Chapter 15. Time Series: Descriptive Analyses, Models, and Forecasting

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Chaper 15 Time Series: Descripive Analyses, Models, and Forecasing

Descripive Analysis: Index Numbers Index Number a number ha measures he change in a variable over ime relaive o he value of he variable during a specific base period Simple Index Number index based on he relaive changes (over ime) in he price or quaniy of a single commodiy In d e x n u m b e r a im e T im e s e r ie s v a lu e a im e T im e s e r ie s v a lu e a b a s e p e r io d 100 I Y Y i 0 100

Descripive Analysis: Index Numbers Laspeyres and Paasche Indexes compared The Laspeyres Index weighs by he purchase quaniies of he baseline period The Paasche Index weighs by he purchase quaniies of he period he index value represens. Laspeyres Index is mos appropriae when baseline purchase quaniies are reasonable approximaions of purchases in subsequen periods. Paasche Index is mos appropriae when you wan o compare curren o baseline prices a curren purchase levels

Descripive Analysis: Index Numbers Calculaing a Laspeyres Index Collec price info for he k price series o be used, denoe as P 1, P 2 P k Selec a base period 0 Collec purchase quaniy info for base period, denoe as Q 10, Q 20..Q k0 Calculae weighed oals for each ime period using he k formula Q P i 0 i k i 1 Q P i 0 i i 1 Calculae he index using he formula I 100 k Q P i 0 i 0 i 1

Descripive Analysis: Index Numbers Calculaing a Paasche Index Collec price info for he k price series o be used, denoe as P 1, P 2 P k Selec a base period 0 Collec purchase quaniy info for every period, denoe as Q 1, Q 2..Q k Calculae he index for ime using he formula I k i 1 k i 1 Q Q i i P P i i 0 100

Descripive Analysis: Exponenial Smoohing Exponenial smoohing is a ype of weighed average, ha applies a weigh w o pas and curren values of he ime series. Exponenial smoohing consan (w) lies beween 0 and 1, and smoohed series E is calculaed as: E Y 1 1 E w Y (1 w ) E 2 2 1 E w Y (1 w ) E 3 3 2 E w Y (1 w ) E 1

Descripive Analysis: Exponenial Smoohing Selecion of smoohing consan w is made by researcher. Small values of w give less weigh o curren value, yield a smooher series Large values of w give more weigh o curren value, yield a more variable series

Time Series Componens 4 componens of ime series models: T secular or long-erm rend C cyclical rend S seasonal effec R residual effec These 4 componens are from a widely used model called he addiive model: Y = T + C + S + R

Forecasing: Exponenial Smoohing Calculaion of Exponenially Smoohed Forecass Calculae exponenially smoohed values E 1, E 2, E for observed ime series Y 1, Y 2, Y. Used las smoohed value o forecas he nex ime series value F E F w Y F 1 Assuming ha Y is relaively free of rend and seasonal componens, use he same forecas for all fuure values of Y : F F F 2 1 F 3 1

Forecasing Trends: The Hol- Winers Model The Hol-Winers model adds a rend componen o he forecas. Calculaing Componens of he Hol-Winers Model 1. Selec exponenial smoohing consan w 2. Selec rend smoohing consan v 3. Calculae he wo componens E and T from ime series Y beginning a ime =2 E 2 =Y 2 T 2 =Y 2 -Y 1 E 3 =wy 3 +(1-w)(E 2 +T 2 ) T 3 =v(e 3 -E 2 )+(1-v)T 2 E =wy +(1-w)(E -1 +T -1 ) T =v(e -E -1 )+(1-v)T -1

Forecasing Trends: The Hol- Winers Model Hol-Winers Forecasing 1. Calculae he exponenially smoohed and rend componens E and T for each observed value of Y ( >2) 2. Calculae he one-sep-ahead forecas using F +1 =E +T 3. Calculae he k-sep-ahead forecas using F +k =E +kt

Measuring Forecas Accuracy: MAD and RMSE Measures of Forecas Accuracy for m Forecass Mean absolue Deviaion (MAD) M A D Mean absolue percenage error (MAPE) Roo mean squared error (RMSE) R M S E m 1 Y m M A P E m 1 m F 1 Y m Y Y F 2 m F 100

