FORECASTING TECHNIQUES ADE 2013 Prof Antoni Espasa TOPIC 1 PART 2 TRENDS AND ACCUMULATION OF KNOWLEDGE. SEASONALITY HANDOUT

FORECASTING TECHNIQUES ADE 2013 Prof Anoni Espasa TOPIC 1 PART 2 TRENDS AND ACCUMULATION OF KNOWLEDGE. SEASONALITY HANDOUT February 2013

MAIN FACTORS CAUSING TRENDS Increases in populaion. Seady inflaion. Technological change. Slowly changes in preferences, habis, aiudes, social regulaions, ec.

Trend and accumulaion of knowledge The rends in economic variables can be relaed o he accumulaion of knowledge by humans beings, Bu he incorporaion of such accumulaion of knowledge in he economic variables can be very differen. In any case he accumulaion of knowledge is no a deerminisic process bu a sochasic one: here is uncerainy around i.

Differen ways of incorporaing he flow of knowledge. The accumulaion of knowledge implies ha knowledge oday incorporaes all he previous knowledge. The incorporaion of his flow of knowledge in an economic variable is no homogenous. I could ake differen forms and so he TRENDS. 1.- Knowledge a ime incorporaes knowledge a ime (-1) plus a conemporaneous random shock wih zero mean.

STOCHACTIC TREND WITH LOCAL OSCILLATIONS IN LEVEL 1 (1) where η is a zero mean sochasic componen. The uni coefficien τ in (1) implies ha pas values are never forgo. This is a possible framework o capure he accumulaion of human knowledge.

The series τ can also be formulaed as he sum of he infinie pas values of ɳ, meaning ha Knowledge oday (rends)is he sum of all he sochasic shocks in knowledge τ = ɳ + ɳ -1 + ɳ -2 + These shocks have zero mean, bu he mean of τ as a sequence of zeros is no idenified. In fac τ does no oscillae around a consan mean, bu around he previous value and consequenly τ shows local oscillaions in level around ime.

A TIME SERIES WITH STOCHASTIC TREND WITH LOL ω χ χ r r r r r r 1 1 1 1 1 1 1 : M odelo final : Comp onene residual ) ( (2) (3)

Accumulaion and uni roo process The former model is a finie difference equaion sysem wih persisence.ha is deermined by a sochasic erm X X w 1. This mahemaic framework is denoed as uni roo process and he resuling rend is said o follow and uni roo process. Series generaed by his ype of rend are said o be inegraed of order on: I(1). espasa@es-econ.uc3m.es

SERIES I(1) The series X can be formulaed as he sum of he infinie pas W They jus show LOL

STOCHASTIC TRENDS: SERIES I(1) Series X in (2) is characerized by he fac ha aking firs differences X X X 1 W The ransformed daa has no evoluiviy. We say ha X is inegraed of order 1 because aking once firs differences he resuling daa is saionary. We denoe X by I(1). Wih he I( ) erminology we indicae ha he rend is sochasic. (3)

Example Thus, in more sangnan producive secors, suchs as mining in he Spanish economy, echnological incremens have zero and rend for hese series show local oscilaions in level bu no sysemaic growh, we say ha hey are inegraed series wih zero mean in is incremen: I(1,0).

2000=100 170 Index of Mining and quarrying oal in Spain 160 150 140 130 120 110 100 90 80 70 60 50 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 Source: EcoWin

VARIABLES IN TERMS OF RATIOS They can follow a srucure of LOL

1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009. Figure 3. RELATIVE PRICES OF CLOTHES WITH RESPECT TO CLOTHES. 1.15 PRECIOS RELATIVOS DEL VESTIDO RESPECTO AL CALZADO 1.10 1.05 1.00 0.95 0.90 Fuene: INE Fecha: Enero de 2010

Figure 23.1 Daily Exchange Rae Yen-Dolar 180 160 140 120 100 80 60 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Period: 2/01/1990-25/02/2000 Source: FRED (Federal Reserve Economic Daa)

Figure 23.2 Daily Variaions in he Yen-Dolar Exchange Rae 4.5 2.5 0.5-1.5-3.5-5.5-7.5 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Period: 2/01/1990-25/02/2000 Source: FRED (Federal Reserve Economic Daa)

DIVERSITY OF TRENDS Trends are presened in differen economic series depending on how he accumulaion of knowledge is incorporaed and changes in habi and social organizaion in he economic magniude ha each series represen.

1 Wih he above scheme he knowledge oday adds o he yeserday knowledge a sochasic facor η which has zero mean, herefore he knowledge accumulaes he pas bu does no grow. Changes in knowledge, η, have zero mean.

