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ISYE 6402 MIDTERM PREP WITH COMPLETE
SOLUTIONS
Question 1: Writing mathematical equation of ARMA -
GARCH
CORRECT ANSWER : see pic
Question 2: confusing fat tail and which direction skewed
CORRECT ANSWER : see pic
Question 3: why Erie wasn't stationary
CORRECT ANSWER : see pic don't be as bold about non-
constant variance as you are w/ non-constant mean and autocovariance
Question 4: how'd you know what ARIMA to choose from this?
CORRECT ANSWER : A seasonal ARIMA model would likely
be preferred. For example, we may want to consider
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ARIMA(1,1,1)(0,1,1), where p and q were selected due to the large first lags in the seasonal-differenced ACF and PACF, and P and Q were selected due to the ACF seasonal lags largely cutting off after the first and the PACF seasonal lags decaying more gradually.
Question 5: where is stationarity required for application of
models?
CORRECT ANSWER : MA models are always stationary, but
there are still spikes in the early lags. Both ARIMA and VAR models can only fit stationary residuals. ARCH and GARCH models can be fit on data with non-constant variance (and therefore they're not stationary), but the data they fit should have constant mean.
Question 6: Checking residuals of an ARIMA model
CORRECT ANSWER: After fitting an ARIMA model, you
must check for constant variance and normality of residuals.How to tell if normally distributed noise? Check Q-Q plot/histogram. Also, can use JB test on normality. We can also
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evaluate whether the residuals have constant variance by applying the arch.test R command, which is a hypothesis testing procedure where the null hypothesis is that the residuals have constant variance.
Question 7: Specifying lag and fitdf in the Box.test
CORRECT ANSWER : The ACF plot suggests no significant
autocorrelation as spikes are all either insignificant or close to 0.The high p-value from the Box test confirms this.Box.test(resid(model.c),lag=24,type="Ljung",fitdf=5) Lag in the Box.test - lag = 2 x seasonal differencing e.g. 12 for monthly.Remember lag must be fitdf +1 at least. For fitdf - it's the sum of Ps + Qs (both little and big Ps and Qs)
Question 8: Is ARCH-GARCH a good candidate for a fat-tailed
QQ plot?
CORRECT ANSWER : Ideally GARCH models are fit to
residuals that don't have fat tails in their Q-Q plot. There are other better models for this case.
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Question 9: Is any MA model stationary?
CORRECT ANSWER : You bet
Question 10: what's the TS function do?
CORRECT ANSWER : make a TS
Question 11: You want to model both the contemporaneous and
lagged relationships among several time series. A VAR model with added deterministic components is best suited for this task.
CORRECT ANSWER : False. We will need to use a SVAR for
this task. The deterministic components capture seasonality or trend, not contemporaneous relationships. Another extension of the VAR model is the so-called structural VAR model. Which is an extension of the VAR model by considering a linear transformation of Yt through the A times Yt on the left. And or a linear transformation of the error term through the term B times epsilon t. The structural VAR models not only models lag temporal relationships, but also contemporaneous dependencies