Monte Carlo Simulation in Time Series Analysis: Cointegration

Transcription

1 Monte Carlo Simulation in Time Series Analysis: Cointegration Hector Zapata Associate Vice Provost for International Programs Past Presidents of the LSU Alumni Association Alumni Professor SOUTH CHINA AGRICULTURAL UNIVERSITY GUANGZHOU, P.R. CHINA NOVEMBER 5, /3/2015 ZAPATA VISIT TO SCAU 1

2 Table of Contents LSU What is Monte Carlo Simulation (MCS)? Why Use in Time Series Econometrics? How to Build a MCS MCS in Cointegration Analysis: Granger Non-Causality Other work in Progress Future Work 11/3/2015 ZAPATA VISIT TO SCAU 2

3 LSU 11/3/2015 ZAPATA VISIT TO SCAU 3

4 Visit the LSU Campus 11/3/2015 ZAPATA VISIT TO SCAU 4

5 Introduction to LSU 11/3/2015 ZAPATA VISIT TO SCAU 5

6 What is Monte Carlo Simulation (MCS)? 11/3/2015 ZAPATA VISIT TO SCAU 6

7 What is Monte Carlo Simulation? Simulation is a numerical technique for experimenting with a digital computer, which often involves a model describing the behavior of an economic system or a business unit over a time a time period (time series analysis interpretation). The Origins date back to Monte Carlo (pic to the right) Source: P. Wilnott: The History of Monte Carlo Simulation Methods in Financial Engineering. 11/3/2015 ZAPATA VISIT TO SCAU 7

8 What is Monte Carlo Simulation? Brief History * E. Fermini s calculation of neutron diffusion atomic bomb. Source: (Monte Carlo Casino in Monaco * S. Ulam: how likely is someone to win once you know the cards on your hand? * J Von Neumann: nuclear fission research at Los Alamos Source: (Monte Carlo Casino in Monaco 11/3/2015 ZAPATA VISIT TO SCAU 8

9 What is Monte Carlo Simulation? Applications: Nuclear physics Financial Engineering Business & Finance (risk analysis; portfolio mgt) Econometric Analysis Agricultural Economics Source: (Monte Carlo Casino in Monaco Source: (Monte Carlo Casino in Monaco 11/3/2015 ZAPATA VISIT TO SCAU 9

10 Who Uses MC Simulation? Retirement Companies Other Financial Planners Wall Street Firms Users of Futures and Options Markets General Motors (inventory mgt.; i.r.; e.r.) Teachers & Researchers Source: (Monte Carlo Casino in Monaco Source: (Monte Carlo Casino in Monaco 11/3/2015 ZAPATA VISIT TO SCAU 10

11 What is Monte Carlo Simulation? Use Random Number Generator to simulate data with from a given PDF or Model MCS uses random number generator, loops and other repetitive tasks to generate the data and compare the performance of different methods. Example: Simulating a histogram from a standard normal distribution: > x=rnorm(100,0,1); # Simple Histogram > hist(x, col= blue, main= N(0,1) Histogram xlab= X ylab= Frequency ) 11/3/2015 ZAPATA VISIT TO SCAU 11

12 What is Monte Carlo Simulation? Simple Examples 11/3/2015 ZAPATA VISIT TO SCAU 12

13 Why Use in Time Series Econometrics? 11/3/2015 ZAPATA VISIT TO SCAU 13

14 Why Use MCS in Time Series Econometrics Small sample properties of estimators of regression coefficients in time series and panel time series Cointegration Gonzalo (1994). Five Alternative Methods of Estimating long-run equilibrium relationships. Journal of Econometrics OLS NLS ML PC CC Typical Ho: All estimators perform the same. Do some behave better in small samples?- --this is very relevant in economics. 11/3/2015 ZAPATA VISIT TO SCAU 14

