1 00:00:01,960 --> 00:00:04,460 - [Lecturer] Hello and welcome to this video lecture 2 00:00:04,460 --> 00:00:08,673 on inference and hypothesis testing. 3 00:00:08,673 --> 00:00:13,390 So today, we're gonna be talking 4 00:00:13,390 --> 00:00:16,270 about how to do hypothesis tests. 5 00:00:16,270 --> 00:00:19,140 And specifically, how do we know 6 00:00:19,140 --> 00:00:24,140 if one of the beta hats that we estimate, 7 00:00:25,720 --> 00:00:30,010 or one or more of them are statistically significant? 8 00:00:30,010 --> 00:00:35,010 So can we say with confidence that given our data, 9 00:00:38,070 --> 00:00:42,500 the beta hat has a real impact on Y. 10 00:00:43,670 --> 00:00:47,810 Does a change in X actually have a change in Y? 11 00:00:47,810 --> 00:00:52,810 And how sure are we about what we are saying? 12 00:00:57,200 --> 00:01:00,213 So again, we're talking about hypothesis tests. 13 00:01:03,320 --> 00:01:07,210 So far, we've been talking about our beta hat, 14 00:01:07,210 --> 00:01:11,830 the OLS estimator, or beta hat OLS, more specifically. 15 00:01:11,830 --> 00:01:15,460 And we learned that if these five assumptions hold, 16 00:01:15,460 --> 00:01:19,990 that OLS is BLUE, it's the best linear unbiased estimator. 17 00:01:19,990 --> 00:01:24,190 And we talked a bunch about under and over-specification, 18 00:01:24,190 --> 00:01:29,190 and variance and variance inflation 19 00:01:29,870 --> 00:01:34,230 by adding an irrelevant variable, 20 00:01:34,230 --> 00:01:39,230 and the bias that ensues by excluding a relevant one. 21 00:01:40,940 --> 00:01:43,700 And now we're gonna look at hypothesis tests. 22 00:01:43,700 --> 00:01:48,200 So what do our data say about the statistical significance 23 00:01:48,200 --> 00:01:53,200 of the betas that we have estimated? 24 00:01:53,550 --> 00:01:58,550 And for this, we need to add a sixth assumption. 25 00:01:59,670 --> 00:02:03,990 So so far, the five that we have learned about 26 00:02:03,990 --> 00:02:08,100 and assumed so far lead to OLS is BLUE. 27 00:02:08,100 --> 00:02:11,010 Now we're going to need a sixth one 28 00:02:11,010 --> 00:02:13,550 to be able to do hypothesis test, 29 00:02:13,550 --> 00:02:18,550 and that is the specific distribution of the error terms. 30 00:02:19,490 --> 00:02:24,310 So we know that its expected value is zero, 31 00:02:24,310 --> 00:02:27,260 and its variance is a constant, 32 00:02:27,260 --> 00:02:31,040 both of which are independent of any value of X. 33 00:02:31,040 --> 00:02:32,650 But we haven't really talked 34 00:02:32,650 --> 00:02:35,620 about the distribution of the error. 35 00:02:35,620 --> 00:02:39,350 So what is sort of the exact shape 36 00:02:39,350 --> 00:02:42,573 of the distribution curve? 37 00:02:44,560 --> 00:02:48,100 So now we're going to add a sixth assumption, 38 00:02:48,100 --> 00:02:50,920 which is the normality of the error. 39 00:02:50,920 --> 00:02:54,310 So we assume that our u's, 40 00:02:54,310 --> 00:02:57,320 our disturbances, our error terms, 41 00:02:57,320 --> 00:03:01,140 have a known distribution. 42 00:03:01,140 --> 00:03:04,900 That we know the basic shape of the bell curve specifically, 43 00:03:04,900 --> 00:03:08,410 that they have a normal distribution. 44 00:03:08,410 --> 00:03:11,070 And the easiest way to think about this 45 00:03:11,070 --> 00:03:13,470 is with a bell curve. 46 00:03:13,470 --> 00:03:18,470 That the shape of the distribution is a bell curve. 47 00:03:19,540 --> 00:03:23,860 So here is a bell curve. 48 00:03:23,860 --> 00:03:27,023 You could see that it's centered on zero. 49 00:03:28,560 --> 00:03:32,150 And so the y-axis here is, 50 00:03:32,150 --> 00:03:37,150 how likely are you going to get that value? 51 00:03:38,260 --> 00:03:41,990 And then the x-axis are the various values. 52 00:03:41,990 --> 00:03:46,738 You can see that in a bell curve that it's most likely 53 00:03:46,738 --> 00:03:49,870 that the closer to zero that you get, 54 00:03:49,870 --> 00:03:53,900 the more likely that those are the error terms. 55 00:03:53,900 --> 00:03:57,050 And the farther away from zero that you get, 56 00:03:57,050 --> 00:03:59,860 the less likely that you're going to get 57 00:03:59,860 --> 00:04:02,363 that value for your error terms. 58 00:04:07,220 --> 00:04:09,160 This is another way of thinking about 59 00:04:09,160 --> 00:04:10,130 what I've been talking about, 60 00:04:10,130 --> 00:04:13,903 the bingo ball distribution. 61 00:04:14,740 --> 00:04:19,740 There's a whole lot of values at or near zero. 62 00:04:20,460 --> 00:04:23,713 And the farther away from zero that you get, 63 00:04:24,670 --> 00:04:28,433 the fewer values of that bingo ball you will have. 64 00:04:32,140 --> 00:04:36,910 So here is the sixth assumption, that in the population, 65 00:04:36,910 --> 00:04:41,910 that the error is not only independent of X1 through Xk, 66 00:04:44,720 --> 00:04:48,940 but it has this normal distribution with a mean zero 67 00:04:48,940 --> 00:04:51,700 and a constant variance sigma squared. 68 00:04:51,700 --> 00:04:53,380 And we denote it like this. 69 00:04:53,380 --> 00:04:58,380 u is distributed in a normal distribution 70 00:04:59,280 --> 00:05:02,800 with mean zero, variance sigma squared. 71 00:05:02,800 --> 00:05:04,473 So that's how we would write it. 72 00:05:07,790 --> 00:05:10,040 Note that this is a fairly strong assumption. 73 00:05:11,960 --> 00:05:16,960 The parts of mean zero and variance of sigma squared, 74 00:05:19,430 --> 00:05:21,020 we already assume that. 75 00:05:21,020 --> 00:05:25,130 That's the assumptions above, 76 00:05:25,130 --> 00:05:27,453 so that really doesn't say much. 77 00:05:30,050 --> 00:05:35,050 But that, to assume a specific distribution is 78 00:05:37,700 --> 00:05:39,460 a fairly strong assumption. 79 00:05:39,460 --> 00:05:42,000 But if we do assume that together, 80 00:05:42,000 --> 00:05:46,470 one through six, that these are often lumped together 81 00:05:46,470 --> 00:05:51,470 and called CLR, classic linear regression assumptions. 82 00:05:57,300 --> 00:06:00,350 Note as well that we can make one more leap, 83 00:06:00,350 --> 00:06:02,910 that if all six of them are true, 84 00:06:02,910 --> 00:06:07,910 that OLS is not only BLUE, it's BEUE. 85 00:06:08,130 --> 00:06:13,130 It is the most efficient, unbiased estimator of any. 86 00:06:15,520 --> 00:06:20,200 And another way that we can say that is that Y, 87 00:06:20,200 --> 00:06:23,330 given all the values of X, 88 00:06:23,330 --> 00:06:25,593 has a distribution 89 00:06:30,100 --> 00:06:35,100 of mean of beta 0 plus beta 1 X1 plus dot dot dot beta k Xk, 90 00:06:37,940 --> 00:06:40,723 and a variance of sigma squared. 