Reading 2: Time-Series Analysis

Page No. 141Date / /ReadingTime - series Analysis*Time Series AndysisFinancial data can be difficult to model using conventionalmultiple regression so what should you do ?Trend of Model:,- Linear trend model- Log- linear trend modelLagged models-Autoregressive model CAR)Time Series :Set of observations on a variables outcomes in differenttime periods used to explain the past and make predictionabout the future.Example:-HodQuarterly sales over the last five 45Monthly CPI over the last 12 yrDaily returns over the last 180 daysLinear Trend Model:Point : Regression with time as the independent variableYE = b. + bit + Ett=1 to nObserved valueTime index: One for the first observation(unemployment)two for the second Page No. 192Date//The predicted change in y is b,Lineartrendisatrendinwhichthe dependentvariablechanges at to constant rate with timeHas a straight lineUpward- sloopingline:-+vetrendDownward - slopingline:--VetrendExample:-Alineartrendmodelhasbo=1.70&b,=3Calculate Yt for t=1 & t=2YE=1.70+3Xt+ E,9,1.7b, represents the absoute change=t3XI= 47042=1.7JA=3in the dependent variable+3x2=7.70Log - Linear Trend ModelLog - Linearregressionassumesthedependentfinancialvariableexhibits @exponential growthYt = e b.4bit(4t) = b. + bitTaking the natural log of bothsides yieldsSlope Coefficient b is the Constant growth!Used for financial time SeriesHas a CuNeConvex Curve : +ve trendConcave Carve : -Ve trendgnDgmgift of out(trampolyment) 143Page No.Date//Example: Ananlystsmodelsquarterlyrevenuedata(inmillionsofus$)forJPNorthfieldusingalog-lineartrend model :In=4.0+0.09tEstrevenueforthe49thquater(t=44)In(499) = 4.0+0.09X49=8.41449 = e 8.41= 4492mLog - Linear Trend Models(Visual)GrowsatConstantlatewithcontinuousCommonding(exportentialgrowth)Linear Trend ModelLog-Linear Trend ModelYo = b. +bit + EtIn (4t) = b. + bat t Et.In (4+)t SerialcorelationtimetimeLimitation of trendmodelsisthattheyarenot useful if theerror termsareserialcorrelationCanbetestedwithDWtestSerialCorrelationinTrendModelsRegressioncrorsinCurrentperiodmustbeuncorrelatedwitherrorsfromallotherperiodsUseDurbinWatsontotestforserialcorelation Page No. 149Date / /Iflinearmodelexhibitsautocorrelation t trylog-linearif log-linearmodelsstillexhibitsautocorrelation t tryautosegreusReqforatimeseriestobeCovariancestationaryAtimeseriesisCovarianceStationaryifitsatisfiesthethree Conditionfollowing]Constantand2]ConstantfiniteExpectedvalueMean(MRL)and3]finitevariance ( same as homoskedesticity)ConstantandfiniteCovarianceoftime series with itself.Xt = b. + b.Xt -(with leading or lagged)= 5 + 0.5Xt-1 =6Xt = 8Xty20Xt =15Xt-1 =8Xt=9Xt-1-15XE=125Xt-l=9Xt-9.5Xt-1 = 12.5Xt =11-25XE-1=10Xt=10IfXt-1=10thenXt=10,Xt+1=10,Xttz=10&SoonThisCalled7Constant&finitemeanMeanofthetimeSeries=b.=5=101- b,1-0.52d Page No. lhsDate / /For amodel to the valid time series must be convaliance StationaryMosteco & financial time Series relationship are not stationaryThe model can be used if the degree of non stationarity isnot Significant*Autoregressive ModelsThe dependent variableis regressed on prior value of itselfA timeseriesregressiononitsownpastvaluesSimplest form : ARCI) model :Xt= b. + b, Xt-1 + Etlag variabl eOr more generally modelARCP) :Xt = bo t b, Xt-1 + bz Xt-2 t +bp Xt P + EtNote: A nonstationarytimesserieswillproducemeaninglessregression results.ForecastingwithanAR(1)ModelOne egAssumeweest the following AR(1) :Xt=1.2 t 0.45 Xt-1 + etQue: If Xo is 5 Calculate a two-stepaheadforecast:One -Step-ahead -forecastXL=1.2+0-45(5)=3.45Two- - step- aheadforecast :X2 = 1.2 -0.45 ( 3.45) = 2-75 test statis calculated as Correlation r-oDabin Watson Test4 = cosselat"Page No. 146Test stat = r- 0=45Tstd errorDate//-Can be use on simple1/VF(multiple lin regressionas well as trend modelHowever it cannot be elationinanARMalelused to test autocrieldfor AR Processdurbin Watsonto test for serial correlationIndividualtonresidual auto correlations with std elsorVEofseriescorreletionexistsmodelisincomplete.