恳请协助推导阶数为(2,1,1)的ARIMAX方程

Hey there! Let's break down how to derive the ARIMAX(2,1,1) equation step by step—starting from the basics so it's clear even if you're new to these models.

ARIMAX is just ARIMA with exogenous (external) variables added, so let's start with the ARIMA(2,1,1) foundation.

First, define the key components:

• $p=2$: 2 autoregressive (AR) terms (we use the 2 most recent past values of the differenced sequence)
• $d=1$: 1st-order differencing (to make the sequence stationary)
• $q=1$: 1 moving average (MA) term (we use the 1 most recent error term)

We use the lag operator $L$ to simplify notation—think of it as a "time-shift button":

• $L Y_t = Y_{t-1}$ (shifts $Y_t$ back 1 period)
• $L^2 Y_t = Y_{t-2}$ (shifts back 2 periods)
• $(1-L)Y_t = Y_t - Y_{t-1}$ (1st-order difference, written as $\Delta Y_t$)

ARIMAX introduces external variables (let's call this set $X_t = [X_{1t}, X_{2t}, ..., X_{kt}]$ where $k$ is the number of external variables) that influence your target sequence $Y_t$. Each $X_{it}$ has a coefficient $\beta_i$, so the combined effect is $\beta^T X_t$ (a dot product of coefficients and variables).

The general ARIMAX formula is:

Φ(L)(1-L)^d Y_t = β^T X_t + Θ(L)ε_t

Where:

• $\Phi(L) = 1 - \phi_1 L - \phi_2 L^2$ (AR polynomial for p=2)
• $\Theta(L) = 1 + \theta_1 L$ (MA polynomial for q=1—note: some textbooks use negative signs here, so double-check your reference!)
• $\varepsilon_t$ = white noise error term (mean 0, constant variance, no autocorrelation)

Let's substitute $d=1$, $\Phi(L)$, and $\Theta(L)$ into the formula:

(1 - φ₁L - φ₂L²)ΔY_t = β^T X_t + (1 + θ₁L)ε_t

Step 1: Expand the left side (AR + differencing) #

Replace $\Delta Y_t$ with $Y_t - Y_{t-1}$, and apply the lag operator to the differenced terms:

ΔY_t - φ₁ΔY_{t-1} - φ₂ΔY_{t-2} = β^T X_t + ε_t + θ₁ε_{t-1}

Substitute each $\Delta$ term:

(Y_t - Y_{t-1}) - φ₁(Y_{t-1} - Y_{t-2}) - φ₂(Y_{t-2} - Y_{t-3}) = β^T X_t + ε_t + θ₁ε_{t-1}

Step 2: Simplify and combine like terms #

Expand the left side and group the $Y$ terms by their time lags:

Y_t - Y_{t-1} - φ₁Y_{t-1} + φ₁Y_{t-2} - φ₂Y_{t-2} + φ₂Y_{t-3} = β^T X_t + ε_t + θ₁ε_{t-1}

Combine coefficients for each lagged $Y$:

Y_t - (1 + φ₁)Y_{t-1} + (φ₁ - φ₂)Y_{t-2} - φ₂Y_{t-3} = β^T X_t + ε_t + θ₁ε_{t-1}

• Exogenous variable stationarity: If your $X_t$ variables are non-stationary, you'll likely need to difference them too, changing the equation to use $\Delta X_t$ instead of $X_t$.
• Sign conventions: Textbooks vary on whether AR/MA polynomials use positive or negative coefficients. Always confirm the notation from your learning resource to avoid confusion.
• Error term assumptions: $\varepsilon_t$ must be white noise—this is a key assumption for the model to be valid.

内容的提问来源于stack exchange，提问作者srinivasa murthy gunturu