Time Series Analysis Final Examination

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1 Dr. Sevap Kesel Time Series Aalysis Fial Examiaio Quesio ( pois): Assume you have a sample of ime series wih observaios yields followig values for sample auocorrelaio Lag (m) ˆ( ρ m) Par a. Calculae he Q-Saisic of Lag by usig he followig formula: ( pois) * Q m T T ( ) = ( + ) m = ˆ ρ T Par b. Wih a sigificace level of 9%, χ -Saisic wih wo degrees of freedom equals o.99. Based o he saisic calculaed i par a, sae he ull hypohesis claimed i his coex ad wrie he coclusio o be made. (0 pois) Quesio (0 pois) Cosider he model X = ϕ X + Z θ Z where Z WN σ (0, ) Par a. Show which codiios assure ha he process iverible. (0 pois) Par b. Show which codiios assures ha he process is saioary. (0 pois) Par c. Obai he forecass for k=,, recursively ad he forecas error for k=,. (0 pois)

2 Quesio ( pois) You have he ime series idusrial producio (ip) o a mohly basis for he period 9:0 uil 000:09. The figures below illusrae he plos of he logarihmic ad of he logdifferece idusrial producio, respecively. Par a. Explai why i is someimes hard o disiguish bewee a red saioary ime series ad a radom walk wih iercep oly by aalysig he graphs. ( pois) Par b. Explai he reasos o ake he log differeces raher ha he differeced origial series i modellig he sochasic erm i he series. ( pois) Par c. Explai how you ca deec he correc lag-srucure by he help of auocorrelaio ad parial auocorrelaio plos. ( pois) Plos of Idusrial Producio.0 LIP DLIP

3 Quesio ( pois): Par a. Wha is he mai purpose of coiegraio aalysis? Discuss also he differeces bewee coiegraio ad correlaio. ( pois) Par b. Assume you have he ime series idusrial producio (ip) ad orders received (ord) which are boh iegraed a he firs differece. You geeraed he residuals series (zha) ou of hose ime series ad esed he residuals for saioariy. The resuls of he ADF-Tes are give i Table below. Based o he oupus give sae if wo series are coiegraed. Sae he reasos. (0 pois) Null Hypohesis: ZHAT has a ui roo Exogeous: Cosa Lag Legh: (Fixed) -Saisic Prob.* Augmeed Dickey-Fuller es saisic Tes criical values: % level -. % level % level -.00 *MacKio (99) oe-sided p-values. Augmeed Dickey-Fuller Tes Equaio Depede Variable: D(ZHAT) Mehod: Leas Squares Dae: 0/0/0 Time: : Sample (adused): 9M0 000M09 Icluded observaios: afer adusmes Coefficie Sd. Error -Saisic Prob. ZHAT(-) D(ZHAT(-)) D(ZHAT(-)) D(ZHAT(-)) C.E R-squared 0. Mea depede var.9e-0 Adused R-squared 0. S.D. depede var 0.0 S.E. of regressio 0.00 Akaike ifo crierio -.0 Sum squared resid 0. Schwarz crierio -. Log likelihood Haa-Qui crier F-saisic.9 Durbi-Waso sa.0090 Prob(F-saisic) 0000

4 Par c. The Error-Correcio model for he ime series idusrial producio (ip) ad orders received (ord) is give i he able below. (0 pois) Wha ca be said abou he log- or shor- ru relaio of hese wo series? Vecor Auoregressio Esimaes Dae: 0/0/0 Time: : Sample (adused): 9M0 000M09 Icluded observaios: afer adusmes Sadard errors i ( ) & -saisics i [ ] DLIP DLORD DLIP(-) (0.09) (0.099) [-.0] [.] DLIP(-) (0.0) (0.09) [-.0] [.000] DLIP(-) 0.0 (0.09) (0.0) [.] [.9] DLORD(-) (0.0) (0.09) [-.] [-.90] DLORD(-) (0.00) (0.099) [-0.09] [-.0] DLORD(-) (0.09) (0.0) [ 0.90] [-.00] ZHAT(-) (0.09) (0.0) [-.] [.009] R-squared Ad. R-squared Sum sq. resids S.E. equaio F-saisic.00.0 Log likelihood Akaike AIC Schwarz SC Mea depede S.D. depede 0.09 Deermia resid covariace (dof ad.).0e-0 Deermia resid covariace.9e-0 Log likelihood.0 Akaike iformaio crierio Schwarz crierio -9.9

