# How to Estimate Growth Rates Between Two Periods from Data with Varying Time Intervals Between Measurements?

I have a dataset of sales records for properties in a local area spanning over 20 years. Each property has one or more historical records of sale, however, as the sales do not occur every year, there are varying time intervals between sales for each property. My goal is to estimate growth in value (i.e. "growth factor") from last sale to the current year for each property. I'll then use this "growth factor" to estimate the current value. I'm not looking to include any other features into this particular model, this work is a baseline that will feed into a more complex model later.

I'm currently conceptualising the data as a set of records with a inputs "current sale year" and "prior year", and output "growth factor".

I think it important that the model have the property that a growth rate calculated for a continuous period from year x to year Z should agree with the multiple growth rates over the same period but when split into smaller time chunks. For instance, the growth rate from year 1 to year 4 should correlate with the growth rates from year 1 to year 2, year 2 to year 3, and year 3 to year 4, when considered collectively.

Here are the steps I've conceptualized so far:

I am looking for suggestions of approaches to accurately estimate the growth rate and subsequently, the current property values. Are there established methods or models for handling such scenarios with sporadic time intervals between matched (i.e. at the property level) data points? Additionally, how can I ensure the correlation of growth rates for continuous periods and their corresponding smaller time chunks within my model?

Have you considered using growth curve models within the multilevel analysis framework (a.k.a. hierarchical linear modeling or random coefficient regression analysis)? See, e.g.,

Kwok, O.-M., Underhill, A. T., Berry, J. W., Luo, W., Elliott, T. R., & Yoon, M. (2008). Analyzing longitudinal data with multilevel models: An example with individuals living with lower extremity intra-articular fractures. Rehabilitation Psychology, 53(3), 370–386. https://doi.org/10.1037/a0012765

Peugh, J. L., & Enders, C. K. (2005). Using the SPSS Mixed procedure to fit cross-sectional and longitudinal multilevel models. Educational and Psychological Measurement, 65, 717-741.

Steele, F. (2008). Multilevel models for longitudinal data. Journal of the Royal Statistical Society Series A: Statistics in Society, 171(1), 5-19. https://academic.oup.com/jrsssa/article/171/1/5/7085119

Also, pretty much any introductory text on multilevel analysis will also cover applications to longitudinal (repeated measures) data:

Heck, R. H. & Thomas, S. L. (2020). An introduction to multilevel modeling techniques (4th edition). Routledge.

Hox, J. J. (2002). Multilevel analysis: Techniques and applications. Mahwah, NJ: Lawrence Erlbaum Associates.

Kreft, I., & de Leeuw, J. (1998). Introducing multilevel modeling. London: Sage.

Luke, D. A. (2004). Multilevel modeling. Thousand Oaks: Sage.

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models (2nd ed.). Thousand Oaks: Sage.

Snijders, T. A. B., & Bosker, R. J. (1999). Multilevel analysis: An introduction to basic and advanced multilevel modeling. London: Sage.