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I have a timeseries (sampled daily, weekdays only) whose volatility clearly shows dependency on day of week. In particular the standard deviation of the differenced series $\Delta y_t$ is smallest on Mondays and peaks on Thursdays.

I have considered GARCH-style models for the volatility, with the respective dummy variables for day of week. However, I am not interested in the volatility of the errors per se, but rather how the mean equation is affected by the day of week. For example if I fit an AR(1) model to $\Delta y$ I observe that its residuals $\varepsilon_t$ on Wednesdays are correlated with $\Delta y_{t-1}$.

In addition, if I assume $\Delta y_t = \phi \Delta y_{t-1} + \varepsilon_t$ but estimate $\phi$ via OLS regression for each weekday separately I get the following for each day of week:

monday: $\phi = 0.68$ (SE = 0.02)
tuesday: $\phi = 0.76$ (SE = 0.04)
wednesday: $\phi = 1.03$ (SE = 0.02)
thursday: $\phi = 0.90$ (SE = 0.02)
friday: $\phi = 0.80$ (SE = 0.018)

Correct me if I'm wrong but to me these effects cannot be captured by a GARCH model for the errors. In light of the errors being correlated with $\Delta y_{t-1}$ I have considered a model which looks something like this: $$ \Delta y_t = \phi \Delta y_{t-1} + \varepsilon_t\\ \varepsilon_t = \gamma 1_{\lbrace t \text{ is Wed} \rbrace} \Delta y_{t-1} + \epsilon_t $$ which can be written as $$ \Delta y_t = (\phi + \gamma 1_{\lbrace t \text{ is Wed} \rbrace}) \Delta y_{t-1} + \epsilon_t $$ but it is unclear to me how to estimate a standard ARIMA-type model in this case.

Are there any known models that deal with these kind of effects? Or maybe I'm missing something and that this is a routinely modeled in a typical ARIMA + GARCH setup?

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  • $\begingroup$ Consider seasonal ARIMA (SARIMA), perhaps without the MA part. If you still get some seasonality in volatility after having applied a satisfactory conditional mean model, consider adding a GARCH equation with weekday dummies. $\endgroup$ Oct 26 '20 at 15:39
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    $\begingroup$ If your objective is to see how day of the week is changing the mean, then it is best to extract seasonal component. For daily series seasonal adjustment is complicated because you need consider many effects (day of the week, day of the month, day of the year, etc.). There is an R package dsa developed by German Central bank that you can explore. This is the paper detailing the method. $\endgroup$
    – Dayne
    Oct 27 '20 at 4:08
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I think you have two options with option one using a SAR(IMA) model. If you have a reoccurring effect every weekday (dummy), then you can already improve the model.

Option 2: As you stated - I would use an AR and then fit a GARCH on the errors. Or even use the SAR(IMA) and then a GARCH.

The combination would result in 3 candidate models. SAR, AR->Garch, SAR->Garch

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