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and Cov( 6;, 6;) = 0, (1 7 ]; 2, ] = 1, . .., IV).
. Theoretical Exercise 4: OLS with 2 regressors.
Consider the following model: yi = Bo + Bidi te;, i = 1,...,5, where y = (5, 2, 3, 2, -2)T
and x = (3/2, 1, 1/2, -1/2, 0). Do this exercise by hand.
1. Find Enix?, Zixiyi, i and y.
2. Compute the OLS estimates Bo and B, for this model3.


3. On a graph, plot the data points and the fitted regression line (33,, = 30 + 3122,).
4. Interpret the parameter estimates, Bo and 31. 5. On your graph in part 3, locate the point of the means, (5, 3}). Discuss what you see.


. Theoretical Exercise 5: OLS estimator in a model with a constant only.
Consider the following model with only the constant as a regressor:
yi = B+ci, i=1, . .., N
6; is iid with E(ci) = 0 and Var(e) = 02. Compute the OLS estimator, Bots, in the two
following ways:
1. Minimize the Sum of Squared Residuals (SSR) with respect to 3 where:
SSR( B) = E ((B) ) = E(yi - B)2
i= 1
2. Let X = [1, ..., 1]" denote the vector of regressors, that is an N dimensional vector of
1s. Compute (XTX)-1XTy.
3. Compare your answer to (a) and (b).
4. Compute the variance of BOLS. How does it depend on the sample size? Hint: Cov(ci, (;) =
0 for i # j since e is iid. Var(azi + bz2) = a Var(21) + b'Var(z2) + 2ab Cov(21, Z2).
5. What does the Law of Large Numbers tell you about limN t. BOLS?
6. What does the Central Limit Theorem tell you about limN-+ VN (BOLS - B)?


4. Bonus points] Let V = (ETWE)-1(ETWSWE)(ETWE)-1. Choose a value of S such
that V can be simplified to V = (E"WE)-1. Hint: S could be a function of E and /or
W. Show the steps you used to arrive at the result.

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