Statistical Method in Economics

Statistical Method in Economics For any help regarding Economics Homework Help Visit : https://www.economicshomeworkhelper.com/ Email - info@economicshomeworkhelper.com or call us at - +1 678 648 4277 Economics Homework Help

Statistical Method in Economics Problems Note: The first four problems are required, and the remaining two are practice problems. If you choose to do the practice problems now, you will receive feedback from the grader. Alternatively, you may use them later in the course to study for exams. Credit may not be awarded for solutions that do not use the methods discussed in this class.

Problem 1 Let E be a continuous random variable with pdf ƒ E (x) , and let a, b and c be constants. a.Prove that for any functions gfi (x) and gX (x) whose expectations exist, E (agfi ( E ) ‡ bgX ( E ) ‡ c) = aE [gfi (E)] ‡ bE [gX (E)] ‡ c. b.

Can you also state that: E (gfi (E) ) = gfi (E [E])?

c.

What can you say about V av (gfi ( E ) ‡ gX (E))?

Problem 2 Suppose that a random variable E f i is distributed uniform [0, fi] , EX is distributed uniform [0, 2] and E 3 is distributed uniform [0, 3]. Assume that they are all independent. d.

Calculate E [(E fi — 2EX ‡ E 3 )] . Σ

Σ

e.

Calculate E (Efi — 2EX ‡ E3)X .

f.

Use Letyour E and result Y befrom two parts random a. and variables, b. to calculate where ƒ V av [(E fi — 2EX ‡ E 3 )] .

fi , ƒ3E,Y (2, fi) = Problem

4

fi

, and ƒE,Y (x, y) = 0 otherwise.

4

a. is the correlation between E and Y ?

What b. Are E and Y independent?

Economics Homework Help

E ,Y(—fi,

fi) = , ƒ

fi X

E ,Y

(0, 2) =

c.

What do you conclude about the relationship between correlation and independence?

Problem 4 Suppose that you2 are considering undertaking a project where the revenue, E f i , has a distribution: ƒ E (x f i ) =

fi

if xfi = 8 . The project cost, , fi

.

Yfi, is distributed: ƒYfi (y fi ) =

. Σ fi if otherwise y c fi, 0 fi X

3

if xfi = 0

4

.

4 , 0 otherwise 3

X . Assume that the revenue

and the cost are independent. a.

What is the expected profit (revenue minus cost) of the project? b.

What is the probability that the project makes a positive profit?

c. d.

What the variance theEprofit of the project?

Now is assume instead of that f i and Yfi are not independent, but in† , if 8xfi = 0 .

3 3 . Σ stead that ƒ E †Y fi fi= 8 2 , 3 xfi|Yfi = X = if x fi = 8 . Now what is the ex0 otherwise pected value and variance of the profit? e. Assume again that the cost and revenue are independent, and sup- pose that now a friend can undertake a different project where the revenue (E X ) is distributed identically to E f i and the the cost (YX) is distributed identically to Yfi. All the variables E f i , Yfi, E X , YX are independent. Before you undertake either project your friend suggests that you equally share the revenue and costs of both projects. What is the expected profit to accepting the suggestion relative to not taking up any project? What is the probabil- ity that you make a positive profit if you accept your friend‘s suggestion? And what is the variance of the your share of the combined profit?

Problem 5 A student is enrolled in n classes. For each class, the probability that the student has an assignment due on a given day is independent, and equal to 0.F. The total time (T ) it takes a student to compete h assignments is t fi— 2h distributed according to the pdf ƒ (t) = e for h > 0. The T student Xh spends no time on homework if nothing is due. a.

If the student is enrolled in 4 classes, what is the expected value of h? What is the variance of h? b.

If h = 2, what is the expected value of T ?

c.

If the student is enrolled in 4 classes, what is the expected value of T ? What is the variance of T ?

Economics Homework Help

Problem 6 Suppose that there are two kinds of stock, and that each of them currently sells for the same price in the stock market. The return on one share of the first stock is a random variable E with E ( E ) = µ and V av ( E ) = aX. The return on one share of the second stock (Y ) also has E (Y ) = µ and V av (Y ) = aX. Consider the following investment plans: Plan fi: Buy fi0 shares of the first stock.

Plan 2: Buy † shares of each stock. a.

Compute the average return for each of the two plans and compare. b.Suppose that the two plans are independent. What is the variance on the return to Plan 2? c.

Now suppose that the two plans are correlated with g =

fi

. What

X

is the variance on the return to Plan 2? d. Now suppose that the two plans are correlated, but g is unknown. If you are risk-averse (i.e. you prefer less variance for a given expected return), which plan do you prefer? Why?

Economics Homework Help

Solutions Problem 1 a.

