1 00:00:11,090 --> 00:00:13,680 So in this lecture, we will be studying Arch. 2 00:00:14,180 --> 00:00:15,750 So why do we call this arch? 3 00:00:16,370 --> 00:00:21,130 Well, Arch stands for Auto Regressive Conditional Heteros Cadastre City. 4 00:00:21,740 --> 00:00:24,260 You can see how this might be a bit difficult to say. 5 00:00:24,440 --> 00:00:26,300 Thus, we simply call it arch. 6 00:00:27,050 --> 00:00:29,200 OK, so let's try to make sense of this name. 7 00:00:29,990 --> 00:00:33,780 Firstly, we've seen that stock returns are heteros get stick. 8 00:00:34,190 --> 00:00:36,780 This means that their variance changes over time. 9 00:00:37,490 --> 00:00:41,180 This is clear because sometimes it's high, but sometimes it's low. 10 00:00:41,960 --> 00:00:47,780 Compare this to completely random idy noise, which would be used to generate a random walk. 11 00:00:48,230 --> 00:00:52,400 We would call this homo sadistic because the variance is the same over time. 12 00:00:52,940 --> 00:00:54,200 Hetero means different. 13 00:00:54,350 --> 00:00:55,500 Homo means same. 14 00:00:56,870 --> 00:00:59,100 OK, so that's why it's hetero sadistic. 15 00:00:59,420 --> 00:01:00,660 What about conditional? 16 00:01:01,520 --> 00:01:07,310 Well, we can see that not only does the variance of the return a change over time, it seems to be 17 00:01:07,310 --> 00:01:09,430 dependent on nearby values. 18 00:01:09,890 --> 00:01:15,440 That is, when we see large returns, they tend to be surrounded by other large returns. 19 00:01:15,830 --> 00:01:20,130 When we see small returns, they tend to be surrounded by other small returns. 20 00:01:20,570 --> 00:01:26,720 So it would make sense to build a model where the current variance is conditioned on previous values. 21 00:01:27,320 --> 00:01:32,870 Furthermore, since it's conditioned on the previous values of the same time series, it's also auto 22 00:01:32,870 --> 00:01:33,560 regressive. 23 00:01:34,220 --> 00:01:39,830 OK, so that explains why we call this auto regressive conditional heteros cadastre. 24 00:01:40,460 --> 00:01:45,110 Basically, it means we're going to build a model for the variance of a time series which can change 25 00:01:45,110 --> 00:01:48,590 over time and its value depends on previous values. 26 00:01:53,150 --> 00:01:58,690 OK, so before we even begin to look at Gargash, let's recall how we typically model a time series. 27 00:01:59,120 --> 00:02:01,220 Most models we build have this form. 28 00:02:01,820 --> 00:02:08,300 We say that Y of T is equal to some function of past values of Y and some model parameters, plus some 29 00:02:08,300 --> 00:02:09,860 noise epsilon of T. 30 00:02:10,890 --> 00:02:15,920 OK, so normally Epsilon of T is assumed to be something like Gaussian white noise. 31 00:02:16,340 --> 00:02:22,940 We typically assume that it's a zero and its variance is constant and that we cannot predict it because 32 00:02:22,940 --> 00:02:23,360 of this. 33 00:02:23,360 --> 00:02:28,160 Our attention typically turns to the first part of this equation, which is what I've called F. 34 00:02:28,970 --> 00:02:34,880 Basically our job is to find theta, which determines that, well, in this section things are a bit 35 00:02:34,880 --> 00:02:35,520 different. 36 00:02:36,020 --> 00:02:40,150 Now we are not concerned with F, but rather we are concerned with Epsilon. 37 00:02:40,820 --> 00:02:44,210 So all throughout this course we've been trying to build a model for F. 38 00:02:44,570 --> 00:02:46,730 Now we are going to build a model for Epsilon. 39 00:02:51,330 --> 00:02:56,820 So another way to think of this is that a time series can be modeled by two separate models are and 40 00:02:56,820 --> 00:03:00,810 Gargash Arima is used to model the mean of the Time series. 41 00:03:01,170 --> 00:03:06,990 This part is assumed to have zero variance, on the other hand, is used to model the variance of the 42 00:03:06,990 --> 00:03:07,860 TIME series. 43 00:03:08,220 --> 00:03:10,140 This part is assumed to have zero mean. 44 00:03:10,830 --> 00:03:16,170 When you put these together, you get both the mean and the variance model by two separate components. 45 00:03:20,750 --> 00:03:25,610 So another thing to consider is why should we build an auto regressive model to begin with? 46 00:03:26,090 --> 00:03:29,160 We'd like to have a bit more evidence to motivate this decision. 47 00:03:30,350 --> 00:03:35,990 So, as you recall, one tool that we use to determine the order of an auto regressive model is the 48 00:03:35,990 --> 00:03:37,670 Akef or Sieff. 49 00:03:38,300 --> 00:03:42,190 Now, recall what happens when you take the akef of stock returns. 50 00:03:42,710 --> 00:03:48,680 We can think of a stock log returns as epsilon of T, since we've seen that in A1 process is often the 51 00:03:48,680 --> 00:03:51,680 best Arima model for a stock price time series. 52 00:03:52,250 --> 00:03:58,100 So if we fit in, I want to the log price then the epsilons or just the log returns. 53 00:03:59,090 --> 00:04:01,190 OK, so what does the AKF look like? 54 00:04:02,030 --> 00:04:05,640 Well, we've seen that there are zero statistically significant lags. 55 00:04:06,020 --> 00:04:11,870 The same thing happens when we look at the PSA and also recall this is the same thing we would see with 56 00:04:11,870 --> 00:04:14,480 completely random Gaussian white noise. 57 00:04:14,840 --> 00:04:19,100 And as such, we've seen that stock returns are very close to a random walk. 58 00:04:23,910 --> 00:04:29,910 But something interesting happens when we plot the Akef for the square of the log returns, when we 59 00:04:29,910 --> 00:04:34,260 do this, we find that there are, in fact, statistically significant lags. 60 00:04:34,740 --> 00:04:40,170 This suggests that the square of the log returns are predictable using pass values. 61 00:04:40,860 --> 00:04:45,540 Now, at this point, you'll just have to take my word that this is true, but we will be demonstrating 62 00:04:45,540 --> 00:04:46,460 this in code. 63 00:04:46,890 --> 00:04:52,170 Otherwise, if you're eager to observe this yourself and as an exercise, please try it on your own.