Time series in ModelRisk

See also: Time series introduction, Time Series window, Wilkie Models windows, Geometric Brownian Motion models, Autoregressive models

A time series model is a stochastic forecast of a variable that varies randomly over time.

ModelRisk contains a number of advanced time series models. All time series can be simulated, parameter estimates can be determined from data, or projections made based on fitting to historic data.

Time series can be inserted by directly inserting (array) functions in spreadsheet cells, or through the univariate time series, multivariate time series, or time series fit windows. These ModelRisk  windows each have their separate page in this help file, while the general use of the functions is explained below.

To generate random values from a time series, use a VoseTimeSeries array function. The general syntax is:

{=VoseTimeSeries([parameters], LastValue,LogReturns)}

where Series is replaced by the name of the time series.

• [parameters] - the time series' parameters separated by commas.

• LastValue - last known historic value. The generated time series values will continue on from this value. Should only be provided if the LogReturn parameter is set to FALSE or omitted.

• LogReturns - optional boolean (TRUE/FALSE) parameter that specifies whether or not the time series are in log returns. Default is FALSE.

For example, to generate 10 random values from a GBM(0.02,0.1) model that start from a value of 100 you would insert

{=VoseTimeGBM(0.02,0.1,100)}

over a range of 10 spreadsheet cells. To generate LogReturns of that same time series you would write

{=VoseTimeGBM(0.02,0.1,100, TRUE)}

As the ModelRisk Time Series functions typically take a lot of parameters, we recommend for these in particular to use the Time Series window to avoid errors.

This topic is about forecasting from time series. For an explanation about time series fitting with Modelrisk see Time series fitting functions.

About the LastValue parameter

Where appropriate, time series functions take a LastValue parameter. This is the last value of the variable from which the time series forecast is to be made, S0.

In a forecast model not fitted to data, LastValue is the value of the variable at time zero, and the forecast projects a series from period one as a departure from this value at time zero. In this situation LastValue is a required parameter.

If the LogReturn option is selected there is no need to specify LastValue, since the forecast then projects the series of log returns rt = LN[St/St-1]. If the LogReturn option is not selected (the default) the forecast needs the LastValue (S0) as it then makes the projection: S1=S0*EXP(r1); S2=S1*EXP(r2); etc.  

The following list describes the time series models available in ModelRisk.

Geometric Brownian Motion (GBM) based models

GBM is usually the default starting point for a time series of a non-negative financial variable - like a stock price, exchange rate or interest rate. It assumes that the fractional changes in the variable between periods are independent, random variables following a Normal distribution. ModelRisk offers the following GBM-related distributions:

VoseTimeGBM - Basic GBM.

VoseTimeGBMJD - GBM with jump diffusion, meaning random discrete jumps can affect the variable.

VoseTimeGBMMR - GBM with mean reversion, meaning the variable is drawn back towards its long-run mean in proportion to its deviation from the mean.

VoseTimeGBMMJDMR - GBM with both mean reversion and jump diffusion.

VoseTimeSeasonalGBM - GBM with seasonal variation. ModelRisk offers a second optional cycle within each period of the first cycle, useful for modelling say week/day or day/hour patterns.

Auto-Regressive Moving Average (ARMA) models

Auto-regression means that the expected fractional change of the variable is proportional (either positively or negatively) to its fractional change in the previous recent periods. Moving average means that the expected fractional change of the variable is different to the long-run mean by a factor that is proportional to its recent variation from its long-run mean.

VoseTimeAR1 - Auto-regressive model with one-period dependence.

VoseTimeAR2 - Auto-regressive model with two-period dependence.

VoseTimeMA1 - Moving-average model with one-period dependence.

VoseTimeMA2 - Moving-average model with two-period dependence.

VoseTimeARMA - Auto-regressive moving average model with one-period dependence.

ARCH-type models

ARCH stands for autoregressive conditional heteroskedasticity. The volatility of the time series is defined as a function of the previous deviations of the variable from its long run mean. ARCH-type models allow periods of higher and lower volatility.

VoseTimeARCH - Basic ARCH model with one-period dependence.

VoseTimeGARCH - Generalized ARCH model with one-period dependence, i.e. ARCH model where the volatility component is an ARMA model.

VoseTimeEGARCH - Exponential general autoregressive conditional heteroskedasticity model, allowing negative values in the linear error variance equation with one-period dependence.

VoseTimeAPARCH - Asymmetric power autoregressive conditional heteroskedasticity with one-period dependence. It is a good one to try because it nests a large number of other models: GARCH, TS-GARCH, NGARCH, GJR-GARCH, TGARCH and log-GARCH.

'Population' Models

These models describe the evolution of a population size.

VoseTimeDeath - Pure Death model: Individuals 'die' independently at the same expected rate. Useful, for example, in modelling the retention of clients, or the timing and number of life insurance claims.

VoseTimeYule - Yule linear growth model: Individuals 'reproduce' by division at the same expected rate, meaning that each individual becomes two. Useful, for example, to model growth in a customer base by word-of-mouth.

Markov Chains

VoseMarkovSample - Markov chains are used to model the change in state of a population of individuals over time. For example, changes in credit ratings of a company, or the health status of life insurance policy holders.

Wilkie Models

The Wilkie Models window

Wilkie Models receive a separate treatment (and window) in ModelRisk . See the Wilkie models topic for a detailed explanation about Wilkie models. The following wilkie models are available:

• price inflation

• wage inflation

• share yields

• share dividends

• long term interest rate

• short term interest rate

Fitting a time series model to data

All time series models can be fitted to spreadsheet data. Fitted time series can be ranked according to different information criteria. See Fitting in ModelRisk for a more detailed explanation.

Multivariate time series

ModelRisk allows you to simulate from a number of multivariate Time Series. This allows for modeling of different quantities that vary in time together, but our connected through some relation: for example the realizations at each point could be correlated, or one component could come about through a regression from current values driven by past values of the other Time Series, etc. The Multivariate Time Series in ModelRisk are:

• VoseTimeMultiAR1 - Multivariate Autoregressive Time Series of order 1

• VoseTimeMultiAR2 - Multivariate Autoregressive Time Series of order 2

• VoseTimeMultiBEKK - Multivariate GARCH Time Series in the so-called BEKK parameterisation

• VoseTimeMultiGBM - Multivariate Geometric Brownian Motion time series

• VoseTimeMultiMA1 - Multivariate Moving-average Time Series of order 1

• VoseTimeMultiMA2 - Multivariate moving-average time series of order 2

All functions for simulating from a multivariate time series function are array functions.