Estimating Stock Market Volatility using Asymmetric GARCH Models

Estimating Stock Market Volatility using Asymmetric GARCH Models

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ESTIMATING STOCK MARKET VOLATILITY USING ASYMMETRIC GARCH MODELS Dima Alberg, Haim Shalit and Rami Yosef Discussion Paper No. 06-10 September 2006 Monaster Center for Economic Research Ben-Gurion University of the Negev P.O. Box 653 Beer Sheva, Israel Fax: 972-8-6472941 Tel: 972-8-6472286 September 11, 2006 Estimating Stock Market Volatility Using Asymmetric GARCH Models. DIMA ALBERG Department of Economics, Ben-Gurion University of the Negev, Beer Sheva, 84105 Israel E-mail: [email protected] HAIM SHALIT Department of Economics, Ben-Gurion University of the Negev, Beer Sheva, 84105 Israel E-mail: [email protected] RAMI YOSEF Department of Business Administration, Ben-Gurion University of the Negev, Beer Sheva, 84105 Israel E-mail: [email protected] Abstract A comprehensive empirical analysis of the return and conditional variance of Tel Aviv Stock Exchange (TASE) indices is performed using GARCH models. The prediction performance of these conditional changing variance models is compared to newer asymmetric GJR and APARCH models. We also quantify the day-of-the-week effect and the leverage effect and test for asymmetric volatility. Our results show that the EGARCH model using a skewed Student-t distribution is the most successful in forecasting the TASE indices. Keywords: GARCH, Leverage Effect, Day-of- Week Effect, Market Volatility. 2 Estimating Stock Market Volatility Using Asymmetric Changing Variance Models 1. Introduction Volatility clustering and leptokurtosis are common observations in financial time series (Mandelbrot (1963)). Another phenomenon often encountered is the socalled “leverage effect” (Black (1976)), which occurs when stock prices changes are negatively correlated with changes in volatility. Observations of this type in financial time series have led to the use of various changing variance models. In his seminal paper, Engle (1982) proposed to model time-varying conditional variance with Auto-Regressive Conditional Heteroskedasticity (ARCH) processes using lagged disturbances. Early empirical evidence shows that a high ARCH order is needed to capture the dynamic behavior of conditional variance. The Generalized ARCH (GARCH) model of Bollerslev (1986) fulfills this requirement as it is based on an infinite ARCH specification which reduces the number of estimated parameters from infinity to two. Both models capture volatility clustering and leptokurtosis, but as their distribution is symmetric, they fail to model the “leverage effect.” To address this problem, many nonlinear extensions of GARCH have been proposed, such as the Exponential GARCH (EGARCH) model by Nelson (1991), the so-called GJR model by Glosten, Jagannathan, and Runkle (1993) and the Asymmetric Power ARCH (APARCH) model by Ding, Granger, and Engle (1993). Another problem encountered when using GARCH models is that they do not always fully embrace the thick tails property of high frequency financial timesseries. To overcome this drawback Bollerslev (1987), Baillie and Bollerslev (1989), Kaiser (1996) and Beine, Laurent, and Lecourt (2000) have used the Student's tdistribution. Similarly to capture skewness Liu and Brorsen (1995) use an asymmetric stable density. But the variance of such a distribution rarely exists. For modelling both skewness and kurtosis Fernandez and Steel (1998) used the skewed Student's t-distribution that was later extended to the GARCH framework by Lambert and Laurent (2000, 2001). Forecasting conditional variance with asymmetric GARCH models has been comprehensively studied by Pagan and Schwert (1990), Brailsford and Faff (1996), Franses, Neele, and Van Dijk (1998) and Loudon, Watt, and Yadav (2000). A 3 comparison of normal density with non-normal ones was made by Hsieh (1989), Baillie and Bollerslev (1989), Peters (2000), Lambert and Laurent (2001). The purpose of this paper is to characterize a volatility model by its ability to forecast and capture commonly held stylized facts about conditional volatility, such as persistence of volatility, mean reverting behavior, and asymmetric impacts of negative versus positive return innovations. We investigate the forecasting performance of GARCH, EGARCH, GJR and APARCH models together with the different density functions: normal distribution, Student's t-distribution, and asymmetric Student's tdistribution. We also compare between symmetric and asymmetric distributions using the three different density functions. We forecast two major Tel-Aviv Stock Exchange (TASE) indices: TA100 and TA25. To compare the results, we use several standard performance measurements. Our results suggest that one can improve overall estimation by using the asymmetric GARCH model with fat-tailed densities for measuring conditional variance. Moreover, we find that the asymmetric EGARCH model is a better predictor than the asymmetric GARCH, GJR and APARCH models. The paper is structured as follows. Section 2 presents the data. In Section 3, we present the methodology and the GARCH models used in the paper. In Section 4, we describe the estimation procedures and present the forecasting results. 4 2. Data The data consist of 3058 daily observations of the TA251 index from 20/10/1992 to 31/5/2005 and 1911 daily observations of the TA1002 index from 02/07/1997 to31/5/2005. To estimate and forecast these indices, we created many calibrating programs3 and use [email protected] 2.0 by Laurent and Peters (2001), a package whose purpose is to estimate and forecast GARCH models and many of its extensions. The code written by Doornik (1999) in the Ox programming language provides a dialog-oriented interface with features that are not available in standard econometric software. Parameters were estimated using the QML technique by Bollerslev and Wooldridge (1992). The optimization algorithm used is the Broyden-FletcherGoldfarb-Shanno (BFGS) quasi-Newton method. 1 2 3 The TA25 Index is a value-weighted index of 25 stocks traded on the Tel Aviv Stock Exchange (TASE). The TA100 Index is a value- weighted index of 100 stocks traded on the TASE. Coded with Visual Basic Applications 5 3. Methodology Early empirical evidence has shown that to capture conditional variance dynamics one needs to select a high ARCH order. The Bollerslev (1986) Generalized ARCH (GARCH) model, which is based on infinite ARCH specifications, allows us to reduce the number of estimated parameters by imposing non-linear restrictions. The GARCH (p, q) model expresses the variance as: q p i =1 j =1 σ t2 = w + ∑ α i ε t2−i + ∑ β jσ t2− j Using the lag operator L, the variance becomes: σ t2 = w + α (L )ε t2 + β (L )σ t2 with α ( L ) = p q j ∑ α i L and β ( L) = ∑ β j L . i j =1 i =1 If all the roots of the polynomial 1 − β ( L) = 0 lie outside the unit circle, we have: σ t2 = w[1 − β (L )]−1 + α (L )[1 − β (L )]−1 ε t2 . This may be envisaged as an ARCH ( ∞ ) process since the conditional variance depends linearly on all previous squared residuals. As such, the conditional variance of yt4 can become larger than the unconditional variance. Then, if past realizations of ε t2 are larger than σ 2 it is given by: σ 2 ≡ E (ε t2 ) = w q p i =1 j =1 1 − ∑α i − ∑ β j Like ARCH, some restrictions are needed to ensure that σ t2 is positive for all t. Bollerslev (1986) shows that imposing w > 0, α i ≥ 0 ( for i = 1, K , q ) and β j ≥ 0 ( for j = 1,K , p ) is sufficient for the conditional variance to be positive. To capture the asymmetry observed in the data, a new class of ARCH models was introduced: the GJR, the exponential GARCH, and the EGARCH (p,q), models : ( ) = a + ∑ (a ln σ q 2 t 0 i =1 4 Mean equation: p i ( ) z t −i + γ i z t −i ) + ∑ b j ln σ t2− j , where z t −i = j =1 ε t −i σ t −i y t = E ( y t Ω t −1 ) + ε