time series project presentation_slides_new (1)

TIME SERIES MODELING OF U.S/U.K FOREIGN EXCHANGE

RATES WITH ARMA-GARCH(1,1)

Xiao XuManshi Li

Siegfried AnyomiCollege of Arts and Science

Department of StatisticsSpring, 2015

Dr. Nao MimotoDepartment of StatisticsSupervisor

2

CONTENT• DEFINITION OF TERMS• INTRODUCTION• DATA DESCRIPTION• DISTRIBUTIONAL PROPERTIES OF RETURNS• CONDTIONAL HETERSCEDASTIC MODELS WITH

ARMA MODEL• SIMULATION STUDIES• 70:30 SPLITTING AND TESTING• ROLLING PREDICTIONS OF 10 DAYS AHEAD• CORRELATION ANALYSIS OF U.S/U.K AND

U.S/EURO EXCHANGE RATE• CONCLUSIONS AND RECOMMENDATIONS• QUESTIONS AND ANSWERS

3

DEFINITON OF TERMS

• Returns: The profit or loss you have on your investments, including income and change in value. For example if you exchange $1.225/£ on Monday and sold it for $1.230/£ the next day, your return is $0.005/£. Same situation for goes for non-dividend paying stocks.

• Brownian Motion and Logarithmic Returns:Let be the price of a financial asset at time 0. Then is a known value. Let be the price of an asset at time . Then is a random variable. Theoretical setup of future price or value of the asset is

4

DEFINITON OF TERMS Cont’d

• is the mean total return on the asset• is the volatility (standard deviation) of returns

on the asset• is the mean dividend yield on the asset• is residuals. A standard normal random variable• is the returns on the asset or financial security. • Case of Exchange rates:• is returns on the exchange rates.

5

INTRODUCTION

• In this project we evaluate the performance of the GARCH model with an ARMA for modeling daily changes in logarithmic exchange rates . In particular, we consider the logarithmic returns of exchange rate sequences of the U.S. dollar / British pound. For logarithmic returns sequence we fit GARCH(1,1) model with varying numbers of parameters assuming the conditional student t-distribution and the condtional skewed student t-distribution, and attempt to replicate the empirical logarithmic exchange rate sequences via simulation. Assessing the fit of GARCH (1,1) model, we fit an ARMA(1,2) and conclude that the family of GARCH(1,1) model assuming the conditional variance follows the skewed student t-distribution with an ARMA(1,2) does adequately reflect the empirical nature of the logarithmic sequences. We further perform a multivariate analysis of the U.S. dollar / British pound and the U.S. dollar /Euro to determine degree of correlations.

6

DATA DESCRIPTION

• Title: U.S/U.K Foreign Exchange Rate (Main Dataset)

• Title: U.S/Euro Foreign Exchange Rate (Auxiliary Dataset)

• Frequency: Daily

• Units: U.S Dollars to One Bristish Pound• Units: U.S Dollars to One Euro

• Date Range: January 1, 2012 to December 31, 2014• 784 trading days in total

7

DISTRIBUTIONAL PROPERTIES OF RETURNS• Original Plot U.S/U.K Exchange Rate

Obvious upward and downward sloping trend towards the end of 2014. The Geometric Brownian Motion

8

DISTRIBUTIONAL PROPERTIES OF RETURNS Cont’d

• ACF of Original Data

Evidence of serial correlations among the observations.

9


• Plot of Logarithmic Exchange Returns

The returns looks much stationary but with random fluctuations

10


• Histogram of the Logarithmic Exchange Returns

• Logarithmic Exchange returns looks normal

11


• ACF of Logarithmic Exchange Returns

• Evidence of serial correlations among returns. GARCH model needed.

12


• PACF of Logarithmic Exchange Returns

• Evidence of short-term to long term dependencies. GARCH model fitting appropriate

13

CONDITIONAL HETEROSCEDASTIC MODELS

• GARCH (1,1) Model using conditional student t- distribution with ARMA(1,2)

14

CONDITIONAL HETEROSCEDASTIC MODELS Cont’d

• Randomness and Residual Analysis of GARCH(1,1) using condition student t-distribution with ARMA(1,2)

• Absence of serial correlations but Jaque-Bera is barely 0. Residuals don’t look normal.