5 Source: Respose Surface esimaios o calculae he criical values of Dickey-Fuller ad Egle-Grager ess wih a cosa. Number of variables M Propabiliy ype of es value β β β wihou cosa wihou red % % % wihou red % % % wih red % % % wihou red % % % wih red % % % wihou red % % % wih red % % % wihou red % % % wih red % % % wihou red % % % wih red % % % wihou red % % % wih red % % % MacKio (99, Table ). Excep for he value i he firs row, all values refer o equaios wih a drif (cosa (iercep)). K = β + β T - + β T -

6 Quesio. Based o he plos of he origial daa ad he ACF, PACF plos of origial daa comme o he model ad he possible compoes i he model (if ay). ( pois) Par a: Mohly Traffic faaliies 0 raffic faaliies 0 0 Idex Auocorrelaio Fucio for raffic faa Auocorrelaio Lag Corr T LBQ Lag Corr T LBQ Lag Corr T LBQ Lag Corr T LBQ Parial Auocorrelaio Fucio for raffic faa Parial Auocorrelaio Lag PAC T Lag PAC T Lag PAC T Lag PAC T Comme o compoes i he model: Type of he model:

7 Par b: Mohly sales 0 sales 00 0 Idex Auocorrelaio Fucio for sales Auocorrelaio Lag Corr T LBQ Lag Corr T LBQ Lag Corr T LBQ Parial Auocorrelaio Fucio for sales Parial Auocorrelaio Lag PAC T Lag PAC T Lag PAC T Comme o compoes i he model: Type of he model: FORMULA SHEET

8 S = wy + ( w) S ; MAD = y ˆ y ; ( ) T SSE = y ˆ y = ; ˆ JB = sˆ + ( k ) = γ ( r, s) = Cov( X, X ) = γ ( r, s) = E[( X µ ( r))( X µ ( s))] ; Cov( X, X ) = γ (0) = Var[ X ] * Q m T T r s x r x s x m ˆ ρ m ( ) = ( + ) ; Q( m) = T ˆ ρ ; T = = r r γ ( h) Cov( X + h, X ) ρh = = ; X = µ + ψ Z γ (0) Cov( X, X ) = X = ϕ X + ϕ X ϕ X + Z + θ Z + θ Z Z X p p q q σ ( + θ ) h = 0 h = 0 θ E Z h h h h θ 0 h + > h > 0 [ ] = 0 γ ( ) = σ θ = m ρx( ) = = m h ( θ ) θ φ = for h > 0 hh ( h+ ) θ θ θ θ θ θ θ φ = φ = φ = θ θ θ ( ) ( ) ( ) h σ ϕ h E[ X ] = 0 γ ( h) = γ ( h) = h > 0 ρ( h) = ϕ ϕ + k l ˆ k AIC = σ + ; k l SIC = l ˆ σ k + ; DW = ; h ρ( h) φ ρ = h, h = φhh = h =,,, L; h φ ρ = = ( ε ε ) ε h, ( ϕ B.. ϕ B )( B) ( λb ) X = ( + θ B θ B )( + ωb ) Z p d s D q s D p q X ( k) = E[ X F ] = Xˆ ( k) = E[ X F ] = ψ Xˆ ( k i) + θ Z ( k i) + k + k i i= = 0 p q e ( l) = X Xˆ ( l) = Z + ψ Z ψ Z ; + l + l + l l + Var[ e ( l)] = σ ( + ψ + ψ ψ l ) Z 0 E[ Z X, X, L + + ] = 0 > 0 ; y c A A y, ε y = c + A A + y, ε ; p y = λ ε + ( a y + a y ) + ε i i i i i= p y = λ ε + ( a y + a y ) + ε i i i i i= K = β + β T + β T

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