91 00:06:48,220 --> 00:06:52,440 So if we think about, how robust is this assumption? 92 00:06:52,440 --> 00:06:55,130 How certain can we be? 93 00:06:55,130 --> 00:06:56,610 Do we feel like we are 94 00:06:56,610 --> 00:07:00,400 on pretty solid ground when we make it? 95 00:07:00,400 --> 00:07:05,400 So the good news is that the central limit theorem says 96 00:07:08,070 --> 00:07:12,920 that as sample size increases, 97 00:07:12,920 --> 00:07:17,920 that the distribution will converge 98 00:07:19,010 --> 00:07:24,010 to a normal distribution, approximately. 99 00:07:24,440 --> 00:07:29,160 And in this class, we basically assume, 100 00:07:29,160 --> 00:07:31,240 and for most applications that you'll do, 101 00:07:31,240 --> 00:07:34,400 that we have a large enough sample 102 00:07:34,400 --> 00:07:37,890 that we can sort of call on the central limit theorem 103 00:07:37,890 --> 00:07:42,890 and make this assumption with a fair amount of certainty. 104 00:07:49,300 --> 00:07:52,240 We're now going to talk about testing 105 00:07:52,240 --> 00:07:55,350 the significance of a single parameter, 106 00:07:55,350 --> 00:08:00,350 a single beta hat that we get from our regression analysis. 107 00:08:01,210 --> 00:08:04,730 So again, we start with our familiar model 108 00:08:04,730 --> 00:08:07,330 with Y on the left side, 109 00:08:07,330 --> 00:08:12,330 and k regressors on the right, and an error term. 110 00:08:13,820 --> 00:08:18,820 And we were assuming that all six of our assumptions hold. 111 00:08:18,860 --> 00:08:23,860 We know that our beta hat OLS is unbiased, 112 00:08:24,020 --> 00:08:28,840 but we can never really know the true population value. 113 00:08:28,840 --> 00:08:31,040 We don't know the real beta. 114 00:08:31,040 --> 00:08:33,840 All we know are our beta hats. 115 00:08:33,840 --> 00:08:37,060 They're unknown and unknowable. 116 00:08:37,060 --> 00:08:42,060 So we wanna know, how sure are we that this beta hat 117 00:08:43,840 --> 00:08:48,413 that we got is close to the true value? 118 00:08:49,410 --> 00:08:54,410 How sure do we need to be in order to say, 119 00:08:54,930 --> 00:08:59,930 yes, this is a true relationship. 120 00:09:01,530 --> 00:09:03,680 And most of the time, as we see, 121 00:09:03,680 --> 00:09:07,750 we're going to test it against a null hypothesis 122 00:09:07,750 --> 00:09:10,993 that this beta equals zero. 123 00:09:16,630 --> 00:09:19,283 So if our assumptions one through six are true, 124 00:09:20,300 --> 00:09:23,253 then we know that the, 125 00:09:25,150 --> 00:09:30,020 we can generate a T-statistic for every beta j. 126 00:09:30,020 --> 00:09:35,020 And this T-statistic is beta j hat minus its true value, 127 00:09:39,780 --> 00:09:44,780 beta j, divided by the standard error of our beta j. 128 00:09:46,770 --> 00:09:49,040 So we're gonna usually test it 129 00:09:49,040 --> 00:09:54,040 against the null hypothesis that beta j equals zero. 130 00:09:54,630 --> 00:09:59,630 So the T-stat that you will get an SPSS is beta hat j 131 00:10:00,850 --> 00:10:04,993 divided by the standard error of beta hat j. 132 00:10:07,990 --> 00:10:12,990 And this has a known probability distribution, 133 00:10:13,720 --> 00:10:18,720 which is a T-distribution, which also has a bell curve. 134 00:10:19,110 --> 00:10:24,110 And note that every T-distribution is specific 135 00:10:24,480 --> 00:10:27,270 for a number of degrees of freedom. 136 00:10:27,270 --> 00:10:32,270 So we're gonna, we know that this has this distribution 137 00:10:37,034 --> 00:10:42,034 for n minus k minus one degrees of freedom. 138 00:10:43,820 --> 00:10:48,820 And basically, any time online or in any book, any textbook, 139 00:10:49,720 --> 00:10:54,720 you can find a table with the T-distribution 140 00:10:56,960 --> 00:11:00,483 with n minus k degrees of freedom. 141 00:11:06,710 --> 00:11:08,230 What this T-test does, 142 00:11:08,230 --> 00:11:12,240 so when we're doing one parameter at a time, 143 00:11:12,240 --> 00:11:14,313 we're you're using T-tests. 144 00:11:16,980 --> 00:11:19,530 And one of the things that we ask is, 145 00:11:19,530 --> 00:11:24,530 how likely are we to get a T-stat of that value? 146 00:11:25,010 --> 00:11:30,010 So recall, our T-stat is the ratio between beta hat j 147 00:11:33,290 --> 00:11:35,923 and the standard error of beta hat j. 148 00:11:37,976 --> 00:11:41,163 And we ask, how likely is it that we would get 149 00:11:42,100 --> 00:11:47,100 that value of a T-stat if the null hypothesis is true? 150 00:11:48,150 --> 00:11:50,010 That this is our p-value. 151 00:11:50,010 --> 00:11:54,193 So if our null hypothesis is true, 152 00:11:55,150 --> 00:12:00,150 beta hat j equals zero. 153 00:12:01,517 --> 00:12:05,780 Now, if it's true, and looking at this bell curve here, 154 00:12:05,780 --> 00:12:07,470 it's pretty darn likely 155 00:12:07,470 --> 00:12:11,113 that we're going to get a very small T value. 156 00:12:14,330 --> 00:12:18,260 Because this curve is centered over zero, 157 00:12:18,260 --> 00:12:23,260 if we get a very large number, either negative or positive, 158 00:12:25,710 --> 00:12:28,920 it's going to be far away from the middle of the bell curve. 159 00:12:28,920 --> 00:12:30,850 It's gonna be way out on the tails. 160 00:12:30,850 --> 00:12:35,850 And it's fairly unlikely that we would get that test stat 161 00:12:36,060 --> 00:12:39,313 if the null hypothesis is true. 162 00:12:44,210 --> 00:12:46,850 I'm going to walk you through how to do 163 00:12:46,850 --> 00:12:51,000 a T-test for a single variable. 164 00:12:51,000 --> 00:12:56,000 So we wanna look specifically if a beta hat j 165 00:12:57,610 --> 00:13:01,020 that we get from our analysis is 166 00:13:01,020 --> 00:13:03,853 statistically different from zero. 167 00:13:04,960 --> 00:13:07,630 So as with any hypothesis tests, 168 00:13:07,630 --> 00:13:09,810 we start with a null hypothesis. 