CFA 2018 Quantitative Analysis Question Bank 04 Time Series Analysis

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Test ID: 7440367

Time-Series Analysis

Question #1 of 106

Question ID: 461854

ᅞ A) ᅞ B) ᅚ C)

Questions #2-7 of 106

Thetable below showstheautocorrelationsofthelagged residualsforthefirst differencesofthenaturallogarithmof quarterly motorcyclesalesthat werefittotheAR(1)model: (lnsales − lnsales ) = b + b (lnsales − lnsales ) + ε . The critical

t-statistic at 5% significanceis2.0, which meansthatthereissignificantautocorrelationforthelag-4residual, indicating the presenceofseasonality. Assuming thetimeseriesis covariancestationary, which ofthefollowing modelsismostlikelyto

CORRECTforthisapparentseasonality?

L

agged

Autocorrelations

of

F

irst

Differences

in

the

L

og

of

M

otorcycle

Sales

L

ag

Autocorrelation

Standard

Error

t-Statistic

1 −

0 .

073

8

0 .

1667

−

0 .

44271

2 −

0 .

1047

0 .

1667

−

0 .

62

8

07

3 −

0 .

02

5

2

0 .

1667

−

0 .

1

5

117

4

0 .55

2

8

0 .

1667

3 .

31614

(ln sales − ln sales ) =b +b (ln sales − ln sales ) +ε .

lnsales = b + b (lnsales ) − b (lnsales ) + ε .

(lnsales − lnsales ) = b + b (lnsales − lnsales ) + b (lnsales − lnsales

) + ε .

Explanation

Seasonality is taken into account inanautoregressive model by adding a seasonal lag variable that corresponds to the seasonality. In the

case ofafirst-differenced quarterly time series, the seasonal lag variable is the first difference for the fourth time period. Recognizing that

the model is fit to the first differences of the natural logarithm of the time series, the seasonal adjustment variable is (ln sales − ln

sales ).

DiemLeisanalyzing thefinancialstatementsofMcDowellManufacturing. He hasmodeled thetimeseriesofMcDowell's gross

marginoverthelast15 years. Theoutputisshown below. Assume 5% significancelevelforallstatisticaltests.

Autoregressive

M

odel

G

ross

M

argin

-

M

cDowell

M

anufacturing

Quarterly

Data:

1 Quarter

1

9

85 to

4 Quarter

2000

t t − 1 0 1 t − 1 t − 2 t

t t −4 0 1 t −1 t −2 t

t 0 1 t − 1 2 t − 4 t

t t − 1 0 1 t − 1 t − 2 2 t − 4 t

− 5 t

t − 4 t − 5

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Question #2 of 106

QuestionID:461796

Regression

Statistics

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260

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0 .

220

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0 .

200

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Thismodelisbest described as:

th

st

nd

rd

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Question #

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QuestionID:461799

ᅚ A) ᅞ B) ᅞ C)

Question #6 of 106

QuestionID:461800

Question #7 of 106

QuestionID:461801

Question #

8 of 106

QuestionID:461864

Noneofthesearestaticallysignificant, so we can concludethatthereisnoevidenceofautocorrelationintheresiduals, and

thereforetheAR modelis properlyspecified. (StudySession3, LOS13.d)

Whatisthe95% confidenceintervalforthe grossmargininthefirst quarterof2004? 0.158 to 0.354.

0.168 to0.240. 0.197to0.305.

Explanation

Theforecastforthefollowing quarteris0.155 + 0.240(0.240) + 0.168(0.260) = 0.256. Sincethestandard erroris0.049and the corresponding t-statistic is2, we can be95% confidentthatthe grossmargin will be within0.256-2 × (0.049)and 0.256 + 2 × (0.049)or0.158 to0.354. (StudySession3, LOS11.h)

With respectto heteroskedasticityinthemodel, we can definitivelysay: nothing.

heteroskedasticityisnota problem becausetheDW statistic isnotsignificant.

anARCH processexists becausetheautocorrelation coefficientsoftheresiduals have differentsigns.

