Econometrics Review Questions

Review Questions
(1) Verify the following identity without expanding the determinant
1 a b c
1 b c a
0
1 c a b
a b c
(2) Use that p q r 25 , to compute the value of
u v w
2a 2c 2b
2u 2w 2v
2p 2r 2q
a b c
(3) Solve the following equation, using the properties of the determinant a x c 0
(4) Find the determinant of the matrix below
3 4 0 0 0
2 5 0 0 0
0 9 2 0 0
0 5 0 6 7
0 0 4 3 4
(5) Whenever possible compute the inverse of the following matrices;
1 0 x
2 1 0 1
4 5 2
x
A x
1
B 0 x 3
2
C 2 2 1
0 0 1
4 1 x
1 1 1
a b x
A11 A12
(6) Assume that A is a partitioned matrix given as A and that matrix B is the
A21 A22
B11 B12
inverse of A , where B . Show that
B21 B22
1 1 1 1 1 1
1 A11 A11 A12 A21A11 A11 A12
1
A
1 1 1
where,
A22 A21 A11 A12
A21A11
1

(7) Using partitioning compute the inverse of the following matrices;
1 0 0 0 1
2 3 0 0
2 0 0 0
0 1 0 0 1
5 2 0 0
B 0 0 1 0 1
C
0 0 0 1
D
0 0 4 3
0 0 1 0
0 0 0 1 0
1 1 1 0 1
0 0 3 2
0 1 0 0
(8) Find the rank of the following matrices
1 1 1
1 1 2 1
A
2 1 1
B 2 1 0 1
4 1 3
1 0 2 1
D 1 2
8 16
(9) Prove that
1 2 0
0 , 1 and
1
1 0 1
are linearly independent
(10) Let A and B be square invertible matrices of the same order. Solve for X in the
following equations;
t
(a) X A B
(b) 1 1
X A A B
(c) Solve for X in the preceding equation when
1 2 1
0 1 2
A 0 1 0
and B 1 0 1
3 1 2
1 1 1
(11)
mx
y
1
Given the system , compute m so that the system has;
x my 2m
1
(a) No solution
(b) Infinitely many solutions
(c) A unique solution
(d) A solution with x 3
(12)
kx y z 1
Consider the system x ky z 1 , use determinants to find those values of k for which
x y kz 1
the system has;
(a) A unique solution
(b) More than one solution
2

(c)
No solution
(13) Given the following matrix:
0 2 1
A 1 1 1
0 0 1
(i) Find the characteristic polynomial and eigenvalues
(ii) Show that the matrix is diagonalizable
(iii) Find the diagonal form of the matrix
(14) Given the following matrix:
0 4 3
B 4 0 0
3 0 0
(i) Find the characteristic polynomial and eigenvalues
(ii) Justify whether the matrix is diagonalizable and if so, find two matrices D and P
such that B PDP 1
(iii) Show how you can compute B 200 (Hint: just write it as the product of 3 matrices)
(15) Given the following matrix:
1 2 2
A
2 1 2
0 0 a
(i) Compute the characteristic polynomial and eigenvalues
(ii) For a 1, check that the matrix A is diagonalizable
(iii) Study for which values a 1the matrix A is diagonalizable
(16) Given the following subsets of
2 ;
2 2
E x, y : y x , y 1 x, x 0
(i)
(ii)
‣
‣
F x y xy y
2
, : 1
‣
2 2 2
G x, y : x 1 y 1, x 1
Sketch the graph of each subset above, its boundary and the interior
Study whether each subset above, is closed, open, bounded and convex, explain
your answer
(17) Let B be a subset of
2 . Discuss which of the following statements is true
3

(i) Int B B B
(ii) B 2 B B
(iii) B
is bounded
(iv)
(v)
B is closed if and only if B is open
B is bounded if and only if B is not bounded
B Bis open
(vi) B is closed if and only if
(vii) B is closed if and only if B
B
(18)
3x 3y z 0
Given the homogenous system 4x 2y mz 0 ,
3x 4y 6z
0
(a) Compute m so that it has no trivial solution and
(b) Solve it for that value
Q x, y, z x 2axy y z 2axz
(19) Given the quadratic form
2 2 2
(a) Determine for what values of a , the quadratic form Q is positive definite
(b) Determine for what values of a , the quadratic form Q is positive indefinite
(20) Solve the differential equation
dy 45
dt 2000 5t
y 80
(21) (i) Solve the differential equation of the logistic type given as P kPM P
where k
and M are positive constants.
(ii) How would the solution in (i) above change given the initial condition 0
P
0
P
3 ln 1, 0, satisfying 1 2
(22) Find the particular solution of xy y x x y
(23) Find the general solution of
xy 2y x
2
(24) Solve the equation
dy
3 , 0
dx
2
x x y x
(25) Find the solution to the differential equation dy xy x which satisfies 3 0
dx y when x
4