#tof ofSol" : Increase order of model by adding more lagged Variab(eg AR (2) from AR (1) or adj for seasonality)Autocorrelations of the error termsIf the error terms have sig serial correlation Cauto correction) themodel used is not the best model to analyze thetime seriesIntercept & slope using linear regressionPredicted valuesTest whether theautocorrelation of error termsError termsautocorrections are sigdiff from zerof the autocrelations are not statif the autocorrections are statsignificant diff from zero ofthe decisions is FTR) : Modelsigilicant diff from 300 C if thedecision is rejected : Model doesfits the time Seriesnot fit the time scriesTest used to Know ttest if the auto corrdations are geneta Sigdff from 0 ; iscalculatedasCorrelationr-otest statcosselat"Page No 146Test stat = r-4==15Tstd errorDate/1/VT/Serial Correlation in an AR MalelCannot usedurbin Watsonto test for serial correlationAR model.IndividualUse a t-test on residual auto correlations with std elrorVEIf Series correletion exists , t model is complete.Sol" : Increase order of model by adding more lagged Variable(eg ARC2) from AR (1) or adj for seasonality)Autocorrelations of the error termsIf the error terms have sig serial correlation Cauto correction) themodel used is not the best model to analyze thetime seriesIntercept & slope using linear regressionPredictedvaluesTest whether theautocorrelation of error termsError termsautocorrections are sigdiff from zeroofthe autocrrelations are not statif the autocorrections are statsignificant diff from zero (ifthe decisions is FTR) : ModelSigificant diff from 340 C if thedecision is rejected : Model doesfits the time seriesnot fit the time scriesTest used to Know if the auto correlations are general Sigdiff from 0 ; t fesst DW test was not15th Jan 2023 SunPage Nimade with assumt"Dateindependent variable & oft-stat ( of :n-2) = Auto correctionvalue & in ARdependent are sopiatemodel independent&Standard errordependent are theStandardyousame Dw fails.errorn = observationbien(e)lesTestingGrit x2forSerialCorrelationinanARmodelSEI0.1419LagAuto corre lationSEt-staton50AI0.18840.14191.3324K2o88420.33630.14142.3784>2014143- -0.12180.1414+0.8614<24- 0.02850.1414+0.2016 <2Rho1 = 0.1414n = 50HoP0ignor -veHa P toAssumingacriticaltivalueof(two 2.0 tail ) themodelisincorrectlyspecifiedToCorrecttryanAR(z)model(i-eaddalog)Atlay2RejectHeCorrelationisnot0SigautocoirelationARmodelExampleTheARC2)modelbetterthedataNocorrelationsXt =significant>betb,Xb2 X4 -2iMPsLagAutocorrationSEstatFTR Ho P=10I-0.01740.1429+0.1217 < 220.09570.14240.0701523-0.14720.1429+1.0301 ( $4-0.15200.1429-1.0641 Multiple R =JRPage No. 148Date / /IMPMean ReversionThe value of the dependent variables tends to fall whenabove its mean and rise when below its meanslopstable RootMean Reverting Level = b.b,<1RuleMRL(1-b)Wewouldhaveb,=1b,unstable mod elproblemifExplosure TootUnit RootofXt > meanrevertinglul,thenXt-1willbelowerthantoIfXt>meanrevertinglul,thenXEHwillbe+higherthanXtCalculatemean - revertinglv)timeseriesgivenb. = 82 137 & b, = - 223formanufacturingcapacityutilizationMRL =82.137= 67.16[1-(-0.223)b,Interceptation : y Current Capacity utilization is above 67.6S got HAit is expected to fall in the next Period i if it isbelow 67.16 it is expected to rise*Comparing Model AccuracySSEIn - Sample : DatausedtodevelopmodelnOut of Sample : Any data outside above range Page No. 149R MSESSGDate//Forecasting acculacy is measured by square root of themeansquared error (RMSE) MSE SEE JMSE = RMSEUsethemodelwiththelawestRMSEbasedonout - of sample forecastingerrorsRegression Coefficient InstabilityPoint : Estregressioncoefficientschange+fromonetimeperiodtoanother.Create a trade-off -betstatisticalreliability of long timeseries andstability of shorttime seriesHas eco process or environmentchanged ?Shorter Sample PeriodMore