169 00:13:09,810 --> 00:13:12,000 Most commonly, again, 170 00:13:12,000 --> 00:13:16,750 we start with our null that beta j equals zero. 171 00:13:16,750 --> 00:13:20,010 Beta j equals zero, if that is true, 172 00:13:20,010 --> 00:13:23,710 that means that the slope of the line is zero. 173 00:13:23,710 --> 00:13:25,940 That it is a horizontal line. 174 00:13:25,940 --> 00:13:30,940 That no matter, that no change in X has any effect on Y. 175 00:13:31,130 --> 00:13:33,560 So to think of our example, 176 00:13:33,560 --> 00:13:37,380 that no change in income has any effect 177 00:13:37,380 --> 00:13:39,690 on local food expenditure. 178 00:13:39,690 --> 00:13:43,500 That changing one's income has no effect 179 00:13:43,500 --> 00:13:46,023 on how much they spend on food. 180 00:13:48,570 --> 00:13:53,263 And we're looking at one at a time, so one of our k betas. 181 00:13:54,570 --> 00:13:56,830 And again, this is a test. 182 00:13:56,830 --> 00:14:00,280 Once all the other X's have been accounted for, 183 00:14:00,280 --> 00:14:03,000 that this Xj has no effect on Y. 184 00:14:03,000 --> 00:14:08,000 So if the null hypothesis is true, Xj has no effect on Y. 185 00:14:16,630 --> 00:14:20,897 When we do SPSS and do a regression, we get a T-stat. 186 00:14:20,897 --> 00:14:25,510 And this T-stat is the ratio of beta hat j 187 00:14:25,510 --> 00:14:29,930 divided by the standard error of beta hat j. 188 00:14:29,930 --> 00:14:33,730 It is its estimated mean 189 00:14:33,730 --> 00:14:37,730 to its estimated standard error. 190 00:14:37,730 --> 00:14:42,730 It is its magnitude divided by its precision. 191 00:14:43,240 --> 00:14:48,240 And since the standard error is always greater than zero, 192 00:14:48,960 --> 00:14:52,760 the T-ratio has the same sign as B. 193 00:14:52,760 --> 00:14:55,220 And a big beta hat 194 00:14:55,220 --> 00:14:59,390 and a small standard error leads to a big T-stat. 195 00:14:59,390 --> 00:15:04,390 So if beta j hat is far away from zero, 196 00:15:05,970 --> 00:15:08,490 and if it's a very precise estimate 197 00:15:08,490 --> 00:15:12,350 along a tall, skinny bell curve, 198 00:15:12,350 --> 00:15:14,973 then we will have a big T-stat. 199 00:15:21,570 --> 00:15:26,570 The beta hat j that we get out of SPSS will rarely, 200 00:15:27,310 --> 00:15:32,310 if ever, be exactly zero, but it might be close to zero. 201 00:15:33,320 --> 00:15:37,130 So basically, what this is asking is, 202 00:15:37,130 --> 00:15:42,130 how likely is it that this beta j hat is far away from zero? 203 00:15:45,720 --> 00:15:48,433 How sure are we that it's far away? 204 00:15:49,280 --> 00:15:54,280 And how sure must we be to reject our null hypothesis? 205 00:15:56,810 --> 00:16:01,430 So to do a test, then we need an alternative hypothesis, 206 00:16:01,430 --> 00:16:02,913 which we will call H1. 207 00:16:07,600 --> 00:16:09,963 We're gonna start with one-sided test. 208 00:16:11,380 --> 00:16:13,450 And not surprisingly, 209 00:16:13,450 --> 00:16:15,950 we're gonna do two-sided test next, 210 00:16:15,950 --> 00:16:17,813 but first, let's do one-sided test. 211 00:16:20,140 --> 00:16:22,240 In a one-sided test, 212 00:16:22,240 --> 00:16:27,240 we say that beta j is strictly greater than zero. 213 00:16:28,500 --> 00:16:32,940 And in this case, we do a one-sided test 214 00:16:32,940 --> 00:16:37,460 if we can assume that beta j less than zero 215 00:16:37,460 --> 00:16:39,020 doesn't make economic sense, 216 00:16:39,020 --> 00:16:41,540 that it's not feasible that you would get this. 217 00:16:41,540 --> 00:16:43,760 It might be in a, there's a lot of examples, 218 00:16:43,760 --> 00:16:48,760 but if some firm never demotes or never gives a pay cut, 219 00:16:50,300 --> 00:16:54,423 then salary will always increase. 220 00:17:00,790 --> 00:17:05,790 The next thing that we need is, how sure do we have to be? 221 00:17:06,320 --> 00:17:11,320 How certain do we have to be to reject the null? 222 00:17:13,140 --> 00:17:15,973 Usually, we choose 5%, 223 00:17:17,680 --> 00:17:22,680 or in some cases, we sort of bat around 1%, 5%, and 10%. 224 00:17:24,990 --> 00:17:26,240 But for right now, 225 00:17:26,240 --> 00:17:29,530 let's assume that we are going to choose 226 00:17:29,530 --> 00:17:34,530 that if there's a 5% chance of a type one error, 227 00:17:39,930 --> 00:17:42,960 anything less than that is good enough. 228 00:17:42,960 --> 00:17:46,900 That for 95% sure or more, 229 00:17:46,900 --> 00:17:49,210 we are sufficiently certain 230 00:17:49,210 --> 00:17:54,210 that we are going to think this is a real relationship. 231 00:17:58,160 --> 00:18:00,950 And what we do is we look at, 232 00:18:00,950 --> 00:18:05,950 we look for a sufficiently large T-stat or T-value 233 00:18:06,010 --> 00:18:10,033 that we can confidently reject this null. 234 00:18:11,147 --> 00:18:16,147 And we call this c, c for, I think, critical value. 235 00:18:19,470 --> 00:18:24,470 So we call c the critical value at the 5% significance value 236 00:18:28,610 --> 00:18:33,610 as the 95th percentile in the T-distribution 237 00:18:33,660 --> 00:18:36,353 with n minus k minus one degrees of freedom. 238 00:18:37,760 --> 00:18:39,440 Another way of thinking of this is, 239 00:18:39,440 --> 00:18:44,440 it's the area under the, it's the part of the curve, 240 00:18:44,610 --> 00:18:46,740 the very tail of the curve 241 00:18:46,740 --> 00:18:51,250 that only has 5% of the area under it. 242 00:18:51,250 --> 00:18:52,290 So in this bell curve, 243 00:18:52,290 --> 00:18:55,340 it's way out on the tail of the curve, 244 00:18:55,340 --> 00:18:59,240 such that only 5% of the area 245 00:18:59,240 --> 00:19:01,003 of that whole curve is under it. 246 00:19:02,470 --> 00:19:07,470 So if our T-stat is greater than c, we reject the null. 247 00:19:12,170 --> 00:19:17,170 Remember, a big T-stat means that beta j hat is big, 248 00:19:17,287 --> 00:19:19,560 and the standard error is small. 249 00:19:19,560 --> 00:19:24,560 If it's less than that critical value, we fail to reject. 250 00:19:27,660 --> 00:19:30,773 Note that the smaller the significance level, 251 00:19:32,040 --> 00:19:34,907 the smaller the critical value, 252 00:19:34,907 --> 00:19:38,110 and the more likely we are to reject it. 253 00:19:38,110 --> 00:19:42,840 So if we reject at the 10% significance value, 254 00:19:42,840 --> 00:19:47,393 we always fail to reject as well at the 1%. 