1. (20 Pts) Concepts (a) 5 pts) In the context of the OLS inodel explain the dif

Question

1. (20 Pts) Concepts (a) 5 pts) In the context of the OLS inodel explain the difference between and , between the residual E, and the regression error, Ei; and between the OLS predicted value Y and EYX (b) (5 pts) What are the four fundamental subspaces of a matrix? What is their formal definition and how are they (c) (5 pts) What is the strict exogeniety assumption in regards to the OLS estimator and what does it gain us in terms (d) 5 pts) Assume we have a linear model, specifically Yi-Ao+, Xits. Propose an estimator for A that s unbiased related to each other? of the properties of the estimator? but inconsistent.

Explanation / Answer

a) Difference between beta hat and beta:

beta correspons to the population parameter that we are interested to calculate

Since collecting the data for the entire population is difficult we take a sample data and estimate the beta hat

This beta hat of the sample is a representative of the beta of the population

Difference between epsilon and epsilon hat:

Epsilon of a data is the difference or the error value from the actual Y value and the estimated Y value from the equation of the population

Epsilon hat is the etimated error from the sample data when we get the y hat values or the estimated values using sample of x data

Epsilon hat is an approximation of epsilon value of the population

Difference between Y hat and E[Y|X]

Y hat is the value that is predicted from the regression equation

E[Y|X] is the expected value or conditional expectation of Y given that X has already occurred

b)

4 fundamental subspaces of a matrix:

1. Column space

2.Row space

3.Null space

4. Transpose null space

1. The column space has all the linear combinations of the columns

2. The Row space has all the linear combinations of the rows

3. The Null space has all the solutions to the equation Bx=0 where B is a matrix

4.Transpose null space has all the solutions to the equation B transpose * x=0 where B is a matrix

Relations :

The Row space of B transpose and Null space of B are orthogonal to each othr

The column space of B and Transpose null space of B are orthogonal

c)

Strict exogeniety assumption in OLS estimator  

E(epsilon|X)=0

This means that the conditional mean value of the function on X is equal to zero

Even when the function of X is non linear the value is zero

Gains from the use of strict exogeneity:

1.The unconditional mean of the error term is considered zero

2.The regressors are uncorrelated with the error terms

d)

We can consider a estimator K= beta1 + Mu

Here the estimator will be unbiased as E(Mu)=0

But the estimator will be inconsitent because

as K tends to beta+Mu but it should only tend towards beta

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