Hypothesis testing and significance for time series A usual test of significance when looking a two populations is the t-test, paired t-test if possible. This assumes that the distribution is normal.
Are there similar simplifying assumptions that produce a significance test for a time series? Specifically we have two fairly small populations of mice that are being treated differently, and we are measuring weight once a week. Both graphs display smoothly increasing functions, with one graph definitely above the other. How do we quantify "definiteness" in this context?
The null hypothesis should be that the weights of the two populations "behave in the same way" as time passes. How can one formulate this in terms of a simple model that's fairly common (just as normal distributions are common) with only a small number of parameters? Once one has done that, how can one measure significance or something analogous to p-values? What about pairing the mice, matching as many characteristics as possible, with each pair having one representative from each of the two populations?
I would welcome a pointer to some relevant well-written and easily understood book or article about time series. I start as an ignoramus. Thanks for your help.
David Epstein
 A: I would suggest identifying an ARIMA model for each mice separately and then review them for similarities and generalization. For example if the first mice has an AR(1) and the second one has an AR(2), the most general (largest) model would be an AR(2). Estimate this model globally i.e. for the combined time series. Compare the error sum of squares for the combined set with the sum of the two individual error sum of squares to generate an F value to test the hypothesis of constant parameters across groups. I you wish you can post your data and I will illustrate this test precisely.
ADDITIONAL COMMENTS:
Since the data set is auto-correlated normality does not apply. If the observations are independent over time then one might apply some of the well-known non-time series methods. IN terms of your request about an easy to read book about time series, I suggest the Wei text by Addison-Wesley. Social scientists will find the non-mathematical approach of Mcleary and Hay (1980) to be more intuitive but lacking rigor.
A: There are many ways to do it if you think of the weight variations as a dynamical process.
For example, it can be modeled as an integrator
$\dot x(t) = \theta x(t)  + v(t)$
where $x(t)$ is the weight variation, $\theta$ relates to how fast the weight changes and $v(t)$ is a stochastic disturbance that may affect the weight variation. You could model $v(t)$ as $\mathcal N(0,Q)$, for a known $Q$ (you can also estimate it).
From here, you can try to identify the parameter $\theta$ for the two populations (and their covariance), using, e.g., a prediction error method. If the Gaussian assumption holds, prediction error methods will give that the estimate of $\theta$ is also Gaussian (asymptotically) and you can therefore build a hypothesis testing to determine whether the estimate of $\theta_1$ is statistically close to that of $\theta_2$.
For a reference, I can suggest this book.
