EF 5070: FinancialEconometrics
Problem Set 4: Mock Final

Due Dec 14th, 2018

Section A (40%)


Attempt ALL questions from this Section


1. (20 pts) Answer briefly the following questions



a) How to understand volatility? How to understand uncertain and risk?



b) Describe one test statistics that can be used to evaluate model adequacy and explain how it works using plain words.



c) Provide two reasons for the observed significant first order autocorrelation of high frequency data.



2. (20 pts) Suppose that simple return of a monthly stock index follows an AR dynamics,



image.png



a) What is the mean of the simple return of this monthly stock?



b) What is the variance of the simple return of this monthly stock?



c) Consider the forecast origin h = 100 with

image.png

. Compute the 1-step-ahead forecast of the simple return at the forecast origin h = 100 and the variance of your forecast error.



d) Compute 2-step-ahead forecast at the forecast origin h=100.



e) Provide two features of AR models.



Section B (60%)



Attempt ALL questions from this Section


1. (30 pts) Consider the daily returns of Apple (AAPL) stock from October 1st, 2009 to September 30, 2017. Let Pt denote daily closing prices. We denote rt as the percentage log returns of daily Apply stock.

a) Provide two reasons why we prefer log returns over price levels in regression analysis.



b) Based on R output Q(b), can you ensure stationarity of the return series? Provide your reasoning.



c) From the R output in appendix Q3 (c), we fit a Gaussian AR (1) model to the rt series. Do you find strong evidence on the predictability ability that lagged returns exhibit on current returns at 5% significance level? Form your hypothesis, write down the test statistics, asymptotic distribution of your test statistics, rejection rule and conclusion.



d) Now, we want to fit an MA model to the rt series. From the R output in appendix Q3 (d) and Figure Q3-1 below, what is the best MA(q) model? Why? Write down the fitted MA(q) model.



案例r语言案例r语言
a) From the Figure Q3-2 below, do you think we should model volatility processes? Why? Now, we fit an GARCH model to the rt series. Please explain why GARCH models can capture the well-known volatility clustering effect.



案例r语言.png

Figure Q3-2



b) Now, we will fit a GARCH(1,1) model to the return series. From R output in appendix Q3 (e)-(f), please write down the fitted model and its equivalent ARMA representation. Is the GARCH(1,1) model estimated in f) adequate?



c) From the R output in appendix Q3 (e)-(f), do you observe significant impact that historical conditional volatility may exhibit on current conditional volatility? Write down your hypothesis, test statistics, rejection rule and conclusion. Does your model satisfy stationary conditions?



1. (30 pts) We will investigate the role nonlinearity plays in our regression analysis.



a) Now, we consider another simple nonlinear model that allows us to ine asymmetric pattern in constant effect as follows,



image.png

From the R output in appendix Q4 (a), write down the fitted model of (Q4-1) and provide interpretations for x .



a) Do you find statistically significant evidence for asymmetry in the constant effect at 10% level? Form your hypothesis, write down the test statistics, rejection rule and conclusion.



b) Do you find statistically significant evidence for the predictability ability that lagged one period return exhibit on current return at 5% level? Form your hypothesis, write down the test statistics, rejection rule and conclusion.



Now, we consider a two-stage Markovian chain Regime-switching model for the market model.

image.png



a) From the R output in appendix Q4 (d)-(f), write down the fitted model.



b) Compute the unconditional volatility of the model in stage 1 and state 2. Which regime has higher uncertainty?



c) From the R output in appendix Q4 (d)-(f), compute the expected duration for stage 1 and stage 2 respectively.



