two terms to the left outside, 53 00:03:36,769 --> 00:03:41,477 pull out Xt, we have polynomial operator we call this phi(B). 54 00:03:41,477 --> 00:03:44,640 So, we write this as phi(B)Xt = Zt. 55 00:03:44,640 --> 00:03:45,516 But what is our phi(B)? 56 00:03:45,516 --> 00:03:50,271 Phi(B) is basically polynomial operator in terms of the backward shift operator. 57 00:03:50,271 --> 00:03:54,894 It is 1- 0.2B- 0.3B squared. 58 00:03:54,894 --> 00:03:59,496 In general, if you look at MA(q) processes, moving average processes, for 59 00:03:59,496 --> 00:04:02,370 their q, assessing there's some drift. 60 00:04:02,370 --> 00:04:06,670 Mu is our drift, our expectation of Xt, in other words. 61 00:04:06,670 --> 00:04:10,490 And Xt is equal to, it's drift plus 62 00:04:12,100 --> 00:04:16,770 basically the linear combination of the random noises. 63 00:04:16,770 --> 00:04:23,980 Q steps back, q lags back instead of Xt, I write Zt instead of Zt -1, we write BZt. 64 00:04:25,090 --> 00:04:28,070 Instead of Zt minus q, we can write this BqZt. 65 00:04:28,070 --> 00:04:30,650 We can pull out Zt, put the drift to the left-hand side. 66 00:04:30,650 --> 00:04:35,760 I have Xt minus mu is equal to polynomial operator acting on Zt. 67 00:04:35,760 --> 00:04:42,410 And this polynomial operator is basically phi 0 + phi 1 B + phi q B q. 68 00:04:42,410 --> 00:04:45,799 Okay, so let me just mention that these phis are actually not phis, 69 00:04:45,799 --> 00:04:49,375 this is a typo, it is our basically Beta 0, Beta 1, Beta q to the end. 70 00:04:51,755 --> 00:04:56,620 Okay, If you have AR(p) processes, basically Xt is 71 00:04:56,620 --> 00:05:02,430 regressed on the previous P steps of the same stochastic process, 72 00:05:02,430 --> 00:05:06,465 plus some random noise, we can use, backwards shifht and 73 00:05:06,465 --> 00:05:12,135 put all these terms to the left, and come up with the format, phi(B)Xt equals Zt. 74 00:05:12,135 --> 00:05:16,432 Where phi(B) is the backwards shift, I'm sorry, it's a polynomial for each ratio. 75 00:05:16,432 --> 00:05:19,855 The back ration is 1-phi 1b and so forth. 76 00:05:22,020 --> 00:05:23,040 So, what have we learned? 77 00:05:23,040 --> 00:05:25,590 We have learned the definition of backward shift operator, 78 00:05:25,590 --> 00:05:29,130 which we will use a lot later on as well. 79 00:05:29,130 --> 00:05:34,009 And we used how to utilize backward shift operator to write MA(q) processes and 80 00:05:34,009 --> 00:05:35,310 AR(p) processes.