Consider a simple nonlinear regression model of the form that yi = ln1xi + ui (0.1) (i) Derive the ordinary least square estimator (OLS) of

Consider a simple nonlinear regression model of the form that yi = lnβ1xi + ui (0.1) 

(i) Derive the ordinary least square estimator (OLS) of ln β1. 

(ii) Discuss whether the sum of the OLS residuals is equal to zero in this model? 

(iii) Show that the OLS estimator of lnβ1 is unbiased in this regression using the assumption that E h u x i = 0. 

(iv) Show that the OLS estimator of β1 is biased in this regression. 

Question 2 

This question aims to assess whether a change in units of measurement of explanatory variables aects the coecient of determination, R2 . First, consider a simple regression model that yi = β0 + β1xi + ui . (0.2) Next, suppose that you change the unit of measurement for xi by scaling up xi by a factor of c. Denote this scaled explanatory variable as x˜i , where x˜i = cxi . The simple regression model with the explanatory variable x˜i can be written as yi = α0 + α1x˜i + ei , (0.3) where ei denotes the error term. 

(i) Let a hat over a parameter denotes the OLS estimator of that parameter. Show that αˆ1 = βˆ 1 c and αˆ0 = βˆ 0. 

(ii) Show that R2 in model (0.2) and in model (0.3) are the same. 

(iii) Interpret your nding from part (ii). 

Question 3 

In a study, you are interested in how the number of visits in a month that someone makes to People's Bread Cafes (Halk Ekmek Kafe) relates to his/her wage in Ankara. You collect data both on the number of visits (visits) and on the wages (wage) for your sample of 100 individuals who are in labor force. Suppose that estimating the regression yields the following results visits = 4.12 − 2.64 lnwage (0.4) 

(i) Interpret the slope estimate in the model. Is the negative slope estimate consistent with your intuition? 

(ii) Find the predicted change in the number of visits due to 50% increase in the wage of an individual. 

(iii) Indeed, how many times a wage earner visits People's Bread Cafes is aected by the proximity of their house to these cafes. However, proximity is not included in the model, and thus, it is among unobserved variables. Suppose that these cafes are located in districts where mostly low-wage families live. By taking this information into account, discuss whether the OLS estimator of the slope parameter above can be unbiased.