Statistics 349(02) Review Questions

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1 Statistics 349(0) Review Questions I. Suppose that for N = 80 observations on the time series { : t T} the following statistics were calculated: _ x = C(0) = 4.99 In addition the sample autocorrelation function, r h, was computed for h = 1,, 3,..., 10 and is tabulated below: h r h a) Compute the sample partial autocorrelation function ^, for = 1,, 3. i) Suppose that this time series is identified as an ARIMA(1,1) time series. Use the method of moments to estimate all the parameters of the model. ii) Suppose that this time series is identified as an AR(3) time series. Use the method of moments to estimate all the parameters of the model. II. Suppose that the time series { : t T} has been identified as an MA(1) time series and satisfies the equation: = + u t + 1 u t-1. Let r 1 the value of the sample autocorrelation function at lag 1. Then the methods of moments estimator of the parameter 1 satisfies the equation: r 1 ^1 - ^ 1 + r 1 = 0 (1) Show that one of the roots of equation (1) is less than or equal to 1 in absolute value while the other root is greater than or equal to 1. III. Consider the AR(1) time series: = -1 + u t. Show that var(x^ T(l)) = 1 - l 1 -. Find lim var(x^ T(l)). l page 1

2 IV. Consider the ARMA(1,1) time series: = -1 + u t + u t-1. Show that var(x^ T(l)) = Find V. Let = lim var(x^ T(l)). l i1 i i + + u t l- 1 + [ ] 1 - denote the inverted form the stationary time series { : t T} and i0 = + iu ti ( with 0 1) denote the random shoc form of the same time series. By defining (s) = x h where (s) = s + s + 3 s s and (s) = 1-1 s - s - 3 s 3-4 s h ( h) s show that: (s) = (s)(1/s) = (s)(1/s) VI. Suppose that for N =100 observations on the time series { : t T} the sample autocorrelation function, r h, was computed for h = 1,, 3,..., 10 and is given below: r 1 = 0.31, r = 0.37, r 3 = -0.05, r 4 = 0.06, r 5 = -0.1 r 6 = 0.11, r 7 = 0.08, r 8 = 0.05, r 9 = 0.1, r 10 = Suggest an ARMA model which may be appropriate. VII. For the AR(1) model { : t T} satisfying the equation: - = (-1 - ) + u t Show that the l-steps ahead forecast of x T+l given observations up to time T is: x^ T(l) = l (x T - ) for l = 1,,3,.... VIII. Let { : t T}, denote an ARMA(1,1) time series satisfying the equation: - = 1 (-1 -)+u t + 1 u t-1 Assume also that {u t : t T} is an independent series with var(u t ) =. Let f() denote the spectral density function of the time series. Determine all global and local maxima of f() on the interval [0,]. page

3 IX. From a series of T = 150 observations the sample autocorrelation function and sample partial autocorrelation function where computed up to lag 10 for: i) the raw data, ii) first differences and iii) second differences The results are tabulated below: i) Autocorrelation Function (AFC) Lag ii) Partial Autocorrelation Function (PAFC) Lag The Sample mean and variance of these series are tabulated below: Mean Variance Identify the time series. Determine preliminary estimates of all the parameters. Write out the equation of the identified model. X. Consider the ARIMA(1,1,1) time series: - -1 = 0.6 ( ) + u t - 0. u t-1. a) Find the Random Shoc form of this time series. b) Find the Inverted form of this time series. page 3

4 XI. It will later be shown that the "Sunspot Data", { : t T}, was identified as an AR() model satisfying the equation (parameters estimated): = ( ) ( ) +u t The estimate of, var(u t ), is the Using these values of the parameters, determine and plot the graph of the spectral density function, f(), of the series. Determine all global and local maxima of f() on the interval [0,]. What is the "period" of the dominant "frequency"? XII. The "Chemical Concentration Data", { : t T}, was identified as an ARMA(1,1) model satisfying the equation (parameters estimated): = ( )+u t - u t-1 The estimate of, var(u t ), is Using these values of the parameters, determine and plot the graph of the spectral density function, f(), of the series. Determine all global and local maxima of f() on the interval [0,]. XIII. Suppose that {u t : t =...,-,-1,0,1,,...} is an independent time series with mean zero, variance u = 4.0. Suppose that the time series { : t =...,-,-1,0,1,,...} satisfies the equation.: = u t -.4 u t-1. a) Determine the autocovariance function and autocorrelation function of the time series. b) Find the random shoc form of the time series. c)suppose that the first four observations of the time series are x 1 =.75, x = 1.10, x 3 = 1.38 and x 4 =0.94 i) Use these observations to compute prediction intervals for the next 4 observations. (Compute both 95% and 66.7% prediction limits) ii) If the fifth observation turns out to be x 5 = 4. use this information to re-compute prediction intervals for the next 3 observations. (Compute both 95% and 66.7% prediction limits) page 4

5 XIV Consider the model for the time series {yt t T} y x v and x x u t t t t t 1 t where {vt t T} and {xt t T} are independent white noise time series with variances and respectively. u u a. Assuming that E x 0 and Var x, determine a means of calculating: i. xˆ t t 1 E xt, yt3, yt, yt 1 ii. xˆ t t E xt, yt3, yt, yt 1, yt iii. xˆ t T Ext, yt, yt 1, yt for t T b. Assuming that 0.9, v.0, 1.5 and 0 15, Suppose that the first T = 15 observations on {yt t T} are given below: t y t t y t Compute i. xˆ t t 1 Ext, yt3, yt, yt 1 t 1,,,15 ii. xˆ t t Ext, yt3, yt, yt 1, yt t 1,,,15 xˆ 15 Ex, y, y, y t 1,,,15 iii. t t T T1 15 XV Consider the following data from a stationary time series t y t t y t t y t Compute and graph iv. The periodogram IT for 0,1,,,14 9 v. The unsmoothed estimate of the spectral density f ˆ for 0,1,,,14 vi. The Danielle estimate of the spectral density ˆ d f for 0,1,,,14 and d 3,5,7 9 page 5

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