This chapter provides facilities for investigating and modelling the statistical structure of series of observations collected at points in time. The models may then be used to forecast the series.
The chapter covers the following models and approaches.
|1.||Univariate time series analysis, including autocorrelation functions and autoregressive moving average (ARMA) models.|
|2.||Univariate spectral analysis.|
|3.||Transfer function (multi-input) modelling, in which one time series is dependent on other time series.|
|4.||Bivariate spectral methods including coherency, gain and input response functions.|
|5.||Vector ARMA models for multivariate time series.|
|6.||Kalman filter models.|
|7.||GARCH models for volatility.|
|8.||Inhomogeneous Time Series.|
Let the given time series be , where is its length. The structure which is intended to be investigated, and which may be most evident to the eye in a graph of the series, can be broadly described as:
|(a)||trends, linear or possibly higher-order polynomial;|
|(b)||seasonal patterns, associated with fixed integer seasonal periods. The presence of such seasonality and the period will normally be known a priori. The pattern may be fixed, or slowly varying from one season to another;|
|(c)||cycles or waves of stable amplitude and period (from peak to peak). The period is not necessarily integer, the corresponding absolute frequency (cycles/time unit) being and angular frequency . The cycle may be of pure sinusoidal form like , or the presence of higher harmonic terms may be indicated, e.g., by asymmetry in the wave form;|
|(d)||quasi-cycles, i.e., waves of fluctuating period and amplitude; and|
|(e)||irregular statistical fluctuations and swings about the overall mean or trend.|
Trends, seasonal patterns, and cycles might be regarded as deterministic components following fixed mathematical equations, and the quasi-cycles and other statistical fluctuations as stochastic and describable by short-term correlation structure. For a finite dataset it is not always easy to discriminate between these two types, and a common description using the class of autoregressive integrated moving-average (ARIMA) models is now widely used. The form of these models is that of difference equations (or recurrence relations) relating present and past values of the series. You are referred to Box and Jenkins (1976) for a thorough account of these models and how to use them. We follow their notation and outline the recommended steps in ARIMA model building for which methods are available.
If the variance of the observations in the series is not constant across the range of observations it may be useful to apply a variance-stabilizing transformation to the series. A common situation is for the variance to increase with the magnitude of the observations and in this case typical transformations used are the log or square root transformation. A range-mean plot or standard deviation-mean plot provides a quick and easy way of detecting non-constant variance and of choosing, if required, a suitable transformation. These are plots of either the range or standard deviation of successive groups of observations against their means.
These may be used to simplify the structure of a time series.
First-order differencing, i.e., forming the new series
will remove a linear trend. First-order seasonal differencing
eliminates a fixed seasonal pattern.
These operations reflect the fact that it is often appropriate to model a time series in terms of changes from one value to another. Differencing is also therefore appropriate when the series has something of the nature of a random walk, which is by definition the accumulation of independent changes.
Differencing may be applied repeatedly to a series, giving
where and are the orders of differencing. The derived series will be shorter, of length , and extend for .
Given that a series has (possibly as a result of simplifying by differencing operations) a homogeneous appearance throughout its length, fluctuating with approximately constant variance about an overall mean level, it is appropriate to assume that its statistical properties are stationary. For most purposes the correlations between terms or separated by lag give an adequate description of the statistical structure and are estimated by the sample autocorrelation function (ACF) , for .
As described by Box and Jenkins (1976), these may be used to indicate which particular ARIMA model may be appropriate.
The information in the autocorrelations, , may be presented in a different light by deriving from them the coefficients of the partial autocorrelation function (PACF) , for . which measures the correlation between and conditional upon the intermediate values . The corresponding sample values give further assistance in the selection of ARIMA models.
Both autocorrelation function (ACF) and PACF may be rapidly computed, particularly in comparison with the time taken to estimate ARIMA models.
The partial autocorrelation coefficient is determined as the final parameter in the minimum variance predictor of in terms of ,
where is the prediction error, and the first subscript of and emphasizes the fact that the parameters will alter as increases. Moderately good estimates of are obtained from the sample autocorrelation function (ACF), and after calculating the partial autocorrelation function (PACF) up to lag , the successive values of the prediction error variance estimates, , are available, together with the final values of the coefficients . If has nonzero mean, , it is adequate to use in place of in the prediction equation.
