1 00:00:00,000 --> 00:00:07,140 So the box plot gives you these kind of robust measures of a distribution. 2 00:00:07,140 --> 00:00:13,037 So here's an illustration of the box plot for the monthly returns on Microsoft. 3 00:00:13,037 --> 00:00:17,609 And so this black line in the middle represents the, the median. 4 00:00:17,609 --> 00:00:23,064 This is roughly the interquartile range. It's not exactly it and then if you look 5 00:00:23,064 --> 00:00:26,550 at the online help for a box plot, you'll see why. 6 00:00:26,550 --> 00:00:31,589 Notice that this is not quite symmetric. So by definition, the median is in the 7 00:00:31,589 --> 00:00:37,565 middle of the interquartile range, but This is so, so that these, these fences 8 00:00:37,565 --> 00:00:41,262 are, the, the, the edges of the box are not quite the, the. 9 00:00:41,262 --> 00:00:47,021 The first and the third quartile, and then you have, You know, essentially these 10 00:00:47,021 --> 00:00:52,434 fences that, illustrate, you know, when you have extreme outliers in the data. 11 00:00:52,434 --> 00:00:57,979 So these points out over here, essentially are representing those values that are, 12 00:00:57,979 --> 00:01:02,158 you know, what, what we characterize as an extreme outlier. 13 00:01:02,158 --> 00:01:06,572 And this would be in the right tail, and this would be in the left tail. 14 00:01:06,572 --> 00:01:10,282 So you have a, the bulk of the distribution is roughly symmetric. 15 00:01:10,282 --> 00:01:14,837 You have two big negative outliers and you have a few big positive outliers. 16 00:01:14,837 --> 00:01:19,405 And again, if you look at the online help for the box plot, ya know, they'll tell 17 00:01:19,405 --> 00:01:22,764 you exactly how this fence and this fence is determined. 18 00:01:22,764 --> 00:01:30,595 From the point of view of interpretation, the big power of box plots comes when you 19 00:01:30,595 --> 00:01:34,664 compare multiple series. Because it allows you to, you know, very 20 00:01:34,664 --> 00:01:39,326 quickly look at the distribution of multiple series in the same plot. 21 00:01:39,326 --> 00:01:44,384 So, here's a, a box plot where I have the box plot of the Gaussian white noise. 22 00:01:44,384 --> 00:01:50,009 This is the simulated data that comes from a normal distribution that has the same 23 00:01:50,009 --> 00:01:55,445 mean invariance as the Microsoft returns. These are the, this is the box plot for 24 00:01:55,445 --> 00:01:59,986 the actual Microsoft returns and this is the box plot for the SMP 500 returns. 25 00:01:59,986 --> 00:02:03,552 And so we see that they're on the same scale which is percent. 26 00:02:03,552 --> 00:02:08,623 So the average return for all of these things is pretty close to zero and a 27 00:02:08,623 --> 00:02:12,048 little bit positive. So, that's the median return. 28 00:02:12,048 --> 00:02:18,151 We see that you know, again, Microsoft and Guassian White Noise, the middle of the 29 00:02:18,151 --> 00:02:23,874 distribution is, is roughly the same. And, what we see with the Microsoft data 30 00:02:23,874 --> 00:02:28,488 is, we have some outliers here. With the Gaussian white noise data we 31 00:02:28,488 --> 00:02:33,441 don't have the outliers because normal distribution doesn't really produce 32 00:02:33,441 --> 00:02:37,552 outliers. And with the S and P 500 data, we see the 33 00:02:37,552 --> 00:02:43,774 smaller spread than Microsoft, and we see the small two negative outliers. 34 00:02:43,774 --> 00:02:49,762 So again, we can get sort of an idea of the shape of the distribution. 