quantitative chapter10

8/19/2019 Quantitative Chapter10

1/27

TIME-SERIES ANALYSIS


2/27

BASIC TIME SERIES

Data on the outcome of a variable or variables in different time periods areknown as time-series data.

• Time-series data are prevalent in finance and can be particularl c!allen"in"

because t!e are li#el t$ vi$late t!e underlin" assumpti$ns $f linear

re"ressi$n%

- Residual err$rs are c$rrelated instead $f bein" unc$rrelated& leadin" t$

inc$nsistent c$efficient estimates%

- T!e mean and'$r variance $f t!e e(planat$r variables ma c!an"e $ver

time& leadin" t$ invalid re"ressi$n results%

• E(ample $f a basic time series #n$)n as an aut$re"ressive pr$cess*

+


3/27

TREN, ANALYSIS

The most basic form of time-series analysis examines trends that aresustained movements in the variable of interest in a specific direction.

• Trend analsis $ften ta#es $ne $f t)$ f$rms*

% Linear trend analsis& in )!ic! t!e dependent variable c!an"es at a

c$nstant rate $ver time%

- E(* if b0=3 and b1=.3& t!en t!e predicted value $f y after t!ree peri$ds is

+% L$"-linear trend analsis& in )!ic! t!e dependent variable c!an"es at an

e(p$nential rate $ver time $r c$nstant "r$)t! at a particular rate

- E(* if b0=.! and b1=1."& t!en t!e predicted value $f y after t!ree peri$ds is

.


4/27

LINEAR /R L/0-LINEAR1

• 2$) d$ )e decide bet)een linear

and l$"-linear trend m$dels1

- Is t!e estimated relati$ns!ip

persistentl ab$ve $r bel$) t!e

trend line1- Are t!e err$r terms c$rrelated1

- 3e can dia"n$se t!ese b

e(aminin" pl$ts $f t!e trend line&

t!e $bserved data& and t!e

residuals $ver time%

4


5/27

TREN, M/,ELS AN, SERIAL C/RRELATI/N

• Are t!e results $f $ur trend m$del estimati$n valid1

- Trend m$dels& b t!eir ver c$nstructi$n& are li#el t$ e(!ibit serial

c$rrelati$n%

- In t!e presence $f serial c$rrelati$n& $ur linear re"ressi$n estimates are

inc$nsistent and p$tentiall invalid%- 5se t!e ,urbin63ats$n test t$ establis! )!et!er t!ere is serial c$rrelati$n in

t!e estimated m$del%

- If s$& it ma be necessar t$ transf$rm $ur data $r use $t!er estimati$n

tec!ni7ues%

8


6/27

A5T/RE0RESSI9E TIME-SERIES M/,ELS

#bbreviated as #$% p& models' the p indicates how many la((ed values ofthe dependent variable are used and is known as the )order* of the model.

• Current values are a functi$n $f pri$r values%

• T!e :$rder; $f t!e AR


7/27


8/27

RESI,5AL A5T/C/RRELATI/N

+e can use the autocorrelation of the residuals from our estimated time-series model to assess model fit.

• T!e aut$c$rrelati$n bet)een $ne time-series $bservati$n and an$t!er $ne at

distance k in time is #n$)n as t!e k t! $rder aut$c$rrelati$n%

• A c$rrectl specified aut$re"ressive m$del )ill !ave residual aut$c$rrelati$ns

t!at d$ n$t differ si"nificantl fr$m er$%

• Testin" pr$cedure*

% Estimate t!e AR m$del and calculate t!e err$r terms


9/27

MEAN RE9ERSI/N

# series is mean revertin( if its values tend to fall when they are above themean and rise when they are below the mean.

• ?$r an AR


10/27

M5LTIERI/, ?/RECASTS

+e can use the chain rule of forecastin( to (ain multiperiod forecasts withan #$% p& model.

• C$nsider an AR


11/27

IN- AN, /5T-/?-SAMLE ?/RECASTIN0

n-sample forecast errors are simply the residuals from a fitted time series'whereas out-of-sample forecast errors are the difference between predicted

values from outside the sample period and the actual values once reali/ed.

• An in-sample f$recast uses t!e fitted m$del t$ $btain predicted values )it!in t!e time

peri$d used t$ estimate m$del parameters%

- An $ut-$f-sample f$recast uses t!e estimated m$del parameters t$ f$recast values$utside $f t!e time peri$d c$vered b t!e sample%

- In b$t! cases& t!e f$recast err$r is t!e difference bet)een t!e f$recast and t!e

realied value $f t!e variable%

- Ideall& )e )ill select m$dels based $n $ut-$f-sample f$recastin" err$r%

• M$del accurac is "enerall assessed b usin" t!e r$$t mean s7uared err$rcriteri$n%

- Calculate all t!e err$rs& s7uare t!em& calculate t!e avera"e& and t!en ta#e t!e

s7uare r$$t $f t!at avera"e%

- T!e m$del )it! t!e l$)est mean-s7uared err$r is ud"ed t!e m$st accurate%


12/27

C/E??ICIENT INSTABILITY

Time-series coefficient estimates can be unstable across time. #ccordin(ly'sample period selection becomes critical to estimatin( valuable models.

• T!is instabilit can als$ affect m$del estimati$n because c!an"es in t!e underlin" time-

series pr$cess can mean t!at different time-series m$dels )$r# better $ver different time

peri$ds%

- E(% A basic AR


13/27

RAN,/M 3ALHS

• An AR


14/27

5NIT R//TS

or an #$%1& time series to be covariance stationary' the absolute value ofthe b1 coefficient must be less than 1.

