ECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates)

Transcription

1 ECON 48 / WH Hong Time Series Daa Analysis. The Naure of Time Series Daa Example of ime series daa (inflaion and unemploymen raes)

2 ECON 48 / WH Hong Time Series Daa Analysis The naure of ime series daa Temporal ordering of observaion (indexed by ) => Canno be arbirarily reordered. Typical feaures: Serial correlaion/nonindependence of observaion. => The pas can affec he fuure. How should we hink abou he randomness in ime series daa? The oucome of economic variables (e.g. GDP, Dow Jones, sock price) is uncerain in a sense ha hey are no foreknown; hus, hey can be viewed as random variables. Randomness does no come from sampling from a populaion unlike cross-secional daa. Time series are sequences of random variables, called sochasic processes (a synonym for random.)

3 ECON 48 / WH Hong Time Series Daa Analysis. Finie Sample Properies of OLS under Classical Assumpions Assumpions for imes series analysis TS.. Linear in parameers: { x,..., xk, y :,..., T} The sochasic process ( ) where y x x u = β0 + β βk k + u TS.. No perfec collineariy: is sequence of errors or disurbance = follows he linear model of: In he sample, no independen variable is consan nor a perfec linear combinaion of ohers. TS.3. Zero condiional mean: E( u X ) = 0 The mean value of he unobserved facors is unrelaed o he values of he explanaory variables in all periods. TS.4. Homoskedasiciy: var( u X ) = var( u ) = σ TS.5. No serial correlaion: ( s ) TS.6. Normaliy: ~ ( 0, ) Corr u, u X = 0 for s u N σ independen of X 3

4 ECON 48 / WH Hong Time Series Daa Analysis More on assumpion TS.3 Denoe X ( x x ) x x =,.., k. Then is -h row of he marix: x x x x x x k = k x x x T Tk Exogeneiy: E( u x ) = 0 The mean of he error erm is unrelaed o he explanaory variables of he same period. Sric Exogeneiy: E( u X ) = 0 The mean of he error erm is unrelaed o he values of he explanaory variables of all periods. Sric exogeneiy is sronger han conemporary exogeneiy TS.3 rules ou feedback from he dependen variable on fuure values of he explanaory variables; his is ofen quesionable especially if explanaory variables adjus o pas changes in he dependen variable. (Exclude he case where pas affec fuure y x + ) 4

5 ECON 48 / WH Hong Time Series Daa Analysis Theorem 0. (Unbiasedness of OLS) Under Assumpion TS. hrough TS.3, he OLS esimaors ˆ β j is unbiased, i.e., ( ˆ j ) E β where = β j T ( ) rˆ rˆ y rˆ y j j j ˆ = = β j = = T T ( rˆ ˆ ) ˆ j rj rj = = T and rˆj is he residual from he regression of x j on oher independen variables. The proof is essenially he same as ha for muliple regression model. (so skipped) Noe ha for unbiasedness, we only need Assumpion TS. hrough TS.3 5

6 ECON 48 / WH Hong Time Series Daa Analysis Discussion on Assumpion TS.4 (Homoskedasiciy) var( u X) var ( u ) σ = = ; The volailiy of he errors mus no be relaed o he explanaory variables in any of he periods. A sufficien condiion is ha he volailiy of he error is independen of he explanaory variables and ha i is consan over ime. Discussion on Assumpion TS.5 (No serial correlaion) ( s ) Corr u, u X = 0 for s; Condiional on he explanaory variables, he unobserved facors mus no be correlaed over ime. Why was such an assumpion no made in he cross-secional case? In he cross-secional case, random sampling is assumed. Therefore, i naurally indicaed ha errors are independen of each oher, and hus cause no correlaion beween errors. This assumpion may easily be violaed if, condiional on knowing he value of independen variables, omied facors are correlaed over ime. 6

7 ECON 48 / WH Hong Time Series Daa Analysis Theorem 0. (OLS sampling variances) Under Assumpion TS. hrough TS.5, he sampling variance of OLS esimaor, ˆ β j, is: var ˆ σ ( β j ) = SSTj ( R j ) X for j =,..., k, where SST = ( x x ) T j j j and = R j is he R-squared from he regression of xj on oher independen variables. As in he unbiasedness, he proof is essenially he same as ha for muliple regression model. (so skipped) Theorem 0.3 (Unbiased esimaion of he error variance) I can be shown ha E ( ˆ σ ) =. = σ where T ˆ σ uˆ n k = Gauss-Markov Theorem Under Assumpion TS. hrough TS.5, he OLS esimaors have he minimal variance of all linear unbiased esimaors of he regression coefficiens. 7

