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Financial Econometric ModelsVincent JEANNIN – ESGF 5IFM

Q1 2012

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Summary of the session (Est. 3h)

• Interim Exam Sum Up• Reminder of Last Session• Generic case AR, MA, ARMA & ARIMA• Heteroscedasticity: Introduction

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Interim Exam Sum-Up

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𝑦=𝑎∗ ln (𝑥 )+𝑏+𝜀

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Two parameters to estimate:• Intercept α• Gradient β

Minimising residuals

𝐸=∑𝑖=1

𝑛

𝜀𝑖❑2=∑

𝑖=1

𝑛

(𝑦 𝑖− (𝑎∗ ln (𝑥𝑖)+𝑏))2

When E is minimal?

When partial derivatives i.r.w. a and b are 0

Attention, logarithms are not additive!

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ln (𝑎+𝑏 )≠ln (𝑎)+ ln (𝑏)

ln (∑ 𝑥𝑖 )≠𝑛 ln (𝑥 )Solution?

Change the variable Z=ln(X)

𝐸=∑𝑖=1

𝑛

𝜀𝑖❑2=∑

𝑖=1

𝑛

(𝑦 𝑖− (𝑎∗ 𝑧𝑖+𝑏))2

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𝑎∗∑𝑖=1

𝑛

𝑧 𝑖+𝑛𝑏=∑𝑖=1

𝑛

𝑦 𝑖

Leads easily to the intercept

𝑎𝑛𝑧+𝑛𝑏=𝑛𝑦

𝑎𝑧+𝑏=𝑦

𝜕𝐸𝜕𝑏

𝑏=𝑦−𝑎𝑧

( 𝑦 𝑖−𝑎𝑥 𝑖−𝑏 )2=𝑦 𝑖2−2𝑎𝑧 𝑖 𝑦 𝑖−2𝑏𝑦 𝑖+𝑎

2𝑧 𝑖2+2𝑎𝑏𝑧 𝑖+𝑏

2

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𝜕𝐸𝜕𝑎

=∑𝑖=1

𝑛

−2 𝑧𝑖 𝑦 𝑖+2𝑎𝑧𝑖❑2+2𝑏𝑧 𝑖=0

y=𝑎𝑧+ 𝑦−𝑎𝑧

y − 𝑦=𝑎(𝑧 −𝑧)

𝑏=𝑦−𝑎𝑧

∑𝑖=1

𝑛

𝑧 𝑖 (𝑦 𝑖−𝑎𝑧 𝑖❑−𝑏)=0

𝜕𝐸𝜕𝑏

=∑𝑖=1

𝑛

−2 𝑦 𝑖+2𝑏+2𝑎𝑧𝑖=0

∑𝑖=1

𝑛

𝑦 𝑖−𝑏−𝑎𝑧𝑖=0

∑𝑖=1

𝑛

𝑦 𝑖− 𝑦+𝑎𝑧−𝑎𝑧 𝑖=0

∑𝑖=1

𝑛

(𝑦 𝑖− 𝑦 )−𝑎(𝑧𝑖− 𝑧)=0

∑𝑖=1

𝑛

𝑧 𝑖 (𝑦 𝑖−𝑎𝑧 𝑖❑− 𝑦+𝑎𝑧 )=0

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0

∑𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

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∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0 ∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖−𝑦 )−𝑎 ( 𝑧𝑖−𝑧 ))

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))−∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

∑𝑖=1

𝑛

(𝑧¿¿ 𝑖−𝑧)(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0¿

𝑎=∑𝑖=1

𝑛

(𝑧¿¿ 𝑖− 𝑧)(𝑦 𝑖− 𝑦 )

∑𝑖=1

𝑛

(𝑧 ¿¿ 𝑖−𝑧)2¿¿

Finally…

We have

and

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𝑎=∑𝑖=1

𝑛

(𝑧¿¿ 𝑖− 𝑧)(𝑦 𝑖− 𝑦 )

∑𝑖=1

𝑛

(𝑧 ¿¿ 𝑖−𝑧)2¿¿ 𝑏=𝑦−𝑎𝑧

Don’t forget…

Z=ln(X)

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No forecast possible (one particular stock against the market)

Hedging is linear…

Accept or reject the regression?

