ESGF 5IFM Q1 2012Financial Econometric Models  Vincent JEANNIN – ESGF 5IFM            Q1 2012                             ...
ESGF 5IFM Q1 2012Summary of the session (est 3h)• Introduction & Objectives• Bibliography• OLS & Exploration              ...
Introduction & Objectives      • What is a model?  =  +    with  being a white noise                                      ...
Bibliography    vinzjeannin@hotmail.com   ESGF 5IFM Q1 20124
OLS  Exploration         OLS: Ordinary Least Square                                                                       ...
Two parameters to estimate:   • Intercept α   • Slope β                                                                   ...
=             2 =              −  +                2   =           −  −       2         =1              =1                ...
Leads easily to the intercept                                                            ∗           +  =                 ...
=  −                y =  +  −                                       y −  = ( −  )                                         ...
We have                                                                    ( −  −   −  ) = 0       and                (  −...
Covariance       =1( −  )( −   ) =                       2            =1( −  )                    Variance                ...
Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2}                                               ...
Let’s asses the quality of the regressionLet’s calculate the correlation coefficient (aka Pearson Product-MomentCorrelatio...
Poor quality                    R=0.62                                                                            R=0.96  ...
What is good quality?                                                                           ESGF 5IFM Q1 2012      Sli...
The regression itself introduces a bias                  Let’s introduce the coefficient of determination R-Squared       ...
In a simple linear regression with intercept 2 =  2                                                                       ...
vinzjeannin@hotmail.com   ESGF 5IFM Q1 201218
Don’t get fooled by numbers!                                                                   ESGF 5IFM Q1 2012    For ev...
ESGF 5IFM Q1 2012Is any linear regression useless?                                                                        ...
First application on financial market     SP / AmEx in 2011                                        ESGF 5IFM Q1 2012      ...
,                          =                      = 0.8501                                                              2 ...
How to use this?                                                                          ESGF 5IFM Q1 2012     • Forecast...
Hedging $1.0M of AmEx Stocks with $1.1046M of SP                                                        ESGF 5IFM Q1 2012 ...
Let’s have a similar approach using a proper statistics and econometrics software                                         ...
hist(Val$AMEX, breaks=20, main=Distribution                               AMEX Returns)                                sd(...
These are obvious negatively skewed distributions                                                                         ...
These are obvious leptokurtic distributions                                                                               ...
Quick check: what are the Skewness and Kurtosis of {1,2,-3,0,-2,1,1}?                                                     ...
ESGF 4IFM Q1 2012                                        vinzjeannin@hotmail.comExcel function KURTR function kurtosis (pa...
By the way, what is the most platykurtic distribution in the nature?                                                    To...
50.01777% rate of success: fair or not fair? Trick coin ?        Can be tested later with a Bayesian approach             ...
Back to our series, a good tool is the BoxPlot                                                                          ES...
Leptokurtic distributionsNegatively skewed distribution                                                               ESGF...
x=seq(-0.2,0.2,length=200)                                     y1=dnorm(x,mean=mean(Val$AMEX),sd=sd(                      ...
ESGF 5IFM Q1 2012                                                Excess kurtosis obvious                                  ...
qqnorm(Val$AMEX)                             qqnorm(Val$SPX) qqline(Val$AMEX)                             qqline(Val$SPX) ...
Can use many tests…•   Kolmogorov-Smirnov•   Jarque Bera•   Chi Square•                                                   ...
ESGF 5IFM Q1 2012x=seq(-4,4,length=1000)plot(ecdf(Val$AMEX),do.points=FALSE, col=red, lwd=3,main=Normal Distribution again...
ks.test(Val$SPX, pnorm)                       ks.test(Val$AMEX, pnorm)        One-sample Kolmogorov-                      ...
vinzjeannin@hotmail.com                                                        1.36        Sample size: 251               ...
Ok, we now know a bit more the 2 series we want to regress                      lm(Val$AMEX~Val$SPX)                     C...
The next important step is no analyse the residuals   Reg-lm(Val$AMEX~Val$SPX)   summary(Reg)                             ...
plot(Reg)                                                   layout(matrix(1:4,2,2))     vinzjeannin@hotmail.com   ESGF 5IF...
