## Weird behavior in high frequency markets

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Using a variety of identification methods and samples, I find that in most cases private spending falls significantly in response to an increase in government spending. These results imply that the average GDP multiplier lies below unity.

## Alan V. Oppenheim’s lectures on Digital Signal Processing (1975)

The MIT has just uploaded Alan V. Oppenheim‘s DSP lectures on youtube.

See more here (and enjoy!)

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## Early time series

The graph (part of a commentary of Macrobius on Cicero’s In Somnium Scipionis) which dates from the tenth, possible eleventh century  was meant to represent a plot of the inclinations of the planetary orbits as a function of the time. The zone of the zodiac is given on a plane with the horizontal (time) axis divided into thirty parts and the ordinate representing the width of the zodiacal belt.

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Source: “Time-Series”, Sir Maurice Kendall

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## Trend extraction and Detrending

Filters may be applied to a time series for a variety of reasons. Suppose that a time series consists of a long-term movement, a trend, upon which is superimposed an irregular component. A moving average filter will smooth the series revealing the trend more clearly.

Assume that $m_{t}$ is the filtered version of the $y_{t}$ series. Then

$m_{t} = M_{n}(L)y_{t} =\sum_{j=-r}^{j=r} w_{j}y_{t-j}$

The weights of a moving average filters add up to one i.e.  $M_{n}(L)=1$. The simplest such filter is the uniform moving average for which:

$w_{j} = \frac{1}{n} \; \; \; \; j=-r,...,r$

The gain of such filter is

$M_{n}(e^{-i \lambda}) = \left| \sum_{j=-r}^{r} \frac{1}{n} e^{-ij\lambda} \right| = \left| \frac{1}{n} \left( 1+2 \sum_{j=1}^{r} cos \lambda_{j} \right) \right| = \left| \frac{sin (n \lambda /2)}{n sin (\lambda /2)} \right|$

Gain of uniform moving average filter

The uniform moving average filter (applied on artificial data)

The Moving Average filter removes a cycle of period n together with its harmonics.

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## Nonlinearities and Thresholds

Jökulsá á Fjöllum nonlinearities

The infamous nonlinearity first observed by  Tong et al. (1985).  This nonlinearity is the effect of the melting of glaciers in the catchment area of Jökulsá á Fjöllum (Jokulsa River) on the latter’s flow.

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## Why is the Hodrick-Prescott filter often inappropriate?

The Hodrick-Prescott (HP) filter is the optimal estimator of the trend component in a smooth trend model with signal-to-noise ratio parameter fixed at 1/1600. It gives the detrended observations, $X_{t}$,  for large samples and t not near the beginning or end of the series

$\displaystyle X_{t}= \left[ \frac{(1-L)^{2}(1-L^{-1})^{2}}{\bar{q}_{\zeta}+ (1-L)^{2}(1-L^{-1})^{2}}\right] Y_{t}$

where $\bar{q}_{\zeta}= \sigma_{\zeta}^{2} / \sigma_{\epsilon}^{2}$.

Bear in mind that if the  smooth trend model was believed to be the true model there would be no reason to apply the HP filter. The filtered data of a smooth trend model contain nothing more than white noise. The belief here is clearly different.

We can easily show that the gain from the detrending filter is given by:

$G(\lambda) = \frac{4(1-cos \lambda)^{2}}{\bar{q}_{\zeta}+4(1-cos \lambda)^{2}} = \frac{16sin^{4}( \lambda /2)}{\bar{q}_{\zeta}+16sin^{4}( \lambda /2)}$

Note that the smaller the ${\bar{q}_{\zeta}}$ the more the filter concentrates on removing low frequencies.

Gain for HP filter

## Some notes on Linear Filters

Let $\{ X_t \}$ and  $\{ Y_t \}$ be two stationary time series related by:

$X_{t} = M_{n}(L)Y_{t} =\sum_{j=- \infty}^{j=\infty} g_{j}Y_{t-j}$

where

$\sum_{j=- \infty}^{j=\infty}|g_{j}| < \infty$ and  $\sum_{j=- \infty}^{j=\infty}|g_{j}|^2 < \infty$

$\{ X_t \}$ is the filtered version of  $\{ Y_t \}$ and $M_{n}(L)$ is the filter. The effect of a linear filter is to change the importance of various cyclical components of the series and/or induce a shift with regard to the position in time.

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## ARIMA spectrum

It is ofter useful to examine time-series from a frequency perspective. Note that  the power spectrum is defined as

$f_{y}(\lambda)=\underbrace{|M(e^{-i\lambda})|^{2}}_{| \Gamma ( \omega) |^{2}}f_{x}(\lambda)$

where

$|M(e^{-i\lambda})|^{2}=g(e^{-i\lambda})$

is the spectral generating function. For ARIMA

$f_{y}(\lambda)= \frac{\sigma}{2 \pi}\frac{|\theta(e^{-i\lambda})|^{2}}{|\phi(e^{-i\lambda})|^{2}}$

The following simple R code generates spectra for ARMA class time series.

spectrums<- function (AR,MA, sigma)
{
n<- length(AR)
m<- length(MA)
AR<- matrix(AR)
AR[-1]<- -AR[-1]
MA<- matrix(MA)
flp<- c()
for (i in 1:100){
TAR<- matrix(exp(0:(n-1)*i*pi/100*1i))
TMA<- matrix(exp(0:(m-1)*i*pi/100*1i))
ARs<- (t(AR)%*%TAR)*Conj(t(AR)%*%TAR)
MAs<- (t(MA)%*%TMA)*Conj(t(MA)%*%TMA)
fl<- 1/(2*pi)*MAs/ARs
flp<-c(flp,fl)
}
Re(c(flp))
}


example:

</pre>
#instruction: spectrums(AR parameters, MA parameters, Variance)

# For AR1 AR=c(1,phi1) MA=c(1), for ARMA(2,2) AR=c(1,phi1,phi2), MA=c(1,theta1, theta2) par(mfrow=c(2,2))

#example ARMA(1,1) phi=0.4, theta=-0.6

plot(spectrums(AR<- c(1,.4),MA<- c(1,-0.6), sigma<-1),ylab="ARMA(0.4,-0.6)", type="l")

#example AR(1) phi=0.5

plot(spectrums(AR<- c(1,.5),MA<- c(1), sigma<-1),ylab="AR1(0.5)", type="l")

#example AR(1) phi=0.8

plot(spectrums(AR<- c(1,.8),MA<- c(1), sigma<-1),ylab="AR1(0.8)", type="l")

#example MA(2) theta1=-0.6, theta2=-0.4

plot(spectrums(AR<- c(1),MA<- c(1,-0.6,-0.4),sigma<-1),ylab="MA2(-0.6,-0.4)", type="l")
<pre>

Note that it is also very  easy to apply a linear filter (or a series of them) to a given time series (or to generate its spectral density function). For example a moving average filter could be applied to a given AR(1) process by executing the following command

2*pi*spectrums(AR<- c(1),MA<- c(1/3,1/3,1/3),sigma<-1)*spectrums(AR<- c(1,-0.8,MA<- c(1),sigma<-1)

and a second difference filter

2*pi*spectrums(AR<- c(1),MA<- c(1,0,-1),sigma<-1)*spectrums(AR<- c(1,-0.8,MA<- c(1),sigma<-1)

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