8 ECON 48 / WH Hong Time Series Daa Analysis Inference under he Classical Linear Model Assumpion Classical Linear Model Assumpions are Assumpion TS. hrough TS.6. Assumpion TS.6 (Normaliy): ~ ( 0, ) u N σ independen of X. Theorem 0.5 (Normal sampling disribuions) Under assumpions TS. hrough TS.6, he OLS esimaors have he usual normal disribuion (condiional on X). This heorem makes he usual - and F-ess valid. Therefore, he inference will be he same as ha of he cross-secional case. 8

9 ECON 48 / WH Hong Time Series Daa Analysis 3. Examples of Time Series Regression Models Saic models In saic imes series models, he curren value of one variable is modeled as he resul of he curren values of explanaory variables. (Example) Phillips curve inf = β + β unem + u 0 There is a conemporaneous relaionship beween unemploymen and inflaion. (Example) Deerminans of murder rae murdre = β + β convre + β unem + β yngmle + u 0 3 The curren murder rae is deermined by he curren convicion rae, unemploymen rae, and fracion of young males in he populaion 9

10 ECON 48 / WH Hong Time Series Daa Analysis Finie disribuion lag models (general case) In finie disribued lag models, he explanaory variables are allowed o influence he dependen variable wih a ime lag Example for a finie disribued lag model The feriliy rae may depend on he ax value of a child, bu for biological and behavioral reasons, he effec may have a lag gf = α0 + δ0pe + δpe + δpe + u gfr : Children born per,000 women in year pe : Tax exempion in year 0

11 ECON 48 / WH Hong Time Series Daa Analysis Inerpreaion of he effecs in finie disribued lag models y = z qz q + u α δ δ Effec of a ransiory shock: y z s = δ s If here is one ime shock in a pas period, he dependen variable will change emporarily by he amoun indicaed by he coefficien of he corresponding lag y y Effec of a permanen shock: = δ δq z z q If here is a permanen shock in a pas period, i.e., he explanaory variable permanenly increase by one uni, he effec on he dependen variable will be he cumulaed effec of all relevan lags. This is a long-run effec on he dependen variable.

12 ECON 48 / WH Hong Time Series Daa Analysis 4. Trend and Seasonaliy () Trend A linear ime rend y α α e y = α = + + <=> ( ) ( ) 0 E Δ y = E y y = α Absracing from random deviaions, he dependen variable increases by a consan amoun per ime uni.

13 ECON 48 / WH Hong Time Series Daa Analysis An exponenial ime rend => Exhibis a consan growh rae. ( ) = <=> log( ) log y α α e ( y y) = α ( Δ ) = E y α Absracing from random deviaions, he dependen variable increases by a consan percenage per ime uni. 3

14 ECON 48 / WH Hong Time Series Daa Analysis Problem of using rending variables in regression analysis If rending variables are regressed on each oher, a spurious relaionship may arise if he variables are driven by a common rend. In his case, i is imporan o include a rend in he regression. (Example) Housing invesmen and prices Esimaion wihou a rend log ( invpc) = log( price) (0.043) (0.38) n = 4, R = 0.08, R = 0.89 invpc : Per capia housing invesmen; price: housing price index I looks as if invesmen and prices are posiively relaed. (Do high prices cause high invesmen??) Esimaion wih a rend ( ) ( ) log invpc = log price n = 4, R = (0.36) (0.679) (0.0035) 0.34, R = There is no significan relaionship beween price and invesmen anymore. 4

15 ECON 48 / WH Hong Time Series Daa Analysis When should a rend be included? If he dependen variable displays an obvious rending behavior If boh he dependen and some independen variables have rends If only some of he independen variables have rend; heir effec on he dependen variable may only be visible afer a rend has been subraced. A derending inerpreaion of regression wih a ime rend I urns ou ha he OLS coefficiens in a regression included a rend are he same as he coefficiens in a regression wihou a rend bu where all he variables have been derended before he regression. 5

16 ECON 48 / WH Hong () Seasonaliy A simple mehod is o includes a se of seasonal dummies: y = β + δ feb + δ mar δ dec + β x β x + u 0 k k Time Series Daa Analysis Similar remarks apply as in he case of a deerminisic rend. Ignoring seasonaliy migh cause spurious relaionships beween variables. The regression coefficien above can be seen as he resul of firs deseasonalizing he dependen and he explanaory varaibles. There are numerous mehods ha adjus seasonaliy on variables. Bu ou of he scope of his course. 6