var group project

7/30/2019 VAR Group Project

1/19

Vector Autoregressive Modeland

Mixture Normal Distribution

Academic Year: 2012/2013

Course Code: ST425

Course Title: Statistical Inference

Professor: Qiwei Yao

Group Members:

Feng Huang, Jinyu Guo,

Oliver Gannon, Shaotong He

1


2/19

Contents

Part I VAR Model

1.1 Introduction

1.2 VAR Model

1.3 Developments in VAR

1.4 Conclusion

Part II Mixture Normal Distribution2.1 R function for generating random numbers from the mixture normal distribution

2.2 Execute the algorithm with p = 1/2 and sigma = 0.25

2.3 R function of algorithm to find the MLE

2.4 Find the value of MLE for an existing data set

Part Example of VAR Model3.1 Example: VAR model in stock markets

List of Tables and Figures

2.1 Descriptive Statistics from data generated in Part 2.1

2.2: Histogram of the Mixture Normal Distribution

2.3: QQ-plot against standard Normal Distribution

2.4 The results of the iterative algorithm found in Part 2.3

3.1: Logarithm plot of weekly closing price

3.2: ADF Stationary Test

3.3: Correlation matrix of residuals

3.4: Diagram of fit and residuals for SP, FTSE and NEKKEI

3.5: One unit impulse response test for SP, FTSE and NEKKEI

2


3/19

Part 1: VAR Model

1.1 Introduction

Complex relationships define the world that we inhabit. Whether it is economic

data or financial data, accurate analysis and forecasting of time series data

requires an account of these relationships and thus should not be confined to

merely univariate autoregressive processes. By restricting attention to only

univariate models, valuable information and analysis may be lost in the

analysis and forecasting of data. Thus, the information that results from the

model may be incorrect and compromise the validity of its findings. Therefore,

a multivariate alternative called the Vector Autoregression model1 has been

developed to combat this issue and has the changed the way complex time

series relationships have been modelled and analysed.

The use of VARs in analysing questions such as the effect of government spending on GDP bears

several advantages that enable a more definitive answer than previously possible. Stock and Watson

(2001) state that VARs capture co-movements in variables that would be left undetected usingunivariate or bivariate models. In addition, the fact that VAR is relatively free from constraining

assumptions enables the VAR model to depict a more realistic picture of complex economic

relationships.

1.2 VAR Model

The VAR model is based on the premise that the current value of a variable is a result of the value

of the variable in the previous period. In this paper, the VAR model takes the following form:

yt

= c + 1y

t1+

2y

t2+ ... +

py

t p+ x

t

Equation (1)

Where c is a constant vector of dimention (nx1), yt is a vector of endogenous variables, yt-1 is a

vector of the lags,

is the coefficient vectors,

is the residual vector,

denotes the exogenous

coefficients and x the exogenous variable. To ensure valid inferences can be accrued from this

1 Henceforth, VAR.3


4/19

process, stationarity, indicating the non-presence of a unit root, must be found in the data.

Stationarity ensues from the following assumptions:

Equation (2)

In order to test the validity of the VAR model, several tests are used to establish whether the VAR

model can be accurately used in analysis and forecasting of data.

As an example, VAR analysis was used to explore the interrelationship between three important

stock markets indices in the world, R is used to build a suitable VAR model for estimation and

further research. The analysis will be presented in the appendix and in the presentation.

1.3 Developments in VAR

VAR is an extremely versatile method that has led to many developments based upon it that has

furthered understanding in many problems confronted by Economists and Statisticians. One of

foremost methods of analysis is the test for causality. The two most prominent methods of analysis

is the use of Granger Causality and Impulse Response Functions.

Granger Causality allows for basic analysis of variable that Granger causes another variable. The

definition of Granger causality differs somewhat from true causality. It states that variations over

time in variable zt can help predict future variations in the variable x t. Whilst it provides an initial

test for causality, the direction of said causality can not be easily ascertained and requires some

interpretation of the results since VAR characterises a joint distribution of the variables and

stochastic interrelationships makes it difficult to find the exact nature of the relationships.Moreover, Granger Causality utilises the information available in the past to make predictions of

the future. Thus, the relationships may not necessarily hold in the future.