The correlation structure in stationary time series may often be represented by a model with a small number of parameters belonging to the autoregressive moving-average (ARMA) class. If the stationary series has been derived by differencing from the original series , then is said to follow an ARIMA model. Taking , the (non-seasonal) ARIMA model with autoregressive parameters and moving-average parameters , represents the structure of by the equation
where is an uncorrelated series (white noise) with mean and constant variance . If has a nonzero mean , then this is allowed for by replacing by in the model. Although is often estimated by the sample mean of this is not always optimal.
A series generated by this model will only be stationary provided restrictions are placed on to avoid unstable growth of . These are called stationarity constraints. The series may also be usefully interpreted as the linear innovations in (and in ), i.e., the error if were to be predicted using the information in all past values , provided also that satisfy invertibility constraints. This allows the series to be regenerated by rewriting the model equation as
For a series with short-term correlation only, i.e., is not significant beyond some low lag (see Box and Jenkins (1976) for the statistical test), then the pure moving-average model is appropriate, with no autoregressive parameters, i.e., .
Autoregressive parameters are appropriate when the autocorrelation function (ACF) pattern decays geometrically, or with a damped sinusoidal pattern which is associated with quasi-periodic behaviour in the series. If the sample partial autocorrelation function (PACF) is significant only up to some low lag , then a pure autoregressive model is appropriate, with . Otherwise moving-average terms will need to be introduced, as well as autoregressive terms.
The seasonal ARIMA model allows for correlation at lags which are multiples of the seasonal period . Taking , the series is represented in a two-stage manner via an intermediate series :
where , are the seasonal parameters and and are the corresponding orders. Again, may be replaced by .
In theory, the parameters of an ARIMA model are determined by a sufficient number of autocorrelations . Using the sample values in their place it is usually (but not always) possible to solve for the corresponding ARIMA parameters.
These are rapidly computed but are not fully efficient estimates, particularly if moving-average parameters are present. They do provide useful preliminary values for an efficient but relatively slow iterative method of estimation. This is based on the least squares principle by which parameters are chosen to minimize the sum of squares of the innovations , which are regenerated from the data using (2), or the reverse of (3) and (4) in the case of seasonal models.
Lack of knowledge of terms on the right-hand side of (2), when , is overcome by introducing unknown series values which are estimated as nuisance parameters, and using correction for transient errors due to the autoregressive terms. If the data is viewed as a single sample from a multivariate Normal density whose covariance matrix is a function of the ARIMA model parameters, then the exact likelihood of the parameters is
The least squares criterion as outlined above is equivalent to using the quadratic form
as an objective function to be minimized. Neglecting the term yields estimates which differ very little from the exact likelihood except in small samples, or in seasonal models with a small number of whole seasons contained in the data. In these cases bias in moving-average parameters may cause them to stick at the boundary of their constraint region, resulting in failure of the estimation method.
Approximate standard errors of the parameter estimates and the correlations between them are available after estimation.
The model residuals, , are the innovations resulting from the estimation and are usually examined for the presence of autocorrelation as a check on the adequacy of the model.
An ARIMA model is particularly suited to extrapolation of a time series. The model equations are simply used for replacing the unknown future values of by zero. This produces future values of , and if differencing has been used this process is reversed (the so-called integration part of ARIMA models) to construct future values of .
Forecast error limits are easily deduced.
This process requires knowledge only of the model orders and parameters together with a limited set of the terms which appear on the right-hand side of the models (3) and (4) (and the differencing equations) when . It does not require knowledge of the whole series.
We call this the state set. It is conveniently constituted after model estimation. Moreover, if new observations come to hand, then the model equations can easily be used to update the state set before constructing forecasts from the end of the new observations. This is particularly useful when forecasts are constructed on a regular basis. The new innovations may be compared with the residual standard deviation, , of the model used for forecasting, as a check that the model is continuing to forecast adequately.
Exponential smoothing is a relatively simple method of short term forecasting for a time series. A variety of different smoothing methods are possible, including; single exponential, Brown's double exponential, linear Holt (also called double exponential smoothing in some references), additive Holt–Winters and multiplicative Holt–Winters. The choice of smoothing method used depends on the characteristics of the time series. If the mean of the series is only slowly changing then single exponential smoothing may be suitable. If there is a trend in the time series, which itself may be slowly changing, then linear Holt smoothing may be suitable. If there is a seasonal component to the time series, e.g., daily or monthly data, then one of the two Holt–Winters methods may be suitable.
