Secion 4 NABE ASTEF 3
APPLIED ECONOMETRICS: TIME-SERIES ANALYSIS 33
Inroducion and Review The Naure of Economic Modeling Judgemen calls unavoidable Economerics an ar Componens of Applied Economerics Specificaion - he model building aciviy Esimaion - fiing he model o daa Verificaion - esing he model Predicion 34
Componens of Time Series Trend (Long-Term) Componen Seasonal Componen Cyclical Componen Random Componen 35
Simple Exrapolaion Models 1. Linear rend model X = b0 + b1 ( fiing he rend line) A ime-series X changes in consan absolue amouns each ime period. Exponenial growh curve The series X changes in consan percenage incremens raher han consan absolue incremens r X = A log X = log A + r e 36
3. Auoregressive rend model X = b + b X ; if b =, hen b represens 0 1 1 0 0 he rae of change in he series X 4. Quadraic rend model X = b 0 + b 1 + b. If b 1, b > 0, X will always be increasing bu even more rapidly as ime goes on. If b 1 < 0 and b > 0, X will a firs decrease bu laer increase. If boh b 1, b < 0, X will always decrease. 37
(5) S-shaped curves X = e ( k ) k1 / Simple exrapolaion mehods are frequenly he basis for making casual long-range forecass of variables. Noe ha he models basically involve regressing X (or log X ) agains a funcion of ime (linear, quadraic, polynomial, exponenial) and/or iself lagged. Alhough hey can be useful as a way of quickly formulaing iniial forecass, hey usually provide lile forecas accuracy. 38
Moving Average Models 1 X = ( X 1 + X + K+ X 1 ) 1 ^ X+ = 1 1 ( X + X 1 + K+ X 11) 1 A Likely Value for X Nex Monh is a Simple Averag of Is Values Over he Pas 1 Monhs 39
EWMA: Exponenially Weighed Moving Average (Exponenial Smoohing) X = ax 1 + α( 1 α) X + α(1 α) X 3 α =.1 X α =.9 X =.1X =.9X 1 1 +.09X +.09X 0 < α < 1 +.081X +.009X 3 3 +.079X +... 4 +... WEIGHTS MUST SUM TO 1 40
To Forecas wih EWMA ( ) ( ) X = ax + a 1 - a X + a 1 - a X + K + 1-1 - Moving Average Models are Adapive Auomaically Adjus o he Available Daa 41
Hol-Winers Smoohing y is he esimaed, or smoohed, level a ime, and F in he esimaed slope a ime. The parameer α conrols smoohing of he level, and β conrols smoohing of he slope. The h-sep-ahead forecas simply akes he esimaed level a ime T and augmens i wih h imes he esimaed slope a ime T. 4
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ARIMA MODELS 44
Box-Jenkins Modeling (ARIMA Models) AR, MA, ARMA, ARIMA models Model specificaion, esimaion, diagnosic checking Forecasing Uni Roos and Uni Roo Tess (Saionariy) Vecor Auoregressions (VARs) Specificaion and Esimaion of VARs Causaliy (Granger Causaliy) Coinegraion Error Correcion Models 45
Definiions Time Series - a series of numerical daa in which each iem is associaed wih a paricular insan in ime. Examples: weekly measures of money supply M, and daily closing prices of sock indices Univariae Time-Series Analysis - an analysis of a single sequence of daa. Muliple Time-Series Analysis - an analysis of several ses of daa for he same sequence of ime periods. Purpose of Time-Series Analysis - sudy he dynamics or emporal srucure of he daa. Since he mid 1970s, he imeseries approach and he economic approach have been converging. 46
A ime-series is a collecion of random variables {X }. Such a collecion of random variables ordered in ime is called a sochasic process. How does one describe a sochasic process? Specify he join disribuion of he variables X. Usually we only define he firs and second momens of he variables X. Saionary sochasic processes and saionary ime series Definiion: A series is said o be sricly saionary if he join disribuion of any se of n observaions X( 1 ), X( ),, X( n ), is he same as he join disribuion of X( 1 +k), X( +k),, X( n +k) for all n and k. ( ) For n = 1 u = µ for all and σ ( ) = σ. ( ) ( ) For n = join disribuion of X andx 1 ( + ) ( + ) he same as ha of X k and X k 1. 47
