Homework 5 (with keys)

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1 Homework 5 (wih keys) 2. (Selecing an employmen forecasing model wih he AIC and SIC) Use he AIC and SIC o assess he necessiy and desirabiliy of including rend and seasonal componens in a forecasing model for Canadian employmen. a. Display he AIC and SIC for a variey of specificaions of rend and seasonaliy. Which would you selec using he AIC? SIC? Do he AIC and SIC selec he same model? If no, which do you prefer? * Remarks, suggesions, hins, soluions: A variey of answers are possible, depending on he specific models fi, bu he upsho is ha rend and seasonaliy are no imporan pars of he dynamics of Canadian employmen. b. Discuss he esimaion resuls and residual plo from your preferred model, and perform a correlogram analysis of he residuals. Discuss, in paricular, he paerns of he sample auocorrelaions and parial auocorrelaions, and heir saisical significance. * Remarks, suggesions, hins, soluions: Because rend and seasonaliy don conribue much o he variaion in Canadian employmen, he residuals from rend+seasonal regressions have properies very similar o he original series. c. How, if a all, are your resuls differen from hose repored in he ex? Are he differences imporan? Why or why no? * Remarks, suggesions, hins, soluions: Any differences are likely unimporan. 1

2 3. Derive auocorrelaion and parial auocorrelaion funcion for a Whie noise process, and graph he correlogram for auocorrelaion and parial auocorrelaion. Soluion: A whie noise process is one wih (virually) no discernible srucure. (i) E(u )=0 (ii) Var(u ) = σ 2 (iii) γ s = 0 if s 0 Tha is, γ 0 = σ 2, γ 1 = γ 2 =. = 0 ( τ> 0) And is ACF and PACF are as follows: 1 j = 0 1 j = 0 ρj = φj j = 0 j 1 0 j 1 Thus he auocorrelaion funcion (ρ j ) will be zero apar from a single peak of 1 a j = 0. 2

3 4. (Simulaing ime series processes) Many cuing-edge esimaion and forecasing echniques involve simulaion. Moreover, simulaion is ofen a good way o ge a feel for a model and is behavior. Whie noise can be simulaed on a compuer using random number generaors, which are available in mos saisics, economerics and forecasing packages. a. Simulae a Gaussian whie noise realizaion of lengh 200. Call he whie noise ε. Compue he correlogram. Discuss. * Remarks, suggesions, hins, soluions: The correlogram should be fla, wih mos sample auocorrelaions inside he Barle bands. b. Form he disribued lag y = ε +. 9ε 1, = 2, 3,..., 200. Compue he sample auocorrelaions and parial auocorrelaions. Discuss. * Remarks, suggesions, hins, soluions: Neiher he sample auocorrelaion nor he sample parial auocorrelaion funcion is fla. For now, he precise paerns are unimporan; wha s imporan is ha he disribued lag series is clearly no whie noise. The illusraes he Slusky-Yule effec: disribued lags of whie noise are serially correlaed. c. Le y 1 =1 and y =.9y +ε, = 2, 3,..., 200. Compue he sample 1 auocorrelaions and parial auocorrelaions. Discuss. * Remarks, suggesions, hins, soluions: Dio. 3

4 5. Derive auocorrelaion and parial auocorrelaion funcion for he firs-order movingaverage process and graph he correlogram for is auocorrelaion and parial auocorrelaion. Soluion: MA(1) model, y = µ + u + θ 1 u -1 (i) Mean: E(y )=µ = µ. (ii) Variance: γ 0 = Var(y ) = Var(µ + u + θ 1 u -1 ) = 0 + σ 2 + θ 2 1 σ 2 = (1 + θ 2 1 )σ 2 (iii) Covariances: γ s = Cov(y, y -s ) γ 1 = Cov(y, y -1 ) = Cov(µ + u + θ 1 u -1, µ + u -1 + θ 1 u -2 ) = E[y - E(y )][y -1 - E(y -1 )] = E[(u + θ 1 u -1 )( u -1 + θ 1 u -2 )] since y - E(y ) = (µ + u + θ 1 u -1 ) - µ = u + θ 1 u -1 = θ 1 σ 2 Thus, τ 1 = γ 1 / γ 0 = θ 1 σ 2 / [(1 + θ 2 1 )σ 2 ] = θ 1 /(1 + θ 2 1 ) γ 2 = Cov(y, y -2 ) = Cov(µ + u + θ 1 u -1, µ + u -2 + θ 1 u -3 ) = E[(u + θ 1 u -1 )( u -2 + θ 1 u -3 )] = 0 γ 3 = E[(u + θ 1 u -1 )( u -3 + θ 1 u -4 )] = 0 γ 4 = 0 Thus, τ s = 0, s = 2, 3, 4.. And is ACF and PACF are as follows: θ j = 0 2 ρ j = 1+ θ 0 j 1 The PACF is dominaed by an exponenial funcion which decreases. 4

5 ACF PACF MA(1): θ>0 Posiive spike a lag 1, ρ s = 0 Oscillaing decay: φ 11 <0 for s 0 MA(1): θ<0 Negaive spike a lag 1, ρ s = 0 Decay: φ 11 < 0 Figure 5: Populaion ACF and PACF for MA(1) 5