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