[ ] ( ) ( ) ( ) γ k = cov X, X + k is he auo cov ariance coefficien a lag k. ( ) ( ) ( ) γ k is called he auo cov ariance funcion acvf ; γ 0 = σ. Auocovariances depend on unis of measuremen of X(). Need o consider auocorrelaions. The auocorrelaion coefficien ρ ρ ( ) ρ ( ) k ( k) ( ) ( ) k a lag k k is he auocorrelaion funcion acf. = γ γ 0 A plo of ρ( k) agains k is called he corre log ram. 48
Sric Saionariy or Sricly Saionary The disribuion of X() is independen of. The mean, variance, and all higher momens are independen of. A very srong assumpion. Weakly Saionary ( ) ( ) 1 µ = µ for all [ + ] = γ ( ) ( ) ( ) ( ) The acvf depends only on he lag cov X, X k k for all. ( ) ( ) 3 γ 0 = σ < If X( 1 ), X( ),, X( n ) follow a mulivariae normal disribuion, he wo conceps of sric saionariy and weak saionariy are equivalen. ( ) ( ) Due o saionariy ρ k = ρ k γ ( k) = cov ( X, X ) = cov ( X, X ) + k k 49
Nonsaionariy Mos of he ime series we encouner are nonsaionary. A sin gle nonsaionary ime series mod el X = µ + e, where he mean is a funcion of ime ( linear rend or quadraic rend) and e is a weakly saionary series. µ 50
Tess for Saionariy Dickey-Fuller Tes Augmened Dickey-Fuller Tes Phillips-Perron Tes 51
Dickey-Fuller Tes y = σ + δ + α y 1 +. H k :α = 0 1 ( ) ( ) 1 = T $α 1 ( is a whie noise process) ( ) 1 = $α 1 SE $α ( ) The criical values for k(1) and (1) are abulaed; hese saisics do no follow he sandard normal or disribuions. 5
Augmened Dickey-Fuller (ADF) Tess Run a regression of he firs difference of he series agains he series lagged once, lagged difference erms and, opionally, a consan and a ime rend. 1 j 1 j= 1 k y = σ + δ + α* y + θ y + ; Re call α* = α 1 You should include sufficien lagged firs differences o secure a whie noise error erm. The es for a uni roo is a es on he coefficien of y 1. If he coefficien α is significanly differen from zero, hen he hypohesis ha y conains a uni roo is rejeced. y Under he null hypohesis of a uni roo, he repored -saisics does no have he sandard -disribuion. For he criical values of he ADF ess, see work by Fuller (1976); Dickey (1976); McKinnon (1991). If he Dickey-Fuller -saisic is smaller in absolue value han he repored criical values you canno rejec he hypohesis of nonsaionariy and he exisence of a uni roo. Your series may no be saionary. 53
LS // Dependen Variable is D(SP500) Dae: 6-06-1994 / Time: 1:0 SMPL range: 1914-1970 Number of observaions: 57 Augmened Dickey-Fuller: UROOT(T,4) SP500 ===================================================================== VARIABLE COEFFICIENT STD. ERROR T-STAT. -TAIL SIG. ===================================================================== D(SP500(-1)) 0.084377 0.1678309 1.419505 0.00 D(SP500(-)) -0.07065 0.179059-0.4176058 0.6780 D(SP500(-3)) -0.0176493 0.1704504-0.1035450 0.9179 D(SP500(-4)) 0.3195438 0.1703978 1.375813 0.0666 SP500(-I) -0.0675839 0.0454305-1.487531 0.1431 C -1.3654933 1.579174-0.8936957 0.3758 TREND 0.11679 0.0618000 1.8815389 0.0657 ===================================================================== R-squared 0.173365 Mean of dependen var 1.303684 Adjused R-squared 0.074169 S.D. of dependen var 4.593806 S.E. of regression 4.40165 Sum of squared resid 976.897 Log likelihood -161.8573 F-saisic 1.747704 Durbin-Wason sa 1.83650 Prob(F-saisic) 0.19384 Augmened Dickey-Fuller: UROOT(T,4) SP500 ===================================================================== Dickey-Fuller -saisic -1.4876 MacKinnon criical values: 1% -4.149 5% -3.4889 54 10% -3.177
Log Yen/Dollar Exchange Rae Changes in Log Yen/Dollar Exchange Rae 55
