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Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge
Edition 6ISBN: 130527010XIntroductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge
Edition 6ISBN: 130527010XTo complete this exercise you need a software package that allows you to generate data from the uniform and normal distributions.
(i) Start by generating 500 observations xi – the explanatory variable – from the uniform distribution with range [0,10]. (Most statistical packages have a command for the Uniform[0,1] distribution; just multiply those observations by 10.) What are the sample mean and sample standard deviation of the xi?
(ii) Randomly generate 500 errors, ui, from the Normal[0,36] distribution. (If you generate a Normal[0,1], as is commonly available, simply multiply the outcomes by six.) Is the sample average of the ui exactly zero? Why or why not? What is the sample standard deviation of the ui?
(iii) Now generate the yi as ![To complete this exercise you need a software package that allows you to generate data from the uniform and normal distributions. (i) Start by generating 500 observations <i>x</i><sub>i</sub><i> </i>– the explanatory variable – from the uniform distribution with range [0,10]. (Most statistical packages have a command for the Uniform[0,1] distribution; just multiply those observations by 10.) What are the sample mean and sample standard deviation of the <i>x</i><sub>i</sub>? (ii) Randomly generate 500 errors, <i>u</i><sub>i</sub>, from the Normal[0,36] distribution. (If you generate a Normal[0,1], as is commonly available, simply multiply the outcomes by six.) Is the sample average of the <i>u</i><sub>i</sub><i> </i>exactly zero? Why or why not? What is the sample standard deviation of the <i>u</i><sub>i</sub>? (iii) Now generate the <i>y</i><sub>i</sub><i> </i>as <i> </i> that is, the population intercept is one and the population slope is two. Use the data to run the regression of <i>y</i><sub>i</sub><i> </i>on <i>x</i><sub>i</sub>. What are your estimates of the intercept and slope? Are they equal to the population values in the above equation? Explain. (iv) Obtain the OLS residuals, <i> </i>, and verify that equation (2.60) hold (subject to Rounding error). (v) Compute the same quantities in equation (2.60) but use the errors <i>u</i><sub>i</sub><i> </i>in place of the residuals. Now what do you conclude? (vi) Repeat parts (i), (ii), and (iii) with a new sample of data, starting with generating the <i>x</i><sub>i</sub>. Now what do you obtain for <i> </i>? Why are these different from what you obtained in part (iii)?](https://d2lvgg3v3hfg70.cloudfront.net/SMCC2709/bf3de869_14be_4369_8bfb_e3496a4d86ec_SMCC2709_11.jpg)
that is, the population intercept is one and the population slope is two. Use the data to run the regression of yi on xi. What are your estimates of the intercept and slope? Are they equal to the population values in the above equation? Explain.
(iv) Obtain the OLS residuals, ![To complete this exercise you need a software package that allows you to generate data from the uniform and normal distributions. (i) Start by generating 500 observations <i>x</i><sub>i</sub><i> </i>– the explanatory variable – from the uniform distribution with range [0,10]. (Most statistical packages have a command for the Uniform[0,1] distribution; just multiply those observations by 10.) What are the sample mean and sample standard deviation of the <i>x</i><sub>i</sub>? (ii) Randomly generate 500 errors, <i>u</i><sub>i</sub>, from the Normal[0,36] distribution. (If you generate a Normal[0,1], as is commonly available, simply multiply the outcomes by six.) Is the sample average of the <i>u</i><sub>i</sub><i> </i>exactly zero? Why or why not? What is the sample standard deviation of the <i>u</i><sub>i</sub>? (iii) Now generate the <i>y</i><sub>i</sub><i> </i>as <i> </i> that is, the population intercept is one and the population slope is two. Use the data to run the regression of <i>y</i><sub>i</sub><i> </i>on <i>x</i><sub>i</sub>. What are your estimates of the intercept and slope? Are they equal to the population values in the above equation? Explain. (iv) Obtain the OLS residuals, <i> </i>, and verify that equation (2.60) hold (subject to Rounding error). (v) Compute the same quantities in equation (2.60) but use the errors <i>u</i><sub>i</sub><i> </i>in place of the residuals. Now what do you conclude? (vi) Repeat parts (i), (ii), and (iii) with a new sample of data, starting with generating the <i>x</i><sub>i</sub>. Now what do you obtain for <i> </i>? Why are these different from what you obtained in part (iii)?](https://d2lvgg3v3hfg70.cloudfront.net/SMCC2709/f9200f96_66f3_4c82_8924_c84a2eba1238_SMCC2709_11.jpg)
, and verify that equation (2.60) hold (subject to Rounding error).
(v) Compute the same quantities in equation (2.60) but use the errors ui in place of the residuals. Now what do you conclude?
(vi) Repeat parts (i), (ii), and (iii) with a new sample of data, starting with generating the xi. Now what do you obtain for ![To complete this exercise you need a software package that allows you to generate data from the uniform and normal distributions. (i) Start by generating 500 observations <i>x</i><sub>i</sub><i> </i>– the explanatory variable – from the uniform distribution with range [0,10]. (Most statistical packages have a command for the Uniform[0,1] distribution; just multiply those observations by 10.) What are the sample mean and sample standard deviation of the <i>x</i><sub>i</sub>? (ii) Randomly generate 500 errors, <i>u</i><sub>i</sub>, from the Normal[0,36] distribution. (If you generate a Normal[0,1], as is commonly available, simply multiply the outcomes by six.) Is the sample average of the <i>u</i><sub>i</sub><i> </i>exactly zero? Why or why not? What is the sample standard deviation of the <i>u</i><sub>i</sub>? (iii) Now generate the <i>y</i><sub>i</sub><i> </i>as <i> </i> that is, the population intercept is one and the population slope is two. Use the data to run the regression of <i>y</i><sub>i</sub><i> </i>on <i>x</i><sub>i</sub>. What are your estimates of the intercept and slope? Are they equal to the population values in the above equation? Explain. (iv) Obtain the OLS residuals, <i> </i>, and verify that equation (2.60) hold (subject to Rounding error). (v) Compute the same quantities in equation (2.60) but use the errors <i>u</i><sub>i</sub><i> </i>in place of the residuals. Now what do you conclude? (vi) Repeat parts (i), (ii), and (iii) with a new sample of data, starting with generating the <i>x</i><sub>i</sub>. Now what do you obtain for <i> </i>? Why are these different from what you obtained in part (iii)?](https://d2lvgg3v3hfg70.cloudfront.net/SMCC2709/08066724_b299_41e2_92db_80b6a3d73fad_SMCC2709_11.jpg)
? Why are these different from what you obtained in part (iii)?
Verified
Step 1 of 7
(i)
Generate 500 observations
, the explanatory variable from the uniform distribution in the range of 0 and 10 using Data Analysis tool pack in Excel and selecting Random Number Generation for Uniform distribution in the range of 0 and 10
The sample mean is 4.931099 and the sample standard deviation is 2.895428
The sample mean is found by dividing the sum total of 500 observations on 
by 499
The sample standard deviation of 
is estimated by dividing the sum of the squared deviation of the 500 observations on the 500 observations on 
from the sample mean by 499 and then taking the square root of it
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