Examples. David Ruppert. April 25, Cornell University. Statistics for Financial Engineering: Some R. Examples. David Ruppert.

Transcription

1 Cornell University April 25, 2009

2 Outline

3 A little about myself BA and MA in mathematics PhD in statistics in 1977 taught in the statistics department at North Carolina for 10 years have been in Operations Research and Information (formerly Industrial) Engineering at Cornell since 1987

4 A little about myself starting teaching Statistics and Finance to undergraduates in 2001 textbook published in 2004 starting teaching Engineering to master s students in 2008 working on revised and expanded textbook now programming exclusively in R

5 Undergraduate Textbook

6 A little about my research have done research in asymptotic theory of splines semiparametric ing measurement error in regression smoothing (nonparametric regression and density estimation) transformation and weighting stochastic approximation biostatistics environmental engineering ing of term structure executive compensation and accounting fraud

7 Three types of regression Linear regression Y i = β 0 + β 1 X i,1 + + β p X i,p + ɛ i, i = 1,..., n regression Y i = m(x i,1,..., X i,p ; β 1,..., β q ) + ɛ i, i = 1,..., n where m is a known function depending on unknown parameters Nonparametric regression Y i = m(x i,1,..., X i,p ) + ɛ i, i = 1,..., n where m is an unknown smooth function

8 Usual assumptions on the noise Usually ɛ 1,..., ɛ n are assumed to be: mutually independent (or at least uncorrelated) homoscedastic (constant variance) normally distributed Much research over the last 50+ years has looked into ways of 1 checking these assumptions 2 statistical methods that require less assumptions

9 Transform-both-sides Ideal (no errors): Y i = f (X i, β) Statistical (first attempt): Y i = f (X i, β) + ɛ i where ɛ 1,..., ɛ n are iid Gaussian TBS : h{y i } = h{f (X i, β)} + ɛ i where ɛ 1,..., ɛ n are iid Gaussian h is an appropriate transformation

10 Estimation of Default Probabilities : ratings: 1=Aaa (best),...,16=b3 (worse) default frequency (estimate of default probability)

11 Some statistical s nonlinear : Pr(default rating) = exp{β 0 + β 1 rating} linear/transformation (in recent textbook): log{pr(default rating)} = β 0 + β 1 rating Problem: cannot take logs of default frequencies that are 0 (Sub-optimal) solution in textbook: throw out these observations

12 A better statistical Transform-both-sides (TBS) see Carroll and Ruppert (1984, 1988): using a power transformation: { } λ { Pr(default rating) + κ = exp(β0 + β 1 rating) + κ } λ λ chosen by residual plots (or maximum likelihood) λ = 1/2 works well for this example log transformations are also commonly used κ > 0 will shift data away from 0

13 The Box-Cox family the most common transformation family is due to Box and Cox (1964): derivative has simple form: h(y, λ) = yλ 1 if λ 0 λ = log(y) if λ = 0 h y (y, λ) = d dy h(y, λ) = yλ 1 for all λ

14 TBS fit compared to others log(default probability) o BOW nonlinear tbs data rating

15 regression residuals absolute residual Theoretical Quantiles Normal QQ plot fitted values Sample Quantiles

16 TBS residuals absolute residual Theoretical Quantiles Normal QQ plot fitted values Sample Quantiles

17 Estimated default probabilities method Pr{default Aaa} as % of TEXTBOOK est TEXTBOOK 0.005% 100% nonlinear 0.002% 40% TBS % 16%

18 A Similar Problem: Challenger # o rings distressed * ** * * * *** * ** *** ** * temp

19 Challenger : Extrapolation to 31 o Logistic regression # o rings distressed all data only data with failures * * * * * * ** temp