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Box-Jenkins Modeling Sochasic Processes Useful in Modeling Time Series (1) a purely random process () a random walk (3) a moving-average (MA) process (4) an auoregressive (AR) process (5) an auoregressive moving-average (ARMA) process (6) an auoregressive inegraed moving-average (ARIMA) process 64
Random Walk Process Purely Random Process A sequence of muually independen idenically disribued random variables consan mean, consan variance A purely random process also is called whie noise. Y = Y + 1 Moving Average Process MA(q) Y = + B + + B l 1 K q q Auoregressive Process AR(p) Y = α Y + α Y + + α Y + 1 1 K p p Auopregressive Moving-Average Process ARMA(p,q) Y = α Y + α Y + K+ α Y + 1 1 p p + B + K+ B l 1 q q 65
ARIMA (p, d, q) In pracice, mos ime series are nonsaionary. One procedure ha is ofen used o conver a nonsaionary series ino a saionary series is successive differencing. The model called an inegraed model because he saionary ARMA model ha is fied o he differenced daa has o be inegraed o provide a model for he nonsaionary daa. 66
The Box-Jenkins Approach One of he mos widely used mehodologies for he analysis of ime-series daa. Basic Seps in he Box-Jenkins Mehodology 1. Differencing he series so as o achieve saionariy. Idenificaion of preliminary model(s) 3. Esimaion of he model(s) 4. Diagnosic checking 5. Using he model for forecasing 67
Differencing To Achieve Saionariy-How? Sudy he graph of he correlogram (i.e., use ACF) of he series. The correlogram of a saionary series drops off as k, he number of lags, becomes large. Common procedure: plo he correlogram of he given series y and successive differences y, y, and look a he correlogram a each sage. Augmened Dickey-Fuller Tes Phillips Perron Tes Orders of AR and MA Componens Examine he correlogram o decide on he appropriae orders of he AR and MA componens. The correlogram of a MA process is zero afer a poin. Tha of an AR process declines geomerically. 68 Use also PACF.
Purely Random Process A purely random process also is called whie noise. A sequence of muually independen idenically disribued random variables. consan mean, consan variance acvf is given by ( ) ( ) γ k = cov X, X = 0 for k 0 k Acf is given by ( ) ρ k = 1 for k = 0 0 for k 0 69
Random Walk Ofen used o describe he behavior of sock prices. Le { ε } be a purely random series wih mean µ and variance σ. Then a process {X } is said o be a random walk if X = X 1 + ε. Assume X 0 = 0. Then In general, X 1 1 X X = = X + = + = 1 1 1 i= 1 i= 1 = E(X ) = µ var(x ) = σ. The mean and variance change wih ; herefore, he process is nonsaionary. Noe, however, ha ε = X X 1. The firs difference of he series is saionary and a whie noise process. Changes in sock prices - a purely random process? i 70
{ } Moving-Average Processes ε Le be a purely random process wih mean zero and variance σ. Than a process defined by { ε } Is called a moving-average process of order m. Denoe by MA(m) X = ε + β 1ε 1+ K+ β m ε m ( ) 0 ( X ) E X m = = ; var σ β i i= 0 icbs γ ( ) k = σ m k i= 0 0 β β i i+ k for k = for k > 0, 1,, K, m m MA(m) process is weakly saionary ρ( k) = m k i= 0 m β β i= 0 0 i β i+ k i for k for k = > 0, 1,, K, m m. 71
Use of lag operaor L j L X = X j MA(m) process wih can be wrien as ( m) ( ) X = + β L + β L + K + β L ε = β L ε. 1 1 β 0 =1 The polynomial in L has m roos. Need o esimae he model and calculae he residuals = [ β ( L) ] 1 X m [ β(l) ] 1 mus converge, he inveribiliy condiion The roos of he equaion m 1+ β1 L + βl +... + βml = 0 mus all be ouside he uni circle o saisfy he inveribiliy condiion. 7
Examples Which of he following MA() processes are inverible? ( a) ( a) ( ) a X X X = ε. 9ε +. ε 1 = ε 18. ε +. 4ε 1 = ε. 8ε +. 4ε 1 Consider (a) and in paricular form he equaion 1 -.9L +.L - 0. Find he roos which saisfy his equaion; from he quadraic formula,.5 and are he roos. Since hey are greaer han 1, hen his MA() process is inverible. 73
