22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 摘要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M *

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1 22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 摘要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M * trista2@mail.stut.edu.tw ** mcchen@finance.nsysu.edu.tw

2 Housing Price Volatility in the Taipei Area and Cobweb Theory I-Chun Tsai* and Ming-Chi Chen** Abstract This paper examines the volatile behavior of real estate prices in the Taipei area. First, cobweb theory is used to explain price volatility and justify our empirical analysis. We use cobweb theory to illustrate inconstant real estate prices and further explain the phenomenon of occasional high and low volatility caused by anticipation. In the empirical test, we use both ARCH and GARCH models to estimate price conditional heteroscedasticity in order to verify a time-varying property of real estate prices. We continue to use the SWARCH model and find that there are at least two states of volatility. The magnitude of the high volatility state is as high as nine times that of low volatility, but low volatility is the normal condition in the market. Because risk is time-varying in the market, we further use the ARCH-M model to observe whether volatility can explain the change in expected returns and find that indeed high risk can bring high return. Keywords: Markov-Switching, Cobweb Theory, GARCH, SWARCH, Real Estate Price * Department of Finance, Southern Taiwan University of Technology, Tainan, trista2@mail.stut.edu.tw ** Department of Finance, National Sun Yat-sen University, Kaohsiung, mcchen@finance.nsysu.edu.tw 2

3 0 Whitehead Hendry Meen 0 Drake Holly and Jones Malpezzi Meen Chen and Patel, Hendry, Meen, 0 2 Hendry Giussani and Hadjimatheou 0 autoregressive conditional heteroscedasticity Engle 2 Bollerslev ad hoc model ARCH GARCH time varying Markov-Switch-ARCH SWARCH ARCH-M. 0 2.

4 ad hoc model ARCH GARCH SWARCH ad hoc model ( ) cobweb theory Whitehead Bowden 0 t HD PY Phs HHN i Ph

5 HD t =f Ph t, PY t, HHN t, i t... HS Ph CC LC HS t g Ph t, CC t, LC t... 2 HD t HS t... 2 HS t g Ph t-, CC t-, LC t-....

6 ( ) Phs i Ph PhGr e PhGr e t γphgr e t-... Ph e t Ph t γ Ph t Ph t-... γ coefficient of adjustment 0<γ 1 HD t f Ph t Ph t, Ph t-i, PY t, HHN t, i t i t... γ Bloomfield and Hales 2002 PhGr γ t γ t PhGr t-i i t γ

7 ARCH SWARCH ARCH-M ( ) ARCH GARCH Engle 2 autoregressive conditional heteroscedasticity model ARCH model q Bollerslev ARCH generalized autoregressive conditional heteroscedasticity model GARCH model Engle, Lilien, and Robins ARCH ARCH in mean model ARCH-M model 1. ARCH model y t first-order autoregression ε t ARCH q y t a 0 a y t- ε t

8 ε t Ω t-1 ~N(0,h t ) h t ω 0 α i ε 2 t-1 h t 2. GARCH model ε t GARCH p,q h t ω 0 α i ε 2 t-1 h t ( ) SWARCH ARCH GARCH Lamoureux and Lastrapes 0 ARCH GARCH Hamilton and Susmel regime switching ARCH SWARCH SWARCH y t ε t SWARCH K, q y t a 0 a y t- ε t ε t Ω t-1 ~N(0,h t ) ε t u t = u t u t h t ω 0 v t gaussian distribution s t, 2,, K s t 2 g g 2 g 2 s t = s t =2 s t transition probabilities P

9 P p ij i j p p 2 p p 2 p 2 p 22 ( ) ARCH in mean Engle, Lilien, and Robins ARCH ARCH-M y t a 0 a y t- a 2 h t- ε t ε t Ω t-1 ~N(0,h t ) h t ω 0 a 2 ( ) Q2 Q ARCH spline SWARCH

10 ( ) Augmented Dickey-Fuller ADF Phillips-Perron PP I ADF PP 2, 2 SWARCH ln Ph ADF test -. PP test -.0 ADF test -. PP test -. linear trend SIC 2 % % 0%

11 a P H b ( ) ARCH AIC akaike information criterion SBC schwartz bayesian criterion R AR. AIC SBC AR AIC 11

12 AR AR AR 2 AR AR AIC SBC Adj-R AR ARCH AR Engle 2 LM ARCH ARCH R p-value AR ARCH ARCH AR -ARCH AR -ARCH 2 AR -GARCH, α 1 α 2 AR AIC SBC R AR ARMA AIC SBC ARR AR 2 AR 12

13 ARCH GARCH y t y t a 0 a y t- a 2 h t- ε t ε t Ω t-1 ~N(0,h t ) h t ω 0 ARCH ARCH 2 GARCH, a a 0.2* * ω * * * α 0.* * * 0.2 α * β * 0. p-value p-value p-value Q Q AIC SBC Centered R Log likelihood β 1 AR log likelihood AIC SBC R 2 ARCH Q white noise 13

