TEST #1 STA 4853 March 6, 2017 Name: Please read the followig directios. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directios This exam is closed book ad closed otes. There are 32 multiple choice questios. Circle the sigle best aswer for each multiple choice questio. Your choice should be made clearly. Always circle the correct respose. (Sometimes the questio has a empty blak or a box, but this is NOT where the aswer goes.) There is o pealty for guessig. The exam has 13 pages. Each questio is worth equal credit. 1
For each of the AR or MA parameters i a time series model, SAS PROC ARIMA reports a estimate, stadard error, t-value, ad p-value. The p-value is the approximate probability of gettig a by chace whose magitude is as large as that observed whe the true value of the is zero. Problem 1. Which of the followig words correctly fills i the first blak above? a) estimate b) t-value c) variace estimate d) Ljug-Box test e) AIC f) SBC g) stadard error h) p-value i) autocorrelatio j) parameter Problem 2. Which of the followig words correctly fills i the secod blak above? a) p-value b) autocorrelatio c) parameter d) variace estimate e) Ljug-Box test f) AIC g) SBC h) estimate i) stadard error j) t-value Problem 3. For a radom shock process, the theoretical ACF a) will have early all values withi the two stadard error bad b) will have a approximate cutoff after lag 1 c) will have a exact cutoff after lag 1 d) is o-statioary e) is idepedet of future values f) is exactly zero for all ozero lags g) decays rapidly to zero Problem 4. defie Suppose a 1, a 2, a 3,... are radom shocks with mea zero ad variace σ 2 a, ad we z 5 = ψ 0 a 5 + ψ 1 a 4 + ψ 2 a 3 z 3 = ψ 0 a 3 + ψ 1 a 2 + ψ 2 a 1. What is the value of E z 5 z 3? a) σ 2 a(ψ 0 ψ 1 + ψ 1 ψ 2 ) b) σ 2 aψ 0 ψ 2 c) σ 2 a(ψ 2 0 + ψ 2 1 + ψ 2 2) d) ψ 2 1 e) ψ 3 1 f) ψ 4 1 g) σ 2 a 1 φ 2 1 h) C 1 φ 1 i) φ 3 1 Problem 5. To geerate a realizatio from a AR(p) process, we eed startig values. a) 0 b) 1 c) 2 d) φ 1 e) φ 11 f) φ p g) φ pp h) p i) p + 1 j) p 1 2
Problem 6. Suppose you fit a regressio model ad the costruct a plot of the residuals versus fitted values (also kow as the predicted values). I this plot the residuals are plotted o the y-axis ad fitted values o the x-axis. If the regressio assumptios are valid, we expect roughly that a) the plot will follow a straight lie b) the fitted values will be ormally distributed c) the residuals will decay to zero d) 95% of the fitted values will be betwee -2 ad 2 e) positive residuals will ted to be followed by positive residuals f) the residuals will form a bad cetered at zero with costat vertical width g) positive residuals will ted to be followed by egative residuals Problem 7. Approximate 95% cofidece itervals for a regressio coefficiet β i use the value 1.96. If the regressio assumptios are valid, a exact 95% cofidece iterval may be costructed by replacig 1.96 by a) a value obtaied from the ormal distributio b) a value obtaied from a chi-square distributio c) a value obtaied from a t-distributio d) 2 e) 1.959964 f) 1.644854 g) 2.326348 Problem 8. I the SAS PROC REG output, the Corrected Total Sums of Squares is a) the total variability i the sample resposes Y i b) the total variability i the sample covariates X i c) the total variability of the residuals ε i d) the total variability of the fitted (predicted) values Ŷi e) equal to R-squared (R 2 ) f) equal to the Error Mea Square g) equal to the Root MSE h) equal to the Adjusted R-squared Problem 9. Suppose z t is a AR(2) process: z t = C + φ 1 z t 1 + φ 2 z t 2 + a t. What is the correlatio betwee z t 2 ad a t? a) φ 2 b) φ 22 c) ρ 2 d) φ 2 1 e) 0 f) 1 g) 1 h) φ 2 2 3
