Econometrics
Department of Economics and Finance
London Business School
Final Examination Examination
Walter Beckert, 12:45 - 15:30 pm, 16 December 2010
Note: This is a closed book, closed notes exam; one A4 sheet with notes is permitted.
Any results pres
Problem Set 6: Cointegration
1) Consider the following example:
yt + xt
= u1t
yt + xt
= u2t
where:
u1t
=
0:2u1t
u2t
=
u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. What is the order of integration of yt and xt ?
ii. Under which conditions are yt and xt coi
Problem Set 1: ARMA processes
1) Consider the following three stochastic processes:
yt
=
+ "t
"t
=
"t
1 + !t
+1
f! t gt= 1
is a White Noise
+1
Show that the autoregressive behaviour of f"t gt= 1 is passed (mutatis
+1
+1
mutandis) on to fyt gt= 1 ; i.e. fy
Problem Set 8: Markov Chains
For the following Markov Chain process:
+1
fst gt= 1 :
p (st = 1=st 1 = 1) =
p (st = 0=st 1 = 0) =
st 2 S = f0; 1g 8t
p
q
1) Compute the conditional and unconditional expectation and variance of st .
2) Compute the expected nu
Problem Set 4: VAR Analysis
1) For the bivariate VAR(2) process in exercise 3 of Problem Set 3 compute the
impulse responses s for s= 1,2 as well as the matrix long run multiplier.
2) Provide an example to show that Granger causality is not transitive.
3)
Problem Set 9: Answer Key
A state representation consists of a state equation and an observation
equation. To obtain it, we must bare in mind that the state variable, t , has to
be an unobservable random variable. In addition, letting vt and ! t represent
Problem Set 5: Answer key
1) The claim is not true. A process that is stationary in the strict sense and
that has nite second moments is covariance stationary. But a process
that is strictly stationary can be non covariance stationary. Lets construct a pr
Problem Set 7: ARCH processes
1) For the following ARCH(1) model:
yt
"t
t=+1
1
fut gt=
=
+ "t
= ut ( 0 +
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
compute the rst through fourth moments for yt both conditional on yt
unconditional.
1
and
2) For the following model:
Problem Set 3: Answer Key
1) Lets write the univariate AR(p) process as:
yt = c +
1 yt 1
+ : +
p yt p
+ "t
Like we did in exercise 6 of Problem Set 0, we shall build a system with
the equation that describes the evolution of yt and the following p 1 ident
Problem Set 2: Box-Jenkins methodology
1) For an AR(1) process we have:
(0)
=
2
"
=
(0)
2
"
1
2
1
2
Hence,
2
p lim R2 =
For a MA(1) process,
(0)
2
"
(0)
=
2
(1 +
1
1+
=
)
2
"
2
Therefore,
p lim R2
=
1
1+
1
2
2
=
1+
2
We can see that a consistent estimator
Problem Set 4: Answer key
1) In exercise 3 of Problem Set 3 we had the following process:
yt =
Let
2
0:02
0:03
(L) = I
0:5 0:1
0:4 0:5
+
yt
2
2L ,
1L
1
+
0
0:25
0
0
yt
2
+ ut
where:
1
=
0:5
0:4
0:1
0:5
2
=
0
0:25
0
0
Consider the VMA(1) representation (wh
Problem Set 1: Answer Key
1) To obtain the process for yt is very simple:
"t
= yt
=
(yt
=
(1
yt
yt
1
) + !t
) + yt + ! t
+1
So, fyt gt= 1 is also a stationary AR(1) process with the same autore+1
gressive coe cient as f"t gt= 1 . However, as the "mutatis
Problem Set 7: Answer key
1) Our model is:
yt
"t
=
+ "t
= ut ( 0 +
t=+1
1
fut gt=
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
We shall begin by computing the conditional moments, as the unconditional ones can then be obtained by way of the Law of Iterated Expectation
Problem Set 8: Answer key
1) We shall begin with computing conditional expectations.
E (st =st
1
= 1)
E (st =st
1
= 0)
= P (st = 1=st
= p
= P (st = 1=st
= 1 q
1
= 1)
1
= 0)
We can combine both in the following formula:
E (st =st
1)
p
=
1
st
st
q
1
1
=1
=0
Problem set 0: Prior notions
+1
Let fyt gt=
1
be a sequence of scalar bounded random variables.