14 Lamoureux and Lastrapes 0 ARCH GARCH SWARCH SWARCH AR -SWARCH 2, SWARCH ( ) SWARCH AR -SWARCH 2, Berndt, Hall, Hall, and Hausman BHHH g g s t s t 2.0 s t 2 s t. Cai Hamilton and Susmel ARCH GARCH GARCH Gray generalised regime-switching GRS GARCH SWARCH ARCH 0 14

15 . ARCH 2. ARCH 2. GARCH, ARCH 15

16 SWARCH y t y t a 0 a 1 y t-1 ε t, ε t Ω t-1 ~N 0,h t ε t u t, u t = u t h t ω 0 α 1 ε 2 t-1 SWARCH 2, a * a 0.00* ω * α 1 0.* 0.0 p * 0.00 p 2 0.2* 0.0 g 2.02* 0.2 AIC -. SBC -2. Centered R 2 0. Log likelihood 2. * 0.0 p ij p p duration p p 2 p 2 p 22 p 0. p i j 16

17 . 2.0 smoothing probability Chen and Patel, Chen and Patel 2002 Chen and Patel

18 AIC SBC R 2 SWARCH ARCH GARCH SWARCH ( ) ARCH in mean ARCH-M a 2 0 SWARCH 2000 Chen and Patel SWARCH SWARCH ARCH-M 18

19 ARCH-M y t y t a 0 a y t- a 2 h t- ε t ε t Ω t-1 ~N 0,h t h t ω 0 α 1 ε 2 t-1 α 2 D t D t 0 2 ARCH-M a a 0.2* 0.0 a 2.2*.2 ω * α * 0. α AIC -.2 SBC -. Centered R Log likelihood 0. *

20 ARCH GARCH time varying Markov-Switch-ARCH SWARCH ARCH-M ARCH GARCH SWARCH

21 Berndt, E. K., B. H. Hall, R. E. Hall, and J. A. Hausman, Estimation and Inference in Nonlinear Structural Models, Annals of Economic and Social Measurement, : -. Bloomfield, R. J. and J. Hales 2002, Predicting the Next Step of a Random Walk: Experimental Evidence of Regime-shifting Beliefs, Journal of Financial Economics, : -. Bollerslev, T., Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, : 0-2. Bowden, R. J. 0, Equilibrium and Disequilibrium in the Housing Market; A Survey, Paper presented at the National Housing Economics Coference, Canberra: Australian Government Publishing Service. Cai, J., A Markov Model of Switching-regime ARCH, Journal of Business and Economic Statistics, 2: 0-. Chen, M. C. and K. Patel, House Price Dynamics and Granger Causality: an Analysis of Taipei New Dwelling Market, Journal of Asian Real Estate Society, : 0-2. Chen, M. C. and K. Patel 2002, An Empirical Analysis of Determination of House Prices in the Taipei Area, Taiwan Economic Review, 0 : -. Drake, L., Modelling UK House Prices Using Cointegration: an Application of the Johansen Technique, Applied Economics, 2: Engle, R. F. 2, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 0: -00. Engle, R. F., D. Lilien, and R. Robins, Estimating Time Varying Risk Premia in the Term Structure: the ARCH-M Model, Econometrica, : -0. Giussani, B. and G. Hadjimatheou 0, House Prices: An Econometrics Model for the UK, The APEX Centre Economics Discussion Paper, No. 0, London: Kingston Polytechnic. Gray, S. F., Modelling the Conditional Distribution of Interest Rates as a Regime-switching Process, Journal of Financial Economics, 2: 2-2. Hamilton, J. D. and R. Susmel, Autoregressive Conditional Heteroskedasticity 21

22 and Change in Regime, Journal of Econometrics, : 0-. Hendry, D. F., Econometric Modelling of House Prices in the UK, in D. F. Hendry and K. F. Wallis, eds., Econometrics and Quantitative Economics, Oxford: Basil Blackwell. Holly, S. and N. Jones, House Prices Since the 0s: Cointegration, Demography and Asymmetries, Economic Modelling, : -. Lamoureux, C. G. and W. D. Lastrapes 0, Persistence in Variance, Structural Change and the GARCH Model, Journal of Business and Economic Statistics, : Malpezzi, S., A Simple Error Correction Model of House Prices, Journal of Housing Economics, : 2-2 Meen, G. P. 0, The Removal of Mortgage Market Constraints and the Implications for Econometric Modelling of UK House Prices, Oxford Bulletin Economics and Statistics, 2 : -2. Meen, G. P. 2002, The Time-Series Behavior of House Prices: A Transatlantic Divide?, Journal of Housing Economics, : -2. Whitehead, C., A Model of the UK Housing Market, Oxford Bulletin of Economics and Statistics, :

1970 Roulac (1996) (shock) (structure change) Barras and Ferguson (1985) Barras (1994) (1990) (1996) (1997) 1

1970 Roulac (1996) (shock) (structure change) Barras and Ferguson (1985) Barras (1994) (1990) (1996) (1997) 1 (1998) 1990 (Unit Root Test) (Cointegration) (Error Correction Model) 1 (1996) 2 (1990) 2 Barras

22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 摘 要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M *

22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 摘要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M *