The following is the covariance matrix for three variables. Age and education ar

Question

The following is the covariance matrix for three variables. Age and education are measured in years. a. Calculate the coefficients in the following OLS regression and interpret the slope in words: log(earnings) = beta + beta_1 age b. Calculate the coefficients in the following OLS regression and interpret the slope in words: log(earnings) = beta_0 + beta_1 education c. Define experience = age - education-5. Calculate the coefficients in the following OLS regression and interpret the slope in words: log(earnings) = beta_0 + beta_1 experience

Explanation / Answer

Back-up Theory

Let, for convenience in explaining and presentation, Y represent log(earnings), X represent age, Z represent education, W represent (X – Z - 5), a represent OLS of 0 and b represent OLS of 1

Now, to work out solution,

Given data set is:

E(Y) = 10, E(X) = 40, E(Z) = 14, V(Y) = 2, V(X) = 160, V(Z) = 9, Cov(X,Y) = 1.5,

Cov(Z,Y) = 1 and Cov(X,Z) = - 1   

Part (a)

Given y = 0 + 1x, b = Cov(X,Y)/V(X) = 1.5/160 = 0.0094 and

a = E(Y) - b E(X) = 10 – (0.0094x40) = 9.624.

Interpretation of slope (b): when age increases by 1 year, the log(earnings) increases by 0.0094.

Part (b)

Given y = 0 + 1z, b = Cov(Z,Y)/V(Z) = 1/9 = 0.1111 and

a = E(Y) - b E(Z) = 10 – (0.1111x14) = 8.4446.

Interpretation of slope (b): when education increases by 1 year, the log(earnings) increases by 0.1111

Part (c)

Now, W = (X – Z - 5) => E(W) = E(X) – E(Z) – E(5) = 160 – 9 – 5 = 146.

V(W) = V(X) + V(Z) – 2Cov(X, Z) = 40 + 14 – (2x-1) = 56.

Cov(W,Y) = E[{W - E(W)}{Y - E(Y)}] = E[{X – E(X)}- {Z – E(Z)}][{Y - E(Y)}]

= E[{X – E(X)}{Y - E(Y)}] - E[{Z – E(Z)}{Y - E(Y)}] = Cov(X, Y) - Cov(Z, Y)

= 1.5 – 1 = 0.5.

Given y = 0 + 1w, b = Cov(W,Y)/V(W) = 0.5/56 = 0.0089 and

a = E(Y) - b E(W) = 10 – (0.0089x146) = 8.7006.

Interpretation of slope (b): when experience increases by 1 year, the log(earnings) increases by 0.0089

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