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I have many sequences (~50) of time series data that I have fit to a non-linear model using a least squares fit. If I had a single sequence of time series data, and I fit a model to it using a least squares fit I would report goodness of fit as the χ² value.

In this case, I would like to report a goodness of fit statistic that describes how well my model fits that data set as a whole (individual fits to each of the recorded sequences of time series data). What would be the appropiate way to do this?

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There is nothing about the $\chi^2$ statistic that prevents you from extending it to all your time series. Concatenate each of your time series' observed, expected values and errors end to end and compute

$$\chi^2 = \sum_{i}^{nN} \frac{(y_i - \hat y_i)^2}{\sigma_i^2} $$

where $n$ is the number of points per series, and $N$ is the number of series' you have fitted.

Overall though, there are several ways to measure goodness of fit, depending on the question you are really asking. If you are asking simply "how well does my model fit the data I have used to fit my model" and you have all the $\sigma_i^2$, then a $\chi^2$ statistic works.

If instead you want to know how well your model is expected to fit or make predictions on time series you have yet to see, then consider using a different model selection/evaluation metric, such as cross validation.

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