📂Statistical Analysis

How to View Time Series Analysis Results Obtained with ARIMA Model in R

Practice

The built-in R dataset AirPassenger consists of monthly airline passenger numbers from 1949 to 1960.

• (1) Model: In fact, if the coefficients can be precisely identified, that’s not the most critical aspect.
  Represents the Seasonal ARIMA model $ARIMA(p,d,q)\times(P,D,Q)_{s}$. For instance, the result of the above analysis ARIMA(0,1,1)(0,1,1)[12] means $ARIMA(0,1,1)\times(0,1,1)_{12}$.
• (2) Coefficients: Represents the coefficients that fit the model.
  ma1 is the moving average process coefficient $\theta_{1}$, and sma1 is the seasonally adjusted coefficient. The first part denotes the type of model, and the latter part denotes the lag. As you can see, the attachment of ’s-’ implies seasonal adjustment.
  • Although not present in the above analysis, ar means the autoregressive process coefficient $\phi$.
  • Drift is something known as drift, which often does not appear significant, but it’s roughly necessary to know about.

• (3) Standard Error: Used for hypothesis testing.
  Typically, if the absolute value of the coefficient is more than double the standard error, it is considered significant. The reason for having double here is usually because the significance level is $\alpha = 0.05$. If you want to be more precise, using 1.96 times is feasible, and it’s natural to use 2.58 times if the significance level is $\alpha = 0.01$, and 1.65 times if it’s $\alpha = 0.1$. [ NOTE: Theoretically, understanding that the sample autocorrelation function and the sample partial autocorrelation function follow a normal distribution is essential. At least, it’s recommended that statistics majors thoroughly study this at least once. ] Let’s look at the analysis above for oil.price.
  • ar1 : $| \phi_{1} | = 0.4574 < 0.2722 \times 2 = 0.5444$, thus not significant.
  • ma1 : $| \theta_{1} | = 0.2400 < 0.2722 \times 2 = 0.5508$, thus not significant.
  • ma2 : $| \theta_{2} | = 0.2563 > 0.0736 \times 2 = 0.1472$, thus significant.
  • drift : $ | b | = 0.1812 < 0.1246 \times 2 = 0.2492$, thus not significant. In other words, it’s virtually the same as saying $ARIMA(0,1,2)$ model and $\theta_{1} = 0$, if put into formula, $$ \nabla y_{t} = e_{t} + 0.2563 e_{t-2} $$

Certainly, this doesn’t mean the analysis is concluded. This is a very simplistic judgment, and it’s necessary to understand the data more deeply to draw proper conclusions. However, this serves to illustrate the method of interpreting the results.