MLR

Paulo Fagandini

Nova SBE

Review

Review

• Matrix/vector SLR
• MLR
• \(\beta_i=\frac{\sum \hat{r}y}{\sum \hat{r}^2}\) with \(\hat{r}\) the residuals of \(x_i=\gamma_0+\sum_{j\neq i} \gamma_j x_j+\varepsilon\)

Exercises

https://moodle.novasbe.pt/mod/folder/view.php?id=4021

Exercise 2.1

On 18^th September of 2015, during apractical class taking place at Nova SBE, a survey was delivered to 52 promising future leaders. This survey collected information on their GPA and on how many hours they usually spend per day in each of the five activities: study, sleep, work, sport, and leisure. Any activity was put into one of the five categories, so that ofr each student the sum of hours in the five activities was 24. Consider the following population regression model:

\[GPA=\beta_0+\beta_1 study+ \beta_2 sleep + \beta_3 work + \beta_4 sport + \beta_5 leisure + u\]

Exercise 2.1

1. Does it make sense to hold sleep, work, sport, and leisure fixed, while changing study?
2. Explain why this model violates assumption MLR.3 (No perfect collinearity). What is the implication for the OLX estimators?
3. How could you reformulate the model so that its parameters have a useful interpretation and it satisfies Assumption MLR.3?

1. By definition \(sleep+study+work+sport+leisure=24\). Changing study requires something else to change.
2. Each indep. var. is a l.c. of all others. It is not possible to obtain estimates.
3. Drop any variable, for example sleep. Then the \(\beta_{study}\) can be interpreted while holding work, sport, and leisure constant.

Exercise 2.1

4. After omitting the variable sleep the model was estimated by OLS. What would your advice to these promising future leaders?

\[ \widehat{GPA}=22.81-0.57 study - 0.20 work - 0.38 sport - 0.51 leisure\]

Studying one more hour a day, implies sleeping one hour less! Thus studying one more hour a day (while sleeping one hour less) reduces GPA by 0.57, on average, certeris paribus.

Exercise 2.3

The following model is a simplified version of the multiple regression model used by Biddle and Hamermesh (1990) to study how total hours work, years of education and age affect sleeping:

\[sleep=\beta_0+\beta_1 totwrk + \beta_2 educ + \beta_3 age + u\]

where sleep and totwrk are measured in minutes per week and educ and age are measured in years.

Exercise 2.3

1. If adults trade off sleep for work what is the sign of \(\beta_1\)?
2. What signs do you think \(\beta_2\) and \(\beta_3\) will have?

1. If there is a trade-off \(\beta_1<0\)
2. It is unclear!

Exercise 2.3

Using real data, the authors have estimated the following relationship:

\[\widehat{sleep}=3,638.25-0.148totwrk-11.13educ+2.20age\]

With \(n=706\) and \(R^2=0.113\).

3. If someone works five more hours per week, by how many minutes is sleep predicted to fall? Is this a large trade-off?
4. Discuss the sign and magnitude of the estimated coefficient on educ.
5. Would you say totwrk, educ, and age explain much of the variation in sleep? What other factors might affect the time spent sleeping? Are these likely to be correlated with totwrk?

3. \(\Delta sleep=-0.148(5\times 60)=-44.4\) minutes. Not a very large trade-off
4. 1 more year of education reduces sleeping time by 11 minutes, on average, ceteris paribus.
5. The three variables only explain 11% of the variability/variation of sleep. Marital stautyes, number/age of children, type of job, health, …, and are likely correlated with total work.