Financial Time Series Analysis: Part II
|
|
- Clarence Martin
- 5 years ago
- Views:
Transcription
1 Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2017
2 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
3 Background 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
4 Background Example 1 Consider the following daily close-to-close SP500 values. S&P 500 Daily Closing Prices and Returns [Jan to Feb ] Return (%) Index 200 day rolling mean Return 22 day rolling mean Index Date
5 Background For example a rolling k = 22 days ( month of trading days) mean m t = 1 k t u=t k+1 r u, (1) and volatility (annualized standard deviation) s t = 252 t (r u m t ) k 2, (2) u=t k+1 t = k,... T give visual idea of volatility dynamics of the returns.
6 Background S&P 500 Daily Closing Prices and Volatility [Jan to Feb ] Index 200 day rolling average Absolute return 22 day rolling volatility Index Volatitliy (%, p.a) Date
7 Background S&P 500 Daily Returns and Volatility [Jan to Feb ] Return 22 day rolling average Absolute return 22 day rolling volatility Return (%) Volatitliy (%, p.a) Date
8 Background Because squared observations are the building blocks of the variance of the series, the results suggest that the variation (volatility) of the series is time dependent. This leads to the so called ARCH-family of models. 3 Note: Volatility not directly observable!! Methods: a) Implied volatility b) Realized volatility c) Econometric modeling (stochastic volatility, ARCH) 3 The inventor of this modeling approach is Robert F. Engle (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50,
9 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
10 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
11 ARCH-modles Consider a time series y t with mean µ = E[y t ] and denote by u t = y t µ the deviation of y t from its mean, t = 1,..., T. The conditional distribution of u t given information up to time point t 1 is denoted as u t F t 1 D(0, σ 2 t ), (3) where F t is the information available at time t (usually the past values of u t ; u 1,..., u t 1 ), D is an appropriated (conditional) distribution (e.g., normal or t-distribution). Then u t follows an ARCH(q) process if its (conditional) variance σ 2 t is of the form σ 2 t = var[u t F t 1 ] = ω + α 1 u 2 t 1 + α 2 u 2 t α q u 2 t q. (4)
12 ARCH-modles Furthermore, it is assumed that ω > 0, α i 0 for all i and α α q < 1. For short it is denoted u t ARCH(q). This reminds essentially an AR(q) process for the squared residuals, because defining ν t = u 2 t σ 2 t, we can write u 2 t = ω + α 1 u 2 t 1 + α 2 u 2 t α q u 2 t q + ν t. (5) Nevertheless, the error term ν t is time heteroscedastic, which implies that the conventional estimation procedure used in AR-estimation does not produce optimal results here.
13 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
14 Properties of ARCH-processes Consider (for the sake of simplicity) ARCH(1) process σ 2 t = ω + αu 2 t 1 (6) with ω > 0 and 0 α < 1 and u t u t 1 N(0, σ 2 t ). (a) u t is white noise: (i) Constant mean (zero): E[u t ] = E[ E t 1 [u t ] ] = E[0] = 0. (7) }{{} =0 Note E t 1 [u t ] = E[u t F t 1 ], the conditional expectation given information up to time t The law of iterated expectations: Consider time points t1 < t 2 such that F t1 F t2, then for any t > t 2 ] [ ] ] ] E t1 [E t2 [u t ] = E E [u t F t2 F t1 = E [u t F t1 = E t1 [u t ]. (8)
15 Properties of ARCH-processes (ii) Constant variance: Using again the law of iterated expectations, we get var[u t ] = E [ ] [ [ ]] ut 2 = E Et 1 u 2 t = E [ ] [ ] σt 2 = E ω + αu 2 t 1 = ω + αe [ ] ut 1 2. = ω(1 + α + α α n ) + α n+1 E [ ut n 1 2 ] }{{} 0, as n = ω ( n lim n i=0 αi) (9) ω = 1 α.
16 Properties of ARCH-processes (iii) Autocovariances: Exercise, show that autocovariances are zero, i.e., E[u t u t+k ] = 0 for all k 0. (Hint: use the law of iterated expectations.) (b) The unconditional distribution of u t is symmetric, but nonnormal: (i) Skewness: Exercise, show that E [ u 3 t ] = 0. (ii) Kurtosis: Exercise, show that under the assumption u t u t 1 N(0, σ 2 t ), and that α < 1/3, the kurtosis E [ ut 4 ] ω 2 = 3 (1 α) 2 1 α 2 1 3α 2. (10) Hint: If X N(0, σ 2 ) then E [ (X µ) 4] = 3(σ 2 ) 2 = 3σ 4.
17 Properties of ARCH-processes Because (1 α 2 )/(1 3α 2 ) > 1 we have that E [ ut 4 ] ω 2 > 3 (1 α) 2 = 3 [var[u t]] 2, (11) we find that the kurtosis of the unconditional distribution exceed that what it would be, if u t were normally distributed. Thus the unconditional distribution of u t is nonnormal and has fatter tails than a normal distribution with variance equal to var[u t ] = ω/(1 α).
18 Properties of ARCH-processes (c) Standardized variables: Write z t = u t σ 2 t (12) then z t NID(0, 1), i.e., normally and independently distributed. Thus we can always write u t = z t σ 2 t, (13) where z t independent standard normal random variables (strict white noise). This gives us a useful device to check after fitting an ARCH model the adequacy of the specification: Check the autocorrelations of the squared standardized series (see, Econometrics I).
19 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
20 Estimation Instead of the normal distribution, more popular conditional distributions in ARCH-modeling are t-distribution and the generalized error distribution (ged), of which the normal distribution is a special case. The unknown parameters are estimated usually by the method of the maximum likelihood (ML).
21 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
22 Generalized ARCH (GARCH) In practice the ARCH needs fairly many lags. Usually far less lags are needed by modifying the model to σt 2 = ω + αut βσt 1, 2 (14) with ω > 0, α > 0, β 0, and α + β < 1. The model is called the Generalized ARCH (GARCH) model. Usually the above GARCH(1,1) is adequate in practice. Econometric packages call α (coefficient of ut 1 2 ) the ARCH parameter and β (coefficient of σt 1 2 ) the GARCH parameter.