1 c Wih he above scheme he knowledge oday adds o he yeserday knowledge: A consan erm c and a sochasic facor η wih zero mean. The change in knowledge, η, has a posiive mean, and herefore i grows along ime. NOTE: ha in his model he level is sochasic bu he growh is deerminisic.

INTEGRATED SERIES WITH SYSTEMATIC GROWTH The previous Series I(1) only exhibis local oscillaions in level. An inegraed series wih sysemaic growh can be represened as Z = Z -1 + b + w (4) Taking firs differences Z = b + w (5) Comparing: Z in (5) wih X in (3) we see ha X has a zero mean and Z a mean wih value b. Therefore X only has local oscillaions in level and Z sysemaic growh. Boh are I(1) bu very differen.

1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 Figure 2.12 Quarerly US Real Gross Domesic Produc (X9 ) 9000 8500 8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 Period: I-1964 / II-1999 Source: BEA A consan 1996 prices. Non Seasonally adjused.

1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 Quarerly variaions US Gross Domesic Produc 5,00 4,00 3,00 2,00 1,00 0,00-1,00-2,00-3,00 Period: I-1964 / II-1999 Source: BEA A consan 1996 prices. Non Seasonally adjused.

I(1,m) TERMINOLOGY To include he fac ha he mean in X can or canno be zero in series I(1) we use he erminology I(1,m) wih m = 0 if mean of X is zero and m = 1 if mean of X is no zero. So X in (2) is I(1,0) and Z in (a) is I(1,1). In I(1,m) τ = 1 + m gives he number of facors in he rend: 1 or 2.

Figure 23.3 Daily Exchange Rae Yen-Dolar 180 160 140 120 100 80 60 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Period: 2/01/1990-25/02/2000 Source: FRED (Federal Reserve Economic Daa)

Figue 23.4 Daily Variaions in he Yen-Dolar Exchange Rae 4.5 2.5 0.5-1.5-3.5-5.5-7.5 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Period: 2/01/1990-25/02/2000 Source: FRED (Federal Reserve Economic Daa)

1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 9000 8500 8000 7500 7000 6500 6000 5500 5000 4500 4000 3500 3000 2500 Figure 23.5 Quarerly US Real Gross Domesic Produc Period: I-1964 / II-1999 Source: BEA A consan 1996 prices. Non Seasonally adjused. Quarerly variaions US Gross Domesic Produc 5.00 4.00 3.00 2.00 1.00 0.00-1.00-2.00-3.00 Period: I-1964 / II-1999 Source: BEA A consan 1996 prices. Non Seasonally adjused.

SERIE I(1,0) EXCHANGE RATE SERIES The differenciaed series has zero mean.

Figura 23.3 Tipo de Cambio Diario Yen-Dólar 180 160 140 120 100 80 60 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Período: 2/01/1990-25/02/2000 Fuene: FRED (Federal Reserve Economic Daa) X

Figura 23.4 4.5 Variaciones Diarias en el Tipo de Cambio Yen-Dólar 2.5 0.5-1.5-3.5-5.5-7.5 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Período: 2/01/1990-25/02/2000 Fuene: FRED (Federal Reserve Economic Daa) X

I(1,1) SERIES GDP The series in difference has mean differen from zero. Therefore, his series show sysemaic grwh and his growh is consan.

TRENDS WITH NOT CONSTANT GROWTH Technological changes have no a consan mean. I is more realisic o assume ha heir mean level has a segmened srucure. We will denoe as I(1,1 s ) he ime series wih hese characerisics. Example: Spanish GDP by Prof.Leandro Prados de la Escosura.

1850 1855 1860 1865 1870 1875 1880 1885 1890 1895 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Series Anuales 100000 Produco Inerior Bruo en España (Miles de millones de peseas) 90000 80000 70000 60000 50000 40000 30000 20000 10000 0 X

TREND WITH TWO UNIT ROOTS In oher cases srucural changes in he mean can be more frequen and hey can follow an uni roo process, ha makes ha series in level follow a process wih wo uni roos and series wih such a feaure are denoed as I (2,0).

Trends wih accumulaion in levels and in growh. The changes in he mean of he growh of knowledge could occur more ofen and hey could also follow a uni roo accumulaion srucure. Then hese rends: show sysemaic growh. he growh is also sochasic. hey have wo uni roos. hey are I(2,0).

