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Econometrie Avansata
Dr. Adrian Codirlasu, CFA
Dr. Bogdan Moinescu
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Definitie
O secventa de valori inregistrate de o
variabila aleatoare specifica intr-o anumita
perioada de timp
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Caracteristici
Frecventa
Populatie vs. esantion
Momente Stationaritate
Sezonalitate
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Frecventa
Reprezinta periodicitatea cu care este observat
variabila.
Functie de specificul seriei de timp, frecventapoate fi zilnica (cum este cazul preturilor
activelor financiare cursurile actiunilor, ratele
de dobanda, cursul de schimb), lunara (de
exemplu, rata inflatiei, salariul mediu peeconomie, rata somajului), trimestriala (cum este
produsul intern brut) sau anuala
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Momentele seriei de timp
Media
Varianta
Coeficientul de asimetrie (skewness)
Kurtosis
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Stationaritate
Conditiile ce trebuie indeplinite pentru ca o
serie de timp s fie stationara sunt:
media seriei de timp sa fie constanta sau cu
alte cuvinte, observatiile trebuie sa fluctuezein jurul mediei.
varianta seriei s fie constanta.
Din punct de vedere economic, o serieeste stationara daca un soc asupra seriei
este temporar (se absoarbe in timp) si nu
permanent.
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Stationaritate
In cazul in care seria nu este stationara,
prin diferentiere, se obtine o serie
stationara.
Ordinul de integrare al seriei reprezinta
numarul de diferentieri succesive
necesare pentru obtinerea unei serii
stationare (sau numarul de radaciniunitare al seriei).
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Sezonalitate
Seriile de timp cu frecventa lunara sau
trimestriala prezinta adesea evoluaii care
au o anumita ciclicitate. De exemplu
activitatea economica se incetineste inlunile de iarna, preturile cresc mai mult n
lunile reci decat in perioada de vara etc.
In analiza econometrica, pentru a eliminaaceste evolutii sezoniere seriile de timp
sunt ajustate sezonier.
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II. Teste statistice
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II.1. Distributii
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Distributia de probabilitate
Este reprezentarea tuturor valorilor pe
care le poate lua o variabil aleatore si a
probabilitii de apariie a acestor valori
Variabile aleatoare
Discrete
Continue
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Distributii
Normala
Lognormala
t
Chi patrat
F
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Distributia normala
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Distributia log-normala
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Distributia t
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Distributia Chi-patrat
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Distributia F
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II.2. Testarea ipotezelor
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Testarea ipotezelor
Definirea ipotezei;
Identificarea testului statistic ce va fi utilizat i a
distribuiei de probabilitate a acestuia;
Specificarea nivelului de relevan al testului;
Specificarea regulii de decizie;
Colectarea datelor i estimarea parametrului;
Luarea deciziei statistice;
Luarea deciziei economice.
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Definirea ipotezei Specificarea ipotezei nule i a ipotezei
alternative
Ipoteza nul, notat cu , reprezint ipoteza ce
este testat, iar ipoteza alternativ, notat cu ,este ipoteza acceptat n cazul n care ipoteza
nul este respins
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Testarea mediei
Esantion mare(n > 30)
Esantion mic(n < 30)
Populatia are o distributie normala Testul tsau testul z Testul t
Populatia nu are o distr ibutie normala Testul t sau testul z Nu se poate testa
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Testarea variantei
cu n 1grade de libertate
varianta esantionului de date utilizat
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III. Analiza seriilor de
timp in EViews
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Crearea unui fisier de lucru
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Definirea seriilor
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Introducerea datelor
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Prelucrea seriilor
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Stationaritatea seriilor de timp
Teste statisticeAugmented Dickey-Fuller (ADF)
Phillips-Perron
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Testarea stationaritatii seriei
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Interpretarea rezultatului
statisticNull Hypothesis: L_EUR has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 3 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -0.981155 0.9448
Test critical values: 1% level -3.962327
5% level -3.411905
10% level -3.127850
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Analiza distributiei seriei
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Serie normal distribuita
Coeficientul de asimetrie (skewness) este zero
distributia normala este simetrica.
Kurtotica (kurtosis) este 3. Dac acest indicator
are o valoare mai mare dect 3, atunci distributiase numete leptokurtotica, iar daca acesta este
mai mic dect 3 atunci distributia se numeste
platikurtotica.