Appendix 1:
R Output Appendix

> library(quantmod)

> library(tseries)

> library(TSA)

> getSymbols(‘AAPL’,from=’2009-10-01′,to=’2017-09-30′)

[1] “AAPL”
> head(AAPL)

AAPL.Open AAPL.High AAPL.Low AAPL.Close AAPL.Volume AAPL.Adjusted



2009-10-01 29.538 29.676 28.797 25.83714 131177900 23.16128
2009-10-02 28.910 29.632 28.900 26.41429 138327000 23.67865
2009-10-05 29.673 29.778 29.366 26.57429 105783300 23.82208
2009-10-06 29.919 30.280 29.848 27.14428 151271400 24.33305
2009-10-07 30.240 30.366 30.124 27.17857 116417000 24.36378
2009-10-08 30.384 30.510 30.102 27.03857 109552800 24.23828
> tail(AAPL)
AAPL.Open AAPL.High AAPL.Low AAPL.Close AAPL.Volume AAPL.Adjusted



2017-09-22 152.085 152.817 151.101 151.89 46645400 151.3459
2017-09-25 150.529 152.376 149.696 150.55 44387300 150.0107
2017-09-26 152.326 154.473 152.235 153.14 36660000 152.5914
2017-09-27 154.353 155.276 154.092 154.23 25504200 153.6776
2017-09-28 154.443 154.835 153.249 153.28 22005500 152.7310
2017-09-29 153.761 154.684 152.546 154.12 26299800 153.5679
> AAPLP=as.numeric(AAPL$AAPL.Close)

>

> #Form percentage log returns

> lnrt=diff(log(AAPLP))*100

> head(lnrt)

[1] 2.2091892 0.6039056 2.1222470 0.1262342 -0.5164430 0.6320077
>



##Q3(b)

> #(b)

> Box.test(lnrt)



Box-Pierce test



data: lnrt

X-squared = 1.307, df = 1, p-value = 0.2529



> adf.test(lnrt,k=20)



Augmented Dickey-Fuller Test



data: lnrt

Dickey-Fuller = -9.2706, Lag order = 20, p-value = 0.01 alternative hypothesis: stationary



##Q3(c)##

> #(c)

> mc=arima(lnrt,order=c(1,0,0))

> mc



Coefficients:

ar1 intercept 0.0255 0.0888

s.e. 0.0223 0.0368



##Q3 (d)

> md=arima(lnrt,order=c(0,0,1))

> md



Coefficients:

ma1 intercept 0.0255 0.0887

s.e. 0.0223 0.0368



#Q3 (e)-(f)

> library(‘fGarch’)

> mf=garchFit(~garch(1,1),data=lnrt,trace=F)

> summary(mf)



Std. Errors:



based on Hessian Error Analysis:



Standardised Residuals Tests:

Statistic p-Value

Jarque-Bera Test R Chi^2 1344.524 0
Shapiro-Wilk Test R W 0.9621468 0
Ljung-Box Test R Q(10) 11.66765 0.3079153
Ljung-Box Test R Q(15) 16.84009 0.3285136
Ljung-Box Test R Q(20) 20.52761 0.4253902
Ljung-Box Test R^2 Q(10) 4.954776 0.8941784
Ljung-Box Test R^2 Q(15) 9.191117 0.8673143
Ljung-Box Test R^2 Q(20) 12.17481 0.9099277
LM Arch Test R TR^2 6.862016 0.8665957


##Q4## ##Q4(a)-(c)##

> y=lnrt[2:T]

> x=lnrt[1:(T-1)]

> idx=c(1:(T-1))[x<0]

> nsp=rep(0,(T-1))

> nsp[idx]=x[idx]

> c1=rep(0,(T-1))

> c1[idx]=1

> xx=cbind(y,x,c1,nsp)

>

> mda=lm(y~c1+x)



> summary(mda) Coefficients:



##Q4(d)-(f)##

> mod<-lm(lnrt ~ 1)

> msm_intercept <- msmFit(mod, k=2, sw=c(T,T,T), p=1)

> summary(msm_intercept) Markov Switching Model



Coefficients:



Regime 1

———

Estimate Std. Error t value Pr(>|t|)





(Intercept)(S) 0.1540 0.0386

lnrt_1(S) 0.0182 0.0328

Residual standard error: 1.063557

—

Regime 2

———



Estimate Std. Error t value Pr(>|t|) (Intercept)(S) -0.0786 0.1169

lnrt_1(S) 0.0205 0.0478

Residual standard error: 2.468642



Transition probabilities:

Regime 1 Regime 2

Regime 1 0.91144868 0.2150418

Re