Result interpretation for Regression With ARIMA Errors

Question

I am new to time-series analysis.

I am struggling to interpret the result I got from auto.arima().

I've read the highly-recommended blog https://robjhyndman.com/hyndsight/arimax/. I also read a similar question on ARIMAX vs. Regression With ARIMA Errors but I still got a few more questions.

My code (I was using auto.arima() to analyze daily ridership data in Boston and I did not do any data manipulation beforehand, like taking log or standard scaling on variables) :

weather_indicators =  df_2019[['Boston Temperature_mean','Boston Precipitation Total','Boston Evapotranspiration','Boston Sunshine Duration','Boston Relative Humidity_mean']]
model2 = pm.auto_arima(df_2019['Total_Daily'], suppress_warnings=True,exogenous = weather_indicators, seasonal=True, m=7)

I used python here but my understanding is that the pmdarima should be very similar to auto.arima() in R.

And I got a result looks like this:

The above result is pretty rough but still I would like to know:

1. Interpretation on coefficients

The coefficients for the exogenous variables (especially the first 3 weather variables) are much larger than coefficients for AR and MA. Does that mean the weather is much more statistically significant to the daily ridership number than the time-series factors are? or is it because the auto_arima() is a Regression with ARMA errors, which renders the coefficient for exogenous variables always being much larger than those for AR,MA?

2. ARIMAX vs multiple linear regression

My focus was on studying the effect of the exogenous variables. If time-series factor (coefficients for AR and MA) are not that statistically significant(very low in this case), maybe a multiple linear regression (between daily ridership and different weather variables) would be more appropriate? Or this SARIMAX result is good enough?

Many thanks.

Answer 1

1. Interpretation on coefficients: First of all, $AR(p)$ and $MA(q)$ terms are bounded between $(-1,1)$ or else the process is not stationary. Second, you judge whether a coefficient is statistically significant based on the relevant $t-test$. For example your third exogenous variable, while large on absolute terms, also has large standard error and cannot be considered statistically significant at the $1\%$ confidence level (i.e. there is a chance $>1\%$ that this coefficient is actually $0$). Third, if your exogenous variables are of different scale it is reasonable to have large discrepancies in absolute terms in the coefficients. So the answer is that I cannot judge which coefficient is more important at this stage; this would probably require applying a method such forward or backward selection in order to assess the impact of each predictor separately.
2. ARIMAX vs multiple linear regression: Good enough is relevant. There are several ways of evaluating a model's performance. Some of them are based on in-sample diagnostics: $R^2$, residual diagnostics and Akaike information criterion among others. In fact I believe that auto.arima() uses AIC as a default for model selection. If forecasting is your goal, you should probably evaluate based on the forecast accuracy on a hold-out set. I would suggest checking the residual diagnostics to get a first assessment of the quality of your model. If your residuals are $i.i.d.$ it would indicate a good fit.