Problem 10. I the regressio output, if the estimate ˆβ i has a t-value t i which is large i magitude, the the associated p-value will be a) greater tha 2 b) greater tha the stadard error c) less tha the stadard error d) greater tha 1 e) less tha 0 f) close to 0 g) close to 1 Problem 11. give by Let X 1, X 2,..., X be a radom sample. The sample stadard deviatio s x is a) 1 (X i X) b) 1 c) 1 (X i X) 1 2 d) 1 1 1 1 1 e) 1 X i f) g) 1 Xi 2 h) 1 (X i X) (X i X) 2 X i X 2 i Problem 12. For a statioary AR(1) process z t = C + φ 1 z t 1 + a t, it is true that Var(z t ) = a) Var(C) + φ 1 Var(z t 1 ) + Var(a t ) b) C + φ 1 Var(z t 1 ) + Var(a t ) c) φ 2 1Var(z t 1 ) + Var(a t ) d) φ 1 Var(z t 1 ) + Var(a t ) e) C 2 + φ 2 1Var(z t 1 ) + a t f) φ 1 Var(z t 1 ) + a t g) Var(C) + φ 1 Var(z t 1 ) + a t h) C + φ 1 Var(z t 1 ) + a t Problem 13. For a statioary AR(1) process, which of the followig is true for all k 1? a) ρ k = C 2 ρ k 1 b) ρ k = σaρ 2 k 1 c) φ kk = φ 1 ρ k 1 d) φ kk = Cρ k 1 e) φ kk = σ a ρ k 1 f) φ kk = φ 2 1ρ k 1 g) φ kk = C 2 ρ k 1 h) φ kk = σaρ 2 k 1 i) ρ k = φ 1 ρ k 1 j) ρ k = Cρ k 1 k) ρ k = σ a ρ k 1 l) ρ k = φ 2 1ρ k 1 4
Problem 14. Suppose you perform a regressio usig SAS PROC REG, ad you specify the DWPROB optio i the MODEL statemet. The, i additio to the Durbi-Watso statistic, SAS will display the first order autocorrelatio of the residuals e 1, e 2,..., e. What is the formula for the first order autocorrelatio of the residuals? d) a) t=2 t=2 e te t 1 t=1 e2 t b) t=1 e2 t t=2 e te t 1 c) 1 (e t ē)(e t 1 ē) e) (e t ē)(e t+1 ē) f) g) 1 e t e t+1 h) t=1 t=1 e 2 t i) t=1 e t e t 1 t=2 (e t ē) 2 t=1 t=1 e2 t 1 t=1 e te t+1 MINIC is a a time series. It fids the values of p ad q of the ARMA(p, q) model which miimizes a estimate of the. Problem 15. Which of the followig phrases correctly fills i the first blak above? a) method for estimatig the parameters of a ARMA(p, q) model for b) method to help idetify the orders p ad q of a appropriate ARMA model for c) test of the ormality of the residuals obtaied by fittig a ARMA(p, q) model to d) test of the sigificace of a ARMA(p, q) model for e) test of the residual ACF whe fittig a ARMA(p, q) model to f) test of the statioarity of a ARMA(p, q) model for Problem 16. Which of the followig choices correctly fills i the secod blak above? a) Likelihood b) Variace Estimate c) Stadard Error Estimate d) SBC e) PACF f) Costat Estimate g) Chi-Square h) ACF Problem 17. Suppose you have a time series z 1, z 2,..., z with a very large value of (say, i the millios) which has bee geerated by some statioary ARMA process. If you use OLS (ordiary least squares) to fit the regressio model the parameter estimate ˆβ 3 will be very close to z t = β 0 + β 1 z t 1 + β 2 z t 2 + β 3 z t 3 + ε t, a) ρ 2 b) ρ 3 c) ρ 4 d) σ 2 a e) φ 22 f) φ 33 g) φ 44 h) µ z i) σ 2 z j) θ 3 k) θ 2 l) θ 4 5