35 00:02:49,762 --> 00:02:56,824 So it's similar to the histogram, but again it's based on robust measures, and 36 00:02:56,824 --> 00:03:05,050 it's quite useful for doing quick comparisons of many series at once. 37 00:03:05,050 --> 00:03:14,802 Now, I've put together a little four graph summary for looking at the distribution of 38 00:03:14,802 --> 00:03:18,619 asset returns. And this four graph summary is borrowed 39 00:03:18,619 --> 00:03:23,863 from a very nice book by Rene Carmona called The Statistical Analysis of 40 00:03:23,863 --> 00:03:27,629 Financial Data. And in this four graph summary there's 41 00:03:27,629 --> 00:03:32,604 going to be a histogram, a box plot, a smooth histogram, and then a QQ plot 42 00:03:32,604 --> 00:03:37,621 relative to a normal. So we have four pictures and it looks like 43 00:03:37,621 --> 00:03:40,403 this. So, say for Microsoft, here we have the 44 00:03:40,403 --> 00:03:45,201 histogram, here we have the smooth density down here so you can get a rough shape of 45 00:03:45,201 --> 00:03:48,605 the distribution. And then, we have the box plot over here, 46 00:03:48,605 --> 00:03:53,817 and the QQ plot relative to the normal. And all of these, these three plots are 47 00:03:53,817 --> 00:03:57,831 giving you similar information. We see kind of a long left tail, a 48 00:03:57,831 --> 00:04:01,412 negative skewness, so we see a little long left tail over here. 49 00:04:01,412 --> 00:04:06,720 We see the negative outliers in the box plot, which is corresponding to these 50 00:04:06,720 --> 00:04:10,479 observations here. And in the Q-Q plot relative to normal, 51 00:04:10,479 --> 00:04:15,189 we're seeing this drop-down relative to the normal graph, and this dipping up a 52 00:04:15,189 --> 00:04:19,477 little bit on the right hand side. So if you look at, you know, a, a p-, a 53 00:04:19,477 --> 00:04:24,170 summary graph like this, you can get an idea that Microsoft returns are. 54 00:04:24,170 --> 00:04:29,278 Kind of approximately normally distributed in terms of the bulk of the distribution 55 00:04:29,278 --> 00:04:32,115 looks normal. But there is some negative skewness and 56 00:04:32,115 --> 00:04:36,937 fatter tales relative to the normal distribution. 57 00:04:36,937 --> 00:04:44,787 Okay. Alright. 58 00:04:44,787 --> 00:04:50,509 So when we have two or more random variables, then, we have, we might want to 59 00:04:50,509 --> 00:04:55,214 look at some descriptive statistics that tell us the relationship between two or 60 00:04:55,214 --> 00:04:58,047 more variables. Now, when we studied probability theory, 61 00:04:58,047 --> 00:05:02,485 when we were looking at the dependence between two random variables, we looked at 62 00:05:02,485 --> 00:05:07,171 covariance and correlation. Now, from the point of view of descriptive 63 00:05:07,171 --> 00:05:11,710 statistics, we can, we have, say we have two random variables X and Y. 64 00:05:11,710 --> 00:05:14,284 And then we observe the sample X1, Y1, X2, Y2. 65 00:05:14,284 --> 00:05:19,569 So, we can think of X as the return on Microsoft, Y as the return on the S and P 66 00:05:19,569 --> 00:05:23,728 500; and then our sample is the data that we download from Yahoo. 67 00:05:23,728 --> 00:05:26,651 Okay? We assume that these returns, you know, 68 00:05:26,651 --> 00:05:30,125 follow a multi-, a bi-variant normal distribution. 69 00:05:30,125 --> 00:05:33,702 For example, as, as being kind of the benchmark. 70 00:05:33,702 --> 00:05:39,012 And then, say we wanna measure the dependence between the Microsoft returns, 71 00:05:39,012 --> 00:05:44,204 and the S and P 500 returns. We'd wanna compute the, sample covariance 72 00:05:44,204 --> 00:05:49,962 and the sample correlation, to get, an idea of what the data say the relationship 73 00:05:49,962 --> 00:05:54,039 between these returns are, okay. Now, in, looking at pairwise 74 00:05:54,039 --> 00:05:59,465 relationships, there's a graphical, diagnost-, a graphical descriptive 75 00:05:59,465 --> 00:06:04,069 statistic called a scatter plot. And a scatter plot is just an x-y plot of 76 00:06:04,069 --> 00:06:08,040 your bi-variant data. So you can plot the returns on the SMP500 77 00:06:08,040 --> 00:06:11,055 on one axis and returns of Microsoft on the other. 