What statistical model should be used? What statistical model should be used when trying to look at a change of good ownership between a sample year set?
 A: From what you wrote, it seems that you are trying to test the interaction effect between age and time, i.e., test whether the relationship between how old someone is and how much they own depends on the year, or similarly, whether changes (trends) in the the number of goods owned by households over time vary by age group.
Regression is the most general approach for modeling interaction effects: https://en.wikipedia.org/wiki/Interaction_(statistics).
The exact specification of your model would depend on your data and whether you have completely independent observations, or whether you have repeated measurements for the subjects in your study, requiring modeling of correlations between observations through fixed effects or random effects.  You would also have to think about the right way to model the dependent variable, since it is likely to be a limited dependent variable which can result in biased estimates if modeled with standard OLS regression methods.  And you would have to think about how to model age, time and their interaction, i.e., whether you should model them in linear form (assuming a constant effect across all values of a variable) or in discrete form (allowing you to test for pairwise differences, say between 2000 and 2005).
There may be other issues that will come up (like confounding), but hopefully this is enough to get you started.
A: Welcome to Cross Validated!
Well, if you are talking in terms of models, an ARIMA (AutoRegressive Integrated Moving Average) might work very well!  Let me explain it a bit for you:
You seem to be wanting a result that is based on time, which would indicate a time series.  Now from this, you would obviously want to use a time series analysis.  Based on what you have said, it would seem to me that an ARIMA model would work very well (or at the very least give you the results that you would prefer!)  A very good model here would be a Box-Jenkins model (named after George Box and Gwilym Jenkins.)  Here is an ARIMA model in a picture format: 

For further reading, I recommend these pages to find out more about time series (and specifically ARIMA/Box-Jenkins models):


*

*https://towardsdatascience.com/time-series-forecasting-arima-models-7f221e9eee06

*https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc445.htm
Hopefully this helped and made things a bit clearer! 
A: If you want to test each age class independently, like running a test per category (ex: testing for the 30yo, and then for the 40yo and so on), I would suggest a simple t-test.
If you want to consider everything at once, so have one variable (number of owned goods) but two grouping factors (age class and year) I think a Chi² test might be what you are looking for.