Problem 18. Suppose we kow the values of a time series {z t } at times 1, 2,..., that is, we kow z 1, z 2, z 3,..., z. The series {z t 5 } (z t lagged by 5) will have a) 4 missig values at the ed of the series b) 5 missig values at the ed of the series c) 4 missig values at the begiig of the series d) 5 missig values at the begiig of the series e) o missig values Problem 19. symbol For a statioary process {z t }, the quatity Cov(z t, z t k ) is represeted by the a) ψ k b) γ k c) φ kk d) θ k e) φ k f) ρ k Problem 20. If the values of X ad Y are obtaied for a radom sample of idividuals: (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ), the sample covariace betwee X ad Y is defied to be a) (X i X)(Y i Ȳ ) (X i X) 2 (Y i Ȳ )2 b) 1 X i Y i c) 1 d) (X i X) 2 (Y i Ȳ )2 1 1 (X i X)(Y i Ȳ ) e) c(x, Y ) s x s y f) (X i X)(Y i Ȳ ) (X i X) 2 (Y i Ȳ )2 Problem 21. A ARMA(2,1) process is defied by a) z t = C + φ 1 z t 1 + φ 2 z t 2 + a t θ 1 a t 1 b) z t = C + φ 1 z t 1 + a t θ 1 a t 1 θ 2 a t 2 c) z t = C + φ 1 z 1 + φ 2 z 2 + a t θ 1 a 1 d) z t = C + φ 1 z 1 + a t θ 1 a 1 θ 2 a 2 e) z t = C + φ 1 z t 1 + φ 2 z t 2 θ 1 a t 1 f) z t = C + φ 1 z t 1 θ 1 a t 1 θ 2 a t 2 g) z t = C + φ 1 z 1 + φ 2 z 2 θ 1 a 1 h) z t = C + φ 1 z 1 θ 1 a 1 θ 2 a 2 6
Problem 22. A ARMA(p, q) process is statioary if ad oly if all the solutios of the uit circle i the complex plae. a) φ(b) = 0 lie strictly iside b) θ(b) = 0 lie strictly iside c) θ(b)/φ(b) = 0 lie strictly iside d) φ(b) = 0 lie strictly outside e) θ(b) = 0 lie strictly outside f) θ(b)/φ(b) = 0 lie strictly outside g) φ(b) = 0 lie o the boudary of h) θ(b) = 0 lie o the boudary of i) θ(b)/φ(b) = 0 lie o the boudary of Problem 23. The process z t = φ 1 z t 1 + a t θ 1 a t 1 θ 2 a t 2 θ 3 a t 3 ca be writte i backshift form as a) (1 θ 1 B θ 2 B 2 θ 3 B 3 )z t = (1 φ 1 B)a t b) (1 θ 1 B θ 2 B 2 θ 3 B 3 )z t = (1 + φ 1 B)a t c) (1 + θ 1 B + θ 2 B 2 + θ 3 B 3 )z t = (1 φ 1 B)a t d) (1 + θ 1 B + θ 2 B 2 + θ 3 B 3 )z t = (1 + φ 1 B)a t e) (1 + φ 1 B)z t = (1 θ 1 B θ 2 B 2 θ 3 B 3 )a t f) (1 + φ 1 B)z t = (1 + θ 1 B + θ 2 B 2 + θ 3 B 3 )a t g) (1 φ 1 B)z t = (1 θ 1 B θ 2 B 2 θ 3 B 3 )a t h) (1 φ 1 B)z t = (1 + θ 1 B + θ 2 B 2 + θ 3 B 3 )a t Problem 24. be used I PROC ARIMA, the ML (maximum likelihood) method of estimatio should a) wheever computatioal speed is very importat b) as a precautio wheever the data are ot ormally distributed c) wheever the CLS method is too slow d) whe the shocks a t are idepedet ad approximately ormally distributed with costat variace e) oly whe the series is o-statioary f) oly whe all the parameters are statistically sigificat 7
Problem 25. For a statioary time series z 1,..., z, suppose the scatterplot of z t versus z t 4 looks like the plot give below. The you kow a) the variace of the series z t teds to icrease with time t b) the variace of the series z t teds to decrease with time t c) the residuals are ot ormally distributed d) the residuals are ormally distributed e) the partial autocorrelatio at lag 4 is is positive f) the partial autocorrelatio at lag 4 is is egative g) the autocorrelatio at lag 4 is positive h) the autocorrelatio at lag 4 is egative i) the mea of the series z t teds to icrease with time t j) the mea of the series z t teds to decrease with time t Problem 26. I regressio, large values of H (the leverage) idetify a) covariates