78 00:06:11,055 --> 00:06:14,081 And then you can see what the data looks like. 79 00:06:14,081 --> 00:06:19,008 So for example. Here's a scatter plot of the monthly 80 00:06:19,008 --> 00:06:21,086 returns on Microsoft versus the S and P 500. 81 00:06:21,086 --> 00:06:26,029 So here, I, I put Microsoft on the X axis. The S and P 500 on the Y axis. 82 00:06:26,029 --> 00:06:30,096 And the black lines here represent. This is the mean for Microsoft, this is 83 00:06:30,096 --> 00:06:33,011 the mean for the S and P 500. Okay? 84 00:06:33,011 --> 00:06:37,066 So remember when we, we studied co-variance and correlation, we, we looked 85 00:06:37,066 --> 00:06:40,088 at these probability scatter plots. And you know? 86 00:06:40,088 --> 00:06:43,092 Essentially, when you look at this data, you know? 87 00:06:43,092 --> 00:06:48,072 What would you say, are, is there a positive or negative relationship between 88 00:06:48,072 --> 00:06:51,676 Microsoft and the S and P 500? Positive, right? 89 00:06:51,676 --> 00:06:57,040 Cuz as Microsoft returns go up, S and P 500 returns tend to go up as well. 90 00:06:57,040 --> 00:07:02,057 As the Microsoft returns go down, the S and P 500 returns are going down. 91 00:07:02,057 --> 00:07:05,051 Okay? So here we see, we see a negative 92 00:07:05,051 --> 00:07:09,017 relationship. And later on, we'll compute the sample 93 00:07:09,017 --> 00:07:13,055 co-variance and correlation. And we, there's a positive sample 94 00:07:13,055 --> 00:07:16,071 co-variance and the sample correlation is.6. 95 00:07:16,071 --> 00:07:22,037 So again, there's a reasonably strong positive linear association between these 96 00:07:22,037 --> 00:07:23,060 two returns. Okay? 97 00:07:26,025 --> 00:07:31,039 Now a. When you have more than two returns, 98 00:07:31,039 --> 00:07:35,010 there's a nifty function in R called pairs, P, A, I, R, S. 99 00:07:35,010 --> 00:07:39,093 And what the pairs function does, it creates all pair y scatter plots. 100 00:07:39,093 --> 00:07:45,025 So here I have three data series, my computer-simulated Gaussian white noise, 101 00:07:45,025 --> 00:07:49,017 the returns on Microsoft, and the returns in S and P 500. 102 00:07:49,017 --> 00:07:52,095 And so now I have what's called a scatter plot matrix. 103 00:07:52,095 --> 00:07:58,097 And, so this graph right here, this is the scatter plot with Gaussian white noise on 104 00:07:58,097 --> 00:08:03,087 this axis and Microsoft on this axis. And we see the scatter plot looks like a 105 00:08:03,087 --> 00:08:07,014 shock and blast. There appears to be no linear association 106 00:08:07,014 --> 00:08:10,025 between them. So we would assume that the covariance is 107 00:08:10,025 --> 00:08:14,019 close to zero and the correlation is close to zero, based on this plot. 108 00:08:14,019 --> 00:08:21,033 This plot represents Gaussian White Noise on this axis, and the S and P 500 on this 109 00:08:21,033 --> 00:08:22,019 axis. Okay? 110 00:08:22,019 --> 00:08:25,687 And, again, this plot also kinda looks like a s-, a shotgun blast, where it 111 00:08:25,687 --> 00:08:30,710 doesn't appear to be any systematic positive or negative relationship in the 112 00:08:30,710 --> 00:08:33,630 data. If anything, you know, there might be what 113 00:08:33,630 --> 00:08:38,102 might, what looks, maybe, to be a slight negative relationship in, in the data. 