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:
Yi = β0 + β1Xi + μi
Where Yi and Xi 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
Yi = β'0 + β'1Xi + 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 Yi 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 Yi sample values. Stating explicitly all required assumptions, prove that the OLS slope coefficient estimator β1 is an unbiased estimator of the slope coefficient β1.
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 Xi ≡ Xi-X'I and Yi ≡ Yi-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 Yi is annual R&D expenditures of the i-th firm (measured in millions of dollars per year) and Xi 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.
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Number of minutes of advertising
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Revenues
|
|
112
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1150
|
|
145
|
180
|
|
208
|
210
|
|
192
|
150
|
|
184
|
180
|
|
223
|
180
|
|
108
|
120
|
|
89
|
60
|
|
47
|
60
|
|
160
|
150
|
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 educi + 0.2 ln(xperi) + 0.1 malei
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 educi + 0.2 ln(xperi) + 0.1 malei
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.
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observation
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Labour-Hours of Work
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Output
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|
1
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10
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11
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|
2
|
7
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10
|
|
3
|
10
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12
|
|
4
|
5
|
6
|
|
5
|
8
|
10
|
|
6
|
8
|
7
|
|
7
|
6
|
9
|
|
8
|
7
|
10
|
|
9
|
9
|
11
|
|
10
|
10
|
10
|
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?