LAW OF ITERATED EXPECTATIONS Law of Iterated Expectations Theorem 1 Law of iterated expectation.s E团=E[E[-] The notation Er[ indicates the expectation over the value of a Example 1 y=bx U0,1 E and a are independent EIy=E[bz + ea E[bx=]+ellul Since e and a are independent, EE=Ee=0, eyal Now, using the law of iterated expectations and E回=5 EIy= Ereli Er[b bEr al Matrix Algebra( Continue) Definition 1 Idempotent Matrit An idempotent matrit is one that is equal to its square, that is, M-= MM= M Example 2 The identity matric I Definition 2 Linear Depende A set of vectors is linearly dependent if any one of the vectors in the set can be written as a linear combination of the othersLAW OF ITERATED EXPECTATIONS 1 Law of Iterated Expectations Theorem 1 Law of Iterated Expectations E [y] = Ex [E [y|x]] The notation Ex [·] indicates the expectation over the value of x. Example 1 y = bx + ε ε ∼ N (0, 1) x ∼ U [0, 1] ε and x are independent. E [y|x] = E [bx + ε|x] = E [bx|x] + E [ε|x] Since ε and x are independent, E [ε|x] = E [ε] = 0, E [y|x] = bx Now, using the law of iterated expectations and E [x] = 1 2 E [y] = Ex [E [y|x]] = Ex [bx] = bEx [x] = 1 2 b Matrix Algebra (Continue) Definition 1 Idempotent Matrix An idempotent matrix is one that is equal to its square, that is, M2= MM = M. Example 2 The identity matrix I I 2 = I · I = I Definition 2 Linear Dependence A set of vectors is linearly dependent if any one of the vectors in the set can be written as a linear combination of the others