Econometrics

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P(X=x,Y=y)=P(X=x)P(Y=y)

Covariance – the extent to which 2 random variables move together

If independent, co-variance is 0. But if co-variance is 0, it does not necessarily mean they are independent.

If positive, data is moving in the same direction

If negative, data is moving in opposite directions

σ_XY=cov(X,Y)=E[(X-E(X))(Y-E(Y))]

σ_XY=E(XY)-E(X)E(Y)

= ∑_(i=1)^k▒∑_(j=1)^l▒〖(x_j-μ_X )(y_i-μ_y ) P(X=x_j,Y=y_i)〗

Correlation – covariance between X and Y, divided by their standard deviation (unitless)

-1≤corr(X,Y)≤1

If the conditional mean of Y does not depend on X, then X and Y are uncorrelated

E(Y│X)=μ_y→cov(Y,X)=0,corr(Y,X)=0

If X and Y are uncorrelated, it does not necessarily mean that the conditional mean of Y given X does not depend on X

Additional properties:

E(X+Y)=E(X)+E(Y)

E(a+bX+cY)=a+bE(X)+cE(Y)

var(a+bY)=b^2 var(Y)

var(aX+bY)

=a^2 var(X)+2abcov(X,Y)+b^2 var(Y)

cov(a+bX,c+dY)=bdcov(X,Y)

E(Y^2 )=σ_Y^2+μ_Y^2

cov(a+bX+cV,Y)=bcov(X,Y)+cov(V,Y)

E(XY)=cov(X,Y)+E(X)E(Y)

var(X+Y)=var(X)+var(Y)+2cov(X,Y)

Bernoulli distribution- outcome is either 0 or 1

p=probability of success

P(x=0) = 1-p P(x=1) = p

Mean: p Variance: p(1-p)

Binomial Distribution – n Bernoulli trials that are independent of each other

f(x)=(n¦x) p^x (1-p)^(n-x)

Mean: np Variance: np(1-p)

Poisson distribution

f(x)= (e^(-λ) λ^x)/x!

Mean and Variance: λ

Continuous Uniform Distribution

f(x)={█(1/(b-a) for a≤x≤b@0 for x<a or x>b)┤

Normal distribution N(µ, σ2)

f(x)=1/(σ√2π) exp⁡(-1/2 (x-μ)^2/σ^2

Standardize the normal distribution using:

Z=(x-μ)/(σ/√n)

Skewness is 0, kurtosis is 3

Lognormal Distribution – a random variable X has a lognormal distribution if its natural logarithm Y=ln X, is normally distributed

Chi-squared Distribution – the distribution of the sum of m squared independent standard normal random variables (m=degrees of freedom)

Student t distribution – with m degrees of freedom is the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom, divided by m

When m is > 30, it is approximately standard normal

F Distribution – with m and n degrees of freedom, distribution of the ratio of a chi-squared random variable with df m, divided by m, to an independently distributed chi-squared random variable with df n, divided by n.

If n is large enough, it can be approximated by Fm,infinity

Random Sample – independently and identically distributed; has joint density of:

fx1,fx2,(x1,x2,.,xn)=f(x1)f(x2)…f(xn)

Sample Mean – an estimator of the mean of the population

X ̅= (X_1+X_2+⋯+X_n)/n=1/n ∑_(i-1)^n▒X_i

Mean: E(X ̅ )=1/n ∑_(i-1)^n▒〖E(X_i)〗=μ

Variance: var(X ̅ )=σ^2/n

Standard Deviation

S^2=1/(n-1) ∑_(i=1)^n▒(X_i-X ̅ )^2 = σ/√n

Sampling from Normal Distributions - Let X be a random sample from a distribution N(µ, σ2), X ̅ has a normal distribution with mean µ and variance σ2/n

Law of Large Numbers (consistency) - X ̅ will be near µ with a very high probability when n is large.

Central Limit Theorem – when n is large, the distribution of X ̅ is approximately normal

Z_n= (∑_(i-1)^n▒〖X ̅_n-μ〗)/(σ/√n)

With mean 0, variance, 1

Estimator Properties:

Unbiased E(μ ̂ )=μ

Consistent μ ̂ → μ

Efficient μ ̂ is more efficient than μ ̃ if var(μ ̂ )<var(μ ̃)

X ̅ is BLUE – Best Liner Unbiased Estimator

Hypothesis Testing

Null hypothesis H_0: θ=θ_0

Alternative hypothesis H_1: θ≠θ_0

p-value – significance probability (exact level of significance), area of the tails in the distribution of X ̅

p-value=2ϕ(-|(X ̅^act-μ_(X,0))/σ_X ̅ |

t-statistic =standardized sample average

t=(X ̅-μ_(X,0))/(SE(X ̅))

p-value=2ϕ(-|t^act |)

Type I Error – reject H_0 when it is true

Type II Error –do not reject H_0 when is is false

Level of Significance – α = probability of type I error

Power of the test – 1- β

β - probability of committing a type II error

Test of Significance Approach

Compute