Self-Entry-Level Test Institute of Economics Institute for Statistics and Econometrics Kiel Universtiy, Germany Dear Candidate, the following questions are not an examination in the usual sense, but merely a voluntary test. The main purpose of this test is to help YOU to find out whether you might be a suc- cessful participant in our master’s degree programme. The knowledge required to answer these questions - without much looking-up and without any (!) learning - is what we expect you to bring with you, as part of the skills acquired in your bachelor studies. In the application process for any of our master’s degree programmes, we can only evaluate you based on your grades and credit points. We cannot take into account your answers to these questions since we do not know the amount of help you might have used. But if you fail to answer 40% of the test questions easily and decide to come to Kiel anyway, the probability of you leaving the programme after the first examinations without success is increased - wasting valuable time and money. So, please be honest to yourself when answering the following questions! Questions of the following sections are relevant for the single programmes: Economics: Mathematics, Econometrics, Statistics, Microeconomics, and Macroeconomics Quantitative Economics: Mathematics, Econometrics, Statistics, Microeconomics, and Macroeconomics Quantitative Finance: Mathematics, Econometrics, and Statistics Environmental and Resource Economics: Mathematics, Econometrics, and Microeconomics 1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Self-Entry-Level Test
Institute of Economics
Institute for Statistics and Econometrics
Kiel Universtiy, Germany
Dear Candidate,
the following questions are not an examination in the usual sense, but merely a voluntary
test. The main purpose of this test is to help YOU to find out whether you might be a suc-
cessful participant in our master’s degree programme. The knowledge required to answer these
questions - without much looking-up and without any (!) learning - is what we expect you to
bring with you, as part of the skills acquired in your bachelor studies.
In the application process for any of our master’s degree programmes, we can only evaluate you
based on your grades and credit points. We cannot take into account your answers to these
questions since we do not know the amount of help you might have used. But if you fail to
answer 40% of the test questions easily and decide to come to Kiel anyway, the probability of
you leaving the programme after the first examinations without success is increased - wasting
valuable time and money.
So, please be honest to yourself when answering the following questions!
Questions of the following sections are relevant for the single programmes:
Economics:
Mathematics, Econometrics, Statistics, Microeconomics, and Macroeconomics
Quantitative Economics:
Mathematics, Econometrics, Statistics, Microeconomics, and Macroeconomics
Quantitative Finance:
Mathematics, Econometrics, and Statistics
Environmental and Resource Economics:
Mathematics, Econometrics, and Microeconomics
1
Mathematical Questions
1. Determine the solution to the following integrals:
(a)∫
2x + ex/8 + cos(2x)dx
(b)∫ √
4x−2/3
+ 4x4dx, x > 0
(c)∫ 10
1exp(− ln(a
x))dx, a > 0
(d)∫ b
01
exp(2x)dx, b > 0
(e)∫ π
0cos(x)dx
2. Determine the solution to the following integrals by partial integration:
(a)∫ 1
1/2ln(4x2)dx
(b)∫x2θ exp(−θx)dx, θ > 0, x > 0
3. Obtain the first and second order derivatives for the following functions at t=0:
(a) f(t) = (pet + 1− p)n, n ∈ N, p ∈ (0, 1)
(b) f(t) = exp(µt+ 12σ2t2), σ > 0
4. Obtain the first order derivatives for the following functions:
(a) f(t) = ln( 3√
(t− a)(t+ a)), |t| > |a|
(b) f(t) = 1(t−1)a
, t > 1
(c) f(t) = cos( 4√t), t > 0
(d) f(t) = sin(cos(t))
(e) f(t) = ln(ln(t)), t > 1
5. Find the limit for n ∈ IN or x ∈ IR, if it exists:
(a) limn→∞
( (12
)n+ 2 · 4n
5 · 4n − 2 · 3n
)
(b) limn→∞
(3√n4 + n2 − 1
(n− 2√n)2 + 2
)
(c) limn→∞
(−1)n2n
2n3 − n2
(d) (using de l’Hopital’s rule) limx→∞
x2e−x
6. Let f(x) = 21−x.
(a) Find the derivative of f(x) up to order 3.
2
(b) Give the n-th derivative f (n)(x) for n ∈ IN0.
(c) Find the Taylor polynomial T 2f (x) of order 2 in x0 = 0.
(d) Give the Taylor series T∞f (x) in x0 = 0.
