Statistically Approximating Clicks From Limited Data Assume a business started in January 2014. 
I have the following daily data (from June 2014 to December 2014):
1. Number of people who joined the website; 
2. Number of people who left the website; 
3. Number of clicks in total the website obtained from people (i.e. theres no way of obtaining clicks other than through people who joined the website at anytime).
Assume I cannot obtain January - May 2014 data. 
Problem:
The number of clicks in total the website obtained may be clicks from people who joined the website before June. 
What I want to approximate:
How many clicks were made by the number of people who joined the website strictly between June - December 2014 so that I can make a better approximation of the total clicks/person. 
I think a statistical time series method to provide some sort of trend needs to be used but I'm not sure what method to research to solve the above problem. Can you help me please? 
 A: What you effectively have are two time-series: Number of clicks per day and Change of users per day (with the total user count being unknown) and you want to estimate the impact of changing the number of users on the number of clicks.
The problem is hard because:
a. you have observational so you can never be really certain about
   the causality between changing the user count and the clicks
b. presumably there are many reasons causing changes in the number of clicks which means that a model based on a change of users will be very weak in its explanatory power
c. time-series models usually involve past values of a single variable, while what you technically want is a time-series model with exogenous variables (change in the number of users) which are a both more complex to estimate and interpret.
If you want to go down this road see: A regression with ARMA errors, which would be my weapon of choice, other options include an ARMAX model or a dynamic linear model (see here for nice tutorial).
If you want to keep it dirty (and simple) my first instinct would be to look at the relationship between the absolute change in daily clicks
$Y_t = \text{#Website clicks day t} - \text{#Website clicks day t - 1}$
and the daily absolute change in users
$X_t = \text{#Users that joined on day t} - \text{#Users that left on day t}$
You could start with a simple linear model and take it from there:
$Y_t = b_1X_t + \epsilon$
where $\epsilon$ follows a Guassian distribution. 
Alternatively, you could also try:
$Y_t = b_1X_{\text{joined},t} + b_2X_{\text{left},t} + \epsilon$ in order to model that the leavers and joiners have a different click rate.
The number of clicks purely by new users over the period June-December would then be: $b_1 (\text{#Users joined})$ in the period June to December.
Also keep in mind the many simplifying assumptions you will be making along the way, such as: clicking behavior doesn't change over time, etc.
Alternative approach:
You could also try to explicitly model the total number of users $Y_t$. You know the number of users that joined $X_{\text{joined},t}$ and left $X_{\text{left},t}$ daily over a period of 6 months. 
You can model $Y_t$ as a Poisson process, or even simpler $X_{\text{joined},t} \sim Pois(\lambda_1)$ and $X_{\text{left},t} \sim Pois(\lambda_2)$ respectively. But the major issue is that this would assume that the rates at which you acquire and loose users is constant which usually doesn't apply for website visits! If you believe the rates $\lambda_1, \lambda_2$ of users joining and leaving are constant, great! 
The maximum likelihood estimate for the rate of users joining each day is just the average of users joined per day. Multiply that with the number of days in total since existence of the website and you have your total user count.
Unfortunately, if you realize the rate is not constant $\lambda_1$ becomes $\lambda_{1}(t)$ and you would have to estimate an inhomogeneous Poisson process. The trouble with that is that since you are missing the preceding part of the data, it would be probably very tricky to "just extrapolate" the rate backwards, so I don't think that approach is a clear win. 
