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 on 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, u_i, 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 u_i?

(iii) Now generate the y_i as

y_i = 1 + 2x_i + u_i ; β₀ + β₁x_i + u_i;

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, u^ i, and verify that equation (2.60) holds (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 β̂₀ and β̂₁? Why are these different from what you obtained in part (iii)?