Explanation

Noneoftheinformationinthe problem providesinformation concerning heteroskedasticity. Notethat heteroskedasticityoccurs whenthe varianceoftheerrortermsisnot constant. When heteroskedasticityis presentinatimeseries, theresidualsappear to comefrom different distributions (modelseemstofit betterinsometime periodsthanothers). (StudySession3, LOS12.k)

Using the provided information, theforecastforthe2nd quarterof2004is: 0.253.

0.250. 0.192.

Explanation

To getthe2nd quarterforecast, weusetheone period forecastforthe1st quarterof2004, which is0.155 + 0.240(0.240) + 0.168(0.260) = 0.256. The4th lag forthe2nd quarteris0.22. Thustheforecastforthe2nd quarteris0.155 + 0.240(0.256) + 0.168(0.220) = 0.253. (StudySession3, LOS12.e)

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ᅞ B) ᅞ C)

Question #12 of 106

QuestionID:461822

Question #13 of 106

QuestionID:461784

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ᅞ C)

Question #14 of 106

QuestionID:461817

can be used to test for a unit root,which exists if the slope coefficient equals one.

cannot beused totestforaunitroot.

can beused totestforaunitroot, which existsiftheslope coefficientislessthanone.

Explanation

Ifyouestimatethefollowing model x = b + b × x + e and get b = 1, thenthe process hasaunitrootand isnonstationary.

The primary concern when deciding uponatimeseriessample period is which ofthefollowing factors? Current underlying economic and market conditions.

Thetotalnumberofobservations.

Thelength ofthesampletime period.

Explanation

There willalways beatradeoff betweentheincreasestatisticalreliabilityofalongertime period and theincreased stabilityof

estimated regression coefficients with shortertime periods. Therefore, theunderlying economic environmentshould bethe deciding factor whenselecting atimeseriessample period.

Rhonda Wilson, CFA, isanalyzing sales datafortheTUV Corp, a currentequity holding in her portfolio. Sheobservesthat

salesforTUV Corp. have grownatasteadilyincreasing rateoverthe pasttenyears duetothesuccessfulintroductionof

somenew products. WilsonanticipatesthatTUV will continuethis patternofsuccess. Which ofthefollowing modelsismost appropriatein heranalysisofsalesforTUV Corp.?

A log-linear trend model,because the data series exhibits a predictable, exponential growth trend.

Alineartrend model, becausethe dataseriesisequally distributed aboveand below

thelineand themeanis constant.

Alog-lineartrend model, becausethe dataseries can be graphed using astraight, upward-sloping line.

Explanation

Thelog-lineartrend modelisthe preferred method fora dataseriesthatexhibitsatrend orfor which theresidualsare predictable. Inthisexample, sales grew atanexponential, orincreasing rate, ratherthanasteadyrate.

Supposethatthetimeseries designated as Y ismeanreverting. If Y = 0.2 + 0.6 Y , the best predictionof Y is:

t 0 1 t-1 t 1

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Questions #1

8 -23 of 106

Explanation

Using the chain-ruleofforecasting,

Forecasted x = −6.0 + 1.1(22) + 0.3(20) = 24.2. Forecasted x = −6.0 + 1.1(24.2) + 0.3(22) = 27.22.

Housing industryanalyst ElaineSmith has beenassigned thetaskofforecasting housing foreclosures. Specifically, Smith is

asked toforecastthe percentageofoutstanding mortgagesthat will beforeclosed uponinthe coming quarter. Smith decides

toemploymultiplelinearregressionand timeseriesanalysis.

Besides constructing aforecastfortheforeclosure percentage, Smith wantstoaddressthefollowing two questions: Research Question

1:

Istheforeclosure percentagesignificantlyaffected byshort-terminterest rates?