23 Generalized ARCH model (GARCH) Applying backward substitution, one easily gets σ 2 t = ω 1 β + α β j 1 ut j 2 (15) j=1 an ARCH( ) process. Thus the GARCH term captures all the history from t 2 backwards of the shocks u t.
24 Generalized ARCH models (GARCH) Example 2 MA(1)-GARCH(1,1) model of SP500 returns estimated with conditional normal, u t F t 1 N(0, σ 2 t ), and GED, u t F t 1 GED(0, σ 2 t ) The model is r t = µ + u t + θu t 1 σ 2 t = ω + αu 2 t 1 + βσ2 t 1. (16)
25 Generalized ARCH models (GARCH) S&P 500 Returns [Jan to Feb 8, 2017]
26 Generalized ARCH models (GARCH) Using R-program with package fgarch (see garchfit(formula = ret ~ arma(2, 0) + garch(1, 1), data = sp1, trace = FALSE) Conditional Distribution: norm Coefficient(s): mu ar1 ar2 omega alpha1 beta Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(> t ) mu e-05 *** ar *** ar omega e-10 *** alpha < 2e-16 *** beta < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized: AIC: BIC:
27 Generalized ARCH models (GARCH) garchfit(formula = ret ~ arma(2, 0) + garch(1, 1), data = sp1, cond.dist = "ged", trace = FALSE) Conditional Distribution: ged Coefficient(s): mu ar1 ar2 omega alpha1 beta1 shape Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(> t ) mu e-09 *** ar e-05 *** ar * omega e-06 *** alpha < 2e-16 *** beta < 2e-16 *** shape < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized: AIC: BIC:
28 Generalized ARCH models (GARCH) Goodness of fit (AIC, BIC) is marginally better for GED. The shape parameter estimate is < 2 (statistically significantly) indicated fat tails. Otherwise the coefficient estimates are about the same.
29 Generalized ARCH models (GARCH) Residual Diagnostics: Autocorrelations of squared standardized residuals: Normal: GED: Lag LB p-value LB p-value JB = , df = 2, p-value = (dropping the outlier JB = 43.96)
30 Generalized ARCH models (GARCH) SP500 Return residuals [MA(1)-GED-GARCH(1,1)] Conditional Volatility Volatility [sqrt(252*h_t)] Days
31 Generalized ARCH models (GARCH) Autocorrelations of the squared standardized residuals pass (approximately) the white noise test. Nevertheless, the normality of the standardized residuals is strongly rejected. Usually this affects mostly to stantard errors. Common practice is to use some sort of robust standard errors (e.g. White).
32 Generalized ARCH models (GARCH) The variance function can be extended by including regressors (exogenous or predetermined variables), x t, in it σ 2 t = ω + αu 2 t 1 + βσ 2 t 1 + πx t. (17) Note that if x t can assume negative values, it may be desirable to introduce absolute values x t in place of x t in the conditional variance function. For example, with daily data a Monday dummy could be introduced into the model to capture the non-trading over the weekends in the volatility.
33 Generalized ARCH models (GARCH) Example 3 Monday effect in SP500 returns and/or volatility? Pulse (additive) effect on mean and innovative effect on volatility y t = φ 0 + φ m M t + φ(y t 1 φ m M t 1 ) + u t σt 2 = ω + πm t + αut βσ2 t 1, (18) where M t = 1 if Monday, zero otherwise.
34 Generalized ARCH models (GARCH) SAS: conditional t-distribution proc autoreg data = tmp; model sp500 = mon/ nlag = 1 garch = (p=1, q=1) dist = t; hetero mon; run;... The AUTOREG Procedure GARCH Estimates SSE Observations 2321 MSE Uncond Var Log Likelihood Total R-Square SBC AIC MAE AICC MAPE Normality Test Pr > ChiSq <.0001 Standard Approx Variable DF Estimate Error t Value Pr > t Variable Label Intercept mon AR ARCH ARCH <.0001 GARCH <.0001 TDFI <.0001 Inverse of t DF HET E E
35 Generalized ARCH models (GARCH) Degrees of freedom estimate: 1/ No empirical evidence of a Monday effect in returns or volatility.
36 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
37 ARCH-M Model The regression equation may be extended by introducing the variance function into the equation y t = x tθ + λg(σ 2 t ) + u t, (19) where u t GARCH, and g is a suitable function (usually square root or logarithm). This is called the ARCH in Mean (ARCH-M) model [Engle, Lilien and Robbins (1987) 5 ]. The ARCH-M model is often used in finance, where the expected return on an asset is related to the expected asset risk. The coefficient λ reflects the risk-return tradeoff. 5 Econometrica, 55,
38 ARCH-M Model Example 4 Does the daily mean return of SP500 depend on the volatility level? Model AR(1)-GARCH(1, 1)-M with conditional t-distribution y t = φ 0 + φ 1 y t 1 + λ σ 2 t + u t σ 2 t = ω + αu 2 t 1 + βσ2 t 1, (20) where u t F t 1 t ν (t-distribution with ν degrees of freedom, to be estimated). proc autoreg data = tmp; model sp500 = / nlag = 1 garch = (p=1, q=1, mean = sqrt) dist = t; /* t-dist */ run;
39 ARCH-M Model The AUTOREG Procedure GARCH Estimates SSE Observations 2321 MSE Uncond Var. Log Likelihood Total R-Square SBC AIC MAE AICC MAPE Normality Test Pr > ChiSq <.0001 Standard Approx Variable DF Estimate Error t Value Pr > t Intercept AR ARCH ARCH <.0001 GARCH <.0001 DELTA TDFI <.0001 Inverse of t DF
40 ARCH-M Model The volatility term in the mean equation (DELTA) is not statistically significant neither is the constant, indicating no discernible drift in SP500 index (again ˆν = 1/ ).