Check correlation and R Squared

Check the normality of residuals

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12𝑦=7.5𝑥=9

𝑦=3+0.5 𝑥𝑟=0.82𝑅2=0.67

For every dataset of the Quarter

Ultimate decider is the normality test on the residuals

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2

Identify

Fit

Forecast

Trend

Seasonality

Residual

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ACF = Auto Correlation in the series

Lag 0, Auto Correlation is 1 𝜌 (𝑋 𝑡 ,𝑋 𝑡)

Lag 1 𝜌 (𝑋 𝑡 ,𝑋 𝑡− 1)

Lag 2 𝜌 (𝑋 𝑡 ,𝑋 𝑡− 2)

Regression of the series against the same series retarded

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PACF = Partial Auto Correlation in the series

Marginal Auto Correlation

Conditional Auto Correlation knowing the Auto Correlation at a lower order

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AR(1)

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Reminder of the last session

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Reminders of the 3 steps

Identify

Fit

Forecast

Exploitation

Auto Correlation Analysis

Estimate the parameters

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Reminders of the 3 components

Trend

Seasonality

Residual

∆ 𝑥𝑡=𝑥𝑡−𝑥𝑡 −1

∆30𝑥𝑡=𝑥𝑡−𝑥𝑡 −30

𝑁 (0.𝜎 )

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There is a correlation between current data and previous data

Main principle

𝑋 𝑡=𝑐+𝜑1𝑋 𝑡 −1+𝜑2𝑋 𝑡 −2+…+𝜑𝑛 𝑋 𝑡−𝑛+𝜀𝑡

𝜑𝑛Parameters of the model

𝜀𝑛White noise

If the parameters are identified, the prediction will be easy

AR(n)

AR

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ACF decreasing

PACF cancelling after order 1

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Decreasing ACF

PACF cancel after order 1

Typically an Autoregressive Process

AR(1)

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Modl<-ar(diff(DATA$Val),order.max=20)plot(Modl$aic)

Let’s try to fit an AR(1) model

The likelihood for the order 1 is significant

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> ar(diff(DATA$Val),order.max=20)

Call:ar(x = diff(DATA$Val), order.max = 20)

Coefficients: 1 2 3 0.5925 -0.1669 0.1385

Order selected 3 sigma^2 estimated as 0.8514

We know the first term of our series

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Need to test the residuals

Box.test(Modl$resid)

Box-Pierce test

data: Modl$resid X-squared = 7e-04, df = 1, p-value = 0.9789

H0 accepted, residuals are independently distributed (white noise)

The differentiated series is a AR(1)

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Stationary series with auto correlation of errors

Main principle

𝑋 𝑡=𝜇+𝑍𝑡+𝜑1𝑍 𝑡−1+𝜑2𝑍 𝑡− 2+…+𝜑𝑛𝑍 𝑡−𝑛

𝜑𝑛Parameters of the model

𝑍𝑛White noise

More difficult to estimate than a AR(n)

MA(n)

MA

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acf(Data,20)pacf(Data,20)

ACF & PACF suggest MA(1)

ACF cancels after order 1

PACF decays to 0

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> Box.test(Rslt$residuals)

Box-Pierce test

data: Rslt$residuals X-squared = 0, df = 1, p-value = 0.9967

It works, MA(1), 0 mean, parameter -0.4621

> arima(Data, order = c(0, 0, 1),include.mean = FALSE)

Call:arima(x = Data, order = c(0, 0, 1), include.mean = FALSE)

Coefficients: ma1 -0.4621s.e. 0.0903

sigma^2 estimated as 0.937: log likelihood = -138.76, aic = 281.52

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The series is a function of past values plus current and past values of the noise

Main principle

ARMA(p,q)

Combines AR(p) & MA(q)

ARMA

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Both ACF and PACF decreases exponentially after order 1

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Generic case AR, MA, ARMA & ARIMA

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ARIMA(p,d,q), AutoRegressive Integrated Moving Average

Combines AR(p) & MA(q)

Non stationary… But can be removed with a differentiation of d

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Non stationary

Typical ARIMA

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Differentiation (d order)

Identification easier

MA(2)

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Integration of the initial differentiation

If the d differentiation is an ARMA(p,q)

Original series is ARIMA(p,d,q)

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Heteroscedasticity: Introduction

AR, MA, ARMA, ARIMA imply stationary series

When there is hetoroscedasticity, not applicable

Conditional heteroscedasticity is the answer

It assumes the current variance of residuals to be a function of the actual sizes of the previous time periods' residuals

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ARCH(q)

AR (q) with heteroscedasticity

GARCH(p,q)

ARMA (p,q) with heteroscedasticity

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Useful for financial series

Variance is very rarely stable