Question

13. Assumption SLR.4 (Zero Conditional Mean) One crucial assumption in the simple linear regression model is that the error term u has a mean of zero, conditional on the value of the explanatory variable Suppose you are using the following simple linear regression model to study the effect of education on salary. In this case, assume that besides education, natural ability level is the only other factor that influences salary, and therefore, u and ability level are equivalent. sal - Be + B,educ + where educ = years of education sal = salary in tens of thousands of dollars u = the normalized value of ability relative to the average ability of all individuals in the population Note: A positive value for u indicates higher than average ability, a negative value for u indicates below average ability, and a value of O foru indicates average ability If u has an expected value of 0, conditional on educ, then Euleduc) = 0, and therefore, (nalledue) - - Though impossible in reality, suppose you have a scatter plot of the expected salary for each level of education in the population E (salſeduc). These expected salaries are denoted by the blue dots on the following graph. Suppose you also have the line representing the population function when the zero conditional mean assumption holds, labeled as Bo + B, educ and depicted as the black line on the graph. Refer to the graph and answer the question that follows.

Drop down: zero or nonzero / holds of does not hold.

True or False: Assuming the model is linear in parameters, and you obtain a random sample of observations with varying values of education, the simple OLS slope and intercept estimates will be unbiased.

Answer 1

$sal = \beta_0 + \beta_1 educ + u$

$E[sal | educ] = E[\beta_0 + \beta_1 educ + u] = E[\beta_0 ] + E[\beta_1 educ] + E[u|sal]$

$E[sal | educ] = \beta_0 + \beta_1 educ$

From the scatter plot, is the distance between actual salary and predicted salary,

= 1 - 1.5 + 1 - 1.5 = -1

Thus,   appears to be nonzero and therefore, the zero conditional mean assumption does not hold.

Assuming the model is linear in parameters, and you obtain a random sample of observations with varying values of education, the simple OLS slope and intercept estimates will be unbiased. This is a True statement if = 0 holds.