114 00:08:38,102 --> 00:08:42,907 But it certainly isn't very strong. So one might expect this sample covariance 115 00:08:42,907 --> 00:08:47,147 to be slightly negative, and the correlation to be a negative number 116 00:08:47,147 --> 00:08:52,469 that's, but pretty close to zero. This plot over here is just, we put 117 00:08:52,469 --> 00:08:57,928 Microsoft on this axis and Gaussian white noise on this axis. 118 00:08:57,928 --> 00:09:03,109 So this plot is a, just slipping the axis from this plot, okay? 119 00:09:03,109 --> 00:09:08,895 And similarly, this, this plot down here is flipping this plot. 120 00:09:08,895 --> 00:09:14,674 And then finally our last plot shows Microsoft on this axis, SMP 500 on this 121 00:09:14,674 --> 00:09:17,337 axis. So that's what I showed you before. 122 00:09:17,337 --> 00:09:22,948 And we see positive linear relationships. So, these two series are positively 123 00:09:22,948 --> 00:09:27,099 correlated positive covariance and a positive correlation. 124 00:09:27,099 --> 00:09:29,524 So again, this is nice if you have ten assets. 125 00:09:29,524 --> 00:09:34,213 You can do this and then you can get a very quick summary of what appears to be 126 00:09:34,213 --> 00:09:36,647 the linear association between the variables. 127 00:09:36,647 --> 00:09:39,655 And you can also I mean again, cause these are plots. 128 00:09:39,655 --> 00:09:43,875 Even if there's a non linear association you know, you would see, you could 129 00:09:43,875 --> 00:09:52,764 possibly see that in, in the plot itself. So, We often summarize the sample 130 00:09:52,764 --> 00:09:59,916 variances and covariances in a matrix. And actually before I get there let me 131 00:09:59,916 --> 00:10:04,488 define the sample statistics. So if you wanna compute the sample 132 00:10:04,488 --> 00:10:09,615 covariance, then you would take your sample of your first data series, the 133 00:10:09,615 --> 00:10:14,876 sample of your second data series. You compute this sample average of x minus 134 00:10:14,876 --> 00:10:19,449 its mean, times y minus its mean. And that's sometimes this is called at, 135 00:10:19,449 --> 00:10:22,276 little s with a subscript xy, or sigma hat xy. 136 00:10:22,276 --> 00:10:27,906 So this is the sample covariance and then the sample correlation is the sample 137 00:10:27,906 --> 00:10:32,989 covariance divided by the product of the sample standard deviations. 138 00:10:32,989 --> 00:10:38,762 So if we're working in R, the bar function computes sample covariance matrix. 139 00:10:38,762 --> 00:10:44,556 So if you so if we have three assets, and we have a three by three matrix, then have 140 00:10:44,556 --> 00:10:49,394 the variances along the diagonals and the covariances on the off-diagonals. 141 00:10:49,394 --> 00:10:55,113 That's what the bar function computes. Notice that there isn't a, actually, there 142 00:10:55,113 --> 00:11:00,655 is a co function, but since with the co function, does it same thing as, as a bar. 143 00:11:00,655 --> 00:11:06,866 The core function COR, gives you the sample correlation matrix. 144 00:11:06,866 --> 00:11:10,429 So, here. So if I take my three data series, my 145 00:11:10,429 --> 00:11:13,783 Gaussian white noise, my Microsoft, and my S and P 500. 