R语言中使用Arima构建AirPassengers的MA(1)模型，公式正确性咨询

Hey there! Let's walk through this step by step to figure out the right model formula for your fitted MA(1) model.

First, let's recap what you did: you used R's Arima() function to fit an MA(1) model to the AirPassengers dataset with this code:

test3=Arima(AirPassengers,c(0,0,1))

Your model output confirms this is an ARIMA(0,0,1) with non-zero mean, with key coefficients:

• ma1 = 0.9642 (standard error: 0.0214)
• mean = 280.6464 (standard error: 10.5788)

Key Background on R's Arima() MA Model Structure #

R's Arima() function uses this specific form for a mean-included MA(1) model:

Yt = μ + et + θ₁*et₋₁

Here:

• μ is the model's mean (your 280.6464 value)
• θ₁ is the MA(1) coefficient (your 0.9642 value)
• et represents the current period error term, while et₋₁ is the error term from the previous period

Comparing the Two Options #

• Option (1): Yt=280.6464+et+0.9642et-1
  This matches perfectly with R's model structure. The mean value, error terms, and MA coefficient (with the correct positive sign) all align exactly with your model's output.

• Option (2): Yt=280.6464+et-0.9642et-1
  This uses a negative sign for the MA term, which doesn't fit R's implementation. It's worth noting some textbooks use an alternative MA formula with a minus sign (Yt = μ + et - θ₁*et₋₁), but R's Arima() outputs coefficients tailored to the plus-sign version—so we go with the sign directly given in the output.

So the correct final model formula is Option (1).

内容的提问来源于stack exchange，提问作者mcxmcx