1 00:00:11,130 --> 00:00:16,530 So in this lecture, we'll be discussing an important topic in Arima theory, which is converting between 2 00:00:16,530 --> 00:00:17,540 different models. 3 00:00:18,270 --> 00:00:24,330 So throughout this course, you've learned about various different models, such as a. m. a va voom 4 00:00:24,330 --> 00:00:25,230 and so forth. 5 00:00:25,680 --> 00:00:30,830 We've discussed these models as if they are different, but in fact, they are the same to some degree. 6 00:00:31,680 --> 00:00:35,340 In this lecture, you learn how to convert a model from one form to another. 7 00:00:35,610 --> 00:00:40,050 For example, how to convert an R to an May answer to a VA. 8 00:00:41,250 --> 00:00:45,870 Now, we could have started this discussion in the arena section, but I think it gives you a better 9 00:00:45,870 --> 00:00:49,640 overall picture when you can incorporate vector models as well. 10 00:00:54,300 --> 00:01:00,270 So let's start with a scale or time series, suppose that we have an air one, this is equal to some 11 00:01:00,270 --> 00:01:08,040 byas term plus five times the previous value of Y plus the error at time t, but note that this equation 12 00:01:08,040 --> 00:01:09,030 is a recurrence. 13 00:01:09,540 --> 00:01:16,290 Therefore, just as we can express Y of T in terms of T minus one, we can express C, T minus one in 14 00:01:16,290 --> 00:01:17,850 terms of T minus two. 15 00:01:18,480 --> 00:01:23,460 So we can plug in the expression for Y of T minus one in terms of Y of T minus two. 16 00:01:25,260 --> 00:01:31,380 The next step is to simplify this expression, so now you can see that we have T minus two now has a 17 00:01:31,380 --> 00:01:34,620 squared coefficient and two different error terms appear. 18 00:01:35,830 --> 00:01:41,590 Of course, we can just do this again and replace Y of T minus two in terms of Y of T minus three. 19 00:01:42,910 --> 00:01:48,640 After we expand this, we see that we get another term, another error term, and the coefficient in 20 00:01:48,640 --> 00:01:52,460 front of Y of T minus three is now five one to the power of three. 21 00:01:53,770 --> 00:01:59,080 The next step is to consider what happens if we repeat this pattern, an infinite number of times. 22 00:02:03,670 --> 00:02:09,190 So when we repeat this pattern, an infinite number of times, we end up with three terms, the first 23 00:02:09,190 --> 00:02:15,040 term involves be multiplied by one over every integer power from zero to infinity. 24 00:02:15,730 --> 00:02:19,600 Note that since B appears in all of these terms, it can be factored out. 25 00:02:20,650 --> 00:02:25,400 The next term I want to look at is the last term, which involves the sum of all past errors. 26 00:02:26,260 --> 00:02:28,900 Now, in a typical model, this would only sum up. 27 00:02:29,020 --> 00:02:30,610 Q But in this case? 28 00:02:30,610 --> 00:02:31,540 Q is infinite. 29 00:02:32,380 --> 00:02:37,210 However, this doesn't change the fact that we still have a weighted sum of past errors. 30 00:02:38,690 --> 00:02:45,590 The final term we have involves the limit of one to the power K times Y of T minus K as K approaches 31 00:02:45,590 --> 00:02:46,350 infinity. 32 00:02:47,420 --> 00:02:53,090 I mentioned this one last because it's kind of a pesky term that we want to go away, but it only goes 33 00:02:53,090 --> 00:02:55,410 away if we make further assumptions. 34 00:02:55,730 --> 00:03:01,250 For example, if the value of the Time series is just zero beyond a certain point in the past, then 35 00:03:01,250 --> 00:03:03,200 clearly this whole thing would be zero. 36 00:03:03,800 --> 00:03:08,480 Or if the magnitude of PHY is less than one, this term would also approach zero. 37 00:03:09,540 --> 00:03:15,090 However, these details are not really too much of a concern, we can just pretend it's zero or constant 38 00:03:15,210 --> 00:03:16,530 without too much trouble. 39 00:03:17,580 --> 00:03:20,120 The main point to realize is that this is an M. 40 00:03:20,190 --> 00:03:21,280 Infiniti model. 41 00:03:21,630 --> 00:03:25,970 We have a bias term and we have a linear combination of past era terms. 42 00:03:26,460 --> 00:03:30,130 Therefore, in error, one is equivalent to an M infinity. 43 00:03:31,230 --> 00:03:35,850 Now, just in case you're curious, in the final line here, I have assumed that the middle term is 44 00:03:35,850 --> 00:03:40,990 zero and we use the summation formula from calculus to simplify the summation. 45 00:03:41,550 --> 00:03:43,280 And clearly, this is just a number. 46 00:03:43,440 --> 00:03:45,150 So it's considered a bias term. 47 00:03:49,760 --> 00:03:56,420 The next model I want to consider is an M1, so let's begin by writing down the formula for M1. 