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Distributie EURRON
-12
-8
-4
0
4
8
12
-.08 -.04 .00 .04 .08
DL_EUR
NormalQuantile
Theoretical Quantile-Quantile
0
20
40
60
80
100
-.04 -.02 .00 .02 .04 .06
DL_EUR
Kernel Density (Epanechnikov, h = 0.0018)
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Functia de autocorelatie
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Functia de autocorelatie
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Trendul seriilor de timp
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Filtrul Hodrick-Prescott
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Ajustarea sezoniera
a seriilor de timp
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Investigarea sezonalitatii
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Proceduri de desezonalizare
Census X12
Census X11
Tramo/Seats
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Proceduri de desezonalizare
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Serie desezonalizata
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IV. Regresia liniara
multipla
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Utilizare Cu ajutorul regresiei liniare multiple, se poate
determina impactul pe care il au mai multe
variabile independente asupra unei anumite
variabile (numita variabila dependenta)
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Ecuatia de regresie
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Determinarea elasticitatilor
Daca variabila dependenta si variabilele
independente sunt specificate in logaritmi
naturali, atunci coeficientii variabilelor
independente pot fi interpretati caelasticitati
Astfel, acesti coeficienti vor arata cu cat la
suta se modifica variabila dependentadaca variabila independenta se modifica
cu 1 la suta
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Ipotezele regresiei liniare
Legtura dintre variabila dependent ivariabilele independente este liniar
Variabilele independente sunt aleatoare.
Intre variabilele independente incluse intr-o regresie nu exista nici o relatie liniara.
Valoarea ateptat a termenului de eroare
este 0
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Ipotezele regresiei liniare
Varianta termenului de eroare esteaceeasi pentru toate observaiile(erori
homoskedastice).
Termenul de eroare este necorelat intreobservatii.
Termenul de eroare este normal distribuit.
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Impactul incalcarii ipotezelor
Heteroskedasticitate Erorile standard ale regresiei sunt incorecte
Corelaieseriala erorilor Erorile standard ale regresiei sunt incorecte
Multicoliniaritate Valori mari ale lui R-patrat si valori mici ale valorilort-statisticale coeficientilor variabilelor independente
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Teste statitistice
pentru regresia liniara
R-patrat
R-patrat ajustat (cu numarul de variabileindependente incluse in regresie)
Criterii informationale
Durbin-Watson
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Teste statitistice
pentru regresia liniara Teste pentru coeficientii obtinuti din
ecuatia de regresie
Testul tpentru testarea individuala a
coeficietilor
Testul Fpentru testarea tuturor coeficientilor
Testul Wald
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Teste statitistice
pentru regresia liniara Teste pentru erorile ecuatiei de regresie
Corlograma erorilorCorelograma erorilor patratice
Testarea distributiei erorilor (testul Jarque-
Berra)
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Regresii cu variabile calitative
Variabile dummyAcestea iau valoarea 1 dac o anumita conditie este
adevarata si valoarea 0 in caz contrar
Numarul de variabile dummyeste cu 1 mai mic decat
numarul de conditii, in caz contrat existandmulticoliniaritate
Variabilele dummypot fi utilizate si pentru captareaimpactului sezonier asupra variabilei independente,
introducand cel mult 11 variabile dummypentrudatele cu frecven lunara sau cel mult 3 variabiledummypentru datele cu frecventa trimestriala, incazul in care datele nu au fost ajustate sezonier inprealabil
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IV.4. Regresii cu serii de
timp in Eviews
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Estimarea functiei de reactie Perioada analizata trim. I 1999 trim. I 2011
Serii de date utilizate: r_eu rata de politic monetar a BCE;
infl_eu inflaiei, msurat prin indicelearmonizat al preurilor, in Uniunea Monetar;
gap_eu output-gap-ul, calculat pe baza unuifiltru Hodrick-Prescott pentru zona euro;
dummy variabil dummy pentru perioada decriza financiara (ia valoarea 1 incepand cu trim. I2009)
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Parametrii regresiei
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Ecuatia estimata
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Indicatori ai regresiei
t-Statistic si probabilitatea asociata,calculat pentru constanta si coeficientul
fiecarei variabile independente
R-Squared, Adjusted R-Squared F-Statistic si probabilitatea asociata
Criteriile informationale (Akaike info
criterion, Schwarz criterion, Hannan-Quinncriter.)
