Applied Statistics Vincent JEANNIN – ESGF 4IFM

Q1 2012

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Summary of the session (est. 4.5h) • Reminders of last session • Multiple regression • Introduction to econometrics • Estimations • Games: beat the statistics

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Reminders of last session

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F 4

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3 Methods

• Historical • Parametrical • Monte-Carlo

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Options: what to look at to calculate the VaR?

4 risk factors: • Underlying price • Interest rate • Volatility • Time

4 answers: • Delta/Gamma approximation knowing the distribution of the underlying • Rho approximation knowing the distribution of the underlying rate • Vega approximation knowing the distribution of implied volatility • Theta (time decay)

Yes but,… Does the underling price/rate/volatility vary independently?

Might be a bit more complicated than expected…

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F 4

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Portfolio scale: what to look at to calculate the VaR?

Big question, is the VaR additive?

NO! Keywords for the future: covariance, correlation, diversification

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VAR 𝑎𝑋 + 𝑏𝑌 = 𝑎2𝑉𝐴𝑅 𝑋 + 𝑏2𝑉𝐴𝑅 𝑌 + 2𝑎𝑏𝐶𝑂𝑉(𝑋, 𝑌)

Parametric VaR on 2 assets?

𝑃 𝑋 ≤ −1.645 ∗ 𝜎 + 𝜇 = 0.05

𝑃 𝑋 ≤ −2.326 ∗ 𝜎 + 𝜇 = 0.01

Asset 1 Mean 0

SD 2.34% Weight 50%

Asset 2 Mean 0

SD 1.50% Weight 50%

Correlation 0.59

What is the VaR (95%)?

2.83%

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F 5

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Linear regression model

Minimize the sum of the square vertical distances between the observations and the linear approximation

𝑦 = 𝑓 𝑥 = 𝑎𝑥 + 𝑏

Residual ε

OLS: Ordinary Least Square

Minimising residuals

𝐸 = 𝜀𝑖2

𝑛

𝑖=1

= 𝑦𝑖 − 𝑎𝑥𝑖 + 𝑏 2

𝑛

𝑖=1

𝑎 =𝐶𝑜𝑣𝑥𝑦

𝜎2𝑥

𝑏 = 𝑦 − 𝑎 𝑥

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𝑟 =𝐶𝑜𝑣𝑥𝑦

𝜎𝑥𝜎𝑦 Value between -1 and 1

Dispersion Regression

Total Dispersion 𝑅2 =

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F 5

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F 5

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F 5

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Differentiation can happen before the OLS

What do you suggest?

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F 5

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𝑌𝐷𝑖𝑓𝑓 = ln(𝑌)

Let’s create a new variable

Magic!

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F 5

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Only one parameters to estimate: • Slope β

Minimising residuals

𝐸 = 𝜀𝑖2

𝑛

𝑖=1

= 𝑦𝑖 − 𝑎𝑥𝑖2

𝑛

𝑖=1

When E is minimal?

When partial derivatives i.r.w. a is 0

New idea… No intercept

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𝐸 = 𝜀𝑖2

𝑛

𝑖=1

= 𝑦𝑖 − 𝑎𝑥𝑖2

𝑛

𝑖=1

𝜕𝐸

𝜕𝑎= −2𝑥𝑖𝑦𝑖 + 2𝑎𝑥𝑖

2

𝑛

𝑖=1

= 0

𝑦𝑖 − 𝑎𝑥𝑖2 = 𝑦𝑖

2 − 2𝑎𝑥𝑖𝑦𝑖 + 𝑎2𝑥𝑖2

Quick high school reminder if necessary…

𝑥𝑖𝑦𝑖 − 𝑎𝑥𝑖2

𝑛

𝑖=1

= 0

𝑎 ∗ 𝑥𝑖2

𝑛

𝑖=1

= 𝑥𝑖𝑦𝑖

𝑛

𝑖=1

𝑎 = 𝑥𝑖𝑦𝑖

𝑛𝑖=1

𝑥𝑖2𝑛

𝑖=1

𝑎 =𝑥𝑖𝑦𝑖

𝑥𝑖2

Any better?

