MATH4022-E1 The Universit\ of Nottingham SCHOOL OF MATHEMATICAL SCIENCES A LEVEL 4 MODULE, SPRING SEMESTER 2019-2020 TIME SERIES AND FORECASTING Suggested time to complete: TWO Hours THIRTY Minutes Paper set: 28/05/2020 - 10:00 Paper due: 04/06/2020 - 10:00 AQVZeU ALL TXeVWiRQV YRXU VROXWiRQV VhRXOd be ZUiWWeQ RQ ZhiWe SaSeU XViQg daUN iQN (QRW SeQciO), RQ a WabOeW, RU W\SeVeW. DR QRW ZUiWe cORVe WR Whe PaUgiQV. YRXU VROXWiRQV VhRXOd iQcOXde cRPSOeWe e[SOaQaWiRQV aQd aOO iQWeUPediaWe deUiYaWiRQV. YRXU VROXWiRQV VhRXOd be baVed RQ Whe PaWeUiaO cRYeUed iQ Whe PRdXOe aQd iWV SUeUeTXiViWeV RQO\. AQ\ QRWaWiRQ XVed VhRXOd be cRQViVWeQW ZiWh WhaW iQ Whe LecWXUe NRWeV. GXidaQce RQ Whe AOWeUQaWiYe AVVeVVPeQW AUUaQgePeQWV caQ be fRXQd RQ Whe FacXOW\ Rf ScieQce MRRdOe Sage: ?iiTb,ffKQQ/H2XMQiiBM;?KX+XmFf+Qm`b2fpB2rXT?T\B/4NNR89Ob2+iBQM@k SXbmiW \RXU aQVZeUV aV a ViQgle PDF ZiWh each Sage iQ Whe cRUUecW RUieQWaWiRQ, WR Whe aSSURSUiaWe dURSbR[ RQ Whe mRdXle¶V MRRdle Sage. 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WheUe QeceVVaU\, PiQRU cOaUificaWiRQV RU geQeUaO gXidaQce Pa\ be SRVWed RQ MRRdOe fRU aOO VWXdeQWV WR acceVV. SWXdeQWV ZiWh aSSURYed accRPPRdaWiRQV aUe SeUPiWWed aQ e[WeQViRQ Rf 3 da\V. The VWaQdaUd UQiYeUViW\ Rf NRWWiQgham SeQalW\ Rf 5% dedXcWiRQ SeU ZRUkiQg da\ Zill aSSl\ WR aQ\ laWe VXbmiVViRQ. MATH4022-E1 TXUQ RYeU MATH4022-E1 Academic Integrit\ in Alternative Assessments The alternative assessment tasks for summer 2020 are to replace e[ams that would have assessed \our individual performance. You will work remotel\ on \our alternative assessment tasks and the\ will all be undertaken in ³open book´ conditions. Work submitted for assessment should be entirel\ \our own work. You must not collude with others or emplo\ the services of others to work on \our assessment. As with all assessments, \ou also need to avoid plagiarism. Plagiarism, collusion and false authorship are all e[amples of academic misconduct. The\ are defined in the Universit\ Academic Misconduct Polic\ at: ?iiTb,ffrrrXMQiiBM;?KX+X mFf+/2KB+b2`pB+2bf[mHBivKMmHfbb2bbK2MiM/r`/bf+/2KB+@KBb+QM/m+iXbTt Plagiarism: representing another person¶s work or ideas as \our own. You could do this b\ failing to correctl\ acknowledge others¶ ideas and work as sources of information in an assignment or neglecting to use quotation marks. This also applies to the use of graphical material, calculations etc. in that plagiarism is not limited to te[t-based sources. There is further guidance about avoiding plagiarism on the Universit\ of Nottingham website. False Authorship: where \ou are not the author of the work \ou submit. This ma\ include submitting the work of another student or submitting work that has been produced (in whole or in part) b\ a third part\ such as through an essa\ mill website. As it is the authorship of an assignment that is contested, there is no requirement to prove that the assignment has been purchased for this to be classed as false authorship. Collusion: cooperation in order to gain an unpermitted advantage. This ma\ occur where \ou have consciousl\ collaborated on a piece of work, in part or whole, and passed it off as \our own individual effort or where \ou authorise another student to use \our work, in part or whole, and to submit it as their own. Note that working with one or more other students to plan \our assignment would be classed as collusion, even if \ou go on to complete \our assignment independentl\ after this preparator\ work. Allowing someone else to cop\ \our work and submit it as their own is also a form of collusion. Statement of Academic Integrit\ B\ submitting a piece of work for assessment \ou are agreeing to the following statements: 1. I confirm that I have read and understood the definitions of plagiarism, false authorship and collusion. 2. I confirm that this assessment is m\ own work and is not copied from an\ other person¶s work (published or unpublished). 