15.1 Stationary vs. Non-stationary time series.

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1 More on AR models and some imporan properies (revised 7/8) 5. Muliple period ahead forecass 5. Muliple period ahead predicion inervals 5.3 Trending series: he rend saionary model 5.4 Forecas errors of he rend saionary model 5.5 Tess for random wal 5.6 Seasonal models 5. Saionary vs. Non-saionary ime series. One imporan aspec of a ime series model is wheher i is mean revering. Tha is, can we expec ha over he long run he process will end o an average value? Saionary processes will end o rever o heir mean value and a non-saionary process will no. Obviously, he models have very differen implicaions abou he long run behavior of a process.

2 Here are 4 simulaions from he model AR() model wih β =. Each series begins a he value 5 and has observaions. Y = + Y + ε Someimes he series wander up, someimes down, someimes up hen down. Where do you hin he process will be in period? There is no force driving he series bac o a mean value.

3 Here are 4 simulaions from he AR() model wih β=.5. Each series begins a he value 5. Y = +.5Y + ε Conrary o he case when β =, when β =.5 he series is araced bac o is mean value of µ = β = β.5 = 5 Clearly, in period, we would expec he process o be somewhere around 5. A lile above, or a lile below. 3

4 Le s loo more closely a where we should expec an AR() model o be in he fuure and how sure we are. To eep he noaion simple, our discussion will use ime as he poin ha we sar looing a fuure values of Y. Laer I ll show you he formula for he general case where we forecas ou periods ino he fuure, saring a an arbirary ime. Y Y Y = β + βy + ε = β + βy + ε ( Y ) = β+ β β + β + ε + ε = β Y + + β β + βε + ε = β + βy + ε Y ( Y ) = β+ β + β β + β + βε + ε + ε3 = β Y + + β + β β + β ε + βε + ε 3 3 Y 4

5 Y = + β + β β + β Y + β ε + βε + ε You can see where his is going. For an arbirary forecas of > periods we can wrie ( ) Y = β Y + + β + + β β + β ε βε + ε and, if β < hen his can be wrien as: Y ( β ) β j = βy + + βε j ( β ) j= Y ( β ) j = βy + β + βε j j= j = βy + ( β ) µ + βε j j= This suff is nown a ime his is (weighed) sum of inervening (fuure) values of ε (The las line comes from µ = ) ( β ) β This is a useful express ha we can use o undersand forecass. β 5

6 Le s find he -sep ahead forecas ha we mae a ime zero. j E ( Y Y) =E βy + ( β ) µ + βε j Y j= j = βy + ( β ) µ + E βε j Y j= ( j ) j= ( ) = β Y + β µ + β E ε Y j = β Y + β µ This expression ells us how o forecas ou muliple periods given an iniial value of Y=Y. Clearly here is nohing special abou saring he forecas a ime period. Here s wha he formula loos lie for a -period ahead forecas made a an arbirary ime. Le Y = E( Y+ Y) denoe he forecas of Y + given Y. Then Y ( ) β Y + β µ = β Y + β µ 6

7 Case : β < If β < hen he -period ahead forecas is given by: Y = β Y + β µ This maes he naure of he forecas perfecly clear. I says ha he forecas is a weighed average of he las value Y and he mean of Y. The furher ou we forecas he more weigh we pu on µ and he less weigh on Y. Now les see wha happens when β = From before, we have: ( ) Y = β Y + + β + + β β + β ε βε + ε so wih β = we ge: Y = Y + β + ε ε + ε = Y + β+ ε j j= nown a ime Inervening (fuure) values of ε 7

8 So, now he forecas is given by: E( Y Y) = E Y + β+ j= ( ε j ) = Y + β + E Y j= = Y + β ε j More generally, when β =, for an arbirary saring poin, and forecas horizon Y = Y + β So as ges large here, he forecas doesn converge o a mean µ. If β < he forecas becomes very negaive. If β > he forecas becomes very large. If β = he forecas is simply Y. 8

