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Let's say we have two cointegrated time-series, $Y_{1}$ and $Y_{2}$, and I want to assess the causal impact of $Y_{1}$ on $Y_{2}$. There is good reason to think that both variables are influenced by a variety of outside factors.

$Y_{1}$ contains some observations equal to 0 (in my case, about 12 from a total 100) that are generated completely independently from the rest of the data (which are never equal to 0). That is, sometimes $Y_1$ is turned off in a quasi-experimental way.

What would be the best way of narrowing down the range of possible impacts of $Y_1$ on $Y_2$?

Context

I am helping a non-profit assess the impact of their advertising spend, $Y_1$ on $Y_2$, their website hits. Both series ramp up exponentially towards the campaign's deadline, so the residuals of $\ln(Y_{1, t}) = \beta_{0} + \beta_{1}\ln(Y_{2, t}) + \epsilon_t$ are stationary. Both series have significant autocorrelation also.

At several stages during the campaign the NFP renegotiated contracts with their media buyer, resulting in exogenous advertising black-out periods.

I'd be very comfortable drawing inference in a cross-section; my experience with economic time-series is that 'causal' parameters are often poorly identified, using bodgy instruments or sign restrictions. But I don't know much about doing quasi-experimental inference in time-series. Any guidance would be greatly appreciated.

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