146 00:11:13,783 --> 00:11:18,442 I use the bar command. And then this is the variance/covariance 147 00:11:18,442 --> 00:11:21,347 matrix. The sample variances are along the 148 00:11:21,347 --> 00:11:24,346 diagonals. And the covariances are on the off 149 00:11:24,346 --> 00:11:27,430 diagonals. So, so notice that we see a negative 150 00:11:27,430 --> 00:11:32,563 sample covariance between Microsoft and Gaussian white noise and a negative 151 00:11:32,563 --> 00:11:37,343 covariance between Microsoft, sorry, between the S and P 500 and the Gaussian 152 00:11:37,343 --> 00:11:40,391 white noise. And we have a positive covariance between 153 00:11:40,391 --> 00:11:44,553 the S and P 500 and Microsoft. Now, covariance is just direction of 154 00:11:44,553 --> 00:11:49,391 association, correlation of strength. So when you look at the correlation 155 00:11:49,391 --> 00:11:54,269 matrix, you have ones along the diagonal because the correlation with each series 156 00:11:54,269 --> 00:11:58,676 with itself by definition is one. And so here's the correlation between the 157 00:11:58,676 --> 00:12:06,135 Gaussian white noise and the Microsoft, that's -.19 Not very strong. 158 00:12:06,135 --> 00:12:10,238 Correlation between Gaussian white noise and S and P 500 is -.24. 159 00:12:10,238 --> 00:12:14,347 Now this is a completely spurious correlation, because the Gaussian white 160 00:12:14,347 --> 00:12:16,421 noise was computer generated. Right? 161 00:12:16,421 --> 00:12:21,196 It has nothing to do with the actual S and P 500 data, but we still had a number 162 00:12:21,196 --> 00:12:24,800 that's, you know, kinda large and negative, and that's just by chance. 163 00:12:24,800 --> 00:12:27,604 Right? If I simulated another Gaussian white 164 00:12:27,604 --> 00:12:32,132 noise, then this correlation, probably close to zero or could be a little 165 00:12:32,132 --> 00:12:36,566 positive, or something like that. The Microsoft and S and P 500 data has a 166 00:12:36,566 --> 00:12:42,032 reasonably strong correlation, that's .6. The last descriptive statistic I want to 167 00:12:42,032 --> 00:12:45,722 talk about is, has to do with time series dependence. 168 00:12:45,722 --> 00:12:51,085 So again we were covering probability theory, we went through the time series 169 00:12:51,085 --> 00:12:56,653 concepts section, and we were trying to think about time dependence in data. 170 00:12:56,653 --> 00:13:00,557 And so we defined what are called autocorrelations. 171 00:13:00,557 --> 00:13:06,305 We defined the covariance between y(t) and y(t-1), the correlation between y(t) and 172 00:13:06,305 --> 00:13:09,798 y(t-1) as well. Well, we can compute sample versions of 173 00:13:09,798 --> 00:13:13,020 these quantities. So given, the time series of data. 174 00:13:13,020 --> 00:13:18,270 If, and, and if we're interested in determining if there's linear dependence 175 00:13:18,270 --> 00:13:21,712 over time. Then we can compute the sample auto 176 00:13:21,712 --> 00:13:26,686 covariance and the sample auto correlation, and see if these things are 177 00:13:26,686 --> 00:13:31,196 different from zero. So the sample autocovariance is just at 178 00:13:31,196 --> 00:13:36,019 lag J, is just the sample covariance between XT and XT minus J. 179 00:13:36,019 --> 00:13:41,762 And the sample auto correlation is just this sample auto co-variance divided by 180 00:13:41,762 --> 00:13:47,331 the sample variance, and these are measures of linear dependence between a 181 00:13:47,331 --> 00:13:53,507 variable and its lags, and then we can do a graphical plot called the sample auto 182 00:13:53,507 --> 00:13:59,745 correlation function, where we just plot this sample auto correlation against the 183 00:13:59,745 --> 00:14:02,783 lag. So, notice that when you calculate this 184 00:14:02,783 --> 00:14:06,465 sum. We start the summation at J plus one, 185 00:14:06,465 --> 00:14:09,965 right, because we're looking at the summation. 186 00:14:09,965 --> 00:14:15,550 So, the first index in this sum is J plus one, so we cut, we look at here. 187 00:14:15,550 --> 00:14:19,341 That's going to be X, J plus one minus the mean. 188 00:14:19,341 --> 00:14:22,429 And then we have XJ + one - J. So that's X1. 