Technology代做、R代写代写、代做data留学生、代写R编程设计

日期：2020-06-23 10:41

South China University of Technology School of Economics and Commerce

Problem Set 3

1. [PHILLIPS.raw] Consider the following Phillips curve equation

???????? = ??0 + ??1?????????? + ????

(a) Obtain the OLS residuals ????? and ?? from the regression ????? on ??????1. Can you find

an evidence of serial correlation?

(b) Calculate the standard error using HAC approach. Use sandwich library in R and

vcovHAC( ) to calculate HAC estimates. Compare the results with (b) explicitly.

2. [FERTIL3.raw] Consider the following problem. The general fertility rate (gfr) is the

number of children born to every 1000 women of child-bearing age. Now for the years 1913

to 1984, let us consider the following regression model:

???????? = ??0 + ??1?????? + ??2???????1 + ??3???????2 + ??4???????3 + ??5???????4 + ??6????2?? + ??7??????????

+ ????

Where pe is the average real dollar value of the personal tax exemption, ww2 is a dummy

variable for World War II (= 1 if t is in the years 1941 through 1945, and pill is unity from

1963 on, when the birth control pill was made available for contraception.

(a) Estimate the above equation by OLS. Are estimated coefficients,

5, significant? If they are not significant, can you say something about the

reason? Test for joint significance of ???????3 and ???????4,

(b) Estimate the long-run propensity and its standard error. Compare these with these

obtained from the model under the null, i.e., ??0

?

: ??4 = ??5 = 0.

Now, let us consider the following regression:

???????? = ??0 + ??1?????? + ??2????2?? + ??3?????????? + ??4?? + ??5??

2

(c) Estimate the above model and report results.

(d) Regress ???????? on ?? and ??

2

and save the residuals. Denote this residuals by ???????

??

.

Then regress ???????

?? on all of the independent variables in the above equation

including ?? and ??

2

. Compare the ??

2 with that from (c). What do you conclude?

(e) Re-estimate the above equation but add ??

3

to the equation. Is this additional variable

statistically significant?

3. This problem is concerned with the relationship between the spot and futures prices of the

S&P 500 index. The data file sp5may.dat has three columns: log(futures price), log(spot

price), and cost-of-carry (x100). The data were obtained from the Chicago Mercantile

Exchange for the S&P 500 stock index in May 1993 and its June futures contract. The time

interval is 1 minute (intraday). Several authors used the data to study index futures arbitrage.

Here we focus on the first two columns. Let ???? and ???? be the log prices of futures and spot,

respectively. Here, what we want to consider is the optimal hedged ratio which can be

obtained from the following regression model:

????? = ?? + ??????? + ????

, ?? = 1,2, ? , ??,

Where ????? = ???? ? ?????1 and ????? = ???? ? ?????1. The error term in the above regression

model, ????

, is assumed to be independent and normally distributed with mean 0 and

variance ??

2

. One can easily check that the ?? in the above regression model minimizes the

variance of the hedged portfolio return ????

?? = ????? ? ???????

. Answer the following subquestions.

You need to provide all details (reasons and analysis results) at each step.

(a) Estimate the unknown parameters by using the OLS estimation method. Based on the

estimates obtain the residual series. Compare histogram of the estimated errors with

the assumed distribution. Are these two distributions similar? Provide some

comments.

(b) One of your friend at the XiongMao University argues that we need to specify other

distribution for ????

. She suggests Student’s distribution with degree of freedom ν of

which p.d.f is given by.

Construct the log-likelihood function to estimate the above regression model based

on the Student’s t distribution with d.o.f ν. Write down this function in R. Note that

the log-likelihood function should be the function of the unknown parameters, ??, ??,

and ν.

(c) Using the data and R obtain the ML estimates. Note that you can use optim in R or

maxLik library to maximize the log-likelihood function given in (b). Compare the

estimates with those in (a). Now obtain the residual series and compare histogram of

the estimated errors with the assumed distribution. Are these two distributions similar?

Provide some comments.