• 3!en t!e abs$lute value $f b1 is 1& t!e time series is said t$ !ave a unit r$$t%

- Because a rand$m )al# is defined as !avin" b G & all rand$m )al#s !ave a

unit r$$t%- 3e cann$t estimate a linear re"ressi$n and t!en test f$r b1 = 1 because t!e

estimati$n itself is invalid%

- Instead& )e c$nduct a ,ic#e6?uller test& )!ic! is available in m$st c$mm$n

statistics pac#a"es& t$ determine if )e !ave a unit r$$t%

4


15/27

5NIT R//TS AN, ESTIMATI/N

- 3e cann$t use linear re"ressi$n t$ estimate parameters f$r a series

c$ntainin" a unit r$$t )it!$ut transf$rmin" t!e data%

- :Differencin(; is t!e pr$cess )e use t$ transf$rm data )it! a unit r$$t it is

perf$rmed b subtractin" $ne value in t!e time series fr$m an$t!er%

- ,ifferencin" als$ !as an :$rder&; )!ic! is t!e number $f time units t!et)$ differenced variables lie apart in time%

- ?$r a rand$m )al#& )e first-difference t!e time series%

- # properly differenced random walk time series will be covariance

stationary with a mean-reversion level of /ero.

•

8


16/27

SM//T2IN0 M/,ELS

These models remove short-term fluctuations by smoothin( out a timeseries.

• An n-peri$d m$vin" avera"e is calculated as

• C$nsider t!e returns $n a "iven b$nd inde( as x 0 = 0.1 ' x -1 = 0.1"' x - = 0.13'

x -3 = 0..

- 3!at is t!e t!ree-peri$d m$vin"-avera"e return f$r $ne peri$d a"$


17/27

M/9IN0-A9ERA0E TIME-SERIES M/,ELS

• Called MA


18/27

,ETERMININ0 T2E /R,ER /? A MA


19/27

AR


20/27

SEAS/NALITY

Time series that show re(ular patterns of movement within a year acrossyears.

• Seas$nal la"s are m$st $ften included as a la""ed value $ne ear bef$re t!e

pri$r value%

- 3e detect suc! patterns t!r$u"! t!e aut$c$rrelati$ns in t!e data%

- ?$r 7uarterl data& t!e f$urt! aut$c$rrelati$n )ill n$t be statisticall er$ if

t!ere is 7uarterl seas$nalit%

- ?$r m$nt!l& t!e +t!& and s$ $n%

- T$ c$rrect f$r seas$nalit& )e can include an additi$nal la""ed term t$

capture t!e seas$nalit%- ?$r 7uarterl data& )e )$uld include a pri$r ear 7uarterl seas$nal la" as

+F


21/27

?/RECASTIN0 3IT2 SEAS/NAL LA0S

• Recall $ur AR


22/27

A5T/RE0RESSI9E M/9IN0-A9ERA0E M/,ELS

t is possible for a time series to have both #$ and 2# processes in it'leadin( to a class of models known as #$2# % p,q & models %and beyond&.

• Alt!$u"! it is an attractive pr$p$siti$n& usin" an ARMA


23/27


24/27

RE,ICTIN0 9ARIANCE

If a series is an ARC2


25/27

C/INTE0RATI/N

Two time series are cointe(rated when they have a financial or economicrelationship that prevents them from diver(in( without bound in the lon( run.

3e )ill $ften f$rmulate m$dels t!at include m$re t!an $ne time series%

- f any time series in a re(ression contains a unit root' the ordinary least

s4uares estimates may be invalid.

- If b$t! time series !ave a unit r$$t and t!e are c$inte"rated& t!e err$r term

)ill be stati$nar and )e can pr$ceed )it! cauti$n t$ estimate t!e

relati$ns!ip via $rdinar least s7uares and c$nduct valid !p$t!esis tests%

- T!e cauti$n arises because t!e re"ressi$n c$efficients represent t!e l$n"-

term relati$ns!ip bet)een t!e variables and ma n$t be useful f$r s!$rt-

term f$recasts%

- 3e can test f$r c$inte"rati$n usin" eit!er an En"le60ran"er $r ,ic#e6

?uller test%

+8


26/27

SELECTIN0 AN AR/RIATE TIME-SERIES M/,EL

ocus 5n6 $e(ression 5utput

• Y$u are m$delin" t!e rate $f

"r$)t! in t!e m$ne suppl $f a

devel$pin" c$untr usin" FF

ears $f annual data% Y$u !ave

estimated an AR


27/27

S5MMARY

M$st financial data are sampled $ver time and& acc$rdin"l& can be m$deled

usin" a special class $f estimati$ns #n$)n as time-series m$dels%

- Time-series m$dels in )!ic! t!e value in a "iven peri$d depends $n values in

pri$r peri$ds& are #n$)n as aut$re"ressive& $r AR m$dels%

- Time-series m$dels in )!ic! t!e value in a "iven peri$d depends $n t!e err$rvalues fr$m pri$r peri$ds& are #n$)n as m$vin" avera"e& $r MA m$dels%

- M$dels )!$se err$r variance c!an"es as a functi$n $f t!e independent

variable are #n$)n as c$nditi$nal !eter$s#edastic m$dels%

- ?$r an AR dependenc& t!ese are #n$)n as ARC2 m$dels%

+@

quantitative chapter10

Documents