Durbin-Watson stat
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Variabila dependenta evectiva
vs estimata
-.015
-.010
-.005
.000
.005
.010
.015
.00
.01
.02
.03
.04
.05
99 00 01 02 03 04 05 06 07 08 09 10 11
Residual Actual Fitted
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Teste asupra termenilor de eroare
Corelograma erorilor
Corelograma erorilor patratice
Testarea tipului de distributie a erorilor
Serial Correlation LM Test
ARCH LM Test
White Heteroskedasticity Test
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Selectare teste termeni de eroare
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Corelograma erorilor
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Corelograma erorilor patratice
T t di t ib ti i l
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Testarea distributiei normale a
erorilor regresiei
0
2
4
6
8
10
12
-0.010 -0.005 0.000 0.005 0.010 0.015
Series: ResidualsSample 1999Q1 2011Q1Observations 49
Mean -6.09e-18Median -0.000688Maximum 0.013773Minimum -0.010947Std. Dev. 0.005747Skewness 0.694016Kurtosis 2.892419
Jarque-Bera 3.957170Probability 0.138265
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Testarea corelatiei seriale
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Testarea termenilor ARCH
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Teste de stabilitate
CUSUM Test
CUSUM of Squares Test
Recursive Coeficients
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Teste de stabilitate
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CUSUM Test
-25
-20
-15
-10
-5
0
5
10
II III IV I II III IV I
2009 2010 2011
CUSUM 5% Significance
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CUSUM of Squares Test
-0.4
0.0
0.4
0.8
1.2
1.6
II III IV I II III IV I
2009 2010 2011
CUSUM of Squares 5% Significance
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Recursive coeficients
.027
.028
.029
.030
.031
.032
.033
.034
II III IV I II III IV I
2009 2010 2011
Recursive C(1) Estimates
2 S.E.
-.8
-.6
-.4
-.2
.0
.2
.4
II III IV I II III IV I
2009 2010 2011
Recursive C(2) Estimates
2 S.E.
.3
.4
.5
.6
.7
.8
II III IV I II III IV I
2009 2010 2011
Recursive C(3) Estimates
2 S.E.
-.020
-.015
-.010
-.005
.000
.005
.010
II III IV I II III IV I
2009 2010 2011
Recursive C(4) Estimates
2 S.E.
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V. Modele ARMA
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Modele ARMA
Modele autoregresive (AR);
Modele cu medii mobile (MA);
ModeleARMA care combina cele doutipuri de procese.
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Estimare modele ARMA
1. Testarea stationaritatii seriei2. Stationarizarea seriei
3. Pe baza coeficienilor de autocorelaie
(funciei de autocorelaie) i a coeficienilorde corelaie parial (funciei de autocorelaie
parial) se determin modelele
autoregresive de start pentru analiza serieide date.
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Estimare modele ARMA
4. Se estimeaza parametri modelelorARMA.
5. Se testeaza caracteristicile modelelor
autoregresive ce au fost estimate n etapaanterioara.
6. Se alege cel mai potrivit model folosind
diverse criterii de analiza.7. Pe baza modelului selectat se fac
diverse analize si prognoze
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V.4. Estimarea modelelor
ARMA in Eviews
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Seria de date
0
50
100
150
200
250
97 98 99 00 01 02 03 04 05 06 07
BUBOR
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Testul de stationaritate ADF
Null Hypothesis: BUBOR has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -5.024900 0.0003
Test critical values: 1% level -4.031899
5% level -3.445590
10% level -3.147710
*MacKinnon (1996) one-sided p-values.
Testul de stationaritate Philips
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Testul de stationaritate Philips-
PerronNull Hypothesis: BUBOR has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 3 (Newey-West using Bartlett kernel)
Adj. t-Stat Prob.*
Phillips-Perron test statistic -5.437216 0.0001
Test critical values: 1% level -4.031899
5% level -3.445590
10% level -3.147710
*MacKinnon (1996) one-sided p-values.
Residual variance (no correction) 364.8682
HAC corrected variance (Bartlett kernel) 462.5140
F ti d t l ti
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Functia de autocorelatie
S ifi ti
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Specificare ecuatie
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Estimare model MA(4)Dependent Variable: BUBOR
Method: Least Squares
Sample (adjusted): 1997M01 2007M08
Included observations: 128 after adjustments
Convergence achieved after 15 iterations
Backcast: 1996M09 1996M12
Variable Coefficient Std. Error t-Statistic Prob.
