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Financial Econometric Models IV
 

Financial Econometric Models IV

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Fourth Session, MSc 5th Year

Fourth Session, MSc 5th Year

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  • As we have b, we can replace it in the equation of the regression line
  • Francis AnscombeWhich is the best linear fit?On what basis?

Financial Econometric Models IV Financial Econometric Models IV Presentation Transcript

  • Q1 2012 ESGF 5IFM Q1 2012 Vincent JEANNIN – ESGF 5IFM vinzjeannin@hotmail.com Financial Econometric Models 1
  • Interim Exam Sum Up Reminder of Last Session Generic case AR, MA, ARMA & ARIMA Heteroscedasticity: Introduction ESGF 5IFM Q1 2012 • • • • vinzjeannin@hotmail.com Summary of the session (Est. 3h) 2
  • ESGF 4IFM Q1 2012 1 vinzjeannin@hotmail.com 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! vinzjeannin@hotmail.com Minimising residuals ESGF 5IFM Q1 2012 Two parameters to estimate: • Intercept α • Gradient β 4
  • Change the variable Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Solution? 5
  • vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Leads easily to the intercept 6
  • 7 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
  • vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 We have and Finally… 8
  • Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Don’t forget… 9
  • Accept or reject the regression? vinzjeannin@hotmail.com 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
  • 11 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
  • Ultimate decider is the normality test on the residuals vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 For every dataset of the Quarter 12
  • Trend Fit Seasonality Forecast Residual vinzjeannin@hotmail.com Identify ESGF 5IFM Q1 2012 2 13
  • Lag 0, Auto Correlation is 1 Lag 1 ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com ACF = Auto Correlation in the series Lag 2 14 Regression of the series against the same series retarded
  • 15 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • Marginal Auto Correlation ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com PACF = Partial Auto Correlation in the series Conditional Auto Correlation knowing the Auto Correlation at a lower order 16
  • vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 3 17
  • 18 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
  • 19 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
  • 20 AR(1) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
  • Exploitation Identify Auto Correlation Analysis Fit Estimate the parameters Forecast vinzjeannin@hotmail.com Reminders of the 3 steps ESGF 4IFM Q1 2012 Reminder of the last session 21
  • Trend Seasonality Residual ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Reminders of the 3 components 22
  • There is a correlation between current data and previous data Parameters of the model White noise vinzjeannin@hotmail.com Main principle ESGF 4IFM Q1 2012 AR AR(n) If the parameters are identified, the prediction will be easy 23
  • 24 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • 25 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • 26 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • PACF cancelling after order 1 ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com ACF decreasing 27
  • Typically an Autoregressive Process vinzjeannin@hotmail.com PACF cancel after order 1 ESGF 4IFM Q1 2012 Decreasing ACF AR(1) 28
  • vinzjeannin@hotmail.com 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$Val),order.max=20) Coefficients: 1 2 0.5925 -0.1669 sigma^2 estimated as 0.8514 vinzjeannin@hotmail.com Order selected 3 3 0.1385 ESGF 4IFM Q1 2012 Call: ar(x = diff(DATA$Val), order.max = 20) We know the first term of our series 30
  • Box-Pierce test data: Modl$resid X-squared = 7e-04, df = 1, p-value = 0.9789 vinzjeannin@hotmail.com Box.test(Modl$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 vinzjeannin@hotmail.com Main principle ESGF 4IFM Q1 2012 MA MA(n) More difficult to estimate than a AR(n) 32
  • 33 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • PACF decays to 0 vinzjeannin@hotmail.com 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 vinzjeannin@hotmail.com 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) vinzjeannin@hotmail.com Main principle ESGF 4IFM Q1 2012 ARMA 36
  • 37 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
  • ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Both ACF and PACF decreases exponentially after order 1 38
  • vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Generic case AR, MA, ARMA & ARIMA 39
  • ARIMA(p,d,q), AutoRegressive Integrated Moving Average vinzjeannin@hotmail.com Non stationary… But can be removed with a differentiation of d ESGF 5IFM Q1 2012 Combines AR(p) & MA(q) 40
  • Typical ARIMA vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Non stationary 41
  • Identification easier vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Differentiation (d order) MA(2) 42
  • Original series is ARIMA(p,d,q) vinzjeannin@hotmail.com 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 vinzjeannin@hotmail.com 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 vinzjeannin@hotmail.com ARCH(q) 45
  • vinzjeannin@hotmail.com Variance is very rarely stable ESGF 5IFM Q1 2012 Useful for financial series 46