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I'm confused about the notation that leads to an $ARMA$ model.

If, 1: Autoregressive model $AR(p)$ is defined as: $Y_t-\mu=\phi*(Y_{t-1}-\mu)+w_t$

Where,

  • $Y_t$: observation in time $t$
  • $Y_{t-1}$: observation in time $t-1$
  • $\mu$: mean
  • $\phi$: autoregressive slope
  • $w_t$: White noise with variance $\sigma^2$ in time $t$

And 2: Moving average model $MA(q)$ is defined as: $Y_t=\mu+w_t+\theta*w_{t-1}$

Where,

  • $\theta$: moving average slope

Why is $1+2=ARMA(p,q)$ defined as: $x_t=\phi*x_{t-1}+w_t+\theta*w_{t-1}$

Specifically, I'm quite confused about the $ARMA$ definition:

a. Where's $\mu$ the mean in 1 and 2. From looking at outputs I know it's now: $w_t$, but I thought that $w_t$ was defined as the white noise error.

b. Sort of silly, why change $Y_t$ (in 1 and 2) for $x_t$ in $ARMA$?

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There are generally, two different versions of AR model used in literature.

1st Type: $$Y_t = \phi_0 + \phi_1 Y_{t-1} +w_t$$

2nd Type (specified by you): $$Y_t - \mu = \phi_1 (Y_{t-1} - \mu) + w_t$$

In both the model, $w_t$ is white noise term. The only difference is, the second type is de-mean model, where $\mu$ is mean of $Y_t$.For the first model, $$E(Y_t) =\mu = \frac{\phi_0}{1-\phi_1}$$ Just replace, $\mu$ with above expression in type-2 AR equation, you will get same type-1 equation.

Now, question arise, why there are two different type of AR models?

  1. Type-1 AR model is more intuitive. It includes intercept and slope coefficient like simple linear regression model. But the problem is that it is difficult to derive autocorrelation and auto covariance from type 1 model.

  2. Since, both correlation and covariance is independent of change of origin, therefore, re-writing your model in demean form simplify the calculation. Like:
    $$y_t = \phi_1 y_{t-1} + w_t $$ where, $y_t = Y_t - \mu$.

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Hi: It seems that they made the substitution that $x_t = Y_t - \mu$ without explicitly stating it. So, that is where $\mu$ went. It's still there but hidden in $x_t$. You can think of $x_t$ as a new variable which represents the "deviation of Y_t from it's mean". So, $x_t$ s just $Y_t$ de-meaned.

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