How to regress a time series of proportions?

Question

Every month, an organization surveys some of its customers (the total number of customers is also known).

The sampled customers answer a survey with a dozen or so questions; sometimes, customers don't answer every question.

A simple average of the proportion of answering customers who gave the top category on Question 1 and the proportion who gave the highest answer to Question 2 is calculated. (The top category for some questions is "Excellent," for other questions it is "Always.")

This average creates a score that is reported at the end of each month.

What command(s) in SPSS could compute the probability/magnitude of improvement in that score over a given historical window?

Also, is it possible to smooth this score? Predict future scores?

I've already investigated an ARIMA, but couldn't figure out how to incorporate the boundaries of the dependent variable (0,1) into the command.

Answer 1

To forecast data that is constrained to the interval $]0,1[$, you can first logistically transform the observations: $ z_t = \ln\big(\frac{y_t}{1-y_t}\big) $. The transformed variables $z_t$ are then unconstrained and can be modeled and forecasted using any method you like (exponential smoothing, state spaces, ARIMA etc.). You then just back-transform the forecasts. However, this is only a first approximation, as the transformation may mess up your underlying distributional assumptions, but it may be a good first step. You may want to take a look at Snyder et al. (2017, IJF), "Forecasting compositional time series: A state space approach".

Unfortunately, I can't help you with SPSS, sorry...