Gilead Sciences, Inc Assignment | Online Assignment

This is a final paper for the course. There is a series of questions, roughly 35 questions, that need to be answered in an essay/paragraph format, not ask-and-answer. We are to research a particular MNC and learn about their organization through the questions the professor provided. My selection was made for Gilead Sciences, Inc. I have attached all of the questions the professor is looking to be researched. If any questions arise please ask and I will answer within the hour

EC2019 Econometrics II: Assignment
This is the assignment that counts 90% of the total mark for this module. All questions must be
answered. All questions carry the same weight, 20% each.
The submission deadline is 3:00pm, Thursday 21st of May 2020. Plagiarism and collusion will not be
tolerated.
1. Consider the following process:
yt = c + 1yt􀀀1 + 2yt􀀀2 + t; 􀀀1 < t < 1 (1)
where c is a constant, and t satises the following assumptions:
E(t) = 0; 8t
E(2t
) = 2; 8t
E(tt􀀀s) = 0; 8s 6= 0
Assume that the roots of the polynomial, 1 􀀀 1z 􀀀 2z2 = 0, are real.
a) Assume that the process of (1) is stationary. Derive the Yule-Walker equations for the
process of (1). [50%]
b) Suppose 1 = 1. Find the range of values for 2 for which the process of (1) is stationary.
[50%]
2. Consider the following process:
yt = c + t 􀀀 1t􀀀1 􀀀 2t􀀀2 􀀀 3t􀀀3; 􀀀1 < t < 1 (2)
where c is a constant, and t is normally distributed with the following assumptions:
E(t) = 0; 8t
E(2t
) = 2; 8t
E(tt􀀀s) = 0; 8s 6= 0
Assume that the process of (2) is invertible.
a) Suppose T consecutive observations, (y1; y2; ; yT ), from the process of (2) were drawn.
Derive the log likelihood function for the conditional maximum likelihood estimation of the
unknown parameters of the process. [50%]
b) Let ^yt+hjt denote the optimal linear prediction of yt+h made at time t. Find ^yt+hjt for
h = 1; ; 4. Demonstrate that the ^yt+hjt = E(yt) for h 4, where E(yt) denotes the
unconditional mean of the process of (2). [20%]
c) Let eh denote the prediction error:
eh = yt+h 􀀀 ^yt+hjt
Find the prediction error variance, V (eh), for h = 1; ; 4. Demonstrate that V (eh) =
V (yt) for h 4, where V (yt) denotes the unconditional variance of the process of (2).
[30%]
1
3. Consider the following process:
yt = yt􀀀1 + t 􀀀 t􀀀1; 􀀀1 < t < 1; jj < 1; jj < 1 (3)
where t satises the following assumptions:
E(t) = 0; 8t
E(2t
) = 2; 8t
E(tt􀀀s) = 0; 8s 6= 0
a) Using the Wold Decomposition Theorem, the process of (3) can be expressed as an MA(1):
yt = 􀀀
1X
i=0
it􀀀i; with 􀀀 0 = 1 (4)
as (3) has no deterministic components. Derive the expressions for parameters i’s in terms
of and . [30%]
b) Given the invertibility condition of jj < 1, this process can also be represented as an
AR(1):
yt =
1X
i=1
iyt􀀀i + t; t iid(0; 2) (5)
Derive the expressions for parameters i’s in terms of and . [30%]
c) Derive
0 and
1, the unconditional variance of yt and its rst-order autocovariance respectively,
in terms of the unknown parameters of the process of (3). [40%]
4. Consider the following process:
yt = c + 1yt􀀀1 + 2yt􀀀2 + t; 􀀀1 < t < 1 (6)
where c is a constant, and t is normally distributed with the following assumptions:
E(t) = 0; 8t
E(2t
) = 2; 8t
E(tt􀀀s) = 0; 8s 6= 0
Assume that the process of (6) is stationary. Suppose we have drawn a sample of T consecutive
observations, (y1; y2; ; yT ), from the process of (6).
Let ~c, ~1, ~2, and ~2 denote the conditional maximum likelihood estimators of c, 1, 2, and 2
respectively. Let ^c, ^1, ^2, and ^2 denote the least squares (OLS) estimators of c, 1, 2, and
2 respectively. Derive ~2. Demonstrate:
~c = ^c
~1 = ^1
~2 = ^2
~2 6= ^2
2
5. Consider the following process:
yt = yt􀀀1 + t 􀀀 t􀀀1; 􀀀1 < t < 1; jj < 1 (7)
where t is normally distributed with the following assumptions:
E(t) = 0; 8t
E(2t
) = 2; 8t
E(tt􀀀s) = 0; 8s 6= 0
a) Assume jj < 1. Show that the process of (7) can be expressed as:
yt = t + ( 􀀀 )
1X
i=1
i􀀀1t􀀀i (8)
[35%]
b) Assume jj > 1. Show that the process of (7) can be expressed as:
yt = 􀀀1t 􀀀 ( 􀀀 )
1X
i=1
􀀀i􀀀1t+i (9)
Does the expression of (9) have any practical purpose? Explain your answer. [35%]
c) Suppose T consecutive observations (y1; y2; ; yT ) were drawn from the process of (7).
Derive the log likelihood function for the exact maximum likelihood estimation of the unknown
parameters of (7). [30%]
[Hint: for parts a) and b), write the process of (7) using the lag operator. Notice that if jj > 1,
j􀀀1j < 1. Then, the formula of the geometric sum to innity can still be used. For part c), use
the results obtained in Q3, part c).]
End of Paper
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