Article type: Focus Article
Transfer Functions Article ID
Stephen Pollock
University of Leicester
Keywords
Impulse response, Frequency response, Spectral density
Abstract
In statistical time-series analysis, signal processing and control engineering, a
tr
EC 7087 Econometric Theory, 2011: A Summary of the Course
1. We began by considering the formula for the conditional expectation of
a variable y , given the value of an associated variable x, under the assumption that these have a bivariate normal distrib
D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS
STATISTICAL FOURIER ANALYSIS
The Fourier Representation of a Sequence
According to the basic result of Fourier analysis, it is always possible to
approximate an arbitrary analytic function dened over a nite i
D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS
ALGEBRAIC POLYNOMIALS
Consider the equation 0 + 1 z + 2 z 2 = 0. Once the equation has been divided
by 2 , it can be factorised as (z 1 )(z 2 ) where 1 , 2 are the roots or
zeros of the equation which are giv
D.S.G. POLLOCK: TOPICS IN ECONOMETRICS
FACTORISING THE THE NORMAL DISTRIBUTION
The joint distribution of x and y can be factored as the product of the marginal distribution
of x and the conditional distribution of y given x:
N (y, x) = N (y |x)N (x).
(1)
3. THE PARTITIONED REGRESSION MODEL
Consider taking a regression equation in the form of
(1)
y = [ X1
X2 ]
1
+ = X1 1 + X2 2 + .
2
Here, [X1 , X2 ] = X and [1 , 2 ] = are obtained by partitioning the matrix X
and vector of the equation y = X + in a confor
D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS
THE FOURIER DECOMPOSITION OF A TIME SERIES
In spite of the notion that a regular trigonometrical function is an inappropriate
means for modelling an economic cycle other than a seasonal uctuation, there
are g
D.S.G. POLLOCK: TOPICS IN ECONOMETRICS
EXPECTATIONS AND CONDITIONAL EXPECTATIONS
The joint density function of x and y is
f (x, y ) = f (x|y )f (y ) = f (y |x)f (x),
(1)
where
f (x) =
f (x, y )dx
and
f (y ) =
f (x, y )dy
y
(2)
x
are the marginal distribut
FILTERING MACROECONOMIC DATA
By D.S.G. Pollock
University of Leicester
Email: stephen [email protected]
This chapter sets forth the theory of linear ltering together with an accompanying frequency-domain analysis. It employs the classical Wiener
K
FILTERS FOR ECONOMETRIC DATA
WienerKolmogorov Filtering of Stationary Sequences
The classical theory of linear ltering was formulated independently by Norbert
Wiener (1941) and Andrei Nikolaevich Kolmogorov (1941) during the Second
World War. They were bo
D.S.G. POLLOCK: TOPICS IN ECONOMETRICS
DIAGONALISATION OF A SYMMETRIC MATRIX
Characteristic Roots and Characteristic Vectors. Let A be an n n symmetric
matrix such that A = A0 , and imagine that the scalar and the vector x satisfy the equation
Ax = x. The
TIME-SERIES ANALYSIS: PASCAL PROGRAM
This program demonstrates alternative ways of characterising a stationary
stochastic process. On the one hand are the time-domain characterisations,
which depend upon the autocorrelation and partial autocorrelation fun
EXERCISE: Fitting Trend Functions to Nonstationary Data
In order to understand the methods for estimating trend functions that are available in
the IDEOLOG program, you are invited to follow the instructions that are given in the
les CONSLOG, SALESLOG, HE
D.S.G. POLLOCK: TOPICS IN ECONOMETRICS 2011
1. EXPECTATIONS AND CONDITIONAL EXPECTATIONS
The joint density function of x and y is
f (x, y ) = f (x|y )f (y ) = f (y |x)f (x),
(1)
where
f (x, y )dx
f (x) =
and
f (y ) =
f (x, y )dy
y
(2)
x
are the marginal d
By following the sequence of commands recorded in this log, you will be able to fit a trend to the
monthly data on U.S. Retail Sales. Using one of the facilities of the program, you will also be
able to extract the seasonal component from the data.
