Financial Econometric Models IV

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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

    1. 1. Q1 2012 ESGF 5IFM Q1 2012 Vincent JEANNIN – ESGF 5IFM vinzjeannin@hotmail.com Financial Econometric Models 1
    2. 2. 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
    3. 3. ESGF 4IFM Q1 2012 1 vinzjeannin@hotmail.com Interim Exam Sum-Up 3
    4. 4. 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
    5. 5. Change the variable Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Solution? 5
    6. 6. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Leads easily to the intercept 6
    7. 7. 7 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    8. 8. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 We have and Finally… 8
    9. 9. Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Don’t forget… 9
    10. 10. 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. 11. 11 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    12. 12. Ultimate decider is the normality test on the residuals vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 For every dataset of the Quarter 12
    13. 13. Trend Fit Seasonality Forecast Residual vinzjeannin@hotmail.com Identify ESGF 5IFM Q1 2012 2 13
    14. 14. 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. 15. 15 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    16. 16. 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
    17. 17. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 3 17
    18. 18. 18 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    19. 19. 19 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    20. 20. 20 AR(1) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    21. 21. 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
    22. 22. Trend Seasonality Residual ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Reminders of the 3 components 22
    23. 23. 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. 24. 24 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    25. 25. 25 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    26. 26. 26 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    27. 27. PACF cancelling after order 1 ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com ACF decreasing 27
    28. 28. Typically an Autoregressive Process vinzjeannin@hotmail.com PACF cancel after order 1 ESGF 4IFM Q1 2012 Decreasing ACF AR(1) 28
    29. 29. 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
    30. 30. > 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
    31. 31. 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
    32. 32. 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. 33. 33 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    34. 34. 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)
    35. 35. > 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
    36. 36. 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. 37. 37 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
    38. 38. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Both ACF and PACF decreases exponentially after order 1 38
    39. 39. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Generic case AR, MA, ARMA & ARIMA 39
    40. 40. 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
    41. 41. Typical ARIMA vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Non stationary 41
    42. 42. Identification easier vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Differentiation (d order) MA(2) 42
    43. 43. 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
    44. 44. 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
    45. 45. GARCH(p,q) ARMA (p,q) with heteroscedasticity ESGF 5IFM Q1 2012 AR (q) with heteroscedasticity vinzjeannin@hotmail.com ARCH(q) 45
    46. 46. vinzjeannin@hotmail.com Variance is very rarely stable ESGF 5IFM Q1 2012 Useful for financial series 46

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