1 00:00:00,000 --> 00:00:04,612 So, we did a bunch of stuff on estimation of parameters in the constant expected 2 00:00:04,612 --> 00:00:07,698 return model. And, so estimation is one part of doing 3 00:00:07,698 --> 00:00:11,238 statistical analysis. We have a model, it has parameters. 4 00:00:11,238 --> 00:00:16,764 You need to estimate those parameters from data, and so went over a pretty extensive 5 00:00:16,764 --> 00:00:21,485 review of the statistical theory of estimation, how to evaluate estimates, how 6 00:00:21,485 --> 00:00:26,548 to compute standard errors, how to use the bootstrap to get standard errors, how to 7 00:00:26,548 --> 00:00:30,172 compute confidence intervals. Now, we're going to look at doing 8 00:00:30,172 --> 00:00:36,064 statistical inference based the model. So, and that, that's the, the name of 9 00:00:36,064 --> 00:00:40,116 hypothesis testing. And, again, to briefly review, when you do 10 00:00:40,116 --> 00:00:45,443 hypothesis testing, there's some maintain view of that, that you want to test. 11 00:00:45,443 --> 00:00:49,930 And, in you know, with respected constant expected return model, we might want to 12 00:00:49,930 --> 00:00:52,805 test hypothesis about certain values of the parameters. 13 00:00:52,805 --> 00:00:58,155 We want to, we want to test hypothesis that the, the expected returns for all the 14 00:00:58,155 --> 00:01:02,539 assets are greater than zero. We might want to test hypothesis that the 15 00:01:02,539 --> 00:01:06,269 correlation between, you know, two pairs of stocks is the same. 16 00:01:06,460 --> 00:01:11,797 Or we might want to test hypothesis that returns really are normally distributed. 17 00:01:12,002 --> 00:01:16,819 Or we could be interested in the hypothesis that the correlation between 18 00:01:16,819 --> 00:01:20,006 two stocks is constant over the entire data sample. 19 00:01:20,006 --> 00:01:25,010 So, there are many different kinds of hypothesis that we could be testing in a 20 00:01:25,010 --> 00:01:28,246 given context. And so, you know, the null hypothesis is 21 00:01:28,246 --> 00:01:32,421 what's being maintained, you know? The correlations are the same between two 22 00:01:32,421 --> 00:01:36,142 pairs of assets, for example. And then, the alternative hypothesis says, 23 00:01:36,142 --> 00:01:38,984 you know, what, what's the alternative view of the world? 24 00:01:38,984 --> 00:01:43,974 For example, the correlations between two pairs of, of returns are not the same. 25 00:01:43,974 --> 00:01:48,450 And, when you do hypothesis testing, you have to make a decision. 26 00:01:48,450 --> 00:01:53,276 Do you reject the null hypothesis or do you not reject the null hypothesis. 27 00:01:53,276 --> 00:01:59,211 And, when we do testing, we specify the level of the test which is the probability 28 00:01:59,211 --> 00:02:04,034 that you reject the null hypothesis given the null hypothesis is true. 29 00:02:04,034 --> 00:02:07,987 And usually, this is some small number like one percent or five%. 30 00:02:08,003 --> 00:02:13,018 And it's the, again, it's the probability of making a particular type of error. 31 00:02:13,314 --> 00:02:18,093 Then you can sort the test statistic from the data. 32 00:02:18,093 --> 00:02:24,075 And, the test statistic, you know, again, summarizes the data evidence in favor or 33 00:02:24,075 --> 00:02:29,020 against the null hypothesis. And typically, when you do hypothesis 34 00:02:29,020 --> 00:02:34,060 testing, if your test statistic is big, that's data evidence against the null 35 00:02:34,060 --> 00:02:40,014 hypothesis and that's indicating that the decision you should make is to reject it. 36 00:02:40,014 --> 00:02:45,048 If the test statistic is small, then, that's data evidence in favor of the null 37 00:02:45,048 --> 00:02:49,052 hypothesis and, and you shouldn't reject in, in that case. 38 00:02:50,084 --> 00:02:56,063 And then, based on the magnitude of the test statistic, you have to make a 39 00:02:56,063 --> 00:02:59,081 decision. And the formal theory of