.
IDEO
LECTURE 1
1. Let y = E (y |x) be the conditional expectation of y given x. Prove that
E cfw_(y y )2 E cfw_(y )2 , where = (x) is any other function of x.
Show that E x(y y ) = 0 and give an interpretation of this condition.
Demonstrate that, if E (y |x)
EXERCISE: Analysis of the Pearson Height Data using GRETL
The Pearson data on the height of fathers v 1 and the height of sons v 2 is in
a plain text (ASCII) le labelled Pearson.txt. These can be brought in gretl
using using the hFile i hOpen Data i hImpo
EXERCISE: Analysis of Panel Data using GRETL
The Data From Greene. The gretl program is already linked to a data
le greene14 1.gdt, which is designed to illustrate the analysis of panel data.
The data le can be loaded via the menu commands hFile ihOpen Da
By following the sequence of commands recorded in this log, you will be able to extract a trendcycle component from the monthly data on U.S. Retail Sales. The end effects are controlled by
inserting an extrapolation between the end and the beginning of th
gretl: http:/gretl.sourceforge.net/
Gnu Regression, Econometrics and Time-series Library
Gretl is a cross-platform software package for econometric analysis, written in
the C programming language. It is is free, open-source software. You may
redistribute
ASTSA :http:/www.stat.pitt.edu/stoffer/tsa2/
ASTSA is a Windows time series package that is distributed as Freeware
and is provided As is without warranty of any kind, either expressed or implied.
ASTSA may not be distributed as a component of any commerc
A GUIDE TO MESOSAUR: A PROGRAM FOR THE
STATISTICAL ANALYSIS OF TIME SERIES
MESOSAUR is a computer program for the statistical analysis of time series,
which was created by a team of programmers and statisticians at the Central
Economics and Mathematics In
ECONOMETRIC THEORY: Exercise 1 (Tutorial)
Matrix Algebra and Manpower Planning
Matrix Multiplication in Microsoft Excel
There is a limited set of matrix operations which may be performed in Excel
with the use of predened commands. The following commands a
EXERCISE: Models with Limited Dependent Variables
A Logistic Model of the Decision to Smoke or not to Smoke
Within the document Smoking and Drinking Amongst Adults 2008, which is
a component of the General Lifestyle Survey 2008, you will nd Table 1.3
Perc
FILTERING MACROECONOMIC DATA
WienerKolmogorov Filtering of Stationary Sequences
The classical theory of linear ltering was formulated independently by
Norbert Wiener (1941) and Andrei Nikolaevich Kolmogorov (1941) during
the Second World War. They were bo
EC3062 ECONOMETRICS
DYNAMIC REGRESSIONS MODELS
Autoregressive Disturbance Processes
Economic variables often follow slowly-evolving trends and they tend to be
strongly correlated with each other. If the disturbance term is compounded
from such variables,
EC3062 ECONOMETRICS
LIMITED DEPENDENT VARIABLES
Logistic Trends
One way of modelling a process of bounded growth is via a logistic
function. See Figure 1. This has been used to model the growth of a
population of animals in an environment with limited foo
EC3062 ECONOMETRICS
MATRIX KRONECKER PRODUCTS
Consider the matrix equation Y = AXB . When all of the factors are 2 2
matrices, this becomes
y11
y21
y12
y22
=
a11
a21
a12
a22
x11
x21
x12
x22
x
x11
, A 12
x21
x22
=
A
=
b11 A
b11
b12
b11
b12
b21
b22
b21
b22
EC3062 ECONOMETRICS
HYPOTHESIS TESTS FOR THE CLASSICAL LINEAR MODEL
The Normal Distribution and the Sampling Distributions
To denote that x is a normally distributed random variable with a mean
of E (x) = and a dispersion matrix of D(x) = , we shall write
EC3062 ECONOMETRICS
IDENTIFICATION OF ARMA MODELS
A stationary stochastic process can be characterised, equivalently, by its
autocovariance function or its partial autocovariance function.
It can also be characterised by is spectral density function, whic