1. Consider the monthly simple returns of the CRSP Decile 1 and 9
portfolios from January 1970 to December 2006. The portfolios consist
of NYSE/AMEX/NASDAQ stocks based on market capitalization and
rebalanced annually. See CRSP (via WRDS) for more information. The
data are in m-dec19.txt with date, Decile-1 return, and Decile-9 return
in three columns.
(a) Compute the first 24 lags of ACF and PACF of the simple return
series of Decile 1 portfolio.
(b) Test the hypothesis that the first 12 lags of ACF are zero. That is,
H0 : 1 = · · · = 12 = 0 versus H1 : i = 0 for some 1 ≤ i ≤ 12. Draw
2. Consider the monthly simple returns of the CRSP Decile 9 portfolio
in Problem 1.
(a) Compute the ACF of the simple returns for the first 12 lags.
(b) Test the hypothesis that all 12 ACFs are zero. That is, H0 : 1 =
· · · = 12 = 0 versus the alternative hypothesis H1 : i = 0 for some i,
1 ≤ i ≤ 12. Draw your conclusion.
3. Consider the monthly U.S. Consumer Price Index Less Engery from
January 1, 1957 to February 2, 2007. The data are seasonally adjusted
and obtained from the Federal Reserve Bank at St Louis. The origin of
the data is U.S. Department of Labors, Bureau of Labot Statistics. The
data file is m-cpileng.txt (in year, mm, dd, cpi format).
Compute the percentage growth rate series of CPI defined as ct = 100[ln(Xt)−
ln(Xt 1)], where Xt denotes the tth observation of CPI.
(a) Compute the 12 lags of ACF and PACF of ct. Test the null hypothesis
that the first 12 lags of ACF are zero. Draw your conclusion.
(b) The patterns of ACF and PACF indicate that the ct series does not
follow a simple AR or MA model. To gain further insight, compute the
first 12 lags of ACF of the diﬀerenced series zt = ct − ct 1.
(c) Based on the ACF of zt, one may fit an ARMA(1,5) model for the ct
series. Write down the fitted model.
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