The major downfall of Granger Causality is that it only considers the effect of one variable on

another in isolation and may fail to encompass the effect of other variables given that the VAR

model is a dynamic system of equations. Therefore, the use of Impulse Response functions allows

us to generate a more complete picture of complex interactions between variables. Thus, Impulse

Response Functions, which can be carried out by any standard analysis software such as STATA,

4


5/19

EViews, and R, allows us to consider the effect of a variable in response to a unit shock to a

variable in a dynamic system. But one major part of the IRF and its correct interpretation is a well

specified model meaning that any time there are variables that are ommitted then this may distort

the results.

In a further, important, development to the VAR methodology is the Structual VAR model.

Typically, different sets of impulses for the impulse response functions can be produced from the

same VAR model it means that, accordining to Lutkepohl (2005), non-sample data may generate

exogenous innovations that could affect the stationary model. SVARs seek to model economic

models with greater precision and accuracy for Statistical Analysis. Economic models can be

thought of as restrictions on stochastic processes where the observables are available to the

researcher and then there are random variables that correspond to the history of shocks and amapping from the shocks where consumer and firm decisions are optimised at every stage of

history.

For example, if we consider the neoclassical growth model. It consists of

optimal investment and labour supply decisions of the households and

historical data for productivity. If the productivity is normally distributed then it

can be linearised by setting yt = D(L)wt , If the roots exist for the matrix D(L)

then it is possible to invert the matrix. The system is represented by A(L)yt =

wt. A(L) is of infinite order. This is a structural VAR representation where a VAR

model is generated from an economic relationship2. This furthers the vision of

Sims (1980) by allowing VAR to analyse causal relations, policy analysis and

the testing of economic theories.

1.4 Conclusion

Therefore, VAR affords researchers the prospect of analysing complex simultaneous relationships

that prevail in the real world with greater precision and accuracy. VAR allows Statistical and

Economic research to go beyond merely analysing the world in a univariate manner. It has allowed

2These restrictions can be even modelled to be long-run and short run as seen in Blanchard

and Quah (1989).5


6/19

breakthroughs in many fields of research that include the effects of monetary policy, the effects of

productivity on hours work and the analysis of Government spending.

Part 2: Mixture Normal Distribution

2.1 Generate random numbers from mixture normal

distributions by R

> mix.norm z=seq(-3,3,0.001)

> lines(z,dnorm(z,mean(m),sqrt(var(m))),col="blue",lwd=2)

6


7/19

Histogram of Mixture Normal Distribution

m

Density

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.30

.4

0.5

0.6

0.7

Figure (2.2): Histogram of the Mixture Normal Distribution.

Draw the graph of the distribution against the standard normal distribution as shown in Figure (2)

> qqnorm(m,xlab="Normal quantiles",ylab="Quantiles of Mixture Normal Distribution")

> qqline(m,col="red")

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Normal Q-Q Plot

Normal quantiles

QuantilesofMixtureNorm

alDistribution

Figure (2.3): QQ-plot against the Standard Normal Distribution.

In the histogram, there are more random numbers that significantly vary from the average giving

more frequent high and low values compared with a standard normal distribution. This indicates a

fat-tail phenomenon.

Furthermore, as it shows in Normal Q-Q Plot, 2 sides of the random numbers line are going away

7


8/19

from the Normal Distribution line. This reflects that the mixture normal distribution is a fat-tail

distribution.

Fat-tail distributions are widely used in finance and insurance industries. The distributions result

in additional risk. Since fat tails can be considered as when events with a low probability actually

do occur, people will tend to overestimate the likelihood that they will occur again. In market

conditions, when taking investment behavior into account, investment strategies often differ from

expected predictions. For instance the distribution of monthly or daily market returns always show

as fat tails distribution; and with it more lower and higher instances of extreme values.

2.3 R function of algorithm to find the for the MLE

Underlying Theory of the algorithm

Since it is typically difficult to find an explicit analytic form for the maximum likelihood estimator,

we build an iterative procedure starting with an initial value and then steadily improve upon it after

each iteration. The estimator is considered to be found when it has become numerically stable. We

use Newton-Raphson algorithm to solve.

Suppose is close to the true value .