For a time series , for , the five smoothing functions are defined by the following:
- Single Exponential Smoothing
- Brown Double Exponential Smoothing
- Linear Holt Smoothing
- Additive Holt–Winters Smoothing
- Multiplicative Holt–Winters Smoothing
and is defined as in the additive Holt–Winters smoothing,
The parameters, , and control the amount of smoothing. The nearer these parameters are to one, the greater the emphasis on the current data point. Generally these parameters take values in the range to . The linear Holt and two Holt–Winters smoothers include an additional parameter, , which acts as a trend dampener. For the trend is dampened and for the forecast function has an exponential trend, removes the trend term from the forecast function and does not dampen the trend.
For all methods, values for , , and can be chosen by trying different values and then visually comparing the results by plotting the fitted values along side the original data. Alternatively, for single exponential smoothing a suitable value for can be obtained by fitting an model. For Brown's double exponential smoothing and linear Holt smoothing with no dampening, (i.e., ), suitable values for and, in the case of linear Holt smoothing, can be obtained by fitting an model. Similarly, the linear Holt method, with , can be expressed as an model and the additive Holt–Winters, with no dampening, (), can be expressed as a seasonal ARIMA model with order of the form . There is no similar procedure for obtaining parameter values for the multiplicative Holt–Winters method, or the additive Holt–Winters method with . In these cases parameters could be selected by minimizing a measure of fit using nonlinear optimization.
In describing a time series using spectral analysis the fundamental components are taken to be sinusoidal waves of the form , which for a given angular frequency , , is specified by its amplitude and phase , . Thus in a time series of observations it is not possible to distinguish more than independent sinusoidal components. The frequency range is limited to the shortest wavelength of two sampling units because any wave of higher frequency is indistinguishable upon sampling (or is aliased with) a wave within this range. Spectral analysis follows the idea that for a series made up of a finite number of sine waves the amplitude of any component at frequency is given to order by
For a series this is defined as
the scaling factor now being chosen in order that
i.e., the spectrum indicates how the sample variance () of the series is distributed over components in the frequency range .
It may be demonstrated that is equivalently defined in terms of the sample ACF of the series as
where are the sample autocovariance coefficients.
If the series does contain a deterministic sinusoidal component of amplitude , this will be revealed in the sample spectrum as a sharp peak of approximate width and height . This is called the discrete part of the spectrum, the variance associated with this component being in effect concentrated at a single frequency.
If the series has no deterministic components, i.e., is purely stochastic being stationary with autocorrelation function (ACF) , then with increasing sample size the expected value of converges to the theoretical spectrum – the continuous part
where are the theoretical autocovariances.
The sample spectrum does not however converge to this value but at each frequency point fluctuates about the theoretical spectrum with an exponential distribution, being independent at frequencies separated by an interval of or more. Various devices are therefore employed to smooth the sample spectrum and reduce its variability. Much of the strength of spectral analysis derives from the fact that the error limits are multiplicative so that features may still show up as significant in a part of the spectrum which has a generally low level, whereas they are completely masked by other components in the original series. The spectrum can help to distinguish deterministic cyclical components from the stochastic quasi-cycle components which produce a broader peak in the spectrum. (The deterministic components can be removed by regression and the remaining part represented by an ARIMA model.)
A large discrete component in a spectrum can distort the continuous part over a large frequency range surrounding the corresponding peak. This may be alleviated at the cost of slightly broadening the peak by tapering a portion of the data at each end of the series with weights which decay smoothly to zero. It is usual to correct for the mean of the series and for any linear trend by simple regression, since they would similarly distort the spectrum.
The estimate is calculated directly from the sample autocovariances as
the smoothing being induced by the lag window weights which extend up to a truncation lag which is generally much less than . The smaller the value of , the greater the degree of smoothing, the spectrum estimates being independent only at a wider frequency separation indicated by the bandwidth which is proportional to . It is wise, however, to calculate the spectrum at intervals appreciably less than this. Although greater smoothing narrows the error limits, it can also distort the spectrum, particularly by flattening peaks and filling in troughs.
The unsmoothed sample spectrum is calculated for a fine division of frequencies, then averaged over intervals centred on each frequency point for which the smoothed spectrum is required. This is usually at a coarser frequency division. The bandwidth corresponds to the width of the averaging interval.
We now consider the context in which one time series, called the dependent or output series, , is believed to depend on one or more explanatory or input series, e.g., . This dependency may follow a simple linear regression, e.g.,
or more generally may involve lagged values of the input
The sequence is called the impulse response function (IRF) of the relationship. The term represents that part of which cannot be explained by the input, and it is assumed to follow a univariate ARIMA model. We call the (output) noise component of , and it includes any constant term in the relationship. It is assumed that the input series, , and the noise component, , are independent.