7. Consider the following Cobb-Douglas production function as given:
f(K,L) = aKbLc, a, b, c > 0;K > 0, L > 0
The partial derivatives are given by f ′K(K,L) = abKb−1Lc and f ′L(K,L) = acKbLc−1.
(a) Calculate the partial differential dfK with respect to K in (K0, L0)=(1,4).
(b) Calculate the total differential df in (K0, L0) = (1, 4) for c = 12.
8. Use the Lagrangian method to determine the critical points of f(x, y) = x2 + y2 subject
to the constraint y = 1x, x > 0.
9. Use the Lagrangian method to solve the critical points of the Cobb-Douglas utility
function U(x, y) = axby1−b, x, y, a > 0, 0 < b < 1 subject to the budget restriction
g(x, y) = 2x+ 3y = 10.
10. Use the Lagrangian method to determine the critical points of the utility function U(x1, x2) =
(x1−1)0.4(x2−2)0.8, x1 > 1, x2 > 2 subject to the budget constraint g(x1, x2) = x1+2x2 =
11.
11. Consider f(x, y) = exp(− 2x
+ y) with x 6= 0.
(a) Determine the first and second order partial derivatives.
(b) Calculate the total differential df in (x0, y0) = (2, 1).
12. Consider f(x, y) = x2 + y2 + 4 and g(x, y) = 2x3 + 2y3 + x2 + y2 with x, y 6= 0. Use
the Lagrangian method to solve for the critical points of f subject to the constraint that
g(x, y) = 0.
13. Calculate the determinants of the following matrices:
(a) A =
1 −3 4
−4 1 3
2 −2 3
(b) B =
1 −3 4
3 2 0
3 1 2
3
(c) C =
2a −1 1
3a 0 3
0 1 1
(d) D =
−1 2 3 4
2 0 0 4
1 1 2 1
4 3 −1 7
14. Determine the eigenvalues of the following matrices:
(a) A =
(0 1
4 0
)
(b) B =
(2 1
1 2
)
(c) C =
7 0 0
0 6 0
0 0 8
(d) D =
0 1 2
0 0 3
0 0 0
(e) E =
2 −2 3
0 3 −2
0 −1 3
15. Let
A =
(1 2
2 4
), B =
(4 −6
−2 3
), C =
(2 1
3 2
),
D =
(−2 7
5 −1
), E =
(1 1
).
Perform the following operations, if possible:
(a) A ·A
(b) A ·B
(c) A ·C
(d) A ·D
(e) ET ·A · E
(f) E ·A · ET
4
16. Find the inverses of the following matrices:
(a) A =
(1 −3
−3 5
)
(b) B =
1 3 2
3 5 0
0 5 7
(c) C =
1 2 4
8 6 3
8 6 2
5
Mathematical Questions: RESULTS
1. (a) = 2x 1ln(2)
+ 8ex/8 + 0.5 sin(2x) + c
(b) = 0.9449x2/3 − 43x−3 + c
(c) = 992a
(d) = 0.5(1− e−2b)
(e) = 0
2. (a) = 0.3863
(b) = −e−θx(x2 + 2xθ
+ 2θ2
)
3. (a) f ′(t)|t=0 = np, f ′′(t)|t=0 = n2p2 + np(1− p)
(b) f ′(t)|t=0 = µ, f ′′(t)|t=0 = µ2 + σ2
4. (a) = 2t3(t2−a2)
(b) = −a(t− 1)−a−1
(c) = 2 sin( 4√t)t−3/2
(d) = − cos(cos(t)) sin(t)
(e) = 1t ln(t)
5. (a) = 0.4
(b) = 1
(c) = 0
(d) = 0
6. (a) f ′(x) = −21−x ln(2), f ′′(x) = 21−x(ln(2))2, f ′′′(x) = −21−x(ln(2))3
(b) = 21−x(− ln(2))n
(c) = 2− 2 ln(2) + (ln(2))2
(d) =∑∞
k=0(− ln(2)x)k
k!