255 00:19:53,310 --> 00:19:56,420 And since we're doing a one-sided test, 256 00:19:56,420 --> 00:20:01,420 any negative T-value will lead to fail to reject the null. 257 00:20:04,490 --> 00:20:09,320 And the rule for our null hypothesis 258 00:20:09,320 --> 00:20:13,280 of beta j being less than zero is the same, 259 00:20:13,280 --> 00:20:17,023 except it's a large negative value, 260 00:20:18,654 --> 00:20:21,640 greater in absolute value than our c. 261 00:20:21,640 --> 00:20:26,640 So way, way out on the left-hand tail of the curve. 262 00:20:32,780 --> 00:20:35,483 More commonly, we do a two-sided test, 263 00:20:36,570 --> 00:20:41,570 where we're honestly not sure what the direction might be. 264 00:20:45,100 --> 00:20:47,440 So an example I thought of was, 265 00:20:47,440 --> 00:20:51,390 does the number of annual trips impact 266 00:20:51,390 --> 00:20:53,810 the total weight of trash? 267 00:20:53,810 --> 00:20:58,810 So if you drive your trash to the dump, 268 00:20:59,440 --> 00:21:04,430 does the annual number of trips impact 269 00:21:07,350 --> 00:21:10,033 how much total weight of trash that you dump? 270 00:21:11,580 --> 00:21:14,250 I think it's fair to say, we wouldn't be sure. 271 00:21:14,250 --> 00:21:16,800 Some people might make a lot of small runs. 272 00:21:16,800 --> 00:21:20,843 Some people might wanna make a few large runs. 273 00:21:20,843 --> 00:21:24,533 It depends on a lot of other factors. 274 00:21:26,710 --> 00:21:30,490 But we have our same null hypothesis, 275 00:21:30,490 --> 00:21:33,430 beta j equals zero. 276 00:21:33,430 --> 00:21:37,770 But our alternative is simply that it does not equal zero. 277 00:21:37,770 --> 00:21:40,410 It could be a positive or a negative number, 278 00:21:40,410 --> 00:21:41,363 and we don't know. 279 00:21:42,540 --> 00:21:45,288 So in this case, we choose our c 280 00:21:45,288 --> 00:21:50,288 such that 2.5% of the curve lies in the two tails. 281 00:21:54,690 --> 00:21:59,360 So it's the 97.5% of each of the two tails. 282 00:21:59,360 --> 00:22:04,360 So that a total of 5% of the area lies in these two tails, 283 00:22:06,810 --> 00:22:09,600 but there's half of it on each side. 284 00:22:09,600 --> 00:22:14,160 And again, a T-ratio of greater than c 285 00:22:14,160 --> 00:22:19,160 or less than negative c results in rejection of the null. 286 00:22:26,300 --> 00:22:29,830 So when we reject the null, 287 00:22:29,830 --> 00:22:34,830 we say we are 95% sure that the true value 288 00:22:38,610 --> 00:22:43,610 of our estimated parameter does not equal zero. 289 00:22:47,147 --> 00:22:50,280 We say that our Xj is 290 00:22:50,280 --> 00:22:55,280 statistically significant at the .05 level. 291 00:23:03,760 --> 00:23:08,760 Note that this T curve is the probability 292 00:23:09,560 --> 00:23:14,560 that you would get a T-stat of that magnitude 293 00:23:14,590 --> 00:23:18,140 if the null hypothesis were true. 294 00:23:18,140 --> 00:23:23,140 And every set of degrees of freedom has its own curve. 295 00:23:24,550 --> 00:23:29,550 And note that as we see here in this diagram, 296 00:23:31,280 --> 00:23:35,470 the greater the degrees of freedom, 297 00:23:35,470 --> 00:23:40,470 the sort of taller and skinnier this curve is. 298 00:23:44,450 --> 00:23:49,450 And also, I had said before, the more degrees of freedom, 299 00:23:50,500 --> 00:23:54,970 the more likely that you're going to find significance. 300 00:23:54,970 --> 00:23:59,520 So you can see that, say our T-value is two. 301 00:23:59,520 --> 00:24:01,930 You could reject the null 302 00:24:01,930 --> 00:24:04,290 with the infinity degrees of freedom, 303 00:24:04,290 --> 00:24:07,913 but not if you only have three degrees of freedom. 304 00:24:10,200 --> 00:24:15,200 So every degrees of freedom value has its own curve. 305 00:24:17,750 --> 00:24:22,750 And the curve in that degree of freedom number is, 306 00:24:24,300 --> 00:24:27,530 how likely is it that you would get 307 00:24:27,530 --> 00:24:32,263 a T-stat of this magnitude if the null is true? 308 00:24:33,390 --> 00:24:37,403 So just to re-emphasize it, 309 00:24:38,880 --> 00:24:41,963 if you get a T-stat near zero, 310 00:24:43,690 --> 00:24:47,410 it's pretty likely that you would get that 311 00:24:47,410 --> 00:24:49,990 if the null is true. 312 00:24:49,990 --> 00:24:53,870 If you get a very large T-stat way out on the tails, 313 00:24:53,870 --> 00:24:56,720 it's very unlikely that you would get that 314 00:24:56,720 --> 00:24:58,573 if the null is true. 315 00:25:00,300 --> 00:25:02,823 If your T-stat is very large, 316 00:25:03,750 --> 00:25:06,820 it means that your beta hat j is large, 317 00:25:06,820 --> 00:25:10,340 it's far away from zero, 318 00:25:10,340 --> 00:25:12,420 and our standard error is small, 319 00:25:12,420 --> 00:25:17,420 that we're pretty darn sure that it lies near that value. 320 00:25:25,920 --> 00:25:30,920 Sometimes we do tests where our null hypothesis is not zero, 321 00:25:34,760 --> 00:25:38,320 but that is beta j equals some constant, 322 00:25:38,320 --> 00:25:42,020 maybe one or some other number. 323 00:25:42,020 --> 00:25:47,020 And then, we simply subtract that from this aj, 324 00:25:50,860 --> 00:25:55,860 some constant that we subtract that from our beta hat j, 325 00:25:56,840 --> 00:25:59,750 divide by the standard error, 326 00:25:59,750 --> 00:26:04,673 and compare that to the critical value. 327 00:26:11,870 --> 00:26:15,130 I wanna emphasize that there's nothing magical, 328 00:26:15,130 --> 00:26:20,130 or certain, or right, or anything about this 5%. 329 00:26:21,250 --> 00:26:24,420 It is, to be honest, arbitrary. 330 00:26:24,420 --> 00:26:27,710 It's sort of, to be honest, it's just a convention. 331 00:26:27,710 --> 00:26:30,163 And it's what we sort of all agree to. 332 00:26:32,550 --> 00:26:36,130 But there's nothing magical about it. 333 00:26:36,130 --> 00:26:40,430 And one of the things that you get 334 00:26:40,430 --> 00:26:45,430 with the SPS output for every beta is a p-value. 335 00:26:47,728 --> 00:26:51,590 And that is, and the p-value measures 336 00:26:51,590 --> 00:26:54,600 what is the smallest significance level 337 00:26:54,600 --> 00:26:57,203 at which we can reject the null? 338 00:26:59,840 --> 00:27:04,840 So if we get a p-value of .04, then we can, 339 00:27:05,280 --> 00:27:10,280 then we're four, we can be 96% sure 340 00:27:10,350 --> 00:27:13,053 that we're not making a type one error. 341 00:27:17,150 --> 00:27:21,440 Let's look at more detail at P, so the p-value. 