I have a time series which contains monthly readings for air pollution in a city. The seasonality from this time series has been removed.

Given two date ranges, for example Jan-Aug 2008 and Jan-Aug 2009, I want to perform a hypothesis test, which tests the claim that the amount of pollution from March-Dec 2009 is no higher than the amount of pollution from March-Dec 2008.

I'm new to time series analysis and have tried to determine the best way of performing this test, but to be honest I'm not sure where to look. I was thinking t-test, but the sample sizes are quite small so I don't know if it can be used? I also was considering using the Wilcoxon Ranked Sum, but I'm unsure if this is suitable.

Could someone suggest what test, those mentioned or any others, would be most appropriate for testing this hypothesis?

(Oh and my preferred statistical software environment is R, just in case you want to link to any packages)

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    $\begingroup$ Your two date ranges appear to comprise eight months each. You then ask to test claims about two different periods of ten months. How do you propose to do that, given you don't have any data at all about the months of September through December of either year?? $\endgroup$ – whuber Dec 13 '18 at 0:09

Estimate an ARIMA model for both time sections making sure that there are no Pulses/Level Shifts/Seasonal Pulses or Local Time trends unaccounted for. Estimate the model globally then use the CHOW Test to test the hypothesis that the parameters are the same for the two periods.

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    $\begingroup$ If you factor out "pulses, level shifts, etc" what would be the meaningfulness of comparing the remaining parameter, which represents an underlying level during the time period? Since the question is whether there was more pollution at one time than during another, it seems to me that discovering any of these sporadic elements would be tantamount to discovering an actual change in the overall pollution--without providing direct information about the question. Thus, at a minimum, it seems some kind of follow-on procedure would be needed to estimate and compare the total pollution amounts. $\endgroup$ – whuber Jan 1 '15 at 20:04
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    $\begingroup$ Upon reflection and correction, if we only adjust for Pulses then the two adjusted series could then be compared using a common model which could include a constant. The two estimated constants would then be proxies for the level of pollutant and then comparable via their standard errors. I stand corrected that the up-front filter should not include level shifts/seasonal pulses or local time trends as their existence would be a prima facie example of difference. $\endgroup$ – IrishStat Jan 1 '15 at 22:18

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