C 41.54023 5.901126 7.039374 0.0000
MA(1) 0.734234 0.051566 14.23862 0.0000
MA(2) 0.495279 0.026993 18.34853 0.0000
MA(3) 0.863535 0.025672 33.63763 0.0000
MA(4) 0.804961 0.050194 16.03709 0.0000
R-squared 0.838087 Mean dependent var 43.94414
Adjusted R-squared 0.832822 S.D. dependent var 42.18206S.E. of regression 17.24715 Akaike info criterion 8.571450
Sum squared resid 36588.11 Schwarz criterion 8.682858
Log likelihood -543.5728 F-statistic 159.1672
Durbin-Watson stat 1.582016 Prob(F-statistic) 0.000000
Inverted MA Roots .42-.90i .42+.90i -.78+.45i -.78-.45i
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Analiza radacini ecuatie
R d i il li l i t i ti
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Radacinile polinomului caracteristic
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
MA roots
Inverse Roots of AR/MA Polynomial(s)
Inverse Roots of AR/MA Polynomial(s)
Specification: BUBOR C MA(1) MA(2) MA(3) MA(4)
Sample: 1997M01 2007M12
Included observations: 128
MA Root(s) Modulus Cycle
0.416806 0.900818i 0.992572 5.524002
-0.783923 0.450021i 0.903910 2.397739
No root lies outside the unit circle.
ARMA model is invertible.
C
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Corelograma erorilor
V l f i i
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Valoarea efectiva vs estimata
-80
-40
0
40
80
0
50
100
150
200
250
97 98 99 00 01 02 03 04 05 06 07
Residual Actual Fitted
E ti d l AR(1)
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Estimare model AR(1)Dependent Variable: BUBOR
Method: Least Squares
Sample (adjusted): 1997M02 2007M08
Included observations: 127 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 4.985180 2.639138 1.888943 0.0612
BUBOR(-1) 0.878633 0.043240 20.32007 0.0000
R-squared 0.767617 Mean dependent var 43.85480
Adjusted R-squared 0.765758 S.D. dependent var 42.33696S.E. of regression 20.49047 Akaike info criterion 8.893420
Sum squared resid 52482.45 Schwarz criterion 8.938210
Log likelihood -562.7322 F-statistic 412.9052
Durbin-Watson stat 1.709848 Prob(F-statistic) 0.000000
C l il
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Corelograma erorilor
V l f ti ti t
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Valoarea efectiva vs estimata
-100
-50
0
50
100
150
0
50
100
150
200
250
97 98 99 00 01 02 03 04 05 06 07
Residual Actual Fitted
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Estimare model ARMA(1,10)Dependent Variable: BUBOR
Method: Least SquaresSample (adjusted): 1997M02 2007M08
Included observations: 127 after adjustments
Convergence achieved after 21 iterations
Backcast: 1996M04 1997M01
Variable Coefficient Std. Error t-Statistic Prob.
BUBOR(-1) 0.974210 0.017718 54.98466 0.0000
MA(5) -0.243541 0.056540 -4.307386 0.0000
MA(6) -0.226437 0.055472 -4.081987 0.0001
MA(7) -0.332302 0.055472 -5.990446 0.0000
MA(10) 0.476373 0.060775 7.838351 0.0000
R-squared 0.867500 Mean dependent var 43.85480
Adjusted R-squared 0.863156 S.D. dependent var 42.33696S.E. of regression 15.66150 Akaike info criterion 8.378862
Sum squared resid 29924.46 Schwarz criterion 8.490837
Log likelihood -527.0577 Durbin-Watson stat 2.297258
Inverted MA Roots .88+.16i .88-.16i .56+.83i .56-.83i
-.02-.90i -.02+.90i -.53+.73i -.53-.73i
-.89+.33i -.89-.33i
C diti d t bilit t
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Conditia de stabilitate
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
MA roots
Inverse Roots of AR/MA Polynomial(s)Inverse Roots of AR/MA Polynomial(s)
Specification: BUBOR BUBOR(-1) MA(5) MA(6)
MA(7) MA(10)
Sample: 1997M01 2007M12
Included observations: 127
MA Root(s) Modulus Cycle
0.558806 0.826275i 0.997494 6.436651
-0.890402 0.332511i 0.950463 2.256737
-0.528267 0.734115i 0.904428 2.863081
-0.022438 0.897521i 0.897802 3.937348
0.882301 0.159193i 0.896548 35.19821
No root lies outside the unit circle.
ARMA model is invertible.
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Corelograma erorilor
Valori efective vs estimate
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Valori efective vs estimate
-80
-40
0
40
80
120
0
50
100
150
200
250
97 98 99 00 01 02 03 04 05 06 07
Residual Actual Fitted
Selectarea specificatiei
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Selectarea specificatiei
MA(4) AR(1) ARMA(1,10)
Adjusted R-squared 0.832822 0.765758 0.863156
Akaike info criterion8.571450 8.893420 8.378862
Schwarz criterion8.682858 8.938210 8.490837
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Prognoze
Dynamic forecast prognozeaza valoarea
in perioada t + 1pe baza datelor efective
pana an momentul t, apoi pentru toateperioadele urmatoare foloseste datele deja
prognozate incepand din momentul t + 1.