Research Question 2:

Istheforeclosure percentagesignificantlyaffected by government

intervention policies?

Smith contendsthatadjustableratemortgagesoftenareused by higherrisk borrowersand thattheir homesareat higherrisk

offoreclosure. Therefore, Smith decidestouseshort-terminterestratesasoneoftheindependent variablestotest Research Question1.

Tomeasuretheeffectsof governmentinterventionin Research Question2, Smith usesa dummy variablethatequals1 whenevertheFederal governmentintervened with afiscal policystimulus packagethatexceeded 2% oftheannualGross

Domestic Product. Smith setsthe dummy variableequalto1forfour quartersstarting with the quarterin which the policyis enacted and extending through thefollowing 3 quarters. Otherwise, the dummy variableequals zero.

Smith uses quarterly dataoverthe past 5 yearsto derive herregression. Smith'sregressionequationis provided in Exhibit1: Exhibit 1: Foreclosure Share Regression Equation

foreclosureshare = b + b (ΔINT) + b (STIM) + b (CRISIS) + ε

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TheresultsofSmith'sregressionare provided in Exhibit2: Exhibit 2: Foreclosure Share Regression Results

Variable Coefficient t-statistic

51 52

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Intercept 3.00 2.40

ΔINT 1.00 2.22

STIM -2.50 -2.10 CRISIS 4.00 2.35

TheANOVAresultsfromSmith'sregressionare provided in Exhibit3: Exhibit 3: Foreclosure Share Regression Equation ANOVA Table

Source Degreesof

Freedom SumofSquares MeanSumofSquares

Regression 3 15 5.0000

Error 16 5 0.3125

Total 19 20

Smith expressesthefollowing concernsabouttheteststatistics derived in herregression: Concern1:Ifmyregressionerrorsexhibit conditional heteroskedasticity, myt-statistics will be

underestimated.

Concern2:Ifmyindependent variablesare correlated with each other, myF-statistic will be overestimated.

Before completing heranalysis, Smith runsaregressionofthe changesinforeclosureshareonitslagged value. Thefollowing regressionresultsand autocorrelations were derived using quarterly dataoverthe past 5 years (Exhibits4and 5,

respectively):

Exhibit4. Lagged Regression Results

Δ foreclosureshare = 0.05 + 0.25(Δ foreclosureshare ) Exhibit 5. Autocorrelation Analysis

Lag Autocorrelation t-statistic

1 0.05 0.22

2 -0.35 -1.53

3 0.25 1.09

4 0.10 0.44

Exhibit6 provides critical valuesfortheStudent'st-Distribution Exhibit 6: Critical Values for Student's t-Distribution

AreainBothTailsCombined

Degreesof Freedom 20% 10% 5% 1%

16 1.337 1.746 2.120 2.921

17 1.333 1.740 2.110 2.898

18 1.330 1.734 2.101 2.878

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Question #1

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QuestionID:479729

Question #1

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QuestionID:479730

ᅞ A)

ᅞ B)

ᅚ C)

19 1.328 1.729 2.093 2.861

20 1.325 1.725 2.086 2.845

Using a1% significancelevel, which ofthefollowing isclosesttothelower bound ofthelower confidenceintervalofthe ΔINT

slope coefficient?

−0.045 −0.296

−0.316

Explanation

Theappropriate confidenceintervalassociated with a1% significancelevelisthe99% confidencelevel, which equals; slope coefficient ± criticalt-statistic (1% significancelevel) × coefficientstandard error

Thestandard errorisnotexplicitly provided inthis question, butit can be derived byknowing theformulaforthet-statistic:

From Exhibit1, the ΔINTslope coefficientestimateequals1.0, and itst-statistic equals2.22. Therefore, solving forthe standard error, we derive:

The critical valueforthe1% significancelevelisfound downthe1% columninthet-tables provided in Exhibit6. The

appropriate degreesoffreedomforthe confidenceintervalequalsn − k − 1 = 20 − 3 − 1 = 16 (kisthenumberofslope estimates = 3). Therefore, the critical valueforthe99% confidenceinterval (or1% significancelevel)equals2.921.

