Financial Econometric Models IV

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Information about Financial Econometric Models IV

Published on February 27, 2014

Author: vinzjeannnin



Fourth Session, MSc 5th Year

Q1 2012 ESGF 5IFM Q1 2012 Vincent JEANNIN – ESGF 5IFM Financial Econometric Models 1

Interim Exam Sum Up Reminder of Last Session Generic case AR, MA, ARMA & ARIMA Heteroscedasticity: Introduction ESGF 5IFM Q1 2012 • • • • Summary of the session (Est. 3h) 2

ESGF 4IFM Q1 2012 1 Interim Exam Sum-Up 3

When E is minimal? When partial derivatives i.r.w. a and b are 0 Attention, logarithms are not additive! Minimising residuals ESGF 5IFM Q1 2012 Two parameters to estimate: • Intercept α • Gradient β 4

Change the variable Z=ln(X) ESGF 5IFM Q1 2012 Solution? 5 ESGF 4IFM Q1 2012 Leads easily to the intercept 6

7 ESGF 5IFM Q1 2012 ESGF 5IFM Q1 2012 We have and Finally… 8

Z=ln(X) ESGF 5IFM Q1 2012 Don’t forget… 9

Accept or reject the regression? Hedging is linear… ESGF 5IFM Q1 2012 No forecast possible (one particular stock against the market) Check correlation and R Squared 10 Check the normality of residuals

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Ultimate decider is the normality test on the residuals ESGF 5IFM Q1 2012 For every dataset of the Quarter 12

Trend Fit Seasonality Forecast Residual Identify ESGF 5IFM Q1 2012 2 13

Lag 0, Auto Correlation is 1 Lag 1 ESGF 5IFM Q1 2012 ACF = Auto Correlation in the series Lag 2 14 Regression of the series against the same series retarded

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Marginal Auto Correlation ESGF 4IFM Q1 2012 PACF = Partial Auto Correlation in the series Conditional Auto Correlation knowing the Auto Correlation at a lower order 16 ESGF 5IFM Q1 2012 3 17

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20 AR(1) ESGF 5IFM Q1 2012

Exploitation Identify Auto Correlation Analysis Fit Estimate the parameters Forecast Reminders of the 3 steps ESGF 4IFM Q1 2012 Reminder of the last session 21

Trend Seasonality Residual ESGF 4IFM Q1 2012 Reminders of the 3 components 22

There is a correlation between current data and previous data Parameters of the model White noise Main principle ESGF 4IFM Q1 2012 AR AR(n) If the parameters are identified, the prediction will be easy 23

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PACF cancelling after order 1 ESGF 4IFM Q1 2012 ACF decreasing 27

Typically an Autoregressive Process PACF cancel after order 1 ESGF 4IFM Q1 2012 Decreasing ACF AR(1) 28 Modl<-ar(diff(DATA$Val),order.max=20) plot(Modl$aic) ESGF 4IFM Q1 2012 Let’s try to fit an AR(1) model 29 The likelihood for the order 1 is significant

> ar(diff(DATA$Val),order.max=20) Coefficients: 1 2 0.5925 -0.1669 sigma^2 estimated as 0.8514 Order selected 3 3 0.1385 ESGF 4IFM Q1 2012 Call: ar(x = diff(DATA$Val), order.max = 20) We know the first term of our series 30

Box-Pierce test data: Modl$resid X-squared = 7e-04, df = 1, p-value = 0.9789 Box.test(Modl$resid) ESGF 4IFM Q1 2012 Need to test the residuals H0 accepted, residuals are independently distributed (white noise) The differentiated series is a AR(1) 31

Stationary series with auto correlation of errors Parameters of the model White noise Main principle ESGF 4IFM Q1 2012 MA MA(n) More difficult to estimate than a AR(n) 32

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PACF decays to 0 ACF cancels after order 1 ESGF 4IFM Q1 2012 acf(Data,20) pacf(Data,20) 34 ACF & PACF suggest MA(1)

> arima(Data, order = c(0, 0, 1),include.mean = FALSE) sigma^2 estimated as 0.937: log likelihood = -138.76, > 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 aic = 281.52 Coefficients: ma1 -0.4621 s.e. 0.0903 ESGF 4IFM Q1 2012 Call: arima(x = Data, order = c(0, 0, 1), include.mean = FALSE) 35

The series is a function of past values plus current and past values of the noise ARMA(p,q) Combines AR(p) & MA(q) Main principle ESGF 4IFM Q1 2012 ARMA 36

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ESGF 4IFM Q1 2012 Both ACF and PACF decreases exponentially after order 1 38 ESGF 5IFM Q1 2012 Generic case AR, MA, ARMA & ARIMA 39

ARIMA(p,d,q), AutoRegressive Integrated Moving Average Non stationary… But can be removed with a differentiation of d ESGF 5IFM Q1 2012 Combines AR(p) & MA(q) 40

Typical ARIMA ESGF 5IFM Q1 2012 Non stationary 41

Identification easier ESGF 5IFM Q1 2012 Differentiation (d order) MA(2) 42

Original series is ARIMA(p,d,q) If the d differentiation is an ARMA(p,q) ESGF 5IFM Q1 2012 Integration of the initial differentiation 43

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 AR, MA, ARMA, ARIMA imply stationary series ESGF 5IFM Q1 2012 Heteroscedasticity: Introduction 44

GARCH(p,q) ARMA (p,q) with heteroscedasticity ESGF 5IFM Q1 2012 AR (q) with heteroscedasticity ARCH(q) 45 Variance is very rarely stable ESGF 5IFM Q1 2012 Useful for financial series 46

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