Alfabetische Lijst

Alfabetische lijst

AR(1)

The AR(1) process is defined as

where Wt is a stationarytime series, et is a white noise error term, and Ft is called the forecasting function.

(figure V.I.1-1)

We can now easily observe how the theoretical ACF of an AR(1) process should look like.

AR(2)

The AR(2) process is defined as

where Wt is a stationarytime series, et is a white noise error term, and Ft is the forecasting function.
The theoretical ACF and PACF are illustrated below. Figure (V.I.1-2) contains two possible ACF and PACF patterns for real roots while figure (V.I.1-3) shows the ACF and PACF patterns when the roots are complex.

(figure V.I.1-2)

(figure V.I.1-3)

ARIMA model

Auto Regressive Integrated Moving Average model. An ARIMA model can be written as the statistical formula that (adequately) describes a non-stationary time series. Hence, it contains lambdatransformations, integration processes, and ARMA models.

Auto Correlation Function (ACF) of a time series

is the series of values that correspond to the correlation of the time series and its own past:

first value = correlation between Yt and Yt-1

second value = correlation between Yt and Yt-2

third value = correlation between Yt and Yt-3

etc...

B

backforecasts

are forecasts of past (unobserved) values of the stationary time series. Backforecasts are used in the estimation algorithm and must converge to zero. If convergence is not achieved, the estimated parameters are biased

Box-Cox transformation (lambda-transform) of a time series: this is a mathematical function that - if properly used - may induce stability of the Standard Deviation (satisfying the second condition of stationarity).

C

Cumulative periodogram

relates the cumulative intensity of independent cyclic waves that are present in the time series to their periods (= length of a cyclic wave).

The spectrum chart provides a visual representation of the relationship between:

the frequency of all sinusoids that are contained in the time series

the intensity at which every sinusoid is present in the time series

The weighted sum of all sinusoids (x-axis) is equal to the original time series. The weights of this sum are equal to the intensities (y-axis) of each respective sinusoid.

The decomposition of a time series into its independent sinusoids is called "spectral analysis".

The cumulative periodogram is the visual representation of the relationship between the period of all sinusoids (x-axis) versus the cumulative intensity of all sinusoids that have a period equal to, or larger than the period of any given sinusoid.

Cyclic waves

are regular (periodic) movements that cause the time series to go up and down. Cyclic waves can be modeled by the use of sinusoids which are weighted sums of a cosine and sine of a given period. The period of a sinusoid is the length (measured in time) that is needed to complete a single cycle: the time that passes between two nearest tops of the sinusoid. The amplitude of a cyclic wave is a measure for the distance between the top and the bottom of the sinusoid.

D

The distribution function

The distribution function of a random variable Z is the function that gives the probability of Z being less than or equal to an real number z:

F(z) = P(Z=<z)

Drifted Random-Walk

a time series where each observation is equal to the previous observation plus a fixed constant number (which may be negative) plus an on-average-zero normal random number (which may be negative). The formula of the Drifted Random-Walk proces is: Yt = Yt-1 + c + et where c is a fixed number, and et is a random value (drawn from a normal distribution with zero mean).

Drifted Seasonal Walk

a time series where each observation is equal to last year's corresponding observation (same month, or same quarter) plus an on-average-zero normal random number (which may be negative) plus a constant term (which may be negative). The formula of the Seasonal Walk proces is: Yt = Yt-s + c + et where c is a constant, and et is a random value (drawn from a normal distribution with zero mean)

E

estimated residuals

are the computed interpolation forecast errors of an ARMA model. The estimated residuals should behave like a (Gaussian) white noise variable.

explicit equation in terms of stochasticinnovations

is the equation that relates future realizations of the time series under investigation Yt to past stochasticinnovations (et).

explicit forecasting function

is the equation that relates the forecast Ft to past observations of Yt.

F

Frequency of a cyclic wave

is the inverse of the period of a wave-like movement. If for instance, a short-term business cycle would exist with an average cycle period of 8 months, then the frequency of that cycle would be equal to 1/8 = 0.125. This means that every single monthly observation of the time series under investigation represents in fact 12.5% of a business cycle that lasts for about 8 months.

implicit ARIMA equation

is written in the form of an ARIMA(p,d,q)(P,D,Q) model with multiplicative seasonality:

Integration process (Integration model)

if a time series exhibits a 'strong' form of non-seasonal (first-order, second-order, third-order, etc...) autocorrelation, and if the trend-like behaviour of the series can be removed by the use of non-seasonal differencing, the time series is said to be generated by an Integration process (or the time series can be modelled by an Integration model).

invertibility of MA processes

implies that the parameters of the MA process lie inside the acceptable region of parameter combinations that result in a non-explosive forecast.

lambda-transform of a time series

this is a mathematical function that - if properly used - may induce stability of the Standard Deviation (satisfying the second condition of stationarity).

MA(1)

The definition of the MA(1) process is given by

(V.I.1-139)

where Wt is a stationarytime series, et is a white noise error component, and Ft is the forecasting function.

The theoretical ACF and PACF for the MA(1) are illustrated in figure (V.I.1-4).

MA(2)

By definition the MA(2) process is

which can be rewritten as

where Wt is a stationarytime series, et is a white noise error component, and Ft is the forecasting function.

These two possible cases are shown in figures (V.I.1-5) and (V.I.1-6).

