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I am using SAS to run an ARIMAX model on two time series where the response series= asthma hospitalizations and the explanatory/input series= influenza hospitalizations. I have gone through and assessed goodness of fit, checked residuals, checked ccf, etc... and I'm finally at my final model with all the estimates. I have log transformed both my x and y series and my final output of conditional least squares estimation comes out to the following:

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I've been trying to understand if there is a way to interpret the estimate on my flu series (xlog) but so far based on the ARIMAX equation it seems as though interpretation of the coefficient would be difficult as it depends on the previous values of my response variable (asthma)? Is that correct or is there an easy interpretation that I'm missing out on?

Also, I'm trying to assess whether influenza predicting asthma or asthma predicting influenza is a stronger so I did the same workflow but changing the input and response series so input=asthma and response=influenza. In doing so I yielded the following estimations:

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With a higher estimate for asthma (ylog) predicting influenza, would I be able to say there is a stronger correlation going the reverse direction? (Although I think this question will be answered from my first question once I understand how to appropriately interpret the coefficient)

thank you so much in advance!

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There's this useful blog post by Hyndman, which talks in detail about different specifications of ARIMAX and similar time series. He's somewhat biased, in my opinion, but has a point.

I find it useful to think in terms of the long run averages. Consider a simple AR(1) model: $$y_t=c+\phi y_{t-1}+e_t$$

The long run average is well known: $$E[y_t]=\frac{c}{1-\phi}$$

Similarly, for ARX(1) model with one exogenous variable you get: $$y_t=c+\phi y_{t-1}+\beta x_t + e_t$$

$$E[y_t]=\frac{c+\beta E[x_t]}{1-\phi}$$

So, one way to look at this is to say that $\beta$ of your exogenous variable impacts the long run average.

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You might want to look at When (and why) should you take the log of a distribution (of numbers)? as any unwarranted transformation can be deadly. Secondly if I were to assess which series was more predictable from the other I would use the mean square errorsfrom the two models that might include pulses/level shifts/local time trends.

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I believe your model selection could be the issue. You should look into VAR because the relationship between your predictor series and response series is potentially bidirectional. "VAR can be used for most multiple time series analysis, and is preferred when causal relationships may be bi-directional. It captures linear interdependencies among multiple time series simultaneously, with one equation computed per variable"

Understanding and using time series analyses in addiction research (Beard et al., 2019).

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