若Y_t为白噪声过程，时间序列X_t=Y_t-0.8Y_{t-1}是否为AR(1)过程？

Hey there! Great question, but let's break this down clearly because your initial guess isn't quite right.

First, let's recap the core definition of an AR(1) process: An AutoRegressive process of order 1 is structured as:
$$X_t = \phi X_{t-1} + \epsilon_t$$
Where:

• $\phi$ is the autoregressive coefficient
• $\epsilon_t$ is a white noise process
• The critical detail: $X_t$ depends on its own past value ($X_{t-1}$), not the lagged values of an external white noise process.

Now look at your time series:
$$X_t = Y_t - 0.8Y_{t-1}$$
Here, $X_t$ is built from the current and lagged values of the white noise $Y_t$, not from lagged values of $X$ itself. This fits the exact definition of a MA(1) (Moving Average of order 1) process, not an AR(1).

To drive this point home, let's try rearranging to see if we can force it into AR(1) form. We know:
$$X_{t-1} = Y_{t-1} - 0.8Y_{t-2}$$
There's no way to substitute this into the equation for $X_t$ to express $X_t$ as a function of $X_{t-1}$ plus white noise—since $X_t$ includes $Y_t$, which has no direct link to $X_{t-1}$.

So to wrap up:

• AR(1) processes rely on their own lagged values
• Your process relies on lagged values of the underlying white noise, making it an MA(1), not AR(1)

内容的提问来源于stack exchange，提问作者user534867