t , where Ω t −1 is the information set at time t-1 6 The parameters allow us to capture the asymmetric effects. For example, if γ 1 = 0 a positive surprise ε t > 0 has the same effect on volatility than a negative surprise ε t < 0 . The presence of a leverage effect can be investigated by testing the hypothesis that γ 1 < 0 . Engle's (1982) ARCH model uses the normal distribution of normalized residuals z t . Bollerslev (1987), on the other hand, proposed a standardized Student's tdistribution with v > 2 degrees of freedom whose density is given by: D(z t ; v ) = Γ((v + 1) / 2 ) z 1 + v−2 Γ(v / 2 ) π (v − 2 ) 2 t − ( v +1) 2 , ∞ where Γ(v ) = ∫ e − x x v −1 dx is the gamma function and v is the parameter measuring the 0 tail thickness. The Student's t-distribution is symmetric around mean zero. For v > 4, the conditional kurtosis equals 3(v − 2)/(v − 4), which exceeds the normal value of 3. The common methodology for estimating ARCH is by maximum likelihood assuming i.i.d. innovations. For D( zt ; v) , the log-likelihood function of {y t (θ )} for the Student's t-distribution is given by: z 2 1T v +1 v 1 ln σ t2 + (1 + v ) ln1 + t , LT ({y t };θ ) = T ln Γ − ln − ln (π (v − 2 )) − ∑ 2 v − 2 2 2 2 t =1 where θ is the vector of parameters to be estimated for the conditional mean, the () conditional variance and, the density function. When v → ∞ we have a normal distribution, so that the lower v is, the fatter are the tails. Recently, Lambert and Laurent (2000, 2001) extended the skewed Student's t-distribution proposed by Fernandez and Steel (1998) to the GARCH framework. Using D( zt ; v) , the loglikelihood function of {y t (θ )} for the skewed Student's t-distribution is given by: ln Γ v + 1 − ln v − 1 ln (π (v − 2 )) + ln 2 + ln(s ) LT ({y t };θ ) = T 1 2 2 2 ξ + ξ − sz + m 1T ∑ ln σ t2 + (1 + v ) ln1 + vt − 2 ξ − It 2 t =1 () , 7 where ξ is the asymmetry parameter, v the degree of freedom of the distribution and v + 1 m Γ v−2 1, if z t ≥ − s 1 1 2 ξ − and s = ξ 2 + 2 − 1 − m 2 It = , m= ξ ξ v − 1 if z < − m π Γ t 2 s (See Lambert and Laurent (2001) for more details.) Maximum likelihood estimates of parameters are usually obtained using the BFGS numerical maximization procedure. In our work instead, we use the quasimaximum likelihood estimator (QMLE). According to Bollerslev and Wooldridge (1992) this estimator is generally consistent, has a normal limiting distribution, and provides asymptotic standard errors that are valid under non-normality. 8 4. Estimation Results 4.1 Descriptive Statistics and the Stationarity Constraint To obtain a stationary series, we use returns rt = 100(log(Pt ) − log(Pt −1 )) where Pt is the closing value of the index at date t. The samples for TA25 and TA100 have means of 0.0433 and 0.0452; standard deviations of 1.4895 and 1.3198; skewness of −0.2106 and -0.4546; and kurtosis of 3.4024 and 5.0898. The sample kurtosis is greater than 3, meaning that return distributions have excess kurtosis for both indices. Excess skewness is also observed, leading to high Jarque-Bera statistics indicating non-normality. Table 1: Descriptive Statistics for Logarithm Differences 100 ⋅ [ln (Pt ) − ln (Pt −1 )] Index Average Min. Max. Std. Dev. Kurtosis Skewness Jarque Bera Stat. TA25 0.0433 -10.1555 7.1408 1.4895 3.4024 -0.2106 43.24 TA100 0.0452 -10.3816 7.6922 1.3198 5.0898 -0.4546 413.56 As daily stock returns may be correlated with the day-of-the-week effect, we avoid the potential calendar effect on the volatility analysis by filtering the daily means and variances using the following two regressions: (1) rt = α 1 SUN t + α 2 MON t + α 3TUE t + α 4WEDt + α 5THU t + δ t , (2) ˆ (rt − rt )2 = β1SUNt + β 2 MONt + β3TUEt + β 4WEDt + β5THUt + ε t , where SUNt, MONt, TUEt ,WEDt and THU t are the dummy variables for Sunday, ˆ Monday, Tuesday, Wednesday and Thursday; and rt is the ordinary least squares (OLS) fitted value of rt from regression (1) at date t. Table 2: Regression