To prove the statement that full independence between X and u implies E[u|X] =0, (1) (i.e.,...
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To prove the statement that "full independence between X and u implies E[u|X] =0", (1) (i.e., full independence between X and u implies that the OLS Assumption 1 holds), we use the following steps. First, we show if X and u are fully independent from each other, then E[u\X] = E[u], (2) i.e., the conditional expectation of u given X is equal to the unconditional expectation of u. Second, we can show that it is always fine to assume E[u] = 0. (3) To see this, suppose for the model Y = Bo+B₁X+u, E[u] = 10. Define a new intercept 30 and a new error term u, so that the new model can be written as: Y = Bo + B1X +ũ, and E[u] = 0, where Y, X, and 3₁ are exactly the same as the original model. What is the relationship be- tween Bo and Bo? Explain by graphs that the original and new models are in fact equivalent to each other. To prove the statement that "full independence between X and u implies E[u|X] =0", (1) (i.e., full independence between X and u implies that the OLS Assumption 1 holds), we use the following steps. First, we show if X and u are fully independent from each other, then E[u\X] = E[u], (2) i.e., the conditional expectation of u given X is equal to the unconditional expectation of u. Second, we can show that it is always fine to assume E[u] = 0. (3) To see this, suppose for the model Y = Bo+B₁X+u, E[u] = 10. Define a new intercept 30 and a new error term u, so that the new model can be written as: Y = Bo + B1X +ũ, and E[u] = 0, where Y, X, and 3₁ are exactly the same as the original model. What is the relationship be- tween Bo and Bo? Explain by graphs that the original and new models are in fact equivalent to each other.
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Auditing and Assurance Services A Systematic Approach
ISBN: 978-1259162343
9th edition
Authors: William Messier, Steven Glover, Douglas Prawitt
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