QQ Plot compares the CDF                                                            ESGF 5IFM Q1 2012A perfect fit is a li...
ESGF 5IFM Q1 2012                                                                            vinzjeannin@hotmail.comResidu...
ESGF 5IFM Q1 2012Nothing suggesting a white noise                                                                         ...
Showing now leverage                        Marginal importance of a point in the regression                              ...
So do we accept the regression?                 Probably not… But let’s check…                 Kolmogorov-Smirnov on resid...
Conclusion                             ESGF 5IFM Q1 2012        OLS        Residuals                             vinzjeann...
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Financial Econometric Models I

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Financial Econometric Models, course I, Busines School MSc level

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Financial Econometric Models I

  1. 1. ESGF 5IFM Q1 2012Financial Econometric Models Vincent JEANNIN – ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 1
  2. 2. ESGF 5IFM Q1 2012Summary of the session (est 3h)• Introduction & Objectives• Bibliography• OLS & Exploration vinzjeannin@hotmail.com 2
  3. 3. Introduction & Objectives • What is a model? = + with being a white noise ESGF 5IFM Q1 2012 • What the point writing models? Describe data behaviour vinzjeannin@hotmail.com Modelise data behaviour Forecast data behaviour• Acquire theory knowledge on Econometrics Statistics• Step by step from OLS to ANOVA on residuals• Usage of R and Excel 3
  4. 4. Bibliography vinzjeannin@hotmail.com ESGF 5IFM Q1 20124
  5. 5. OLS Exploration OLS: Ordinary Least Square ESGF 5IFM Q1 2012 Linear regression model Minimize the sum of the square vertical distances between the observations and the linear approximation vinzjeannin@hotmail.com = = + Residual ε 5
  6. 6. Two parameters to estimate: • Intercept α • Slope β ESGF 5IFM Q1 2012Minimising residuals = 2 = − + 2 vinzjeannin@hotmail.com =1 =1 When E is minimal? When partial derivatives i.r.w. a and b are 0 6
  7. 7. = 2 = − + 2 = − − 2 =1 =1 =1 Quick high school reminder if necessary… ESGF 5IFM Q1 2012 − − 2 = 2 − 2 − 2 + 2 2 + 2 + 2 vinzjeannin@hotmail.com = −2 + 2 2 + 2 = 0 = −2 + 2 + 2 = 0 =1 =1 − + 2 + = 0 − + + = 0=1 =1 ∗ 2 + ∗ = ∗ + = =1 =1 =1 =1 =1 7
  8. 8. Leads easily to the intercept ∗ + = =1 =1 ESGF 5IFM Q1 2012 + = + = vinzjeannin@hotmail.com = − The regression line is going through ( , ) The distance of this point to the line is 0 indeed 8
  9. 9. = − y = + − y − = ( − ) ESGF 5IFM Q1 2012 = −2 + 2 2 + 2 = 0 = −2 + 2 + 2 = 0 =1 =1 vinzjeannin@hotmail.com − − = 0 − − = 0 =1 =1 − − + = 0 =1 − + − = 0 =1 ( − − − ) = 0 ( − ) − ( − ) = 0 =1 =1 9 ( − − − ) = 0 =1
  10. 10. We have ( − − − ) = 0 and ( − − − ) = 0=1 =1 ESGF 5IFM Q1 2012 ( − − − ) = ( − − − ) =1 =1 vinzjeannin@hotmail.com ( − − − ) − − − − =0 =1 =1 ( − )( − − − ) = 0 =1 Finally… =1( − )( − ) 10 = 2 =1( − )
  11. 11. Covariance =1( − )( − ) = 2 =1( − ) Variance ESGF 5IFM Q1 2012 = 2 vinzjeannin@hotmail.com = − You can use Excel function INTERCEPT and SLOPE 11
  12. 12. Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2} ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 12 You can use Excel function VAR.P, COVARIANCE.P and STDEV.P
