ECON3002 Econometrics

What is Econometrics

Lecture Notes for ECON3002

Multiple Linear Regression

ECON3002 Question List

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Self Textbook notes

2.3 Two Random Variables

Joint and Marginal Distributions

The Joint Probability Density Distribution and the Marginal Probability Distribution

Conditional Distributions

The distribution of a random variable $Y$ conditional on another random variable $X$ taking on a specific value is called the conditional distribution of $Y$ given $X$, it is written as $Pr(Y=y|X=x)$

In the table 2.2, the $Pr(Y=y|X=x)$ could be expressed that what is the prob of a long commute if I already know it is raining?

That is asking $Pr(Y=0|X=0)$ , and it is equals to $0.15/0.30=50\%$

$$Pr(Y=y|X=x)=\frac{Pr(X=x,Y=y)}{Pr(X=x)}$$

Remember the second part is the event that you already know.

Expected value and variance of a linear function

The basic knowledge: Variance

Prove: if $Y=a+bX$ then $Var(Y)=b^{2}Var(X)$ , why?

Since $Var(X)=E[(x-E(X){)}^2]$ so that $Var(Y)=E(a+bx-E(a+bx){)}^2$

$Var(Y)=E[(b(x-\overline{x}){)}^2]$ (the right part is the $Var(X)$)

So $Var(Y)=b^{2}Var(X)$

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Prove: $Var(X)=E(X^2)-[E(X{)}^2]$:

$Var(X)=E[(X-E(X){)}^2]=E[X^2-2XE(X)+E(X{)}^2]$

$=E[X^2]-2E(XE(X))+E(X{)}^2]$

==Remember, $E(E(X))=E(X)$, so:==

$=E[X^2]-2E(X{)}^2+E(X{)}^2$

$=E[X^2]-E(X{)}^2$

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Moments

Both Mean and Variance belong to a class of measures, called Moments

It is divided by the Raw Moments and the Central Moments

Calculate the skewness:

$Skewness:=\frac{{\sigma }^{(3)}}{{\sigma }^3}$

$Kurtosis:=\frac{{\sigma }^{(4)}}{{\sigma }^4}$ (measure the thickness of the tails)

$ExcessKurtosis:=Kurtosis-3$

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Standard Deviation

The Standard Deviation of $X$ is denoted as ${\sigma }_X$, also the $\sqrt{Var(X)}$

Joint and Marginal Distributions

The Joint Probability Density Distribution is the probability that two Discrete Random Variable $X$ and $Y$ simultaneously take on particular values. (say $x$ and $y$)

The law of iterated expectations

Law of Iterated Expectations is the mean of Y is the weighted average of the conditional expectation of $Y$ given X, weighted by the prob distribution of $X$

$$E(Y)=\sum _{i=1}^{l}E(Y|X=x_i)Pr(X=x_i)$$

So that the expectation of $Y$ is the expectation of the conditional expectation of $Y$ given $X$,

$$E(Y)=E[E(Y|X)]$$

And this is the core of Law of Iterated Expectations

The Independence

The Discrete Case

Independence Two random variables $X$ and $Y$ are said to be independently distributed if

$$Pr(Y=y|X=x)=Pr(Y=y)$$

Since the conditional distribution implies that

$$Pr(Y=y|X=x)=\frac{Pr(X=x,Y=y)}{Pr(X=x)}$$

That is

$$Pr(X=x,Y=y)=Pr(X=x)Pr(Y=y)$$

Then the $X$ and $Y$ are independent.

The Continuous Case

$Y$ is independent of $X$:

$f_{y|x}(y|x)=f_y(y)$

$f_{yx}(y,x)=f_y(y)f_x(x)$

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We use Correlation Coefficient to estimate the strength of the 2 variables.

It is rescaled to be between $-1$ to $1$ , and it is also unit free.

$$\rho =\frac{cov(X,Y)}{{\sigma }_X{\sigma }_Y}$$

How to compute $cov(X,Y)$? See Covariance

If $\rho =1$, it is perfect positive linear relation

If $\rho =-1$, it is perfect negative linear relation

(Most of the time the Correlation Coefficient would be between when estimate econometrics problem.)

That is, we use absolute value $|\rho |$ to reflects the strength of linear association between the 2 variables.

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Independence vs. Uncorrelation

It is claimed that Independence ⇒ $\rho =0$ , but not vice versa. (Think of the question $Y=X^2$)

This is why we have to highlight the “linear” in

That is, we use absolute value $|\rho |$ to reflects the strength of linear association between the 2 variables.

The independence only hold if $X$ has no relationship (neither linear nor nonlinear) of $Y$!

ECON3002 Lecture2

ECON 3002 Question List

1. What is lagged value/ forward value?
2. What is ${\Delta }^k{Y}_t={\Delta }^{k-1}{Y}_t-{\Delta }^{k-1}{Y}_{t-1}$

Reference