3. I confirm that I have not worked with others to complete this work. 4. I understand that plagiarism, false authorship, and collusion are academic offences and I ma\ be referred to the Academic Misconduct Committee if plagiarism, false authorship or collusion is suspected. MATH4022-E1 TXUQ RYeU 1 MATH4022-E1 1. Throughout this question |ু~ is a white noise process with variance ౨3 . (a) Identif\ each t\pe of ARIMA)- - * model in the following list, stating the values of- - and the values of the parameters of the model. i) ি > 1/7িѿ2 , ু ii) ি > 1/236িѿ4 , ু ѿ 1/6ুѿ2 iii) ি > 2/6িѿ2 ѿ 1/6িѿ3 , ু ѿ 1/3ুѿ3 iv) ি > ѿ1/86িѿ2 , 3িѿ3 , ু where 3 > 1/** and ** are the final two digits of \our Student ID number. For e[ample if \our Student ID is 64367234 then ** is 34 and hence 3 > 1/**> 1/34. [10 marks] (b) State whether or not each of the models in part (a) is weakl\ stationar\, giving \our reasons. [10 marks] (c) Consider an AR(2) process |ি~ with parameters 2 > 1/7 and 3 > ѿ1/1:. E[plain wh\ the process is weakl\ stationar\ and derive the MA(ҋ) representation of |ি~, i.e. ি > ҋา>1 পুѿ where প are to be determined. [10 marks] (d) Consider again an AR(2) process with parameters 2 > 1/7 and 3 > ѿ1/1:. Using the Yule-Walker equations, derive an e[pression for the autocorrelation function ౦)υ* forυ ? 1 and compute the correlation between the realisations of the process at lag 1. [10 marks] MATH4022-E1 2 MATH4022-E1 2. Throughout this question |ু~ is a white noise process with variance ౨3 , and is the backward shift operator. (a) In the figure below there is a plot of a time series, its sample autocorrelation function and its sample partial autocorrelation function. i) Identif\ an appropriate t\pe of ARIMA model for the time series and provide reasons for \our choice. ii) Write down the model in the form)*)ি ѿ ౡ* > ౝ)*ু- where e[pressions for the operators )* and ౝ)* should be provided. iii) Suggest appro[imate values for the parameters of the model, giving \our reasons. iv) How would \ou check if the model is a good fit? Time x 0 200 400 600 800 96 98 10 0 10 2 10 4 0 5 10 15 20 25 30 −0 .2 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Lag AC F Series x 0 5 10 15 20 25 30 −0 .4 −0 .2 0. 0 0. 2 0. 4 Lag Pa rti al AC F Series x [20 marks] (b) Let ি > ু ѿ 1/:ুѿ2 , 1/3ুѿ3/ What order of ARIMA model is |ী~, where ী > ি ѿ িѿ2? Provide the conditions required for invertibilit\ of the model for ী (but there is no need to check whether or not the series ী is invertible). [10 marks] (c) An MA(1) model ি > )2 ѿ 1/6*ু༚ has been fitted to a time series. If the residuals of a time series look like an AR(2) of the form)2 ѿ 1/4 ѿ 1/53*ী > - where |~ is a white noise process, what model should be fitted ne[t? Give \our reasons. [10 marks] MATH4022-E1 TXUQ OYeU 3 MATH4022-E1 3. Throughout this question |ু~ is a white noise process with variance ౨3 , and is the backward shift operator. (a) In \our own words briefl\ discuss the use of the Akaike and Ba\esian information criteria in fitting time series models. [10 marks] (b) Consider the following time seriesি2 > 374/2-Ϳ -িৎѿ4 > 373/9- িৎѿ3 > 372/9- িৎѿ2 > 373/3- িৎ > 373/2 and an AR(1) model is assumed with fitted parametersȠ > 1/: - Ƞౡ > 374/1 - Ƞ౨3 > 1/5/ Obtain a prediction for the time series at three time steps ahead at time > ৎ , 4 and provide an appro[imate 95% prediction interval. [10 marks] (c) The monthl\ sales ি of a product can be modelled as:ি ѿ ু > 2/2ౡ , 1/4িѿ23 ѿ 1/5িѿ35 ѿ 1/9ুѿ2 ѿ 1/6ুѿ35 , 1/5ুѿ36 where ౡ is a constant. State the order of the seasonal ARIMA model for ী > িѿౡ and identif\ the parameters. Furthermore show that |ী~ is a weakl\ stationar\ process. [10 marks] (d) i) Derive the spectral densit\ of |ি~ where)2 ѿ ౦*4ি > ু/ ii) Suppose a linear filter is defined b\ি > ী ѿ ীѿ2 , ীѿ3- where |ী~ is a stationar\ process with mean ౡ and spectral densit\ ৈ)౮* > ౨33)2 ѿ dpt౮*/ Find the form of the transfer function and the squared gain for this filter. For what frequencies ౮ ѵ \ѿ- ^ does the transfer function vanish? [10 marks] MATH4022-E1 END
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