Moving-average processes arise in economerics mosly hrough rend eliminaion mehods. One procedure ofen used for rend eliminaion is ha of successive differencing of he ime series X. X = a + a + 0 1. X = a + a ( 1) +. 1 0 1 1 X X = a + a ( 1) +. 1 1 1 1 X X = a + 1 1 1. X = a 1 + 1. A moving average process of order 1. 74
Auoregressive (AR) Process Le { ε } be a purely random process wih mean zero and variance σ. The process {X } given by X = α X + α X + + α X + 1 1 K r r Is called an auoregressive process of order r. Denoe by AR(r). An AR(r) process is a regression of X on is own pas values. ( r α α α ) 1 K r X = L + L + L X + Equivalenly X ( 1 α α α ) 1 K = L L L X = 1 r ( r α L α L Kα L ) 1 1 r For saionariy, we require ha he roos of 1 - a 1 L - a L - -a r L r = 0 lie ouside he uni circle. 75
Examples Which of he following AR() processes are sable? (a) (b) (c) X X X = = =.9X.8X X 1 1 1.X.4X.8X + ε + ε + ε Consider (c) and in paricular form he equaion 1 - L +.8L - 0 Using he quadraic formula, he roos of his equaion are.65 ±.97i. The modulus of his expression is 1.5; herefore, his AR() process is sable! 76
For he AR(r) process and under he assumpion of weak saionariy, icbs ( ) ( ) ( ) ρ k = α 1ρ k 1 + K+ α kρ k r, k = 1,, K, r. Subsiuing k = 1,,, r and noing we ge equaions o deermine he r parameers a 1, a,, a r. These expressions are called he Yule-Walker equaions. For an AR() process ρ ( 1) ( ) ( ) ( ) ρ k = α ρ k 1 + α ρ k 1 ( ) ( ) ( ) ρ 0 = α ρ 1 + α ρ 1 ( ) ( ) ( ) ρ( k) = ρ( k), ρ 0 = α ρ 1 + α ρ = 1 1 ( ) ( ) ( ) ( ) ρ 1 = α ρ 0 + α ρ 1 = α + α ρ 1 = α 1 1 α 1 1 ( ) ( ) ( ) ( ) ρ = α ρ 1 + α ρ 0 = α + ρ 1 + α ρ ( ) = 1 1 α 1 α 1 + α α + 1 α 1 α 77
Example Consider he AR() process X = X. 5X +. 1 X X +. 5X =. 1 ( 1 5 ) L +. L X =. Examine he roos of.5l - L + 1 = 0 equivalen o L - L + = 0. 1 ± i are he roos of he polynomial. In modulus 1 ± i is equal o = 1414. which is clearly greaer han 1. Therefore his process is saionary. Because he roos are complex, ρ(k) will be a sinusoidal funcion. ICBST ρ(1) =.666 ρ(7) =.041 ρ() =.166 ρ(8) =.06 ρ(3) =.166 ρ(9) =.041 ρ(4) =.5 ρ(10) =.010 ρ(5) =.166 ρ(11) =.010 ρ(6) =.041 ρ(1) =.015 78
Auoregressive Moving-Average (ARMA) Process Models ha are combinaions of he AR and MA models. An ARMA (p, q) model is defined as X α X α X β β = 1 1 + + p p + + 1 + K+ q q. { ε } is a purely random process wih mean zero and variance ( L) X ( L) More compacly, Φ = Θ, where are polynomials of orders p and q, respecively ( ) 1 Φ L = α L α L α L 1 Θ L = 1+ β L + β L + K+ β L ( ) 1 φ p σ. ( L) and θ ( L) For saionariy, we require ha he roos of φ( L) = 0 lie ouside he uni circle. For inveribiliy of he MA componen, we require ha he roos of θ(l) lie ouside he uni circle. 79 p q q
ARMA (1, 1) X X (1 α Example = α1x 1+ +β1 1 1 L)X 1+ β = 1 α 1 L L 1 = (1 + β The acvf and acf of an ARMA model are more complicaed han for an AR or MA model. In he case of an ARMA (1, 1) model, ICBST 1+ β + α β var( X ) = σ 1 α γ ( 1) cov ( X, X ) = = ( ) ρ 1 = ρ ( ) ( ) ( α + β ) ( 1+ α β ) 1+ β + α β k = α ρ k 1 for k 1 L) ( α + β ) ( 1+ α β ) 1 α 1 σ 80
Auoregressive Inegraed Moving-Average (ARIMA) Process In pracice, mos ime series are nonsaionary. One procedure ha is ofen used o conver a nonsaionary series ino a saionary series is successive differencing. Define he operaor =1 L X ( L) = 1 X = X X. 1 ( ) ( ) X = 1 L X = 1 L + L X = X X 1 + X. Suppose ha d X is a saionary series ha can be represened by an ARMA (p, q) model. Then we can say ha X can be represened by an ARIMA (p, d, q) model. The model is called an inegraed model because he saionary ARMA model ha is fied o he differenced daa has o be inegraed o provide a model for 81 he nonsaionary daa.