Coefficients for Day-of-the-Week Effect – TA25 Regression Mean Sunday S.E. Variance S.E. (2) Tuesday Wednesday Thursday 0.12** 0.020 0.041 -0.0395 0.074 0.06 (1) Monday 0.06 0.06 0.06 0.06 3.41** 1.45** 2.19** 1.91** 2.12** 0.205 0.205 0.207 0.208 0.207 9 Table 3: Regression Coefficients for Day-of-the-Week Effect – TA100 Regression Sunday Monday Tuesday Wednesday Thursday Mean 0.178** 0.089 -0.040 -0.097 0.090 S.E. 0.0663 0.0663 0.0673 0.0673 0.0671 Variance (2) 2.732** 0.878** 1.658** 1.757** 1.626** S.E. 0.231 0.231 0.235 0.235 0.235 (1) The OLS estimates of the two regressions on Tables 2 and 3 show that the TA25 and TA100 indices have significantly positive daily means on Sunday and significant daily variations for Sunday through Thursday5. To eliminate the daily effects, we “standardize” the daily returns using ˆ ˆ ˆ ˆ yt = (rt − rt ) / η t , where η t is the fitted value of (rt − rt )2 from regression (2) at date t. We now substitute the original daily return rt with yt and referred to it as the daily return at date t. Table 4: Descriptive Statistics for "Standardized" Returns yt Index Average Min. Max. Std.Dev. Kurtosis Skewness Jarque Bera Stat. TA25 0 -6.7131 5.30431 0.9999 3.1528 -0.2179 27.17 TA100 0 -8.0305 5.8756 1.00026 4.569 -0.4561 262.27 Returns yt are thus normalized to zero mean and unit variance. The sample skewness and kurtosis of yt’s are -0.2179 and 3.1528;-0.4561 and 4.569 for the two indices. 4.2 Choosing a Volatility Model For the TA25 index, convergence could not be reached with the EGARCH model and a Student's t-distribution. Therefore we turn to the other three models where all asymmetric coefficients are significant at standard levels. Moreover, the Akaike information criteria (AIC) and the log-likelihood values indicate that the EGARCH, APARCH or GJR models better estimate the series than traditional GARCH. These models are estimated by the approximate quasi-maximum likelihood estimator assuming normal, Student-t or skewed Student-t errors. Note that it is quite 5 * and ** - means significance at 5% and 1% levels, respectively. 10 evident that the recursive evaluation of maximum likelihood is conditional on unobserved values and therefore the estimation cannot be considered to be perfectly exact. To solve the problem of unobserved values, we set these quantities to their unconditional expected values. When we analyzed the densities we found that the two Student's t-distributions (symmetric and skewed) clearly outperform the normal distribution. Indeed, the loglikelihood function increases when using the skewed Student's t-distribution, leading to AIC criteria of 2.701 and 2.730 for the normal density versus 2.665 and 2.697 for the non normal densities, for the TA 100 and the TA 25 respectively. Table 5: Comparison between the Models for the TA1006 Normal Student's t Skewed t APARCH APARCH EGARCH Q(20) 20.739 20.731 20.313 Q2(20) 24.051 27.762 24.630 P(50) 72.909 48.472 46.326 Prob[1] 0.015 0.494 0.582 Prob[2] 0.002 0.168 0.196 AIC 2.730 2.697 2.697 Log-Lik -2600.690 -2568.060 -2566.940 TA100 The skewed Student's t-distribution shows results that are superior to the symmetric Student-t distribution when modeling the TA 25 and TA100. A possible explanation for this result is that, if skewness is significant in both series, its magnitude will be inferior in both indices. It may therefore be necessary to add two asymmetric parameters (asymmetric GARCH + asymmetric distribution). 6 In Tables 5 and 6 Q(20) and Q2(20) are respectively the Box-Pierce statistics at lag 20 of the standardized and squared standardized residuals. P(50) is the Pearson goodness-of-fit with 50 cells. AIC is the Akaike information criterion. Log-Lik is the log-likelihood value. 