Static forecast prognozeaza o observatieinainte numai pe baza datelor efective.
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Realizarea de prognoze
Prognoza dinamica a seriei
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Prognoza dinamica a seriei
-60
-40
-20
0
20
40
60
80
2007M09 2007M10 2007M11 2007M12
BUBORF
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VI. Modele cu date panel
Modele cu date panel
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Modele cu date panel
Constau in estimarea de ecuatii de
regresie in care sunt folosite date care sunt
in acelasi timp atat serii de timp ct si datecrosssectionale
Utilizari
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Utilizari
Rezumarea printr-un singur coeficient al impactului unei
variabile asupra unui grup de serii de timp variabile
dependente (grup de companii, de tari, etc.).
Estimarea de coeficienti specifici (constanta sau
coeficienti ai variabilelor independente) pentru fiecareserie de timp considerata ca variabila dependenta
efecte fixe.
Gruparea variabilelor dependente in categorii si estimarea
impactului categoriei din care face parte variabiladependenta asupra evolutiei acesteia
D fi i d l l i
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Definirea modelului
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Definirea indentificatorilor
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Definirea seriilor
E ti d i
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Ecuatia de regresie
Rezultate regresie
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Rezultate regresieDependent Variable: DLOG(HICP?)
Method: Pooled EGLS (Cross-section SUR)
Included observations: 188 after adjustments
Cross-sections included: 4
Total pool (balanced) observations: 752
Linear estimation after one-step weighting matrix
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000698 0.000445 1.566998 0.1175
DLOG(HICP_EU(-1)) 0.612692 0.104768 5.848058 0.0000DLOG(HICP?(-1)) 0.553629 0.030088 18.40025 0.0000
DLOG(ER?(-1)) 0.083851 0.012757 6.572889 0.0000
Weighted Statistics
R-squared 0.447658 Mean dependent var 0.640863
Adjusted R-squared 0.445443 S.D. dependent var 1.207505
S.E. of regression 0.928331 Sum squared resid 644.6258
F-statistic 202.0780 Durbin-Watson stat 2.104933Prob(F-statistic) 0.000000
Unweighted Statistics
R-squared 0.602410 Mean dependent var 0.007916
Sum squared resid 0.080251 Durbin-Watson stat 1.864487
Coeficienti indi id ali
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Coeficienti individuali
Rezultate regresie
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Rezultate regresieDependent Variable: DLOG(HICP?)
Method: Pooled EGLS (Cross-section SUR)
Sample (adjusted): 1996M03 2011M10
Included observations: 188 after adjustmentsCross-sections included: 4
Total pool (balanced) observations: 752
Linear estimation after one-step weighting matrix
Variable Coefficient Std. Error t-Statistic Prob.
C 0.002113 0.000471 4.489158 0.0000
DLOG(HICP_EU(-1)) 0.631560 0.102724 6.148098 0.0000
DLOG(HICP?(-1)) 0.492847 0.031776 15.50997 0.0000
DLOG(ER?(-1)) 0.077977 0.012530 6.223189 0.0000
Fixed Effects (Cross)
_CZ--C -0.002185
_HU--C -0.000839
_PO--C -0.001681
_RO--C 0.004706
Effects Specification
Cross-section fixed (dummy variables)
Weighted Statistics
R-squared 0.466068 Mean dependent var 0.648656
Adjusted R-squared 0.461768 S.D. dependent var 1.221242
S.E. of regression 0.928200 Sum squared resid 641.8587
F-statistic 108.3850 Durbin-Watson stat 2.054515
Prob(F-statistic) 0.000000
Unweighted Statistics
R-squared 0.614957 Mean dependent var 0.007916
Sum squared resid 0.077719 Durbin-Watson stat 1.802623
Impactul anticipatiilor inflationiste
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Impactul anticipatiilor inflationiste
Rezultate regresie
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Rezultate regresieDependent Variable: DLOG(HICP?)
Method: Pooled EGLS (Cross-section SUR)
Sample (adjusted): 1996M03 2011M10
Included observations: 188 af ter adjustmentsCross-sections included: 4
Total pool (balanced) observations: 752
Linear estimation after one-step weighting matrix
Variable Coefficient Std. Error t-Statistic Prob.
