3.고봉현

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1 Price Volatility, Seasonality and Day-of-the Week Effect for Aquacultural Fishes in Korean Fishery Markets ARCH-LM Abstract.,,,. 2000,, ,382 3, %, (2008). * (Corresponding author : , kbh0225@jdi.re.kr) 49

2 29.8%., (, 2002).....,. (uncertainty) (risk). 1).., (price level) (asymmetry), (volatility spillover), (volatility clustering). (2004) (,, 7 ), GARCH t (nonlinear dynamics). (2008) (, ), ARCH., (2007) Binh. et. al(2008).,., 1) (2007),,. 50

3 ) (kg ). 2,613,. (seasonal anomalies), (,, ).,,,, (,,, ).,. Bollerslev(1986) GARCH (Generalized Autoregressive Conditinal Heteroskedasticity). GARCH, 2).. 3),,. GARCH-M, GARCH-M.. 51

4 . Bollerslev(1986) ARCH, GARCH(p, q). y t X t θ ε t ε t σ t μ t, ε t Ψ t 1 N(0,σ 2 t) σ 2 t α 0 p Σ i 1 α i ε 2 t 1 q Σ j 1 β j σ 2 t 1 α i 0, β j 0 q 0 ARCH(p), p q 0 ε t (white noise). ARCH GARCH AR ARMA. α β, α β. p q 1 GARCH(1,1) ( ) (stationarity condition) α 1 β 1 1. α 1 β 1 (persistency parameter), α 1 β 1 1 Integrated GARCH(IGARCH) 4). GARCH(1,1) (fat tailed) (leptokurtic). ( ). GARCH(1,1). r t c m Σ i 1 b ir t i d 1D 1t d 2D 2t d 3D 3t d 4D 4t d 5D 5t 2πx 2πx 2πx 2πx e 1 sin ( ) e 2sin ( ) e 3cos ( ) e 4cos ( ) ε t Y 1 Y 2 Y 1 Y 2 ε t σ t μ t, ε t Ψ t 1 N(0,σ 2 t) σ 2 t θ αε 2 t 1 βσ 2 t 1 γ 1 D 1t γ 2 D 2t γ 3 D 3t γ 4 D 4t γ 5 D 5t 2πx 2πx 2πx 2πx δ 1sin ( Y ) δ2sin ( ) δ3cos ( ) δ4cos ( 1 Y 2 Y 1 Y 2 ) 4) α 1. λ( α 1 β 1) 1 ( ) (, 2004, pp ). 52

5 , r t (, ), (AR) (lag) AIC(Akaike Information Criterion). Ψ t 1 t 1, μt 0 1 i.i.d.. D 1t D 6t. t D 1t 1, 0. (D 6t ) (perfect multicollinearity) (dummy variable trap)., sin cos sine cosine, (spectral analysis) 5). sine cosine. π , x 1 1, Y 1 1, Y 2. F. GARCH(1,1) n student t GARCH(1,1) t 6). t GARCH (log likelihood function). GARCH(1,1) n T 1 1 ε 2 t L ( ) ln (2π) ( ) Σ T lnσ 2 t 1 t ( ) Σ T ln t 1 σ 2 t GARCH(1,1) t v 1 v 2 2 L Γ ( ) Γ ( ) 1 π 1/2 [(v 2)σ 2 t] 1/2 [1 ε 2 tσ 2 t(v 2) 1 ] (v 1) 2 t v (v 20),, v fat tailed. fat tailed 5),., sine cosine 2π cosine 0 cos t cos( t), sine sin t sin( t). sine cosine 1 1 ( 1998, p. 933). 6), 2004, pp

6 r t. BHHH, BFGS, Marquart DFP (numerical optimization) (algorithm) (θ, α, β) 7).. (price level) (change rate of price),. (non stationary) I(1)., (price level) (raw data). t P t, P t P t 1 Rt P t 1 [t 1, t] (change rate of price).. P t P t 1 P t P t 1 Rt log(1 ) P t 1 P t 1 P t log logpt logp t 1 Pt ,. 7) EViews 5.0v. Marquart (option). 54