  13. 13. Let’s asses the quality of the regressionLet’s calculate the correlation coefficient (aka Pearson Product-MomentCorrelation Coefficient – PPMCC): ESGF 5IFM Q1 2012 = Value between -1 and 1 = 1 vinzjeannin@hotmail.com Perfect dependence ~0 No dependence Give an idea of the dispersion of the scatterplot 13 You can use Excel function CORREL
  14. 14. Poor quality R=0.62 R=0.96 High quality vinzjeannin@hotmail.com ESGF 5IFM Q1 201214
  15. 15. What is good quality? ESGF 5IFM Q1 2012 Slightly discretionary… vinzjeannin@hotmail.comIf 3 ≥ = 0.8666 … 2 It’s largely admitted as the threshold for acceptable / poor 15
  16. 16. The regression itself introduces a bias Let’s introduce the coefficient of determination R-Squared ESGF 5IFM Q1 2012Total Dispersion = Dispersion Regression + Dispersion Residual vinzjeannin@hotmail.com 2 2 2 − = − + − Dispersion Regression 2 = Total Dispersion In other words the part of the total dispersion explained by the regression 16 You can use Excel function RSQ
  17. 17. In a simple linear regression with intercept 2 = 2 ESGF 5IFM Q1 2012Is a good correlation coefficient and a good coefficient ofdetermination enough to accept the regression? vinzjeannin@hotmail.com Not necessarily! Residuals need to have no effect, in other word to be a white noise! 17
  18. 18. vinzjeannin@hotmail.com ESGF 5IFM Q1 201218
  19. 19. Don’t get fooled by numbers! ESGF 5IFM Q1 2012 For every dataset of the Quarter = 9 = 7.5 vinzjeannin@hotmail.com = 3 + 0.5 = 0.82 2 = 0.67 Can you say at this stage which regression is the best? 19Certainly not those on the right you need a LINEAR dependence
  20. 20. ESGF 5IFM Q1 2012Is any linear regression useless? vinzjeannin@hotmail.com Think what you could do to the series Polynomial transformation, log transformation,… 20 Else, non linear regressions, but it’s another story
  21. 21. First application on financial market SP / AmEx in 2011 ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 21
  22. 22. , = = 0.8501 2 = 2 = 0.7227 ESGF 5IFM Q1 2012 Oups :-o Is Excel wrong? vinzjeannin@hotmail.com R-Squared has different calculation methodsLet’s accept the following regression then as the quality seems pretty good = 0.06% + 1.1046 ∗ 22
  23. 23. How to use this? ESGF 5IFM Q1 2012 • Forecasting? Not really… Both are random variables vinzjeannin@hotmail.com • Hedging? Yes but basis risk Yes but careful to the residuals… In theory, what is the daily result of the hedge? Let’s have a try! 23
  24. 24. Hedging $1.0M of AmEx Stocks with $1.1046M of SP ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com It would have been too easy… Great differences… Why? Sensitivity to the size of the sample 24 Heteroscedasticity
  25. 25. Let’s have a similar approach using a proper statistics and econometrics software ESGF 5IFM Q1 2012 • Free • Open Source • Developments shared by developers vinzjeannin@hotmail.com Let’s begin with statistical exploration to get familiar with the series and the software Val-read.csv(file=C:/Users/Vinz/Desktop/Val.csv,head=TRUE,sep=,) summary(Val) SPX AMEX Min. :-0.0666344 Min. :-0.0883287 1st Qu.:-0.0069082 1st Qu.:-0.0094580 Median : 0.0010016 Median : 0.0013007 25 Mean : 0.0001249 Mean : 0.0005891 3rd Qu.: 0.0075235 3rd Qu.: 0.0102923 Max. : 0.0474068 Max. : 0.0710967
  26. 26. hist(Val$AMEX, breaks=20, main=Distribution AMEX Returns) sd(Val$AMEX) [1] 0.01915489 ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com hist(Val$SPX, breaks=20, main=DistributionSPXX Returns) sd(Val$SPX)[1] 0.01468776 26