Model Specificaion, Esimaion, and Diagnosic Saionariy and Inveribiliy Checking p q 1 L αl... αpl )X = (1 + B1L + BL +... + BqL ) ( 1 α ε The model is saionary if all roos of he equaion 1 1 K α L α L α L are larger han 1 in modulus. Any MA process is saionary. p p A series saisfies he inveribiliy condiion if all roos of he equaion q 1+ B L + B L + L + B L = 0 1 are larger han 1 in modulus. The inveribiliy condiion ensures ha X may be expressed in erms of ε and an infinie weighed sum of previous X s. q 8
AR Models Esimaion of AR, MA, and ARMA Models OLS minimizes ε he only problem is he choice of he degree of auoregression; also a loss in he number of observaions used as he lag lengh increases. MA or ARMA Models Canno wrie he error sum of squares ε as simply a funcion of he observed X s and he parameers in he AR models. (1) can wrie down he covariance marix of he moving-average error and assuming normaliy, use he maximum likelihood mehod of esimaion. () use a grid-search procedure (Box and Jenkins, 1976) o compue by successive subsiuion for each value of B 1, B,..., B q, iniial values. Wih successive values of generaed, ε ε can be compued for all permuaions of B 1, B,..., B q. given some 83 ε
In addiion, we have o check he serial correlaion paern of he residuals need o be sure ha here is no serial correlaion. Box and Pierce (1970) sugges looking a no jus he firs-order auocorrelaion bu auocorrelaion of all orders of he residuals. M k k= 1 Calculae Q = N r, where r k is he auocorrelaion of lag k, and N is he number of observaions in he series. If he model fied is appropriae, Q &~ X m p q where p and q are he orders of he AR and MA componens. 84
Ljung and Box (1978) sugges a modificaion of he Q-saisic for moderae sample sizes. m k= 1 1 Q* = N( N + ) ( N k) r k 85
Forecasing We esimae he model wih n observaions. Need o forecas X n+k, a k- period ahead forecas. 86
Key Poins (1) wrie ou he expression for X n+k () replace all fuure values X n+j ( j> 0, j < k) by heir forecass + (3) replace all n j(j > 0) by zero (4) replace all n j (j > 0) by OLS residuals from he esimaion sage 87
Time-Series Idenificaion Inspec daa-derived esimaes of wo funcions: auocorrelaion funcion (ACF), rˆ (j) = corr(x, x j ) ˆ( π j) parial auocorrelaion funcion (PACF), parial correlaion coefficien a lag j ACF For auocorrelaion r$( j), assign a variance r$ ( i), where n i runs form -j+1 o j-1. The sandard error is he square roo of his variance. 88
ICBST For an auoregression of order p, he coefficiens $π j are 0 for all j > p. The PACF is mos useful for idenifying AR processes because for an AR(p), he PACF is 0 beyond lag p. An approximae sandard error for he esimaed parial auocorrelaion is n -1/. 89
Summary of Model Specificaion A he idenificaion sage, you compue he ACF and PACF. Behavior of he esimaed funcions is he key o model idenificaion. The behavior of funcions for differen processes is summarized below. MA(q) AR(p) ARMA(p,q) Whie Noise ACF D(q) T T 0 PACF T D(p) T 0 Where, D(q) means he funcion drops off o 0 afer he lag q. T means he funcion dampens. 0 means he funcion is 0 a all nonzero lags. Use he ACF and he PACF o limi your search o a few plausible models. 90