stabiditybutlessStatreliabilityLonger Sample periodLess stabalitybut morestatisticalreliabilityData must also beCovariance stationaryformodeltobe validRandom Walkswithout driftAtime Series inwhichpredictedvalueofadependentvariablein oneperiodisequaltothevalueofdependentvariable inpreviousperiodplusan error termbo=0XE = XE-1 + E6b= 1Xt =It1Xt-1+EEEt isXt - Xtyl Page No. 150Date//With draftA time series in which predicted value of a dependent Vacialin one period is equal to the value of dependent variablein previous period plus or mias a constant amt and anerror termbo toXt = be + Xty + EtdriftTwotypes:Nodrift&dift.-Definingcharactersitic : bi=lUnitUnit RootsRandom walk with driftIn anAR(1) modelunit rootcoefficient must be<1if Coefficient =1Unit rootsUnit root is defining feature of landom walksor decease over time (1990 Stock market)are common in Series than consistently0Not Covariance Statanot have finite mean0To use standardregression analysis, we must convert this datato called First differencing!Covariance stat This Conversion isWhy are Unit Roots ProblematicIfbi=1, then mean reverting lul is undefinedb.bobo1-b,1-10 Page No. 151Date//without a mean reverting lul t the time series is nonstalionaryQue: How do we test for a unit root ?aliableUnit foot test ofnon stationarityAutocorrectet. on Approchfautocorrelationsdonotexhibitthesecharacteristics , itissaidtobeanonstationarytimeseries:Autocorrectionsatalllagsarestatnotdifffrom0ORAstheno.oflagsincreasestheautocorrelationsdropsdowntotnon stationarily test Dickey-Fuller Test for Unit- frootBased on a transformed version of the AR(1) model:More definitive than autocorredation approach(-)0XE- from both sides:XE - Xtyb.+ (b-d)Xt-t+EtorXt - Xt-1=b.tJ,X+-+Etwhereg,=(b,-1)llfthereisanunitrootintheAR(1)modelthenbe0bi-1bl=1RODTERT(FgwillUnitTR)DFtest:He:gtimeserieshasaunitrootandisnonstatHa: g, 0 time series does not have a unit rootbirl STABLETest is conducted by calculating a t-stat for g, & usingarevisedSetofcriticalvaluescomputedbyDF.Usingrevisedsetof+-statstifthenullisrejectedthereisno evidence of a unit root. Page No. 152linear frentRNo MRLb=1?Date/First DifferencingUse first Differencing to remove unit roots / nonstationarityCreate a new dependent variable Y defined as the changein X:MRL=OYt = Xt - Xt-1&b. &b, =0Yt = bo t b, ( 4t-1) + Ety time ER Series is a random walk then we must convert thisdata to covariance stat This conversioniscalledfirstdiff*Sea sonalityAtime series Shows Consistent seasonal patternsObj : Incorporate seasonal component (Xt-4 in quarterly or Xtv inmonthly models )In the following lvl example an AR(1) model is applied to qroccupancyforaresorthotel chain.Pre dictquarterly occupancy /vl for a resort hotel chainusingthefellowingAR(1)modelbasedon10yrquarterly data (n 40) ARI & determine if Seasonality existsofInXE = be tb,In Xt-1+ EtR2 - 0.7929ConflicientsStandard Ecrert-statStand aid error - 0.1952Intercept0.03750.02742>1.369observations - 39lay0.53180.1635 2 3.25 26(terr Page No. 153Date//Does This Suggest Seasonality ?LagAutocorrection No Use SEt-statI0.06150.1601-0-38412-0-01210.1601-0-07563-0.02120.1601-0-1324240.87190.16015.4460Significant residual autocorrelationat lag 4 indicates seasonality"Seasonality Can be detected by plotting the values on a0 Seasonality is present if the autocorrection of error term isgraph or calculating autocorrectionig significantly diff from ZeroCorrecting SeasonalityGiven a significant by 4 autocorrelation (t-stat > approx 2)we opt for dn AR (1) model incorporatingaSeasonal log:-In tt = be + b, (In Xt-1) tbz (In Xt-W+ EtCorrection : Adding a log of dependent variable (corres pondingto the Same period in previous yr) to the modewindas another independent variable. Page No. 155Date//CoefficientStd errort-statR - 0.94898Intercept0.00441-73470.0085Std error - 0.37540-05274.9298lagl0.2598observation - 36lag40.79210.21663.6570LagAutocorrelationSEt-statNo sig residual