Lecture 10. Applied Econometrics. Time Series Filters. Jozef Barunik ( utia. cas. cz

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1 Applied Econometrics Lecture 10 Time 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 IE Jozef Barunik ( utia. cas. cz

2 2 Lecture8.nb 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 Filters Frequency, Period, Sampling Fourier Transforms Filters in Practice Problems with Filters

3 Lecture8.nb Linear Filters - Motivation A discrete time series as a sequence of observations ordered by a time t, 8x t < t=- we apply filter to extract certain features from time series x t : trend seasonalities business cycles noise 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?)

4 4 Lecture8.nb Linear Filters in Time Domain Linear filter simply converts a time series x t into another time series y t by a linear transformation x t Ø FILTER Ø y t The output y t is result of convolution of the input x t with a coefficient vector w t : Hw t are also called filter coefficients) y t = i=- w i x t-i where convolution is formally defined as w * x t = i=- w i x t-i This might be a problem, as future realizations are needed, thus we may restrict convolution to: y t = i=0 w i x t-i, where only past is utilized (Casual or physically realizable filter)

5 Lecture8.nb 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

6 6 Lecture8.nb Infinite Impulse Response (IIR) Filters Consider a form of constant coefficient linear difference equation: y t = L i=1 a i y t-i + M i=0 w i x t-i, where the current value of the output is determined by L lagged values of output y t and M lagged values of input x t, as well as current input value. Consider this first-order difference equation: y t = a y t-1 + x t. does it look familiar to you?? But it does not have solution without previous information about y t by recursive substitution ( y 1 = a y 0 + x 1, y 2 = a y 1 + x 2... ) we get general solution y t = a t y 0 + t-1 i=0 a i x t-i

7 Lecture8.nb Finite Impulse Response (FIR) Filters General form of an FIR filter is: M y t = i=-n w i x t-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?

8 w * x t 8 Lecture8.nb Finite Impulse Response (FIR) Filters cont. YES! CONVOLUTION

9 Lecture8.nb Finite Impulse Response (FIR) Filters cont. very common example is Simple Centered Moving Average: y t = 1 M+N+1 Hx t-m x t-1 + x t + x t x t+n L The impulse response of this filter is finite: 1 M+N+1 w i = :, if i = -N,..., -1, 0, 1,..., M 0 otherwise

10 10 Lecture8.nb 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 y t = M i=0 w i x t-i, This form of FIR filter is more common, consider for example Simple Moving Average (SMA): y t = 1 M+1 i=0 M x t-i

11 Lecture8.nb Finite Impulse Response (FIR) Filters cont. - EMA Of course we can upgrade SMA to have i.e. linearly declining weights, example of SMA(6): y t = 2 H6L H7L H6 x t + 5 x t x t x t x t-4 + x t-5 L Exponential Moving Average (EMA) brings idea of weighting lagged observations exponentially. y t = a x t-1 + H1 - al y t-1, where a is smoothing factor a = 2 N+1, N number of lags included, x t is observation in time, and y t is EMA in time. alternativelly, EMA t = EMA t-1 + ahprice - EMA t-1 L BUT these filters has quite large lag (see next slide for example)

12 12 Lecture8.nb Finite Impulse Response (FIR) Filters cont. - SMA vs. EMA Comparison of Simple Moving Average and Exponential Moving Average MA1HML 50 EMA2HML 50 zoom interval Hin daysl 1130 Show whole period from - to 1

13 Lecture8.nb 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 = 2 µ EMAHML - EMAH2 M - 1L, where M is the period of EMA. Of course there are many other possibilities

14 14 Lecture8.nb 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 from - to 1

15 Lecture8.nb 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 much simpler sinusoidal components thus we are approaching time series as a weighted sum of harmonic functions (sines and cosines

16 16 Lecture8.nb 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

17 Lecture8.nb Sampling The process of converting a signal into a numeric sequence (in other words, continuous time or space to discrete time or space) The Nyquist frequency is equal to one-half of the sampling frequency w NYQUIST = T-1 2 The Nyquist frequency is the highest frequency that can be measured in a signal

18 18 Lecture8.nb Sampling - An Undersampled Signal sinh2 p f tl Consider signal with frequency f = 7kHz, and use sampling rate of f s = 8 khz This is an undersampled signal Use following interactive example to sample your own signal: frequency 6877 sampling rate f s 8 khz 16 khz 48 khz sinh2 p f tl Ê Ê Ê Ê Ê Ê Ê Ê Ê thsl Author: Carsten Roppel Source:

19 Lecture8.nb Sampling cont. According to the sampling theorem, following may occur: f f s ê 2: the samples uniquely represent the sine wave of frequency f. f > f s ê 2: aliasing occurs, because the replicated spectra begin to overlap. 0 f f s ê 2: a spectral line appears at the frequency f - f s. (Again try to use the interactive example from previous slide to see this)

20 20 Lecture8.nb 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?

21 Lecture8.nb Fourier Theory - Mathematical Prerequisities First Definitions are following (these has very nice property, that they are infinite): sin HxL = x - x3 3! + x5 5! - x7 7! +... = n=0 cos HxL = 1 - x2 2! + x4 4! - x6 6! +... = n=0 now consider definition of e x : e x x = n n=0 n! = 1 + x + x2 2! + x3 3! +... H-1L n x 2 n+1 H2 n+1l! H-1L n x 2 n H2 nl!

22 22 Lecture8.nb Fourier Theory - The Complex Exponential The complex exponential e i x is defined to be the complex variable whose real and imaginary parts are cos(x) and sin(x): Also known as Euler relationship: e i x = coshxl + i sinhxl

23 Lecture8.nb Fourier Transform Any infinite sequence x t may be viewed as a combination of an infinite number of sinusoids with different amplitudes and phases Discrete Fourier Transform (DFT) is defined as: XH f L = t=- x t e -i2p f t where x t are fourier coefficients and can be obtained from the inverse Fourier transfrom: x t = 1 p 2 p Ÿ -pxh f L e i 2 p f t f,

24 24 Lecture8.nb Fourier Series at Work Nice example of approximation of five different periodic functions by Fourier Series convergence See 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 Author : Alain Goriely Source : http : // demonstrations.wolfram.com/fourierseriesofsimplefunctions/

25 Lecture8.nb

26 26 Lecture8.nb Filters in Practice - EMA Volatility Estimation EMA filter plays important role in the risk management l 0.08 S&P standard deviation estimate with EMA Hl=0.9L

27 Lecture8.nb 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 properties of the original data. It does so by finding the values of t that minimise the function: L = T t=1 H y t - t t L 2 + l T-1 t=2 HHt t+1 - t t L - Ht t - t t-1 LL 2 where the weight on smoothness (l). main drawback of HP filter is that it

28 28 Lecture8.nb Filters in Practice - Hodrick-Prescott (HP) Filters l 2162 zoom interval Hin daysl 347 Show whole period Jan Jul Jan from - to 1

29 Lecture8.nb 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 information we can lose if we extract what appears to be "noise"

30 30 Lecture8.nb 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 Methods in Finance and Economics, Academic Press, ISBN Percival D.B., Walden A.T. : Wavelet Methods for Time Series Analysis, Cambridge University Press, ISBN

31 Lecture8.nb Questions Other examples during seminar THANK YOU FOR ATTENTION!

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