So, the99% confidenceintervalforthe ΔINTslope coefficientis:

1.00 ± 2.921(0.450):lower bound equals1 − 1.316and upper bound 1 + 1.316

or (−0.316, 2.316).

(StudySession3, LOS11.e)

Based on herregressionresultsin Exhibit2, using a 5% levelofsignificance, Smith should concludethat: both stimulus packages andhousingcrises have significant effects on

foreclosure percentages.

stimulus packages havesignificanteffectsonforeclosure percentages, but housing crises donot havesignificanteffectsonforeclosure percentages.

stimulus packages donot havesignificanteffectsonforeclosure percentages, but

housing crises do havesignificanteffectsonforeclosure percentages.

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Question #20 of 106

QuestionID:479731

ᅞ A) ᅚ B) ᅞ C)

Question #21 of 106

QuestionID:479732

Question #22 of 106

QuestionID:479733

Theappropriateteststatistic fortestsofsignificanceonindividualslope coefficientestimatesisthet-statistic, which is provided in Exhibit2foreach regression coefficientestimate. Thereported t-statistic equals-2.10fortheSTIM slopeestimateand equals2.35 fortheCRISISslopeestimate. The criticalt-statistic forthe 5% significancelevelequals2.12 (16 degreesof freedom, 5% levelofsignificance).

Therefore, theslopeestimateforSTIM isnotstatisticallysignificant (thereported t-statistic, -2.10, isnotlargeenough). In contrast, theslopeestimateforCRISISisstatisticallysignificant (thereported t-statistic, 2.35, exceedsthe 5% significance level critical value). (StudySession3, LOS10.a)

Thestandard errorofestimateforSmith'sregressionisclosestto:

0.53 0.56

0.16

Explanation

TheformulafortheStandard Errorofthe Estimate (SEE)is:

TheSEE equalsthestandard deviationoftheregressionresiduals. Alow SEE impliesa high R . (StudySession3, LOS10.h)

IsSmith correctorincorrectregarding Concerns1and 2? Correct on both Concerns.

Incorrecton both Concerns.

Only correctonone concernand incorrectontheother.

Explanation

Smith'sConcern1isincorrect. Heteroskedasticityisa violationofaregressionassumption, and referstoregressionerror variancethatisnot constantoverallobservationsintheregression. Conditional heteroskedasticityisa casein which theerror varianceisrelated tothemagnitudesoftheindependent variables (theerror varianceis "conditional" ontheindependent

variables). The consequenceof conditional heteroskedasticityisthatthestandard errors will betoolow, which, inturn, causes

thet-statisticsto betoo high. Smith'sConcern2alsoisnot correct. Multicollinearityreferstoindependent variablesthatare correlated with each other. Multicollinearity causesstandard errorsfortheregression coefficientsto betoo high, which, inturn, causesthet-statisticsto betoolow. However, contrarytoSmith's concern, multicollinearity hasnoeffectontheF-statistic. (StudySession3, LOS11.k)

Themostrecent changeinforeclosureshare was +1 percent. Smith decidesto base heranalysisonthe dataand methods provided in Exhibits4and 5, and determinesthatthetwo-step ahead forecastforthe changeinforeclosureshare (in percent)

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ᅚ A)

ᅞ B) ᅞ C)

Question #23 of 106

QuestionID:479734

Question #24 of 106

QuestionID:461823

ᅞ A)

ᅚ B)

ᅞ C)

Smith is correct on the two-step ahead forecast for change in foreclosure share only.

Smith is correcton both theforecastand themeanreverting level.

Smith is correctonthemean-reverting levelforforecastof changeinforeclosure shareonly.

