Final Examination 7/6/2011

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The Islamic University of Gaza Faculty of Commerce Department of Economics & Applied Statistics Time Series Analysis - Dr. Samir Safi Spring Semester 211 Final Examination 7/6/211 Name: ID: INSTRUCTIONS: 1. Write your name and student ID. 2. You have 2 hours 3. This exam must be your own work entirely. You cannot talk to or share information with anyone. 4. Show all your work. Partial credit will only be given where sufficient understanding of the problem has been demonstrated and work is shown. DON'T WRITE ON THIS TABLE Q UESTION #1 #2 #3 #4 TOTAL P OINTS 1

Question #1: [2 Points] Consider an AR(1) model Yt Yt 1 et e t is a white noise process such that et is a white noise process. Let Y be a random variable with mean µ and 2 variance σ. 2 t 1 t a. (5 Points) Show that for t > we can write Y = e + φ e + φ e + L + φ e + φ Y = φ +. Where φ can be any number and { } t t t 1 t 2 1 b. (3 Points) Derive an expression for the mean of Y t for t >. 2

c. (6 Points) Derive an expression for the variance of Y t for t >. d. (3 Points) Suppose that µ =. Argue that, if { } Y is stationary, we must have φ 1 t e. (3 Points)Suppose that µ =. Show that, if { } Y is stationary, we must have φ < 1 t 3

Question #2: [2 Points] a- (5 Points) Consider the ARMA(1,1) process: Y = φ Y t t + e θ e 1 t t. Find the variance of the 1 process { Y t }. b- (5 Points) Identify as specific ARIMA model for Yt =.5Yt 1.5Yt 2 + et.5et 1 +.25et 2. Determine p, d, and q and the values of the Parameters, the φ's and θ 's 4

c- (1 Points) Consider the model Yt = et 1 et 2 +.5et 3. Find the autocorrelation function for this process. 5

Question #3: [3 Points] Consider the data named " robot" that contains a time series obtained from an industrial robot. The robot was put through a sequence of maneuvers, and the distance from a desired ending point was recorded in inches. This was repeated 324 times to form the time series.. (a) (3 Points) Interpret the time series plot of the data. (b) (3 Points) Interpret the sample ACF for these data (c) (3 Points) Based on KPSS test of Stationarity, do these data appear to come from a stationary or nonstationary process? (d) (3 Points) Interpret the time series plot of the differences of these data. (e) (3 Points) Interpret the sample ACF for the differences of the logarithms of these data. 6

(f) (3 Points) Based on KPSS test of Stationarity, do the differences of these data appear to come from a stationary or nonstationary process? (g) (3 Points) Consider fitting an IMA(1,1) model for these data. Does the model have statistically significant parameter estimates? Interpret (h) (9 Points) Discuss the fit of an IMA(1,1) model for these data in terms of : - h1: The plot of the Standradized Residuals - h2: Normality of the Residuals - h3: Autocorrelation of the Residuals 7

Question #4: [1 Points] The following graphs represent the sample autocorrelation function (ACF), sample partial autocorrelation function (PACF), and Extended PACF for six different time series. Based on this empirical evidence decide if the series are AR(p), MA(q), ARMA(p,q) and specify its order. Justify your answer. Series A: Series B: Series C: Series D: Series E: 8

Output for Question #3: Parts A & B: Time Series Plot of Robot End PositionPlot- Part A Sample ACF of Robot Data - Part B Robot End Position -.5..5 ACF -.1..1.2.3 Part C: 5 1 15 2 25 3 Time KPSS Test for Level Stationarity 5 1 15 2 25 data: robot KPSS Level = 2.4518, Truncation lag parameter = 4, p-value =.1 Parts D & E: Plot for Part (D) Sample ACF of Difference of Robot Data - Part E Difference of Robot End Position -.5..5.1 ACF -.4 -.3 -.2 -.1..1 Part F: 5 1 15 2 25 3 Time KPSS Test for Level Stationarity 5 1 15 2 25 data: diff(robot) KPSS Level =.138, Truncation lag parameter = 4, p-value =.1 Part G: Coefficients: ma1 -.8713 s.e..389 9

Part H: S.Residuals from IMA(1,1)Model- Part H Quantile-Quantile Plot- Part H IMA(1,1) Residuals -3-2 -1 1 2 3 Sample Quantiles -.5..5 5 1 15 2 25 3 Time -3-2 -1 1 2 3 Theoretical Quantiles Shapiro-Wilk normality test data: residuals(mode) W =.9969, p-value =.799 Sample ACF of Residuals from IMA(1,1) Model for Robot Data ACF of Residuals-IMA(1,1) -.15 -.1 -.5..5.1 5 1 15 2 25 Box-Ljung test data: residuals from mod2 X-squared = 17.88, df = 11, p-value =.155 1

Output for Question #4: Series A: Series a Series a ACF -.2..2.4.6.8 Partial ACF -.2..2.4.6.8 5 1 15 2 5 1 15 2 Series B: Series b Series b ACF -.5 -.4 -.3 -.2 -.1..1.2 Partial ACF -.5 -.4 -.3 -.2 -.1..1.2 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 11

Series C: Series c Series c ACF -.2 -.1..1.2 Partial ACF -.2 -.1..1.2 Series D: 5 1 15 2 5 1 15 2 AR/MA 1 2 3 4 5 6 7 8 9 1 11 12 13 x x x o o o o o o o o o o o 1 x x x o x o o o o o o o o o 2 x o o o o o o o o o o o o o 3 x o o o o o o o o o o o o o 4 x o o o o o o o o o o o o o 5 x o o x o o o o o o o o o o 6 x o o x x o o o o o o o o o 7 x x x x o x o o o o o o o o Series E: 12

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