G14TFG-E1 The Universit\ of Nottingham SCHOOL OF MATHEMATICAL SCIENCES A LEVEL 4 MODULE, SPRING SEMESTER 2017-2018 TIME SERIES AND FORECASTING Time allowed TWO Hours THIRTY Minutes Candidates ma\ complete the front coYer of their ansZer book and sign their desk card bXt mXst NOT Zrite an\thing else Xntil the start of the e[amination period is annoXnced. CUedLW ZLOO be gLYeQ fRU Whe beVW THREE aQVZeUV Onl\ silent, self-contained calcXlators Zith a Single-Line Displa\ or DXal-Line Displa\ are permitted in this e[amination. Dictionaries are not alloZed Zith one e[ception. Those Zhose first langXage is not English ma\ Xse a standard translation dictionar\ to translate betZeen that langXage and English proYided that neither langXage is the sXbject of this e[amination. SXbject specific translation dictionaries are not permitted. No electronic deYices capable of storing and retrieYing te[t, inclXding electronic dictionaries, ma\ be Xsed. DO NOT WXUQ e[aPLQaWLRQ SaSeU RYeU XQWLO LQVWUXcWed WR dR VR ADDITIONAL MATERIAL: NHaYH¶V SWaWLVWLFV TaEOHV G14TFG-E1 TXrn oYer 1 G14TFG-E1 1. Throughout this question |ু~ is an IID white noise process with mean 0 and variance ౨3 . (a) Consider the following processes:ি ѿ 1/8িѿ3 > ু ѿ 4ুѿ2 , 4ুѿ3 ѿ ুѿ4- (1)ি ѿ 45িѿ2 , 56িѿ3 > ু ѿ 1/:ুѿ2- (2)ি ѿ 1/:িѿ2 , 1/3িѿ3 > ু ѿ 1/8ুѿ2 , 1/5ুѿ3/ (3) i) In each case, state the t\pe of ARMA process. ii) Which of (1), (2) and (3) are weakl\ stationar\? Justif\ \our answer. iii) Which of (1), (2) and (3) are invertible? Justif\ \our answer. [15 marks] (b) i) Suppose that |ি~ is the MA(2) processি > ౡ , ু ѿ ౝ2ুѿ2 ѿ ౝ3ুѿ3/ (4) Derive from first principles the autocovariance function and autocorrelation function of the process. ii) Suppose now that ౝ2 > 25 ѿ ౠ and ౝ3 > 3ౠ in (4). Determine for what values of ౠ the process is invertible. iii) Now consider general ౝ2 and ౝ3 in (4) and suppose that the process is invertible. Derive the AR(ҋ) representation of ু. You ma\ assume that2 ѿ ౝ2 ѿ ౝ33 > )2 ѿ ౙ2*)2 ѿ ౙ3*/ iv) Discuss wh\ invertibilit\ of a fitted model is important when assessing goodness-of-fit. [19 marks] (c) Suppose |ি~ is a weakl\ stationar\ time series with autocovariance function ౘ)υ*, and suppose that we observe the sequence ি2-Ϳ -িৎ. Calculate an e[pression for the variance of Ȣি > ৎѿ2Ѿৎ>2ি in terms of the ౘ)υ*, and e[press \our answer in a form which is as simple as possible. [6 marks] G14TFG-E1 2 G14TFG-E1 2. Throughout this question |ু~ is an IID white noise process with mean 0 and variance ౨3 . (a) Define the Akaike Information Criterion (AIC), e[plain what it is used for, and discuss wh\ it is preferable to use AIC rather than the ma[imised log likelihood. [6 marks] (b) After observing a sequence of observations from a weakl\ stationar\ time series |ি~, it was decided to fit an ARMA)3- 2* model. The fitted model is given b\)*ি > ౝ)*ু༚ - (5) where )* > 2 ѿ ౖ2 ѿ ౖ33 and ౝ)* > 2 ѿ 2. i) Write down an e[pression for the residuals of the fitted model (5). What is the ke\ assumption that is required for calculating the residuals? ii) Suppose that the correct model is the MA)3* modelি > ু ѿ ౝ2ুѿ2 ѿ ౝ3ুѿ3/ What model do the residuals |ু༚ ~ in the fitted model (5) follow? [8 marks] (c) The general seasonal ARIMA model is sometimes written in abbreviated form as ARIMA)- - *)ষ -ফ-স*υ. i) Write down the model e[plicitl\ in terms of backshift and difference operators, and e[plain what , , , ষ, ফ, স and υ represent. ii) Consider the modelি > িѿ2 , 23িѿ5 ѿ 23িѿ6 , ু ѿ 34ুѿ2 ѿ 34ুѿ5 , 5:ুѿ6/ Determine , , , ষ, ফ, স and υ e[plicitl\. [10 marks] (d) Suppose that the time series |ি~ is seasonal with period υ > 23. It is assumed that the seasonal effects satisf\ > ѿ23 for all , and there is also a quadratic trend > ূ , ৃ , ৄ3, where ূ, ৃ and ৄ are constants. Consider the additive modelি > , , ু- (6) where |ু~ is white noise. i) E[plain wh\ the model (6) is non-stationar\ in general. ii) Show that the difference operator Ѵ323 acting on ি in (6) reduces the series to a stationar\ process. iii) Consider