E (agfi ( E ) ‡ bgX ( E ) ‡ c) = = =

a

∫ œ (agfi (x) ‡ bgX (x) ‡ c) ƒ E (x) dx ∫ —œ ∫ œ œ ag (x)fi ƒ (x) Edx ‡ bgX (x) ƒ E (x) dx ‡ —œ —œ ∫ œ ∫ œ g (x) ƒ (x) g (x)X ƒ (x)Edx ‡ c fi dx ‡E b

—œ

—œ

∫

œ

cƒ (x)E dx

—œ

∫

œ

—œ

ƒ (x) E dx

= aE [gfi (E)] ‡ bE [gX (E)] ‡ c b.

No, because gfi (x) is not necessarily linear. For example take ∫ X E (g (E)) = g (x) = x , E ~ fiU [0, fi]. Here: x dx = , but: gfi (E [E]) = O fi X fi 3 fi . ∫ fi ΣX xdx = O. Thus: E (g4 ( E))fi > g (E [E]). More generally, fi fi Jensen‘s Inequality states that for a strictly convex function gfi (E [E]) (such as gfi (x) = xX): E (gfi ( E)) > gfi (E [E]). For a strictly concave function gfi (E [E]) : E (gfi (E) ) c gfi (E [E]) . c.

V av (gfi ( E ) ‡ gX (E)) =

.

Σ

E (g f i ( E ) ‡ gX (E)) X — (E (gfi (E) ) ‡ E (gX (E))) X .

Σ

= E g f i (E) X — (E (gfi (E))) X .

Σ

‡ E g X (E) X — (E (gX (E))) X ‡2 (E (gfi ( E ) gX (E) ) — E (gfi (E) ) E (gX (E))) =

V av (gfi (E)) ‡ V av (gX (E)) ‡ 2Cov (gfi ( E ) , gX (E) )

which aligns with the general formula for the variance of a sum of random variables.

Problem 2 a.

E [(E fi — 2EX ‡ E 3 )] = E [Efi]—2E [E X ]‡E [E 3 ] = 2 (0.† — 2 (fi) ‡ fi.†) = 0

Economics Homework Help

2

Problem b.

.

We first find ƒ E f i , E 2 , E 3 (xfi , xX, x 3 ) =

fi 6

if x cfi [0, fi] , x cX [0, 2] , x c3 [0, 3] 0 otherwise

.

X

We can then define g (E f i , EX , E 3 ) = (E f i — 2EX ‡ E 3 ) . It follows that Σ

E (Efi — 2EX ‡ E3)X

Σ

= E [g (E f i , E∫ X , 3E∫ 3 )]X∫ = = = = =

O 3∫

O X

fi

fi ∫

X

Σ6 fi

X

(x fi — 2xX ‡ x 3 ) dxfidxXdx3

Σ X ‡ 4xX ‡X x3 — 2xX — 4xXx3 ‡ x3 dxXdx3 O O Σ ∫ 3Σ 22 3 X 6 ‡ 2x3 — 6x3 dx3 O fi 3 (22 ‡ fi8 — 2F) 6 6 fi3 6 Σ (E [Z]) , Xso: V av [(E — 2Efi‡ E )]X = E3 ( E — 2E ‡fi E ) X—

Σ Σ c.

V av [Z ] = E Z — X (E [(E — = X— 0 fi3 =6 fi 2E ‡X E )]) 3 fi X

O

fi

fi3 6

Problem 3 a.

E [E] = 0, E [Y ] = → E †4[E] E [Y ] = 0. Now: E [EY ] = fi = 0 → Cov(E, Y ) = E [EY ] — E [E] E [Y ] = 0 — 0 = 0 → (—fi) ‡ (2) 4

Cov(E,Y )

Covv(E, Y ) = ,2Y a v ( E ) Y av(Y ) = 0 so E and Y are uncorrelated. . fiΣb. . 3Σ E and Y are not independent, for example: ƒE,Y (0, fi) = 0 ƒ= = ƒ E (0) ƒY (fi) .

4 4 c. You saw in class that independence implies uncorrelatedness. This exercise proves that two uncorrelated random variables are not necessarily independent.

Problem 4 a.

E [E —fiY ] =fi E [ E ] — fi E [Y ]fi=

8—

. Σ . fi ‡fi4 = fi .X

3Σ X

3 4

fi 4

b.

Pr [E —fiY > 0] fi = Pr [E = fi8] = .

c.

V av (E —fi Y ) fi= V av (E )fi ‡ V av (Y fi) — 2Cov (Efi , fiY ) = fi2 ‡ —

fi6 .

0 = fi93

d. The expected value of the profit is unaffected (because profit is a linear function of revenue and cost). However, the variance will change because the covariance term in the above expression is no longer equal to

Economics Homework Help

Σ 3

X

zero. To find the covariance, we will first need to compute the joint distrib- ution of E f i and Yfi. We calculate the joint distribution by multiplying the conditional pdf of Efi |Yfi by the marginal distribution of Yfi. So we have , † for .0, 3Σ ... fi63 for .8, X3Σ fi6 X F ƒ E Yfi(x fi fi , y fi ) = fi6for (0, fi) fi fi6 for (8, fi) ., 0 otherwise , . ,

Note that we used the law of total probability to calculate ƒ F 8 if x = fi0 fi 8 if x fi = 8 . Then we calculate the covariance: 0 otherwise Cov (E f i , Yfi) = E (E fi Y fi ) — E (E f i ) E (Yfi) † 3 F = 0‡ fi2 ‡ 0‡ fi6 fi6 . Σ . fi6Σ . Σ . fi = 4 Thus the variance of the profit is now equal to

fi8†

.