FULLY STOCHASTIC TRENDS In X = X -1 + b + w he level facor X -1 is sochasic bu he incremen facor b is deerminisic A model wih sochasic incremen facor is X = X -1 + (X -1 - X -2 ) + w (6) Now aking firs differences X = X - X -1 = (X -1 - X -2 ) + w, (7) so in (7) X sill has evoluiviy and in fac is I(1,0). Differencing again 2 X = (X - X -1 ) - (X -1 - X -2 ) = w, (8) w is saionary. Thus is (6) X ~ I(2,0) In (6) we need uni coefficiens(roos) X ~ I(1,0) 2 X ~ I(0,0)

EXAMPLES: PRICE INDEXES AND INFLATION PRICE INDEXES,OR MORE SPECIFICALLY ITS LOGARITMIC TRANSFORMATION TYPICALLY ARE I(2,0). THEREFORE, INFLATION FIRST DIFFERENCES OF THE LOGARITHM OF PRICES- ARE I(1,0).I(1,0). INFLATION EVOLVES THROUGH TIME AS GENERATED BY A PROCESS WITH LOCAL OSCILATIONS IN LEVEL. The saionary ransformaion IS OBTAINED BY APPLAYING TWO DIFFERENCES TO THE LOG OF PRICES.

1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Figure 23.6 200 180 160 140 120 100 80 60 40 20 0 Monhly Consumer Price Index in US, excluding food and energy prices Period: 1958.01-2000.01 Source: BLS

1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Figure 23.7 US Core Inflaion 16 14 12 10 8 6 4 2 0 Period: 1958.01-2000.01 Source: BLS * Core inflaion has been defined as he rae of growh of a consumer price index obained by excluding from he global consumer price index he prices corresponding o food and energy. Here we use he year-on-year rae of growh o measure Core inflaion.

ELIMINATING TRENDS BY DIFFERENCING In series I(d,m) differencing d imes we eliminae he rend. In series T(2), also I(0,2), X = a + b + w (9) X = b + (w - w -1 ), (10) X is also saionary, bu he residual erm (W - W -1 ) has bad sochasic properies. This is due o he fac ha in (9) he proper eliminaion of he rend is by regression. Neverheless differencing also removes deerminisic rends. Unless we observed a very sable deerminisic rend we will use I(d,m) models wih d 0.

STOCHASTIC TRENDS: SUMMARY Tha a rend is sochasic means ha i is affeced by random shocks which are unforecasable. I is useful o associae rends o accumulaion of he pas. This implies ha he presen of he rend incorporaes he pas wih a uni coefficien (denoed UNIT ROOT in he erminology of finie difference equaions).

A LINEAR FORMULATION FOR A STOCHASTIC TREND τ = τ -1 + { growing facor} + ɳ. Possibiliies for he growing facor: 1 o be zero 2 o be deerminisic, a fixed quaniy 3 o be sochasic, accumulaing previous growh. In he firs case he corresponding ime series have no sysemaic growh. Only LOL. In he oher wo cases he series show sysemaic growh. In case 2 he growh has a consan mean. In case 3 he growh shows local oscillaions in level.

INTEGRATION IN TRENDS τ = τ -1 + { growing facor} + ɳ. (1) Suppose ha he growh facor (gf) is zero. Then τ = ɳ + ɳ -1 + ɳ -2 + ɳ -3 + ɳ -4 + (2) equals o he sum or INTEGRATION of all he previous shocks. If in equaion (1) he gf is non-zero hen equaion (2) will also include he inegraion of all previous gf s. In his case here is sysemaic growh in he rend. In (1) here is only one uni roo and herefore here is only one inegraion process. The rends are hen I(1), bu wo differen formulaions are possible. I(1) rends wih no growh denoed rends I(1.0)- and rends in which he growh has a consan mean denoed I(1,1) rends.

DOUBLE INTEGRATION IN TRENDS τ = τ -1 + { growing facor} + ɳ. (1) The gf in (1) could also include a uni roo, ie anoher accumulaion process. τ = τ -1 + {τ -1 - τ -2 } + ɳ. (3) In (3) here is a uni roo in he levels he coefficien of τ -1.Bu passing τ - 1 o he RHS of (1) we ge τ - τ -1 = {τ -1 - τ -2 } + ɳ. (4) And in (4) sill we have anoher uni roo on he growh. Meaning ha in one we have wo uni roos: - On he levels and - Anoher in he growh. - These rends are I(2). - In ime series wih rends I(d), differencing d ime we ge he saionary ransformaion, ie an I(0) series.

STOCHASTIC SEASONALITY

Summary of sochasic rends The model for he observed variable is: X = τ + w, (1) where τ is a sochasic rend, which can be seen as relaed on how he incorporaion of knowledge affecs o economic variables. MODELS FOR STOCHASTIC TRENDS: τ = τ -1 + η (2) This rend only shows LOL. τ = τ -1 + c + η (3) This rend shows sysemaic growh wih consan mean in growh. τ = τ -1 + (τ -1 - τ -2 ) + η (4) This rend shows fully sochasic sysemaic growh.