Apply Taylor expansion,

Therefore,

Then use iterative estimators

for , where is a prescribed initial value. We define

if and differ by a small amount.

From the algorithm, we developed two methods to estimate MLE in R.

Version 1

MLE=function(p,init,Tiny){

i=0

y0=init+Tiny

8


9/19

y1=init

while(abs(y1-y0)>Tiny){

y0=y1

n


10/19

S0=A/C

S1=(B*C-A^2)/(C^2)

s0=mean(S0)

s1=mean(S1)

y1=y0-s0/s1

i=i+1

cat(i,"iteration:",y1,"\n")

}

cat("MLE:",y1,"\n","No. of iterations:",i,"\n")

}

Avoiding loops, the second method can work more efficiently. Therefore, Version 2 is preferred to

the Version 1.

2.4 Find the value of MLE for an existing data set

The iterative algorithm found in Question 3 was implemented below and the results are shown in

Table (2).

> source("E:\\st425\\project\\Q3_2.R")> x x1 s MLE(0.5,s,0.01)

Iteration Number Result of MLE

1 1.648325

2 1.859769

3 1.916065

4 1.919343

MLE 1.919343

Table (2.4): Results of the iterative algorithm found in question 3.

10


11/19

Appendix

Example: VAR model in stock markets

Data: weekly closing price of S&P 500, FTSE 100, and NEKKEI 225 from 02/04/1984 to

22/10/2012 (from Yahoo Finance)

Figure (3.1): Logarithm plot of weekly closing price

To explore the interrelationship between these three important stock markets index in the world, R

is used to build a suitable VAR model for estimation and further research.

> library("vars")

> logprice=read.csv("E:/LSE/Inference/group project/logprice.csv",header=T,row.names=1)> attach(logprice)

11


12/19

From the data plot, original data are not stationary. After data pretreatment, we use ADF stationary

test is applied to test stationarity. The results (Table 1) show that the stationarity only exists when

using the return (first difference of the logarithm of prices) in all three markets, but not the

logarithm of prices. Statistics imply that all returns are at 99% confidence level that these series are

stationary. As a result, the return series will be used to build VAR model for further analysis.

Table 1 ADF Stationary Test

Variable lag True ValueCritical value

1% 5% 10%

SP 2 -1.718 -3.96 -3.41 -3.12

SP 1 -26.327 -3.43 -2.86 -2.57

FTSE 2 -2.225 -3.96 -3.41 -3.12

FTSE 1 -25.286 -3.43 -2.86 -2.57

NEKKEI 2 -2.927 -3.96 -3.41 -3.12

NEKKEI 1 -25.456 -3.43 -2.86 -2.57

Table (3.2): ADF Stationary Test

> adfsp1 |t|)

(Intercept) 2.312e-02 1.185e-02 1.951 0.0512 .z.lag.1 -3.718e-03 2.165e-03 -1.718 0.0861 .tt 3.414e-06 3.406e-06 1.002 0.3164

z.diff.lag1 -4.021e-02 2.593e-02 -1.551 0.1212z.diff.lag2 5.573e-02 2.592e-02 2.150 0.0317 *

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.02347 on 1483 degrees of freedomMultiple R-squared: 0.0081, Adjusted R-squared: 0.005425F-statistic: 3.028 on 4 and 1483 DF, p-value: 0.01684

Value of test-statistic is: -1.7176 3.461 2.377

Critical values for test statistics:

1pct 5pct 10pcttau3 -3.96 -3.41 -3.12

phi2 6.09 4.68 4.03

phi3 8.27 6.25 5.34

As the following five tests are similar to the above one, the results are left out.

12


13/19

> adfftse1 adfnekkei1 #FIRST DIFFERENCE> adfsp2 adfftse2 adfnekkei2 #LAG SELECTION USING RETURN> VARselect(return, lag.max = 8, type = "both")$selection

AIC(n) HQ(n) SC(n) FPE(n)2 1 1 2

$criteria1 2 3 4 5

AIC(n) -2.307133e+01 -2.307818e+01 -2.307348e+01 -2.306934e+01 -2.306699e+01HQ(n) -2.305133e+01 -2.304618e+01 -2.302947e+01 -2.301334e+01 -2.299899e+01SC(n) -2.301768e+01 -2.299233e+01 -2.295543e+01 -2.291910e+01 -2.288457e+01