(xfi|Yfi = fi) =

E fi†Yfi =fi

fi fi6Σ

8— 2

.

fi6

e.

The expected value of your share of the combined profit is EΣ(E 3 .

The probability of making a positive profit is now Pr fiX (E f i — Yfi ‡ EX — YX) > 0 = 4 Pr [E . fi=Σ8. UΣE = X8] = Pr [E = 8] ‡fi Pr [E = 8] — X Pr [E = E fi = 8]X= ‡ fi fi fi F And because all of the variables are 4 4 = 4 — . independent, Σ fi6 . the variance of your share of the profit is V av ( E — Y ‡ E fiX— Y )fi = fi X X fi fi fi fi fi93 .

4 V av (E f i ) ‡ 4 V av (Yfi ) ‡4 V av (E X ) ‡ 4 V av (YX) = 3X Problem 5 There are three variables in this problem: N

=

number of classes a student is enrolled in

K

=

number of assignments due

T

=

total time to complete K assignments

We know that K has a binomial distribution for a given value of N, with —t 2K p = 0.F and that T |K has the pdf e forXK tf i≤ 0. a.

We are given that N = 4, and want to find the probability that K = 2. We use the binomial distribution . Σ 4 X X Pr ( K = 2|N = 4) = 2 (0.F) (0.3) = 0.2646

Economics Homework Help

Σ † 4

Σ fi

fi

X fi 4

— ΣYfi ‡ E X— YX) =

Σ

4

b.

Now we want the expected value of T , given that K = 2. ∫ œ t — te 2K dt E (T |K = 2) = 2K O ∫ œ t e— t 4 dt = O ∫ œ —t Σ 4—t Σœ 4 = —te 4 ‡ e dt O O Σ Σ œ —t —t 4 4e 4 = —te — O = 4

c.

I t will be most straight-forward to calculate the expected value of T by using the law of iterated expectations.

E (T ) = E K (E T (T |K)) We can generalize from part b to see that E (T | K) = 2K. We use the fact that N is . Σ fixed at 4 and get 4 Σ 4 = 2K (0.F)K (0.3)4—K K E (T ) = E K (2K)K =O = †.6 We will use the conditional variance identity to find V av (T ) V av (T ) = E(V av(T |K)) ‡ V av(E(T |K)) X ∫ But first,. we need to find V av(T |K).

First we need E(T |K). Σ œ tX 2—tK e dt E T X|K = O ∫ œ Σ X 2K—t Σœ —t 2K = —t e 2K O ‡ 2te dt O ∫ œ —t Σ X —t Σœ Σ —t Σœ 2K = —t e 2K O ‡ —4Kte 2K O ‡ 4K e dt O Σ X —t Σœ Σ Σ Σ œ —t —t Σœ = —t e 2K O ‡ —4Kte 2K O ‡ —8K e X 2K =

O

8KX

So V av(T | K) = E(T X |K) — (E(T |K)) X = 4KX And now V av (T ) = E(4K X ) ‡ . V av(2K) Σ X 4 (E (K) ) = 8E K — = 8 (8.68) — 4 (2.8) = 38.08

Economics Homework Help

X

X

Problem 6 In parts b-d, plans (with a lower-case p) should be replaced with stocks.

Sorry about the typo. a. b. c.

Plan fi: E (fi0E) = fi0E ( E ) = fi0µ Plan 2: E (†E ‡ †Y ) = †E ( E ) ‡ †E (Y ) = fi0µ Thus, both plans yield the same expected return.

V av (†E ‡ †Y ) = 2†V av ( E ) ‡ 2†V av (Y ) — †0Cov (E, Y ) = †0aX First note that g =

Cov(E,Y ) ¸ implies ¸ a

2

a

2

Cov (E, Y ) =

for g =

X.

2 aX

fi X

a 2 V av X(†E ‡ †Y ) = 2†V av ( E ) ‡ 2†V av (Y ) — †0Cov (E, Y ) = †0a ‡ Thus, †0 X= F†a . d. Since both plans have the same expected return, a risk-averse in- vestor will prefer the one with the lower variance of the return. For Plan fi, we have V av (fi0E) = fi00aX. For Plan 2, we have V av (†E ‡ †Y ) = 2†V av ( E ) ‡ 2†V av (Y ) ‡ †0Cov (E, Y ) = †0aX ‡ †0gµX. Because —fi ≤ g ≤ fi, the variance of the return to Plan 2 is weakly less than the variance of the return to Plan fi. Thus even for an unknown g, Plan 2 is preferred.

Economics Homework Help