20 Variance stabilizing transformation: how it works y Pop 2 transformed Pop 1 transformed log(x) Pop 1 Pop

21 Strength of Box-Cox family Take a < b Then h y (b, λ) h y (a, λ) = ( ) b λ 1 a which is increasing in λ and equals 1 when λ = 1 λ = 1 is the dividing point between concave and convex transformations h(y, λ) becomes a stronger concave transformation as λ decreases from 1 also, h(y, λ) becomes a stronger convex transformation as λ increases from 1

22 Strength of Box-Cox family, cont. Example: b/a = 2 Derivative ratio α 1 concave convex α

23 Maximum likelihood L(β, λ, σ) = n log(σ) + n [h(y i + κ, λ) h { f (X i, β) + κ, λ }] 2 i=1 n (λ 1) log(y i ) i=1 } {{ } from Jacobian 2σ 2 can maximize over σ analytically: σ 2 = n 1 n i=1 [h(y i + κ, λ) h { f (X i, β) + κ, λ }] 2 they maximize over (β, λ) with optim, for example κ is fixed in advance

24 Reference for TBS Transformation and Weighting in by Carroll and Ruppert (1988) Lots of examples But none in finance _

25 1-Year Treasury Constant Maturity Rate, daily data rate year Source: Board of Governors of the Federal Reserve System

26 R t versus year change in rate year

27 R t versus R t 1 change in rate lagged rate

28 R 2 t versus R t 1 change in rate lagged rate

29 Drift function Discretized diffusion : R t = µ(r t 1 ) + σ(r t 1 )ɛ t µ(x) is the drift function σ(x) is the volatility function (as before)

30 Estimating Volatility Parametric : (Common in practice) Nonparametric : Var{( R t )} = β 0 R β 1 t 1 Var{( R t )} = σ 2 (R t 1 ) where σ( ) is a smooth function will be ed as a spline In these s: no dependence on t

31 Spline Software The penalized spline fits shown here were obtained using the function spm in R s SemiPar package author is Matt Wand

32 Comparing parametric and nonparametric volatility fits diff_rate^ nonpar par lag_rate lag_rate

33 Comparing parametric and nonparametric volatility fits: zooming in near lag_rate nonpar par

34 Spline fitting Estimation of drift function diff_rate lag_rate lag_rate 1st der lag_rate 2nd der lag_rate

35 Residuals for diffusion residual t := R t µ(r t 1 ) E(residual t ) = 0 std residual t := residual t σ(r t 1 ) E(std residual 2 t ) = 1

36 Question Are the drift and volatility functions constant in time?

37 Residual plots: ordinary residuals residual year

38 Residual plots: standardized residuals Lag ACF autocorrelation function Normal Q Q Plot Sample Quantiles Theoretical Quantiles

39 Residual plots: Squared standardized residuals Squared Std residual year

40 Residual plots: Squared standardized residuals autocorrelation function ACF Lag

41 GARCH(p, q) The GARCH(p, q) is a t = ɛ t σ t, where q p σ t = α0 + α i at i 2 + β i σt i 2. i=1 i=1 and ɛ t is an iid (strong) white noise process a t is weak white noise uncorrelated but with volatility clustering

42 GARCH(1,1) fit using garch in tseries Call: garch(x = std_drift_resid^2, order = c(1, 1)) Model: GARCH(1,1) Coefficient(s): Estimate Std. Error t value Pr(> t ) a <2e-16 *** a <2e-16 *** b <2e-16 *** Box-Ljung test data: Squared.Residuals X-squared = 0.13, df = 1, p-value =

43 GARCH: estimated conditional standard deviations GARCH conditional std year

44 GARCH: squared residuals with lowess smooth year GARCH squared residuals

45 GARCH residuals GARCH residuals ACF Lag

46 AR(1)/GARCH(1,1) Call: garchfit(formula = ~arma(1, 0) + garch(1, 1), data = std_drift_resid) Mean and Variance Equation: data ~ arma(1, 0) + garch(1, 1) [data = std_drift_resid] Conditional Distribution: norm Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(> t ) mu ar < 2e-16 *** omega e-13 *** alpha < 2e-16 *** beta < 2e-16 ***

