Essay About Economic Data And Rth Moment Of A Random Variable X

Econometrics
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1.0 Econometrics – the science and art of using economic theory and statistical techniques to analyze economic data
Multiple regression model – provides a mathematical way to quantify how a change in one variable affects another variable, holding other things constant

Causality – specific action leads to a specific, measurable consequence
Randomized controlled experiment – treatment is assigned randomly thus eliminating the possibility of a systematic relationship between the control group (receives no treatment) and the treatment group (received the treatment.

Causal effect – the effect of an outcome of a given action or treatment as measured in an ideal randomized controlled experiment
Two sources of data in Econometrics:
Experimental data – come from experiments designed to evaluate or investigate a causal effect.
Observational data – actual behavior outside experimental setting
Cross-sectional data – data for different entities for a single time period
Time series data – data for single entity collected at different time periods.
Panel data (Longitudinal data) – for multiple entities which each entity is observed at two or more time periods
2.0 Outcomes – mutually exclusive potential results of a random process
Probability of an outcome – proportion of time that the outcome occurs in the long run
Sample space –set of all possible outcomes
Event – a subset of the sample space
Random variable – numerical summary of a random outcome
Properties of probability
0 ≤ P(A) ≤ 1
If A, B, C, …, are exhaustive set of events, P(A+B+C+…) = 1
Conditional probability
P(A│B)=P(A⋂B)/(P(B))
Bayes Theorem
Probability Distribution of a Discrete Random Variable – list of all possible values of the variable and the probability that each value will occur
Discrete Density Function
If X is a discrete random variable with values x1, x2,..,xn, then the function
f(x)=P(X=xi) for i=1,2,…n
is defined to be the discrete density function of X
Cumulative distribution function (cdf) – probability that a random variable is less than or equal to a particular value
F(x)=P(X≤x)
Probability Density Function of a Continuous Random Variable – area under the pdf between 2 points is the probability that the random variable falls between these 2 points.

Probability that X is an exact number is 0
f(x) is the pdf of X if the following conditions are satisfied:
f(x)≥0
∫_(-∞)^∞▒〖f(x)dx=1〗
∫_a^b▒〖f(x)dx=P(a≤X≤b)〗
Mean/Expected Value
Discrete: μ_X=E(X)= ∑_x▒〖xf(x)〗
Continuous: E(X)= ∫_(-∞)^∞▒xf(x)dx
Variance
Standard Deviation σ_x= √(var(X))
Expectation
Discrete: E[g(X)]= ∑_x▒〖g(x)f(x)〗
Continuous: E[g(X)]= ∫_(-∞)^∞▒g(x)f(x)dx
Moments – rth moment of a random variable X is defined as E(Xr)
Skewness – how much a distribution deviates from symmetry
0 skewness means the graph is symmetric
Positive skew, tail is longer at the right
Negative skew, tail is longer at the left
γ_1= (E〖(X-μ)〗^3)/σ^3
Kurtosis – measure of how much mass is in its tails; a measure of how much of the variance arises from extreme values.
Leptokurtic – kurtosis > 3 (heavy tailed)
γ_2= (E〖(X-μ)〗^4)/σ^4
Joint Probability Distribution – probability that 2 random variables simultaneously take on certain values
Marginal Probability Distribution – distribution of one variable in a joint distribution with another variable
Marginal distribution of X
f(x)= ∑_y▒〖f(x,y)〗
Marginal distribution of Y
f(y)= ∑_x▒〖f(x,y)〗
Conditional Density Function
f(x ┤|Y=y)=P(X=x│Y=y)
= (P(X=x,Y=y))/(P(X=x))
Conditional Expectation – mean value of x when Y=y
E(X│Y=y)= ∑_x▒〖xf(x|Y=y)〗
Law of Iterated Expectation – the mean of Y is the weighted average of the conditional expectation of Y given X, weighted by the probability distribution of X.

E(Y)=E(E(Y│X))
Conditional Variance – variance of the conditional distribution of Y given X
var(Y│X=x)=
∑_y▒〖〖[y-E(Y│X=x)]〗^2 f(y|X=x)〗
Independence – X and Y are independent if the conditional distribution of Y given X equals the marginal distribution of Y
P(Y=y│X=x)=P(Y=y)