Part one. The coefficient on Bs is about minus 0.52 Okay, it's usual. Standard error is 0.110 Please give a T statistic of minus 4.7. Remember, the teen statistic is calculated by taking beta hat divided by its standard error. So you will take minus 0.52 divided by 0.110 Yeah. Even these t statistic, right? For the case of usual standard error, we can conclude that beta hat of BS is highly significant. This T value is much larger than the critical value of 2.33 So you may see in your, um, statistical program statistical software. This variable should be significant at the 1% level just for the usual standard error. Okay, When we estimate the robust than that error, we get a much larger number. We get point to a three, and that reduces the T statistic. The new T statistic is about minus 1.8 and even the critical value the smallest, smallest critical value of one. That is, for the 10% level of significance. We can conclude that the coefficient of be as variable is marginally significant. If you use a statistical software. You may see that this variable is significant at the 10% level. So because the standard error has changed by a lot, when we move to the option of robust, we may conclude that there is serious hetero scad elasticity in the data. This is the symbol for exist. The estimate of this beta is not small. Yeah, by economic sense. Although it is not close to the hypothesized value of one. Yeah, should be minus one. Okay, let's move. Thio Part three So I switched part two and three by Mystic. We will get to part to you. After this we will include four dummies in the regression. Okay? And we can see that after we include the dumbest. Yeah, the estimates are still the same. Okay, Before dummies do not change our result. And looking at the coefficients of the dummies, we see that all the observation 15 08 only this observation has a significant coefficient. It is. Statistic is very large. Minus 6.1. You may see that none of the other three dummies is significant even at the 10% level. Okay, Back to part two. We will drop four observations with variable B s greater than 40.5. Then we will re estimate the initial regression equation. And we look for the change in the beta hat on BS. The beta head we got is minus one point. Sorry. Minus pulling 86 And the robust T statistic is minus 1.27 So the practical significance of the coefficient is much lower. And given this T statistic, this coefficient is not significant at while the 5% level, you should check that again. So this T value is still larger than one in absolute value, but it is very close to one. Would you like to hear from you? That's why I guess it is not statistically significant at the 5% level, but maybe at the 10% level. But we use the 5% level as the standard. Okay, let's move to part four. And in this part, we win. Estimate four regression in each regression we win. Drop one. Observation One influential observation from the sample Recall that initially before dropping any observations the beta had of bs is minus 0.52 Now, if we drop observation 15 08 only we will get beta head off Bs as minus 0.20 If we drop observation 68 only beta head is minus 680.5. Next observation, 11 27. Beta head is minus 0.54 And the last observation 16 07 We get minus point 53 So the only big change happens when we drop observation. 15 08 We can conclude that the estimate is Onley sensitive to the inclusion of observation 15 08 part five. Yeah, The results we got so far reveal that if an observation is extreme enough, it can really influence the O. L s estimate we have a large sample in this problem are simple is about 1800 observations. However, the old L s estimates has shown earlier are not immune to the influential observations. Let me write that down. Extreme observation. Yeah, can influence all L s estimates substantially even when the sample is large. In this problem, the influential observation 15 08 happens to be the observation with the lowest average salary and the highest Bs ratio. So including this observation clearly ships the estimate toward minus one part six In Part six, we will adopt a new estimation method. We will use L A D. Which represents least absolute deviation. This is a substitute method for our l s in the case, we have influential observations. We don't need to drop these observations from our simple when we use thes method. Unlike O l s, uh, s maximize. Sorry. Minimize the sum of square of errors L a de minimis the sum of absolute values of the residuals. The L A D. Estimate of beta head on B s with the foreign. Simple is minus 0.109 And the T statistic is minus 1.1 When we drop observation 15 08 from the sample. An estimate The regression equation with L I d. Again, we get beta hat on B s as minus 0.97 and the T statistic is minus point 81 Looking at these numbers, you can see that the change in the estimate is modest and based on the value of the T statistic, very close to one. Beta hat on bs is statistically insignificant. In both cases,