189 00:14:22,429 --> 00:14:24,205 Right? Minus the mean. 190 00:14:24,205 --> 00:14:29,586 So we're looking at the relationship between XJ and J lags from XJ. 191 00:14:29,586 --> 00:14:33,945 So from the first observation to the Jth observation. 192 00:14:33,945 --> 00:14:40,012 Then the next term in the sum is XJ + one. And then this would be X2, and so on. 193 00:14:40,012 --> 00:14:45,780 So we're looking at the relationship between, you know, X at time T and its 194 00:14:45,780 --> 00:14:51,215 lag. So if I guess if you draw a picture. 195 00:14:51,215 --> 00:14:54,841 Right? So we have you know, one, two, three, 196 00:14:54,841 --> 00:15:01,372 four, you know, up to J, and so this would be XJ, and then we have X1, and then we 197 00:15:01,372 --> 00:15:06,404 have XJ plus one X2. So when we're calculating this sum, so 198 00:15:06,404 --> 00:15:12,958 we're looking at the relationship between this variable and this variable, this 199 00:15:12,958 --> 00:15:18,558 variable and that variable, and in terms of the computation of this. 200 00:15:18,558 --> 00:15:22,856 If we started this at zero. Then this would be XO. 201 00:15:22,856 --> 00:15:27,299 Or say, if we started this at one, then this would be X1. 202 00:15:27,299 --> 00:15:31,776 But this would be X1-J. And we don't have data, for that. 203 00:15:31,776 --> 00:15:38,277 So, this, this notation is to emphasize that we have to do this computation based 204 00:15:38,277 --> 00:15:42,507 on the data sample that we actually observed. 205 00:15:42,507 --> 00:15:50,115 Okay, so here's an example of sample auto correlations. 206 00:15:50,115 --> 00:15:56,333 So this is, these are plots that are measuring the estimated time dependence in 207 00:15:56,333 --> 00:16:00,052 the data. And the top graph is for the Gaussian 208 00:16:00,052 --> 00:16:04,055 white noise. And the middle graph is for Microsoft. 209 00:16:04,055 --> 00:16:10,535 The bottom graph is for the S and P 500. And this vehicular is being plotted here. 210 00:16:10,535 --> 00:16:15,361 So, this is I guess, lag one, lag two, lag three, lag four. 211 00:16:15,361 --> 00:16:22,485 So for Gaussian white noise, the estimated correlation between xt and xt minus one 212 00:16:22,485 --> 00:16:27,222 is, is actually zero. It seems, you're just, because you're not 213 00:16:27,222 --> 00:16:31,603 seeing here. The estimated correlation between xt and 214 00:16:31,603 --> 00:16:37,866 xt minus two is a small negative number. So the scale here this is minus 0.15. 215 00:16:37,866 --> 00:16:44,045 So this value is you know like minus 0.05. So this is a very small number. 216 00:16:44,045 --> 00:16:48,486 Now on these graphs are blue dotted lines, okay. 217 00:16:48,486 --> 00:16:54,782 These blue dotted lines are thresholds to determine whether or not these values are 218 00:16:54,782 --> 00:16:57,643 statistically different from zero. Alright. 219 00:16:57,643 --> 00:17:03,026 After the midterm I'll explain where these blue dotted lines come from. 220 00:17:03,026 --> 00:17:08,986 They're essentially based upon 95 percent confidence interval for these estimates. 221 00:17:08,986 --> 00:17:15,129 So when you look at this graph, how you're supposed to read it, is if any of the 222 00:17:15,129 --> 00:17:20,742 estimated auto correlations extend beyond the blue line, then those auto 223 00:17:20,742 --> 00:17:25,727 correlations are thought to be statistically different from zero at the 224 00:17:25,727 --> 00:17:30,016 at with, with 95 percent confidence. And so for the Gaussian white noise, we 225 00:17:30,016 --> 00:17:34,291 see none of these auto- correlations lie outside of the blue dotted lines. 