The part of which is explained by the input is called the input component :
The eventual aim is to model both these components of on the basis of observations of and . In applications to forecasting or control both components are important. In general there may be more than one input series, e.g., and , which are assumed to be independent and corresponding components and , so
In a similar manner to that in which the structure of a univariate series may be represented by a finite-parameter ARIMA model, the structure of an input component may be represented by a transfer function (TF) model with delay time , autoregressive-like parameters and moving-average-like parameters :
If this represents an impulse response function (IRF) which is infinite in extent and decays with geometric and/or sinusoidal behaviour. The parameters are constrained to satisfy a stability condition identical to the stationarity condition of autoregressive models. There is no constraint on .
An important tool for investigating how an input series affects an output series is the sample cross-correlation function (CCF) , for between the series. If and are (jointly) stationary time series this is an estimator of the theoretical quantity
The sequence , for , is distinct from , though it is possible to interpret
When the series and are believed to be related by a transfer function (TF) model, the CCF is determined by the impulse response function (IRF) and the autocorrelation function (ACF) of the input .
In the particular case when is an uncorrelated series or white noise (and is uncorrelated with any other inputs):
and the sample CCF can provide an estimate of :
where and are the sample standard deviations of and , respectively.
In theory the IRF coefficients determine the parameters in the TF model, and using to estimate it is possible to solve for preliminary estimates of , .
In general an input series is not white noise, but may be represented by an ARIMA model with innovations or residuals which are white noise. If precisely the same operations by which is generated from are applied to the output to produce a series , then the transfer function relationship between and is preserved between and . It is then possible to estimate
The procedure of generating from (and from ) is called prewhitening or filtering by an ARIMA model. Although is necessarily white noise, this is not generally true of .
The term multi-input model is used for the situation when one output series is related to one or more input series , as described in [Linear Lagged Relationships Between Time Series]. If for a given input the relationship is a simple linear regression, it is called a simple input; otherwise it is a transfer function input. The error or noise term follows an ARIMA model.
Given that the orders of all the transfer function models and the ARIMA model of a multi-input model have been specified, the various parameters in those models may be (simultaneously) estimated.
The procedure used is closely related to the least squares principle applied to the innovations in the ARIMA noise model.
The innovations are derived for any proposed set of parameter values by calculating the response of each input to the transfer functions and then evaluating the noise as the difference between this response (combined for all the inputs) and the output. The innovations are derived from the noise using the ARIMA model in the same manner as for a univariate series, and as described in [ARIMA models].
In estimating the parameters, consideration has to be given to the lagged terms in the various model equations which are associated with times prior to the observation period, and are therefore unknown. The method descriptions provide the necessary detail as to how this problem is treated.
Also, as described in [ARIMA model estimation] the sum of squares criterion
is related to the quadratic form in the exact log-likelihood of the parameters:
Here is the vector of appropriately differenced noise terms, and
where is the innovation variance parameter.
The least squares criterion is therefore identical to minimization of the quadratic form, but is not identical to exact likelihood. Because may be expressed as , where is a function of the ARIMA model parameters, substitution of by its maximum likelihood (ML) estimator yields a concentrated (or profile) likelihood which is a function of
is the length of the differenced noise series , and .
Use of the above quantity, called the deviance, , as an objective function is preferable to the use of alone, on the grounds that it is equivalent to exact likelihood, and yields estimates with better properties. However, there is an appreciable computational penalty in calculating , and in large samples it differs very little from , except in the important case of seasonal ARIMA models where the number of whole seasons within the data length must also be large.
You are given the option of taking the objective function to be either or , or a third possibility, the marginal likelihood. This is similar to exact likelihood but can counteract bias in the ARIMA model due to the fitting of a large number of simple inputs.
Approximate standard errors of the parameter estimates and the correlations between them are available after estimation.
The model residuals are the innovations resulting from the estimation, and they are usually examined for the presence of either autocorrelation or cross-correlation with the inputs. Absence of such correlation provides some confirmation of the adequacy of the model.
A multi-input model may be used to forecast the output series provided future values (possibly forecasts) of the input series are supplied.
