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Using R for Econometrics and Statistics

Take Home Exam

The due date is 2019-11-19

For the exam, you are required to submit a Rmarkdown file and a html file

generated from your Rmarkdown file.

1. See ”rates.doc” for a description of the data file. For all questions, use

1962: 1 through 2012: 6 as the sample period. Use the first 24 observations

(1960 : 1 through 1961 : 12) for initial conditions and differencing

transformations. You are to calculate the following. You should write

your own code (use R), but can borrow from pre-existing code where you

feel comfortable doing so.

(a) Start by plotting the unemployment rate against time. Is the series

trending? Cyclical?

(b) Estimate an AR(4) model (always include an intercept!) by leastsquares.

Report coefficient estimates, robust standard errors, and a

one-step point forecast for July 2012.

(c) Estimate a set of autoregressions (always include an intercept!) by

least-squares, AR (1) through AR (24). For each, calculate the Cross

Validation information criterion. Also calculate the BIC, AIC, AICc

Mallows, Robust Mallows information criteria. Create a table for

your results.

(d) Based on the CV criteria, select an AR model.

(e) Use this model to make a one-step point forecast for July 2012 .

(f) Report coefficient estimates and robust standard errors.

(g) Now consider the other variables in the data set. After making suitable

transformations, include these variables in your model. Using

the information criteria, select a forecasting model.

(h) Use this forecasting model to make a one-step point forecast for July

2012 .

2. In this problem you’ll use ridge regression the lasso to estimate the salary

of various baseball players based on a bunch of predictor measurements.

This data set is taken from the ”ISLR” package, and R package that

accompanies the ”Introduction to Statistical Learning” textbook. You

should now have the objects x, y, the former being a 263 × 20matrix of

predictor variables, and the latter a 263 dimensional vector of salaries.

(For more information, download and install the ISLR package and type

1 Hitters.) Download and install the g1mnet package from the CRAN

repository. We’ll be using this package to perform ridge regression and

the lasso. Finally, define

gr id =10ˆseq(10 , −2 ,1 ength =100)

This is a large grid of λ values, and we’ll eventually instruct the glmnet

function to compute the ridge and lasso estimates at each one of these

values of λ.

(a) The ‘glmnet‘ function, by default, internally scales the predictor variables

so that they will have standard deviation 1, before solving the

ridge regression or lasso problems. This is a result of its default setting

‘standardize=TRUE‘. Explain why such scaling is appropriate

in our particular application.

(b) Run the following command

r i d . mod = glmnet ( x , y , lambda=grid , alpha =0)

l a s . mod = glmnet ( x , y , lambda=grid , alpha =1)

This fits ridge regression and lasso estimates, over the whole sequence

of λ values specified by grid. The flag ”alpha=0” notifies g1mnet to

perform ridge regression, and ”alpha=1” notifies it to perform lasso

regression. Verify that, for each model, as λ decreases, the value of

the penalty term only increases. That is, for the ridge regression

model, the squared `2 norm of the coefficients only gets bigger as λ

decreases. And for the lasso model, the `1 norm of the coefficients

only gets bigger as λ decreases. You should do this by producing a

plot of λ (on the x-axis) versus) versus the penalty (on the y-axis)

for each method. The plot should be on a log-log scale.

(c) Verify that, for a very small value of λ, both the ridge regression

and lasso estimates are very close to the least squares estimate. Also

verify that, for a very large value of λ, both the ridge regression and

lasso estimates approach 0 in all components (except the intercept,

which is not penalized by default).

(d) For each of the ridge regression and lasso models, perform 5 -fold

cross-validation to determine the best value of λ. Report the results

from both the usual rule, and the one standard error rule for choosing

λ. You can either perform this cross-validation procedure manually,

or use the ”cv.glmnet” function. Either way, produce a plot of the

cross-validation error curve as a function of λ, for both the ridge and

lasso models.

(e) From the last part, you should have computed 4 values of the tuning

parameter:

λ

ridge

min , λridge

1se , λlasso

min , λlasso

1se

These are the results of running 5-fold cross-validation on each of

the ridge and lasso models, and using the usual rule (min) or the

one standard error rule (1se) to select λ. Now, using the predict

function, with type: ”coef”, and the ridge and lasso models fit in

part (b), report the coefficient estimates at the appropriate values

of λ. That is, you will report two coefficient vectors coming from

ridge regression with λ = λ

rige

min and λ = λ

rige

1se , and likewise for the

lasso. How do the coefficient estimates from the usual rule compare

to those from the one standard error rule? How do the ridge estimates

compare to those from the lasso?

(f) Suppose that you were coaching a young baseball player who wanted

to strike it rich in the major leagues. What handful of attributes

would you tell this player to focus on? (That is, how to measure

variable importance?)

3. Value at Risk (VaR) is a statistical measure of downsiden current position,

It estimates how much a set of investments might lose given normal

market conditions in a set time period. A vaR statistic has three

components.al time period b) confidence level. c) loss ammount (or loss

percentage). For 95% confidence level, we can say that the worst daily

loss will not exceed VaR estimation. If we use historical data, we can

estimate vaR by the quantile value. For our data this estimation is:

quant i le ( s p5 0 0 r e t , 0 . 0 5 )

Delta-normal approach assumes that all stock returns are normally distributed.

This method consists of going back in time and computing the

variance of returns. Value at Risk can be defined as:

V aR(α) = µ + σ ∗ N−1(α)

where µ is the mean stock return, σ is the standard deviation of returns,

α is the selected confidence level and N −1

is the inverse CDF function,

generating the corresponding quantile of a normal distribution given α.

The results of such a simple model often disappointing and are rarely

used in practice today. The assumption of normality and constant daily

variance is usually wrong and that is the case for data as well.

Previously we observed that returns exhibit time-varying volatility. Hence

for the estimation of VaR we use the conditional variance given by GARCH

model. For the underlined asset’s distribution properties we can use the

student’t-distribution. For this method Value at Risk is expressed as:

VaRt(α) = µ + ˆσt|t−1 ∗ F−1

(α)

where ˆσt|t−1 is the conditional standard deviation given the information

at t − 1 and F−1

is the inverse CDF function of t−distribution.

(a) For the ”sp500price” data set we have used in class, you need to specify

an appropriate form of GARCH model for the return of ’sp500’

with information criterion method and rolling estimation method.

For rolling method, the window size is 2500 and the estimation is

implemented every 100 observations (To save the computation cost,

moving window method is recommended.)

(b) Forward looking risk management uses the predicted quantiles from

the GARCH estimation. Using ” ugarchroll” method to estimate the

best GARCH model you have obtained from (a) by rolling method

to compute VaRt(α) for α = 0.05. The setting of rolling estimation

is the same as (a).

(c) A VaR exceedance occurs when the actual return is less than the predicted

value-at-risk: Rt < V aRt

. Plot the scattered points of actual

returns, the predicted VaRt(α), and highlight the VaR exceedance

points by different color.

(d) The frequency of VaR exceedances is called the VaR coverage. A

valid prediction model has a coverage that is close to the probability

level α used. If coverage > α : too many exceedances: the predicted

quantile should be more negative. Risk of losing money has been

underestimated.If coverage < α : too few exceedances, the predicted

quantile was too negative. Risk of losing money has been overestimated.

Compute the VaR coverage of the AR(1)-GJR-GARCH with

skew-t distribution, AR(1)-GJR-GARCH with t distribution, AR(1)-

GARCH with t distribution, and AR(1)-GJR-GARCH with skew-t

distribution with rolling estimation implemented every 1000 observations

instead of 100. You can create a table to display your results.

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