Economics 2P91 Fall 2022: Practice problems Question 1 Consider the following multiple regression model: Yi=0+1X1i+2X2i+3X3i+ui (a) What problem does multiple regression help us address? (b) What is the minimum number of observations (sample size) required to obtain the OLS estimates ^0,^1,^2 and ^3 ? (c) Suppose that the regressors X1i and X3i are related as follows: X1i=2X3i. Briefly discuss the problem which this presents to the OLS estimation of this model. (d) What is the interpretation of the slope coefficients in multiple regression models? Question 2 We saw that one way to account for an omitted variable is to estimate our regression of interest for each value (subsample) of the omitted variable (like we did for socioeconomic status in lecture 5). We also saw that we can the same thing by including the omitted variable as another independent variable. What are the advantages of using multiple regression over the subsample approach? Question 3 Imagine that you are working with a police department and are asked to investigate the relationship between intimate partner violence (IPV) and whether or not their local hockey team lost. You collect data for cities (i) with a hockey team in 2011 and are thinking about the following regression model: IPVi=0+1teamlosti+ui (a) Your police liason suggests that alcohol consumption is often a factor in IPV, do you think that alcohol consumption per capita is an omitted variable? Explain why or why not. (HINT: What two things do we need for omitted variable bias to be a concern? Are they present here?) (b) Would excluding it result in a positive or negative bias for your estimate of 1 ? Explain. (HINT: Have a look at slides 42-44 in Lecture 5. Think about the signs (positive or negative) of 2 and cov (teamlost, alcoholconsumption) and explain why those signs are they directions they are.) (c) What are some additional omitted variables you might want to control for? Suppose that you are investigating which factors determine someone's wages. You want to know how the length of time a worker has been with the same firm (something we call 'tenure' and which here is measured in months) and how a firm's size (number of employees) are related to wages. That is, you are interested in the following regression model: wagesi=0+1tenurei+2firmsizei+ui and your estimates give you the following table: Table 1: Heteroskedastic-robust standard errors in parentheses. (a) What is the interpretation of 1 and 2 ? (b) How many degrees of freedom do we have in this regression? (c) What is the t-critical value you would use to test H0:1=0H1:1=0 at the 5% significance level? Is this a one-tailed or two-tailed test? (d) What is the 95% confidence interval for 1 ? (e) You want to know if either of these two variables influence wages. Write out the null and alterantive hypothesis you would be testing. (f) After conducting the F-test, you compute a p-value of 0.00000000000000022. What can you conclude about the joint effect of tenure and firm size on wages? Formulas t=SE(^1)^11,0CI1=[^1t1,dfSE(^1),^1+t1,dfSE(^1)] Economics 2P91 Fall 2022: Practice problems Question 1 Consider the following multiple regression model: Yi=0+1X1i+2X2i+3X3i+ui (a) What problem does multiple regression help us address? (b) What is the minimum number of observations (sample size) required to obtain the OLS estimates ^0,^1,^2 and ^3 ? (c) Suppose that the regressors X1i and X3i are related as follows: X1i=2X3i. Briefly discuss the problem which this presents to the OLS estimation of this model. (d) What is the interpretation of the slope coefficients in multiple regression models? Question 2 We saw that one way to account for an omitted variable is to estimate our regression of interest for each value (subsample) of the omitted variable (like we did for socioeconomic status in lecture 5). We also saw that we can the same thing by including the omitted variable as another independent variable. What are the advantages of using multiple regression over the subsample approach? Question 3 Imagine that you are working with a police department and are asked to investigate the relationship between intimate partner violence (IPV) and whether or not their local hockey team lost. You collect data for cities (i) with a hockey team in 2011 and are thinking about the following regression model: IPVi=0+1teamlosti+ui (a) Your police liason suggests that alcohol consumption is often a factor in IPV, do you think that alcohol consumption per capita is an omitted variable? Explain why or why not. (HINT: What two things do we need for omitted variable bias to be a concern? Are they present here?) (b) Would excluding it result in a positive or negative bias for your estimate of 1 ? Explain. (HINT: Have a look at slides 42-44 in Lecture 5. Think about the signs (positive or negative) of 2 and cov (teamlost, alcoholconsumption) and explain why those signs are they directions they are.) (c) What are some additional omitted variables you might want to control for? Suppose that you are investigating which factors determine someone's wages. You want to know how the length of time a worker has been with the same firm (something we call 'tenure' and which here is measured in months) and how a firm's size (number of employees) are related to wages. That is, you are interested in the following regression model: wagesi=0+1tenurei+2firmsizei+ui and your estimates give you the following table: Table 1: Heteroskedastic-robust standard errors in parentheses. (a) What is the interpretation of 1 and 2 ? (b) How many degrees of freedom do we have in this regression? (c) What is the t-critical value you would use to test H0:1=0H1:1=0 at the 5% significance level? Is this a one-tailed or two-tailed test? (d) What is the 95% confidence interval for 1 ? (e) You want to know if either of these two variables influence wages. Write out the null and alterantive hypothesis you would be testing. (f) After conducting the F-test, you compute a p-value of 0.00000000000000022. What can you conclude about the joint effect of tenure and firm size on wages? Formulas t=SE(^1)^11,0CI1=[^1t1,dfSE(^1),^1+t1,dfSE(^1)]