226 00:17:34,291 --> 00:17:37,557 So we see no evidence of time dependence in the data. 227 00:17:37,557 --> 00:17:41,339 And again, that should be the case. Because the Gaussian white noise is 228 00:17:41,339 --> 00:17:44,787 computer simulated with no time dependence in the data. 229 00:17:44,787 --> 00:17:49,648 On the other hand, if we look at the Microsoft returns, we see that the first 230 00:17:49,648 --> 00:17:54,483 return is negative, and it's about -.2. So this is the correlation between the 231 00:17:54,483 --> 00:17:57,587 return in month T and the return in month T -one. 232 00:17:57,587 --> 00:18:02,342 And notice that it's negative. And it extends beyond, beyond the blue 233 00:18:02,342 --> 00:18:05,235 dotted line. So we would view this as being 234 00:18:05,235 --> 00:18:09,290 statistically significant. So there appears to be a negative 235 00:18:09,290 --> 00:18:13,740 correlation between the return this month and the return last month. 236 00:18:13,740 --> 00:18:16,115 Okay? Now, the other auto-correlations. 237 00:18:16,115 --> 00:18:20,595 So this is the, return, the autocorrelation between the return at 238 00:18:20,595 --> 00:18:23,658 month T and the return at month T minus two. 239 00:18:23,658 --> 00:18:29,202 So this is the lag two autocorrelation, the lag three autocorrelation and so on. 240 00:18:29,202 --> 00:18:34,838 Here, notice that the lag two is not beyond the blue dotted line but the lag 241 00:18:34,838 --> 00:18:39,943 three is slightly positive, 'kay. So if we run through the Microsoft data 242 00:18:39,943 --> 00:18:45,158 and we view this graph we would say there appears to be some evidence for time 243 00:18:45,158 --> 00:18:50,146 dependence in the data, you know. But if we, then, if you look at the S and 244 00:18:50,146 --> 00:18:52,542 P 500 data. We see none of the sample 245 00:18:52,542 --> 00:18:55,710 auto-correlations are outside of the blue dotted line. 246 00:18:55,710 --> 00:19:01,885 So for the S and P 500 data, there appears to be no evidence for time-dependence in, 247 00:19:01,885 --> 00:19:05,154 in the date. In statistical analysis you should never 248 00:19:05,154 --> 00:19:09,890 say there is, right, because you're, in statistics you're never certain, you're 249 00:19:09,890 --> 00:19:12,420 never 100 percent certain of anything. Right? 250 00:19:12,420 --> 00:19:17,388 All you can say, and you want to think of yourself as, like, sitting on the jury 251 00:19:17,388 --> 00:19:21,209 evaluating evidence, right? Is the evidence in favor of, or is the 252 00:19:21,209 --> 00:19:25,675 evidence against, something, right? So you would interpret this as, there's 253 00:19:25,675 --> 00:19:30,564 data evidence in favor of some time dependence in the Microsoft returns. 254 00:19:30,564 --> 00:19:35,360 When we look at the Microsoft data, just how do you interpret this negative 255 00:19:35,360 --> 00:19:38,316 correlation? And it's literally saying that, if the 256 00:19:38,316 --> 00:19:43,063 return this month is positive. Then, there's a tendency for the return 257 00:19:43,063 --> 00:19:46,981 next month to be negative. Right, so there's kinda of a reversal 258 00:19:46,981 --> 00:19:49,007 that's going on here. >> And vice versa. 259 00:19:49,007 --> 00:19:53,076 >> And vice versa, and if the return is negative this month there is a tendency 260 00:19:53,076 --> 00:19:57,726 for the return to be positive next month. Right, so, one could ask the question, 261 00:19:57,726 --> 00:20:03,075 what could be causing such a reversal? Now, in finance there is some explanation 262 00:20:03,075 --> 00:20:09,032 for the reversal effect in asset returns that's known as the bid-ass bounce. 