15 Why Use MCS in Time Series Econometrics Selected Readings Rubinstein, R. Y. Simulation and the Monte Carlo Method. John Wiley & Sons, Inc., New York, 1981 Johnson, M. E. Multivariate Statistical Simulation. John Wiley & Sons, Inc., New York, Scott, D. W. Multivariate Density Estimation: Theory, Practice and Visualization. John Wile & Sons, Inc., New York, Gentle, J. E. Random Number Generation and Monte Carlo Methods, 2 nd Edition, Springer, New York, Dagpunar, J. S. Simulation and Monte Carlo: with Applications in Finance and MCMC. John Wiley & Sons, Ltd., England, Barreto, H. and F. M. Howland, Introductory Econometrics: Using Monte Carlo Simulation with Microsoft Excel. Cambridge University Press, New York, Handbook of Econometrics, Ch. 16 Monte Carlo Experimentation in Econometrics. Elsevier, B. V. 11/3/2015 ZAPATA VISIT TO SCAU 15

16 How to Build a MCS 11/3/2015 ZAPATA VISIT TO SCAU 16

17 Example 1: How to Build a MCS STEPS Step 1. Step 2. Step 3. Step 4. Step 5. What Question(s) do you plan to investigate? In Non-Causality: How Granger Non-Causality Tests Perform in a) small samples, b) model dimension, c) degree of cointegration, direction of causality, and various dynamic structures? Construct a simulated universe Identify the model (process that you want to simulate) Model dimension (1, 2, 3, ) variables and Complexity Define the inputs (parameters of the model; sample sizes+100) Random Number Generation (decide on distribution functions to use; pre-testing) Run a small scale of the experiment and fine-tune the design Simulate and Analyze Output Step 6. Report Results based on questions in Step 1. 11/3/2015 ZAPATA VISIT TO SCAU 17

18 MCS in Cointegration Analysis: Granger Non-Causality 11/3/2015 ZAPATA VISIT TO SCAU 18

19 Example 1: MCS in Cointegration Analysis A Monte Carlo Experiment designed to study the performance of two Wald and a likelihood ratio tests for Granger non-causality in bivariate and cointegrated systems 11/3/2015 ZAPATA VISIT TO SCAU 19

20 Example 1: MCS in Cointegration Analysis In Compact Matrix Notation: 11/3/2015 ZAPATA VISIT TO SCAU 20

21 Example 1: MCS in Cointegration Analysis In the Zapata & Rambaldi Experiment the hypothesis that r < p is formulated as: 11/3/2015 ZAPATA VISIT TO SCAU 21

22 Example 1: MCS in Cointegration Analysis Zapata & Rambaldi investigated Granger Non-Causality in Model (3) for 2 and 3 variable models, that is, in: WALD TEST Z t =[ y t X 1t ] Bivariate Case: Cointegration if r=1 Z t = [ y t X 1t X 2t ] Trivariate Case: cointegration if r=1 or r=2 Non-Causality means from X to Y: The Wald Statistics for Testing Ho: where R is NxP2k, rank(r)=n, and Σφ is the var-cov of φ. 11/3/2015 ZAPATA VISIT TO SCAU 22

23 Example 1: MCS in Cointegration Analysis Likelihood Ratio Test Cointegration literature has suggested that there may be efficiency gains by imposing the cointegrating restrictions under both the null and alternative hypotheses in testing non-causality. This means that testing y -/-> X must consider not only equation (4): and restrictions on Γ and Π as: Short & Long Run Non-Causality Long Run Non-Causality Non-Causality conditional on Cointegration Chi-Square(DF1 (k=1) or DF2 (k>2) 11/3/2015 ZAPATA VISIT TO SCAU 23

24 Example 1: MCS in Cointegration Analysis WALD TEST in Augmented VAR 11/3/2015 ZAPATA VISIT TO SCAU 24

25 Example 1: MCS in Cointegration Analysis The DGPs for Non-Causality Testing in Cointegration 11/3/2015 ZAPATA VISIT TO SCAU 25