FPE(n) 9.555370e-11 9.490137e-11 9.534923e-11 9.574476e-11 9.596949e-116 7 8

AIC(n) -2.306183e+01 -2.306401e+01 -2.305867e+01HQ(n) -2.298183e+01 -2.297200e+01 -2.295467e+01SC(n) -2.284721e+01 -2.281719e+01 -2.277966e+01

FPE(n) 9.646635e-11 9.625715e-11 9.677221e-11

The estimation of this VAR(1) model is as follows:

> returns varreturn summary(varreturn)

VAR Estimation Results:=========================

Endogenous variables: SP, FTSE, NEKKEI

Deterministic variables: bothSample size: 1489

Log Likelihood: 10849.391

Roots of the characteristic polynomial:0.1877 0.1042 0.1042

Call:VAR(y = returns, p = 1, type = "both")

Estimation results for equation SP:===================================SP = SP.l1 + FTSE.l1 + NEKKEI.l1 + const + trend

Estimate Std. Error t value Pr(>|t|)SP.l1 -3.903e-02 3.706e-02 -1.053 0.2925

13


14/19

FTSE.l1 2.989e-02 3.630e-02 0.823 0.4104NEKKEI.l1 -4.716e-02 2.427e-02 -1.943 0.0522 .

const 3.033e-03 1.222e-03 2.482 0.0132 *trend -2.066e-06 1.418e-06 -1.457 0.1453

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.02349 on 1484 degrees of freedomMultiple R-Squared: 0.005853, Adjusted R-squared: 0.003174F-statistic: 2.184 on 4 and 1484 DF, p-value: 0.06856

Estimation results for equation FTSE:=====================================

FTSE = SP.l1 + FTSE.l1 + NEKKEI.l1 + const + trend

Estimate Std. Error t value Pr(>|t|)SP.l1 2.164e-01 3.748e-02 5.774 9.41e-09 ***

FTSE.l1 -1.428e-01 3.671e-02 -3.889 0.000105 ***

NEKKEI.l1 -7.695e-02 2.454e-02 -3.135 0.001751 **const 2.358e-03 1.236e-03 1.908 0.056576 .trend -1.916e-06 1.434e-06 -1.337 0.181544---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.02376 on 1484 degrees of freedom

Multiple R-Squared: 0.02688, Adjusted R-squared: 0.02426F-statistic: 10.25 on 4 and 1484 DF, p-value: 3.473e-08

Estimation results for equation NEKKEI:=======================================

NEKKEI = SP.l1 + FTSE.l1 + NEKKEI.l1 + const + trend

Estimate Std. Error t value Pr(>|t|)SP.l1 2.114e-01 4.517e-02 4.680 3.13e-06 ***

FTSE.l1 1.749e-02 4.424e-02 0.395 0.692618NEKKEI.l1 -1.133e-01 2.958e-02 -3.830 0.000133 ***const 1.183e-03 1.489e-03 0.794 0.427157trend -2.243e-06 1.728e-06 -1.298 0.194412

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.02863 on 1484 degrees of freedomMultiple R-Squared: 0.0278, Adjusted R-squared: 0.02518F-statistic: 10.61 on 4 and 1484 DF, p-value: 1.778e-08

Covariance matrix of residuals:SP FTSE NEKKEI

SP 0.0005519 0.0003961 0.0003146FTSE 0.0003961 0.0005645 0.0003102

NEKKEI 0.0003146 0.0003102 0.0008197

14


15/19

SP FTSE NEKKEI

SP 1.0000 0.7097 0.4677

FTSE 0.7097 1.0000 0.4561

NEKKEI 0.4677 0.4561 1.0000

Table(3.3): Correlation matrix of residuals

The correlation matrix tells that all three markets are positively correlated, consistent with thefinancial facts. However, the connection between markets is not so strong. The diagram of fit the

residuals for each market are presented below. The large fluctuation marked is caused by the 2008

global financial crisis. From the graphs, we can tell that the fluctuation on the negative side seems

have heavier tail than that on the positive side.

> plot(varreturn,names="SP")> plot(varreturn,names="FTSE")> plot(varreturn,names="NEKKEI")

-0.