C 0.001447 0.000443 3.264956 0.0011
DLOG(HICP_EU(-1)) 0.572149 0.102100 5.603786 0.0000
DLOG(ER?(-1)) 0.065592 0.012463 5.262752 0.0000
_CZ--DLOG(HICP_CZ(-1)) 0.193372 0.066868 2.891863 0.0039
_HU--DLOG(HICP_HU(-1)) 0.406401 0.052893 7.683466 0.0000
_PO--DLOG(HICP_PO(-1)) 0.326507 0.063476 5.143756 0.0000
_RO--DLOG(HICP_RO(-1)) 0.686025 0.035300 19.43402 0.0000
Weighted Statistics
R-squared 0.472755 Mean dependent var 0.644691
Adjusted R-squared
0.468508
S.D. dependent var
1.220327
S.E. of regression 0.924597 Sum squared resid 636.8850
F-statistic 111.3341 Durbin-Watson stat 2.058521
Prob(F-statistic) 0.000000
Unweighted Statistics
R-squared 0.623347 Mean dependent var 0.007916
Sum squared resid 0.076025 Durbin-Watson stat 2.116026
Impactul cursului de schimb
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Impactul cursului de schimb
Rezultate regresie
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Rezultate regresieDependent Variable: DLOG(HICP?)
Method: Pooled EGLS (Cross-section SUR)
Sample (adjusted): 1996M03 2011M10
Included observations: 188 af ter adjustmentsCross-sections included: 4
Total pool (balanced) observations: 752
Linear estimation after one-step weighting matrix
Variable Coefficient Std. Error t-Statistic Prob.
C 0.001175 0.000383 3.064805 0.0023
DLOG(HICP_EU(-1)) 0.513687 0.090404 5.682137 0.0000
DLOG(HICP?(-1)) 0.405206 0.028875 14.03333 0.0000
_CZ--DLOG(ER_CZ(-1)) 0.053506 0.026337 2.031585 0.0426
_HU--DLOG(ER_HU(-1)) 0.006168 0.017179 0.359047 0.7197
_PO--DLOG(ER_PO(-1)) 0.020524 0.011008 1.864565 0.0626
_RO--DLOG(ER_RO(-1)) 0.417841 0.030630 13.64161 0.0000
Weighted Statistics
R-squared 0.506046 Mean dependent var 0.689472
Adjusted R-squared
0.502068
S.D. dependent var
1.354914
S.E. of regression 0.995840 Sum squared resid 738.8147
F-statistic 127.2065 Durbin-Watson stat 2.027024
Prob(F-statistic) 0.000000
Unweighted Statistics
R-squared 0.708343 Mean dependent var 0.007916
Sum squared resid 0.058869 Durbin-Watson stat 1.943093
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VII. Modele GARCH
Ti i d l tilit t
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Tipuri de volatilitate
Istorica calculata pe baza preturiloristorice ale activelor
Exponentialy weighted moving average -
EWMA Estimata prin modele econometrice
(Generalised Autoregressive Conditional
Heteroskedasticity - GARCH) Implicita calculata din preturile optiunilor
Calcul volatilitate istorica
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Calcul volatilitate istorica
1. Observatii curs spot S0, S1, . . . , Sn laintervale de ani
2. Calcul randament in timp continuu:
3. Calculul deviatiei standard, s,pentru randamentele ui
4. Estimarea volatilitatii istorice ca:
u
S
Sii
i=
ln
1
=
s
Modelul EWMA
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Modelul EWMA
Conform acestui model, volatilitatea dinziua neste o medie ponderata intre
volatilitatea din ziua anterioara si
randamentul la patrat u2
din ziuaanterioara
RiskMetrics (JP Morgan, Reuters)foloseste = 0.94 pentru calculul
volatilitatii zilnice
2
1
2
1
2)1( += nnn u
Modele GARCH
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Modele GARCH
In modelul GARCH, volatilitatea depindede volatilitatile anterioare si de
randamentele patratice anterioare ale
activului
Coeficientii variabielor sunt extimati prin
diverse proceduri econometrice
2
1
2
1
2in
q
i
ikn
p
k
kn u =
= ++=
Tipuri de modele GARCH
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Tipuri de modele GARCH
ARCH
GARCH
GARCH in Mean
Treshold ARCH - TARCH
Exponential GARCH - EGARCH
Integrated GARCH - IGARCH
Specificare model
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Specificare model
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Estimare GARCH(1 1)
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Estimare GARCH(1,1)Dependent Variable: DL_EUR
Method: ML - ARCH (Marquardt) - Normal distribution
Sample (adjusted): 2 2148Included observations: 2147 after adjustments
Convergence achieved after 19 iterations
Variance backcast: ON
GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1)
Coefficient Std. Error z-Statistic Prob.
