Problem 4. For Ye vectors of dimension n x 1 and B an n x n matrix, consider the model:I-B)Y...

Question

Problem 4. For Ye vectors of dimension n x 1 and B an n x n matrix, consider the model:
I-B)Y -p)=¢
such that:
(S1) (I — B) is nonsingula
(S2) €~ N(0,A), where A is diagonal with positive elements
(83) bi; =0 Vi
(a) Write the expression of the above model in the form ¥; =... to show that the model specifies
that each element of Y depends on the other elements of ¥ according to the elements of B,
except that the i" element does not depend on itself.
(b) Find E[Y].
(c) Find Var(Y).
(d) Show that Var(Y) is positive definite for any B defined as above.

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Ajay · Accepted Answer

Question a
 Y = μ + (IB)⁻¹€
 
Question b
 E[Y] = μ
 
Question c
Var(Y) = A
 
Question d
 Var(Y) is positive definite for any B defined as above.
Explanation:
Question a
· Y = μ + (IB)⁻¹€
· The model stipulates that each component of Y is dependent on the other components of Y in accordance with the components of B, with one exception: the ith component is not dependent on itself. To illustrate this point, let's rewrite the model so that it looks like Y = + (IB)1€.

Problem 4. For Ye vectors of dimension n x 1 and B an n x n matrix, consider the model: I-B)Y -p)=¢ such that: (S1) (I — B) is nonsingular (S2) €~ N(0,A), where A is diagonal with positive...

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