41 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
42 Asymmetric ARCH: TARCH, EGARCH, PARCH A stylized fact in stock markets is that downward movements are followed by higher volatility than upward movements. A rough view of this can be obtained from the cross-autocorrelations of z t and z 2 t, where z t defined in (12).
43 Asymmetric ARCH: TARCH, EGARCH, PARCH Example 5 Cross-autocorrelations of z t and z 2 t from Ex 4 Cross-correlations of z and z^2 ACF Lag Some autocorrelations signify possible leverage effect.
44 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
45 The TARCH model Threshold ARCH, TARCH (Zakoian 1994, Journal of Economic Dynamics and Control, , Glosten, Jagannathan and Runkle 1993, Journal of Finance, ) is given by [TARCH(1,1)] σ 2 t = ω + αu 2 t 1 + γu 2 t 1d t 1 + βσ 2 t 1, (21) where ω > 0, α, β 0, α γ + β < 1, and d t = 1, if u t < 0 (bad news) and zero otherwise. The impact of good news is α and bad news α + γ. Thus, γ 0 implies asymmetry. Leverage exists if γ > 0.
46 The TARCH model Example 6 SP500 returns, MA(1)-TARCH model (EViews,
47 The TARCH model Dependent Variable: SP500 Method: ML - ARCH (Marquardt) - Generalized error distribution (GED) Sample (adjusted): 1/04/2000 3/27/2009 Included observations: 2321 after adjustments Convergence achieved after 28 iterations MA Backcast: 1/03/2000 Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) + C(6)*GARCH(-1) ============================================================== Variable Coefficient Std. Error z-stat Prob C MA(1) ============================================================== Variance Equation ============================================================== C RESID(-1)^ RES(-1)^2*(RES(-1)<0) GARCH(-1) ============================================================== R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) ==============================================================
48 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
49 The EGARCH model Nelson (1991) (Econometrica, ) proposed the Exponential GARCH (EGARCH) model for the variance function of the form (EGARCH(1,1)) log σ 2 t = ω + β log σ 2 t 1 + α z t 1 + γz t 1, (22) where z t = u t / σ 2 t is the standardized shock. Again, the impact is asymmetric if γ 0, and leverage is present if γ < 0.
50 The EGARCH model Example 7 MA(1)-EGARCH(1,1)-M estimation results.
51 The EGARCH model Dependent Variable: SP500 Method: ML - ARCH (Marquardt) - Generalized error distribution (GED) Sample (adjusted): 1/04/2000 3/27/2009 Included observations: 2321 after adjustments Convergence achieved after 25 iterations MA Backcast: 1/03/2000 Presample variance: backcast (parameter = 0.7) LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(5) *RESID(-1)/@SQRT(GARCH(-1)) + C(6)*LOG(GARCH(-1)) ============================================================= Variable Coefficient Std. Error z-statistic Prob C MA(1) ============================================================= Variance Equation ============================================================= C(3) C(4) C(5) C(6) ============================================================= GED PARAM =============================================================... ============================================================= Inverted MA Roots.07 =============================================================
52 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
53 Ding, Granger, and Engle (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance. PARC(1,1) σ δ t = ω + βσ δ t 1 + α( u t 1 γu t 1 ) δ, (23) where γ is the leverage parameter. Again γ > 0 implies leverage.
54 Power ARCH (PARCH) Example 8 R: fgarch::garchfit MA(1)-APARCH(1,1) results for SP500 returns. R parametrization: MA(1), mu and theta gfa <- fgarch::garchfit(sp500r~arma(0,1) + aparch(1,1), data = sp500r, cond.dist = "ged", trace=f) Mean and Variance Equation: data ~ arma(0, 1) + aparch(1, 1) Conditional Distribution: ged Error Analysis: Estimate Std. Error t value Pr(> t ) mu ma *** omega *** alpha e-11 *** gamma < 2e-16 *** beta < 2e-16 *** delta e-09 *** shape < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Log Likelihood: normalized:
55 Power ARCH (PARCH) The leverage parameter ( gamma1 ) estimates to unity. ˆδ = ( ) does not deviate significantly from unity (standard deviation process).
56 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
57 Integrated GARCH (IGARCH) Often in GARCH ˆα + ˆβ 1. Engle and Bollerslev (19896). Modelling the persistence of conditional variances, Econometrics Reviews 5, 1 50, introduce integrated GARCH with α + β = 1. σ 2 t = ω + αu 2 t 1 + (1 α)σ 2 t 1. (24) Close to the EWMA (Exponentially Weighted Moving Average) specification σ 2 t = αu 2 t 1 + (1 α)σ 2 t 1 (25) favored often by practitioners (e.g. RiskMetrics). Unconditional variance does not exist [more details, see Nelson (1990). Stationarity and persistence in in the GARCH(1,1) model. Econometric Theory 6, ].
58 Predicting Volatility 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
59 Predicting Volatility Predicting Volatility Predicting with the GARCH models is straightforward. Generally a k-period forward prediction is of the form σt k 2 = E [ ] [ ] t u 2 t+k = E u 2 t+k F t (26) k = 1, 2,....
60 Predicting Volatility Predicting Volatility Because u t = σ t z t, (27) where generally z t i.i.d(0, 1). Thus in (26) E t [ u 2 t+k ] [ = E t σ 2 t+k zt+k 2 ] [ ] [ ] = E t σ 2 t+k Et z 2 t+k (z t are i.i.d(0, 1)) (28) = E t [ σ 2 t+k ]. This can be utilize to derive explicit prediction formulas in most cases.