24000 24000 20000 20000 16000 16000 12000 12000 8000 8000 4000 4000 0 500 1000

-50-200 -100-300 -150 500 1000 1500 2000 2500 500 1000 1500 2000 2500 (c) (d)

7 (a) (b) (c) (d) 1,. 2.,,., , )., , ) (2007, pp ),. 55

8 2 (5 ) 4, , ,. (KMI, ). 1,., (skewness) 0,, (kurtosis) 3 9)., (R t R) 3 (R t R) 4 σ 3 σ 4 (skewness), left-skewed, right-skewed. (kurtosis) 3 (leptokurtic), (playkurtic),, leptokurtic (, 2004, pp ). 9) E [ ], E [ ] 56

9 (skewness) (kurtosis) Jarque Bera (p value) ( ) ( ) 1,795, ,256 Q (10) (0.000) (0.000) Q (20) (0.000) (0.000) Q 2 (10) (0.000) (0.000) Q 2 (20) (0.000) (0.000) Q(T) T Ljung Box Q, Q 2 (T) T Ljung Box Q, ( ) (significance level). (a) (b) 3 (leptokurtic) ( 3 ). 0, 1,.,. (H 0 ) Jarque Bera., 1%. 57

10 leptokurtic., T. (H 0 ) T Ljung Box Q 10), 1%.,. Ljung Box Q. Ljung Box Q ARCH. (stationarity), ARCH (ARCH Lagrange Multiplier : ARCH LM)., DF, ADF, PP.,, 1% (stationary)., ARCH LM (1, 5, 10 ) ARCH 1% 2 ARCH LM ARCH LM DF ADF PP ARCH(1) ARCH(5) ARCH(10) * * * * * * * * * * * * 1) * 1%. 2) 1% , (lagged differences) 5. 3) ARCH(T) T ARCH. ρ^2 j 10) Q T(T 2) Σ q ~χ2 j 1 (q) (T j) 58

11 . ARCH.,,., GARCH.. II GARCH 3., AIC, 2, 4. n (10% ) 3 GARCH 2πx 2πx 2πx 2πx r t c Σ m i 1 b i r t i d 1 D 1t d 2 D 2t d 3 D 3t d 4 D 4t d 5 D 5t e 1 sin( ) e 2 sin( ) e 3 cos( ) e 4 cos( ) ε t Y1 Y 2 Y 1 Y 2 c b1 b2 b3 b4 d1 d2 d3 d4 d5 e1 e2 e3 e4 n t n t 0.47( 1.86)*** 0.65( 27.71)* 0.38( 13.99)* 1.27(0.92) 8.22(1.41) 0.85( 0.01) 5.85(0.35) 0.002(0.66) 0.65(0.94) 0.84(0.40) 1.42(0.04) 0.43( 1.11) (0.9820) (0.9327) 0.007(0.04) 0.62( 28.75)* 0.38( 16.73)* 3.85(1.31) 5.22(3.40)* 1.21( 0.79) 2.57(1.07) 1.69( 1.84)** 1.23(2.36)** 1.03(0.85) 1.58(3.99)* 0.06( 0.72) (0.0006) (0.0023) 0.19(0.54) 0.66( 30.09)* 0.53( 21.66)* 0.43( 16.80)* 0.35( 13.38)* 12.36(0.55) 7.16(1.85)*** 3.54( 0.05) 1.99(0.45) 1.32(1.12) 1.76(0.25) 1.63(0.32) 2.51(0.02) 0.52(0.61) (0.6785) (0.7735) 0.32(0.98) 0.67( 32.50)* 0.52( 20.76)* 0.43( 17.15)* 0.35( 13.75)* 6.03( 1.12) 4.26( 10.02)* 0.006(0.33) 1.62(2.32)** 3.27(4.51)* 0.37(1.12) 0.14(1.24) 0.86(0.31) 0.68( 0.43) (0.0000) (0.8989) 1) *, **, *** 1%, 5%, 10%, ( ) z. 2) H 0 d 1 d 2 d 3 d 4 d 5 0, H 0 e 1 e 2 e 3 e 4 0 F, ( ) p value. 59