9 5. Forecas errors. How big will our forecas errors be? To answer his quesion we wan o now he difference beween he acual value of Y + and he forecased value. We again use Y denoe he = E( Y+ Y) forecas of Y + given Y. Using his noaion, we are ineresed in he error associaed wih he -sep ahead forecas error e = Y Y ( + ) In his secion we find he variance of for. e Jus lie in he one-sep ahead forecas, his ells us abou our uncerainy. 9

10 Le s sar wih case i.e. β <. We can wrie Y + as j f j + = β + β µ + βε+ j= + βε+ j j= j= Y again, his is (weighed) sum of inervening values of ε j e = ( Y+ Y ) = βε + j j= Y Y Y Les find he forecas error variance: j Var ( e ) = Var βε + j j= bu he ε are iid. So ( β j j ) Var ( e ) = βvar ( ε+ j) = σ β = σ β For large his becomes Var e j= j= σ = β ( )

11 Does his mae sense? The size of he forecas error variance is deermined by σ and β and he forecas horizon. The size of he surprise in each period is deermined by σ. The larger β is, he larger he swings are ha Y can ae away from he mean. The forecas error variance is increasing as we forecas furher ino he fuure. The forecas error variance converges o a fixed number as becomes large. The fixed number is jus he variance of Y. Case i.e. β = Here so Y= Y + β + ε+ j = Y + ε+ j j= j= Y = + = + j j= e e Y Y ε The variance of is hen: Var ( e ) = Var ε+ j = j= σ This is compleely differen from he case where β <. The forecas error variance is proporional o he forecas horizon,. The variance only depends on, no.

12 Of course, in pracice we don now he rue parameers so we plug in our bes guesses: The -sep ahead forecas when β <is given by: ( b ) b ˆ Y = by + = by + ( b ) y ( b ) where is he sample average of Y. y The bes guess for he -sep ahead forecas error variance when β < is given by: ( b ) ( ˆ EY+ Y ) s b ( ) Hence when β < he 95% predicion inerval for is given by: ˆ Y ± ( b ) ( b ) s Y +

13 For he case where β = we have Var ( e ) = Var ε+ j = σ j= So he 95% predicion inerval for he sep ahead forecas is given by: Y ± s (remember ha Y = Y ) Forecas for β =.8 and β =.5 σ=

14 Forecas for β = and β = σ= Saionary vs. non-saionary summary Forecass Y Saionary models β < Mean rever = β Y + β µ Non-saionary models β = Trend up or down depending on sign of β Y = Y + β Forecas errors Iniially increase wih he forecas horizon. Var e = ( β ) ( β ) σ Increases wih he forecas horizon. Var e ( ) = σ 4

15 5.3 Trending series: he rend saionary models Someimes ime series can rend up or down, bu no in he same way as a random wal. If a process is saionary afer removing a rend hen i is called a rend saionary process. Hence is a rend saionary process if: Where Y ( β δ ) Y + = Y consan + rend Y is a mean zero saionary process. Hence he rend saionary model says ha he deviaions of Y from he rend line have mean zero and, mean rever. 5

16 Trend line rend β δ I drew his wih δ>, bu d could also be less han zero which means he series rends down. Here s a simulaion from he model Y = Y

17 Here s he rend line rend (.5.5 ) = Here s (Y -rend), a saionary AR() Y rend = Y ( )

18 Forecasing a rend saionary model The rend saionary model says ha he deviaions of Y from a rend line follow an saionary model. Le Y ~ denoe he - period ahead forecas of Y If Y follows an AR() we now he -sep ahead forecas is given by: Y = β Y + β µ = β Y (µ=, since β = righ?) 8