now the multiplicative modelি > , ু/ (7) Show b\ direct calculation that the operator Ѵ323 does not reduce (7) to a stationar\ process. iv) Find a difference operator that does reduce ি in (7) to a stationar\ process. [16 marks] G14TFG-E1 TXrn OYer 3 G14TFG-E1 3. Throughout this question |ু~ is an IID white noise process with mean 0 and variance ౨3 . (a) Suppose that |ি~ is a weakl\ stationar\ time series with MA(ҋ) representation ি >Ѿҋ>1 ৃুѿ where Ѿҋ>1 ৃ3 = ҋ. Let Ƞিৎ,ো)ৎ* denote the optimal (in the mean square sense) ো-step-ahead predictor of িৎ,ো given knowledge of the time series up to and including time > ৎ. i) Prove that Ƞিৎ,ো)ৎ* > ҋา>1 ৃ,োুৎѿ/ ii) As a b\-product of \our calculations on part (a)(i), write down an e[pression for the corresponding ো-step-ahead mean square prediction error, denoted ౨3ো. [10 marks] (b) Consider the modelি ѿ িѿ2 > ু ѿ ౝুѿ2 (8) where }} = 2 and }ౝ} = 2. i) Derive the MA(ҋ) representation of ি in (8). ii) Deduce the AR(ҋ) representation of ু in (8). [8 marks] (c) If we wish to calculate Ƞিৎ,ো)ৎ* for a weakl\ stationar\ and invertible ARMA)- *model, we ma\ make use of the formula)* Ƞিৎ,ো)ৎ* > ౝ)*\ুৎ,ো^/ (9) i) E[plain how \ু^ is defined. ii) Illustrate the use of (9) b\ appl\ing it to model (8). Specificall\, use (9) to derive a formula for Ƞিৎ,ো)ৎ* in terms of িৎ, ুৎ, and ౝ. iii) Using \our e[pression in (c)(ii) combined with the AR(ҋ) representation for ু in (b)(ii), e[press Ƞিৎ,ো)ৎ* in terms of িৎ- িৎѿ2-Ϳ . Simplif\ \our answer as much as possible. [10 marks] continXed on ne[t page G14TFG-E1 4 G14TFG-E1 (d) Suppose the annual profits (in thousands of pounds) ma\ be modelled as a weakl\ stationar\ ARMA)2- 2* modelি ѿ 911 > 1/7)িѿ2 ѿ 911* , ু ѿ 1/5ুѿ2 where the white noise variance ౨3 > 2. Profits for the previous 5 \ears are as follows: Year Profits 2013 810 2014 800 2015 820 2016 800 2017 790 i) Provide forecasts for the profits in 2018 and 2019. ii) Provide appro[imate 95% prediction intervals for 2018 and 2019. [12 marks] G14TFG-E1 TXrn OYer 5 G14TFG-E1 4. (a) The partial autocorrelation function (PACF) is defined via the s\stem of equations৪্্ > ಠ্- (10) where ্ Ӓ 2 and ৪্ > ֢֠֡֡֡ 2 ౦2 ౦3 ՜ ౦্ѿ2౦2 2 ՞౦3 ՞ ՞ ՞՛ ՞ ՞ ౦2౦্ѿ2 ౦2 2 ֣֤֤ ֤֥ - ্ > ্2՛্্ - ಠ্ > ౦2՛౦্ - and )౦*Ӓ2 is the autocorrelation sequence for a weakl\ stationar\ process |ি~. i) How is the PACF at lag ্ defined? ii) Derive the s\stem of equations (10) b\ considering the minimisation ofফ্ > ব\)ি ѿ ԑ্৲্*3^, where ৲্ > )িѿ2-Ϳ -িѿ্*ԑ. iii) E[plain how the PACF behaves in the case of a weakl\ stationar\ AR() process. [Note: proofs of the properties \oX state are not reqXired.] [12 marks] (b) i) Given a weakl\ stationar\ time series |ী~ with autocovariance function ౘ)υ*, define the autocovariance generating function ɵ)*. ii) Define the spectral densit\ function ৈ)౮* of the same process |ী~ in terms of the autocovariance function ౘ)υ*. iii) Provide justification for the formula ৈ)౮* > 23ɵ)fѿj౮*/ iv) Supppose now that |ী~ is a weakl\ stationar\ AR(1) process. Find the spectral densit\ ৈ)౮* in this case. v) Provide a rough sketch of the spectral densit\ ৈ)౮* in the AR(1) case, treating fundamentall\ different cases separatel\. Comment on where the spectral densit\ attains its ma[imum in each different case \ou identif\. [14 marks] continXed on ne[t page G14TFG-E1 6 G14TFG-E1 (c) Let |ী~ denote a weakl\ stationar\ time series with mean 0, and consider the filtered time series ি > ҋา>ѿҋ ৃীѿ- (11) where it is assumed that Ѿҋ>ѿҋ ৃ3 = ҋ. i) Define the transfer function, υ)౮*, of the linear filter in (11). Also, write down the squared gain associated with υ)౮*. ii) Suppose that ৃ > ৄ when Ӓ 1 and ৃ > 1 when = 1, where }ৄ} = 2. Calculate the squared gain of this linear filter. iii) Write down an e[pression for the spectral densit\ of |ি~ in terms of the spectral densit\ of |ী~ and the transfer function υ)౮*. [Note: an\ resXlts that \oX Xse shoXld be clearl\ stated bXt not proYed.] [14 marks] G14TFG-E1 EQG
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