hypotheses 40 00:02:59,081 --> 00:03:05,040 testing, we usually have values of the test statistic that lie in the rejection 41 00:03:05,040 --> 00:03:07,246 region. So, if the test statistic isn't big 42 00:03:07,246 --> 00:03:12,526 enough, then you should reject. And so, the rejection region is a cut off 43 00:03:12,526 --> 00:03:17,620 point typically of how big the test statistic needs to be in order for you to 44 00:03:17,620 --> 00:03:20,064 reject. And this cut-off region is typically 45 00:03:20,064 --> 00:03:24,524 called a critical value, and the critical value has the property such that if say, 46 00:03:24,524 --> 00:03:28,968 the absolute value of the test statistic is bigger than the critical value you 47 00:03:28,968 --> 00:03:32,967 reject, but if the absolute value of the test statistic is less than the critical 48 00:03:32,967 --> 00:03:37,072 value you don't reject. Now, where this critical value comes from 49 00:03:37,072 --> 00:03:42,500 depends upon the test statistic that you use and the hypothesis that you're 50 00:03:42,500 --> 00:03:46,078 testing. And, so we'll go through several examples 51 00:03:46,078 --> 00:03:55,089 to illustrate how this works. Now, in any decision making context and a 52 00:03:55,089 --> 00:04:00,171 hypothesis testing context is a decision making context. 53 00:04:00,171 --> 00:04:04,452 So, there are two decisions that you can make. 54 00:04:04,452 --> 00:04:10,728 You either reject the null hypothesis or you don't reject the null hypothesis, 55 00:04:10,728 --> 00:04:13,445 okay? And, there, there is reality, what's 56 00:04:13,445 --> 00:04:18,317 actually true in the world. What's true is, the null hypothesis, it, I 57 00:04:18,317 --> 00:04:23,014 mean, it could be the state of the world that the null hypothesis that you're 58 00:04:23,014 --> 00:04:26,070 testing is really true. But, it could also be that the null 59 00:04:26,070 --> 00:04:32,002 hypothesis you're testing is really false. And again, I'd like to think of, you know, 60 00:04:32,002 --> 00:04:37,139 the criminal justice system example. You're sitting on a jury, you're trying to 61 00:04:37,139 --> 00:04:41,550 decide, you know, the fate of, you know, someone's on trial for murder. 62 00:04:41,766 --> 00:04:47,334 You have to decide whether or not, and, and then the maintain hypothesis that you 63 00:04:47,334 --> 00:04:51,723 are innocent until proven guilty. So, the null hypothesis is this, this 64 00:04:51,723 --> 00:04:55,702 person is innocent. And, the decision you're going to make are 65 00:04:55,702 --> 00:04:59,535 you reject the null hypothesis. So, you reject that the person is 66 00:04:59,535 --> 00:05:04,083 innocent, you conclude that they're guilty, or you decide not to reject the 67 00:05:04,083 --> 00:05:07,746 null hypothesis. That is, you, you conclude this person 68 00:05:07,746 --> 00:05:11,396 really is innocent. Now, in reality, this person is a killer 69 00:05:11,396 --> 00:05:15,173 or he's not a killer, right? So, in reality, you know, the person is 70 00:05:15,173 --> 00:05:18,041 really innocent. And then, if you reject the null 71 00:05:18,041 --> 00:05:22,506 hypothesis when the person is really is, then you make what's called a type one 72 00:05:22,506 --> 00:05:25,075 error, okay? So, that's sending the innocent person to 73 00:05:25,075 --> 00:05:28,002 the gas chamber. That's a bad thing to do. 74 00:05:28,174 --> 00:05:32,027 On the other hand, if the person is innocent and you don't reject the null 75 00:05:32,027 --> 00:05:36,092 hypothesis, you conclude that the innocent person is innocent and the person gets to 76 00:05:36,092 --> 00:05:39,030 go free, and you've made the right decision. 