留学生考试辅导 ECON3206/5206: Financial Econometrics - Week-1: A Review of Random Varia

ECON3206/5206: Financial Econometrics – Week-1: A Review of Random Variables and Distributions

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

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School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

ECON3206/5206: Financial Econometrics
Week-1: A Review of Random Variables and Distributions

School of Economics1

1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

May 20, 2020

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Random Variables: Definitions

Topic 1. Features of Some Financial Time Series

• RV and probability distribution
– Two types of RV

• Discrete RV: it takes a finite or countable number of values.

• Continuous RV: it may take any value in a interval.

– Probability distribution (how likely the values occur)
• Discrete RV

– Probability mass (pmf):

– Cumulative distribution (cdf): ( ) = $ * ≤ ) = ∑ ,#-./-

• Continuous RV

– Probability density (pdf): 0(1)
– Cumulative distribution (cdf):

( ) = $ * ≤ ) = 2 0 1 31

Area under 0(1) curve up to )

* )5 )5 )6 …
$(* = )#) ,5 ,5 ,6 …

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Slides-01, Financial Econometrics 12

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Random Variables: Unconditional (marginal) Expectations (moments)

Topic 1. Features of Some Financial Time Series

• Characterise RVs
– Mean or expected value of *

• The weighted average of all possible values, a measure of location

• Discrete RV: 7 * = ∑ )#$(* = )#)#
• Continuous RV: 7 * = 2 10 1 31

– Mean of 8 * (eg. *5, 9:, …)
• Discrete RV: 7 8 * = ∑ 8 )# $(* = )#)#
• Continuous RV: 7 8 * = 2 8 1 0 1 31

– Variance of *:
• A measure of the amount of variation in all possible values

Var * = 7{ * − 7 * 5} = 7 *5 − 7 * 5

– Covariance between * and @
• A measure of association

Cov *, @ = 7{ * − 7 * @ − 7 @ }

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Slides-01, Financial Econometrics 13

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Random Variables: Conditioning

Topic 1. Features of Some Financial Time Series

• Conditioning

– Conditional distribution of @ given *
• The distribution of @ when * is “known”.
• It depends on the value of *.
• It is denoted as @|*.
eg. the distribution of wage for age = 20,

the distribution of wage for age = 30, …
wage|age depends on age.

– Conditional expectation
• It is calculated with the conditional distribution,

treating * as “known” or “fixed”.
• It depends on * and, hence, is also a RV.

bhpt|bhpt-1:

dist. of bhpt
when bhpt-1 is

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School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Random Variables

Topic 1. Features of Some Financial Time Series

• Conditioning

– Conditional mean of @ given *
• generally depends on *.

eg. Linear regression @ = G + H* + I.
7 @ * = G + H* + 7(I|*) = G + H*.

– Conditional variance of @ give *
• Var @ * generally depends on *.

eg. Linear regression @ = G + H* + I.
Var @ * = Var(I|*).

UNSW Business School,

Slides-01, Financial Econometrics 15

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Properties of Expectation operator

Topic 1. Features of Some Financial Time Series

• Rules for expectation and variance operations
Let * and @ be RVs and G be a constant.
– Expectation is a linear operator

7 G = G, Var G = 0;
7 G* = G7(*);
7 * + @ = 7 * +
7 8 * * = 8(*) for any function 8(⋅);
7 @ = 7[7 @ * ] (iterated expectations);
7 @ * = if @ is independent of *;

– Variance in a nonlinear operator
Var G* = G5Var * ;
Var * + @ = Var * + Var @ + 2Cov(*, @);
Var @ = 7 Var @ * + Var[7 @ * ].

UNSW Business School,

Slides-01, Financial Econometrics 16

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Sample moments

Topic 1. Features of Some Financial Time Series

• Descriptive statistics

– Suppose we have (time series) observations
*R 5L = *5, *5, … , *L and @R 5L = @5, … , @L}.

– Sample mean

• a measure of location (central tendency)

• an estimator of population mean 7(*)
– Sample variance and standard deviation

∑ *R − *M 5LR%5 , NO: = NO:5

• a measure of variation

• an estimator of the population variance Var(*)

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Slides-01, Financial Econometrics 17

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions Random Variable: Expectations (first and second moments) Conditioning Properties of the ’E’xpectation operator Sample moments Behold the summation operator

Behold the summation operator

Linear operator:

Summation of a constant:

Nonlinear relations:

How to use summation operator ∑

1 1 1 1= = = =

+ + = + +∑ ∑ ∑ ∑

a bx cy a b x c y

1 1 1= = =

Topic 1. Features of Some Financial Time Series

UNSW Business School,

Slides-01, Financial Econometrics 18

School of Economics ECON3206/5206: Financial Econometrics

Random Variable: Definitions
Random Variable: Expectations (first and second moments)
Conditioning
Properties of the ‘E’xpectation operator
Sample moments
Behold the summation operator

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