Forecasing Trends: Simple Linear Regression Simple Linear Regression is he simples inferenial forecasing model Afer fiing he regression line o exising daa, he leas squares model can be used o forecas fuure values of he dependen variable Two problems are associaed wih using a LSM o forecas ime series: 1. Forecasing falls ouside of he experimenal region, increases widh of predicion inervals 2. Cyclical effecs are no buil in o he model, and inroduce he problem of correlaed error

Seasonal Regression Models Use of muliple regression model wih dummy variables o describe seasonal differences is common In he following example, dummy variables are se up for Quarers 1, 2 and 3 Model ha reflecs seasonal componen and expeced growh in usage is: E ( Y ) Q Q Q 0 1 2 1 3 2 4 3

Seasonal Regression Models Daa o be used is in he following able: Quarerly Power Loads (megawas) for a Souhern Uiliy Company, 1992-2003 Year Quarer Power Load Year Quarer Power Load 1992 1 68.8 1998 1 130.6 2 65.0 2 116.8 3 88.4 3 144.2 4 69.0 4 123.3 1993 1 83.6 1999 1 142.3 2 69.7 2 124.0 3 90.2 3 146.1 4 72.5 4 135.5 1994 1 106.8 2000 1 147.1 2 89.2 2 119.3 3 110.7 3 138.2 4 91.7 4 127.6 1995 1 108.6 2001 1 143.4 2 98.9 2 134.0 3 120.1 3 159.6 4 102.1 4 135.1 1996 1 113.1 2002 1 149.5 2 94.2 2 123.3 3 120.5 3 154.4 4 107.4 4 139.4 1997 1 116.2 2003 1 151.6 2 104.4 2 133.7 3 131.7 3 154.5 4 117.9 4 135.1

Seasonal Regression Models Resul of regression analysis is:

Seasonal Regression Models Forecas resuls and acual values for 2004 are:

Seasonal Regression Models Use of muliplicaive models ofen provides a beer forecasing model when he ime series is changing a an increasing rae over ime. Muliplicaive model for Power Load problem would be: ln ( ) E Y Q Q Q 0 1 2 1 3 2 4 3 Taking anilogarihm of boh sides shows muliplicaive naure: Y e x p e x p e x p Q Q Q e x p 0 1 2 1 3 2 4 3 Consan Secular Seasonal Componen Residual rend Componen

Auocorrelaion and he Durbin- Wason Tes A residual paern as illusraed here suggess ha auocorrelaion may be an issue. Auocorrelaion is he correlaion beween ime series residuals a differen poins in ime. Correlaion beween neighboring residuals is called firs-order auocorrelaion

Auocorrelaion and he Durbin- Wason Tes Durbin-Wason d saisic is calculaed o es for he presence of firs-order auocorrelaion n 2 R R 1 2 d R a n g e o f d : 0 d 4 n 2 R 1 If residuals are uncorrelaed, hen d 2. If residuals are posiively auocorrelaed, hen d<2, and if auocorrelaion is very srong, d 0. If residuals are negaively auocorrelaed, hen d>2, and if auocorrelaion is very srong, d 4.

Auocorrelaion and he Durbin- Wason Tes One-Tailed Tes H 0 :No firs-order auocorrelaion H a :Posiive firs-order auocorrelaion (or H a :Negaive firs-order auocorrelaion) Two-Tailed Tes H 0 :No firs-order auocorrelaion H a :Posiive or negaive firs-order auocorrelaion Tes Saisic d n 2 R n 1 R R 2 1 2 Rejecion region: d< d L, [or (4-d)< d L, if H a :Negaive firs-order auocorrelaion] Where d L, is he lower abled value corresponding o k independen variables and n observaions. The corresponding upper value d U, defines a possibly significan region beween d L, and d U, Rejecion region: d< d L,/2 or (4-d)< d L,/2 Where d L,/2 is he lower abled value corresponding o k independen variables and n observaions. The corresponding upper value d U,/2 defines a possibly significan region beween d L,/2 and d U,/2

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