20

-0.

10

0.

00

0.

10

Diagram of fit and residuals for SP

0 500 1000 1500

-0.

20

-0.

10

0.0

0

0.1

0

0 2 4 6 8 10 12

0.0

1.

0

ACF Residuals

2 4 6 8 10 12-0.

06

PACF Residuals

-0.

2

-0.

1

0.

0

0

.1

Diagram of fit and residuals for FTSE

0 500 1000 1500

-0.

20

-0.

10

0.0

0

0.1

0

0 2 4 6 8 10 12

0.0

1.

0

ACF Residuals

2 4 6 8 10 12-0.

10

PACF Residuals

-0.2

-0.1

0.0

0.1

Diagram of fit and residuals for NEKKEI

0 500 1000 1500

-0.2

-0.1

0.

0

0.

1

0 2 4 6 8 10 12

0.0

1.0

ACF Residuals

2 4 6 8 10 12

-0.

05

PACF Residuals

Figure (3.4): Diagram of fit and residuals for SP, FTSE and NEKKEI

Having the VAR(1) model, we can do the Granger Causality Test. Results imply that only S&P 500Granger-cause the other markets, while neither FTSE 100 nor NEKKEI 225 has influence on other

markets. Possible reasons may be as follows-- These three indexes represent the stock market in

15


16/19

US, UK, and Japan, who play significant roles of global economy. Basically, these three markets are

connected with each other at different level of interdependence relationship. It makes the global

financial systems work efficiently, and fits the trend of the economy. Comparing the other two

markets, US stock exchange is the most mature market in the world, and S&P 500 contains more

international stocks, which plays a significant role in the home markets. When a significant change

happens in US markets, the international stocks are being influenced, which will affect the

performance of these stocks in their home market, like UK market. As long as these leading stocksbeing affected, the whole UK market will be influenced. US stock market is more powerful than

any markets in the world. As a result, the impact of the collapse of US market is more serious than

that of UK or Japan markets, considering the dominate status of US market.

> #GRANGER CAUSALITY TEST> causality(varreturn,cause="SP")$Granger

Granger causality H0: SP do not Granger-cause FTSE NEKKEIdata: VAR object varreturn

F-Test = 19.3138, df1 = 2, df2 = 4452, p-value = 4.449e-09

$Instant

H0: No instantaneous causality between: SP and FTSE NEKKEI

data: VAR object varreturnChi-squared = 515.7039, df = 2, p-value < 2.2e-16

> causality(varreturn,cause="FTSE")$Granger

Granger causality H0: FTSE do not Granger-cause SP NEKKEI

data: VAR object varreturn

F-Test = 0.339, df1 = 2, df2 = 4452, p-value = 0.7125

$InstantH0: No instantaneous causality between: FTSE and SP NEKKEI


> causality(varreturn,cause="NEKKEI")$Granger

Granger causality H0: NEKKEI do not Granger-cause SP FTSE

data: VAR object varreturnF-Test = 4.9951, df1 = 2, df2 = 4452, p-value = 0.006809

$Instant

H0: No instantaneous causality between: NEKKEI and SP FTSE


16


17/19

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of SP to SP

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10

Response of FTSE to SP

-.1

.0

.1

.2

.3

.4

1 2 3 4 5 6 7 8 9 10

Response of NEKKEI to SP

Response to Nonfactorized One Unit Innovations 2 S.E.

-.08

-.04

.00

.04

.08

.12

1 2 3 4 5 6 7 8 9 10

Response of SP to FTSE

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of FTSE to FTSE

-.08

-.04

.00

.04

.08

.12

1 2 3 4 5 6 7 8 9 10

Response of NEKKEI to FTSE


-.10

-.08

-.06

-.04

-.02

.00

.02

1 2 3 4 5 6 7 8 9 10

Response of SP to NEKKEI

-.16

-.12

-.08

-.04

.00

.04

1 2 3 4 5 6 7 8 9 10

Response of FTSE to NEKKEI

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of NEKKEI to NEKKEI


Figure (3.5): One unit impulse response test for SP, FTSE and NEKKEI

Results from the one unit impulse response tests are shown above. The responses of all three

markets when S&P 500, FTSE 100 and NEKKEI 225 receive an external unit shock are presented

in each column. When the shock just be given (first period), the impact remains in the own market

only. During the second period, the influences of the shock begin to spread to the other markets.