12 , (n t ) 1%.,, n. t, ( ), ( )., n, t,,. t ( ), ( ). t., t,., 11).,, t 1 sine cosine. (H 0 e 1 e 2 e 3 e 4 0) 0 t 1%. 4. sine cosine 1. 12)., cosine sine 1... sine cosine 11). 12),. sine cosine. 60

13 sin12 cos12 sin12 sin6 cos6 cos12 sine cosine 1 sine cosine , α β 1%, (λ α β) ,. α β 1%, 61

14 4 GARCH 2πx 2πx 2πx 2πx σ 2 t θ αε 2 t 1 βσ 2 t 1 γ 1 D 1t γ 2 D 2t γ 3 D 3t γ 4 D 4t γ 5 D 5t δ 1 sin( ) δ 2 sin( ) δ 3 cos( ) δ 3 cos( ) Y 1 Y 2 Y 1 Y 2 θ α β υ γ 1 γ 2 γ 3 γ 4 γ 5 δ 1 δ 2 δ 3 δ 4 (λ α β) Log likelihood n t n t 32.92(2.38)** 0.01(5.70)* 0.97(84.78)* 56.47(2.05)** 20.52( 1.18) 9.38(0.20) 71.17( 1.19) 46.51( 2.75)* 0.29(0.40) 0.39( 0.54) 0.99(0.67) 1.24( 1.09) (0.0012) ( ) (4.24)* 0.03(11.88)* 0.93(49.50)* 2.27(36.33)* 33.98(3.24)* 17.18( 1.21) 17.59(0.66) 27.88( 2.45)** 25.50( 2.87)* 3.37(2.93)* 3.62(2.91)* 1.64( 1.81)*** 0.29( 0.34) (0.0002) (0.0000) ( 0.13) 0.09(4.80)* 0.78(17.20)* (1.93)*** 88.12(1.57) 25.42(0.36) 38.59(0.54) 21.45(0.32) 0.30( 1.37) 0.07(0.02) 10.14(1.75)*** 0.90( 0.26) (0.0035) (0.0012) (0.19) 0.11(5.69)* 0.77(20.41)* 4.55(21.88)* 98.33(2.54)** 41.54(1.21) 44.80(1.33) 3.72( 0.12) 27.95(0.75) 1.67(2.18)** 0.63(0.22) 5.68(1.86)*** 1.51( 0.49) (0.0017) (0.0000) ) *, **, *** 1%, 5%, 10%, ( ) z. 2) H 0 γ 1 γ 2 γ 3 γ 4 γ 5 0, H 0 δ 1 δ 2 δ 3 δ 4 0 F, ( ) p value )., t v , 1%., (fat tailed)., t , 225. t fat tail. ( ) GARCH t GARCH(1,1) t. 13) (λ) 1 (volatility clustering) ( (2004), (2007) ). 62

15 .,,.,,,,,.. (n t ).,, 1%., n. t sine cosine 1 sine., cosine 1.. Bollerslev(1986) GARCH., ,. 63

16 , JB,.. ( ) (leptokurtic)..,., Ljung Box Q ARCH., 14)., (stationarity) (unit root test), (stationary). ARCH (ARCH Lagrang Multiplier ; ARCH LM) ARCH. GARCH.,,. ( ), ( ), ( ), ( ), ( )., ( ) ( )., 14) (2001),. 64

17 ...,,., sine cosine 1 sine, cosine 1..., (λ) (volatility clustering).,, GARCH(1,1) n student t GARCH(1,1) t., fat tailed t GARCH(1,1) t.,., 5%, 15)..,., 15) 2004 DB. 65