C 0.000198 8.77E-05 2.260467 0.0238
Variance Equation
C 2.50E-07 5.08E-08 4.926018 0.0000
RESID(-1)^2 0.138819 0.009128 15.20872 0.0000
GARCH(-1)
0.868095
0.008286
104.7609
0.0000
R-squared -0.001355 Mean dependent var 0.000427
Adjusted R-squared -0.002757 S.D. dependent var 0.006208
S.E. of regression 0.006216 Akaike info criterion -7.718359
Sum squared resid 0.082811 Schwarz criterion -7.707792
Log likelihood 8289.659 Durbin-Watson stat 1.848831
Estimare EGARCH(2,1,1)
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Estimare EGARCH(2,1,1)Dependent Variable: DL_EUR
Method: ML - ARCH (Marquardt) - Generalized error distribution (GED)
Sample (adjusted): 2 2148
Included observations: 2147 after adjustments
Convergence achieved after 25 iterations
Variance backcast: ON
LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) +
C(5)*ABS(RESID(-2)/@SQRT(GARCH(-2))) + C(6)*RESID(-1)
/@SQRT(GARCH(-1)) + C(7)*LOG(GARCH(-1))
Coefficient Std. Error z-Statistic Prob.
@SQRT(GARCH) 0.112635 0.039554 2.847620 0.0044
C -0.000457 0.000156 -2.932430 0.0034
Variance Equation
C(3) -0.284286 0.056336 -5.046247 0.0000
C(4) 0.399824 0.049811 8.026844 0.0000
C(5) -0.170567 0.049394 -3.453202 0.0006
C(6) -0.026267 0.013602 -1.931131 0.0535
C(7) 0.989348 0.004282 231.0737 0.0000
GED PARAMETER 1.293394 0.047198 27.40373 0.0000
R-squared 0.002000 Mean dependent var 0.000427
Adjusted R-squared -0.001266 S.D. dependent var 0.006208
S.E. of regression 0.006212 Akaike info criterion -7.788111
Sum squared resid 0.082534 Schwarz criterion -7.766977
Log likelihood 8368.537 F-statistic 0.612409
Durbin-Watson stat 1.855328 Prob(F-statistic) 0.746122
Corelograma erorilor patratice
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Corelograma erorilor patratice
V l tilit t diti t
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Volatilitatea conditionata
.000
.004
.008
.012
.016
.020
.024
.028
250 500 750 1000 1250 1500 1750 2000
Conditional standard deviation
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VIII. Modele cu vectori
autoregresivi (VAR)
Definitie si utilizare
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Definitie si utilizare
Un model VAR(Vector Autoregression) permite
tratarea simetrica a tuturor variabilelor din model,
in sensul ca nu presupune implicit exogeneitatea
unei anumite variabile (cum se intampla in cazul
OLS).
Construire model VAR
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Construire model VAR
Alegere numar de lag-uri
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ege e u a de ag u
Alegere numar lag-uri
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Alegere numar lag uri
Stabilitate model VAR
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Stabilitate model VAR
Stationaritate model VAR
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Stationaritate model VAR
Stationaritate model VAR
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Stationaritate model VAR
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_RO),DLOG(HICP_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_RO),DLOG(ER_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_RO),DLOG(HICP_EU)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(ER_RO),DLOG(HICP_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(ER_RO),DLOG(ER_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(ER_RO),DLOG(HICP_EU)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_EU),DLOG(HICP_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_EU),DLOG(ER_RO)(-i))
-.6
-.4
-.2
.0
.2
.4
.6
1 2 3 4 5 6 7 8 9 10 11 12
Cor(DLOG(HICP_EU),DLOG(HICP_EU)(-i))
Autocorrelations with 2 Std.Err. Bounds
Testarea autocorelatiei
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VAR Residual Portmanteau Tests for Autocorrelations
Null Hypothesis: no residual autocorrelations up to lag hSample: 1996M01 2011M12
Included observations: 186
Lags Q-Stat Prob. Adj Q-Stat Prob. df
1 0.497749 NA* 0.500439 NA* NA*
2 1.487910 NA* 1.501363 NA* NA*
3 4.163052 NA* 4.220360 NA* NA*
4 22.25030 0.1353 22.70513 0.1218 16
5 35.55546 0.0786 36.37783 0.0661 25
6 52.90164 0.0204 54.30223 0.0150 34
7 64.09821 0.0201 65.93665 0.0138 43
8 81.99966 0.0050 84.64266 0.0028 52
9 89.82131 0.0096 92.86202 0.0053 61
10 95.09797 0.0247 98.43849 0.0141 7011 99.91440 0.0561 103.5577 0.0334 79
12 154.0077 0.0000 161.3815 0.0000 88
*The test is valid only for lags larger than the VAR lag order.