which ca be dropped b) covariates with serial correlatio c) covariates which should be retaied d) serial correlatio i the residuals e) cases with uusual respose values f) ifluetial cases i the data g) cases with uusual covariate values h) ifluetial covariates i the model Problem 27. Suppose you are aalyzig a time series z 1, z 2,..., z usig SAS PROC ARIMA. I the output produced by the IDENTIFY statemet, the values of the Ljug-Box test statistics Q(6), Q(12), Q(18), Q(24) are give alog with their correspodig P -values. If {z t } is actually just a radom shock sequece, the we expect that a) H 0 : ρ 1 = ρ 2 = = ρ 6 = 0 will be rejected b) H 0 : Q(6) = Q(12) = Q(18) = Q(24) = 0 will be rejected c) all four P -values will be large d) all four P -values will be small e) Q(6), Q(12), Q(18), Q(24) will all be large f) most of the ACF values will lie outside the bad g) most of the PACF values will lie outside the bad 8
Suppose that, based o a radom sample of 400 observatios, you fit a regressio model Y = β 0 + β 1 X 1 + β 2 X 2 + ε for a respose variable Y o two covariates X 1 ad X 2. Use the iformatio i the table below to aswer the ext two questios. Parameter Estimates Parameter Stadard Variable DF Estimate Error t-value Pr > t Itercept 1 4.0 0.5 <.0001 X1 1 40.0 10.0 <.0001 X2 1 24.0 16.0.1336 Problem 28. The t-value etries i the table have bee left blak. What is the t-value for the X1 row of the table? a) 1.5 b) 4.0 c) 16.0 d).25 e).0625 f) 2.0 g) 0.5 Problem 29. Assumig all the regressio assumptios are valid, compute a approximate 95% cofidece iterval for β 1. a) (20.4, 59.6) b) (38.04, 41.96) c) (39.951, 40.049) d) (9.804, 10.196) e) (8.04, 11.96) f) ( 68.4, 88.4) g) (39.02, 40.98) h) (6.08, 13.92) Problem 30. A process is geerated by z t = C + φ 1 z t 1 + φ 2 z t 2 + a t θ 1 a t 1. What are the requiremets for this process to be statioary? a) φ 1 < 1, θ 1 < 1 b) φ 2 < 1, θ 1 < 1 c) φ 1 < 1 d) θ 1 < 1 e) φ 2 < 1, φ 2 + φ 1 < 1, φ 2 φ 1 < 1, θ 1 < 1 f) φ 1 < 1, φ 1 + φ 2 < 1, φ 1 φ 2 < 1, θ 1 < 1 g) φ 2 < 1, φ 2 + φ 1 < 1, φ 2 φ 1 < 1 h) φ 1 < 1, φ 1 + φ 2 < 1, φ 1 φ 2 < 1 9
The last two problems use the three pages of SAS output attached to the ed of this exam. Problem 31. The first page of output has the title IDENTIFY A GOOD MODEL. This page gives the usual output produced by the IDENTIFY statemet i PROC ARIMA for a series z1 of legth 200. Based o this output, a reasoable model for this series is a) MA(2) b) MA(3) c) MA(4) d) AR(1) e) AR(2) f) ARMA(1,1) g) ARMA(2,2) h) ARMA(1,3) Problem 32. The last two pages of output have the title TRY AR(1) ad MA(3) MODELS. SAS PROC ARIMA was used to fit a AR(1) ad a MA(3) model to a time series. (This series is ot the series from the previous problem.) These pages give parts of the output produced by the ESTIMATE statemet for these two models. The first page gives some of the output from fittig the AR(1) model; the secod page gives output from the MA(3) model. Suppose you relied o the AIC or SBC to choose betwee these two models. Which of the followig statemets is true? a) AIC selects AR(1); SBC selects MA(3) b) AIC selects MA(3); SBC selects AR(1) c) Both AIC ad SBC select MA(3) d) Both AIC ad SBC select AR(1) e) Both AIC ad SBC reject the ull hypothesis f) Neither AIC or SBC reject the ull hypothesis 10