77 00:05:39,030 --> 00:05:44,036 Now, the other state of the world could be is that the null hypothesis is false, so 78 00:05:44,036 --> 00:05:49,041 that the guy's really a killer. And, you reject the null hypothesis, so 79 00:05:49,041 --> 00:05:54,535 you found that the killer is guilty. So, okay, you've, you've sent the killer 80 00:05:54,535 --> 00:05:58,734 to the gas chamber. And depending upon your politics that 81 00:05:58,932 --> 00:06:03,109 could be no error. And the other situation is for the, for 82 00:06:03,109 --> 00:06:06,818 the guilty person. You don't reject the null hypothesis, so 83 00:06:06,818 --> 00:06:10,359 you found out that the guilty person is really innocent. 84 00:06:10,359 --> 00:06:14,823 So, you've decided this person innocent when they're in fact, guilty, that's 85 00:06:14,823 --> 00:06:18,616 called a type two error. So, this is the error where you allow the 86 00:06:18,616 --> 00:06:22,946 serial killer back out into the public. And, that's a bad thing to do. 87 00:06:22,946 --> 00:06:28,322 And you would like to make sure that you don't make those kinds of errors. 88 00:06:28,322 --> 00:06:33,973 Now, when you do hypothesis testing, the level of the test is the probability of 89 00:06:33,973 --> 00:06:38,667 type one error. It's the probability that you reject an 90 00:06:38,667 --> 00:06:43,839 hypothesis when it's true, okay? So again, this is the, the probability of, 91 00:06:43,839 --> 00:06:47,951 of find, finding that an innocent person is, is guilty. 92 00:06:47,951 --> 00:06:53,560 And, the goal of hypothesis testing is, is to make the level of the test small. 93 00:06:53,560 --> 00:06:56,589 So, you'd like to minimize this type of level. 94 00:06:56,589 --> 00:07:02,862 And so, we usually set our test such that we have a pre-specified probability of 95 00:07:02,862 --> 00:07:06,932 committing this kind of error, five%, one percent are typical values. 96 00:07:06,932 --> 00:07:13,251 Now, something called the power of the test is one minus the probability of type 97 00:07:13,251 --> 00:07:15,972 two error. So, the power of the test is the 98 00:07:15,972 --> 00:07:20,672 probability of rejecting the null hypothesis, given the null hypothesis is 99 00:07:20,672 --> 00:07:23,968 false. So, you'd like to have a test to be able 100 00:07:23,968 --> 00:07:27,337 to determine if the null hypothesis isn't true. 101 00:07:27,337 --> 00:07:32,375 So, in the courtroom example, you know, a legal system that has good power is a 102 00:07:32,375 --> 00:07:35,585 legal system that convicts guilty people, right? 103 00:07:35,585 --> 00:07:42,127 And, a good legal system is a system that does not convict innocent people. 104 00:07:42,127 --> 00:07:50,026 Now, there's a problem in the statistical testing context is that it's impossible to 105 00:07:50,026 --> 00:07:56,062 simultaneously have the level of test approximately zero, and have power close 106 00:07:56,062 --> 00:08:01,102 to 100%, okay? Because if you make the level of test go 107 00:08:01,102 --> 00:08:08,305 down to zero, that is, if you never falsely, you know, convict somebody, then, 108 00:08:08,305 --> 00:08:12,598 in, in some respects, you're making a decision that you're never going to 109 00:08:12,598 --> 00:08:18,055 convict anybody, which means that this probability would get close to zero. 110 00:08:18,275 --> 00:08:24,065 And similarly, if you want to have high power, then you tend to push up the, the 111 00:08:24,065 --> 00:08:28,309 probability of type one error. If you want to, you know, increase, you 112 00:08:28,309 --> 00:08:34,085 know, try to increase this probability of rejecting null hypothesis when the null 113 00:08:34,085 --> 00:08:41,033 hypothesis is false, you know, then, then you tend to also commit higher probability 114 00:08:41,033 --> 00:08:45,390 of type one error. So, these, these, two properties of test 115 00:08:45,390 --> 00:08:50,018 have conflicting views. And the goal in, in hypothesis testing is 116 00:08:50,018 --> 00:08:55,033 given a probability of type one error construct a test that has the higher 117 00:08:55,033 --> 00:08:58,076 power. And so, the kinds of test statistics that 118 00:08:58,076 --> 00:09:04,019 we're going to be looking at, have the property that they tend to have you know, 119 00:09:04,019 --> 00:09:08,065 high power in a decision context.