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1. The following sample on the level of education (measured by the number of years of schooling) and wages (hourly) earned by 15 individuals is as follows:

Education    Wages ($)        Education         Wages ($)
6                  4.45                 10                 7.31
7                  5.57                 12                  7.82
8                  5.97                 14                 11.02
9                  7.33                 15                 10.67
10                7.31                 16                 10.83
11                6.58                 17                 13.61
6                 4.45                  15                10.67
18              13.53

According to the human capital theory education increases a worker’s productivity and thus leads to higher wages. Consider the economic model Wages β₀ + β₁ Education.

a. What sign would you expect β₀ and β₁ to take? Draw a rough scatter graph of the data with wages on the vertical axis and education on the horizontal axis. Does it look like the data support the human capital theory?

b. Estimate Wages β₀ + β₁Education + u. Give a plot of Wagesi. Interpret the intercept and the slope coefficient.

c. find out the standard error of β₀ and β₁

d. find out the 99% confidence interval for β₀ and β₁.

e. Test the null hypothesis β1 = 0 at 1% level of significance.

f. find out and interpret R2

g. describe the assumption of homoscedasticity.

2. Under the OLS estimation the following hold:

∑ u_i = 0, ∑ u_iX_i = 0, and ∑ u_i Y_i = 0.

Prove and describe the above properties.

3. Show that the OLS estimate of β₀ and β₁  is unbiased

4. Prove that the cov (β₀, β₁) =−X σ²/∑x_i²

5. For the following model derive the least square estimator of α, Y = α + u

Show that the OLS estimator for α is BLUE under Gauss Markov Assumptions.

Which assumption is important for the estimator to be unbiased?