## Using n = 19 and B = 100 obtain the statistical size and power of the t-test for testing H0 : = 0 against H1 : = 0:1 and against H1 : = 0:4 with signicance level = 5%. Explain which distribution is employed to calculate the size of the test.

Instructions
1. Provide a typed report on the following questions. Include the Stata code, Stata outputs (tables and graphs) and the relevant econometric explanation.
2. Hand written solutions will not be accepted. The overall quality of the answers and explanation the points below in adequate manner will be taken into account when allocating the grades for each part of the questions.
3. Submit your assignment on Blackboard by Monday 2 December 2019.

Question 1. Labour cost is an important indicator of employment and economic conditions. Belgium has one of the highest labour costs among European countries. We use the dataset Labour to examine the wage costs of workers in a set of Belgium rms using the following four variables.
capital, which is the total xed assets, end of 1995 (in 1,000,000 euro) labour which is the number of workers (in employment) output which is the value added (in 1,000,000 euro) wage which is the wage costs per worker (in 1000 euro).
A. Estimate the Linear Regression Model in Stata given by
wagei = + labouri + ui; for i = 1;:::;n (1)
Report the estimated OLS equation (estimated parameters and standard errors), the sample size and R-squared of the model. Give an economic interpretation of the model estimates ^ and ^ [5 points].
B. Estimate the Multiple Linear Regression Model by rst demeaning all variables.
wagei = + 1labouri + 2capitali + 3outputi + ui; (2)
Report the estimated OLS equation (estimated parameters and standard errors), the sample size and R-squared of the model. Give an economic interpretation of the model estimates ^ , ^ 1, ^ 2, ^ 3 [15 points]. C. Conduct a t-test for the null hypothesis = 0 in (1). [15 points]. D. Conduct a t-test for the null hypothesis 1 = 0 in (1). In which way is this hypothesis test diāerent that that of part C? [15 points]

Question 2. In this question we conduct a small simulation study.
A. Presentation Table the four decisions of hypothesis testing and give the probabilistic statements which represent the Type I and Type II errors. What is the statistical size and power of a test? [10 points]. B. Use the model yi = i + ui, so that xi = i and ui i.i.d N(0;1) for i = 1;:::;n. Write a short program in Stata for a simulation study using the following parameters. (i) n = 19, = f0;0:1;0:4g, and B = 100. (ii) n = 100, = f0;0:1;0:4g, and B = 100. where B is the number of replications. In each case store the values of ^ , s:e:(^ ) and the t-statistic
T =
^ s:e:(^ )
: (3)
Report your code, summary statistics and histogram for both of these statistics. Comment on the distribution of the estimates. How does these results relate to the distributional results we covered in lectures? [20 points].
C. Using n = 19 and B = 100 obtain the statistical size and power of the t-test for testing H0 : = 0 against H1 : = 0:1 and against H1 : = 0:4 with signicance level = 5%. Explain which distribution is employed to calculate the size of the test [20 points].
Hint. For the purpose of this simulation study you can use the build-in function of Stata simulate.