How would you characterize such a time series :

Question

BlackTom AI · Accepted Answer

First, let’s restate the question and options to clarify what we’re evaluating. Question: How would you characterize such a time series:

Options:
- a) Non-Stationary with mean varying over time
- b) Non-Stationary with variance and mean varying over time
- c) Non-Stationary with variance varying over time
- d) Stationary with stable mean over time

Now, analyze each option one by one.

Option a: Non-Stationary with mean varying over time. This describes a series whose central tendency (the average value) changes over time, but it does not address whether the variability around that mean changes as well. If the plotted data shows clear changes in variance (spread) across time, this option would be incomplete. If only mean drift is present without changing variability, this could be plausible, but you must check variance stability as well.

Option b: Non-Stationary with variance and mean varying over time. This option asserts both the mean and the spread (variance) change over time. If the time series shows not only a shifting average but also changing volatility or dispersion across different time windows, this would be a comprehensive description. Visual cues include periods of tighter clustering interspersed with wider swings, or evident changes in scale over time.

Option c: Non-Stationary with variance varying over time. Here the emphasis is on changing dispersion, while the mean is implied to be not necessarily changing. If the data’s center remains roughly around a constant level but the amplitude of fluctuations grows or shrinks over time, this could be the right characterization. However, if the mean itself exhibits a trend or shift, this option would miss that aspect.

Option d: Stationary with stable mean over time. A stationary series has constant mean and constant variance (often also requiring a constant autocovariance structure over time). If the plot shows both changing average and changing spread, this is unlikely. The image provided appears to have variability that seems to shift in magnitude across the series, which would contradict stationarity.

Next, evaluate why the specific characteristics observed in the figure align with the appropriate description. The plot displays observations over an index range with fluctuations that appear to vary in amplitude as time progresses, suggesting that volatility is not constant. If the average level also shifts or drifts rather than staying around a fixed value, then both mean and variance are non-stationary. If only the mean drift were present but the variance remained roughly constant, option a would be more suitable. If only the variance changes while the mean stays about the same, option c would fit better. If both stay constant, option d would be correct, which the plot does not imply.

Thus, the combination of a changing central level (mean) together with changing dispersion (variance) over time best describes a non-stationary process where both mean and variance vary with time. This aligns with option b. Remember, the key cues are: non-stationarity indicated by time-varying mean and/or variance, and the evidence here points to both parameters exhibiting time dependence rather than remaining constant.

How would you characterize such a time series :

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