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Statistics 4861b/9861b Midterm Example Questions
发布时间：2022-03-05
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Statistics 4861b/9861b Midterm Example Questions
1. Define the following:
a) A time series { Yt } is weakly stationary.
b) Partial autocorrelation function of a stationary time series { Yt }.
c) A stationary time series { Yt } has short memory.
d) A time series { Yt } is MA(q).
e) The characteristic equation of an AR(p) time series.
f) A time series { Yt } is ARMA(d,q).
2. Answer following questions.
a) Does strict stationarity imply weak stationarity? Give your reason. For which type of time series, strict stationarity and weak stationarity are equivalent?
b) Is an AR(p) time series always stationary? Is a MA(q) time series always stationary? Show your reasons. If it is not, give a simple time series example.
c) If a stationary time series { Yt } is either an AR(p) or a MA(q) and p or q is unknown, find a proper way to identify p or q respectively. Explain why it works.
3. Circle one (only one) choice for each of the following multiple-choice questions. Multiple circles or answer in your exam booklet will not be counted.
a) A MA(q) time series that is not invertible means
A It is not stationary. B It is explosive. C It is stationary. D It is random walk.
b) [2] The following two plots are the sample ACF and sample PACF plots of a time series y. Based on the plots, you would choose which model to fit x?
Series x Series x
0 5 10 15 20
Lag
5 10 15 20
Lag
A MA(2). B AR(2). C MA(3). D AR(3).
c) For an ARMA(1, 1) time series { Yt }, it might
A be stationary. B be invertible. C be explosive. D All of above
4. Determine which of the following ARMA processes are stationary and which of them are invertible. For each process, identify p and q and write down a specific AR(p), MA(q), or ARMA(p,q) form. For example, if you identify it as AR(2), then p=2 and q=0, and write it down as AR(2). For those processes that are invertible, compute the first three coefficients 
0 ,
1 ,
2 in the AR( 
) form.
a) [4] Yt 
Yt 
1 
et 
1.5et 
1 .
b) [4] Yt 
0.8Yt 
1 
et 
0.4et 
1 .
c) [4] Yt 
0.3Yt 
1 
et 
0.7et 
1 
0. 1et 
2 .
d) [4] Yt 
et 
0.25et 
1 
1.25et 
2
5. For an AR(1) time series, do the following.
a) Find the MA( 
) form, that is, find all 
k in term of 
1 when | 
1 |
1 . Compute its ACF from its MA( 
) form, i.e., use 
k to computer ACF.
b) When 
1 
1, AR(1) become nonstationary. Can it be represented by MA( 
) form? Assuming Zt 
0, t = 0, -1, -2, …, find each Zt (t=1,2…., n) in terms of at . How do you describe the
statistical behave of Zn when n is large (you can assume at is iid)?
