Applied Econometrics

Lecture 9Time Series Filters

Please note that for interactive manipulation you need Mathematica 6 version of this .pdf. Mathematica 6 is available at all Lab's Computers at IES

http://staff.utia.cas.cz/barunikJozef Barunik ( barunik @ utia. cas . cz )

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Outline

Why Filters? - they has its very interesting area of use in analysis of time series

Linear Filters

Time Domain Filters Infinite Impulse Response (IIR)Finite Impulse Response (FIR): MA, EWMA, Zero-LagFilter

Frequency Domain FiltersFrequency, Period, SamplingFourier Transforms

Filters in PracticeProblems with Filters

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Linear Filters - Motivation

A discrete time series as a sequence of observations ordered by a time t, 8xt<t=-¶¶

we apply filter to extract certain features from time series xt :trendseasonalitiesbusiness cyclesnoise

of course this is very tricky, as we have to choose to what extent we want to "filter" the series(Do we really filter only noise? Or some important infomration? How do we know?)

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Linear Filters in Time Domain

Linear filter simply converts a time series xt into another time series yt by a linear transformation

xt Ø FILTER Ø yt

The output yt is result of convolution of the input xt with a coefficient vector wt : Hwt are also called filter coefficients)

yt =⁄i=-¶¶ wi xt-i

where convolution is formally defined asw *xt =⁄i=-¶

¶ wi xt-i

This might be a problem, as future realizations are needed, thus we may restrict convolution to:yt =⁄i=0

¶ wi xt-i, where only past is utilized (Casual or physically realizable filter)

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Linear Filters in Time Domain cont.

We classify linear filters according to their response to an impulse signal

if the impulse response of filter is finite, we have finite impulse response filter - FIR filter

if the impulse response of filter is infinite, we have IIR filter

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Infinite Impulse Response (IIR) Filters

Consider a form of constant coefficient linear difference equation:

yt =⁄i=1L ai yt-i +⁄i=0

M wi xt-i,where the current value of the output is determined by L lagged values of output yt and Mlagged values of input xt , as well as current input value.

Consider this first-order difference equation: yt = a yt-1 + xt .

does it look familiar to you?? But it does not have solution without previous information aboutytby recursive substitution ( y1 = a y0 + x1, y2 = a y1 + x2 ... ) we get general solutionyt = at y0 +⁄i=0

t-1 ai xt-i

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Finite Impulse Response (FIR) Filters

General form of an FIR filter is:

yt =⁄i=-NM wi xt-i,

where filter processes M future and M past values as well as the current value of the input.thus filter is noncausual

Anyone remember how is this equation called?

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Finite Impulse Response (FIR) Filters cont.

YES !

CONVOLUTION

w *xt¢ | £

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Finite Impulse Response (FIR) Filters cont.

very common example is Simple Centered Moving Average:

yt = 1M+N+1 Hxt-M + ... + xt-1 + xt + xt+1 + ... + xt+NL

The impulse response of this filter is finite:

wi = :1

M+N+1 ,

0

if i = -N , ..., -1, 0, 1, ..., Motherwise

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Finite Impulse Response (FIR) Filters cont. - SMA

General form migh be reduced to causal filter by imposing the restriction N=0. Future values will be ignored by filter, which is inevitable for applications like forecasting

yt =⁄i=0M wi xt-i,

This form of FIR filter is more common, consider for example Simple Moving Average (SMA):

yt = 1M+1 ⁄i=0

M xt-i

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Finite Impulse Response (FIR) Filters cont. - EMA

Of course we can upgrade SMA to have i.e. linearly declining weights, example of SMA(6):

yt = 2H6L H7L H6 xt + 5 xt-1 + 4 xt-2 + 3 xt-3 + 2 xt-4 + xt-5L

Exponential Moving Average (EMA) brings idea of weighting lagged observations exponentially. yt = a xt-1 + H1 - aL yt-1,

where a is smoothing factor a = 2N+1 , N number of lags included, xt is observation in time, and yt

is EMA in time.alternativelly, EMAt = EMAt-1 + aHprice - EMAt-1L

BUT these filters has quite large lag (see next slide for example)

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Finite Impulse Response (FIR) Filters cont. - SMA vs. EMAComparison of Simple Moving Average and Exponential Moving Average

MA1HML 50

EMA2HML 50

zoom interval Hin daysL 1130

Show whole period

2004 2005 2006 2007 200820

25

30

35

from - to 1

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Finite Impulse Response (FIR) Filters cont. - Zero Lag Filters

Well SMA/EMA filters do not use future values, they are casual, but in cost of quite large lag,thus they react very slowly as you can see from demonstration.

Solution might be simple idea of differencing 2 EMA:

EMAHML = 2µEMAHML- EMAH2 M - 1L,where M is the period of EMA.

