1.1.md

Lecture Notes on Econometrics

by Qiang Gao, updated at March 26, 2018

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Chapter 1 Finite-Sample Properties of OLS

Section 1 The Classical Linear Regression Model

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The Big Picture of Econometrics

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:

1. (from data to model) Parameter Estimation.
2. (from model to data) Hypothesis Test.

Assumption 1.1 (linearity)

1. It's a tautology

Because the error term $$ \varepsilon_i $$ is defined as

$$ \varepsilon_i = y_i - \mathbf{x}_i \cdot \boldsymbol\beta, $$

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.

Assumption 1.2 (strict exogeneity)

Joint Distribution

$$ f_{Y,X}(y, x) \qquad \oint f_{Y,X}(y, x),dx,dy = 1 $$

Marginal Distribution

$$ \begin{align} f_{Y} (y) \equiv \oint f_{Y,X}(y, x) , dx && \oint f_{Y} (y) , dy = 1 \\ f_{X} (x) \equiv \oint f_{Y,X}(y, x) , dy && \oint f_{X} (x) , dx = 1 \end{align} $$

Conditional Distribution

(Unconditional) Expectation

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.

Law of Total Expectations

$$ \mathrm{E} ( \mathrm{E} (y | x) ) = \mathrm{E} (y). $$

Law of Iterated Expectations

$$ \mathrm{E} ( \mathrm{E} (y | x, z) | z ) = \mathrm{E} (y | z). $$

Moment

The $$k$$-th order moment of a random variable $$x$$ is defined as

$$ \mathrm{E}(x^k) $$

Variance

$$ \begin{align} \mathrm{Var}(x) &= \mathrm{E} [ (x - \mathrm{E} (x))^2 ] && \text{(definition)} \\ &= \mathrm{E}(x^2)- E(x)^2 && \text{(formula)} \end{align} $$

Covariance

$$ \begin{align} \mathrm{Cov} (x, y) &= \mathrm{E} [ (x - \mathrm{E} (x) )( y - \mathrm{E} (y)) ] && \text{(definition)} \\ &= \mathrm{E} (xy) - \mathrm{E}(x) \mathrm{E} (y) && \text{(formula)} \end{align} $$

Correlation Coefficient

$$ \rho_{x,y} = \frac{\mathrm{Cov} (x,y)}{\sqrt {\mathrm{Var} (x) \mathrm{Var} (y)}} \in [-1, 1] $$

Linearity of Expectation

$$ \mathrm{E} (ax + b) = a \mathrm{E} (x) + b $$

Nonlinearity of Variance

$$ \mathrm{Var} (ax + b) = a^2 \mathrm{Var} (x). $$

Assumption 1.3 (no multicollinearity)

• perfect multicollinearity can occur in rare conditions as long as its measure is zero.

Assumption 1.4 (spherical error variance)

$$ \mathbf{x} \mathbf{x}' \equiv \begin{bmatrix} x_1^2 & \cdots & x_1 x_n \\ \vdots & \ddots & \vdots \\ x_n x_1 & \cdots & x_n^2 \end{bmatrix} $$

$$ \mathrm{E} \begin{bmatrix} a_{11} & \cdots & a_{n1} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix} \equiv \begin{bmatrix} \mathrm{E} (a_{11}) & \cdots & \mathrm{E} (a_{n1}) \\ \vdots & \ddots & \vdots \\ \mathrm{E} (a_{m1}) & \cdots & \mathrm{E} (a_{mn}) \end{bmatrix}, $$

$$ \mathrm{Var} ( \mathbf{x} ) \equiv \mathrm{E} [ ( \mathbf{x} - \overline{\mathbf{x}} ) ( \mathbf{x} - \overline{\mathbf{x}} )' ] $$

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Copyright ©2018 by Qiang Gao