Time series analysis

Time Series Analysis, Market Models and Forecasting
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March 31, 2010

Individual Assignment

1. Visual inspection of a simple plot of the three data series (FTSE 100, LLOY and RBS) over the time period February 3, 1997 – February 8, 2006 suggests that all three data series are non-stationary. Also, the AR(I) coefficient is very close to 1 for all three series, suggesting the series arenon-stationary. In order to formally assess whether the three data series are stationary an Augmented Dickey-Fuller test was performed on each of the series. A summary of the test results appears in the table below:

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As shown in the table above, in all three instances the test statistic was below even the 10% confidence level, so it can be concluded that all three series areindeed non-stationary, and represent random walks. As expected, the time series for the returns of all three data sets were stationary.

2. A simple autoregressive model for the returns of each stock was specified with the following equation: D(STOCK) c AR(I). The AR(I) coefficient was significant for the LLOY data set, but insignificant for RBS. The model output appears below:

[pic]It’s worth noting that the t-statistic for both autoregressive models is very close to 2, and that the RBS model has almost the same significance as the LLOY model. In both cases the AR(I) coefficient is less than 1, implying the series is mean-reverting, which is expected since we are looking at differences of returns. In attempt to improve the models, they were first tested for trend-stationarity.However, neither model exhibited this type of behaviour at a significant level. Looking at the correlogram of residuals for LLOY, partial autocorrelation appears significant at lags 2, 3 and 5. Similarly, for RBS, lags 3 and 5 show significant partial autocorrelation. In order to remove autocorrelation, the models were adjusted by introducing additional lags. The results appear on thefollowing page.
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Note that each of the lag coefficient in the resulting models shows much stronger significance than the AR(I) coefficients in the initial test. Also, the correlogram for the above models do not show any significant levels of autocorrelation or partial autocorrelation.

3. The first “rough” test for ARCH effects consisted of looking at the squared residuals correlogramfor each of the two models that are shown above. The correlogram for the first 10 lags of each stock return series appears below:

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Additionally, the histogram-normality test yields kurtosis of 5.411 for the LLOY model and 6.529 for the RBS model, also suggesting the presence of ARCH effects. For this reason, more formal ARCH testing needed to be conducted. The table below shows theresults of the Lagrange Multiplier test. Several iterations of the test yielded that ARCH effects are significant for 8 lag specifications for LLOY and for 4 lag specifications for RBS. Note that the p-values for both models are zero and the n*R2 statistics are very large. This test confirms that ARCH effects are present in both models, which means that there is volatility clustering withinboth time series.

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Adding a GARCH (1,1) model to the analysis makes it necessary to re-specify both models. With a GARCH (1,1) component added to the LLOY model, the coefficients at lags 2, 3 and 5 become insignificant, and so do all the residual lags, except for the first one. For the RBS model, several iterations yield that a GARCH (2,1) model, together with autoregressivecoefficients at lags 3 and 5, actually produces the most significant results. Under this specification, the AR(5) t-statistic is still slightly below 2, however, removing it makes the AR(3) t-statistic drop below 2 and become insignificant. Below is the summary of the two GARCH models:

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For both models above, the corellograms of their squared residuals no longer show any significant...