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Its a bridge between model and data. Model is some theoretic mathematical formulation. Data is collected/measured from the real world according to the definition of variables.
The bridge is in two ways:
(from data to model) Parameter Estimation.
(from model to data) Hypothesis Test.
Assumption 1.1 (linearity)
1. It's a tautology
Because the error term $$ \varepsilon_i $$ is defined as
the equality in (1.1.1) trivially holds true by definition.
Equation (1.1.1) only restricts a linear functional relationship between $$y$$ and $$ \mathbf{x} $$, nothing more.
2. Nonlinearity can be linearized
The linearity assumption is not so much restrictive, because any nonlinear function can be easily linearized.
3. The knowns are $$( y_i, \mathbf{x}_i $$) and the unknowns are $$( \boldsymbol\beta, \varepsilon_i )$$
4. $$ \boldsymbol\beta $$ is of primary interest
$$ \beta $$ means marginal separate effects.
5. $$ \varepsilon_i $$ is of primary concern
$$\varepsilon_i$$ should not depend on $$ \mathbf{x} $$
6. marginal separate effect relies on total differentiation
explicit equation
implicit equation
differential vs. elasticity
7. variables are usually transformed (in log)
By the rules of differentiation
$$
\frac{d \ln x}{dx} = \frac{1}{x},
$$
we can write it in total differential form as
$$
d \ln x = \frac{dx}{x}.
$$
Similarly,
$$
d \ln y = \frac{dy}{y}.
$$
So
$$
\frac{d \ln y}{d \ln x} = \frac{d y / y}{d x / x}
$$
coincides with the definition of elasticity. It is of this reason that
in economics, variables are often expressed in logs rather than in
levels in equations.
The (unconditional) expectation $$\mathrm{E}(x)$$ is defined as
$$
\mathrm{E}(x) = \int x f(y, x) , dy dx
$$
Conditional Expectation
If $$(y, x)$$ are jointly distributed random variables, where their joint p.d.f. is expressed as $$f(y, x)$$, then $$\mathrm{E} (y | x)$$ is defined
as
$$
\mathrm{E} (y|x) = \int_{-\infty}^{+\infty} y \frac{ f(y, x) }{ \int_{-\infty}^{+\infty} f(y, x) dy } dy,
$$
where $$\int_{-\infty}^{+\infty} f(y, x) dy$$ is the definition of the marginal distribution of $$x$$. In words, the expectation of $$y$$ conditional on $$x$$ is the weighted average of $$y$$, where the weighting is the conditional probability density.