47 AR(1)/GARCH(1,1) residuals AR(1)/GARCH(1,1) residuals ACF Lag

48 AR(1)/GARCH(1,1) residuals - QQ plot Normal Q Q Plot Sample Quantiles Theoretical Quantiles

49 AR(1)/GARCH(1,1) Call: garchfit(formula = ~arma(1, 0) + garch(1, 1), data = std_drift_resid, cond.dist = "std") Mean and Variance Equation: data ~ arma(1, 0) + garch(1, 1) [data = std_drift_resid] Conditional Distribution: std Error Analysis: Estimate Std. Error t value Pr(> t ) mu ar < 2e-16 *** omega ** alpha < 2e-16 *** beta < 2e-16 *** shape < 2e-16 ***

50 AR(1)/GARCH(1,1) residuals - QQ plot QQ plot using t(3.91) Theoretical quantiles Sample quantiles

51 Final for the interest rate s R t = µ(r t 1 ) + σ(r t 1 )a t 1 Model was fit in two steps: 1 estimate µ() and σ() spm in SemiPar 2 a t as AR(1)/GARCH(1,1) garchfit in fgarch 2 Could the two step be combined? 3 Would combining them change the results?

52 Reference for spline ing Semiparametric by Ruppert, Wand, and Carroll (2003) Lots of examples. But most from biostatistics and epidemiology

53 statistics analysis allows the use of prior information hierarchical priors can: specify knowledge that a group of parameters are similar to each other estimate their common distribution WinBUGS can be run from inside R using the R2WinBUGS package there is a similar BRugs package that runs OpenBugs BRugs is no longer on CRAN

54 midcapd.ts in fecofin package 500 daily on: 20 stocks market

55 Goal The goal is to use the first 100 days to estimate the mean for the next 400 days Four possible estimators: sample means Bayes estimation (shrinkage) mean of means (total shrinkage) CAPM ( return) = beta ( market return)

56 Who won? Estimate Sum of squared errors sample means 1.9 Bayes 0.17 mean of means 0.12 CAPM CAPM Squared estimation errors are summed over the 20 stocks CAPM 1: use mean of first 100 market CAPM 2: use mean of last 400 market

57 Why does shrinkage help? sample means Bayes mean mean estimate target estimate target

58 Likelihood and prior r i,t = tth return on i stock Likelihood: IN = Independent Normal Hierarchical Prior: r i,t = µ i + ɛ i,t ɛ i,t IN (0, σ 2 ɛ ) µ i IN (α, σ 2 µ) Diffuse (non-informative) priors on α, σ 2 ɛ, σ 2 µ Auto and cross-sectional correlations are ignored (treated as 0)

59 -driven shrinkage Hierarchical Prior: µ i IN (α, σ 2 µ) the µ i are shrunk towards α α should be (approximately) the mean of the means σµ/σ 2 ɛ 2 controls the amount of shrinkage large σµ/σ 2 ɛ 2 less shrinkage data-driven shrinkage because σµ 2 and σɛ 2 are estimated

60 WinBUGS output > print(means.sim,digits=3) Inference for Bugs at "midcap.bug", fit using WinBUGS, 3 chains, each with 5100 iterations (first 100 discarded) n.sims = iterations saved mean sd 2.5% 25% 50% 75% 97.5% Rhat n.eff mu[1] 1.1e e e e e mu[2] 1.2e e e e e mu[3] 7.7e e e e e mu[4] 4.5e e e e e mu[18] 8.3e e e e e mu[19] 5.1e e e e e mu[20] 4.8e e e e e sigma_mu 1.5e e e e e sigma_eps 4.3e e e e e alpha 8.8e e e e e deviance 1.2e e e e e For each parameter, n.eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor (at convergence, Rhat=1). DIC info (using the rule, pd = Dbar-Dhat) pd = 4.1 and DIC = DIC is an estimate of predictive error (lower deviance is better).