Econometrics notes

Statistical background

Random variable of probabilities

Random variables cannot be observed as such; what we observe are realizations of random variables. The probability distribution formula gives the probabilities for different values of the random variable X in the case of discrete random variables.

Properties of probability distributions

• All probabilities lie between 0 and 1 (non-negative).
• The null set is assigned probability 0, and the full set of possibilities is assigned probability 1.
• The probability assigned to an event that is the union of two disjoint events is the sum of the probabilities assigned to those disjoint events.

Probability density function

This formula gives the probabilities for different values of the random variable X for continuous random variables.

• Non-negative.
• ∫ f(x) dx = 1
• P(a < x < b) = ∫ f(x) dx

Normal distribution

The normal distribution is a bell-shaped distribution. Its density function is:

f(x) = ¹/_σ√2π exp ^{-(x- μ)2}/_2σ², -∞ < x < +∞

Where μ = mean and σ² = variance. When x has a normal distribution with mean μ and variance σ², we write x ~ N(μ, σ²).

The expected value of X is:

E(X) = μ = ∫ x f(x) dx

The variance is the expected value of (X - μ)²:

Var(X) = ∫ (x - μ)² f(x) dx

Standard normal distribution

We say that X has a standard normal distribution if its probability density function is:

f(x) = ¹/_√2π exp ^-x2/₂, -∞ < x < +∞

The expected value E(X) is:

E(X) = 0

For the variance, we have that E[X²] is:

E[X²] = 1

We can write x ~ N(0,1).

Χ distribution

If X₁, X₂, …, X_n are i.i.d., then Z = ∑ X_i² has a χ² distribution with n degrees of freedom.

T distribution

If X ~ N(0,1) and X and Z are independent, then t = X/√(Z/n) ~ t distribution with n degrees of freedom.

F distribution

If Z₁ ~ χ²_(d1) and Z₂ ~ χ²_(d2) and they are independent, then F = (Z₁/d₁)/(Z₂/d₂) ~ F distribution with d₁ and d₂ degrees of freedom.

Sampling distribution

Suppose we have a sample y₁, y₂, …, y_n of n independent observations from a normal population with mean μ and variance σ².

• An estimate (W, S²) is a number calculated according to a rule or formula from the sample.
• An estimator (W, S²) is the rule or formula which yields the estimate.

What is the relationship between mean μ and the estimator W?