342 00:27:21,440 --> 00:27:25,890 Again, you always get a p-value in the output of SPSS 343 00:27:25,890 --> 00:27:29,990 or any regressions software, I imagine. 344 00:27:29,990 --> 00:27:30,890 It's a number. 345 00:27:30,890 --> 00:27:32,590 It's a probability. 346 00:27:32,590 --> 00:27:34,963 It is strictly between zero and one, 347 00:27:35,950 --> 00:27:38,360 where zero, of course is, there is no chance, 348 00:27:38,360 --> 00:27:40,213 and one is certain. 349 00:27:42,820 --> 00:27:46,980 When you get the output from SPSS, 350 00:27:47,920 --> 00:27:50,840 the standard is that the null 351 00:27:50,840 --> 00:27:55,110 of that beta equals zero against a two-tailed test, 352 00:27:55,110 --> 00:27:56,410 the two-sided alternative. 353 00:27:59,511 --> 00:28:02,340 And it is the probability where the T 354 00:28:11,220 --> 00:28:16,220 with a n minus k minus one degree of freedom. 355 00:28:18,160 --> 00:28:23,160 So that place on the curve is greater than the test stat. 356 00:28:29,530 --> 00:28:32,850 More broadly, P is the likelihood 357 00:28:32,850 --> 00:28:37,850 of a type one error, of a false positive. 358 00:28:37,850 --> 00:28:39,930 It's the probability 359 00:28:39,930 --> 00:28:44,930 that you reject the null when you shouldn't. 360 00:28:56,910 --> 00:28:59,750 The p-value is also the probability 361 00:28:59,750 --> 00:29:02,910 of seeing a T-stat that large. 362 00:29:02,910 --> 00:29:06,430 So how likely is it that you would get that magnitude 363 00:29:06,430 --> 00:29:10,980 of a T-stat if the null hypothesis were true? 364 00:29:10,980 --> 00:29:15,770 So if you have a p-value of .02, 365 00:29:15,770 --> 00:29:20,290 there's only a 2% chance that the T-stat 366 00:29:20,290 --> 00:29:25,290 that you got would occur if the null is true. 367 00:29:26,480 --> 00:29:29,450 So you can be 98% sure 368 00:29:29,450 --> 00:29:32,670 that there's a real relationship there, 369 00:29:32,670 --> 00:29:36,553 or that your beta does not equal zero. 370 00:29:37,420 --> 00:29:42,420 So a small p-value is evidence against the null. 371 00:29:45,050 --> 00:29:49,380 So having a small p means you are likely to reject it. 372 00:29:49,380 --> 00:29:51,830 And a large p gives you 373 00:29:51,830 --> 00:29:56,390 a little evidence to reject the null. 374 00:29:56,390 --> 00:30:01,390 A very large p-value, you will fail to reject the null. 375 00:30:05,310 --> 00:30:08,890 And just a bit here of nomenclature, 376 00:30:08,890 --> 00:30:13,890 that we would say we fail to reject the null at the 5%, 377 00:30:15,470 --> 00:30:17,280 not we accept the null. 378 00:30:17,280 --> 00:30:20,903 That's just sort of the way that you phrase it. 379 00:30:25,430 --> 00:30:27,840 I wanna make an aside on significance, 380 00:30:27,840 --> 00:30:30,530 that just because something is statistically significant 381 00:30:30,530 --> 00:30:32,870 doesn't mean that it's economically 382 00:30:32,870 --> 00:30:35,740 or socially or otherwise important. 383 00:30:35,740 --> 00:30:40,450 That, again, statistical significance is 384 00:30:42,420 --> 00:30:45,310 strictly a statistical measure. 385 00:30:45,310 --> 00:30:49,360 It's the function of the beta's magnitude, 386 00:30:49,360 --> 00:30:51,623 and its variance or standard error. 387 00:30:52,600 --> 00:30:55,560 And there may be times when something 388 00:30:55,560 --> 00:30:57,370 could be statistically significant, 389 00:30:57,370 --> 00:31:01,080 but not economically important, say. 390 00:31:01,080 --> 00:31:05,350 And an example here is if your Y is 391 00:31:05,350 --> 00:31:08,340 local food expenditure in dollars 392 00:31:08,340 --> 00:31:11,563 and our X1 is annual income. 393 00:31:12,520 --> 00:31:17,113 And if beta is significant and its value is .01, 394 00:31:19,940 --> 00:31:23,320 that means that for every thousand dollars of income, 395 00:31:23,320 --> 00:31:26,130 folks spend one cent more per year. 396 00:31:26,130 --> 00:31:31,110 It might be, clearly, we reject our null, 397 00:31:31,110 --> 00:31:34,130 and it is a statistically significant measure, 398 00:31:34,130 --> 00:31:38,373 but it's not very economically important. 399 00:31:45,960 --> 00:31:49,120 Note as well that as n increases, 400 00:31:49,120 --> 00:31:52,070 so our sample size increases, 401 00:31:52,070 --> 00:31:54,190 our variance decreases 402 00:31:54,190 --> 00:31:59,190 and the likelihood of significance increases. 403 00:31:59,800 --> 00:32:04,800 So some researchers choose as n gets larger 404 00:32:05,080 --> 00:32:10,080 to require a stricter standard. 405 00:32:10,210 --> 00:32:15,210 So they'll use maybe .05 if n is in the hundreds, 406 00:32:15,790 --> 00:32:19,023 but .01 if it's in the thousands, 407 00:32:27,520 --> 00:32:30,033 Some quick rules for empirical work. 408 00:32:31,200 --> 00:32:33,350 First thing you do when you get the result is 409 00:32:33,350 --> 00:32:35,300 you look for significance, 410 00:32:35,300 --> 00:32:38,170 and then you sort of look at its magnitude and think about, 411 00:32:38,170 --> 00:32:40,930 is it economically, or socially important? 412 00:32:40,930 --> 00:32:44,300 Does it really say anything 413 00:32:44,300 --> 00:32:47,950 about the phenomenon that we are studying? 414 00:32:47,950 --> 00:32:51,220 And if it is significant in the wrong sign, 415 00:32:51,220 --> 00:32:53,513 it's often a sign of specification error. 416 00:33:00,020 --> 00:33:01,660 Another thing that we can do is 417 00:33:01,660 --> 00:33:04,238 to create a confidence interval 418 00:33:04,238 --> 00:33:08,500 for a given confidence level for every beta j. 419 00:33:08,500 --> 00:33:13,500 So we've been saying the 95%, so that would be a p of .05. 420 00:33:18,240 --> 00:33:23,240 The confidence interval is beta j plus or minus c 421 00:33:24,320 --> 00:33:27,290 times the standard error of beta j, 422 00:33:27,290 --> 00:33:32,290 where c is the constant at the 97.5 percentile. 423 00:33:32,870 --> 00:33:37,870 So that point out in the two tails of the curve 424 00:33:38,000 --> 00:33:41,890 with 2.5% underneath it, 425 00:33:41,890 --> 00:33:46,890 and the T curve with n minus k minus one degrees of freedom. 426 00:33:49,040 --> 00:33:53,170 So this tells us two things. 427 00:33:53,170 --> 00:33:57,433 First, how precise is our estimate. 