Explanation

Forecastsare derived bysubstituting theappropriate valueforthe period t-1lagged value.

So, theone-step ahead forecastequals0.30%. Thetwo-step ahead (%)forecastis derived bysubstituting 0.30intothe equation.

ΔForeclosureShare = 0.05 + 0.25(0.30) = 0.125 Therefore, thetwo-step ahead forecastequals0.125%.

(StudySession3, LOS11.d)

Assumeforthis questionthatSmith findsthattheforeclosureshareseries hasaunitroot. Underthese conditions, she can mostreliablyregressforeclosureshareagainstthe changeininterestrates (ΔINT)if:

ΔINT has unit root and is cointegratedwith foreclosure share.

ΔINT hasunitrootand isnot cointegrated with foreclosureshare. ΔINT doesnot haveunitroot.

Explanation

Theerrortermsintheregressionsfor choicesA, B, and C will benonstationary. Therefore, someoftheregression assumptions will be violated and theregressionresultsareunreliable. If, however, both seriesarenonstationary (which will happenifeach hasunitroot), but cointegrated, thentheerrorterm will be covariancestationaryand theregressionresultsare

reliable. (StudySession3, LOS11.n)

Themainreason whyfinancialand timeseriesintrinsicallyexhibitsomeformofnonstationarityisthat:

most financial and time series have a natural tendency to revert toward their means.

mostfinancialand economic relationshipsare dynamic and theestimated regression coefficients can vary greatly between periods.

serial correlation, a contributing factortononstationarity, isalways presenttoa certain degreeinmostfinancialand timeseries.

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Question #2

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QuestionID:461806

Question #26 of 106

QuestionID:461802

Question #27 of 106

QuestionID:461856

ᅞ A) ᅚ B) ᅞ C) Explanation

Becauseallfinancialand timeseriesrelationshipsare dynamic, regression coefficients can vary widelyfrom period to period. Therefore, financialand timeseries willalwaysexhibitsomeamountofinstabilityornonstationarity.

Considertheestimated model x = -6.0 + 1.1 x + 0.3 x + ε thatisestimated over 50 periods. The valueofthetimeseries

forthe49 observationis20and the valueofthetimeseriesforthe 50 observationis22. Whatistheforecastforthe 51

observation? 23. 30.2. 24.2.

Explanation

Forecasted x = -6.0 + 1.1 (22) + 0.3 (20) = 24.2.

Themodel x = b + b x + b x + b x + b x + ε is:

an autoregressive conditional heteroskedastic model, ARCH.

anautoregressivemodel, AR(4). amoving averagemodel, MA(4).

Explanation

Thisisanautoregressivemodel (i.e., lagged dependent variableasindependent variables)oforder p=4 (thatis, 4lags).

Supposeyouestimatethefollowing modelofresidualsfromanautoregressivemodel:

ε = 0.25 + 0.6ε + μ , where ε = ε

Iftheresidualattimetis0.9, theforecasted variancefortimet+1is:

0.850.

0.736. 0.790.

Explanation

The varianceatt = t + 1is0.25 + [0.60 (0.9) ] = 0.25 + 0.486 = 0.736. Seealso, ARCH models.

t t-1 t-2 t

th th st

51

t 0 1 t-1 2 t-2 3 t-3 4 t-4 t

t2 2t-1 t ^

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Question #2

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QuestionID:461858

Question #2

9 of 106

QuestionID:461855

ᅞ A) ᅚ B)

ᅞ C)

Supposeyouestimatethefollowing modelofresidualsfromanautoregressivemodel:

ε = 0.4 + 0.80ε + μ , where ε = ε

Iftheresidualattimetis2.0, theforecasted variancefortimet+1is:

3.2. 3.6. 2.0.

Explanation

The varianceatt=t+1is0.4 + [0.80 (4.0)] = 0.4 + 3.2. = 3.6.

The data below yieldsthefollowing AR(1)specification: x = 0.9-0.55x + E , and theindicated fitted valuesand residuals.