61 Predicting Volatility Predicting Volatility ARCH(1): σ 2 t+1 = ω + αu 2 t (29) σ 2 t k = ω(1 αk 1 ) 1 α + α k 1 σ 2 t+1 = σ 2 + α k 1 (σ 2 t+1 σ2 ), where σ 2 = var[u t ] = ω 1 α Recursive fromula: σ σt k 2 = t+1 2 for k = 1 ω + ασt k 1 2 for k > 1 (30) (31) (32)
62 Predicting Volatility Predicting Volatility GARCH(1,1): σ 2 t+1 = ω + αu 2 t + βσ 2 t. (33) where σ 2 t k = ω(1 (α+β)k 1 ) 1 (α+β) + (α + β) k 1 σ 2 t+1 = σ 2 + (α + β) k 1 (σ 2 t+1 σ2 ), σ 2 = (34) ω 1 α β. (35) Recursive fromula: σ σt k 2 = t+1 2 for k = 1 ω + (α + β)σt k 1 2 for k > 1 (36)
63 Predicting Volatility Predicting Volatility IGARCH: σ 2 t+1 = ω + αu 2 t + (1 α)σ 2 t. (37) σ 2 t k = (k 1)ω + σ2 t+1. (38) Recursive fromula: σ σt k 2 = t+1 2 for k = 1 ω + σt k 1 2 for k > 1 (39)
64 Predicting Volatility Predicting Volatility TGARCH: σ 2 t+1 = ω + αu 2 t + γu 2 t d t + βσ 2 t. (40) with σ 2 t k = ω(1 (α+ 1 2 γ+β)k 1 ) 1 (α+ 1 2 γ+β) + (α γ + β)k 1 σ 2 t+1 = σ 2 + (α γ + β)k 1 (σ 2 t+1 σ2 ) σ 2 = (41) ω 1 α 1 2 γ β. (42) Recursive fromula: σ σt k 2 = t+1 2 for k = 1 ω + (α γ + β)σ2 t k 1 for k > 1 (43)
65 Predicting Volatility Predicting Volatility EGARCH and APARCH prediction equations are a bit more involved. Recursive formulas are more appropriate in these cases. The volatility forecasts are applied for example in Value At Risk computations.
66 Predicting Volatility Predicting Volatility Evaluation of predictions unfortunately not that straightforward! (see, Andersen and Bollerslev (1998) International Economic Review.)
67 Realized volatility 1 Volatility Models Background ARCH-models Properties of ARCH-processes Estimation of ARCH models Generalized ARCH models (GARCH) ARCH-M Model Asymmetric ARCH: TARCH, EGARCH, PARCH The TARCH model The EGARCH model Power ARCH (PARCH) Integrated GARCH (IGARCH) Predicting Volatility Realized volatility
68 Realized volatility Realized volatility Under certain assumptions, volatility during a period of time can be estimated more and more precisely as the frequency of the returns increases. Daily (log) returns r t are sums of intraday returns (e.g. returns calculated at 30 minutes interval) r t = m r t (h) (44) h=1 where r t (h) = log P t (h) log P t (h 1) is the day s t intraday return in time interval [h 1, h], h = 1,..., m, P t (h) is the price at time point h within the day t, P t (0) is the opening price and P t (m) is the closing price.
69 Realized volatility Realized volatility The realized variance for day t is defined as m ˆσ t,m 2 = rt 2 (h) (45) and the realized volatility is ˆσ t,m = h=1 ˆσ 2 t,m which is typically presented in percentages per annum (i.e., scaled by the square root of the number of trading days and presented in percentages). Under certain conditions it can be shown that ˆσ t,m σ t as m, i.e., when the intraday return interval 0. For a resent survey on RV, see: Andersen, T.G. and L. Benzoni (2009). Realized volatility. In Handbook of Financial Time Series, T.G. Andersen, R.A. Davis, J-P. Kreiss and T. Mikosh (eds), Springer, New York, pp SSRN version is available at: id= &rec=1&srcabs=903659
Donald Trump's Random Walk Up Wall Street
Donald Trump's Random Walk Up Wall Street Research Question: Did upward stock market trend since beginning of Obama era in January 2009 increase after Donald Trump was elected President? Data: Daily data
More informationFinancial Time Series Analysis (FTSA)
Financial Time Series Analysis (FTSA) Lecture 6: Conditional Heteroscedastic Models Few models are capable of generating the type of ARCH one sees in the data.... Most of these studies are best summarized
More informationResearch Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and Its Extended Forms
Discrete Dynamics in Nature and Society Volume 2009, Article ID 743685, 9 pages doi:10.1155/2009/743685 Research Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and
More informationAmath 546/Econ 589 Univariate GARCH Models: Advanced Topics
Amath 546/Econ 589 Univariate GARCH Models: Advanced Topics Eric Zivot April 29, 2013 Lecture Outline The Leverage Effect Asymmetric GARCH Models Forecasts from Asymmetric GARCH Models GARCH Models with
More informationMODELING EXCHANGE RATE VOLATILITY OF UZBEK SUM BY USING ARCH FAMILY MODELS
International Journal of Economics, Commerce and Management United Kingdom Vol. VI, Issue 11, November 2018 http://ijecm.co.uk/ ISSN 2348 0386 MODELING EXCHANGE RATE VOLATILITY OF UZBEK SUM BY USING ARCH
More informationARCH and GARCH models
ARCH and GARCH models Fulvio Corsi SNS Pisa 5 Dic 2011 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 1 / 21 Asset prices S&P 500 index from 1982 to 2009 1600 1400 1200 1000 800 600 400 200
More informationANALYSIS OF ECONOMIC TIME SERIES Analysis of Financial Time Series. Nonlinear Univariate and Linear Multivariate Time Series. Seppo PynnÄonen, 2003
ANALYSIS OF ECONOMIC TIME SERIES Analysis of Financial Time Series Nonlinear Univariate and Linear Multivariate Time Series Seppo PynnÄonen, 2003 c Professor Seppo PynnÄonen, Department of Mathematics
More informationLecture Note of Bus 41202, Spring 2017: More Volatility Models. Mr. Ruey Tsay
Lecture Note of Bus 41202, Spring 2017: More Volatility Models. Mr. Ruey Tsay Package Note: We use fgarch to estimate most volatility models, but will discuss the package rugarch later, which can be used
More informationBrief Sketch of Solutions: Tutorial 2. 2) graphs. 3) unit root tests
Brief Sketch of Solutions: Tutorial 2 2) graphs LJAPAN DJAPAN 5.2.12 5.0.08 4.8.04 4.6.00 4.4 -.04 4.2 -.08 4.0 01 02 03 04 05 06 07 08 09 -.12 01 02 03 04 05 06 07 08 09 LUSA DUSA 7.4.12 7.3 7.2.08 7.1.04
More informationVOLATILITY. Time Varying Volatility
VOLATILITY Time Varying Volatility CONDITIONAL VOLATILITY IS THE STANDARD DEVIATION OF the unpredictable part of the series. We define the conditional variance as: 2 2 2 t E yt E yt Ft Ft E t Ft surprise
More informationFinancial Econometrics
Financial Econometrics Volatility Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) Volatility 01/13 1 / 37 Squared log returns for CRSP daily GPD (TCD) Volatility 01/13 2 / 37 Absolute value
More informationLecture 5a: ARCH Models
Lecture 5a: ARCH Models 1 2 Big Picture 1. We use ARMA model for the conditional mean 2. We use ARCH model for the conditional variance 3. ARMA and ARCH model can be used together to describe both conditional
More informationFinancial Econometrics: Problem Set # 3 Solutions
Financial Econometrics: Problem Set # 3 Solutions N Vera Chau The University of Chicago: Booth February 9, 219 1 a. You can generate the returns using the exact same strategy as given in problem 2 below.
