The following are a collection of questions asked in STA 137 midterms over the past 15 years. These questions are meant to be for practice. The actual midterm will look differently.

1. Let (St : t e N0) be a random walk with drift given by the equations

S0 = 0,        St = µ + St_1 + Zt,        t e N,

where µ is a real number and (Zt : t e N) ~ WN(0, σ2 ).

(a) Is (St : t e N0) weakly stationary?

(b) Give the definition of the difference operator V.

(c) Is the differenced process (VSt : t e N) weakly stationary?

2. Let (Xt : t e Z) be a stochastic process given by the equations Xt  = mt + Zt, where mt  = b1t and (Zt : t e Z) ~ WN(0, σ2 ).

(a) Define the two-sided moving average filter Wt that can be used to estimate mt . (b) Compute E[Wt] and Var(Wt) based on (Xt : t e Z).

3. Let (Yt : t e Z) be a weakly stationary process with mean zero. Define the process (Xt : t e Z) by

Xt = st + Yt,        t e Z,

where st is a seasonal component of period d.

(a) Give the definition of the lag-d difference operator Vd used to eliminate st .

(b) Apply Vd to the process (Xt : t e Z) and compute Cov(VdXt+1 , VdXt).

4. Let (Xt : t e Z) be a stochastic process given by the equations

Xt = Zt + Zt_1 ,        t e Z,

where (Zt : t e Z) ~ WN(0, σ2 ).

(a) What is the limit distribution of the correlations ρZ (h), h 0, for the white noise (Zt : t e Z)?

(Restate Theorem 1.2. 1!)

(b) For h e Z, compute the correlation ρX (h) = Corr(Xt+h, Xt).

(c) Would you suggest to give a confidence interval for ρX (2) based on Theorem 1.2. 1? Explain why or why not.

5.    (a) Give the definitions of an ARMA(p, q) process and a linear process.

0             5            10           15           20           25

0             5             10           15           20           25

Figure 1:  The sample ACF (left) and the sample PACF (right) of a weakly stationary process.

(b) Under which conditions are ARMA(p, q) processes (1) causal and (2) invertible;

(c) Which ARMA(p, q) processes can be represented as linear processes?

6. Which of the following processes is causal and/or invertible? If the process is invertible, determine the π -weights π0 , π 1 and π2 .

(a) Xt = Xt_1 _ .5Xt_2 + .7Xt_3 + Zt _ Zt_1 + .5Zt_2 _ .7Zt_3 ;

(b) Xt = _.3Xt_1 + Zt + .8Zt_1 + .15Zt_2 .

7.    (a) Give the definition of the PACF of a weakly stationary stochastic process;

(b) Inspect the ACF and PACF in Figure 1 and suggest an appropriate ARMA(p, q) model to fit to the data.;

(c) Compute the PACF of a causal AR(2) process at lag 3, that is φ33 .

8.    (a) If (Xt : t e Z) is a causal AR(3) process of which we have observed X1 , . . . , Xn, determine the forecast Xˆn+1 without using any of the prediction algorithms;

(b) If (Xt : t e Z) is an invertible MA(1) process, determine the truncated one-step predictor that is

given by the equation

n

Xn(_)+1 = _      πjXn+1_j ,

j=1

that is, determine the π -weights π0 , π 1 , . . . , πn .

(c) Compute the prediction error Pn(_)+1 = E[(Xn+1 _ Xn(_)+1)2] that is made in part (b).

9.    (a) Give the definition of a weakly stationary stochastic process.

(b) Give an example of a nonstationary stochastic process and explain why it is nonstationary.

10.    (a) Let (Xt : t e Z) be a stochastic process given by the equations

Xt = b0 + b1t + b2t2 + Zt ,        (Zt: t e Z) ~ WN(0, σ2 ).

Is the process weakly and/or strictly stationary?

(b) Compute (V2Xt : t e Z). Note that V2Xt = V(VXt).

11.    (a) Let (Xt :  e Z) be the AR(1) process given by

Xt = φXt_1 + Zt ,         t e Z,

(d) Larger p decreases both bias and variance; larger q decreases both bias and variance; (e) None of the above.