428 00:33:58,360 --> 00:34:02,690 And you can imagine that you might have seen 429 00:34:03,900 --> 00:34:08,070 data presented with little error bars around it, 430 00:34:08,070 --> 00:34:10,330 like limits a lot like that. 431 00:34:10,330 --> 00:34:15,330 And if zero lies within it, what does that mean? 432 00:34:17,330 --> 00:34:20,623 And we'll discuss it in class. 433 00:34:24,570 --> 00:34:26,980 So far, we've been talking about T-tests, 434 00:34:26,980 --> 00:34:31,980 and conducting, looking at one parameter at a time, 435 00:34:33,730 --> 00:34:37,523 a single hypothesis that we're looking at, 436 00:34:39,730 --> 00:34:43,560 pulling out a single beta j and testing 437 00:34:43,560 --> 00:34:46,840 the null hypothesis that it's equal to zero. 438 00:34:46,840 --> 00:34:50,760 What about, though, if we want to test more than one? 439 00:34:50,760 --> 00:34:53,710 And when this is often used 440 00:34:53,710 --> 00:34:57,420 in sort of nested models where we wanna see, 441 00:34:57,420 --> 00:35:00,537 is there a group of parameters 442 00:35:02,790 --> 00:35:07,790 that jointly have an effect on our Y? 443 00:35:08,270 --> 00:35:10,630 And for this, we use an F-test. 444 00:35:10,630 --> 00:35:15,440 And this is, the procedure is called 445 00:35:15,440 --> 00:35:18,620 testing multiple restrictions. 446 00:35:18,620 --> 00:35:23,620 So looking at more than one beta j at a time, 447 00:35:24,160 --> 00:35:26,590 and doing a test, if jointly, 448 00:35:26,590 --> 00:35:29,483 they are significantly different than zero. 449 00:35:34,780 --> 00:35:39,040 So here's a model that we are now familiar with. 450 00:35:39,040 --> 00:35:42,590 We have five regressors, k equals five. 451 00:35:42,590 --> 00:35:45,909 And we wonder, do X3, X4, 452 00:35:45,909 --> 00:35:50,909 and X5 jointly have an effect on our Y? 453 00:35:57,570 --> 00:36:01,520 So under this test, we have a null hypothesis 454 00:36:01,520 --> 00:36:06,020 that beta3 equals beta4 equals beta5 equals zero, 455 00:36:06,020 --> 00:36:09,990 or each one is equal to zero. 456 00:36:09,990 --> 00:36:14,733 Our alternative hypothesis is if any of these are not true. 457 00:36:15,810 --> 00:36:19,660 And we call this exclusion restrictions 458 00:36:19,660 --> 00:36:20,690 that we're sort of pulling, 459 00:36:20,690 --> 00:36:24,560 when we do the test, we pull those out of the model. 460 00:36:24,560 --> 00:36:28,960 Or we are restricting their value to zero. 461 00:36:28,960 --> 00:36:33,960 And then we do a multiple hypothesis test, 462 00:36:34,520 --> 00:36:35,713 which is an F-test. 463 00:36:42,170 --> 00:36:44,313 When we test multiple restrictions, 464 00:36:46,210 --> 00:36:51,210 the null is that all of them are jointly equal to zero. 465 00:36:51,690 --> 00:36:55,890 The alternative is that it's not true. 466 00:36:55,890 --> 00:36:57,520 So we reject the null 467 00:36:57,520 --> 00:37:02,253 if any of the three B's is not equal to zero. 468 00:37:03,770 --> 00:37:06,210 Why don't we do all three? 469 00:37:06,210 --> 00:37:09,540 Just do each one singly. 470 00:37:09,540 --> 00:37:14,540 So the problem here, the biggest one is 471 00:37:15,810 --> 00:37:20,810 that three 5% significance tests, that .95 cubed is .86. 472 00:37:27,067 --> 00:37:31,240 So if any of them, it doesn't hold 473 00:37:31,240 --> 00:37:35,270 that the other one is equal to zero, 474 00:37:35,270 --> 00:37:38,490 which is what we're doing in an F-test. 475 00:37:38,490 --> 00:37:43,490 It takes out, that the F-test takes out all three singly, 476 00:37:43,880 --> 00:37:48,623 whereas the T-test only takes out one. 477 00:37:55,480 --> 00:37:57,530 The intuition of the test is, 478 00:37:57,530 --> 00:38:01,540 how much does the goodness of fit change 479 00:38:01,540 --> 00:38:06,540 with and without these regressors in the equation? 480 00:38:07,840 --> 00:38:12,570 Recall that adding regressors 481 00:38:12,570 --> 00:38:16,090 always improves the goodness of fit. 482 00:38:16,090 --> 00:38:17,750 It always increases R squared. 483 00:38:17,750 --> 00:38:21,700 It always decreases the sum of squared residuals. 484 00:38:21,700 --> 00:38:25,040 And taking them out always increases 485 00:38:25,040 --> 00:38:27,650 the sum of squared residuals. 486 00:38:27,650 --> 00:38:32,640 And actually, this is what we use to do the test. 487 00:38:32,640 --> 00:38:35,190 So we look at the SSR. 488 00:38:35,190 --> 00:38:37,490 And we run the two models. 489 00:38:37,490 --> 00:38:41,040 We run the k equals five, which is our full model, 490 00:38:41,040 --> 00:38:42,620 and the k equals two 491 00:38:42,620 --> 00:38:45,253 where we leave in two and take out three. 492 00:38:51,690 --> 00:38:53,750 We run both regressions, 493 00:38:53,750 --> 00:38:57,050 and we note the sum of squared residuals of each one, 494 00:38:57,050 --> 00:38:59,620 which you get in SPSS. 495 00:38:59,620 --> 00:39:03,010 Note that the SSR from the restricted model is 496 00:39:03,010 --> 00:39:06,720 always greater than in the full model, 497 00:39:06,720 --> 00:39:08,773 and think about why that is. 498 00:39:18,750 --> 00:39:23,750 So in our general case, we make queue restrictions. 499 00:39:24,490 --> 00:39:27,760 So we start with k regressors. 500 00:39:27,760 --> 00:39:29,940 We remove q of them. 501 00:39:29,940 --> 00:39:34,173 So in our restricted model, we have k minus q. 502 00:39:35,280 --> 00:39:40,280 And we can just order them so that the ones 503 00:39:40,320 --> 00:39:44,023 that we're going to take out are the last ones. 504 00:39:45,360 --> 00:39:47,803 So here is the F step. 505 00:39:49,770 --> 00:39:54,770 And it is, the best way to think of it is, 506 00:39:54,820 --> 00:39:59,820 it is the relative change of the sum of squared residuals 507 00:40:00,120 --> 00:40:03,840 weighted by the degree of freedom. 508 00:40:03,840 --> 00:40:08,840 So you look at the SSR of the restricted model 509 00:40:10,470 --> 00:40:14,030 and the SSR of the unrestricted model. 510 00:40:14,030 --> 00:40:19,030 And the biggest driver of the magnitude of that stat is, 511 00:40:19,170 --> 00:40:21,310 how much does that change? 