7. (a) = ab4c∆K
(b) = 2ab∆K + a4∆L
8. x = y = ±1, λ = ±2
9. x = 5b, y = 103
(1− b)
10. x1 = 3, x2 = 4, λ = 0.4595
6
11. (a) f ′x(x, y) = 2f(x, y)x−2, f ′y(x, y) = f(x, y)
f ′′xx(x, y) = 4f(x, y)(x−4 − x−3), f ′′yy(x, y) = f(x, y), f ′′xy(x, y) = f ′x(x, y)
(b) = 0.5∆x+ ∆y
12. x = y = −0.5, λ = 2
13. (a) = −21
(b) = 10
(c) = 0
(d) = 108
14. (a) λ1,2 = ±2
(b) λ1 = 3, λ2 = 1
(c) λ1 = 8, λ2 = 7, λ3 = 6
(d) λ1,2,3 = 0
(e) λ1 = 2, λ2,3 = 3±√
2
15. (a) =
(5 10
10 20
)
(b) =
(0 0
0 0
)
(c) =
(8 5
16 10
)
(d) =
(8 5
16 10
)(e) not defined
(f) = 9
16. (a) = −14
(5 3
3 1
)
(b) =
17.5 −5.5 −5
−10.5 3.5 3
7.5 −2.5 −2
(c) =
−0.6 2 −1.8
0.8 −3 2.9
0.0 1 −1.0
7
Econometric Questions
1. What is a cumulative distribution function, a probability distribution function (for dis-
crete random variables), and a probability density function (for continuous random vari-
ables)?
2. What is an expected value, a variance and a standard deviation of a random variable Y ?
3. Characterize a joint and a conditional probability distributions.
4. What is the normal distribution, the t distribution, the F distribution, and the χ2 distri-
bution?
5. What means convergence in probability and convergence in distribution?
6. What is simple random sampling? What means that the observations Y1, ..., Yn of a
random sample are independently and identically distributed (i.i.d.)?
7. How is the sample average Y computed?
8. Show that if Y1, ..., Yn are i.i.d. with mean µY and variance σ2Y then:
(a) Y is an unbiased and consistent estimator of the population mean,
(b) the sampling distribution of Y has mean µY and variance σ2Y
= σ2Y /n,
(c) the law of large numbers says that Y converges in probability to µY , and
(d) the central limit theorem says that the standardized version of Y ,(Y − µY )/σY , has
a standard normal distribution when n is large.
9. What is the difference between population and sample regression line?
10. What is this: β = (X′X)−1X′y?
11. Explain the use of PX = X(X′X)−1X′ and MX = In −PX.
12. Show that if (1) the regression errors, ui, have a mean of zero conditional on the regres-
sors Xi, (2) the sample observations are i.i.d. random draws from the population, (3)
the regressors are not linearly dependent, and (4) large outliers are unlikely, then the
OLS estimators of the population model Yi = Xiβ + ui are unbiased, consistent, and
asymptotically normally distributed.
13. What is the purpose and meaning of R2, adjusted R2, and the standard error of the
regression (SER)?
14. Show that the squared SER can be written a s2u = U′MXU
n−k−1.
8
15. How can single and joint hypotheses be tested using t and F -statistics? What is the
p-value?
16. Construct a 95% confidence interval for a regression coefficient.
17. What is the difference between traditional and heteroscedasticity-consistent standard er-
rors?
18. State the Gauss-Markov theorem.
19. Show that if the assumptions of question 12 hold and the regression errors are additionally
homoscedastic and normally distributed, then:
(a) U|X ∼ N (0n, σ2uIn).
(b) β|X ∼ N (β, σ2u(X
′X)−1).
(c) n−k−1σ2u× σ2
u ∼ χ2n−k−1, and
(d) tj =βj−βj
σu√
[(X′X)−1]jj∼ t(n− k − 1)
20. Why can multicollinearity matter?
21. What is an omitted variable bias?
22. Explain other forms of regressor endogeneity such as the choice of an incorrect functional
form, measurement error, and simultaneous causality.
23. Interpret the regression coefficients for different functional forms: linear-linear, log-linear,
linear-log, and log-log.
24. What is the use of interaction terms?
9
Econometric Questions: RESULTS
1. The cumulative probability distribution function of a random variable evaluated
at a particular value is the probability that the random variable is less than or equal to
that particular value.
The probability distribution of discrete random variables is the list of all possible
values of the variable and the probability that each value will occur. These probabilities
sum to 1.
Because a continuous random variable can take on a continuum of possible values, the
probability distribution used for discrete variables, which lists the probability of each
possible value of the random variable, is not suitable for continuous variables. Instead,
the probability is summarized by the probability density function. The area under the
probability density function between any two points is the probability that the random
variable falls between those two points.
2. The expected value of a random variable Y , denoted E(Y ), is the long-run average value
of the random variable over many repeated trials or occurrences. The expected value of
a discrete random variable is computed as a weighted average of the possible outcomes
of that random variable, where the weights are the probabilities of that outcome. The
expected value of Y is also called the expectation of Y or the mean of Y and is denoted
µY .