Multiple regressions

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𝑦 = 𝑏0 + 𝑏1𝑋1+𝑏2𝑋2+…+𝑏𝑛𝑋𝑛 + ε

More than one explanatory variables

Choosing factors can be difficult

Much tougher without software

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Variables may not be dependent form each other

Financial methods such APT (Arbitrage Pricing Theory) tries to have pure and independent factors

Used a lot in economics

R-Square is very often very poor

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Ratio Investment / GDP , World Bank, developing countries

𝑅 = 19.5 −5.8𝐶𝑜𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛 + 6.3𝐶𝑜𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛 + 2𝑆𝑐ℎ𝑜𝑜𝑙 − 1.1𝐺𝐷𝑃 − 2𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛

Let’s discuss…

• Corruption: current corruption • CorruptionPrediction: future corruption • School: level of education • GDP: GDP • Distortion: how badly policies are run

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F 4

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Opposite effect of corruption variables

Any logic with this?

The current level of corruption decreases investment

The future level of corruption increases investment

Investors learn how to live with corruption…

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R-Squared is 0.24, very poor…

• General to specific: this starts off with a comprehensive model, including all the likely explanatory variables, then simplifies it.

• Specific to general: this begins with a simple model that is easy to understand, then explanatory variables are added to improve the model’s explanatory power.

How to find the right model?

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Golden rules

Be logic

Have the best R-Squared

Not over complicate

Introduction to econometrics

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3 steps

Identify

Fit

Forecast

𝑂𝑏𝑠 = 𝑀𝑜𝑑𝑒𝑙 + 𝜀 with 𝜀 being a white noise What is a model?

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3 components

Trend

Seasonality

Residual

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F 4

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Stationary series are easier to forecast… Transform it!

A series is stationary if the mean and the variance are stable

Which one is more likely to be stationary?

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Properties of stationary series

(𝑌1, 𝑌2, 𝑌3, … , 𝑌𝑛)

(𝑌2, 𝑌3, 𝑌4, … , 𝑌𝑛+1)

Same distribution of the following

Distribution not time dependent

Rare occurrence

Stationarity accepted if

𝐸(𝑌𝑡) = 𝜇 Constant in the time

𝐶𝑜𝑣(𝑌𝑡 , 𝑌𝑡−𝑛) Depends only on n

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About the residuals…

White noise!

Normality test

Have an idea with

Skewness

Kurtosis

Proper tests: KS, Durbin Watson, Portmanteau,…

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eps<-resid(TReg)

ks.test(eps, "pnorm")

layout(matrix(1:4,2,2))

plot(TReg)

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F 4

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lag.plot(DATA$Val, 9, do.lines=FALSE)

Differentiation seems to be interesting

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F 5

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Check ACF/PACF for autocorrelation

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𝑋𝑡 = 𝑐 + 𝜑1𝑋𝑡−1 + 𝜑2𝑋𝑡−2 + ⋯+ 𝜑𝑛𝑋𝑡−𝑛 + 𝜀𝑡

𝜑𝑛 Parameters of the model

𝜀𝑛 White noise

Auto Regressive model

AR(n)

Estimations

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F 4

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Small sample: Binomial Distribution

Large sample: Normal Distribution

)()1()!(!

!)( xnx pp

xnx

nxf

)1(, pnpnpN

n is the size of the sample, x, the number individuals with the particular characteristic

𝐸 𝑋 = 𝑛𝑝

𝑉 𝑋 = 𝑛𝑝(1 − 𝑝)

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Binomial Distribution

𝐸 𝑌 = 𝑝 𝑉 𝑌 =𝑝(1 − 𝑝)

𝑛

Normal approximation

𝑌~𝑁 𝑝,𝑝(1 − 𝑝)

𝑛 Standardisation possible

𝑌∗~𝑁 0,1

𝑌∗ =𝑌 − 𝑝

𝑝(1 − 𝑝)𝑛

Normal approximation works only if

𝑛𝑝 ≥ 5 𝑛(1 − 𝑝) ≥ 5

Estimate a proportion 𝑌 =

𝑋

𝑛

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𝑃 𝑝1 < 𝑝 < 𝑝2 = 0.95 Let’s look for p with a 95% confidence interval

Easy solve!

𝑃 𝜇 − 1.96 ∗ 𝜎 ≤ 𝑋 ≤ 𝜇 + 1.96 ∗ 𝜎 = 0.95

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F 4

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52 Heads out of 100 toss…

𝑌~𝑁 0.52,0.04996

95% confidence interval

𝑝1 = 0.62

𝑌~𝑁 ? , ?

𝑝2 = 0.42

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Mean estimation

Problem

The SD of the actual population is unknown

Mean has a Student’s distribution

Similarity with normal

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Student’s properties

• It is symmetric about its mean • It has a mean of zero • It has a standard deviation and variance greater than 1. • There are actually many t distributions, one for each degree of freedom • As the sample size increases, the t distribution approaches the normal distribution. • It is bell shaped. • The t-scores can be negative or positive, but the probabilities are always positive.

