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# Anl Time Ser ECON 677

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This 7 page Class Notes was uploaded by Leopoldo Ritchie DDS on Thursday October 29, 2015. The Class Notes belongs to ECON 677 at University of Michigan taught by Edward Ionides in Fall. Since its upload, it has received 30 views. For similar materials see /class/231653/econ-677-university-of-michigan in Economcs at University of Michigan.

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Date Created: 10/29/15

Statistics 531Econ 677 Winter 2005 Ed Ionides SECTION A We investigate some data from neurophysiology An electrode implanted painlessly into a monkey7s brain records a sequence of ring events for an individual neuron cell neurons communicate by ring pulses of electrical charge Suppose the ring times are F IUF27FTH7 measured in milliseconds 1ms is 10 3s We take as our time series xi FHL 7 E with t 17 7T This is the series of times intervals between ring events The data7 with T 4157 are plotted in Fig 1 We wish to model st in order to quantify the behavior of the neuron7 to later compare it with other neurons and investigate the effects of experimental treatments The estimated ACE and PACF of st are shown in Fig 2 200 150 l 100 I I I I I o 1 00 200 300 400 t Figure 1 Time series x of intervals between neuron ring events a b Pamal ACF Figure 2 a Estimated autocorrelation function of zt b Estimated partial autocorrelation function of xi A1 2 pts What ARMA model for x does Fig 2 suggest and why There is no single clear cut answer The PACF appears to cut o at lag Z The ACF is also within the dashed lines after lag 1 but shows some indication of decreasing like a damped oscillation an AR2 property The PACF also has some resemblance to a damped oscillation an MA property Anything between AR1 and ARMA 22 can be defended by inspecting Fig 2 A2 2 pts Another way to select a model is by comparing AlC values A table of A10 values is shown in Table 1 What ARMA model does this suggest and why p q 0 1 2 3 0 NA 39615 39627 39647 1 39610 39626 39646 39666 2 39626 39605 39597 39617 3 39646 39654 39626 39683 Table 1 A10 values from tting ARMApq models to x ARMA 22 has the lowest AIC so is favored by this criterion ARMA 21 and ARMA 10 are simpler models which also have promising AIC values A3 2 pts Find the log likelihood of an ARMA21 model and explain your calculation AIC 72A 2k where A is the mamimized log likelihood and h is the number of parameters Here h 5 1 261u02 so A 7197525 A4 2 pts Does the table of A10 values contain any evidence for or against the claim that the likelihood is correctly calculated and maximized Explain The table is inconsistent 7 adding a parameter can only increase the mamimized log likelihood ie the A10 can only increase by S 2 Compare ARMA 33 to ARMA This can only come about by imperfect 7 b r quot andor 39 39 quot SECTION B Fitting an ARMA22 model gives the following R printout Call arimaX X order C2 0 2 Coefficients arl ar2 mal ma2 intercept 16009 06445 14982 05219 264163 se 01886 01839 02104 02094 07954 sigma 2 estimated as 7917 log likelihood 197388 aic 395976 B1 4 pts Write out the tted model carefully stating all the model assumptions xi 264 160dt1 7 264 7 064t2 7 264 7 wt 7150wtijL 05sz where wt is white noise with standard deviation 28 This calculation also assumes that wt is Gaussian ie wt N02812 ACF o 5 1o 15 20 25 Figure 3 Estimated autocorrelation function of the residuals from tting an ARMA22 model to zt B2 3 pts Fig 3 shows the ACF of the residuals from tting an ARMA22 model Com ment on which modeling assumptions this gure supports and which it does not support It supports the assumptions that the driving noise process is uncorrelated and has no trend It does not reveal anything about the assumptions that the driving noise is constant variance or ii Gaussian B3 3 pts Find the roots of the AR and the MA polynomials for the tted ARMA22 model ls there evidence for parameter redundancy AR polynomial 1 71602 064522 has roots 160 124 i 0102 MA polynomial 1 7 1502 05222 has roots 106181 So the AR roots ar not very close to the MA roots 7 there is no strong suggestion of parameter redundancy B4 2 pts Simulations from the tted ARMA22 model were computed as follows arma22lt arimax orderc 2 O 2 Ntlt lengthx simlt repONt wlt rnormNt mO sdsqrt arma22sigma2 fornt in 3Nt simntlt arma22coef quotarlquot simnt 1arma22coef quotar2quot simnt 2 arma22coefquotma1quotwnt 1arma22coefquotma2quotwnt 2wnt simlt simarma22coef quot intercept quot Sample output is shown in Fig 4 What does a comparison of Fig 4 with Fig 1 say about ARMA modeling of xi Almost all well considered and comprehensible answers were accepted here The plots look quite di erent Fig 1 is always positive and appears to have some regularity to the peaks Fig 4 has less pronounced peaks is more symmetric about its mean and occasionally becomes negative The data do not resemble a Gaussian ARMA 22 process 3 53 Figure 4 A simulation from the tted ARMA22 model B5 2 pts ls the random process generated in B4 and plotted in Fig 4 stationary Answer yes or no and explain No The initial values contradict having a constant variance the initial variance is zero However the process is asymptotically stationary