More informationConditional Heteroscedasticity
1 Conditional Heteroscedasticity May 30, 2010 Junhui Qian 1 Introduction ARMA(p,q) models dictate that the conditional mean of a time series depends on past observations of the time series and the past
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationModelling Stock Market Return Volatility: Evidence from India
Modelling Stock Market Return Volatility: Evidence from India Saurabh Singh Assistant Professor, Graduate School of Business,Devi Ahilya Vishwavidyalaya, Indore 452001 (M.P.) India Dr. L.K Tripathi Dean,
More informationFinancial Time Series Lecture 4: Univariate Volatility Models. Conditional Heteroscedastic Models
Financial Time Series Lecture 4: Univariate Volatility Models Conditional Heteroscedastic Models What is the volatility of an asset? Answer: Conditional standard deviation of the asset return (price) Why
More informationFinancial Econometrics Lecture 5: Modelling Volatility and Correlation
Financial Econometrics Lecture 5: Modelling Volatility and Correlation Dayong Zhang Research Institute of Economics and Management Autumn, 2011 Learning Outcomes Discuss the special features of financial
More informationLecture Note of Bus 41202, Spring 2008: More Volatility Models. Mr. Ruey Tsay
Lecture Note of Bus 41202, Spring 2008: More Volatility Models. Mr. Ruey Tsay The EGARCH model Asymmetry in responses to + & returns: g(ɛ t ) = θɛ t + γ[ ɛ t E( ɛ t )], with E[g(ɛ t )] = 0. To see asymmetry
More informationVolatility Analysis of Nepalese Stock Market
The Journal of Nepalese Business Studies Vol. V No. 1 Dec. 008 Volatility Analysis of Nepalese Stock Market Surya Bahadur G.C. Abstract Modeling and forecasting volatility of capital markets has been important
More informationIndian Institute of Management Calcutta. Working Paper Series. WPS No. 797 March Implied Volatility and Predictability of GARCH Models
Indian Institute of Management Calcutta Working Paper Series WPS No. 797 March 2017 Implied Volatility and Predictability of GARCH Models Vivek Rajvanshi Assistant Professor, Indian Institute of Management
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationFinancial Econometrics Jeffrey R. Russell. Midterm 2014 Suggested Solutions. TA: B. B. Deng
Financial Econometrics Jeffrey R. Russell Midterm 2014 Suggested Solutions TA: B. B. Deng Unless otherwise stated, e t is iid N(0,s 2 ) 1. (12 points) Consider the three series y1, y2, y3, and y4. Match
More informationDoes currency substitution affect exchange rate uncertainty? the case of Turkey
MPRA Munich Personal RePEc Archive Does currency substitution affect exchange rate uncertainty? the case of Turkey Korap Levent Istanbul University Institute of Social Sciences, Besim Ömer Paşa Cd. Kaptan-ı
More informationLAMPIRAN. Null Hypothesis: LO has a unit root Exogenous: Constant Lag Length: 1 (Automatic based on SIC, MAXLAG=13)
74 LAMPIRAN Lampiran 1 Analisis ARIMA 1.1. Uji Stasioneritas Variabel 1. Data Harga Minyak Riil Level Null Hypothesis: LO has a unit root Lag Length: 1 (Automatic based on SIC, MAXLAG=13) Augmented Dickey-Fuller
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (34 pts) Answer briefly the following questions. Each question has
More informationGARCH Models for Inflation Volatility in Oman
Rev. Integr. Bus. Econ. Res. Vol 2(2) 1 GARCH Models for Inflation Volatility in Oman Muhammad Idrees Ahmad Department of Mathematics and Statistics, College of Science, Sultan Qaboos Universty, Alkhod,
More informationEconometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland
Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2018 Part IV Financial Time Series As of Feb 5, 2018 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio
More informationGARCH Models. Instructor: G. William Schwert
APS 425 Fall 2015 GARCH Models Instructor: G. William Schwert 585-275-2470 schwert@schwert.ssb.rochester.edu Autocorrelated Heteroskedasticity Suppose you have regression residuals Mean = 0, not autocorrelated
More informationForecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models
The Financial Review 37 (2002) 93--104 Forecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models Mohammad Najand Old Dominion University Abstract The study examines the relative ability
More informationChapter 4 Level of Volatility in the Indian Stock Market
Chapter 4 Level of Volatility in the Indian Stock Market Measurement of volatility is an important issue in financial econometrics. The main reason for the prominent role that volatility plays in financial
More informationModelling Stock Returns Volatility on Uganda Securities Exchange
Applied Mathematical Sciences, Vol. 8, 2014, no. 104, 5173-5184 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46394 Modelling Stock Returns Volatility on Uganda Securities Exchange Jalira
More informationModeling the volatility of FTSE All Share Index Returns
MPRA Munich Personal RePEc Archive Modeling the volatility of FTSE All Share Index Returns Bayraci, Selcuk University of Exeter, Yeditepe University 27. April 2007 Online at http://mpra.ub.uni-muenchen.de/28095/
More informationAn Empirical Research on Chinese Stock Market Volatility Based. on Garch
Volume 04 - Issue 07 July 2018 PP. 15-23 An Empirical Research on Chinese Stock Market Volatility Based on Garch Ya Qian Zhu 1, Wen huili* 1 (Department of Mathematics and Finance, Hunan University of
More informationVolatility Model for Financial Market Risk Management : An Analysis on JSX Index Return Covariance Matrix
Working Paper in Economics and Development Studies Department of Economics Padjadjaran University No. 00907 Volatility Model for Financial Market Risk Management : An Analysis on JSX Index Return Covariance