512 00:40:21,310 --> 00:40:26,100 How much does the SSR change 513 00:40:26,100 --> 00:40:30,350 when we take those regressors out? 514 00:40:30,350 --> 00:40:35,200 So if there's a big change, we will see that this, 515 00:40:35,200 --> 00:40:37,320 the part of the numerator, 516 00:40:37,320 --> 00:40:40,340 SSR, we restricted minus SSR. 517 00:40:40,340 --> 00:40:42,420 Unrestricted will be a big number, 518 00:40:42,420 --> 00:40:45,213 which leads to a big F-stat. 519 00:40:50,900 --> 00:40:53,220 So under our null hypothesis, 520 00:40:53,220 --> 00:40:57,603 and if our CLR assumptions all hold, then we, 521 00:40:57,603 --> 00:41:02,603 in the same way that we know the distribution of the T-stat, 522 00:41:03,310 --> 00:41:08,040 we now know the distribution of the F-stat, 523 00:41:08,040 --> 00:41:10,423 and it has an F-distribution 524 00:41:13,290 --> 00:41:17,470 with q and n minus k minus one degrees of freedom. 525 00:41:17,470 --> 00:41:20,670 q being the number of restrictions, 526 00:41:20,670 --> 00:41:23,960 how many regressors we take out, 527 00:41:23,960 --> 00:41:27,710 n is the size of the sample, 528 00:41:27,710 --> 00:41:31,610 k is the total regressors in the full model, and one. 529 00:41:31,610 --> 00:41:36,243 So it's q, and n minus k minus one degrees of freedom. 530 00:41:41,470 --> 00:41:46,470 As with a T-test, we reject the null if the F-stat is big. 531 00:41:47,340 --> 00:41:51,113 If it's sufficiently large to be way out in the tail. 532 00:41:52,560 --> 00:41:55,780 and a large F-stat implies a big change 533 00:41:55,780 --> 00:42:00,780 in the residual when we make these restrictions. 534 00:42:01,440 --> 00:42:06,440 Again, we choose a significance level, like 5%. 535 00:42:07,520 --> 00:42:10,940 And we find the critical value 536 00:42:10,940 --> 00:42:15,557 for some c for the F-distribution, 537 00:42:16,740 --> 00:42:20,283 with that number of degrees of freedom. 538 00:42:23,310 --> 00:42:26,640 We reject the null if the F-stat is 539 00:42:26,640 --> 00:42:28,960 bigger than c, the critical value, 540 00:42:28,960 --> 00:42:32,210 same as with a T-test. 541 00:42:32,210 --> 00:42:35,860 This is especially valuable in testing restrictions 542 00:42:35,860 --> 00:42:39,680 of highly correlated regressors, 543 00:42:39,680 --> 00:42:42,140 because it tests them as a group. 544 00:42:42,140 --> 00:42:46,460 And it's like all together, are they significant? 545 00:42:46,460 --> 00:42:51,460 Not, is one significant holding all the others constant? 546 00:42:51,910 --> 00:42:56,660 So we're gonna do an example in SPSS 547 00:42:56,660 --> 00:43:00,450 that looks at major league baseball batting stats. 548 00:43:00,450 --> 00:43:02,610 But I wanna talk to you here about another one. 549 00:43:02,610 --> 00:43:04,340 And I don't have data for this. 550 00:43:04,340 --> 00:43:06,910 This is just kind of a thought experiment, 551 00:43:06,910 --> 00:43:08,460 but I think it would hold true, 552 00:43:09,600 --> 00:43:13,370 of farming practices and thinking 553 00:43:13,370 --> 00:43:17,390 about climate change best management practices. 554 00:43:17,390 --> 00:43:22,390 So three practices that we know are sort of good for soil 555 00:43:23,620 --> 00:43:25,790 and good for keeping phosphorus 556 00:43:25,790 --> 00:43:29,280 out of the lake are cover cropping, 557 00:43:29,280 --> 00:43:34,280 buffer strips around a stream, and reduced tillage. 558 00:43:36,640 --> 00:43:40,200 So if our Y is how much phosphorus is 559 00:43:40,200 --> 00:43:42,010 leaking off of the farm, 560 00:43:42,010 --> 00:43:44,210 and we might have a bunch of regressors 561 00:43:44,210 --> 00:43:47,160 about farmer experience and age 562 00:43:47,160 --> 00:43:49,630 and what they grow and all kinds of things like that. 563 00:43:49,630 --> 00:43:52,770 But looking at these three practices 564 00:43:54,080 --> 00:43:56,510 that maybe each one of them 565 00:43:56,510 --> 00:44:00,370 wouldn't be significant from a T-test, 566 00:44:00,370 --> 00:44:05,370 that having a buffer strip holding cover crop 567 00:44:06,430 --> 00:44:08,660 and reduced tillage constant 568 00:44:08,660 --> 00:44:13,510 might not have a significant effect on phosphorus load, 569 00:44:13,510 --> 00:44:17,420 but the three of them together do. 570 00:44:17,420 --> 00:44:18,840 And I invite you to maybe think 571 00:44:18,840 --> 00:44:22,103 of another example of how this, 572 00:44:23,330 --> 00:44:25,850 where this might come into play. 573 00:44:25,850 --> 00:44:28,170 And again, we'll see it when we do 574 00:44:28,170 --> 00:44:31,123 an SPSS exercise with baseball stats. 575 00:44:36,670 --> 00:44:39,750 The cool thing about T and F-stats 576 00:44:39,750 --> 00:44:43,830 and this sort of intuition of how they work is 577 00:44:43,830 --> 00:44:46,663 I think all test stats work the same. 578 00:44:47,800 --> 00:44:51,920 They all work from this same intuition. 579 00:44:51,920 --> 00:44:56,920 So you start with some sort of hypothesis test of something, 580 00:44:57,730 --> 00:45:00,420 or many somethings equal zero, 581 00:45:00,420 --> 00:45:05,420 that the one thing has no effect on another. 582 00:45:09,830 --> 00:45:11,913 If you get a big stat, 583 00:45:12,820 --> 00:45:14,980 it means it's going to be way out 584 00:45:14,980 --> 00:45:18,003 in the tail of the distribution. 585 00:45:19,202 --> 00:45:24,080 And so you can think of this as being far away from zero, 586 00:45:24,080 --> 00:45:26,630 far away from the null hypothesis. 587 00:45:26,630 --> 00:45:30,190 This test stat is sort of far away from zero, 588 00:45:30,190 --> 00:45:34,290 it's far away from the idea of the null being true. 589 00:45:34,290 --> 00:45:38,590 And if that's true, when we get a big F and a big T-test, 590 00:45:38,590 --> 00:45:43,290 it's very unlikely that the null hypothesis is true. 591 00:45:43,290 --> 00:45:47,150 We have a high confidence to reject our null. 592 00:45:47,150 --> 00:45:51,970 And that is associated with a small p-value 593 00:45:51,970 --> 00:45:54,230 and a small significance. 594 00:45:54,230 --> 00:45:56,460 So a big test stat means 595 00:45:56,460 --> 00:46:01,460 that we can usually reject the null with high confidence, 596 00:46:02,010 --> 00:46:05,093 and it's associated with a small p-value. 597 00:46:12,910 --> 00:46:17,910 Doing one hypothesis test at a time, 598 00:46:18,150 --> 00:46:20,120 one beta j at a time, 599 00:46:20,120 --> 00:46:25,120 it makes sense to just use an F-test. 600 00:46:25,650 --> 00:46:27,440 They're easier to compute, 601 00:46:27,440 --> 00:46:29,500 you get the output automatically. 