Normal-ish distribution in a discrete environment with a confidence interval

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Student’s Statistic

S=𝑛

𝑛−1𝜎

𝑃 𝑥 −𝑆

𝑛∗ 𝑡𝛼/2 < 𝜇 < 𝑥 +

𝑆

𝑛∗ 𝑡𝛼/2 = 0.95

Degree of freedom

n-1

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IPO Premiums IPO1 / 12% IPO2 / 15% IPO3 / 13% IPO4 / 18% IPO5 / 20% IPO6 / 5%

SD: 𝜎=4.81%

DF: 𝐷𝐹=5

S: 𝑆=5.27%

t: 𝑡=2.571

𝜇1: 𝜇1=19.36%

𝑥 : 𝑥 =13.83%

𝜇2: 𝜇2=8.30%

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Is a frequency difference significant?

𝑌1~𝑁 𝑝1,𝑝1(1 − 𝑝1)

𝑛1 𝑌2~𝑁 𝑝2,

𝑝2(1 − 𝑝2)

𝑛2

𝑍 = 𝑌1 − 𝑌2

𝐸(𝑍) = 𝐸(𝑌1) − E(𝑌2)

𝑉(𝑍) = 𝑉(𝑌1) + V(𝑌2) Assumption of independence

𝑍~𝑁 𝑝1 − 𝑝2,𝑝1(1 − 𝑝1)

𝑛1+

𝑝2(1 − 𝑝2)

𝑛2

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Observations 100 Friendly Takeover, 80 success 60 Hostiles Takeover, 50 success

Is the difference significant? 95% confidence

Friendly 80%

Hostiles 83%

Global frequency

𝑝 =𝑛1𝐹1 + 𝑛2𝐹2

𝑛1 +𝑛2 𝑝 =

80 + 50

100 + 60= 81.25%

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𝑡∗ =𝐹1 − 𝐹2

𝑝 (1 − 𝑝 )1𝑛1

+1𝑛2

𝑡∗ = −0.52298

If 𝑃(−1.96 < 𝑡∗ < 1.96) = 0.95the frequencies are the same

with a 95% confidence interval

The frequencies are equal

Their difference is not significant

Actual difference due to fluctuation of samples

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Is a SD difference significant?

Fisher Snedecor distribution

𝑆𝑥 2

𝑆𝑦 2

𝜎𝑝 2

𝜎𝑞 2

Total variance

Total variance

Sample variance

Sample variance

𝑆𝑥 2

𝑆𝑦 2∗𝜎𝑝 2

𝜎𝑞 2~𝐹(𝑛𝑝 − 1, 𝑛𝑞 − 1)

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𝜎𝑝 2 = 𝜎𝑞 2 You want to test

𝑆𝑥 2

𝑆𝑦 2~𝐹(𝑛𝑝 − 1, 𝑛𝑞 − 1)

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𝑆𝑥 2

𝑆𝑦 2~𝐹(5,4)

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95% confidence interval F-Table

𝑆𝑥 2

𝑆𝑦 2< 6.26 If SD are equals (at 95% CI)

Games: Beat the Statistics

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Is Martingale safe?

Bet on 2:1, double when you lose…

Risk of ruin?

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Bet on 2:1

Is this really 2:1? 18

37= 0.4865

Obvious how casino is making money!

The probability of the casino to win is always bigger than the probability of the player to win!

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You’ll be right with a martingale… Eventually! But when?

The 2011 recorded record series is 26 reds in Las Vegas, Nevada

You were on the black and hoping the reversal, you begun with $2

At the 27 round you need

227 = $134,217,728

And don’t forget you lost already

21 + 22 + ⋯+ 226 = $134,217,726

Casino limit stakes

Your pocket may not be deep enough anyway!

And if you win at the 27th roll, you made…

$2 Quite risky…

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“No one can possibly win at roulette unless he steals money from the table while the

croupier isn’t looking.” — Albert Einstein

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Binomial approach

𝑃 𝑥 = 𝐶𝑥𝑛𝑝𝑥(1 − 𝑝)𝑛−𝑥

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$255, $1 flat bet

$255, $1 start, martingale double when you lose

Ruin in 255 times for flat bet

Ruin in 8 times for martingale

1,000,000 times comparison, 100 rounds maximum

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Conclusion

Multiple Regression

Econometrics

Estimations

Statistics & Games