Circle the correct answer(s).

23.  Show that the one-sided moving average filter gives exponentially decaying weights to past observations. [Recall that mˆ1 = Y1 and mˆ t = aYt + (1 _ a)mˆ t_1 for some a e (0, 1) and t = 2, . . . , n.]

24. Assume that you have given the following quarterly data over three years:

10,  15,  8,  15;      x,  22,  12,  16;      11,  12,  11,  10,

where x signifies a missing observation. Suppose you fit the trend-plus-seasonality model Yt  = mt + st + Xt with d = 4 to the data using the small trend method. If mˆ2 = 15, what is the value of sˆ1 ?

25.  Suppose we are considering average temperatures in the northwest hemisphere. As one would expect, temperatures drop in the winter and increase in the summer. Additionally, due to climate change, there is a trend towards increasing temperatures.  But there is also another seasonal trend:  El Nio, where temperatures increase and decrease in a cyclical pattern over multiple years. Describe an algorithm to deseasonalize this model.

In other words, given the model Yt = mt+ s1t+ s2t+ Xt for the data, describe an algorithm to estimate both s1t and s2t with (known) seasonal periods d1 and d2 .

26. Which of the following relations between strictly and weakly stationary time series are true?

(a)  Strict stationarity and weak stationarity are identical concepts;

(b) Weak stationarity and finite second moments imply strict stationarity;

(c)  Strict stationarity and finite second moments imply weak stationarity;

(d) Weak stationarity and Gaussianity imply strict stationarity;

(e) A weakly stationary process cannot be strictly stationary.

Circle the correct answers. (There may be more than one correct choice.)

27. Figure 2 shows the time series plot and the sample ACF of an AR(1) process Xt  = φXt_1 + Zt . Which parameter φ has been used to generate these observations?

(a) φ = _0.9              (b) φ = _0.3              (c) φ = _0.1              (d) φ = 0.3              (e) φ = 0.9

Circle the correct answer.

28. Let the process (Xt : t e Z) be given by

Xt = 5 + Zt _ .5Zt_1 + .25Zt_2 ,

Compute the ACF ρ(h) = Corr(Xt, Xt+h) for this process.

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Figure 2: Time series plot and ACF of an AR(1) process.

29. Figure 3 displays the time series plot, the ACF and the qq plot of a series of residuals that has been obtained after detrending and deseasonalizing a data set of size n = 100.

(a) Based on the ACF alone would you suggest that the residuals are dependent?          (b) Based on the time series plot, would you trust your answer in (a)? Why or why not?

(c) Do the time series plot and the qq plot support the claim of normally distributed residuals? Give precise arguments for each of your choices.

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Time

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Lag

-2                    - 1                     0                      1                      2

Theoretical Quantiles

Figure 3: Time series plot, ACF and qq plot of a residual series.

30. Is the MA process

Xt = Zt + 3Zt_1 + 3Zt_2 + Zt_3 ,        t e Z,

causal and/or invertible? If the process is causal, determine the ψ-weights (ψj : j e N0) in the represen- tation Xt  = ψ(B)Zt  = ψjZt_j . [Recall that ψ(z) = ψjzj  = θ(z)/φ(z).] item Circle the correct statements. (There may be more than one correct choice.)

(a) MA(q) processes are always causal;

(b) AR(p) processes are always causal;

(c) ARMA(p, q) processes are not necessarily causal but always invertible;

(d) Causality is determined by the zeros of the moving average polynomial;

(e) Invertibility is determined by the zeros of the moving average polynomial.

31. Figure 4 shows the sample ACF and sample PACF computed from n = 100 observations X1 , . . . , X100 . Which of the following is the most likely candidate model to generate these plots?

(a) MA(3)            (b) AR(2)            (c) ARMA(2,4)            (d) AR(3)            (e) none of the above Circle the correct answer.

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Lag

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Figure 4: Sample ACF (left) and sample PACF (right) of an observed time series.

