Advanced Econometrics 2020 - Homework 2
Deadline: 23:59:59, Friday November 27, 2020
Instructions1
:
• Form groups of three yourself.
• As a solution, provide one Jupyter Notebook with R source-code. Code should be properly
commented, interpretations of results as well as theoretical derivations2
should be written
in markdown cells. This is the only file you need to send.
• Use “set.seed()“ function, so I can replicate your results.
• Be concise (no lengthy essays please). Although, be sure to include all important things as
I cannot second-guess your work.
• The problem set is due on 27th November. Late submission automatically means
0 points.
• If you have any questions concerning the homework, do contact me by mail. Do it rather
sooner than later.
1The contact person for this homework is Matěj Nevrla, the same mail as for submission of
homeworks.
2
If you prefer not to write formulas in LATEX, you can send PDF with your derivations and
interpretations in additional file and R code in Jupyter Notebook.
1
Problem 1 Bootstrap
(1.5 points)
Consider dataset city from the package boot. Data consists of random sample
of size n = 10 from the population of big cities in the US and each observation
corresponds to the number of inhabitants of a given city (in thousands) in year 1920
(variable u) and number of inhabitants of a given city in year 1930 (variable x).
Let (U, X) be a random vector corresponding to the number of inhabitants in the
years 1920 and 1930, and (ui
, xi), i = 1, . . . , 10, be our random sample.
Consider the following ratio
R ≡
E(X)
E(U)
.
a) Draw the dependence between xi and ui using the dataset.
b) Estimate the value of R from the random sample.
c) Compute the bootstrap standard error and the bootstrap bias of the estimate of
R. Use both ”brute force” bootstrapping and function boot from the respective
package.
d) Compute the 95% confidence intervals using both percentile and normal approximation
method. Argue whether the normal approximation is valid in this case.
e) Test (without refinement) H0 that R = 1 against the alternative HA : R ̸= 1.
Calculate bootstrap t-statistic and compare it with critical values of the standard
normal distribution. Use also the percentile method for testing H0 and compare
the results.
2
Problem 2 Endogeneity
(2 points)
Let us follow the idea of the first exercise from Seminar 6 but for now we create
another artificial dataset containing 300 observations (note that although variance
of RVs is specified below, R commands often require to specify standard deviation
instead):
z1 ∼ N(2, 3
2
), z2 ∼ N(2, 1.5
2
), z3 ∼ N(0, 2
2
), z4 ∼ N(1.8, 2.5
2
)
ϵ1 ∼ N(0, 1.5
2
), ϵ2 ∼ N(0, 1.5
2
), ϵ3 ∼ N(0, 1.5
2
)
x1 = 0.3z1 − z2 + 0.9z4 + 0.75ϵ1
x2 = 0.75z2 + 0.75ϵ2
x3 ∼ N(0, 1)
y = 1 + 2.5x1 − x2 + 0.45x3 + ϵ3
On the dataset, we should estimate the following model:
y = const + β1x1 + β3x3 + ϵ
a) Discuss the nature of the endogeneity problem in the system above. You might
check important correlations and you should explain the difference between x1
and x3. Do you expect to observe any bias within the OLS estimation? Explain
why.
b) Estimate the model by OLS and interpret.
c) The data set includes some potential IV candidates: z1; z2; z3; z4. What assumptions
need to be satisfied in order to have a ‘good’ instrument? Which of these
candidates seem to be ‘good’ instruments and why? Test their relevancy statistically.
Is there any invalid, irrelevant, or weak instrument?
d) Based on section d), choose the best instrument and run the IV regression. Run
also 2SLS regression using all ‘good’ instruments. Compare coefficient estimates
and standard errors
e) Finally, test for the endogeneity using the Hausman test. Report and interpret
the results.
f) Using extended dataset (simulate more data from the data generating process),
show that OLS is not consistent estimator of β1 and β3. Show that 2SLS provides
consistent results.
3
Problem 3 GMM
(1.5 points)
In the dataset hw_data.csv, you have a time series which comes from Moving Average
process with q lags - MA(q) process. You will need libraries gmm and tseries
to answer the following questions. Note that if you do not answer correctly point a),
all consecutive questions will be wrong, hence no points can be earned.
a) How would you identify the lag q? What is the lag?
b) Derive the moment conditions function for the process you identified previously
and write a corresponding function in R. Use more moment conditions than is
the number of coefficients that you want to estimate.
c) Estimate the model using gmm and both identity and optimal weighting matrix.
Provide the output and interpret the coefficient significance and the J-test statistics.
4
Problem 4 CAPM beta of Apple Inc.
(3 points)
You are going to estimate CAPM betas for Apple Inc. on two different time periods
and compare the results. Inference on the estimated parameters will be performed
using bootstrap.
CAPM beta for company i can be estimated by regressing its returns on market
returns using simple OLS
is risk-free interest rate, all at time t.
1. Download adjusted daily prices of Apple Inc. (ticker 'aapl') for year 2008
and 2017, separately. Download daily data for Nasdaq Composite Index (ticker
'^ixic'), which will be used as a proxy variable for market. Chicago Board
Options Exchange (CBOE) 10y interest rate T-note (ticker ’^tnx’) will be used
as a risk-free rate.
2. Compute log-returns for all series.
3. Draw histogram and kernel density approximation of Apple log returns. Compare
it with normal distribution with the same mean and standard deviation.
Does samples considerably deviate from normality? Which noticeable features
do you observe in the data? Perform this whole analysis on both time periods
separately.
4. Compute CAPM beta for Apple using simple OLS on each time period separately.
Compare the obtained results.
5. Use the nonparametric bootstrap to compute bootstrap standard error of CAPM
beta estimate based on 1000 bootstrap replications and bootstrap sample size
equal to the size of the original sample. Use boot command. Perform this
whole analysis on both time periods separately.
5
6. Graphically compare histogram and kernel density approximation (use gaussian
kernel only) of the boot-based set of 1000 bootstrapped CAPM betas with normal
distribution. Compute 95% percent confidence intervals for CAPM beta
for both time periods using percentile method and method based on normal
approximation. Which method is in this case probably more suitable. Based
on the estimated confidence intervals, do the estimates of CAPM beta significantly
differ over these 2 periods?
7. Another approach how to use bootstrap is instead of resampling the observations
is to resample the estimated residuals. Estimate CAPM beta by OLS,
save the estimated residuals, and perform bootstrap by resampling the residuals
and generating new values of dependent variables as
are resampled from empirical distribution function. In each bootstrap
replication, calculate values of dependent variable for all values of independent
variable. The rest of the procedure is the same as in the case of resampling the
observations. Perform this analysis on one time period of your choice. Compare
the results obtained from this approach and from the traditional approach from
above. Why this method may not be the optimal in this case?