R outputs for Question 6. (1) Data input > library(tseries) > P = get. hist. quote (instrument = "GE", start="2021-01-01", + end= "2024-03-31", + quote = c("Adj Close"), provider = "yahoo", compression = "d") time series starts 2021-01-04 time series ends 2024-03-28 > r = diff (as.matrix (log(P))) >rna.omit(r) > length (r) [1] 813 > r[809:813] [1] -0.0085 -0.0095 0.0003 0.0372 -0.0258 (2) Model 1: ARCH (2) model with Student-t innovations > model = garchFit (~ garch (2,0),data=r, cond.dist="std", trace=FALSE) > summary (model) Title: GARCH Modelling Call: garchFit (formula = garch (2, 0), data = r, cond.dist = "std", trace = FALSE) Coefficient (s): mu omega alpha1 alpha2 shape 0.00222 0.00125 0.13373 0.28380 3.60533 Error Analysis: Estimate Std. Error t value Pr(>/t/) mu 0.00222 omega 0.00125 alpha1 0.13373 alpha2 0.28380 0.0770380 0.0008828 2.525 0.01156 * 0.0001510 8.285 2.22e-16 *** 0.0470310 2.843 0.00446 ** 3.684 0.00023 *** shape 3.60533 0.4081007 8.834 < 2e-16 *** Signif. codes: 0*** 0.001 ** 0.01 * 0.05 0.1 1 Log Likelihood: 2454.99 normalized: 1.86125 Standardised Residuals Tests: Statistic p-value Jarque-Bera Test R Chi-2 396.094 0 Shapiro-Wilk Test R W 0.96440 0 Ljung-Box Test R Q(10) 14.3254 0.158659 Ljung-Box Test R Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R Q(15) 19.718 Q(20) 29.618 0.18302 R 2 Q(10) 15.6807 R^2 Q(15) 34.3686 R^2 Q(20) 46.0913 R TR 2 24.3045 0.0762852 0.109145 0.00302187 0.000783077 0.018485 Information Criterion Statistics: AIC BIC SIC HQIC -3.71492 -3.69527 -3.71495 -3.70755 > model1@sigma.t[809:813] [1] 0.0188 0.0185 0.0186 0.0183 0.0209 > model1@residuals [809:813] [1] -0.0098 -0.0109 -0.0010 0.0358 -0.0272 (3) Model 2: GARCH(1, 1) model with Student-t innovations > model2 = garchFit (~ garch (1,1),data=r, cond.dist="std", trace=FALSE) > summary(model2) Title: GARCH Modelling Call: garchFit (formula = garch (1, 1), data = r, cond.dist = "std", trace = FALSE) Mean and Variance Equation: data garch (1, 1) [data = r] Conditional Distribution: std Coefficient (s): mu omega 1.6222e-03 alpha1 betal shape 6.5063e+00 1.9317e-06 2.1377e-02 9.7241e-01 Std. Errors: based on Hessian Error Analysis: mu Estimate 1.622e-03 Std. Error t value Pr(>|t|) 5.954e-04 omega 1.932e-06 2.093e-06 alpha1 2.138e-02 9.904e-03 2.724 0.00644 ** 0.923 0.35599 2.158 0.03090 * beta1 shape 6.506e+00 1.329e+00 9.724e-01 1.345e-02 72.312 <2e-16 *** 4.895 9.83e-07 *** Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 0.1 1 Log Likelihood: 2104.16 normalized: 2.58814 Standardised Residuals Tests: Statistic p-value Jarque-Bera Test R R W Chi-2 209.655 0 0.979421 2.75269e-09 Shapiro-Wilk Test Ljung-Box Test Ljung-Box Test Ljung-Box Test R Q(10) 13.0809 0.219186 R Q(15) 15.3775 0.42458 R Q(20) 21.8284 0.349893 Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R2 Q(10) 7.29837 0.697008 R^2 Q(15) 8.53473 0.900563 R^2 Q(20) 11.9514 0.91774 TR 2 8.58235 0.738128 R Information Criterion Statistics: AIC BIC SIC HQIC -5.16398 -5.13507 -5.16405 -5.15288 > model2@sigma.t[809:813] [1] 0.0145 0.0144 0.0144 0.0142 0.0150 > model2@residuals [809:813] [1] -0.0101 -0.0111 -0.0013 0.0355 -0.0274 (4) Model 3: GARCH(2, 2) model with Student-t innovations > model3 = garchFit( garch (2,2),data=r, cond.dist="std", trace=FALSE) > summary(model3) Title: GARCH Modelling Call: garchFit (formula trace = FALSE) = garch (2, 2), data = r, cond.dist = "std", Mean and Variance Equation: data garch (2, 2) [data = r] Conditional Distribution: std Coefficient (s): mu omega 1.7597e-03 1.9533e-05 alpha1 8.5937e-02 2.0207e-03 Std. Errors: based on Hessian alpha2 beta1 beta2 shape 1.0000e-08 8.5846e-01 6.2003e+00 Error Analysis: mu Estimate Std. Error t value Pr(>/t/) 1.760e-03 5.898e-04 omega 1.953e-05 alpha1 8.594e-02 