The variance and standard deviation measure the dispersion or the “spread” of a probabil-
ity distribution. The variance of a random variable Y , denoted var(Y ), is the expected
value of the square of the deviation of Y from its mean: var(Y ) = E[(Y −µY )2]. Because
the variance involves the square of Y , the units of the variance are the units of the square
of Y , which makes the variance awkward to interpret. It is therefore common to measure
the spread by the standard deviation, which is the square root of the variance and is
denoted σY . The standard deviation has the same unit as Y .
3. The joint probability distribution of two discrete random variables, say X and Y , is
the probability that the random variables simultaneously take on certain values, say x and
y. The probabilities of all possible (x, y) combinations sum to 1. The joint probability
distribution can be written as the function Pr(X = x, Y = y).
The distribution of a random variable Y conditional on another random variable X taking
on a specific value is called the conditional distribution of Y given X. The condi-
tional probability that Y takes on the value y when X takes on the value x is written
Pr(Y = y|X = x).
10
4. A continuous random variable with a normal distribution has the familiar bell-shaped
probability density. The function defining the normal probability density is given by:
fY (y) =1
σ√
2πe−
12
( y−µσ
)2
The normal density with mean µ and variance σ2 is symmetric around its mean and
has 95% of its probability between µ − 1.96σ and µ + 1.96σ. The Normal distribution
with mean µ and variance σ2 is expressed concisely as N(µ, σ2). The standard normal
distribution is the normal distribution with mean µ = 0 and variance σ2 = 1 and is
denoted by N(0, 1).
The Student t distribution with m degrees of freedom is defined to be the distribution
of the ratio of a standard normal random variable, divided by the square root of an
independently distributed chi-squared random variable with m degrees of freedom divided
by m.
The F distribution with m and n degrees of freedom, denoted Fm,n is defined to be
the distribution of the ratio of a chi-squared random variable with degrees of freedom m
divided by m, to an independently distributed chi-squared random variable with degrees
of freedom n, divided by n.
The chi-squared distribution is the distribution of the sum of m squared independent
standard normal random variables. The distribution depends on m, which is called the
degrees of freedom of the chi-squared distribution. A chi-squared distribution with m
degrees of freedom is denoted by χ2m.
5. Let S1, S2, ..., Sn be a sequence of random variables. For example, Sn could be the sample
average Y of a sample of n observations of the random variable Y . The sequence of random
variables {Sn} is said to converge in probability to a limit, µ (that is, Snp−→ µ), if the
probability that Sn is within ±δ of µ tends to 1 as n −→ ∞, as long as the constant δ is
positive. That is, Snp−→ µ if and only if Pr(|Sn − µ| ≥ δ) −→ 0 as n −→∞ for every δ > 0.
If the distributions of a sequence of random variables converge to a limit as n goes to
infinity, then the sequence of random variables is said to converge in distribution.
The central limit theorem says that, under general conditions, the standardized sample
average converges in distribution to a normal random variable. Let F1, F2, ..., Fn be a se-
quence cumulative distribution functions corresponding to a sequence of random variables,
S1, S2, ..., Sn. For example, Sn might be the standardized sample average, (Y − µY )/σY .
Then the sequence of random variables Sn is said to converge in distribution to S (denoted
Snd−→ S) if the distribution functions {Fn} converge to F , the distribution of S. That is,
Snd−→ S if and only if lim
n→∞Fn(t) = F (t) where the limit holds at all points t at which the
limiting distribution F is continous.
11
6. Random sampling is one of the most popular types of random or probability sampling.
In a simple random sample, n objects are selected at random from a population and each
member of the population is equally likely to be included in the sample. The value of
the random variable Y for the ith randomly drawn object is denoted Yi. Because each
object is equally likely to be drawn and the distribution of Yi is the same for all i, the
random variables Y1, ..., Yn are independently and identically distributed (i.i.d.); that is,
the distribution of Yi is the same for all i = 1, ..., n and Y1 is distributed independently
of Y2, ..., Yn and so forth.
7. The sample average or sample mean, Y , of the n observations Y1, ..., Yn is obtained as
follows:
Y =1
n(Y1 + Y2 + ...+ Yn) =
1
n
n∑i=1
Yi
8. (a) Let µY denote some estimator of µY , such as Y or Y1. The estimator µY is unbi-
ased if E(µY ) = µY , where E(µY ) is the mean of the sampling distribution of µY ;
otherwise, µY is biased.