263 00:20:09,054 --> 00:20:12,085 I'll come back to that a little bit later on. 264 00:20:13,007 --> 00:20:18,087 But that's, that's one explanation. It's, it's not a very good explanation for 265 00:20:18,087 --> 00:20:22,083 monthly data. It's, it's a better explanation if, if you 266 00:20:22,083 --> 00:20:26,036 have say, asset returns computed every minute. 267 00:20:26,058 --> 00:20:30,032 But for monthly data, it's a bit of a mystery. 268 00:20:30,032 --> 00:20:34,000 You know, why is this negative correlation here? 269 00:20:36,055 --> 00:20:43,063 Okay, so to summarize, I like to call these stylized facts. 270 00:20:43,063 --> 00:20:47,010 So in, in this section we looked at three assets. 271 00:20:47,010 --> 00:20:52,073 We looked at Microsoft, the S and P five, actually we only looked at two assets. 272 00:20:52,073 --> 00:20:56,077 We looked at Microsoft and we looked at the S and P 500. 273 00:20:56,077 --> 00:21:02,077 And of course, one is always tempted to try to make sweeping generalizations based 274 00:21:02,077 --> 00:21:07,088 upon, or, maybe I should rephrase this. You should be careful not to make sweeping 275 00:21:07,088 --> 00:21:11,019 generalizations based on the analysis of two assets. 276 00:21:11,019 --> 00:21:16,396 But, you know, I've, I've analyzed a lot more than two assets, and others have 277 00:21:16,396 --> 00:21:21,040 analyzed a lot more than two assets. And, so when you look at monthly 278 00:21:21,040 --> 00:21:27,002 continuously compounded returns, these are the results that you tend to find when you 279 00:21:27,002 --> 00:21:30,084 look at, you know, lots and lots and lots of different assets. 280 00:21:30,084 --> 00:21:36,060 At the monthly basis, returns appear to be approximately normally distributed, okay? 281 00:21:36,060 --> 00:21:42,307 They don't follow the normal distribution exactly, But the normal distribution is 282 00:21:42,307 --> 00:21:44,723 not a horrible mistake. All right? 283 00:21:44,723 --> 00:21:50,187 There's some noticeable negative skewness and excess kurtosis in typical assets, 284 00:21:50,187 --> 00:21:53,079 right? If you look at bivariate relationships, 285 00:21:53,079 --> 00:21:58,091 many assets are contemporaneously correlated, that is, and they tend to be 286 00:21:58,091 --> 00:22:04,002 positively correlated. And so, and that's one of the things as 287 00:22:04,002 --> 00:22:08,015 well, if you look at temporal dependence in the data. 288 00:22:08,015 --> 00:22:14,027 At the monthly level there is not a whole lot of evidence for strong temporal 289 00:22:14,027 --> 00:22:19,018 dependence, so assets are approximately uncorrelated over time. 290 00:22:19,041 --> 00:22:25,053 Then again when you look at many assets, the Microsoft data showed there was some 291 00:22:25,053 --> 00:22:31,057 negative dependence on one lag, but when you looked at, you know, other lags there 292 00:22:31,057 --> 00:22:36,071 was not much that was there. So these are broad, what I call stylized 293 00:22:36,071 --> 00:22:41,456 facts and you want to keep these sort of in the back of your mind. 294 00:22:41,456 --> 00:22:45,402 Stylized facts are useful at the model building phase. 295 00:22:45,402 --> 00:22:50,409 So if you want to build a probability model for asset returns, we want our 296 00:22:50,409 --> 00:22:54,856 probability model to capture the basic stylized facts of the data, okay. 297 00:22:54,856 --> 00:23:00,378 If our model doesn't capture the basic stylized facts, it's not a good model. 298 00:23:00,378 --> 00:23:07,518 And we're always gonna keep that in mind as we look at particular models in this 299 00:23:07,518 --> 00:23:11,108 class. We always wanna ask ourselves, you know, 300 00:23:11,108 --> 00:23:16,510 if we you know, simulate data from our model, does it look like real data?