alpha2 2.021e-03 3.912e-05 2.983 0.00285 ** 0.499 0.61758 6.281e-02 1.368 0.17123 3.805e-02 1.145e-01 beta1 1.000e-08 beta2 8.585e-01 9.204e-02 shape 6.200e+00 1.287e+00 0.053 0.95765 0.000 1.00000 9.327 <2e-16 *** 4.816 1.46e-06 *** Signif. codes: O *** 0.001 ** 0.01 * 0.05 0.1 1 Log Likelihood: 2106.63 normalized: 2.59118 Standardised Residuals Tests: Statistic p-value 373.896 0 0.974125 8.11435e-11 Jarque-Bera Test R Chi-2 Shapiro-Wilk Test R W R Q(10) 10.656 0.384935 Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R Q(15) 12.3735 0.65057 R Q(20) 18.8254 0.5332 R^2 Q(10) 3.55055 0.965354 R^2 Q(15) 4.49121 0.995628 R^2 Q(20) 6.15067 0.998678 R TR 2 4.06828 0.982172 Information Criterion Statistics: AIC BIC SIC HQIC -5.16515 -5.12467 -5.16530 -5.14961 > mode13@sigma.t [809:813] [1] 0.0177 0.0144 0.0173 0.0141 0.0196 > model3@residuals [809:813] [1] -0.0102 -0.0113 -0.0014 0.0354 -0.0276 6 (Model Selection, Estimation and Prediction of GARCH) Consider the daily returns rt of General Electric Company stock (ticker: "GE") from "2021-01-01" to "2024-03-31", comprising a total of 813 daily returns. Using the "fGarch" package of R, outputs of fitting three GARCH models to the returns are given at the end of this question. Model 1 ARCH (1) with standard normal innovations; Model 2 Model 3 GARCH (1, 1) with Student-t innovations; GARCH (2, 2) with Student-t innovations; Based on the outputs, answer the following questions. (a) What can be inferred from the Standardized Residual Tests conducted on Model 1? (b) Which model do you recommend for prediction between Model 2 and Model 3? Why? (c) Write down the fitted model for the model that you recommended in Part (b). (d) Using the model recommended in Part (b), predict the conditional volatility in the next trading day, specifically trading day 814.

R outputs for Question 6. (1) Data input > library(tseries) > P = get. hist. quote (instrument = "GE", start="2021-01-01", + end= "2024-03-31", + quote = c("Adj Close"), provider = "yahoo", compression = "d") time series starts 2021-01-04 time series ends 2024-03-28 > r = diff (as.matrix (log(P))) >rna.omit(r) > length (r) [1] 813 > r[809:813] [1] -0.0085 -0.0095 0.0003 0.0372 -0.0258 (2) Model 1: ARCH (2) model with Student-t innovations > model = garchFit (~ garch (2,0),data=r, cond.dist="std", trace=FALSE) > summary (model) Title: GARCH Modelling Call: garchFit (formula = garch (2, 0), data = r, cond.dist = "std", trace = FALSE) Coefficient (s): mu omega alpha1 alpha2 shape 0.00222 0.00125 0.13373 0.28380 3.60533 Error Analysis: Estimate Std. Error t value Pr(>/t/) mu 0.00222 omega 0.00125 alpha1 0.13373 alpha2 0.28380 0.0770380 0.0008828 2.525 0.01156 * 0.0001510 8.285 2.22e-16 * 0.0470310 2.843 0.00446 3.684 0.00023 * shape 3.60533 0.4081007 8.834 < 2e-16 * Signif. codes: 0* 0.001 0.01 * 0.05 0.1 1 Log Likelihood: 2454.99 normalized: 1.86125 Standardised Residuals Tests: Statistic p-value Jarque-Bera Test R Chi-2 396.094 0 Shapiro-Wilk Test R W 0.96440 0 Ljung-Box Test R Q(10) 14.3254 0.158659 Ljung-Box Test R Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R Q(15) 19.718 Q(20) 29.618 0.18302 R 2 Q(10) 15.6807 R^2 Q(15) 34.3686 R^2 Q(20) 46.0913 R TR 2 24.3045 0.0762852 0.109145 0.00302187 0.000783077 0.018485 Information Criterion Statistics: AIC BIC SIC HQIC -3.71492 -3.69527 -3.71495 -3.70755 > model1@sigma.t[809:813] [1] 0.0188 0.0185 0.0186 0.0183 0.0209 > model1@residuals [809:813] [1] -0.0098 -0.0109 -0.0010 0.0358 -0.0272 (3) Model 2: GARCH(1, 1) model with Student-t innovations > model2 = garchFit (~ garch (1,1),data=r, cond.dist="std", trace=FALSE) > summary(model2) Title: GARCH Modelling Call: garchFit (formula = garch (1, 1), data = r, cond.dist = "std", trace = FALSE) Mean and