Construction of the forecasts requires knowledge only of the model orders and parameters, together with a limited set of the most recent variables which appear in the model equations. This is called the state set. It is conveniently constituted after model estimation. Moreover, if new observations of the output series and of (all) the independent input series become available, then the model equations can easily be used to update the state set before constructing forecasts from the end of the new observations. The new innovations generated in this updating may be used to monitor the continuing adequacy of the model.
In many time series applications it is desired to calculate the response (or output) of a transfer function (TF) model for a given input series.
Smoothing, detrending, and seasonal adjustment are typical applications. You must specify the orders and parameters of a TF model for the purpose being considered. This may then be applied to the input series.
Again, problems may arise due to ignorance of the input series values prior to the observation period. The transient errors which can arise from this may be substantially reduced by using ‘backforecasts’ of these unknown observations.
Multi-input modelling represents one output time series in terms of one or more input series. Although there are circumstances in which it may be more appropriate to analyse a set of time series by modelling each one in turn as the output series with the remainder as inputs, there is a more symmetric approach in such a context. These models are known as vector autoregressive moving-average (VARMA) models.
As in the case of a univariate time series, it may be useful to simplify the series by differencing operations which may be used to remove linear or seasonal trends, thus ensuring that the resulting series to be used in the model estimation is stationary. It may also be necessary to apply transformations to the individual components of the multivariate series in order to stabilize the variance. Commonly used transformations are the log and square root transformations.
Multivariate analogues of the autocorrelation and partial autocorrelation functions are available for analysing a set of time series, , for , thereby making it possible to obtain some understanding of a suitable VARMA model for the observed series.
It is assumed that the time series have been differenced if necessary, and that they are jointly stationary. The lagged correlations between all possible pairs of series, i.e.,
are then taken to provide an adequate description of the statistical relationships between the series. These quantities are estimated by sample auto- and cross-correlations . For each these may be viewed as elements of a (lagged) autocorrelation matrix.
Thus consider the vector process (with elements ) and lagged autocovariance matrices with elements of where . Correspondingly, is estimated by the matrix with elements where is the sample variance of .
For a series with short-term cross-correlation only, i.e., is not significant beyond some low lag , then the pure vector model, with no autoregressive parameters, i.e., , is appropriate.
The correlation matrices provide a description of the joint statistical properties of the series. It is also possible to calculate matrix quantities which are closely analogous to the partial autocorrelations of univariate series (see [Partial autocorrelations]). Wei (1990) discusses both the partial autoregression matrices proposed by Tiao and Box (1981) and partial lag correlation matrices.
In the univariate case the partial autocorrelation function (PACF) between and is the correlation coefficient between the two after removing the linear dependence on each of the intervening variables . This partial autocorrelation may also be obtained as the last regression coefficient associated with when regressing on its lagged variables . Tiao and Box (1981) extended this method to the multivariate case to define the partial autoregression matrix. Heyse and Wei (1985) also extended the univariate definition of the PACF to derive the correlation matrix between the vectors and after removing the linear dependence on each of the intervening vectors , the partial lag correlation matrix.
Note that the partial lag correlation matrix is a correlation coefficient matrix since each of its elements is a properly normalized correlation coefficient. This is not true of the partial autoregression matrices (except in the univariate case for which the two types of matrix are the same). The partial lag correlation matrix at lag also reduces to the regular correlation matrix at lag ; this is not true of the partial autoregression matrices (again except in the univariate case).
Both the above share the same cut-off property for autoregressive processes; that is for an autoregressive process of order , the terms of the matrix at lags and greater are zero. Thus if the sample partial cross-correlations are significant only up to some low lag then a pure vector model is appropriate with . Otherwise moving-average terms will need to be introduced as well as autoregressive terms.
Under the hypothesis that is an autoregressive process of order , times the sum of the squared elements of the partial lag correlation matrix at lag is asymptotically distributed as a variable with degrees of freedom where is the dimension of the multivariate time series. This provides a diagnostic aid for determining the order of an autoregressive model.
The partial autoregression matrices may be found by solving a multivariate version of the Yule–Walker equations to find the autoregression matrices, using the final regression matrix coefficient as the partial autoregression matrix at that particular lag.
The basis of these calculations is a multivariate autoregressive model:
where are matrix coefficients, and is the vector of errors in the prediction. These coefficients may be rapidly computed using a recursive technique which requires, and simultaneously furnishes, a backward prediction equation:
(in the univariate case ).
The forward prediction equation coefficients, , are of direct interest, together with the covariance matrix of the prediction errors . The calculation of these quantities for a particular maximum equation lag involves calculation of the same quantities for increasing values of .