18 .,.,.,. 66

19 ,,, 32 1, 2001, pp ,,, 45 1, 2004, pp ,,, 34 2, 2007, pp , GARCH,, 22 2, 2007, pp , ( 2 ),, 2004., : GARCH,, 7 4, pp ,,, 38 2, 2007, pp ,,,, 35 1, 2008, pp ,,,, 45 2, 2004, pp ,,, 37 2, 2006, pp , ( ),, 1998.,,,,, 2002.,,. Bollerslev, T., Generalized Autoregressive Conditional Heteroskedasticity, Journal of Economaetrics, No.31, 1986, pp Bollerslev, T., A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return, Review of Economics and Statistics, No. 69, 1987, pp Bollerslev, T., Engle, R. and Nelson, D., ARCH Model, Handbook of Econometrics, V4, edited by Engle and Mcfadden, Enders. W., Applied Econometric Time Series(Second Edition), New York : John Wilely & Sons, Inc., Engle, R. F., Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U. K. Inflation, Econometrica, 1982, pp Engle, R. F. and Bollerslev, T., Modeling the persistence of conditional variances : comments, Econometric Reviews, No.5, Engle, R. F., and Chowdhury Mustafa, Implied ARCH Models from Options Prices, Journal of Econometrics, 1992, pp Eview 5.0 User s Guide, Quantitative Micro Software, LLC.,

20 Glosten, L. R., Jaganathan, R., Runkle, D., On the Relation between the Expected Value and the Volatility of the Normal Excess Return on Stocks, Journal of Finance, No.48, Jiang, J., Zhao, Q., and Hui, Y. V., Robust Modelling of ARCH Models, Journal of Forecasting, No.20, Lamoureux, C. G., Lastrapes, W. D., Heteroskedasticity in Stock Return Data : Volume versus GARCH Effects, Journal of Finance, No.45, Nelson, D. B., Conditional Heteroscedasticity in Asset Returns, Econometrica, No.59,

21 Price Volatility, Seasonality and Day-of-the Week Effect for Aquacultural Fishes in Korean Fishery Markets Abstract This study proviedes GARCH model(bollerslev, 1986) to analyze the structural characteristics of price volatility in domestic aquacultural fish market of Korea. As a case study, flatfish and rock fish are analyzed as major species with relatively high portion in an aspect of production volume among fish captured in Korea. For analyzing, this study uses daily market data (dating from Jan to June 30, 2008) published by the Noryangjin Fisheries Wholesale Market which is located in Seoul of Korea. This study performs normality test on trading volume and price volatility of flatfish and rock fish as an advanced empirical approach. The normality test adopted is Jarque Bera test statistic. As a result, first, a null hypothesis that an empirical distribution follows normal distribution was rejected in both fishes. The distribution of daily market data of them were not only biased toward positive( ) direction in terms of kurtosis and skewness, but also characterized by leptokurtic distribution with long right tail. Secondly, serial correlations were found in data on market trading volume and price volatility of two species during very long period. Thirdly, the results of unit root test and ARCH LM test showed that all data of time series were very stationary and demonstrated effects of ARCH. These statistical characteristics can be explained as a reasonable ground for supporting the fitness of GARCH model in order to estimate conditional variances that reveal price volatility in empirical analysis. From empirical data analysis above, this study drew the following conclusions. First of all, from an empirical analysis on potential effects of seasonality and the day of week on price volatility of aquacultural fish, Monday effects were found in both species and Thursday and Friday effects were also found in 69

22 flatfish. This indicates that Monday is effective in expanding price volatility of aquacultural fish market and also Monday has higher effects upon the price volatility of fish than other days of week have since it has more new information for weekend. Secondly, the empirical analysis led to a common conclusion that there was very high price volatility of flatfish and rock fish. This points out that the persistency parameter(λ), an index of possibility for current volatility to sustain similarly in the future, was higher than 0.8 equivalently nearly to 1 in both flatfish and rock fish, which presents volatility clustering. Also, this study estimated and compared and model that hypothesized normal distributions in order to determine fitness of respective models. As a result, the fitness of GARCH(1,1) t model was better than model where the distribution of error term was hypothesized through distribution due to characteristics of fat tailed distribution, was also better than model, as described in the results of basic statistic analysis. In conclusion, this study has an important mean in that it was introduced firstly in Korea to investigate in price volatility of Korean aquacultural fishery products, although there was partially a limited of official statistic data. Therefore, it is expected that the results of this study will be useful as a reference material for making and assessing governmental policies. Also, it is looked forward that the results will be helpful to build a fishery business plan as and aspect of producer, and also to take timely measures to potential price fluctuations of fishery products in market. Hence, it is advisable that further studies related to such price volatility in fishery market will extend and evolve into a wider variety of articles and issues in near future. 70

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