# Strategies for analyzing the functional relationship between two time series?

Suppose we have time-dependent survey data about name recognition for a political campaign. We're interested in learning how campaign spending effects that name recognition. My interest is in strategies for defining the relationship between spend and name recognition.

My thoughts so far are that we can model whether a survey respondent recognized the name as follows:

$$Y_{itc} \sim Bern(\theta_{tc})$$

where i indexes the person, t indexes the time the question was asked, and c indicates the candidate asked about. Then:

$$\theta_{tc} = f(S_{1...t})+\gamma_c+\epsilon$$

where $$S_{1...t}$$ is the entire spend history prior to t (could be in monthly spend for example), $$\gamma_c$$ is an effect specific to candidate c and $$\epsilon$$ is random error.

Where I get hung up is parameterizing $$f(S_{1...t})$$. My primary interest is how the relationship between spend and name recognition. Does spend immediately bump name recognition and then fade? Does it take a few months before results are seen? Is there a decay over time to the effect of money spent? I want to be able to understand the relationship of spend and recognition without being too rigid about what that effect has to look like.

Just to clarify, I'm only interested in the effect of a candidate's spend on their own campaign. Effects from any other campaign's spend would be lumped into random error.