df is degrees of freedom for (approximate) chi-square distribution
Testarea autocorelatiei
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VAR Residual Serial Correlation LM Tests
Null Hypothesis: no serial correlation at lag order hSample: 1996M01 2011M12
Included observations: 186
Lags LM-Stat Prob
1 9.676269 0.3773
2 6.781290 0.6599
3 17.34069 0.0436
4 21.62957 0.0101
5 14.02825 0.1213
6 18.97320 0.0254
7 12.64418 0.1794
8 20.40055 0.01569 8.294159 0.5048
10 5.512769 0.7875
11 5.240046 0.8129
12 72.10294 0.0000
Probs from chi-square with 9 df.
Testarea distributiei normale
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VAR Residual Normality Tests
Orthogonalization: Cholesky (Lutkepohl)
Null Hypothesis: residuals are multivariate normal
Sample: 1996M01 2011M12
Included observations: 186
Component Skewness Chi-sq df Prob.
1 1.088627 36.73835 1 0.0000
2 0.763380 18.06523 1 0.0000
3 -0.043528 0.058734 1 0.8085
Joint 54.86231 3 0.0000
Component Kurtosis Chi-sq df Prob.
1 5.840031 62.50978 1 0.0000
2 12.63516 719.4818 1 0.0000
3 4.804536 25.23672 1 0.0000
Joint 807.2283 3 0.0000
Component Jarque-Bera df Prob.
1 99.24813 2 0.0000
2 737.5470 2 0.0000
3 25.29545 2 0.0000
Joint 862.0906 6 0.0000
Testarea heteroskedasticitatii
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VAR Residual Heteroskedasticity Tests: No Cross Terms (only levels and squares)
Sample: 1996M01 2011M12
Included observations: 186
Joint test:
Chi-sq df Prob.
420.2497 108 0.0000
Individual components:
Dependent R-squared F(18,167) Prob. Chi-sq(18) Prob.
res1*res1 0.359985 5.218402 0.0000 66.95714 0.0000
res2*res2 0.458122 7.843734 0.0000 85.21061 0.0000
res3*res3 0.410657 6.464790 0.0000 76.38214 0.0000
res2*res1 0.414220 6.560566 0.0000 77.04500 0.0000
res3*res1 0.137523 1.479350 0.1029 25.57924 0.1098
res3*res2 0.715462 23.32867 0.0000 133.0759 0.0000
Definitia impulsului
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p Definitia impulsurilor conteaza deoarece:
in cazul descompunerii Cholesky conteaza
ordonarea variabilelor daca acestea sunt
corelate intre ele;
in cazul descompunerii generalizate nuconteaza ordonarea variabilelor;
in cazul descompunerii structurale, aceasta
poate fi utilizata doar daca a fost specificat
anterior un model structural cu restrictiile
necesare.
Functii de impuls-raspuns
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Functii de impuls-raspuns
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p p
-.010
-.005
.000
.005
.010
.015
.020
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(HICP_RO) to DLOG(HICP_RO)
-.010
-.005
.000
.005
.010
.015
.020
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(HICP_RO) to DLOG(ER_RO)
-.010
-.005
.000
.005
.010
.015
.020
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(HICP_RO) to DLOG(HICP_EU)
Response to Cholesky One S.D. Innovations 2 S.E.
-.02
-.01
.00
.01
.02
.03
.04
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(ER_RO) to DLOG(HICP_RO)
-.02
-.01
.00
.01
.02
.03
.04
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(ER_RO) to DLOG(ER_RO)
-.02
-.01
.00
.01
.02
.03
.04
1 2 3 4 5 6 7 8 9 10 11 12
Response of DLOG(ER_RO) to DLOG(HICP_EU)
Response to Cholesky One S.D. Innovations 2 S.E.
Descompunerea variantei
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0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10 11 12
Percent DLOG(HICP_RO) variance due to DLOG(HICP_RO)
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10 11 12
Percent DLOG(HICP_RO) variance due to DLOG(ER_RO)
80
100Percent DLOG(HICP_RO) variance due to DLOG(HICP_EU)
Variance Decomposition
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10 11 12
Percent DLOG(ER_RO) variance due to DLOG(HICP_RO)
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10 11 12
Percent DLOG(ER_RO) variance due to DLOG(ER_RO)
80
100Percent DLOG(ER_RO) variance due to DLOG(HICP_EU)
Variance Decomposition