The quantities may be viewed as generalized variance ratios, and provide a measure of the efficiency of prediction (the smaller the better). The reduction from to which occurs on extending the order of the predictor to may be represented as
where is a multiple squared partial autocorrelation coefficient associated with degrees of freedom.
Sample estimates of all the above quantities may be derived by using the series covariance matrices , for , in place of . The best lag for prediction purposes may be chosen as that which yields the minimum final prediction error (FPE) criterion:
An alternative method of estimating the sample partial autoregression matrices is by using multivariate least squares to fit a series of multivariate autoregressive models of increasing order.
The cross-correlation structure of a stationary multivariate time series may often be represented by a model with a small number of parameters belonging to the VARMA class. If the stationary series has been derived by transforming and/or differencing the original series , then is said to follow the VARMA model:
where is a vector of uncorrelated residual series (white noise) with zero mean and constant covariance matrix , are the autoregressive (AR) parameter matrices and are the moving-average (MA) parameter matrices. If has a nonzero mean , then this can be allowed for by replacing by in the model.
A series generated by this model will only be stationary provided restrictions are placed on to avoid unstable growth of . These are stationarity constraints. The series may also be usefully interpreted as the linear innovations in , i.e., the error if were to be predicted using the information in all past values , provided also that satisfy what are known as invertibility constraints. This allows the series to be generated by rewriting the model equation as
The method of maximum likelihood (ML) may be used to estimate the parameters of a specified VARMA model from the observed multivariate time series together with their standard errors and correlations.
The residuals from the model may be examined for the presence of autocorrelations as a check on the adequacy of the fitted model.
Forecasts of the series may be constructed using a multivariate version of the univariate method. Efficient methods are available for updating the forecasts each time new observations become available.
The relationship between two time series may be investigated in terms of their sinusoidal components at different frequencies. At frequency a component of of the form
has its amplitude and phase lag estimated by
and similarly for . In the univariate analysis only the amplitude was important – in the cross analysis the phase is important.
This is defined by
It may be demonstrated that this is equivalently defined in terms of the sample cross-correlation function (CCF), , of the series as
where is the cross-covariance function.
The cross-spectrum is specified by its real part or cospectrum and imaginary part or quadrature spectrum , but for the purpose of interpretation the cross-amplitude spectrum and phase spectrum are useful:
If the series and contain deterministic sinusoidal components of amplitudes and phases at frequency , then will have a peak of approximate width and height at that frequency, with corresponding phase . This supplies no information that cannot be obtained from the two series separately. The statistical relationship between the series is better revealed when the series are purely stochastic and jointly stationary, in which case the expected value of converges with increasing sample size to the theoretical cross-spectrum
where . The sample spectrum, as in the univariate case, does not converge to the theoretical spectrum without some form of smoothing which either implicitly (using a lag window) or explicitly (using a frequency window) averages the sample spectrum over wider bands of frequency to obtain a smoothed estimate .
If there is no statistical relationship between the series at a given frequency, then , and the smoothed estimate , will be close to . This is assessed by the squared coherency between the series:
where is the corresponding smoothed univariate spectrum estimate for , and similarly for . The coherency can be treated as a squared multiple correlation. It is similarly invariant in theory not only to simple scaling of and , but also to filtering of the two series, and provides a useful test statistic for the relationship between autocorrelated series. Note that without smoothing,
so the coherency is at all frequencies, just as a correlation is for a sample of size . Thus smoothing is essential for cross-spectrum analysis.
If is believed to be related to by a linear lagged relationship as in [Linear Lagged Relationships Between Time Series], i.e.,
then the theoretical cross-spectrum is
is called the frequency response of the relationship.
Thus if were a sinusoidal wave at frequency (and were absent), would be similar but multiplied in amplitude by and shifted in phase by . Furthermore, the theoretical univariate spectrum
where , with spectrum , is assumed independent of the input .
Cross-spectral analysis thus furnishes estimates of the gain
and the phase
From these representations of the estimated frequency response , parametric transfer function (TF) models may be recognized and selected. The noise spectrum may also be estimated as
a formula which reflects the fact that in essence a regression is being performed of the sinusoidal components of on those of over each frequency band.
Interpretation of the frequency response may be aided by extracting from estimates of the impulse response function (IRF) . It is assumed that there is no anticipatory response between and , i.e., no coefficients with or are needed (their presence might indicate feedback between the series).
