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I know the "average theoretical cost per impression" for Jan 13 - Dec 13. I have other monthly time series for "total # of impressions", "total # of clicks" and "total number of conversions" for Jan 13 - Dec 13.

The link is: I pay per impression, impressions (largest number) cause clicks (always smaller then the impressions number) which cause conversions (smallest number, a subset of clicks). Note that the data gives me the 'additional gain in impressions, clicks, conversions' due to the spend (i.e. if spend = 0 we have 0 impressions, clicks or conversions).

If I use 'ratios' i.e. lets say 80% of impressions convert to clicks over 1 year, 50% of clicks convert to conversions, and the average cost per impression is 1 dollar then you are paying 1x80%x50% = $0.40 per conversion (in theory). I can compare this number to the average "actual spend" to determine how much you actually paid per conversion (total actual spend/total conversions). The problem with this method is 1. it is not accounting for a lag in gains (ie spend today gain tomorrow) 2. we are assuming independence of impressions, clicks and conversions which is not true.

I use a time series model (similar process to this: Time Series Function - Constant vs Piecewise) to figure out a model that tells me the link between "actual spend" (not theoretical cost per click) and impressions/clicks/conversions.

I want to 'scale' the cost per clicks monthly time series in order to figure out what the time series model should be for 1) clicks 2) conversions in dollar terms (i.e. theoretical cost per click scaled), and use the "expected" value for the to determine an expected cost per click (and also do some predictions for the next 12 months).

When I look at the density plot (per month) of the conversion/clicks or clicks/impressions ratio it looks log normal. Do I need a time series method or could I assume a log normal for both ratios and 'multiply them together' (I'm assuming I need to test independence and I reckon it will fail).

How can I approach this? Is it even possible?

(any R code or an example would be great! also i have daily data for everything except theoretical cost per click therefore converted the other data to monthly is that ok?)

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