From the test results, we can conclude that the impact approximately lasts for four periods (four

weeks) after different level of fluctuation. The impact of US market on others is the deepest, since

the fluctuations are largest when S&P 500 being shocked. This finding is identical to the Grangercausality test above, that US stock market has the most powerful effect on the global finance.

AT THE END

17


18/19

In this report, we introduced VAR model, including model background, model structure, model

development and model application. Whats more, we conducted further research on mixture

normal distribution, including generating random numbers form mixture normal distribution,

estimate parameter of the distribution by MLE and fat tail phenomenon in real financial market

world.

We would like to present our sincere thanks to Professor Qiwei Yao for what he taught in this

course and suggested for the project. At the same time, we would also thank the assistant Helen for

her help in R learning and solving problem.

At the end, thanks for the efforts every member made for the project. Please let us know if there is

anyone interested in more about our project.

Thanks.

Bibliography

Lutkephol, H. (2005) New Introduction to Multiple Time Series Analysis, 1st Ed.,

Springer: New York

Juselius, K. (2009) The Cointegrated VAR Model, 1st Ed., Oxford University Press:

Oxford.

Hamilton, J (1994) Time Series Analysis.,1st Ed., Princeton University Press:

Princeton, New Jersey

Stock, J. and Watson, M. (2001) Vector Autoregressions., The Journal of Economic

Perspectives, 15(4), pp. 101-115

Watson, M. (1994) Vector Autoregressions and Cointegration In. Handbook of

Econometrics, Volume IV. Ed. R. Engle and D.McFadden, pp. 2843-2910, Elsevier:

Amsterdam.

Granger, C. (1969) Investigating Causal Relations by Econometric Models and Cross-

Spectral Methods, Econometrica, 37, pp. 424-438

Fernndez-Vilaverde, J. and Rubio-Ramirez J., No Date, Structural Vector Autoregressions.

[Online] Available at: [Accessed 7thDecember 2012].

18
http://economics.sas.upenn.edu/~jesusfv/svars_format.pdfhttp://economics.sas.upenn.edu/~jesusfv/svars_format.pdf


19/19

Barro, R. (2006) Rate Disasters and asset markets in the 20th Century, Quarterly Journal of

Economics121, pp. 823-866

Nordhaus, W. (2011) Elementary Statistics of Tail Events, [Online] Available at:

[Accessed 9th December

2012]

King, R., Plosser, C., Stock J. and Watson, M. (1991) Stochastic Trends and Economic

Fluctuations, American Economic Association, 81(4), pp. 819-840

Gali, J. (1999) Technology, Employment and the Business Cycle: Do Technology Shocks Explain

Aggregate Fluctuations, American Economic Association, 89(1), pp. 249-271

Sims, C. (1980) Macroeconomics and Reality, Econometrica, 48(1), pp. 1-48

Blanchard, O. and Quah, D. (1989) The Dynamic Effects of Aggregate Demand andSupply Disturbances, American Economic Review, 79(4), pp. 655-673Browning, E.S.(2007-10-15). "Exorcising Ghosts of Octobers Past". The Wall Street Journal(Dow Jones & Company):

pp. C1C2. Retrieved 2007-10-15.

"Dow Jones biggest percentage declines". South Florida Sun-Sentinel. 2008-09-30.

19
http://nordhaus.econ.yale.edu/documents/statisticsoftailevents.pdfhttp://online.wsj.com/article/SB119239926667758592.html?mod=mkts_main_news_hs_hhttp://www.sun-sentinel.com/business/sfl-flzdowbox0930sbsep30,0,7864544.storyhttp://www.sun-sentinel.com/business/sfl-flzdowbox0930sbsep30,0,7864544.storyhttp://nordhaus.econ.yale.edu/documents/statisticsoftailevents.pdfhttp://online.wsj.com/article/SB119239926667758592.html?mod=mkts_main_news_hs_hhttp://www.sun-sentinel.com/business/sfl-flzdowbox0930sbsep30,0,7864544.story

var group project

Documents