as in Shumway and Sto er Problem 22 One could make the simulations e ectively stationary by throwing away some number say 100 values at the start of the simulation SECTION C We now investigate a logarithmic transformation of the data Below is the R printout from tting an ARMA22 model to log0 zt Call arimax log10x order C2 0 2 Coefficients ar1 ar2 ma1 ma2 intercept 16975 07250 14647 04647 12925 se 00740 00718 00941 00939 00021 sigma 2 estimated as 007637 log likelihood 5679 aic 12559 WWW I I I I o 1 00 200 300 400 t Figure 5 Time plot of log10 zt F amal ACF r Figure 6 a Estimated autocorrelation function of logl0 zt b Estimated partial autocor relation function of loglO zt Cl 2 pts ls there any indication from Fig 57 Fig 6 and the tted model printouts in Sections B and C that ARMA modeling is more successful after a log transformation or less Explain The model printouts do not tell us whether the transformation is appropriate 7 the A10 val ues are not comparable The estimated ACF does not say anything about the transformation The time plot looks more symmetric 7 symmetry is a property of Gaussian ARMA models so this is encouraging ACF Lag Figure 7 Estimated autocorrelation function of the residuals from tting an ARMA22 model to logl0 zt Statistics 531Ec0n 677 Winter 2009 We investigate the monthly number mumps cases reported in New York City from January 1928 to June 1972 During this period before the introduction of a vaccine mumps was a common childhood disease to which almost all children were exposed Because mumps displays a character istic rash it is fairly easily diagnosed Mumps is a reportable disease meaning that doctors have a legal obligation to report any cases they encounter This dataset therefore gives an opportunity to study disease transmission and maybe learn lessons relevant to diseases of current concern such as bird u SARS or HIVAIDS The data which we shall denote by ztt 12 are graphed in Fig 1 1000 1500 2000 500 1930 1940 1950 1960 1970 Figure 1 Monthly mumps reports mt in New York City from January 1928 to June 1972 Section A Spectral analysis We seek to interpret the estimated spectum in Fig 2 and in particular the features labeled 1 through 1e06 3 sp ectru m 1e04 1e02 I I I I 0 1 2 3 4 5 6 frequency Figure 2 An estimated spectral density for mt calculated via spectrumxspansc 35 7 A1 2 points A2 3 points One might expect mumps to have annual seasonality One might also expect mumps to have long term cycles as the population of suscpetible children those without immunity replenishes after previous outbreaks Discuss the interpretation of the 5 spectral peaks labeled 1 through 5 in Fig 2 You do not have to discuss here whether these peaks are statistically signi cant which is question A3 below What are the units of frequency in Fig 2 Explain how you reach your answer A3 2 points Comment on the statistical signi cance of these ve peaks You are not expected to present formal tests but you should say what your opinion is and why Section B ARIMA analysis We try tting an ARIMA3 00 x 0 1112 model Call this model M1 The output from M1arimax orderc 3 O O seasonalc 0 1 1 is ar1 ar2 ar3 sma1 12032 03025 00632 08841 se 00439 00674 00442 00231 sigmaquot2 estimated as 12881 log likelihood 322076 aic 645151 Another possibility is to model logmt again using ARIMA30 0 x 01112 Call this model M2 The output from M2arimalogx orderc 3 O O seasonalc O 1 1 is ar1 ar2 ar3 sma1 09197 01577 01710 08080 se 00434 00592 00438 00285 sigmaquot2 estimated as 003632 log likelihood 11748 aic 22496 B1 2 points Can the above analysis determine whether a log transformation is appropriate Explain A table comparing AlC values for various ARIMA 0j X 01112 models for logzt is given below AR MA 0 1 2 3 4 0 NA 3127628 927453 4291403 1318598 1 2139453 2119458 2244315 22735447 2255215 2 2119459 2125350 2368260 22370594 2247305 3 2249618 2379834 2361537 23441349 2324061 4 2298224 2211222 2364941 23521621 2397320 B2 2 points The software gave no error messages while computing this table ls there any reason to suspect that the numeric maximization of the likelihood is less than adequate B3 4 points Discuss brie y what you learn from the AlC table shown in terms of developing a suitable model for these data Explain brie y why AlC may not be the only criterion considered when selecting a model and list some other analyses that you would carry out to determine and defent a choice of model Section C Diagnostic analysis Fig 3 contains six diagnostic plots three for each of models M1 and M2 C1 2 points Explain carefully the meaning of the dashed line in sample ACF plots produced by R for example in Fig 3a1 Here you are asked to explain the statistical method later parts will ask you to interpret the results in the context of the data and models under investigation C2 2 points Compare a1 and b1 in Fig 3 What does this tell you about models M1 and M2 C3 2 points Compare a2 and b2 in Fig 3 What does this tell you about models M1 and M2 C4 2 points Compare a3 and b3 in Fig 3 What does this tell you about models M1 and M2 In particular what do you learn about the appropriateness of an assumption that the white noise process driving the ARlMA model is independent and identically distributed

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