Time Series Note 9 & 10 & 11
what is a wavelet
A Wavelet is a wave-like oscillation that is localized in time
Discrete Fourier Transform (DFT)
A formula that reduces any periodic waveform, no matter how complex, to a series of simple sine waves each of which has its own amplitude and frequency.
periodogram definition and formula for odd and even sample sizes
A graphical representation of harmonic information in a data set. Often taken from Fourier analysis of the data
what is ARMA vs ARCH used for
ARMA model for the conditional mean We use ARCH (autoregressive conditional heteroscedasticity) model for the conditional variance
if a series is stationary what does it look like on the frequency domain
All frequency components exist at all times
why do we need conditional variance
An asset is risky if its return is volatile (changes with a wide range over time) We know that variance is a measure volatility (dispersion), and so the risk
what is a fourier series
Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient
periodic signal
Any periodic signal can be defined by three parameters: ➢Amplitude ➢Frequency/ Period ➢Phase
Fourier transformation formula
Converts data from time domain into frequency domain
what is volatility clustering
Financial markets tend to exhibit volatility clustering, which means that periods of high volatility tend to be followed by more volatile periods or are grouped together. The occurrence of significant news in the market tends to cause volatility, and it may take several periods for the market to fully process and react to the news, resulting in increased volatility
what does a root near 1 of the autoregressive polynomial mean
For example, a root near 1 of the autoregressive polynomial suggests that the data should be differenced before fitting an ARMA model
what is frequency domain analysis
Frequency domain analysis, also known as spectral analysis, studies how periodic components at different frequencies describe the evolution of a time series. Thus, the main concern of this approach is to determine the periodic components embedded in the time series
what does a root near 1 of the moving average polynomial mean
However, a root near 1 of the moving-average polynomial indicates that the data were overdifferenced
when should we use ARCH(p)
Note that ARCH(p) should only ever be applied to a series that has already had an appropriate model fitted sufficient to leave the residuals looking like white noise. Since we can only tell whether ARCH is appropriate or not by squaring the residuals and examining the ACF, we also need to ensure that the mean of the residuals is zero.
period and frequency
Period (T) refers to the amount of time, a signal needs to complete one cycle. Frequency (f) refers to the number of periods per time unit. Frequency and Period are the inverse of each other:
why do we sometimes want to use lagged effects as a predictor in our dynamic regression models
Sometimes, the impact of a predictor which is included in a regression model will not be simple and immediate. For example, an advertising campaign may impact sales for some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months. Similarly, a change in a company's safety policy may reduce accidents immediately, but have a diminishing effect over time as employees take less care when they become familiar with the new working conditions
why do we give af about wavelets
The basic idea is to compute how much of a wavelet is in a signal for a particular scale and location
amplitude
The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
what does the prescence of volatility clustering imply
The presence of volatility clustering in financial markets implies that the conditional variance changes over time, meaning that high volatility today can lead to high volatility in the future. The ARCH process can capture the phenomenon of volatility clustering as it models time-varying conditional variance
phase
The term phase describes the position of the waveform relative to time zero. That is, it measures the horizontal displacement of the basic sine or cosine function.
when does the unit root problem arise in time series
The unit root problem in time series arises when either the autoregressive or moving-average polynomial of an ARMA model has a root on or near the unit circle
what is time domain analysis
Time domain analysis examines how a time series process evolves through time. Its main concern is to explore whether a time series has a trend
how do we make forecasts with dynamic regression models
To forecast using a regression model with ARIMA errors, we need to forecast the regression part of the model and the ARIMA part of the model, and combine the results. As with ordinary regression models, in order to obtain forecasts we first need to forecast the predictors. When the predictors are known into the future (e.g., calendar-related variables such as time, day-of-week, etc.), this is straightforward. But when the predictors are themselves unknown, we must either model them separately, or use assumed future values for each predictor
Inverse discrete Fourier transform (IDFT)
Transforms multiple OFDM subcarriers from the frequency domain to the time domain.
properties of wavelets and define them
Wavelets have two basic properties: scale and location. Scale (or dilation) defines how "stretched" or "squished" a wavelet is. This property is related to frequency as defined for waves. Location defines where the wavelet is positioned in time (or space).
when is dynamic regression with Fourier terms useful
When there are long seasonal periods, a dynamic regression with Fourier terms is often better than other models. For example, daily data can have annual seasonality of length 365, weekly data has seasonal period of approximately 52, while half-hourly data can have several seasonal periods, the shortest of which is the daily pattern of period 48
advantages and disadvantages of short term fourier transform
Wide windows do not provide good localization at high frequencies ➢ So use narrower windows at high frequencies Narrow windows do not provide good localization at low frequencies ➢ Use wider windows at low frequencies
how do we make forecasts using croston's method
alpha is a smoothing parameter
how do we analyze time series in time domain
autocorrelation function or correlogram
what property would an ARMA model of log returns have
conditional variance of sigma^2is independent of t and past returns
how does short time fourier transform work
dennis derived this stuff during ww2 - nice
if the sample acf of the square of the ARMA residuals is not white noise what does this show
if acf of residuals is white noise but acf of squared residuals is not white noise it shows that the residuals are white noise but are NOT independent -> we need a garch model
why do we use a garch model
it has been found that better fits to the data are obtained by relaxing the Gaussian assumption of the residuals {e} and supposing instead that the distribution of r_t given { r_s, s < t} has a heavier tailed zero-mean distribution such as Student's t-distribution. To incorporate such distributions we can define a general GARCH(p, q) process as a stationary process { r_t} satisfying the following generalized forms of error distribution
when n is odd, how can we write the frequencies in a fourier series
let n = 2k+1 -> then the frequencies are of the form 1/n, 2/n, ...,k/n and are called the fourier frequencies
what is dynamic regression
like a normal linear regression but the errors (n_t) are allowed to contain autocorrelation
what process does log return of stock price follow
log return (or simply return) for day t, has sample-paths resembling those of white noise
if a series has trend or seasonal effects should we use arch?
no. ARCH should only ever be applied to series that do not have any trends or seasonal effects, i.e. that has no (evident) serially correlation. ARIMA is often applied to such a series, at which point ARCH may be a good fit
can we say log return is independent white noise?
nope!
find E(r_t r_{t-1}) for an ARCH(1) model
shows that r_t and r_{t-1} are uncorrelated
another name for frequency domain analysis what fields is it used in
spectral analysis has been found to be especially useful in acoustics, communications engineering, geophysical science, and biomedical science, for example
what process do log prices of a stock follow
stock closing prices appear to be non stationary but log price shows sample paths like those of a random walk
when does using time series of counts not differ from the continuous models we have been studying so far
when we have at least 100 counts
is cosine a shifted sine function?
yes
what is a downside of using the fourier transform to determine if a series is nonstationary
you cannot see it in the time domain in the frequency domain it looks the same but in the time domain it might be different