How to form a single index?

Question

My goal is to order companies based on their profitability in a specified period (from the most profitable, to the least profitable). For that I need a single index.

I have got a dataset similar to the one below:

Company	Ratio 1 (2020)	Ratio 2 (2020)	Ratio 1 (2021)	Ratio 2 (2021)	Index
Company A	X%	X%	X%	X%	?
Company B	X%	X%	X%	X%	?
Company C	X%	X%	X%	X%	?

The difference is that I have 300 companies, 6 different profitability ratios (operational, gross profit, net profit etc.) and 3 years. Based on the financial analysis literature, I believe it is sound to assume that the profitability ratios measure the same underlying construct (i.e. overall profitability).

Is it methodologically valid to simply pool the data across time points, i.e. combine the 6 ratios at 3 years into 6*3=18 data points for each company and then do e.g. PCA? I would not hesitate if I had a single-period data (one year, six ratios), however I am not sure if I can apply this procedure for a time-series (there might be a temporal dependency where for each ratio: t depends on t-1, and t-1 depends on t-2).

Also, in the case PCA is not the right approach, what other procedure could I apply? I use SPSS.

Answer 1

In general, index construction is very complex and should strongly depend on the subject matter background. Institutions like the United Nations have a long history of revising, readjusting, and redefining indexes (such as human development index) over years.

I wouldn't generally say that it's not correct to run PCA on 18 variables as suggested; this is not too bad as an initial idea. Some possible issues with it are:

1. Dependence over time works in a different way from dependence between the 6 ratios (as you correctly state), so it may be more appropriate to somehow aggregate the three time points first and then run PCA on the six ratios only, with aggregated times. To what extent this makes sense depends on how the time series actually look like, and also subtleties of the intended interpretation, i.e., what exactly the index is supposed to measure (I know you wrote "profitability" but there may be several intuitive ways to aggregate your numbers and subject matter expertise would be required to say whether implications of whatever approach taken are appropriate here or not; a statistician may understand the implications but not necessarily know whether these are desirable or not in the field).

2. It may be a consideration to logarithmise the ratios, as logs of ratios are actually differences and may behave friendlier with linear approaches such as PCA. Whether and to what extent this applies here requires knowledge of the data and the field, once more.

3. You may want to think about weighting and standardisation; do all the six ratios have the same importance? Are the value ranges and distributions so that if you simply aggregate them, they will be balanced appropriately (if not, what kind of standardisation would be appropriate)? Do you want every company regardless of size have the same impact on your computations? What about outliers and the like? (You should certainly do some outlier diagnostics, because even though PCA is a good tool in principle, it can be strongly affected by outliers.)