Time Series Function - Constant vs Piecewise I have daily data for online marketing $ spend and the number of clicks to the website gained.
I want to determine a function that 'maps' the two together. I cannot use normal linear regression because of the timely nature of the data (e.g. spend today may result in more clicks today because of spammers, tomorrow because of interested parties, the next day because of loyal customers and so on). 
I kind of 'know' it should look like a log curve starting from (0,something). Why? No spend = some organic clicks. And for each additional click you need to spend a little more but the payoff eventually will cut off as only so many people will be interested in clicking your website. 
I am unsure how to actually find the function that correlates the two using the data I have. I have no missing data.
Do you have any ideas? 
Would the derivative of that function give me the $ cost per additional click?
Bonus point: how do I prove that constant spend is better then ad-hoc spend? Would I calculate the area under the function above (ie constant spend) vs the sum of spend and sum of clicks and show that the sum of clicks is lower when ad-hoc bursts of investment are made? 
Edit: sample data here: http://pastebin.com/Ggjg8zR8
First row = date, Column B/C map together, Column D/E map together, Column F is what the company spent marketing $ on directly (so E = organic + billed + additional gained from investment). 
 A: What you appear to be describing is often referred to as a lag model where the # of clicks in period t responds to spend in period t , t-1 , t-2 etc.. This is also called a Transfer Function or a Dynamic Regression. The function is called an equation. Oftentimes hourly models can be built which can include day-of-the-week and other temporal effects. Data is often impacted by anomalies/level shifts etc which need to be identified and accounted for. You might be able to get help if you actually posted your data or some coded version of your data via dropbox and readers of the list could actually suggest ideas and even use their favorite tricks/approaches to helping you. 
AFTER RECEIPT OF DATA:
I used 200 days of daily data for the spend and click where spend is the X variable and click is the dependent Y variable. This is a plot of daily clicks against time  . Adding the spend series to the graph  . The 200 observations are partially shown here  . Using a commercailly available piece of software a reasonable model was found which included not only the correct lag structure for Y versus X but a number of unusual values as depicted here  . The program found that the error process was not constant (visually obvious from graphs) and incorporated weights to stabilize the error variance.  . The residuals from the model appear informationless and are free of evidented memory . The model is presented here  and 
 Finally a presentation of an ARMAX model  The actual/fit and forecasts graph summarizes the analysis