Variance Equation: data garch (1, 1) [data = r] Conditional Distribution: std Coefficient (s): mu omega 1.6222e-03 alpha1 betal shape 6.5063e+00 1.9317e-06 2.1377e-02 9.7241e-01 Std. Errors: based on Hessian Error Analysis: mu Estimate 1.622e-03 Std. Error t value Pr(>|t|) 5.954e-04 omega 1.932e-06 2.093e-06 alpha1 2.138e-02 9.904e-03 2.724 0.00644 ** 0.923 0.35599 2.158 0.03090 * beta1 shape 6.506e+00 1.329e+00 9.724e-01 1.345e-02 72.312 <2e-16 * 4.895 9.83e-07 * Signif. codes: 0 * 0.001 0.01 * 0.05 0.1 1 Log Likelihood: 2104.16 normalized: 2.58814 Standardised Residuals Tests: Statistic p-value Jarque-Bera Test R R W Chi-2 209.655 0 0.979421 2.75269e-09 Shapiro-Wilk Test Ljung-Box Test Ljung-Box Test Ljung-Box Test R Q(10) 13.0809 0.219186 R Q(15) 15.3775 0.42458 R Q(20) 21.8284 0.349893 Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R2 Q(10) 7.29837 0.697008 R^2 Q(15) 8.53473 0.900563 R^2 Q(20) 11.9514 0.91774 TR 2 8.58235 0.738128 R Information Criterion Statistics: AIC BIC SIC HQIC -5.16398 -5.13507 -5.16405 -5.15288 > model2@sigma.t[809:813] [1] 0.0145 0.0144 0.0144 0.0142 0.0150 > model2@residuals [809:813] [1] -0.0101 -0.0111 -0.0013 0.0355 -0.0274 (4) Model 3: GARCH(2, 2) model with Student-t innovations > model3 = garchFit( garch (2,2),data=r, cond.dist="std", trace=FALSE) > summary(model3) Title: GARCH Modelling Call: garchFit (formula trace = FALSE) = garch (2, 2), data = r, cond.dist = "std", Mean and Variance Equation: data garch (2, 2) [data = r] Conditional Distribution: std Coefficient (s): mu omega 1.7597e-03 1.9533e-05 alpha1 8.5937e-02 2.0207e-03 Std. Errors: based on Hessian alpha2 beta1 beta2 shape 1.0000e-08 8.5846e-01 6.2003e+00 Error Analysis: mu Estimate Std. Error t value Pr(>/t/) 1.760e-03 5.898e-04 omega 1.953e-05 alpha1 8.594e-02 alpha2 2.021e-03 3.912e-05 2.983 0.00285 0.499 0.61758 6.281e-02 1.368 0.17123 3.805e-02 1.145e-01 beta1 1.000e-08 beta2 8.585e-01 9.204e-02 shape 6.200e+00 1.287e+00 0.053 0.95765 0.000 1.00000 9.327 <2e-16 * 4.816 1.46e-06 * Signif. codes: O * 0.001 ** 0.01 * 0.05 0.1 1 Log Likelihood: 2106.63 normalized: 2.59118 Standardised Residuals Tests: Statistic p-value 373.896 0 0.974125 8.11435e-11 Jarque-Bera Test R Chi-2 Shapiro-Wilk Test R W R Q(10) 10.656 0.384935 Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test Ljung-Box Test LM Arch Test R Q(15) 12.3735 0.65057 R Q(20) 18.8254 0.5332 R^2 Q(10) 3.55055 0.965354 R^2 Q(15) 4.49121 0.995628 R^2 Q(20) 6.15067 0.998678 R TR 2 4.06828 0.982172 Information Criterion Statistics: AIC BIC SIC HQIC -5.16515 -5.12467 -5.16530 -5.14961 > mode13@sigma.t [809:813] [1] 0.0177 0.0144 0.0173 0.0141 0.0196 > model3@residuals [809:813] [1] -0.0102 -0.0113 -0.0014 0.0354 -0.0276 6 (Model Selection, Estimation and Prediction of GARCH) Consider the daily returns rt of General Electric Company stock (ticker: "GE") from "2021-01-01" to "2024-03-31", comprising a total of 813 daily returns. Using the "fGarch" package of R, outputs of fitting three GARCH models to the returns are given at the end of this question. Model 1 ARCH (1) with standard normal innovations; Model 2 Model 3 GARCH (1, 1) with Student-t innovations; GARCH (2, 2) with Student-t innovations; Based on the outputs, answer the following questions. (a) What can be inferred from the Standardized Residual Tests conducted on Model 1? (b) Which model do you recommend for prediction between Model 2 and Model 3? Why? (c) Write down the fitted model for the model that you recommended in Part (b). (d) Using the model recommended in Part (b), predict the conditional volatility in the next trading day, specifically trading day 814.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 25EQ

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