(figure V.I.1-5)

(figure V.I.1-6)

Most Probable Change due to Chance

is equal to the standard error of the 1-step ahead forecast of a statistical model that adequately describes the time series. In a non-drifted non-seasonal random-walk time series for instance, the most probable change due to chance is equal to the standard error of the first order non-seasonal difference (d=1) of the time series.

Partial Auto Correlation Function (PACF)

is a special kind of Auto Correlation Function. The correlation coefficients are computed such that the all correlations are independent of each other. This implies that the correlations in the PACF are not mutually correlated (unlike in the ordinary ACF).

Probability function

The probability function of the random variable Z, denoted by f (z) is the function that gives the probability of Z taking the value z, for any real number z:

f (z) = P (Z=z)

Random-Walk

a time series where each observation is equal to the previous observation plus an on-average-zero normal random number (which may be negative). The formula of the Random-Walk process is: Yt = Yt-1 + et where et is a random value (drawn from a normal distribution with zero mean).

estimated residuals

are the computed interpolation forecast errors of an ARMA model. The estimated residuals should behave like a (Gaussian) white noise variable.

Seasonal Integration process

if a time series exhibits a 'strong' form of seasonal autocorrelation, and if the(strongly) seasonal behaviour of the series can be removed by the use of seasonal differencing, the time series is said to be generated by a SeasonalIntegration process (or the time series can be modelled by a SeasonalIntegration model).

Seasonal Walk

a time series where each observation is equal to last year's corresponding observation (same month, or same quarter) plus an on-average-zero normal random number (which may be negative). The formula of the Seasonal Walk proces is: Yt = Yt-s + et where et is a random value (drawn from a normal distribution with zero mean).

Sinusoid

is a regular wave-like movement with a fixed period (or frequency), and a fixed amplitude. In fact it can be computed as a "weighted sum" of a sine and a cosine function, at a given frequency.

Spectrum of a time series

relates the intensity (= amplitude in %) of independent cyclic waves that are present in the time series to their frequencies (= 1 / length of a cyclic wave). The spectrum can be interpreted as a "fingerprint" of the time series, and it enables the researcher to identify the strongest (c.q. most important) cyclic waves of the time series under investigation.

stability of AR processes

implies that the parameters of the AR process lie inside the acceptable region of parameter combinations that result in a non-explosive forecast.

Standard Deviation-Mean Plot (SMP)

is the scatter plot of the Standard Deviation versus Mean of sequential sections of the time series. Each section contains the same number of observations, preferably s periods (s = 12 for monthly time series). The regression line of the SMP illustrates how the Standard Deviation is related to the level (=mean)of the time series.

Stationary Time Series

is a time series that satisfies the following two conditions:

first condition: the time series is NOT Integrated; its variance cannot be reduced by applying any combination of seasonaland non-seasonal differencing. (Note: this is only a "rule of thumb", not a formal or scientific description!!!)

second condition: the most probable change due to chance is constant over time. This implies a constant standard error over the whole time period of the time series. This condition is imposed in order to be able to easily differentiate between changes that are due to chance, and changes that can be attributed to some important exogenous event.

Stationary Time Series (revised definition): is a time series that satisfies the following two conditions:

first condition: the time series is NOT Integrated. This implies that the Auto Correlation Function MUST NOT contain a series of slowly decreasing:

non-seasonal autocorrelation coefficients (at time lag 1, 2, 3, etc...)
seasonal autocorrelation coefficients (at time lag s, 2s, 3s, ... with s = seasonal period)

Alternatively this condition implies that:

the Spectrum does not show evidence of strong (important) cyclic waves of very low frequency (long periods)
the Spectrum does not show any evidence of strong cyclic waves of seasonal frequency (or periods s, s/2, s/3, s/4, ... with s = seasonal period)

stochastic innovations

are unsystematic (random) resultants of an unlimited number of (independent) causal impulses. By definition, stochastic innovations are unpredictable; only the probability density function is (assumed to be) known. Under general (and weak) conditions it can be assumed that stochastic innovations are normally distributed (cfr. Gaussian white noise).

Stochastic process

A statistical time series can be considered to be the result from some underlying statistical stochasticprocess. The processis representedby a mathematical model. The time series is a single realization of the generating process.

Time series

A time series is a set of observations ordered in time. In most cases (and in the context of this course) it is assumed that time series are equi-distant.
An equi-distant time series is discrete, with observations Yt at times t = 1, 2, 3, ..., T, where T is the length of the time series (= number of observations).

Example 1: Monthly sales data areassumed to beequi-distant (even though not every month has the same number of working days.

Example 2: Quotes on the stock market are not equi-distant because the interval between two sequential quotes is always different (this may range from a small fraction of 1 second to 30 seconds or even more). Since equi-distant time series are much easier to use, most stock market data providers compute aggregated quotes and prices: most commonly they offer 1, 2, 5, 10, 20, 60-minute data, daily averages, or daily closing prices.

Variance Reduction Matrix

a table containing the variances of the time series under investigation after various combinations of various degrees of non-seasonal and seasonal differencing. This table is used to identify the differencing combination that yields the lowest variance (c.q. explains the time series best).

(Gaussian) white noise

is a zero-mean, homoskedastic, uncorrelated, and normally distributed time series.

Wold's decomposition theorem

The most fundamental justification for time series analysis (as described in this course) is due to Wold's decomposition theorem, where it is explicitly proved that any (stationary) time series can be decomposed into two different parts. The first (deterministic) part can be exactly described by a linear combination of its own past, the second part is a MA component of a finite order.