1)
Let L and 1 be the "lag" and "identity" operators respectively,
dened as follows:
Lyt
Lk yt
L0 yt
L0
=
=
=
=
yt 1
L(Lk
yt
1
1
)yt = yt
k
Show that L is a li
Problem Set 3: Introduction to VAR processes
1) Show that the companion form of a scalar AR(p) process is a p-variate
VAR(1) process
2) For a stationary n-variate VAR(p) process:
i. show that its companion form is an np-variate VAR(1) process.
3) Consider
Problem Set 9: Kalman lter
1) Provide the state-space representation for the following (stationary) arma
processes:
a. yt =
+ ut + u t
1
b. yt = c + yt
1
+ ut
c. yt = c + yt
1
+ ut + u t
1
2) For the series rf in mpeppext.w and the series cons in weldat.w
Problem Set 2: Box-Jenkins methodology
1) Recall the denition of R2 :
T
X
t=1
T
X
R2 = 1
(yt
b
"2t
y)2
t=1
+1
For a consistent estimator of the parameters of the process fyt gt=
p lim R2 = 1
1
:
2
"
0
+1
Compute the last expression when fyt gt= 1 follows
Econometrics
Department of Economics and Finance
London Business School
Final Examination Examination
Walter Beckert, 12:15 - 15:00 h, 07 December 2010
Note: This is a closed book, closed notes exam; one A4 sheet with notes is permitted.
Any results prese
ECONOMETRICS
Walter Beckert
Department of Economics and Finance
London Business School
Autumn 2012
Venue: TBC.
Time: Thu 12:45 - 15:30.
Contact: w.beckert@bbk.ac.uk
Oce Hours: Birkbeck College, University of London, Malet Street, oce 724, by appointment.
Econometrics, London Business School
Suggested Analytical and Applied Problems
Walter Beckert, Autumn term 2011
The exercises below are suggested for independent study. They constitute both analytical
and computational (applied) problems. The TA is the pr
Econometrics, London Business School
Walter Beckert, Autumn term 2011
1. Multivariate Normal Distribution
Denition: Let y RN be a realization of the vector valued random variable Y , with
support1 RN . Then, the N components of the random vector Y have a
London Business School, Department of Economics and Finance
Walter Beckert
1
Mathematics Review
Autumn term 2010
Single Variable Calculus
References: Simon and Blume, chapters 2 to 5.
This section studies real functions of one variable, that is, functions
Econometrics
What is Econometrics?
Statistics in Economics
Economic analysis typically proceeds in terms of
(parametric) economic models. Econometric methods are used to estimate (the parameters of) such
models.
Important distinction to Statistics:
Stati
Normal Distribution Theory
So far: conditional moment assumptions; induced
conditional moments of LS estimators.
Now: add assumptions about conditional distribution of y given X; induces conditional distribution
of LS estimators.
See handout for useful di
Linear Simultaneous Equations Systems
Recall: Econometricians treat RHS regressors as
stochastic and condition on them; statisticians often treat them as predetermined (or deterministic,
exogenous).
This leaves possibility that, in econometric analyses LH
Maximum Likelihood Estimation
Key ingredient: Parametric probability model for
actual and potential observations.
Assumption 1: (Distribution) Consider the bivariate random variable (U, V ) and an i.i.d. sample
cfw_(Un , Vn), n = 1, , N drawn from the di
Limited Dependent Variable Models
Limited dependent variables typically are
(i) qualitative dependent variables;
(ii) dependent variables having limited support.
Models for such data are often derived from latent
variable models.
1. Probabilistic Choice M
Panel Data Analysis
1. Motivation (1): Omitted variable problem
Consider cross section data y, X = [x1 , , xk ] and
omitted (unobserved) regressor , in the linear regression model
E[yi |xi, i] = i + x a.s.
i
yi = i + x + i, E[i|xi, i] = 0, i, a.s.
i
Two c
Quantile Regression
Least Squares Regression: summarizes average (expectation) of response variable, given regressors.
Quantile Regression: computes dierent regression
curves corresponding to precentage points (quantiles) of the conditional distribution o
Problem Set 6: Answer key
1) The model is:
yt + xt = u1t
u1t = 0:2u1t
yt + xt = u2t
u2t = u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. The order the integration can be found by noting that u1t is I(1):
1
0:2z
P (z)
=
1
0:2z
0:8z 2
P (1)
=
1
0:2
0:8 = 0
0:
Problem Set 5: Unit Root Processes
1) Prove the following claim or nd a counterexample. "Every Strict Stationary
process is Covariance Stationary"
t=+1
1:
2) Consider the following two alternative processes for fyt gt=
A)
B)
t=+1
1
where f"t gt=
yt
yt
= a