Trend models for X Combining (1) wih (2), (3) and (4) we ge hree differen models for economic ime series wih rends: X = X -1 + w (2 ) daa wih LOL I(1,0) X = X -1 + c + w (3 ) daa wih consan mean growh I(1,1). X = X -1 + (X -1 - X -2 ) + w daa wih fully sochasic sysemaic growh I(2,0).

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 Monhly ime series wih sochasic rends and seasonaliy. 5000 4500 4000 3500 3000 2500 2000 1500 1000 Ingreso por Turismo en España (Millones de euros) 500 X

Seasonaliy is deermined by sochasic facors as meeorological cyclical regulariies along a naural year. When his ype of seasonaliy appears joinly wih a sochasic rend hen One of he uni roos is no over he inmediae pas (-1) Bu over he same period of he previous year (-s)

I(1,0) EE SERIES WITH I(1,0) TREND AND STOCHASTIC SEASONALITY. I shows soochasic local oscillaions in level and sochasic seasonaliy: X X s (5) In (5) here is an uni roo (coefficien for X -1 )bu on is own seasonal lagged value. This is why we denoe his series as I(1,0) EE Taking one difference of X i becomes saionary: s X X X s (6)

DECOMPOSITION OF THE SEASONAL LAG X s X [ X 1 X 2... X s 1] 1 (1) In (1):- X-1 capure he level in (-1): TREND FACTOR and - he erm beween brackes [..], capure he sum wih negaive of he differenciaed series a all ime periods of he year: FACTOR SEASONAL FACTOR (1) poins a he obvious fac ha each value in a disan pas (-s) is equal o each value in a recen pas (-1) less he incremens beween he wo momens.

SERIES I(1,1) WITH STOCHASTIC SEASONALITY SS These series have a rend wih sysemaic growh in which level is sochasic, bu growh is deerminisic. Besides i has sochasic seiasonaliy : X X s b ω (7) Yearly differences: s X X X s b * (8) are saionary bu hey have an average growh ha is differen from zero

SERIES I(2,0)SS They have sysemaic growh wih sochasic level and slope. Besides i has sochasic seasonaliy: X X ( X X ) 1 s s1 ω (9) X : I(1,0) EE X : I(1,0) wih no seasonaliy s s X W esacionaria

DECOMPOSITION OF THE SEASONAL LAG OF THE DIFFERENCED SERIES. X s X 2 2 1 [ X... X ] 1 s 1 (2) where facor. X 1 is he growing facor and he erm in brackes he seasonal The resuling model for X is: X X 1 X 1 2 2 [ X 1... X s1 ] w

STOCHASTIC SEASONALITY AND STOCHASTIC GROWTH Now he equivalen equaion (24) is X X )X W, s1 ( L... L 1 1 (28) (1 L) X ( L... L s1 ) (1 L) X W, (29) 2 (1 L... L s1 ) X W (30) U s 1( L) X X W s W (31) (32) Equaion (32) can also be wrien as X = X -1 + X -1 - {LU s-1 (L)X -X -1 } + W. (32 ) Componen associaed wih rend Componen associaed wih seasonal cycle

REMOVING TREND AND SEASONALITY A B C D E Series wih local oscillaions in levels I(1,0) X = X -1 + W (33) X = W (34) Series wih local oscillaions in levels and sochasic seasonaliy I(1,0)SS X = X -1 - X -1 + + X -s + W (35) X = X -s + W (36) s X = W (37) Series wih sysemaic growh in which he growh has a consan mean I(1,1) X = X -1 + b + W (38) X = b + W (39) As (C) wih sochasic seasonaliy I(1,1)SS X = X -1 + s b - X -1 + + X -s + W, (40) X = X -s + s b +W (41) s X = s b + W (42) Series wih sochasic growh and sochasic seasonaliy I(2,0)SS X = X -1 + X -1 - {LU s-1 (L) X -1 - X -1 } + W (43) s X = W (44)

DEVIATIONS FROM THE EVOLUTIVE MEAN OF A TIME SERIES A ime series wih rend and seasonaliy could be represened as: where T is he rend facor X = T S W (4) S is he seasonal facor and W is a facor which capures he deviaions of X from he pah of he evoluive facors T and S. By consrucion W does no show evoluive behaviour. Business cycles oscillaions an shor-erm flucuaions are he componens of W. They resul from he ime dependency on he W daa.