48 00:03:57,200 --> 00:04:00,200 You'll notice that I've written the parameters with these prime symbols. 49 00:04:00,740 --> 00:04:05,410 This is because the trick here is to write them a one in a different form. 50 00:04:06,050 --> 00:04:11,900 Specifically, we're going to bring all the terms to the other side in order to express Epsilon T in 51 00:04:11,900 --> 00:04:13,250 terms of everything else. 52 00:04:14,360 --> 00:04:20,180 Now we have a normal SI and a normal theta one which are just negative C prime and negative theta one 53 00:04:20,180 --> 00:04:20,750 prime. 54 00:04:21,380 --> 00:04:24,320 So the trick is to use this recurrence instead. 55 00:04:25,800 --> 00:04:31,260 For the next step, we can replace Epsilon at T minus one with an expression in terms of Epsilon of 56 00:04:31,260 --> 00:04:36,720 T minus two, and we can do this because we just found a recurrence that allows us to do this. 57 00:04:37,350 --> 00:04:43,320 And this should seem familiar because it's exactly the same kind of substitution we just used after 58 00:04:43,320 --> 00:04:44,070 expanding this. 59 00:04:44,070 --> 00:04:49,430 We now have to y terms to see terms and one error term a time T minus two. 60 00:04:50,430 --> 00:04:55,590 The next step, of course, is to replace the Epsilon at T minus two term with an expression in terms 61 00:04:55,590 --> 00:04:57,270 of Epsilon at T minus three. 62 00:04:58,570 --> 00:05:04,240 After expanding this expression, we get three wide terms, three terms and the error term at time T 63 00:05:04,240 --> 00:05:05,110 minus three. 64 00:05:05,980 --> 00:05:09,310 OK, so clearly this is the same kind of pattern we had before. 65 00:05:14,140 --> 00:05:19,800 So, again, if we extend this an infinite number of times, we get an expression with three terms, 66 00:05:20,290 --> 00:05:22,990 the first term is an infinite weighted sum of was. 67 00:05:23,500 --> 00:05:28,530 The second term is just the constant, since theta one is constant and see is also constant. 68 00:05:29,080 --> 00:05:31,420 Note that I've used the infinite summation formula. 69 00:05:31,420 --> 00:05:37,780 Once again, the final term is the limit, which again for simplicity will assume a zero. 70 00:05:39,280 --> 00:05:43,010 Now, remember that our Time series is Y of T, not Epsilon of T. 71 00:05:43,540 --> 00:05:47,930 So in order to get back, we have T we can simply pull it out of the summation. 72 00:05:48,490 --> 00:05:52,170 Now the summation starts at K equals one instead of equals zero. 73 00:05:53,670 --> 00:05:59,700 The next step is to move everything to the other side into a form we can more easily recognize, as 74 00:05:59,700 --> 00:06:05,050 you can see, we have a constant term, plus an infinite sum of lag terms, plus the usual error. 75 00:06:05,520 --> 00:06:07,350 Thus, this is in r infinity. 76 00:06:12,100 --> 00:06:19,300 OK, so we've just shown the equivalence between ARMM, the next step is to think of vector auto regressions. 77 00:06:19,780 --> 00:06:22,870 In fact, for part of this, there's really no more work to do. 78 00:06:23,290 --> 00:06:29,230 We can just use the same kind of mathematical substitutions to show that VAR one is the same as Vermes 79 00:06:29,230 --> 00:06:29,950 Infinity. 80 00:06:30,670 --> 00:06:34,690 Likewise, we can show that PVM one is the same as VAR Infinity. 81 00:06:35,970 --> 00:06:41,040 The only difference is that instead of multiplying scalars by scalars, you'll be multiplying matrices 82 00:06:41,040 --> 00:06:46,380 by vectors, but intuitively you should be able to see that the same math applies. 83 00:06:51,110 --> 00:06:58,070 The interesting part, I think, is to show that an AARP can be expressed as a VA, want to see how 84 00:06:58,070 --> 00:06:58,920 this can be done? 85 00:06:58,940 --> 00:07:01,850 It's easiest to start with an actual process. 86 00:07:02,480 --> 00:07:08,330 In this case, we have to coefficients five one in five to note that for the following slides, you'll 87 00:07:08,330 --> 00:07:10,560 notice that these formulas look a little different. 88 00:07:11,030 --> 00:07:16,230 This is because the program I normally use to make equations does not work with matrices. 89 00:07:16,580 --> 00:07:21,450 So in order to have matrices, I had to use a different program, which is what you're seeing here. 90 00:07:22,010 --> 00:07:27,170 In fact, if you read a lot of papers, this will probably look more similar to what you're used to. 91 00:07:28,900 --> 00:07:35,440 So suppose we create a new vector, which is just a vector containing both YFC and YFC minus one. 92 00:07:36,190 --> 00:07:41,310 We can then write the VA one model in terms of this whole vector instead of just Y of T. 93 00:07:42,130 --> 00:07:48,370 So as an exercise, I recommend going backwards and expanding this matrix equation on paper for the 94 00:07:48,370 --> 00:07:50,240 top row when you multiply this out. 95 00:07:50,500 --> 00:07:55,390 You should get back to the original R two for the bottom row when you multiply it out. 96 00:07:55,630 --> 00:07:57,550 You should get back to the trivial equation. 