The estimate of the cross-spectrum is calculated from the sample cross-variances as
The lag window extends up to a truncation lag as in the univariate case, but its centre is shifted by an alignment lag usually chosen to coincide with the peak cross-correlation. This is equivalent to an alignment of the series for peak cross-correlation at lag , and reduces bias in the phase estimation.
The selection of the truncation lag , which fixes the bandwidth of the estimate, is based on the same criteria as for univariate series, and the same choice of and window shape should be used as in univariate spectrum estimation to obtain valid estimates of the coherency, gain, etc., and test statistics.
The computations are exactly as for smoothing of the univariate spectrum except that allowance is made for an implicit alignment shift between the series.
Kalman filtering provides a method for the analysis of multidimensional time series. The underlying model is:
where is the unobserved state vector, is the observed measurement vector, is the state noise, is the measurement noise, is the state transition matrix, is the noise coefficient matrix and is the measurement coefficient matrix at time . The state noise and the measurement noise are assumed to be uncorrelated with zero mean and covariance matrices:
If the system matrices , , and the covariance matrices are known then Kalman filtering can be used to compute the minimum variance estimate of the stochastic variable .
The estimate of given observations to is denoted by with state covariance matrix while the estimate of given observations to is denoted by with covariance matrix .
The update of the estimate, , from time to time , is computed in two stages.
First, the update equations are
where the residual has an associated covariance matrix , and is the Kalman gain matrix with
The second stage is the one-step-ahead prediction equations given by
These two stages can be combined to give the one-step-ahead update-prediction equations
The above equations thus provide a method for recursively calculating the estimates of the state vectors and and their covariance matrices and from their previous values. This recursive procedure can be viewed in a Bayesian framework as being the updating of the prior by the data .
The initial values and are required to start the recursion. For stationary systems, can be computed from the following equation:
which can be solved by iterating on the equation. For the value can be used if it is available.
To improve the stability of the computations the square root algorithm is used. One recursion of the square root covariance filter algorithm which can be summarised as follows:
where is an orthogonal transformation triangularizing the left-hand pre-array to produce the right-hand post-array, is the lower triangular Cholesky factor of the state covariance matrix , and are the lower triangular Cholesky factor of the covariance matrices and and is the lower triangular Cholesky factor of the covariance matrix of the residuals. The relationship between the Kalman gain matrix, , and is given by
To improve the efficiency of the computations when the matrices and do not vary with time the system can be transformed to give a simpler structure. The transformed state vector is where is the transformation that reduces the matrix pair to lower observer Hessenberg form. That is, the matrix is computed such that the compound matrix
is a lower trapezoidal matrix. The transformations need only be computed once at the start of a series, and the covariance matrices and can still be time-varying.
If the state space model contains unknown parameters, , these can be estimated using maximum likelihood (ML). Assuming that and are normal variates the log-likelihood for observations , for , is given by
Optimal estimates for the unknown model parameters can then be obtained by using a suitable optimizer method to maximize the likelihood function.
Once the model has been fitted forecasting can be performed by using the one-step-ahead prediction equations. The one-step-ahead prediction equations can also be used to ‘jump over’ any missing values in the series.
Many commonly used time series models can be written as state space models. A univariate model can be cast into the following state space form:
The representation for a -variate series (VARMA) is very similar to that given above, except now the state vector is of length and the and are now matrices and the 1s in , and are now the identity matrix of order . If or then the appropriate or matrices are set to zero, respectively.
Since the compound matrix
is already in lower observer Hessenberg form (i.e., it is lower trapezoidal with zeros in the top right-hand triangle) the invariant Kalman filter algorithm can be used directly without the need to generate a transformation matrix .
Rather than modelling the mean (for example using regression models) or the autocorrelation (by using ARMA models) there are circumstances in which the variance of a time series needs to be modelled. This is common in financial data modelling where the variance (or standard deviation) is known as volatility. The ability to forecast volatility is a vital part in deciding the risk attached to financial decisions like portfolio selection. The basic model for relating the variance at time to the variance at previous times is the autoregressive conditional heteroskedastic (ARCH) model. The standard ARCH model is defined as
where is the information up to time and is the conditional variance.
In a similar way to that in which autoregressive (AR) models were generalized to ARMA models the ARCH models have been generalized to a GARCH model; see Engle (1982), Bollerslev (1986) and Hamilton (1994)
This can be combined with a regression model:
where and where , for , are the exogenous variables.