  27. 27. These are obvious negatively skewed distributions ESGF 5IFM Q1 2012 Reminders 3 − − 3 = = − 2 3/2 vinzjeannin@hotmail.com• Negative skew: long left tail, mass on the right, skew to the left• Positive skew: long right tail, mass on the left, skew to the right skewness(Val$AMEX) [1] -0.2453693 skewness(Val$SPX) 27 [1] -0.4178701
  28. 28. These are obvious leptokurtic distributions ESGF 5IFM Q1 2012 Reminders 4 − − 4 = = − 2 2 vinzjeannin@hotmail.com library(moments) kurtosis(Val$AMEX) What is their K?[1] 5.770583 (excess kurtosis) kurtosis(Val$SPX)[1] 5.671254 28 Subtract 3 to make it relative to the normal distribution…
  29. 29. Quick check: what are the Skewness and Kurtosis of {1,2,-3,0,-2,1,1}? ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Excel function SKEW R function skewness (package moments) 29
  30. 30. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.comExcel function KURTR function kurtosis (package moments) 30
  31. 31. By the way, what is the most platykurtic distribution in the nature? Toss it! ESGF 4IFM Q1 2012 Head = Success = 1 / Tail = Failure = 0 vinzjeannin@hotmail.com require(moments) library(moments) toss-rbinom(10000000,1,0.5) mean(toss)[1] 0.5001777 kurtosis(toss)[1] 1.000001 kurtosis(toss)-3[1] -1.999999 hist(toss, breaks=10,main=Tossing acoin 10 millions times,xlab=Resultof the trial,ylab=Occurence) 31 sum(toss)[1] 5001777
  32. 32. 50.01777% rate of success: fair or not fair? Trick coin ? Can be tested later with a Bayesian approach ESGF 4IFM Q1 2012On a perfect 50/50, Kurtosis would be 1, Excess Kurtosis -2: the minimum!This is a Bernoulli trial (, ) with 1 and 0 1 ∈ ℝ and integer vinzjeannin@hotmail.com Mean SD (1 − ) Skewness 1 − 2 (1 − ) Kurtosis 1 −3 (1 − ) 32 Easy to demonstrate if p=0.5 the Kurtosis will be the lowest Bit more complicated to demonstrate it for any distribution
  33. 33. Back to our series, a good tool is the BoxPlot ESGF 5IFM Q1 2012TooManyOutliers! vinzjeannin@hotmail.comThere should be 2 maxTo be normalFatter tails than thenormal distribution 33 boxplot(Val$AMEX,Val$SPX, main=AMEX SP BoxPlots, names=c(AMEX,SPX),col=blue)
  34. 34. Leptokurtic distributionsNegatively skewed distribution ESGF 5IFM Q1 2012 Are they normal distributions? vinzjeannin@hotmail.com Let’s compare them to normal distributions with same standard deviation and mean and make the QQ Plots 34
  35. 35. x=seq(-0.2,0.2,length=200) y1=dnorm(x,mean=mean(Val$AMEX),sd=sd( Val$AMEX)) hist(Val$AMEX, breaks=100,main=AmEx Returns / Normal ESGF 5IFM Q1 2012 Distribution,xlab=Return,ylab=Occ urence) lines(x,y1,type=l,lwd=3,col=red) vinzjeannin@hotmail.comx=seq(-0.2,0.2,length=200)y1=dnorm(x,mean=mean(Val$SPX),sd=sd(Val$SPX))hist(Val$SPX, breaks=20,main=SP Returns/ NormalDistribution,xlab=Return,ylab=Occurence)lines(x,y1,type=l,lwd=3,col=red) 35
  36. 36. ESGF 5IFM Q1 2012 Excess kurtosis obvious vinzjeannin@hotmail.comFatter and longer tails 36Let’s have a look to their CDF through QQPlot
  37. 37. qqnorm(Val$AMEX) qqnorm(Val$SPX) qqline(Val$AMEX) qqline(Val$SPX) ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com Fatter tails 37 Let’s properly test the normality
  38. 38. Can use many tests…• Kolmogorov-Smirnov• Jarque Bera• Chi Square• ESGF 5IFM Q1 2012 Shapiro WilkLet’s try Kolmogorov-Smirnov It compares the distance between the empirical vinzjeannin@hotmail.com CDF and the CFD of the reference distribution 38