19 Y is he -sep ahead forecas of he deviaion of Y from he rend line. Tha is, i s he expecaion of Y -rend. Hence, o ge he forecas of Y we simply add bac in he rend line a ime period + o our forecas of he deviaion Y. ( ) β δ E Y Y Y + = Expeced rend line deviaion For he rend saionary AR() model: Y = β Y + β + δ + The forecas of a rend saionary model is composed of he forecas of he deviaion of Y from he rend plus he rend line. The long run forecas of Y does no converge o µ, bu raher o he rend line (so his is a non-saionary model). Hence in he long run our forecas jus revers o he rend line! 9

20 5.4 Wha are he forecas errors? = Y ( + ) ( Y β δ( ) ) ( Y + β δ( ) ) e = Y Y = Y This is jus he forecas error associaed wih he deviaion of Y + from he rend, i.e. he forecas error associaed wih he AR par! We already now wha hese loo lie: ( β ) Var ( e ) = σ β ( ) (.5.5 ) where.8 Y + = Y Y = Y + ε

21 Trend saionary summary Forecass converge o he rend line. Forecas errors do no diverge, bu behave lie a saionary model wih a variance ha increases iniially hen reaches a fixed value. Lie he random wal model, he rend saionary model does no mean rever o a fixed number (mean). Unlie he random wal model, he rend saionary model forecas errors do no increase wihou bound. Esimaion In a firs sep, esimae he rend line by running he regression: Y = β + δ+ ε Nex, creae he series Y = Y b d where b and d are he esimaes of he inercep and he rend slope respecively. Nex, model he de-rended series wih an ARMA model. ~

22 5.5 Tess for a random wal We saw ha here is a big difference beween he properies of a saionary β < and non-saionary β = model. Someimes i is difficul o ell he difference in a given sample beween he wo. One plo is β = and he oher β =

23 Clearly, we are ineresed in running he regression Y = β + βy and esing he null ha β =. Why can we es he null of saionariy? The -sa from his regression doesn follow a -disribuion. Neiher he mean or variance of Y exiss! Forunaely some smar guys in he 7 s and 8 s figured ou how o do he es correcly. They figured ou he correc p-values for he usual -es. The criical value is no and depends on he sample size. The acual disribuion is called he Dicey-Fuller disribuion. Forunaely, mos sofware does his es for us. 3

24 Here is he es for series P-value Since he p-value is small we can rejec he null of β = a he 6% level Uni roo es for series P-value Since he p-value is large we don rejec he null of β = 4

25 5.6 Seasonal Models. Many ime-series daa exhibi some sor of seasonaliy (e.g., he beer producion daa, January effec in soc reurns). Include indicaor variables in he model o conrol for seasonaliy. Example: Domesic Beer Producion Regression of beer producion on monh indicaor variables: MTB > regr 'Beer' 'Jan'-'Nov' Predicor Coef SDev T P Consan Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov December is he excluded caegory. S =.5887 R-Sq = 9.% R-Sq(adj) = 9.5% 5

26 MTB > le = /sqr(7) MTB > prin K.357 MTB > acf 'RESID' ACF of RESID XXXX -.66 XXX 3.6 XXX XXXXXXX XXXXXX XXX XXXXX X 9.33 XXXXXXXXX.59 XXXXX.95 XXX.4 XXXX XXXX XX XXXXX XXXXXXXX 7.6 XX 8.9 XXX This acf loos a lo beer han he one we go from he AR() model residuals. Le s include lagged beer producion as well: MTB > regr 'Beer' 'Beer(-)' 'Jan'-'Nov' Predicor Coef SDev T P Consan Beer( Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov S =.5949 R-Sq = 9.% R-Sq(adj) = 9.4% Pas producion doesn seem o be significan once seasonaliy is conrolled for. 6

27 The acf loos prey much he same: MTB > acf 'RESID' ACF of RESID X -.89 XXX 3.97 XXX XXXXXX XXXXXX XX XXXXX XX 9.35 XXXXXXXXX.5 XXXX.67 XXX.3 XXXX XXXX 4 -. X XXXX XXXXXXXX 7.48 XX 8.79 XXX 7