Univariate Time Series Fit

See also: ModelRisk functions and windows, Time series fitting functions, Time series in ModelRisk, Common elements of ModelRisk windows

Introduction

The Time Series Fit window from ModelRisk

The Time Series available in ModelRisk can be fitted to a given set of spreadsheet data.

The fitted Time Series (or explicitly the fitted parameter values) can then be used for time series forecasting in your spreadsheet model.

The fitted distributions are ranked according to the SIC, AIC (Akaike) and HQIC information criteria. For these holds: the lower an information criterion, the better the fit. To avoid confusion the negatives of these criteria are displayed in the list. This means that:

the higher the value shown in the list, the better the fit.

AIC and the other Information Criteria are superior goodness of fit statistics to other fit ranking criteria (e.g. chi-squared), because they take into account the number of parameters estimated, and penalize for overfitting: a model that has a good fit using fewer parameters is preferred over one that needs more parameters.

The AIC is the least strict of the three in penalizing for more parameters, while SIC is the strictest. More information on these can be found here.

Source data can be either actual observations of the variable (e.g. stock price) or the log return of that variable. If the source data are log returns, the parameter LogReturns should be set to TRUE (by marking the check box), and a forecast will correspondingly generate a stochastic log returns series.

Depending on the time series model being fit, some of the function input parameters may not be present. PastTimeStamps and FutureTimeStamps only appear where the mathematical model allows variations between the time intervals of the observations (and the projections). The Geometric Brownian Motion (GBM) and the GBM with Mean Reversion models, for example, can jump from one observation to any future point in time. The GBM with Jump Diffusion model, however, cannot do so because it would have to allow any number of jumps to occur within a time period leading to an intractable model. When providing an array of TimeStamps please make sure it is in ascending order.

The set of values in Source data may not contain the last observation you have (e.g. the series may not allows TimeStamps but you know what the variable value is now. In this case you can provide the last observed value as LastValue. More generally, LastValue allows you to construct a series from some particular point that is based on an analysis of past behavior. If LogReturns is set to TRUE, LastValue becomes redundant and is ignored.

Both historical data and data generated from the fitted model are displayed, optionally with more than one line for the forecast.

You can read more about the theory behind time series here.

To see the output functions of this window, click here.

Window elements

When opening Time Series Fit you are first presented a screen to select Time Series to fit. In the fitting window itself, this can be changed at any time (i.e. new Time Series added and removed for fitting).

In the Source data region you can select the location of the data to fit to in your spreadsheet.

In the Time Series list on the left, you can add and remove Time Series to fit by pressing the Add or Remove buttons. Click on a Time Series in the list to toggle it for the preview graph. Click on one of the three information criteria to sort the fitted models according to this criterion (lower value is better fit).

In the Time Series Fit Function Parameters window, extra data can be filled in - this depends on the Time Series currently selected from the list.

• To display a preview graph of the Log Returns rather than the values themselves, mark the LogReturn check box.

• In the LastValue field, you can specify the last data value in time you have. Random values generated from the fitted time series model will start from this value.

• To take into account the uncertainty that (unavoidably) exists about the fitted parameters, mark the Uncertainty check box. The smaller the dataset, the larger the uncertainty on the fitted parameters will be. To read the motivation behind this parameter click here.

  The uncertainty parameter is common to all ModelRisk fitting functions. It allows the inclusion of uncertainty about the fitted model parameter estimates. Unfortunately, it is common practice in risk analysis to use just the maximum likelihood estimates (MLEs) for a fitted distribution, copula or time series. However, when there are relatively few data available or when the model needs to be precise, omitting the uncertainty about the true parameter values can lead to significant underestimation of the model output uncertainty. The Uncertainty parameter is set to FALSE by default (i.e. returns MLEs or projections based on MLEs) to coincide with common practice, but we strongly recommend setting it to TRUE. Uncertainty values are then generated for the fitted parameters using parametric bootstrapping techniques, which has the great advantage of allowing correlation structure between uncertain parameters and non-normal marginal uncertainty distributions, the latter being an important constraint of more classical methods based on asymptotic results (i.e. when the amount of data approaches infinity).

By default, the vertical scale of the preview graph is automatically rescaled according to the (historical/generated) data. To keep the vertical scale fixed, check Fix Y-scale.