Assignment part 1:

1. All questions pertain to the simple (two-variable) linear regression model for which the population regression equation can be written in conventional notation as:

Y_i = β₀ +  β₁X_i + μ_i

Where Y_i and X_i are observable variables, β₀ and β₁ are unknown (constant) regression coefficients, and ui is an unobservable random error term. The Ordinary Least Squares (OLS) sample regression equation corresponding to regression equation (1) is

Y_i =  β'₀ +  β'₁X_i + μ_'i ( i= 1, ...., N)

where β₀ is the OLS estimator of the intercept coefficient β₀,  β'₀ is the OLS estimator of the slope coefficient Y_i is the OLS residual for the i - th sample observation, and N is sample size (the number of observations in the sample).

a) Show that the OLS slope coefficient estimator β₂ is a linear function of the Y_i sample values. Stating explicitly all required assumptions, prove that the OLS slope coefficient estimator β₁ is an unbiased estimator of the slope coefficient β_1.

b) Stating explicitly all required assumptions, prove that the OLS slope coefficient estimator β₁ is an unbiased estimator of the slope coefficient β₁

c) State the Ordinary Least Squares (OLS) estimation criterion. State the OLS normal equations. Derive the OLS normal equations from the OLS estimation criterion.

2. Explain what is meant by each of the following statements about the estimator θ of the population parameter θ. Show your working

a)  θ' is an unbiased estimator of θ

b) θ' is an efficient estimator of θ

3. A researcher is using data for a sample of 25 business schools that offer MBA degrees to investigate the relationship between annual salary gain of graduates Y₁ (measured in thousands of dollars per year) and annual tuition fees X₁ (measured in thousands of dollars per year). Preliminary analysis of the sample data

Where X_i ≡ X_i-X'_I and Y_i ≡ Y_i-Y'_i for i = 1, ......, N. Use the above sample information to answer all the following questions. Show explicitly all formulas and calculations

a) Use the above information to compute OLS estimates of the intercept coefficient β₀ and the slope coefficient β₁

b) Interpret the slope coefficient estimate you calculated in part (a) - i.e., explain in words what the numeric value you calculated for β₁ means.

c) Calculate an estimate of σ², the error variance

d) Compute the value of R², the coefficient of determination for the estimated OLS sample regression equation. Briefly explain what the value you have calculated for R² means

4. You have been commissioned to investigate the relationship between annual R&D expenditures (Y) and total annual sales revenues (X) for chemical firms. You have assembled data for a sample of 32 chemical firms, where Y_i is annual R&D expenditures of the i-th firm (measured in millions of dollars per year) and X_i is total annual sales revenues of the i-th firm (measured in millions of dollars per year). Your research assistant has used the sample data to estimate the following OLS sample regression equation, where the figures in parentheses below the coefficient estimates are the estimated standard errors of the coefficient estimates:

a) Compute the two-sided 95% confidence interval for the slope coefficient β.

b) Perform a test of the null hypothesis H₀: β₂ = 0 against the alternative hypothesis H₁: β₂ ≠ 0 at the 1% significance level (i.e., for significance level α = 0.01). Show how you calculated the test statistic. State the decision rule you use, and the inference you would draw from the test. Briefly explain what the test outcome means.

Assignment part 2:

1. Consider the following data on FIFA Sales Consultancy. This consultancy organization provides advice on the amount of media advertising firms should embark on to achieve a certain level of revenue. The economic consultancy firm collects data on the number of minutes required to advertise per month in order to achieve desired revenue in millions of kwacha.

a) Is the linear regression model appropriate? If so, why or why not?

b) If it is appropriate, find the fitted line and interpret it.

c) Forecast the Revenues if the firm's number of advertising minutes are increased to 300

2. The classical linear regression model CRLM is the mainstay of econometrics

a)

 i. Explain the CLRM assumptions discussed in class

 ii. Why is OLS appropriate when all these assumptions hold?

b) When all CRLM assumptions hold, it is important to also check the asymptotic properties. True or false? Justify your answer

3. Suppose you use OLS to estimate

In(wage) = 0.45 + 0.07 educ_i + 0.2 ln(xper_i) + 0.1 male_i

Where

wage: hourly in wage in dollars, educ: Years of education, xper: Years of work experience and male: 1 if person is male, =0 if female. You obtain the following fitted model

In (wage) = 0.45 + 0.07 educ_i + 0.2 ln(xper_i) + 0.1 male_i

a) What effects does education have on the wage, according to the estimates obtained?

b) How much dose an extra years of experience boosts the wage?

c) Again considering the above setup, suppose gender is correlated with years of education and has an effect on the wage. Adding a variable female which equals 1 if the person is female and equals 0 if the person is male will have which of the following effects

4.       

a) Is the linear regression model appropriate? Is your finding sensible? Why or why not?

b) Estimate the linear regression equation

c) Give an interpretation of the fitted line

d) What will be the value of the output if labour-hours of work are 11?