Of course there are many other possibilities¢ | £

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Finite Impulse Response (FIR) Filters cont. - Zero Lag Filters

Comparison of Zero-Lag and EMA filter (choose number of lags to include in filter again)

M 50

zoom interval Hin daysL 1130

Show whole period

2004 2005 2006 2007 200820

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from - to 1

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Linear Filters in Frequency Domain

until now, we discussed filters with time domain only

let's draw attention to frequencies (in the Fourier space) of time series instead of quantities

the idea is very simple: each time series can be decomposed into a weighted sum of muchsimpler sinusoidal components

thus we are approaching time series as a weighted sum of harmonic functions (sines and cosines)

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Frequency, Period

f is frequency of cycles per second, (c.p.s., Hz)

w = 2 p f - angular frequency (radians per second)

T = 1 ê f - period

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Sampling

The process of converting a signal into a numeric sequence (in other words, continuous time orspace to discrete time or space)

The Nyquist frequency is equal to one-half of the sampling frequency

wNYQUIST =T-12

The Nyquist frequency is the highest frequency that can be measured in a signal

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Sampling - An Undersampled Signal

sinH2 p f tL

Consider signal with frequency f = 7kHz, and use sampling rate of fs = 8 kHzThis is an undersampled signal

Use following interactive example to sample your own signal:

frequency

f @HzD 6877

sampling rate

fs 8 kHz 16 kHz 48 kHz

Ê

ÊÊ

Ê

Ê

ÊÊ

Ê

Ê

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

tHsL

sinH2 p f tL

Author: Carsten RoppelSource: http://demonstrations.wolfram.com/SamplingTheorem/

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Sampling cont.

According to the sampling theorem, following may occur:

f § fs ê 2: the samples uniquely represent the sine wave of frequency f .

f > fs ê 2: aliasing occurs, because the replicated spectra begin to overlap.

0 § f § fs ê 2: a spectral line appears at the frequency f - fs .

(Again try to use the interactive example from previous slide to see this)¢ | £

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Fourier Theory - Mathematical Prerequisities First

A transform takes one function (or signal) and turns it into another function (signal)

Anyone remember how the functions sin(x) and cos(x) are defined ?

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Fourier Theory - Mathematical Prerequisities First

Definitions are following (these has very nice property, that they are infinite):

sin HxL = x - x33! +

x55! -

x77! + ... =⁄n=0

¶ H-1Ln x2 n+1H2 n+1L!

cos HxL = 1 - x22! +

x44! -

x66! + ... =⁄n=0

¶ H-1Ln x2 nH2 nL!

now consider definition of ex:

ex =⁄n=0¶ xn

n! = 1 + x + x22! +

x33! + ...

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Fourier Theory - The Complex Exponential

The complex exponential ei x is defined to be the complex variable whose real and imaginaryparts are cos(x) and sin(x): Also known as Euler relationship:

ei x = cosHxL+ i sinHxL¢ | £

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Fourier Transform

Any infinite sequence xt may be viewed as a combination of an infinite number of sinusoidswith different amplitudes and phases

Discrete Fourier Transform (DFT) is defined as:

XH f L =⁄t=-¶¶ xt e-i2p f t

where xt are fourier coefficients and can be obtained from the inverse Fourier transfrom:

xt = 12 p Ÿ-p

p XH f L ei 2 p f t „ f ,

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Fourier Series at Work

Nice example of approximation of five different periodic functions by Fourier Series conver-genceSee how Fourier series approximate continuous function vs. "step" function

function step sawtooth parabola cubic half-wave rectifier

number of terms 17

x range 2 p

-6 -4 -2 2 4 6

-1.0

-0.5

0.5

1.0

Author : Alain GorielySource : http : // demonstrations.wolfram.com/FourierSeriesOfSimpleFunctions/

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Filters in Practice - EMA Volatility Estimation

EMA filter plays important role in the risk management

l

S&P standard deviation estimate with EMA Hl=0.9L

1980 1990 20000.00

0.02

0.04

0.06

0.08

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Filters in Practice - Hodrick-Prescott (HP) Filters

HP filter was developed basically fo business cycles analysis, but can also be used to other series.The standard (unvariate) HP filter finds a smoothed series based only on the time series proper-ties of the original data. It does so by finding the values of t that minimise the function:

L = ⁄t=1T H yt - ttL2 + l⁄t=2

T-1 HHtt+1 - ttL- Htt - tt-1LL2

where the weight on smoothness (l).

main drawback of HP filter is that it ¢ | £

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Filters in Practice - Hodrick-Prescott (HP) Filters

l 2162

zoom interval Hin daysL 347

Show whole period

Jan Jul Jan19

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Problems with Filters

filters are very good tool for extraction of certain features, i.e. cyclical parts, etc.

they can not be used for forecasting ! (look at the construction)

denoising - it is often difficult to find "the best" treshold, we never now what kind of informa-tion we can lose if we extract what appears to be "noise"

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Further Readings

Other type of powerfull Nonlinear filters is Wavelets

students interested in filters are adviced to use following literature:

Gencay R., Selcuk F., Whitcher B. : An Introduction to Wavelets and Other Filtering Methodsin Finance and Economics, Academic Press, ISBN 0122796705

Percival D.B., Walden A.T. : Wavelet Methods for Time Series Analysis, Cambridge UniversityPress, ISBN 0521685087

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Questions

Other examples during seminar

THANK YOU FOR ATTENTION !

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