QUANTITATIVE RESEARCH METHODS: An example of volatility analysis

LONDON METROPOLITAN UNIVERSITY BUSINESS SCHOOL

RESEARCH METHODS

AN EXAMPLE OF VOLATILITY ANALYSIS

by

John Coshall

AN EXAMPLE OF VOLATILITY ANALYSIS

Page

1 Introduction3

2 Derivation of the mean equation 6

3 Computation of logarithms and the mean equation in EViews8

4 The variance equation in EViews 10

5 Forecasting from volatility models16

AN EXAMPLE OF VOLATILITY ANALYSIS

1 Introduction

On the university file server, you will find the SPSS data file TOURISMEARNINGS(VOL).SAV. The file contains UK monthly earnings from international tourism (000s, current prices, SPSS variable name EARNINGS) from January 1980 to March 2006. The purpose of this exercise is to fit a volatility model to the data from January 1980 to February 2005 inclusive, which is called the historical period. We hold back data from March 2005 to March 2006 to see how good the volatility model has been in generating forecasts. The latter period of time is called the holdback period. There are two research questions:

1) Is the concept of volatility relevant to these data and

2) Does volatility modelling offer an improvement on competing forecasting methods?

The raw data are plotted below:

Seasonality, an upward trend and increasing variation are clearly evident. There seems to be a drop in earnings associated with events leading up to the Gulf War of 1991 and with the events of September 11th 2001.

Natural logarithms have to be taken before any differencing is considered. This is achieved in SPSS via:

Transform

Compute variable

in SPSS, by which means a variable called LOGEARN is created.

In the data file, each reading has been labelled from 1st to 315th via a variable called CASENUM. In order to select the period before March 2005 as the historical period, we need to access the Select Cases dialogue box by means of:

Data

Select cases

and the dialogue box below appears:

We need to select cases for which CASENUM is less than or equal to 302. Choose the option If condition is satisfied from the previous dialogue box and click the IF button:

Type in or paste the condition as shown above. When returning to the original data file, you will notice that readings not selected for model building i.e. the holdback period are crossed out in the left had margin.

2 Derivation of the mean equation

EViews is used to fit volatility models and the package permits the mean equation to be either a regression model or an ARIMA model. Here we shall use the latter. Having computed LOGEARN, we clearly need to take seasonal differences of this variable in SPSS via:

Transform

Create Time Series

and selecting seasonal differences of order 1. This new variable was given the name SEASDIFF. A plot of the first order, seasonally differenced data suggested that SD = 1 is adequate.

This leaves the trend to be dealt with. Using the above method to take ordinary first differences of SEASDIFF, suggested that D = 1 is adequate. The plot below is the EARNINGS after taking logarithms, 1st order seasonal differencing and 1st order ordinary differencing:

It only remains to use the autocorrelation function (ACF) and the partial autocorrelation function (PACF) to establish a final ARIMA model. The usual considerations led to an AR(1), AR(2), MA(12) model with no intercept, to act as the mean equation.

3 Computation of logarithms and the mean equation in EViews

It is necessary to define the historical period as 1980M1 to 2005M2 in EViews before defining the mean and variance equations that are necessary for volatility analysis, since the default in EViews is that all readings should be used. In EViews, click:

Quick

Sample

to generate the dialogue box below. EViews uses the notation @all to represent that all the readings should be used. I have replaced this by the first and last dates of the historical period:

Next we have to compute the natural logarithm of the variable EARNINGS, which is achieved via:

Quick

Generate Series

and in EViews, the LOG operator represents logarithms to the base of e:

If the user wishes to run the AR(1), AR(2), MA(12) model as the mean equation, click:

Quick

Estimate equation

and complete the dialogue box as shown overleaf:

EViews uses the d or difference operator, which has the form d(variable name, order of ordinary differences, order of seasonal differences) and the order typed in for the seasonal differences here is the number of months i.e. 12. Clicking the OK button will give the ARIMA parameter values and their statistical significance. This is not overly important here, since we wish to add a variance equation to the above mean equation in order to produce a volatility model.