11 Table 6: Comparison between the Models for the TA25 Normal Student's t Skewed t TA25 GJR APARCH GJR APARCH EGARCH Q(20) 32.290 32.363 32.080 32.157 32.689 Q2(20) 14.875 17.177 18.705 26.229 24.349 P(50) 58.710 65.806 46.055 50.241 45.564 Prob[1] 0.161 0.055 0.593 0.424 0.613 Prob[2] 0.045 0.008 0.271 0.129 0.218 AIC 2.701 2.699 2.669 2.665 2.665 Log-Lik -4122.670 -4118.770 -4073.290 -4065.650 -4065.430 All the models describing the dynamics of the first two moments of the series are shown by Box-Pierce statistics for residuals and squared residuals. All are nonsignificant at the 5% level. The stationary constraints are observed for every model and for every density. The values (ranging from 0.831 to 0.982) suggest long persistence of the volatility for the indices. 12 4.3 Forecasting The forecasting ability of GARCH models has been comprehensively discussed by Poon and Granger (2001). However Andersen and Bollerslev (1997) pointed out that the squared daily returns may not be the proper measure to assess the forecasting performance of the different GARCH models for conditional variance. Thus, we consider the following five measures to assess forecasting ability: 1. Mean squared error (MSE): MSE = 1 S +h 2 ∑ σˆ t − σ t2 h + 1 t =S ( 2 ) 2. Median squared error (MedSE): ( ˆ MedSE = Inv( f Med (et )), where et = σ t2 − σ t2 ) 2 and t ∈ [S , S + h ] 3. Mean absolute error (MAE): MAE = 1 S +h 2 ∑ σˆ t − σ t2 h + 1 t =S 4. Adjusted mean absolute percentage error (AMAPE): AMAPE = ˆ 1 S + h σ t2 − σ t2 ∑ σˆ 2 + σ 2 , h + 1 t =S t t ˆ where h is the number of lead steps, S is the sample size, σ t2 is the forecasted variance and σ t2 is the “actual” variance. 5. Theil's inequality coefficient (TIC)7: TIC = 1 S +h ˆ ∑ ( y t − y t )2 h + 1 t=S 1 S +h 1 S +h ˆ ∑ ( y t )2 − h + 1 ∑ ( y t )2 h + 1 t=S t=S The forecasting ability is reported by ranking the different models with respect to the five measures. This is done in tables 7 and 8 where we compare the distributions for the TA 25 and TA 100 indexes. 7 The Theil inequality coefficient is a scale invariant measure that lies always between zero and one, where zero indicates a perfect fit. 13 Table 7: Forecasting Analysis for the TA25 Index: Comparing between Densities GARCH EGARCH GJR APARCH TA25 Student-t Skewed-t Skewed-t Student-t Skewed-t Skewed-t MSE(1) 0.187 0.187 0.188 0.187 0.187 0.188 MSE(2) 0.344 0.343 0.269 0.407 0.415 0.347 MedSE(1) 0.029 0.029 0.033 0.033 0.033 0.033 MedSE(2) 0.223 0.227 0.128 0.309 0.316 0.224 MAE(1) 0.274 0.274 0.272 0.273 0.272 0.272 MAE(2) 0.500 0.499 0.397 0.568 0.575 0.501 RMSE(1) 0.432 0.432 0.433 0.433 0.433 0.434 RMSE(2) 0.587 0.586 0.519 0.638 0.644 0.589 AMAPE(2) 0.782 0.782 0.758 0.795 0.796 0.781 TIC(1) 0.938 0.944 0.983 0.968 0.976 0.981 TIC(2) 0.559 0.559 0.565 (1)- Mean Equation, (2)-Variance Equation 0.565 0.566 0.560 For the TA 25 index shown on Table 7 the results support the use of the asymmetric EGARCH model. For most measures in the variance equation, the EGARCH model outperforms the APARCH model. The GARCH model provides much less satisfactory results and the GJR model provides the poorest forecasts. For the TA 100 index shown on Table 8, the EGARCH model gives better forecasts than the GARCH model while the APARCH and GJR models give the poorest forecasts. The skewed Student-t distribution is the most successful in forecasting the TA100 conditional variance, contrary to the TA25 where the results conflict. Therefore we were unable to draw a general conclusion. The skewed Student-t distribution seems to be the best for forecasting series showing higher skewness. In fact Lambert and Laurent (2001) found that the skewed Student-t density is more appropriate for modeling the NASDAQ index than symmetric densities. 