Annotated Equations

A quick reference for the models introduced in the R Tutorial.

Time-series linear model (TSLM)

A TSLM is similar to a standard regression.

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#319795}{\beta_1 x_t}}_{\textstyle\color{#319795}{\text{covariate}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

• \(y_t\) — the observed value at time \(t\)
• \(c\) — a constant, the intercept
• \(\beta_1 x_t\) — the covariate effect: how strongly the predictor \(x_t\) is related to \(y_t\)
• \(\epsilon_t\) — random error at time \(t\)

Limitation: Regression assumes independent observations, so autocorrelation in the residuals limits the model’s statistical inferences.

Adding trends

We can explicitly model trends that are not captured by the covariates by fitting them directly.

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#319795}{\beta_1 x_{1,t}}}_{\textstyle\color{#319795}{\text{covariate 1}}} \;+\; \underbrace{\color{#059669}{\beta_3 t}}_{\textstyle\color{#059669}{\text{trend}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

• \(y_t\) — the observed value at time \(t\)
• \(c\) — a constant, the intercept
• \(\beta_1 x_{1,t}\) — the covariate’s effect
• \(\beta_3 t\) — the trend term: \(t\) is time and serves as a predictor for long-term increases or decreases the other covariates don’t capture
• \(\epsilon_t\) — random error at time \(t\)

Reading it: adding trend() doesn’t explain why there’s a long-term change — it just accounts for the fact that a long-term change exists

ARIMAX: ARIMA with external predictors

ARIMA() can include exogenous covariates directly, combining regression with AR and MA structure in one model.

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#319795}{\beta_1 x_{1,t}}}_{\textstyle\color{#319795}{\text{covariate}}} \;+\; \underbrace{\color{#c05621}{\beta_2 y_{t-1}}}_{\textstyle\color{#c05621}{\text{AR: lag-1}}} \;+\; \underbrace{\color{#7c3aed}{\theta_1 \epsilon_{t-1}}}_{\textstyle\color{#7c3aed}{\text{MA: lag-1}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

• \(y_t\) — the observed value at time \(t\)
• \(c\) — a constant
• \(\beta_1 x_{1,t}\) — the covariate effect
• \(\beta_2 y_{t-1}\) — AR: the effect of the previous value
• \(\theta_1 \epsilon_{t-1}\) — MA: the effect of the previous error
• \(\epsilon_t\) — random error at time \(t\)

Why this helps: AR and MA terms capture autocorrelation that a plain TSLM leaves behind, while the covariates allow inclusion of external drivers.

The differenced ARIMAX model

Once ARIMA() decides differencing is needed, every term in the model — including the covariate — gets differenced too, and the constant drops out (as in the differencing section of the previous lesson).

\[ y_t' = \underbrace{\color{#319795}{\beta_1 x_{1,t}'}}_{\textstyle\color{#319795}{\text{covariate (differenced)}}} \;+\; \underbrace{\color{#c05621}{\beta_2 y_{t-1}'}}_{\textstyle\color{#c05621}{\text{AR: lag-1}}} \;+\; \underbrace{\color{#7c3aed}{\theta_1 \epsilon_{t-1}}}_{\textstyle\color{#7c3aed}{\text{MA: lag-1}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

• \(y_t'\) — the differenced response
• \(\beta_1 x_{1,t}'\) — the differenced covariate effect
• \(\beta_2 y_{t-1}'\) — AR: the effect of the previous differenced value
• \(\theta_1 \epsilon_{t-1}\) — MA: the effect of the previous error
• \(\epsilon_t\) — random error at time \(t\)

Regression with ARIMA errors

fable doesn’t literally fit the equations above — it fits a plain regression on the covariate, then models the leftover structure in that regression’s error with an ARIMA model.

\[ y_t = \underbrace{\color{#319795}{\beta_1 x_{1,t}}}_{\textstyle\color{#319795}{\text{covariate effect}}} \;+\; \underbrace{\color{#d53f8c}{\eta_t}}_{\textstyle\color{#d53f8c}{\text{ARIMA-structured error}}} \]

\[ \eta_t = \underbrace{\color{#c05621}{\beta_2 \eta_{t-1}}}_{\textstyle\color{#c05621}{\text{AR: lag-1 on the error}}} \;+\; \underbrace{\color{#7c3aed}{\theta_1 \epsilon_{t-1}}}_{\textstyle\color{#7c3aed}{\text{MA: lag-1}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

• \(y_t\) — the response
• \(\beta_1 x_{1,t}\) — the covariate’s effect modeled as a linear regression term
• \(\eta_t\) — the time-series structured regression error
• \(\beta_2 \eta_{t-1}\) — the AR part of that structured error
• \(\theta_1 \epsilon_{t-1}\) — the MA part of that structured error
• \(\epsilon_t\) — the remaining Normally distributed error left after the time-series components

Why split it this way: it keeps the covariate’s coefficient directly interpretable (a plain regression slope), while letting the ARIMA machinery handle whatever autocorrelation the regression alone couldn’t.