STATISTICS 730: APPLIED TIME SERIES ANALYSIS, Fall 2004, Professor Dickey
HOMEWORK 10:
DIRECTIONS: Complete the problems below. Show all of your work to receive credit.
Make sure to include output from the SAS output window and graphs from the SAS graphics window
WHEN YOU NEED TO SUPPORT YOUR ANSWERS!!!
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A topic of much interest lately is that of global warming. Because one needs a very long record to detect the levels
of warming that are expected and because the methods of taking temperature measurements change over time, the time
series are generally constructed as anomalies or deviations from some level. Because stations with different mean levels
have different average tempertaures, seasonality, and record lengths, simple averages are not appropriate, hence
the calculation of these "anomalies". Here we look at the yearly anomalies for the northern (North) and southern
(South) hemispheres.
[1] Plot the DATA. Consider whether to model the data on the original or on the
log scale. Note that from the partial autocorrelations, it seems that about 4 lags should suffice for augmenting terms.
For ease of grading, let's all use the (ADF=(4)) specification. Which tau test would be testing for stationarity
around a linear trend? What are the results for both series? Briefly explain.
[2] Taking differences of both series and modeling the Northern hemisphere data with an ARIMA(3,1,1), compute
the cross correlations of South with North, prewhitened (as PROC ARIMA will automatically do). Note that there
seems to be lagged dependence both ways at lag +1 and -1.
[3] One way to fit a bivariate model is to regress each variable on lags of itself and the other variable.
This is called a vector autoregression or VAR. Try this with 2 PROC REG runs and list the lag
2 bivariate model.
[4] Johansen's method involves regressing the first differences X(t)-X(t-1) and Y(t)-Y(t-1) let us say, on
the lagged levels X(t-1) Y(t-1) and lagged differences Y(t-1)-Y(t-2) and X(t-1)-X(t-2). The first 2x2 matrix
in the resulting model has eigenvalues that determine the cointegration status of the series. Run the two
regressions, including an intercept, and write down the resulting bivariate model.
5] Turn in your COMPLETE SAS program from the enhanced editor window.
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LINKS: SAS Online Documentation
[A] Use your vector model from either [3] or [4] ignoring any intercept terms. Make a sequence where North and
South are 0 up to time 10 then at time 11 North is 1 and South 0. At the next time point, if North is X, you
would have X(t-1)=1, Y(t-1)=0, X(t-2)=Y(t-2)=0. Apply the two coefficient matrices to the two lag vectors
just described to predict the next vector ( X(t), Y(t) ). Keep going to t+1, t+2, ... t+10. This is the
impulse response function for a pulse in X. Plot X(t) and Y(t) (versus t) on the same plot.
[B] The real work of cointegration analysis follows from the result in [4]. One must compute some eigenvalues
and test these against some special tables in Johansen's paper. PROC VARMAX will do this for you. Read the
cointegration section of the book and apply the method to this data.
[C] Recall the construction materials sales in our book. We differenced all those series and modeled the
differences. We encountered some convergence problems and had some indications that we might have
overdifferenced. Perhaps our series are cointegrated. Apply PROC VARMAX to cehck this out.