More informationModeling Exchange Rate Volatility using APARCH Models
96 TUTA/IOE/PCU Journal of the Institute of Engineering, 2018, 14(1): 96-106 TUTA/IOE/PCU Printed in Nepal Carolyn Ogutu 1, Betuel Canhanga 2, Pitos Biganda 3 1 School of Mathematics, University of Nairobi,
More informationApplied Econometrics with. Financial Econometrics
Applied Econometrics with Extension 1 Financial Econometrics Christian Kleiber, Achim Zeileis 2008 2017 Applied Econometrics with R Ext. 1 Financial Econometrics 0 / 21 Financial Econometrics Overview
More informationVolatility Clustering of Fine Wine Prices assuming Different Distributions
Volatility Clustering of Fine Wine Prices assuming Different Distributions Cynthia Royal Tori, PhD Valdosta State University Langdale College of Business 1500 N. Patterson Street, Valdosta, GA USA 31698
More informationRisk Management. Risk: the quantifiable likelihood of loss or less-than-expected returns.
ARCH/GARCH Models 1 Risk Management Risk: the quantifiable likelihood of loss or less-than-expected returns. In recent decades the field of financial risk management has undergone explosive development.
More informationA market risk model for asymmetric distributed series of return
University of Wollongong Research Online University of Wollongong in Dubai - Papers University of Wollongong in Dubai 2012 A market risk model for asymmetric distributed series of return Kostas Giannopoulos
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (42 pts) Answer briefly the following questions. 1. Questions
More informationARIMA ANALYSIS WITH INTERVENTIONS / OUTLIERS
TASK Run intervention analysis on the price of stock M: model a function of the price as ARIMA with outliers and interventions. SOLUTION The document below is an abridged version of the solution provided
More informationGraduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay. Midterm
Graduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay Midterm GSB Honor Code: I pledge my honor that I have not violated the Honor Code during this examination.
More informationThe Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis
The Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis WenShwo Fang Department of Economics Feng Chia University 100 WenHwa Road, Taichung, TAIWAN Stephen M. Miller* College of Business University
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationMODELING ROMANIAN EXCHANGE RATE EVOLUTION WITH GARCH, TGARCH, GARCH- IN MEAN MODELS
MODELING ROMANIAN EXCHANGE RATE EVOLUTION WITH GARCH, TGARCH, GARCH- IN MEAN MODELS Trenca Ioan Babes-Bolyai University, Faculty of Economics and Business Administration Cociuba Mihail Ioan Babes-Bolyai
More informationModelling Stock Indexes Volatility of Emerging Markets
Modelling Stock Indexes Volatility of Emerging Markets Farhan Ahmed 1 Samia Muhammed Umer 2 Raza Ali 3 ABSTRACT This study aims to investigate the use of ARCH (autoregressive conditional heteroscedasticity)
More informationInternational Journal of Business and Administration Research Review. Vol.3, Issue.22, April-June Page 1
A STUDY ON ANALYZING VOLATILITY OF GOLD PRICE IN INDIA Mr. Arun Kumar D C* Dr. P.V.Raveendra** *Research scholar,bharathiar University, Coimbatore. **Professor and Head Department of Management Studies,
More informationFin285a:Computer Simulations and Risk Assessment Section 7.1 Modeling Volatility: basic models Daníelson, ,
Fin285a:Computer Simulations and Risk Assessment Section 7.1 Modeling Volatility: basic models Daníelson, 2.1-2.3, 2.7-2.8 Overview Moving average model Exponentially weighted moving average (EWMA) GARCH
More informationBrief Sketch of Solutions: Tutorial 1. 2) descriptive statistics and correlogram. Series: LGCSI Sample 12/31/ /11/2009 Observations 2596
Brief Sketch of Solutions: Tutorial 1 2) descriptive statistics and correlogram 240 200 160 120 80 40 0 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 Series: LGCSI Sample 12/31/1999 12/11/2009 Observations 2596 Mean
More informationVOLATILITY. Finance is risk/return trade-off.
VOLATILITY RISK Finance is risk/return trade-off. Volatility is risk. Advance knowledge of risks allows us to avoid them. But what would we have to do to avoid them altogether??? Imagine! How much should
More informationNotes on the Treasury Yield Curve Forecasts. October Kara Naccarelli
Notes on the Treasury Yield Curve Forecasts October 2017 Kara Naccarelli Moody s Analytics has updated its forecast equations for the Treasury yield curve. The revised equations are the Treasury yields
More informationAppendixes Appendix 1 Data of Dependent Variables and Independent Variables Period
Appendixes Appendix 1 Data of Dependent Variables and Independent Variables Period 1-15 1 ROA INF KURS FG January 1,3,7 9 -,19 February 1,79,5 95 3,1 March 1,3,7 91,95 April 1,79,1 919,71 May 1,99,7 955
More informationFinancial Econometrics Notes. Kevin Sheppard University of Oxford
Financial Econometrics Notes Kevin Sheppard University of Oxford Monday 15 th January, 2018 2 This version: 22:52, Monday 15 th January, 2018 2018 Kevin Sheppard ii Contents 1 Probability, Random Variables
More informationAnalysis of the Influence of the Annualized Rate of Rentability on the Unit Value of the Net Assets of the Private Administered Pension Fund NN
Year XVIII No. 20/2018 175 Analysis of the Influence of the Annualized Rate of Rentability on the Unit Value of the Net Assets of the Private Administered Pension Fund NN Constantin DURAC 1 1 University
More informationProperties of financail time series GARCH(p,q) models Risk premium and ARCH-M models Leverage effects and asymmetric GARCH models.