14 Table 8: Forecasting Analysis for the TA100 Index: Comparing between Densities GARCH EGARCH GJR APARCH Skewed-t Skewed-t Student-t Student-t MSE(1) 0.218 0.220 0.219 0.220 MSE(2) 0.366 0.275 0.444 0.393 MedSE(1) 0.093 0.092 0.093 0.092 MedSE(2) 0.376 0.292 0.466 0.409 MAE(1) 0.379 0.381 0.38 0.381 MAE(2) 0.562 0.480 0.624 0.586 RMSE(1) 0.466 0.469 0.468 0.469 RMSE(2) 0.605 0.525 0.666 0.627 AMAPE(2) 0.648 0.621 0.664 0.654 TIC(1) 0.954 0.969 0.965 0.970 TIC(2) 0.538 0.561 0.509 (1)- Mean Equation, (2)-Variance Equation 0.547 TA100 5. Conclusion We compared the forecasting performance of several GARCH models using different distributions for two Tel Aviv stock index returns. We found that the EGARCH skewed Student-t model is the most promising for characterizing the dynamic behavior of these returns as it reflects their underlying process in terms of serial correlation, asymmetric volatility clustering, and leptokurtic innovation. The results also show that asymmetric GARCH models improve the forecasting performance. Among the tested models, the EGARCH skewed Student-t model outperformed GARGH, GJR and APARCH models. This result further implies that the EGARCH model might be more useful than the other three models when applying risk management strategies for Tel Aviv stock index returns. 15 References Baillie, R.T. and T. Bollerslev (1989): “Common Stochastic Trends in a System of Exchange Rates,” Journal of Monetary Economics, 44, 167-181. Black, F. (1976): “Studies of Stock Market Volatility Changes,” Proceedings of the American Statistical Association, Business and Economic Statistics Section, 177– 181. Bollerslev, T. (1986): “Generalized Autoregressive Conditional Heteroskedasticity,” Journal of Econometrics, 31, 307–327. Bollerslev, T. (1987): “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return,” Review of Economics and Statistics, 69, 542–547. Bollerslev, T., and J. Wooldridge (1992): “Quasi-Maximum Likelihood Estimation Inference in Dynamic Models with Time-varying Covariance,” Econometric Theory, 11, 143–172. Brailsford, T., and R. Faff (1996): “An Evaluation of Volatility Forecasting Techniques,” Journal of Banking and Finance, 20, 419–438. Ding, Z., Granger and R. Engle (1993): “A Long Memory Property of Stock Returns and a New Model,” Journal of Empirical Finance, 1, 83-106. Doornik, J. A. (1999): “An Object Oriented Matrix Programming Language” Timberlake Consultant Ltd., 3rd ed. Engle, R. (1982): “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50, 987–1007. Engle, R., and T. Bollerslev (1986): “Modeling the Persistence of Conditional Variances,” Econometric Reviews, 5, 1–50. 16 Fernandez, C., and M. Steel (1998): “On Bayesian Modeling of Fat Tails and Skewness,” Journal of the American Statistical Association, 93, 359–371. Glosten, L., R. Jagannathan, and D. Runkle (1993): “On the Relation between Expected Return on Stocks,” Journal of Finance, 48, 1779–1801. Jarque, C., and A. Bera (1987): “A Test for Normality of Observations and Regression Residuals,” International Statistical Review, 55, 163–172. Laurent, S., and J.-P. Peters (2001): [email protected] 2.0 : An Ox Package for Estimating and Forecasting Various ARCH Models,” Proceedings 8th Forecasting Financial Markets, London, May 2001. Lambert, P., and S. Laurent (2000): “Modelling Skewness Dynamics in Series of Financial Data,” Discussion Paper, Institut de Statistique, Louvain-la-Neuve. Lambert, P., and S. Laurent (2001): “Modelling Financial Time Series Using GARCH-Type Models and a Skewed Student Density,” Mimeo, Université de Liège. Mandelbrot, B. (1963): “The Variation of Certain Speculative Prices,” Journal of Business, 36, 394–419. Nelson, D. (1991): “Conditional Heteroskedasticity in Asset Returns: a New Approach,” Econometrica, 59, 349–370. Palm, (1996): “GARCH Models of Volatility”, Handbook of Statistics, Vol.14, 209239 Pagan, A. and Schwert, (1990): “Alternative Models for Conditional Stock Volatility,” Journal of Econometrics, 45, 267-290. Peters, J. (2000): “Development of a Package in the Ox Environment for GARCH Models and Two Empirical Applications,” Mimeo, Université de Liège. 17 ...
View Full Document

This note was uploaded on 02/28/2010 for the course ECO 211 taught by Professor Gilo during the Spring '10 term at Young Harris.

Ask a homework question - tutors are online