# Taylor rule and government spending rules in different frequency

Hi all,

I would like to know how to model the Taylor rule and the fiscal rules in a weekly model. My intuition for fiscal rules is that spending does not react immediately to movements in output since it is necessary to pass laws to increase spending, so instead of having a standard rule like this:

\text{ln}(\frac{G_{t}}{G_{ss}}) = \rho_{G} \text{ln}(\frac{G_{t-1}}{G_{ss}}) + (1 - \rho_{G})[\varphi_{D}\text{ln}(\frac{D_{t}}{D_{ss}}) + \varphi_{Y}\text{ln}(\frac{Y_{t}}{Y_{ss}}) ]

In a weekly model I would have something like this:

\text{ln}(\frac{G_{t}}{G_{ss}}) = \rho_{G} \text{ln}(\frac{G_{t-1}}{G_{ss}}) + (1 - \rho_{G})[\varphi_{D}\text{ln}(\frac{D_{t-12}}{D_{ss}}) + \varphi_{Y}\text{ln}(\frac{Y_{t-12}}{Y_{ss}}) ]

But I’m not sure. Also, would the size of the coefficients change?

On the other hand, something similar happens in the Taylor rule, the interest rate reacts to the inflation of 12 periods ago, that is:

\text{ln}(\frac{R_{t}}{R_{ss}}) = \rho_{R} \text{ln}(\frac{R_{t-1}}{R_{ss}}) + (1 - \rho_{R})[\psi_{\Pi}\text{ln}(\frac{\Pi_{Q,t}}{\Pi_{Q,ss}}) + \psi_{Y}\text{ln}(\frac{Y_{t-12}}{Y_{ss}}) ]

where, \Pi_{Q,t} is the quarterly inflation rate, that is:

\Pi_{Q,t} =\overset{11} { \underset{j=0}{\prod}}\Pi_{t+j}

Is my intuition correct or do these rules stay the same regardless of the pattern frequency?

This is tricky, because in there is usually time aggregation. Your specification for G would assume that government spending reacts to debt exactly 12 weeks ago. Similarly, you would have monetary policy react to inflation over the last quarter, but not quarter in the calendar time sense, but just the last 12 weeks.

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I see, so I think it’s better to use the standard policy rules. Appreciate your help.