15


• GARCH(1,1) Model for skewed student t-distribution with ARMA (1,2)

16


• Randomness and Residual Analysis of GARCH(1,1) using skewed student t-distribution with ARMA(1,2)

• Obvious disappearance of serial correlations• Jaque-Bera test shows residuals are normal

17

SIMULATION STUDIES FOR EXCHANGE RETURNS OF FITTED GARCH MODELS Cont’d

• Parameter Estimates From Model vs. Simulation for conditional t-distribution:

Model parameters Simulation parameters

Omega: 8.0849*e-(07) Omega: 8.4122*e-(07)

Alpha1: 8.6439*e-(03) Alpha1: 8.31121*e-(03)

Beta1: 9.5263*e-(01) Beta1: 8.50192*e-(01)

Shape: 3.6043 Shape: 3.5934

AR1: 1.6965*e(-02) AR1: 1.3677*e(-03)

MA1: -4.4912*e(-02) MA1: -5.1439*e(-02)

MA2: 2.2117*e(-02) MA2: 3.4512*e(-02)

18

SIMULATION STUDIES FOR EXCHANGE RETURNS OF FITTED GARCH MODELS Cont’d

• Parameter Estimates From Model vs. Simulation for conditional skewed t-distribution:

• Simulated parameter estimated using the skewed conditional student t-distribution looks more like their actual model parameters

Model parameters Simulation parametersOmega: 7.7053 *e-(07) Omega: 7.65801*e(-07)Alpha1: 8.3826*e-(03) Alpha1: 8.2649*e(-03)Beta1: 9.5522*e-(01) Beta1: 8.85961*e(-01)Shape: 3.5275 Shape: 3.4921Skewness: 8.9866*e(-01) Skewness: 9.2107*e(-01)AR1: 3.0412*e(-02) AR1:3.1239*e(-02) MA1: -5.5686*e(-02) MA1: -5.5481*e(-02)MA2: 1.4140*e(-02) MA2: 1.3775*e(-02)

19

MONTE CARLO SIMULATION STUDIES FOR EXCHANGE RETURNS OF FITTED ARMA-GARCH MODELS Cont’d

• Estimated rMSE of Parameters from Simulation

• The rMSE of simulated parameters from GARCH(1,1) using conditional skewed student t-distribution looks smaller than conditional student t-distribution

Conditional t-Dsitribution with ARMA

Skewed Conditional – t-Distribution with ARMA

Omega: 2.3113*e(-05) Omega: 2.1451*e(-05)Alpha1: 2.534*e(-01) Alpha1: 1.8390*e(-01)Beta1: 4.6822*e(-01) Beta1: 4.7334*e(-01)Shape: 15.7102*e(00) Shape: 14.8657*e(00)AR1: 1.8923*e(00) AR1:1.4819*e(00)MA1: 2.3471*e(00) MA1: 2.206*e(00)MA2: 9.7426*e(-01) MA2:8.239*e(-01)

Skewness: 1.13008*e(-01)

20

DOES THE DATA TRULY COMES FROM SSTD??

• Kolmogorov Sminorv Test Analysis

• Compared with empirical observations, sstd fits well.

21

MONTE CARLO SIMULATION STUDIES FOR EXCHANGE RETURNS OF FITTED ARMA-GARCH MODEL Cont’dSimulation Studies and Kolmogorov Smirnov Test Results

ARMA(1,2)-GARCH(1,1) from skewed conditional student t-distribution is better than the conditional student t-distribution case, so it is the appropriate and chosen model!

22

70:30 Splitting and Rolling Prediction from GARCH(1,1) skewed student t-distribution with ARMA(1,2)• Volatility Prediction

23

70:30 Splitting and Rolling Prediction from ARMA(1,2)-GARCH(1,1) skewed student t-distribution• CONDITIONAL MEAN PREDICTION

24

70:30 Splitting and Rolling Prediction from GARCH(1,1) skewed student t-distribution Cont’d

• Randomness and Residuals of Rolling Prediction

• Jaque-Bera Test looks great!!! 70% of the dataset seemingly captures volatility behavior

25

ROLLING PREDICTION OF 10 DAYS AHEAD

• Our ARMA(1,2)-GARCH(1,1) Model using conditional skewed student t-Distribution

26

CORRELATION ANALYSIS OF U.S/U.K AND U.S/EURO

• Plot of Returns of U.S/U.K and U.S/EU

• Both returns show similar stationary trending relationship

27

CORRELATION ANALYSIS OF U.S/U.K AND U.S/EURO Cont’d

• ACF Plot of the Combined Returns

• Both returns are highly correlated• Profit making in pounds and Euro highly

depends on the U.S. economy.

28

CONCLUSIONS

• ARMA(1,2)-GARCH(1,1) adequately fit the U.S/U.K Exchange rates.

• Dollar/Pound and Dollar/Euro are highly correlated.

• The performance of the Pound and the Euro may depend on the U.S. dollar and possibly the U.S. economy.

29

RECOMMENDATIONS

• In the context of time series of financial returns, the modeling of the lower tail of their distributions is the primary focus of risk measures. Value at Risk is the probability the one might lose with a given level of probability .

• Future work will seek to estimate the Value at Risk based the the ARMA(1,2)-GARCH(1,1) model.

Thank You!

Questions

30

time series project presentation_slides_new (1)

Documents