automcorrection\-0-05260.1667-0.3156and improvement in R22007150.16670.4240indicate the new model is3-0.02410.1667-0.1446Correctly specified40.04350.1667-0-2610Forecasting with an AR modelGiven /vl for 2015 bo==0.0085 b, = =0.2598 & be =0.792occupancyforecast hotel occupanty for 2016QuaterOccupancyIn Xt = 0.0085 to 2598 (In 600,00) +20 F15 91250,00092750,0000.7921 ( In 250,000) 2 13.310393450,00013703ah600,000Xt = eF1691C771= 603,378.52Autoregressive Conditional Hete roskedasticity (ARCH) ModelsHeteroskedasticity occurs when variance of error is:X1 CroorVariame- Non constant ; &- Conditional on independent variable solRe-estimate the model willtoGeneralized least squaresPage No 155ARCMDate//Standard errors t-stats + & conclusions are incorrect If ARch existSolution:Usegeneralizedleastsquaresorothermethodstocorrect for heteroskedosticityARCHTotestwhetheratimeserviesisARCH (1)Squaredresiduals,Dareregressedonfirstlegofsquaredresiduals:2PredictedEta2=+azEt-1+MtError term of errorerror termInterreptofcurrentPeriod error termperiodslopeof last perioda sig, the time feries(2)is ARCH [1>a1Ifla,issignificant ,thetimeSeriesisARCH a)dis GitAlsoifa,issignificantfuturevarianceoferrorscanbeapredictedusing:Ttao+a2EEssentially , a velatility forecasting model0ARCH exists if the variance of error terms in one periodisldsidependent onthevarianceoferrortermsinprevious period0Testing : Squared errors from the model are regressed onthefirstlagofthesquaredresidualsExample: Determine if below results indicate ARCK (1) & if sopredict next period's variance given current periodsquared error of0.5625 Page No.156If R SignifichRejectDate//to bi = 0VariableaiSbit-statP-valueConstant5.90681.0863)5.4375<0.001lag 10.45150.095584.7238>2<0.001The high t-stat Clow p-value) on the lag \ codficient indicatea, is significant & so the time series is ARCH (1)Nxt period est variance of error is :Forcastederror tek TH = 5.9068 + 0.4515 X 0.5625 = 6.1608Multiple Time SeriesCan we use regression on time series constanting unit roots ?- Example : EstStock beta- Regress a time series of stock returns on a time seriesof market returns.Either or both time Series could have a unit rootHow reliable is the beta est ?CointegrationCointegration : Two time series are related to same macio variablesor follow same trendIf time Series are cointegrated , ellor , H term from regressingoneontheother is convariance stat & t. test are reliablePoint : If Cointegrated t model can be used Page No. 157Date//Howand /or cointegrationtime - servies variables should be analyzed for nonstationarity,to test whether the two time series have unit roots, a Dicky -Fuller test is usedPossible scenarios :2)1Both time series are Covariance stationary (linear regression)2Only the dependent variable time Series is covariance stationary(not use linear regression)3/Only the independent variable time series in Covariance stat(not use linear regression)4 Neither time series is covariance stat and the two seriesare not cointegrated (not use linear regression)5Neither time series is Covariance stat and the two seriesare cointegrated C use linear regression)Appropriate time -Series modd to analyze a given invest" problemUnderstand the invest problem you have and make a choice ofmodelIf you have decided to use a time - Series model plot the valuesto see whether the time series looks covariance stationary.Use a trend model, if there is no seosonality or structual shiftIf you find significant serial Correlation in the error terms,use a complex model such as AR modelIfto the data has Series corfelation, examine the data forstationarity before last running an AR modelIf you find significant serial correlation in the residuals,use an AR(2) model. Page No. 158Date / /Checkfor seasonalityTestwhethererrorterms have ARCHPerformtestsofmodel'sout -of - -sampleforecastingperformance(RMSE)4085for)tashaogabagrtsw237792 smit921791batargartion37.0forbrebeaterbalLabornleben boart