Simulate Time Series Data 

Any time series process in Mathematica can be used to simulate random time series data. Use ARProcess, MAProcess, or ARMAProcess to simulate a weakly stationary time series.

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X RandomFunction[ARMAProcess[{.1, .2}, {.3, -.2}, 1], {0, 250}]; ListPlot[%, Filling -> Axis]

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Use ARIMAProcess to simulate a time series with a trend.

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X RandomFunction[ARIMAProcess[{-.7}, 2, {.2}, 1], {0, 250}]; ListPlot[%, Filling -> Axis]

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Use SARMAProcess or SARIMAProcess to simulate seasonal time series.

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X RandomFunction[ SARIMAProcess[{.4}, 1, {-1.3}, {12, {.7, .3}, 2, {.7, -.3}}, 1], {0, 250}]; ListPlot[%, Filling -> Axis]

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Use FARIMAProcess to simulate a long memory time series.

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X RandomFunction[FARIMAProcess[{.1}, .45, {.2}, 1], {0, 250}]; ListPlot[%, Filling -> Axis]

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