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Original series I(2,0)SS 3000000 EXPORTACIONES DE ALIMENTOS, BEBIDAS Y TABACO EN ESPAÑA Miles de euros 2500000 2000000 1500000 1000000 500000 0 Fuene: Aduanas

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Logarimic ransformaion I(2,0)SS 15 EXPORTACIONES DE ALIMENTOS, BEBIDAS Y TABACOEN ESPAÑA Serie en logarimos 15 14 14 13 13 12 Fuene: Aduanas

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Firs differences of logs, I(1,0)SS 0.80 EXPORTACIONES DE ALIMENTOS, BEBIDAS Y TABACO EN ESPAÑA Primeras diferencias de logarimos 0.60 0.40 0.20 0.00-0.20-0.40-0.60 Fuene: Aduanas

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Annual differences of he logarihms, I(1,0) wihou seasonaliy 0.60 0.50 0.40 0.30 0.20 0.10 0.00-0.10-0.20-0.30 EXPORTACIONES DE ALIMENTOS, BEBIDAS Y TABACO EN ESPAÑA Diferencias anuales de logarimos Fuene: Aduanas

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Regular and seasonal differences of logs, I(0) 0.40 0.30 0.20 0.10 0.00-0.10-0.20-0.30-0.40-0.50 EXPORTACIONES DE ALIMENTOS, BEBIDAS Y TABACO EN ESPAÑA Diferencias regular y anual de logarimos Source: Aduanas

CONCLUSION Applying regular and seasonal differences we can remove rend and seasonaliy. Laer in he course we will see saisical ess o decide he ype of differences ha we need o apply o a given ime series o eliminae possible rend and seasonaliy. The properies of he regular and seasonal raes of growh of X depend on he model for X.

DEVIATTONS OF X FROM ITS EVOLUTIVITY PATH: DEPENDENCY ON THEIR PAST.

DEVIATIONS OF THE EVOLUTIVITY MEAN FOR A TIME SERIES A ime series coulb be represened wih rend and seasonaliy according o he following model: where T is he rend facor S is he seasonal facor, and X = T S W (4) W is a facor ha capures he deviaion of X from T y S. By consrucion, W does no show evoluiviy behavior. The componens of W are business and shor run cyclical oscillaions. They are he resul from he ime dependence of W.

STATIONARY DEPENDENCE (CYCLICAL OSCILLATIONS AND SHORT RUN PERTURBATIONS),

DEVIATIONS OF X OVER TREND AND SEASONALITY, THAT WE WILL CALL W, SHOW CORRELATION WITH RESPECT TO PREVIOUS DEVIATIONS. WE NEED TO SPECIFY A MODEL FOR W CONDITIONING OVER THE PAST DEVITATIONS. THIS WILL BE DONE BY SETTING A GENERAL ASSUMPTION FOR THIS CORRELATION.

ORIGINAL TIME SERIES AND INCREMENTS OR GROWTH RATES OF THE ORIGINAL TIME SERIES.

MONTHLY, QUARTERLY OR ANNUAL RATES OF GROWTH. EXAMPLES IN SERIES OF PRODUCTION, EMPLOYMENT, AND PRICES.

1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Monhly Consumer Price Index in US, excluding food and energy prices (X7 ) 200 160 120 80 40 0 Period: 1958.01-2000.01 Source: BLS

1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Figure 2.6 Core Inflaion (X3 ) 16 14 12 10 8 6 4 2 0 Period: 1958.01-2000.01 Source: BLS * Core inflaion has been defined as he rae of growh of a consumer price index obained by excluding from he global consumer price index he prices corresponding o food and energy. Here we use he year-on-year rae of growh o measure Core inflaion.

SOME BASIC MODEL TO CAPTURE TREND AND SEASONAL EVOLUTION 1.- Models wih local oscillaions and wihou seasonal oscillaions I I 0,1 1,0 X s Also obain he model for And he properies of his variable

2.- Oscilaciones locales de nivel con esacionalidad deerminisa (ED). I 0,1 s ED I1,0) ED Derivaciones para X 3.- Oscilaciones locales de nivel con esacionalidad esocásica (EE). I,0) EE sx 1 Derivaciones para

4.- Sysemaic growh wih no seasonaliy I I I I 0,2 0,2 1,1 1,1 s s Derivaions for X I 2,0 Derivaions for X SX and SX

5.- Sysemaic growh wih deerminisic seasonaliy. Proceed as in he previous poin 6.- Sysemaic growh wih sochasic seasonaliy: I 1,1 EE s I1,1 EE Derivaion for X I 2,0 EE Derivaion for X SX y SX

DEVELOP EXERCISES OVER THESE MODELS

END OF TOPIC 1