The above models assume that the change in variance, , is symmetric with respect to the shocks, that is, that a large negative value of has the same effect as a large positive value of . A frequently observed effect is that a large negative value often leads to a greater variance than a large positive value. The following three asymmetric models represent this effect in different ways using the parameter as a measure of the asymmetry.
Type I AGARCH()
Type II AGARCH()
GJR-GARCH(), or Glosten, Jagannathan and Runkle GARCH (see Glosten et al. (1993))
where if and if .
The first assumes that the effects of the shocks are symmetric about rather than zero, so that for the effect of negative shocks is increased and the effect of positive shocks is decreased. Both the Type II AGARCH and the GJR GARCH (see Glosten et al. (1993)) models introduce asymmetry by increasing the value of the coefficient of for negative values of . In the case of the Type II AGARCH the effect is multiplicative while for the GJR GARCH the effect is additive.
|Type II AGARCH|
(Note that in the case of GJR GARCH, needs to be positive to inflate variance after negative shocks while for Type I and Type II AGARCH, needs to be negative.)
A third type of GARCH model is the exponential GARCH (EGARCH). In this model the variance relationship is on the log scale and hence asymmetric.
where and denotes the expected value of .
Note that the terms represent a symmetric contribution to the variance while the terms give an asymmetric contribution.
Another common characteristic of financial data is that it is heavier in the tails (leptokurtic) than the Normal distribution. To model this the Normal distribution is replaced by a scaled Student's -distribution (that is a Student's -distribution standardized to have variance ). The Student's -distribution is such that the smaller the degrees of freedom the higher the kurtosis for degrees of freedom .
The models are fitted by maximizing the conditional log-likelihood. For the Normal distribution the conditional log-likelihood is
For the Student's -distribution the function is more complex. An approximation to the standard errors of the parameter estimates is computed from the Fisher information matrix.
If we denote a generic univariate time series as a sequence of pairs of values , for where the 's represent an observed scalar value and the 's the time that the value was observed, then in a standard time series analysis, as discussed in other sections of this introduction, it is assumed that the series being analysed is homogeneous, that is the sampling times are regularly spaced with for some value . In many real world applications this assumption does not hold, that is, the series is inhomogeneous.
Standard time series analysis techniques cannot be used on an inhomogeneous series without first preprocessing the series to construct an artificial homogeneous series, by for example, resampling the series at regular intervals. Zumbach and Müller (2001) introduced a series of operators that can be used to extract robust information directly from the inhomogeneous time series. In this context, robust information means that the results should be essentially independent of minor changes to the sampling mechanism used when collecting the data, for example, changing a number of time stamps or adding or removing a few observations.
The basic operator available for inhomogeneous time series is the exponential moving average (EMA). This operator has a single parameter, , and is an average operator with an exponentially decaying kernel given by:
This gives rise to the following iterative formula:
The value of depends on the method of interpolation chosen. Three interpolation methods are available:
Given the EMA, a number of other operators can be defined, including:
A discussion of each of these operators, their use and in some cases, alternative definitions, are given in Zumbach and Müller (2001).
|(i)||-Iterated Exponential Moving Average, defined as
|(ii)||Moving Average (MA), defined as
|(iii)||Moving Norm (MNorm), defined as
|(iv)||Moving Variance (MVar), defined as
|(v)||Moving Standard Deviation (MSD), defined as
|(vi)||Differential ( ), defined as
|(vii)||Volatility, defined as
Akaike H (1971) Autoregressive model fitting for control Ann. Inst. Statist. Math. 23 163–180
Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity Journal of Econometrics 31 307–327
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica 50 987–1008
Gentleman W S and Sande G (1966) Fast Fourier transforms for fun and profit Proc. Joint Computer Conference, AFIPS 29 563–578
Glosten L, Jagannathan R and Runkle D (1993) Relationship between the expected value and the volatility of nominal excess return on stocks Journal of Finance 48 1779–1801
Hamilton J (1994) Time Series Analysis Princeton University Press
Heyse J F and Wei W W S (1985) The partial lag autocorrelation function Technical Report No. 32 Department of Statistics, Temple University, Philadelphia
Tiao G C and Box G E P (1981) Modelling multiple time series with applications J. Am. Stat. Assoc. 76 802–816
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley
Zumbach G O and Müller U A (2001) Operators on inhomogeneous time series International Journal of Theoretical and Applied Finance 4(1) 147–178