32.  Suppose you are considering the AR(2) process given by parameter values φ1 = .8 and φ2 = _.5.

(a) What is the value of the ACF ρ at lag h = 2, that is, what is ρ(2) = Corr(X2, X0 )?           (b) What is the value of the PACF φhh at lag h = 2, that is φ22 = Corr(X2 _ X2(1), X0 _ X0(1))?

33. Let (Xt : t e Z) be the MA(2) process given by the equations Xt  = Zt + θ 1Zt_1 + θ2Zt_2 . Suppose you know X1 . Compute the prediction Xˆ2 of X2 based on X1 .

34. Using n = 100 observations, a researcher fitted an AR(2) model to the data. The fitted parameter values were

φ 1 = .7,        φ2 = _.3,        σ2 = 1.

Then all but the last two observations X99  =  1 and X100  =  .8 were deleted.  Based in the available information, please provide predictions Xˆ101 and Xˆ102 for the next two observations.

35. In Figure 5, n = 100 observations from a model of the form Yt  = mt + Xt are displayed. Choose the order p of the polynomial b0 + b1t + . . . + bptp that adequately captures the trend mt in this plot.

(a) p = 0                  (b) p = 1                  (c) p = 2                  (d) p = 3                  (e) none of (a)–(d).

0                                        20                                       40                                       60                                       80                                       100

Figure 5:  Realization from a time series model of the form Yt = mt + Xt .

36. Let V be the difference operator and specify the quadratic “trend” mt = b0 + b2t2 for some constants b0 and b2 . What is Vmt? [Note: Vmt = mt _ mt_1 ]

(a) 0                    (b) b0                            (c) 2b2t _ b2                            (d) b2                            (e) none of the above

37. In Figures 6–8, n = 100 observations simulated from time series (Xt), (Yt) and (Zt) are shown. Which of the three processes is not stationary?

(a) (Xt)

(b) (Yt)                    (c) (Zt)                    (d) none of the processes is stationary

100

Time

Figure 6:  Realization of (Xt)

38. Let (Zt : t e Z) ~ WN(0, σ2 ) and let (Xt : t e Z) be given by the equations

Xt = Zt _ Zt_1 ,        t e Z.

What are the values of the ACVF γ(h) = Cov(Xt, Xt+h) for h = 0, 1, 2?

(a)    γ(0) = 2σ2 ,        γ(1) = _σ 2 ,        γ(2) = 0;

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Figure 7:  Realization of (Yt).

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Figure 8:  Realization of (Zt).

(b)    V(0) = 2σ2 ,        V(1) = σ 2 ,           V(2) = 0;

(c)    V(0) = σ 2 ,          V(1) = _2σ2 ,      V(2) = σ 2 ;

(d)    V(0) = _2σ2 ,     V(1) = σ 2 ,           V(2) = 0;

(e)    None of the above.

39. The sample ACF of three processes (X), (X) and (X) are given in Figure 9.  Which of the following statements is the most credible?

(a) (X) ~ WN(0, σ2 ), (X) ~ MA(2), (X) ~ AR(1);

(b) (X) ~ MA(2), (X) ~ WN(0, σ2 ), (X) ~ AR(1);

(c) (X) ~ WN(0, σ2 ), (X) ~ AR(1), (X) ~ MA(2);

(d) (X) ~ AR(1), (X) ~ MA(2), (X) ~ WN(0, σ2 ); (e) None of the above statements is credible.

40. After an inspection of the time series plot in Figure 10, a researcher decides to fit an AR(1) process Xt  = φXt_1 + Zt to the data. As supporting evidence she provides the scatter plots of Xt versus Xt_1 (left) and Xt _ φXt_1 versus Xt_2 _ φXt_1 (right) in Figure 11. Which of the following statements is true?

(a) The researcher should have fit an AR(2) process instead;

(b) The researcher’s selection of an AR(1) process is supported by the plots;

(c) There is no need to fit a time series model, the observations are white noise;

(d) The observations seem to come from an MA process not from an AR process;

X_t^(1)

20

X_t^(2)

20

X_t^(3)

20

Figure 9: The sample ACF of the processes in Problem 39

(e) None of the above statements is credible.

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Figure 10: Time series plot of the observations.

2

X_{t- 1}