  39. 39. ESGF 5IFM Q1 2012x=seq(-4,4,length=1000)plot(ecdf(Val$AMEX),do.points=FALSE, col=red, lwd=3,main=Normal Distribution against AMEX - CFDs, xlab=x,ylab=P(X=x))lines(x,pnorm(x,mean=mean(Val$AMEX),sd=sd(Val$AMEX)),col=blue,type=l,lwd=3) vinzjeannin@hotmail.comx=seq(-4,4,length=1000)plot(ecdf(Val$SPX),do.points=FALSE, col=red, lwd=3,main=Normal Distribution against SP - CFDs, xlab=x,ylab=P(X=x))lines(x,pnorm(x,mean=mean(Val$SPX),sd=sd(Val$SPX)),col=blue,type=l,lwd=3) 39
  40. 40. ks.test(Val$SPX, pnorm) ks.test(Val$AMEX, pnorm) One-sample Kolmogorov- One-sample Kolmogorov-SmirnovSmirnov test testdata: Val$SPX data: Val$AMEXD = 0.4811, p-value 2.2e-16 D = 0.4742, p-value 2.2e-16alternative hypothesis: two-sided alternative hypothesis: two-sided ESGF 5IFM Q1 2012 The 0 hypothesis is the distribution is normal vinzjeannin@hotmail.com Do we accept or reject the hypothesis 0 with a 95% confidence interval? The hypothesis regarding the distributional form is rejected if the test statistic, D, is greater than the critical value obtained from a table 40
  41. 41. vinzjeannin@hotmail.com 1.36 Sample size: 251 = 0.086 251 Rejected or not? 41 P-Value was givingRejected! Series aren’t fitting a normal distribution the answer
  42. 42. Ok, we now know a bit more the 2 series we want to regress lm(Val$AMEX~Val$SPX) Call: lm(formula = Val$AMEX ~ Val$SPX) ESGF 5IFM Q1 2012 Coefficients: (Intercept) Val$SPX 0.0004505 1.1096287plot(Val$SPX,Val$AMEX, main=SP / AmEx, xlab=SP, ylab=AmEx,col=red) vinzjeannin@hotmail.comabline(lm(Val$AMEX~Val$SPX), col=blue) = 110.96% ∗ + 0.045% 42
  43. 43. The next important step is no analyse the residuals Reg-lm(Val$AMEX~Val$SPX) summary(Reg) ESGF 5IFM Q1 2012 Call: lm(formula = Val$AMEX ~ Val$SPX) Residuals: Min 1Q Median 3Q Max -0.030387 -0.006072 -0.000114 0.006624 0.027824 vinzjeannin@hotmail.com Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) 0.0004505 0.0006365 0.708 0.48 Val$SPX 1.1096287 0.0434231 25.554 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.01008 on 249 degrees of freedom Multiple R-squared: 0.7239, Adjusted R-squared: 0.7228 F-statistic: 653 on 1 and 249 DF, p-value: 2.2e-16 43They need to be a white noise, you can have a first assessment with quartiles
  44. 44. plot(Reg) layout(matrix(1:4,2,2)) vinzjeannin@hotmail.com ESGF 5IFM Q1 201244
  45. 45. QQ Plot compares the CDF ESGF 5IFM Q1 2012A perfect fit is a line vinzjeannin@hotmail.com Left tail noticeably different 45
  46. 46. ESGF 5IFM Q1 2012 vinzjeannin@hotmail.comResiduals should be randomly distributed around the 0 horizontal lineYou don’t want to see a trend, a dependenceTo accept or reject the regression you need residuals to be a white noise 46 Their mean should be 0
  47. 47. ESGF 5IFM Q1 2012Nothing suggesting a white noise vinzjeannin@hotmail.com • Square root of the standardized residuals as a function of the fitted values • There should be no obvious trend in this plot 47
  48. 48. Showing now leverage Marginal importance of a point in the regression ESGF 5IFM Q1 2012 vinzjeannin@hotmail.comFar points suggest outlier or poor model 48
  49. 49. So do we accept the regression? Probably not… But let’s check… Kolmogorov-Smirnov on residuals ESGF 5IFM Q1 2012 1.36 Higher bound value for the = = 0.086 251 H0 to be accepted vinzjeannin@hotmail.com Resid-resid(Reg) ks.test(Resid, pnorm) One-sample Kolmogorov-Smirnov test data: Resid D = 0.4889, p-value 2.2e-16 alternative hypothesis: two-sidedRejected! Regression between 2 different asset are very often poor 49 Heteroscedasticity Basis risk if you hedge anyway
  50. 50. Conclusion ESGF 5IFM Q1 2012 OLS Residuals vinzjeannin@hotmail.com Normality Heteroscedasticity 50
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