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