Chapter 19 The normal linear model

$$y_t = \beta_0 + \beta_1 x_t + \varepsilon_t$$

19.1 Assumptions of the linear model

• Relationship between predictor $x$ and predictand $y$ is linear.

• Both $x$ and $y$ are known, observed without error.

• Errors have mean zero.

• Errors are independent of each other.
  

• Errors are uncorrelated with predictor variables $x_t$.

Often, assume stronger additional conditions that errors are independent, identically normally distributed: for all $t$, $\varepsilon_t \sim N(0, \sigma^2)$. for a constant $\sigma^2$.

In compact vector and matrix notation, we may write:

$$Y = X \beta + \varepsilon$$ $$\varepsilon \sim N(0, \sigma^2 I_T)$$ Readings: FPP, Section 7.1

19.2 Examples of the normal linear model