Another desirable property of an estimator µY is that, when the sample size is large,
the uncertainty about the value of µY arising from random variations in the sample is
very small. Stated more precisely, a desirable property of µY is that the probability
that it lies within a small interval of the true value µY approaches 1 as sample size
increases, that is, µY is consistent for µY .
(b) The mean of Y is given by:
E(Y ) =1
n
n∑i=1
E(Yi) = µY .
The variance of Y is given as follows:
σ2Y = var
(1
n
N∑i=1
E(Yi)
)
=1
n2
N∑i=1
var(Yi) +1
n2
N∑i=1
N∑j=1,j 6=i
cov(Yi, Yj)
=σ2Y
n
where Y1, ..., Yn are i.i.d. and Yi and Yj are independently distributed for i 6= j, so
cov(Yi, Yj) = 0.
12
(c) The law of large numbers states that, under general conditions, Y will be near
µY with very high probability when n is large. This is sometimes called the “law
of averages.” When a large number of random variables with the same mean are
averaged together, the large values balance the small values and their sample average
is close to their common mean. The sample average Y converges in probability to
µY (or, equivalently, Y is consistent for µY ) if the probability that Y is in the range
(µY−c) to (µY +c) becomes arbitrarily close to 1 as n increases for any constant c > 0.
The convergence of Y to µ in probability is written, Yp−→ µY . The law of large
numbers says that if Yi, i = 1, .., n are are independently and identically distributed
with E(Yi) = µY and if large outliers are unlikely (technically if var(Yi) = σ2Y <∞,
then Yp−→ µY .
(d) Recall that the mean of Y is µY and its variance is σ2Y
= σ2Y /n. According to the
central limit theorem, when n is large, the distribution of Y is approximately
N(µY , σ2Y
). The distribution of Y is exactly N(µY , σ2Y
) when the sample is drawn
from a population with the normal distribution N(µY , σ2Y ). Therefore, the distri-
bution of the standardized version of Y , (Y − µY )/σY is well approximated by a
standard normal distribution N(0, 1) when n is large.
9. The population regression line is given by Yi = β0 +β1Xi+ui. This is the relationship
that holds between Y and X on average over the population. Thus, if you knew the value
of X, according to this population regression line you would predict that the value of the
dependent variable, Y , is β0 + β1X. Since the coefficients β0 and β1 of the population
regression line are unknown we must use sample data to estimate them. This can be
achieved using the ordinary least squares (OLS) estimators for β0 denoted by β0 and
the OLS estimator for β1 denoted by β1. The OLS regression line, also called sample
regression line, is the straight line constructed using the OLS estimators: β0 + β1X.
The predicted value of Yi given Xi, based on the OLS regression line is Yi = β0 + β1Xi.
The residual for the ith observation is the difference between Yi and its predicted value:
ui = Yi − Yi. Concluding, one can say that the OLS estimators, β0 and β1 are sample
counterparts of the population coefficients, β0 and β1. Similarly, the OLS regression line
β0 + β1X is the sample counterpart of the population regression line β0 + β1X, and the
OLS residuals ui are sample counterparts of the population errors ui.
10. The formula β = (X ′X)−1X ′y calculates the OLS estimators β0 and β1 for our
true unknown population coefficients β0 and β1, where (X ′X) is non-singular matrix.
The OLS estimator minimizes the sum of squared prediction mistakes,∑n
i=1(Yi − b0 −b1X1i−· · ·− bkXki)
2. The above formula for the OLS estimator is obtained by taking the
derivative of the sum of squared prediction mistakes with respect to each element of the
coefficient vector, setting these derivatives to zero, and solving for the estimator β.
13
11. The algebra of OLS in the multivariate model relies on the two symmetric n x n matrices,
PX = X(X ′X)−1X ′ and MX = In − PX . A matrix C is idempotent if C is square
and CC = C. Because PX = PXPX and MX = MXMX and because PX and MX are
symmetric, PX and MX are symmetric idempotent matrices. The matrices PX and MX
can be used to decompose an n-dimensional vector Z into two parts: a part that is
spanned by the columns of X and a part orthogonal to the columns of X. In other words,
PXZ is the projection of Z onto the space spanned by the columns of X, MXZ is the
part of Z orthogonal to the columns of X, and Z = PXZ +MXZ.
12. The least squares estimator is unbiased in every sample. To show this, write