4 The variance equation in EViews

In order to run various volatility models, it is first necessary to type in the mean equation via:

Quick

Estimate equation

which generates the dialogue box at the top of the previous page. However, we do not require Least Squares as the Model, so click the downward pointing arrow and choose the ARCH class of volatility models. This gives rise to the following dialogue box:

The mean equation appears at the top of this dialogue box. Note that the default model in EViews is the GARCH(1,1) scheme which occurs often in financial analyses. Click the Options tab at the top of this dialogue box and choose the settings overleaf:

There are two algorithms available and if a solution does not converge via one of them, try the other. The maximum number of iterations used by either algorithm is 500, but may be increased.

Returning to the GARCH dialogue box on the previous page, you can change the numbers of ARCH and GARCH terms in your model. I would suggest looking at all combinations of ARCH = 0, 1, 2 with GARCH = 0, 1, 2 (ignoring the ARCH = 0, GARCH = 0 combination). Most models reported in the literature fall within these parameter values. When the Threshold order is set to zero, the volatility model is symmetric i.e. the impact of good news equals the impact of bad news. If the user requires an asymmetric model, change the Threshold order to have value 1.

If you click the downward pointing arrow associated with the Model box, you will find the EGARCH and PARCH models become available. Again try the combinations of ARCH terms = 0, 1, 2 and GARCH terms = 0, 1, 2. When the Asymmetric order is set to 1, the user is generating an asymmetric EGARCH or PARCH model; when the Asymmetric order is set to zero, the model is symmetric.

Clearly, more than one of the ARCH, GARCH, EGARCH and PARCH models may have significant parameters and also obey any restrictions placed on the model e.g. non-negativity in the cases of the (G)ARCH models. If two or more significant models are generated, then the optimal model is that with the minimum Schwarz Bayesian Criterion (SBC) or with the maximum Log Likelihood criterion (LL), with the latter seemingly being the preferred reporting method in the literature.

All combinations of symmetric and asymmetric volatility models were run. The symmetric EGARCH(0,1) model was applied via the dialogue box shown overleaf:

The results below apply to the symmetric EGARCH(0,1) model (SBC = -2.186, LL = 327.89). This was marginally superior to a symmetric GARCH(0,1) model (SBC = -2.185, LL = 327.84).

The coefficients of the mean equation are all significant. By default, EViews always enters a constant term [C(4) in the above dialogue box]. It does not matter that this term is not significantly different from zero. We conclude that UK earnings from international tourism can be volatile. Further, the impact of a bad news shock to tourism earnings has the same impact as a good news shock of the same magnitude. Consequently, the inclusion of the volatility concept should improve forecasts.

5 Forecasting from volatility models

Click the forecast button in the above dialogue box, firstly to generate forecasts for the historical period. There is the option of forecasting LOGEARN or the differenced values for this variable. Note that the default EViews name for the forecasts is LOGEARNF.

Static forecasts have been chosen as opposed to dynamic forecasts. Static forecasts use the previously observed values of the Y variable when generating forecasts. This is fine for the historical period, because previous values of Y are known. Dynamic forecasts use the previously forecasted values of Y in order to generate further forecasts. This would apply to the holdback period, since in reality, future values of Y are unobserved.

The results below were obtained:

The top graph is of LOGEARN and LOGEARNF and suggests an excellent fit. The values of the RMSE, MAE and MAPE confirm this. The bottom graph is a plot of the conditional variance over time for the historical period. Volatility peaked around 1984 and thereafter remained at a relatively constant level.

Of course, we are interested in forecasting the EARNINGS variable into the holdback period 2005M3 to 2006M3. Run through the forecasting procedure again, but this time selecting the time period as 2005M3 to 2006M3 and choose dynamic forecasts. The forecasts are generated, but we must convert the logarithms back to earnings data via:

Quick

Generate Series

and using EARNINGF to represent forecasted earnings from tourism, then EARNINGF = EXP(LOGEARNF) as shown below:

Open just EARNINGS and ERARNINGF in one EViews sheet and plot them over the holdback time period as shown on the next page. You use the option:

Quick

Plot

to achieve this. The variable names are entered for you and the dates pertaining to the holdback period have to be set. The question is whether other models e.g. exponential smoothing can beat the adequacy measures such as MAPE over the holdback period.

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