5 III Properties of financail time series GARCH(p,q) models Risk premium and ARCH-M models Leverage effects and asymmetric GARCH models 1 ARCH: Autoregressive Conditional Heteroscedasticity Conditional
More informationAmath 546/Econ 589 Univariate GARCH Models
Amath 546/Econ 589 Univariate GARCH Models Eric Zivot April 24, 2013 Lecture Outline Conditional vs. Unconditional Risk Measures Empirical regularities of asset returns Engle s ARCH model Testing for ARCH
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationTime series: Variance modelling
Time series: Variance modelling Bernt Arne Ødegaard 5 October 018 Contents 1 Motivation 1 1.1 Variance clustering.......................... 1 1. Relation to heteroskedasticity.................... 3 1.3
More informationCross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period
Cahier de recherche/working Paper 13-13 Cross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period 2000-2012 David Ardia Lennart F. Hoogerheide Mai/May
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationFinancial Econometrics Jeffrey R. Russell Midterm 2014
Name: Financial Econometrics Jeffrey R. Russell Midterm 2014 You have 2 hours to complete the exam. Use can use a calculator and one side of an 8.5x11 cheat sheet. Try to fit all your work in the space
More informationShort-selling constraints and stock-return volatility: empirical evidence from the German stock market
Short-selling constraints and stock-return volatility: empirical evidence from the German stock market Martin Bohl, Gerrit Reher, Bernd Wilfling Westfälische Wilhelms-Universität Münster Contents 1. Introduction
More informationModelling financial data with stochastic processes
Modelling financial data with stochastic processes Vlad Ardelean, Fabian Tinkl 01.08.2012 Chair of statistics and econometrics FAU Erlangen-Nuremberg Outline Introduction Stochastic processes Volatility
More informationGraduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay. Final Exam
Graduate School of Business, University of Chicago Business 41202, Spring Quarter 2007, Mr. Ruey S. Tsay Final Exam GSB Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationForecasting the Volatility in Financial Assets using Conditional Variance Models
LUND UNIVERSITY MASTER S THESIS Forecasting the Volatility in Financial Assets using Conditional Variance Models Authors: Hugo Hultman Jesper Swanson Supervisor: Dag Rydorff DEPARTMENT OF ECONOMICS SEMINAR
More informationStudy on Dynamic Risk Measurement Based on ARMA-GJR-AL Model
Applied and Computational Mathematics 5; 4(3): 6- Published online April 3, 5 (http://www.sciencepublishinggroup.com/j/acm) doi:.648/j.acm.543.3 ISSN: 38-565 (Print); ISSN: 38-563 (Online) Study on Dynamic
More informationVolatility Forecasting Performance at Multiple Horizons
Volatility Forecasting Performance at Multiple Horizons For the degree of Master of Science in Financial Economics at Erasmus School of Economics, Erasmus University Rotterdam Author: Sharon Vijn Supervisor:
More informationVolatility in the Indian Financial Market Before, During and After the Global Financial Crisis
Volatility in the Indian Financial Market Before, During and After the Global Financial Crisis Praveen Kulshreshtha Indian Institute of Technology Kanpur, India Aakriti Mittal Indian Institute of Technology
More informationForecasting Volatility of USD/MUR Exchange Rate using a GARCH (1,1) model with GED and Student s-t errors
UNIVERSITY OF MAURITIUS RESEARCH JOURNAL Volume 17 2011 University of Mauritius, Réduit, Mauritius Research Week 2009/2010 Forecasting Volatility of USD/MUR Exchange Rate using a GARCH (1,1) model with
More informationVolatility Forecasting on the Stockholm Stock Exchange
Volatility Forecasting on the Stockholm Stock Exchange Paper within: Authors: Tutors: Civilekonom examensarbete/master thesis in Business Administration (30hp), Finance track Gustafsson, Robert Quinones,
More informationOil Price Effects on Exchange Rate and Price Level: The Case of South Korea
Oil Price Effects on Exchange Rate and Price Level: The Case of South Korea Mirzosaid SULTONOV 東北公益文科大学総合研究論集第 34 号抜刷 2018 年 7 月 30 日発行 研究論文 Oil Price Effects on Exchange Rate and Price Level: The Case
More informationImplied Volatility v/s Realized Volatility: A Forecasting Dimension
4 Implied Volatility v/s Realized Volatility: A Forecasting Dimension 4.1 Introduction Modelling and predicting financial market volatility has played an important role for market participants as it enables
More informationMarket Risk Management for Financial Institutions Based on GARCH Family Models
Washington University in St. Louis Washington University Open Scholarship Arts & Sciences Electronic Theses and Dissertations Arts & Sciences Spring 5-2017 Market Risk Management for Financial Institutions
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay. Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay Final Exam Booth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationEconometric Models for the Analysis of Financial Portfolios
Econometric Models for the Analysis of Financial Portfolios Professor Gabriela Victoria ANGHELACHE, Ph.D. Academy of Economic Studies Bucharest Professor Constantin ANGHELACHE, Ph.D. Artifex University
More informationLecture 5: Univariate Volatility
Lecture 5: Univariate Volatility Modellig, ARCH and GARCH Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015 Overview Stepwise Distribution Modeling Approach Three Key Facts to Remember Volatility
More informationINFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE
INFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE Abstract Petr Makovský If there is any market which is said to be effective, this is the the FOREX market. Here we
More information12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006.