602 00:46:29,500 --> 00:46:32,490 It will yield the same result 603 00:46:32,490 --> 00:46:35,273 as an F-test in a two-tailed test. 604 00:46:37,310 --> 00:46:42,310 And a one-tailed test is the best, 605 00:46:42,614 --> 00:46:47,340 or a T-test is best for a single restriction, 606 00:46:47,340 --> 00:46:49,510 so there's really no point 607 00:46:49,510 --> 00:46:53,460 in doing the q equals one F-test. 608 00:46:53,460 --> 00:46:57,693 You can simply do a T-test and you'll get the same result. 609 00:47:05,130 --> 00:47:08,090 Last time, we talked at length about specification. 610 00:47:08,090 --> 00:47:10,890 What to include, and what not. 611 00:47:10,890 --> 00:47:13,450 So one way that this can be used, 612 00:47:13,450 --> 00:47:14,780 because you run a full model, 613 00:47:14,780 --> 00:47:19,393 you include things that you really think are important. 614 00:47:21,700 --> 00:47:26,260 You run this full model, and then you look at the output, 615 00:47:26,260 --> 00:47:31,260 and you can look at those with a p of greater than .10. 616 00:47:33,010 --> 00:47:38,010 Or maybe a T-stat of less than one in absolute value, 617 00:47:38,080 --> 00:47:40,500 and use an F-test. 618 00:47:40,500 --> 00:47:45,325 Or also, if you do collinearity diagnostics, 619 00:47:45,325 --> 00:47:49,980 you can do an F-test of the most colinear variables 620 00:47:49,980 --> 00:47:54,263 and use that to determine which is your best model. 621 00:47:58,550 --> 00:48:03,320 There's also a way to use R squared to calculate an F-stat. 622 00:48:03,320 --> 00:48:05,720 You'll get the same results. 623 00:48:05,720 --> 00:48:08,200 It's just a change in R squared 624 00:48:09,670 --> 00:48:12,700 adjusted for degrees of freedom. 625 00:48:12,700 --> 00:48:15,813 This may be a little easier in some stat packages. 626 00:48:23,330 --> 00:48:27,420 Another thing that you get sort of for free in SPSS is 627 00:48:27,420 --> 00:48:31,720 the overall significance of a regression. 628 00:48:31,720 --> 00:48:36,350 So here, your null is, none of the X's matter. 629 00:48:36,350 --> 00:48:39,860 All of them are jointly equal to zero. 630 00:48:39,860 --> 00:48:44,610 The restricted model is just an intercept and an error. 631 00:48:44,610 --> 00:48:46,790 And you could run this. 632 00:48:46,790 --> 00:48:49,420 And since the R squared from the rejected, 633 00:48:49,420 --> 00:48:54,040 from the restricted model is zero, here is the F-stat. 634 00:48:59,220 --> 00:49:03,100 Almost always, you will find 635 00:49:03,100 --> 00:49:06,900 that the overall regression is significant. 636 00:49:06,900 --> 00:49:10,083 If you don't, you really just have a very poor model. 637 00:49:14,310 --> 00:49:17,130 For right now, just keep these in mind. 638 00:49:17,130 --> 00:49:19,150 Here are other hypothesis tests 639 00:49:19,150 --> 00:49:24,150 that you use in other models, not in OLS, 640 00:49:24,390 --> 00:49:25,790 and we're gonna revisit them. 641 00:49:25,790 --> 00:49:29,543 It's just to sort of have them in your mind for right now. 642 00:49:35,320 --> 00:49:37,110 I wanted to touch briefly 643 00:49:38,330 --> 00:49:43,080 on some thoughts that the Kennedy textbook. 644 00:49:43,080 --> 00:49:47,520 And as always, keep in mind that all models are wrong, 645 00:49:47,520 --> 00:49:51,640 some models are useful, as George box said. 646 00:49:51,640 --> 00:49:56,640 So models are just lenses into the world. 647 00:49:57,060 --> 00:50:00,583 And how do we make them more useful? 648 00:50:02,620 --> 00:50:07,170 So recognize first 649 00:50:07,170 --> 00:50:11,687 that the specification can be a challenge, 650 00:50:13,410 --> 00:50:17,540 that you're not always control of data generation. 651 00:50:20,580 --> 00:50:23,960 That in many cases, it's more art than science. 652 00:50:23,960 --> 00:50:25,110 And also in many cases, 653 00:50:25,110 --> 00:50:30,110 there may not be a single universally accepted best way. 654 00:50:35,300 --> 00:50:38,790 So a good place to start is 655 00:50:38,790 --> 00:50:43,790 to use a theory and prior studies, 656 00:50:47,150 --> 00:50:49,390 maybe some common sense, 657 00:50:49,390 --> 00:50:52,563 and it include things, but the goal being, 658 00:50:52,563 --> 00:50:56,110 what the text, what author calls a clean economic story. 659 00:50:56,110 --> 00:51:00,360 And note again that models with a low overall F-stat, 660 00:51:00,360 --> 00:51:03,420 where it's almost all error or noise, 661 00:51:03,420 --> 00:51:05,573 may just be misspecified. 662 00:51:12,260 --> 00:51:17,260 That it's always good to use these kind of tests. 663 00:51:18,180 --> 00:51:20,120 In some cases, it might make sense 664 00:51:20,120 --> 00:51:22,560 to set aside a few observations 665 00:51:22,560 --> 00:51:27,560 and see if your model holds in those. 666 00:51:31,080 --> 00:51:34,736 And if there are many (indistinct), 667 00:51:34,736 --> 00:51:39,736 it's often worthwhile to compare and contrast them, 668 00:51:41,400 --> 00:51:44,311 do it a few ways, see what you get, 669 00:51:44,311 --> 00:51:47,400 and discuss strengths and weaknesses, 670 00:51:47,400 --> 00:51:50,180 and probably choose which one. 671 00:51:50,180 --> 00:51:52,700 The most important point here in this slide is, 672 00:51:52,700 --> 00:51:55,940 be prepared to explain the path that you chose. 673 00:51:55,940 --> 00:51:58,960 That there may not be one single way, 674 00:51:58,960 --> 00:52:00,730 but you really have to make a good case 675 00:52:00,730 --> 00:52:04,453 of this is what I did and why I chose this. 676 00:52:10,550 --> 00:52:12,480 So again, some of the, 677 00:52:12,480 --> 00:52:15,410 this is an approach that I have used. 678 00:52:15,410 --> 00:52:17,760 So starting with one. 679 00:52:17,760 --> 00:52:21,550 Start with a fairly large number of regressors. 680 00:52:21,550 --> 00:52:24,160 Ones that you really think have an impact 681 00:52:24,160 --> 00:52:29,160 based on theory, common sense, and previous studies. 682 00:52:31,240 --> 00:52:33,110 Look at the overall F-test. 683 00:52:33,110 --> 00:52:36,700 And then run a restricted model 684 00:52:36,700 --> 00:52:41,080 with only those variables that are significant, 685 00:52:41,080 --> 00:52:44,910 where T value in absolute value greater than one. 686 00:52:44,910 --> 00:52:48,860 And then use an F-test to choose 687 00:52:48,860 --> 00:52:50,963 sort of what is your final model. 688 00:52:52,000 --> 00:52:55,020 So I hope you have a better understanding 689 00:52:55,020 --> 00:52:58,823 of hypothesis tests now, and thank you.