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;
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 y
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 pa
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 expe
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 examp
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 ran
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 stationar
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
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 de
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:
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
gres
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 the
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
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
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 comp
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
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
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;
ECONOMETRICS
Walter Beckert
Department of Economics and Finance
London Business School
Autumn 2012
Venue: TBC.
Time: Thu 12:45 - 15:30.
Contact: [email protected]
Oce Hours: Birkbeck College, Univer
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
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
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 stu
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) s
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 d
Linear Simultaneous Equations Systems
Recall: Econometricians treat RHS regressors as
stochastic and condition on them; statisticians often treat them as predetermined (or deterministic,
exogenous).
T
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
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
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
Quantile Regression
Least Squares Regression: summarizes average (expectation) of response variable, given regressors.
Quantile Regression: computes dierent regression
curves corresponding to precenta