12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. Autoregressive Conditional Heteroscedasticity with Estimates of Variance
More informationModeling Volatility of Price of Some Selected Agricultural Products in Ethiopia: ARIMA-GARCH Applications
Modeling Volatility of Price of Some Selected Agricultural Products in Ethiopia: ARIMA-GARCH Applications Background: Agricultural products market policies in Ethiopia have undergone dramatic changes over
More informationPortfolio construction by volatility forecasts: Does the covariance structure matter?
Portfolio construction by volatility forecasts: Does the covariance structure matter? Momtchil Pojarliev and Wolfgang Polasek INVESCO Asset Management, Bleichstrasse 60-62, D-60313 Frankfurt email: momtchil
More informationUniversité de Montréal. Rapport de recherche. Empirical Analysis of Jumps Contribution to Volatility Forecasting Using High Frequency Data
Université de Montréal Rapport de recherche Empirical Analysis of Jumps Contribution to Volatility Forecasting Using High Frequency Data Rédigé par : Imhof, Adolfo Dirigé par : Kalnina, Ilze Département
More informationMODELING VOLATILITY OF BSE SECTORAL INDICES
MODELING VOLATILITY OF BSE SECTORAL INDICES DR.S.MOHANDASS *; MRS.P.RENUKADEVI ** * DIRECTOR, DEPARTMENT OF MANAGEMENT SCIENCES, SVS INSTITUTE OF MANAGEMENT SCIENCES, MYLERIPALAYAM POST, ARASAMPALAYAM,COIMBATORE
More informationFinancial Times Series. Lecture 8
Financial Times Series Lecture 8 Nobel Prize Robert Engle got the Nobel Prize in Economics in 2003 for the ARCH model which he introduced in 1982 It turns out that in many applications there will be many
More informationThe Effect of 9/11 on the Stock Market Volatility Dynamics: Empirical Evidence from a Front Line State
Aalborg University From the SelectedWorks of Omar Farooq 2008 The Effect of 9/11 on the Stock Market Volatility Dynamics: Empirical Evidence from a Front Line State Omar Farooq Sheraz Ahmed Available at:
More informationARCH modeling of the returns of first bank of Nigeria
AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 015,Science Huβ, http://www.scihub.org/ajsir ISSN: 153-649X, doi:10.551/ajsir.015.6.6.131.140 ARCH modeling of the returns of first bank of Nigeria
More informationEstimating and forecasting volatility of stock indices using asymmetric GARCH models and Student-t densities: Evidence from Chittagong Stock Exchange
IJBFMR 3 (215) 19-34 ISSN 253-1842 Estimating and forecasting volatility of stock indices using asymmetric GARCH models and Student-t densities: Evidence from Chittagong Stock Exchange Md. Qamruzzaman
More informationChapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29
Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting
More informationForecasting Value at Risk in the Swedish stock market an investigation of GARCH volatility models
Forecasting Value at Risk in the Swedish stock market an investigation of GARCH volatility models Joel Nilsson Bachelor thesis Supervisor: Lars Forsberg Spring 2015 Abstract The purpose of this thesis
More informationInvestment Opportunity in BSE-SENSEX: A study based on asymmetric GARCH model
Investment Opportunity in BSE-SENSEX: A study based on asymmetric GARCH model Jatin Trivedi Associate Professor, Ph.D AMITY UNIVERSITY, Mumbai contact.tjatin@gmail.com Abstract This article aims to focus
More informationTime series analysis on return of spot gold price
Time series analysis on return of spot gold price Team member: Tian Xie (#1371992) Zizhen Li(#1368493) Contents Exploratory Analysis... 2 Data description... 2 Data preparation... 2 Basics Stats... 2 Unit
More informationDaniel de Almeida and Luiz K. Hotta*
Pesquisa Operacional (2014) 34(2): 237-250 2014 Brazilian Operations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/pope doi: 10.1590/0101-7438.2014.034.02.0237
More informationGlobal Volatility and Forex Returns in East Asia
WP/8/8 Global Volatility and Forex Returns in East Asia Sanjay Kalra 8 International Monetary Fund WP/8/8 IMF Working Paper Asia and Pacific Department Global Volatility and Forex Returns in East Asia
More informationLampiran 1 : Grafik Data HIV Asli
Lampiran 1 : Grafik Data HIV Asli 70 60 50 Penderita 40 30 20 10 2007 2008 2009 2010 2011 Tahun HIV Mean 34.15000 Median 31.50000 Maximum 60.00000 Minimum 19.00000 Std. Dev. 10.45057 Skewness 0.584866
More informationLecture 6: Non Normal Distributions
Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015 Overview Non-normalities in (standardized) residuals from asset return
More informationTHE INFLATION - INFLATION UNCERTAINTY NEXUS IN ROMANIA
THE INFLATION - INFLATION UNCERTAINTY NEXUS IN ROMANIA Daniela ZAPODEANU University of Oradea, Faculty of Economic Science Oradea, Romania Mihail Ioan COCIUBA University of Oradea, Faculty of Economic
More informationEvidence of Market Inefficiency from the Bucharest Stock Exchange
American Journal of Economics 2014, 4(2A): 1-6 DOI: 10.5923/s.economics.201401.01 Evidence of Market Inefficiency from the Bucharest Stock Exchange Ekaterina Damianova University of Durham Abstract This
More informationUNIVERSITÀ DEGLI STUDI DI PADOVA. Dipartimento di Scienze Economiche Marco Fanno
UNIVERSITÀ DEGLI STUDI DI PADOVA Dipartimento di Scienze Economiche Marco Fanno MODELING AND FORECASTING REALIZED RANGE VOLATILITY MASSIMILIANO CAPORIN University of Padova GABRIEL G. VELO University of
More informationPer Capita Housing Starts: Forecasting and the Effects of Interest Rate
1 David I. Goodman The University of Idaho Economics 351 Professor Ismail H. Genc March 13th, 2003 Per Capita Housing Starts: Forecasting and the Effects of Interest Rate Abstract This study examines the
More information