26 % Rejection of Non-Causality in Cointegration in Contemporaneous Bivariate Models, Wald and LR Tests, 5% Level. Note that for DGP 1, Y does not cause X so when testing that y-/-> X, we expected the rejection percent to be abournd 0.05%. This is highlighted in the Table. Note at the smaller the sample size (n=25), the higher the frequency of rejection, and that this frequency is the highest for MWALD. WALD LR MWALD 11/3/2015 ZAPATA VISIT TO SCAU 26

27 Results % Rejection of Non-Causality in Cointegration in Contemporaneous Bivariate Models, Wald and LR Tests, 5% Level. Note that for DGP 1, X causes Y so when testing that y-/-> X, we expected the rejection percent to be around 100%. This is highlighted in the Table. Note at the smaller the sample size (n=25), the smaller the frequency of rejection; as expected, rejection is 100% at n=400. WALD LR MWALD 11/3/2015 ZAPATA VISIT TO SCAU 27

28 Draw Main Conclusions WALD, LR, MWALD work the same when sample size n=100 or higher, As sample size gets smaller, the LR test works best, With very small samples (n=25), LR is best, With very small samples, MWALD is worst, 11/3/2015 ZAPATA VISIT TO SCAU 28

29 Other Applications 11/3/2015 ZAPATA VISIT TO SCAU 29

30 OTHER APPLICATIONS Ran, T. and H. Zapata. Mixed Unit Roots and Deterministic Trends in Noncausality Tests. Selected Paper, Southern Agricultural Economics Association, Annual Meeting, Dallas, TX, February Tan, Y. and Zapata, H. Hog Price Transmission in Global Markets: China, EU and U.S. Selected Paper, Southern Agricultural Economics Association, February Chen, R. and Zapata, H. Dynamics of Price Volatility in the China-U.S. Hog Industries. Selected Paper, Southern Agricultural Economics Association, February (Simulation) 11/3/2015 ZAPATA VISIT TO SCAU 30

31 Future Work 11/3/2015 ZAPATA VISIT TO SCAU 31

32 FUTURE RESEARCH Abstract: This article examines hog price linkages and volatility between the United States and China using monthly data from June 1996 to December Both countries are among the main producers and consumers of pork in the world, with market trends and price volatility being of daily interest to worldwide market participants. U.S. Hog futures rallied in 2013 when Smithfield Foods Inc., the world s biggest pork processor and hog producer, agreed to an almost $5 billion takeover by a Chinese company (the biggest Chinese takeover of a U.S. company). Liquidity, market activity and price volatility are closely analyzed in this market because of the increasing financial interest. This article specifically analyzes volatility and spillover effects using a MGARCH-BEKK model with error-correction mechanism. Overall, it is found that volatility in Chinese hog prices is explained by own-price volatility and past unexpected events (shocks). American hog price volatility, however, is mostly explained by its own past shocks (events in the U.S. market). One common aggregate linkage between the two markets is unidirectional volatility spillover effects from China to U.S. hog prices, paralleling the flow of hog-pork exports from the U.S. to China. 11/3/2015 ZAPATA VISIT TO SCAU 32

33 FUTURE RESEARCH The VEC-MGARCH model is represented in Equations (3)-(5): PC t = c 1 + θ 1 ecm t 1 + PA t = c 2 + θ 2 ecm t 1 + m i=1 m i=1 δ 1i PC t i + δ 2i PC t i + m i=1 m i=1 ε t = ε 1t ε ~N 0, H t 2t H t = B 0 B 0 + A ε t 1 ε t 1 A + G H t 1 G γ 1i PA t i + ε 1t γ 2i PA t i + ε 2t H t = h PC,t h PC PA,t B h PC PA,t h 0 = b 11 b 12 A = a 11 a 12 PA,t 0 b 22 a 21 a G = g g g 21 g 22 11/3/2015 ZAPATA VISIT TO SCAU 33

34 FUTURE RESEARCH THANK YOU! 11/3/2015 ZAPATA VISIT TO SCAU 34