ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 3: Univariate Time Series – II
The point of this question is to suggest a general “road map” for analyzing univariate time series with ARMA models.
1. The file Merck.csv contains daily data of stock prices of Merck & Co., Inc. (MRK) during 2001-2013. In what follows, we use yt to denote the adjusted closing prices (adjclose in the data) in time t.
(a) LoadthedatatoStata,generateadatevariable,declarethedataastimeseries, and keep only observations during January 1, 2011 – January 31, 2012.
(b) Construct the following variables:
• Changes in prices: ∆yt = yt − yt−1
• Log returns: rt = log(yt/yt−1)
(c) Draw time series plots of yt and ∆yt and comment on their stationarity1.
(d) Compute and plot (using either ac/pac or corrgram) ACF and PACF of yt and ∆yt. Comment on your findings.
(e) Based on the ACF and PACF of ∆yt you obtained in (d), propose and estimate at least three ARMA(p, q) models for ∆yt.
(f) Use AIC and BIC to select an ARMA(p, q) model. Estimate the AR and MA parameters of this model and report estimation results.
(g) Draw a time series plot of the residuals you obtain via estimating the ARMA model selected in (f). Comment on your findings. Run the Ljung-Box test (at significance level α = 5%) for the white noises hypothesis and report test results. Note that you will need to adjust the degree(s) of freedom as you are analyzing estimation residuals.
(h) Forecast MRK stock prices in January, 2012. Compare your predicted prices with real prices in the data.
(i) Repeat (c) – (h) for the log returns rt. Note that here you forecast the daily returns (yt − yt−1)/yt−1 in January, 2012. Hint: Recall that (yt − yt−1)/yt−1 ≈ rt.
1You should use only 2011 data for parts (c)-(g). 1