97 00:07:57,550 --> 00:08:04,750 Y of T minus one equals y t minus one and of course recognize that this new vector equation is of our 98 00:08:04,750 --> 00:08:05,200 one. 99 00:08:09,900 --> 00:08:13,980 One way to see this, if it's not already clear, is the use substitution. 100 00:08:14,700 --> 00:08:21,450 So suppose that we let the Vector ZT be equal to a vector containing YFC and Y of T minus one. 101 00:08:22,350 --> 00:08:25,740 We can let the prime be A vector containing B and zero. 102 00:08:26,400 --> 00:08:33,220 We can let capital thig be a matrix containing five one five to one and zero, and we can let eight 103 00:08:33,260 --> 00:08:36,270 to be a vector containing Epsilon C and zero. 104 00:08:37,260 --> 00:08:42,090 So when we do this, the equation becomes precisely the definition of a var one. 105 00:08:42,900 --> 00:08:49,770 The trick is seeing that Z of T minus one is a vector containing Y of T minus one and Y of T minus two. 106 00:08:54,510 --> 00:09:01,500 So let's now try this strategy again for three, so this time we'll create a vector with three components 107 00:09:01,500 --> 00:09:05,790 containing YFC, YFC minus one and YFC minus two. 108 00:09:06,630 --> 00:09:12,270 Again, we do the same thing where we express the top row as the original Air Three and the next two 109 00:09:12,270 --> 00:09:14,210 rows as dummy equations. 110 00:09:15,270 --> 00:09:19,500 So if you multiply out the bottom two rows, you should get the trivial equations. 111 00:09:19,500 --> 00:09:24,360 YFC minus one equals YFC minus one and YFC minus two equals YFC minus two. 112 00:09:25,230 --> 00:09:28,590 And again, note that this vector model is of our one. 113 00:09:33,320 --> 00:09:38,600 So an even better trick is to show that we can do the same thing, not just with an AARP, but with 114 00:09:38,600 --> 00:09:39,320 a VIP. 115 00:09:40,550 --> 00:09:42,240 So let's start with a VA three. 116 00:09:42,980 --> 00:09:47,240 This is just like what we had before, except now with matrices and vectors. 117 00:09:47,930 --> 00:09:54,560 So the first step, again, is to concatenate Y of T, Y of C minus one and Y of T minus two into a 118 00:09:54,560 --> 00:09:55,400 bigger vector. 119 00:09:56,120 --> 00:10:00,500 So this new vector would be three times the size as each individual y. 120 00:10:02,540 --> 00:10:07,700 The next step follows the same pattern where the top row becomes the original VARE three and the next 121 00:10:07,700 --> 00:10:09,640 two rows are dummy equations. 122 00:10:10,280 --> 00:10:16,850 But note that now instead of having ones, we have eyes, as you recall, any matrix times eye, which 123 00:10:16,850 --> 00:10:20,840 is the identity matrix, just gives you back the same original matrix. 124 00:10:22,100 --> 00:10:28,220 So what we have shown is any VPE can be expressed as of our one, except that the vector has become 125 00:10:28,220 --> 00:10:29,420 P times larger. 126 00:10:33,990 --> 00:10:35,470 So what have we just shown? 127 00:10:35,970 --> 00:10:43,050 We've shown that both AARP and VA can be expressed as VA one and of course, as we saw earlier, this 128 00:10:43,050 --> 00:10:45,060 can be expressed as VA infinity. 129 00:10:45,870 --> 00:10:51,030 Now, one thing we didn't go through is to show that McHugh and the VA make you can be expressed as 130 00:10:51,030 --> 00:10:51,840 VA one. 131 00:10:52,590 --> 00:10:57,450 So as an exercise, you can show that this is true by applying the same tricks we learned earlier in 132 00:10:57,450 --> 00:10:58,200 this lecture. 133 00:10:59,070 --> 00:11:02,980 But we also know that PVM one can be expressed as VA infinity. 134 00:11:03,630 --> 00:11:08,300 So we've sort of completed the cycle between VA to FEMA and back again. 135 00:11:12,930 --> 00:11:17,690 Now, again, this lecture is optional, so if you don't care about this sort of thing, that's OK. 136 00:11:18,150 --> 00:11:22,200 But to be clear, this is a typical part of any college level time series course. 137 00:11:22,470 --> 00:11:23,970 So it's something worth knowing. 138 00:11:24,810 --> 00:11:29,730 Just keep in mind that this leads to a lot more math that we could do, which eventually leads to practical 139 00:11:29,730 --> 00:11:30,840 applications. 140 00:11:31,350 --> 00:11:35,990 At this time, we won't look into these things since they require a lot more linear algebra. 141 00:11:36,210 --> 00:11:39,350 But this at least exposes you to more possibilities.