Supplements to Section 1.1

by Qiang Gao, updated at Mar 13, 2017

Taylor's Linearization of Multivariate Functions

Any general nonlinear function $y = f(x_1, x_2)$ can be approximately linearized around a point $\bar y = f(\bar x_1, \bar x_2)$ as

$y - \bar y = \frac{\partial f(\bar x_1, \bar x_2)}{\partial x_1} (x_1 - \bar x_1) + \frac{\partial f(\bar x_1, \bar x_2)}{\partial x_2} (x_2 - \bar x_2)$

Total Differential

See the wiki page.

Differential vs. Elasticity

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,

$d \ln y = \frac{dy}{y}.$

$\frac{d \ln y}{d \ln x} = \frac{d y / y}{d x / x}$

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.

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

$\mathrm{Var}(x) = \mathrm{E} [ (x - \mathrm{E} (x))^2 ]$

Covariance

$\mathrm{Cov} (x, y) = \mathrm{E} [ (x - \bar x )( y - \bar y) ] = \mathrm{E} (xy) - \mathrm{E}(x) \mathrm{E} (y)$

Correlation Coefficient

$\rho_{x,y} = \frac{\mathrm{Cov} (x,y)}{\sqrt {\mathrm{Var} (x) \mathrm{Var} (y)}}$

Linearity of Expectation

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

Nonlinearity of Variance

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

Section 1.1