2
(26) Given the nonlinear differential equation y y 6y
5 . Construct the phase diagram
and test the dynamic stability using arrows and the derivative test.
(27) Given the demand function Qd
c bP and the supply function Qs
g hP . Assume
dP
that m Q
d
Q
s, for m 0 , find the conditions for dynamic price stability in the
dt
market.
(28) Assume that Q P,
Q P and P P Q Q
such that price is no
dt t st t t1
t st dt
longer determined by a market-clearing mechanism but by the level of inventory Q Q .
Furthermore, assume that 0
Qst
Qdt
will tend to reduce
price and a depletion of inventory Qst
Qdt
will cause prices to rise.
(a) Find the price P
t
for any period
(b) Comment on the stability conditions of the time path
since a build-up in inventory
(29) Given the demand function D( t) a0 a1p( t)
and the supply function S( t) b0 b1P ( t)
and assuming the rate of change of price is proportional to excess demand such that
dP
k D S , for k 0 , Show that if at 0, (0),
dt t p p the solution is given by
ka1b 1t
a0 b0
p p p 0
p e , where p
b1
a1
(30) Explain any three uses of differential equations in economics.
(31) State the rules for the integrating factor
(32) Using an example, explain the following;
(i) Second order, fourth degree differential equation
(ii) Partial differential equation
st
dt
(33) Find the general solution of
xy 2y x
2
(34) Solve the differential equation
y
x y
e
(35) Find y given that
and y
2
y x xy
0 1
2 0 1
(36) Find y given that y xy x and y
(37) Prove that
ˆ
ˆ
ˆ
2
MSE Var Bias
5

(38) The joint density function of two continuous random variables x and
c 2 x y 2 x 6, 0 y 5
f x,
y
0
otherwise
Find the following;
(i) The value of the constant c
(ii) The marginal distribution functions for x and y
(iii) The marginal density functions for x and y
(iv) P3 x 4, y 2
(v) Px
3
(vi) Px
y 4
(39) The joint density function of two continuous random variables x and
cxy 0 x 4, 1 y 5
f x,
y
0 otherwise
Find the following;
(i) The value of the constant c
P 1 x 2, 2 y 3
(ii)
(iii) Px
3, y 2
(iv) The marginal distribution function for X if 0x
4
(v) The marginal distribution function for Y if 1 y
5
(40) The joint density function of the random variables X and Y is given by
8 xy 0 x 1, 0 y x
f x,
y
0 otherwise
Find: (a) The marginal density of X
(b) The marginal density of Y
(c) The conditional density of X
(d) The condition density of Y
y is given by
y is given by
(41) The joint pdf of two random variables X and Y is given by
k 0 y x1
f x,
y
0 otherwise
Where k is a constant.
(a) Determine the value of k
(b) Find the marginal pdfs of X and Y
P 0 x 1 , 0 y
1
(c) Find
2 2
6

(42) Let X and Y be two random variables with the pdf given by;
2 2
2 2
x y
x y
2
f x, y
e x , - y
4
Show that X and Y are not independent but are uncorrelated
(43) The joint probability function of 2 discrete random variables X and Y is given by
f x, y c 2x y , where x and y can assume all integers such that 0 x 2, 0 y
3
and , 0
f x y otherwise
Find
(a) E(X)
(b) E(Y)
(c) E(XY)
(d) E(X 2 )
(e) E(Y 2 )
(f) Var(X)
(g) Var(Y)
(h) Cov(X,Y)
(44) A random variable X has density function given by
2x
2 e x
0
f x
0 x 0
Find (a) the moment generating function (b) the first four moments about the origin
(45) Find the first four moments (a) about the origin, (b) about the mean, for a random variable
X having density function
2
4x9 x 81, 0 x 3
f x
0
otherwise
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