BUA Time Series Statistic Questions

I’m working on a statistics question and need an explanation and answer to help me learn.

For statistical inference purposes you can use an??(significance) level of 0.05.For

each case, please clearly state your hypotheses, rejection criteria, and conclusion
when needed.

1. Consider a random walk model with no drift as follows

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

where ??(????)=0,??(????)=??2 and ?????? are uncorrelated.

Show how you can obtain the autocorrelation function at lag k, denoted by ???? (Recall
that we have obtained the autocovariance function for a random walk in–class).

2. Consider an AR model of order 2 as follows

???? =??1?????1 +??2?????2 +????

where ??(????)=0,??(????)=??2 and ?????? are uncorrelated.

– Show how you can obtain ??(????)
– Show how you can obtain the autocovariance and autocorrelation functions at lag
k as a function of ??1, ??2 autocovariances, and autocorrelations at lags k–1 and k–
2. (Hint: you can take the above AR(2) equation and multiply both sides by ???????
and take the expectation to obtain ??(????,???????)).

3. Write a simulation sampler for a first order AR process for ??1 =0.5 (similar to our
random walk sampler from Set 3, rwsim.R). Please use a seed of 1 for your sampler.
Obtain the time series, acf and pacf plots of the simulated series. Estimate an AR(1)
model using the simulated series and investigate if the residuals are white noise.

4. Consider the monthly price of gas in the US (Jan 1974–Dec 1982) given by gas.txt.

a) Obtain the time series plot. Does this look like a stationary series?
b) Obtain the ACF and the PACF of the series. Does this look like a random walk or
an AR process?
c) Obtain the ACF and the PACF of the first difference of the series. Does this look
like a random walk or an AR process?
d) Estimate a random walk model and a suitable AR model on the differenced series.
Test if the residuals are white noise for each case. Discuss the significance of the

model parameters for each model. Compare the AIC estimates. What do you
observe?
e) Obtain a plot of the actual data versus your fitted random walk and AR models
from part d.