Time x fitted

values residuals

1 1 -

-2 -1 0.35 -1.35

3 2 1.45 0.55

4 -1 -0.2 -0.8

5 0 1.45 -1.45

6 2 0.9 1.1

7 0 -0.2 0.2

8 1 0.9 0.1

9 2 0.35 1.65

Thefollowing setsof dataareordered fromearliesttolatest. TotestforARCH, theresearchershould regress:

(1, 4, 1, 0, 4, 0, 1, 4) on (1, 1, 4, 1, 0, 4, 0, 1)

(1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01)on (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225).

(-1.35, 0.55, -0.8, -1.45, 1.1, 0.2, 0.1, 1.65)on (0.35, 1.45, -0.2, 1.45, 0.9, -0.2, 0.9, 0.35)

Explanation

ThetestforARCH is based onaregressionofthesquared residualsontheirlagged values. Thesquared residualsare (1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225). So, (1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01)isregressed on (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225). If coefficienta in: isstatistically differentfrom zero, the

timeseriesexhibitsARCH(1).

t2 t-12 t ^

t t-1 t

t

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Question #30 of 106

QuestionID:461815

Questions #31-36 of 106

Theregressionresultsfromfitting anAR(1)toamonthlytimeseriesare presented below. Whatisthemean-reverting levelfor themodel?

M

odel

:

Δ

Ex

p

=

b

+

b

Δ

Ex

p

+ ε

Coefficien

t

s

St

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Error

t-St

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00

6

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3

0 .

012 5 <

0 .

0001

1.6258.

0.6151.

7.3220.

Explanation

The mean-reverting level is b / (1 − b) = 1.3304 / (1 − 0.1817) = 1.6258.

YolandaSeerveld isananalyststudying the growth ofsalesofanew restaurant chain called Very Vegan. Theincreaseinthe public'sawarenessof healthfuleating habits has had a very positiveeffecton Very Vegan's business. Seerveld has gathered quarterly datafortherestaurant'ssalesforthe pastthreeyears. Overthetwelve periods, sales grew from $17.2millioninthe

first quarterto $106.3millioninthelast quarter. Because Very Vegan hasexperienced growth ofmorethan 500% overthe

threeyears, theSeerveld suspectsanexponential growth modelmay bemoreappropriatethanasimplelineartrend model. However, she begins byestimating thesimplelineartrend model:

(sales) = α + β × (Trend) + ε

Whe

r

e

t

he

Tr

e

n

d is

1 ,

2 ,

3 ,

4 , 5,

6 ,

7 , 8,

9 ,

10 ,

11 ,

12 .

RegressionStatistics

Multiple R 0.952640

R 0.907523

Adjusted R

0 .8

9

8

2

7

5

Standard Error 8.135514

Observations

12

1storderautocorrelation coefficientof theresiduals: −0.075

ANOVA

df SS

Regression 1 6495.203

t 0 1 t-1 t

0 1

t t t

2

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Question #31 of 106

QuestionID:485703

Residual 10 661.8659

Total 11 7157.069

Coefficients StandardError Intercept

10 .

001

5

5.

00

7

1

Trend

6 .

74

00

0 .

6

8

0

3

Theanalystthenestimatesthefollowing model:

(naturallogarithmofsales) = α + β × (Trend) + ε

RegressionStatistics

Multiple R

0 .

9

5

202

8

R 0.906357

Adjusted R 0.896992

Standard Error 0.166686

Observations

12

1storderautocorrelation coefficientof theresiduals: −0.348

ANOVA

df SS

Regression 1 2.6892 Residual 10 0.2778 Total 11 2.9670

Coefficients StandardError

Intercept

2 .

9

8

0

3

0 .

102

6

Trend 0.1371

0 .

01

4

0

Seerveld comparestheresults based upontheoutputstatisticsand conductstwo-tailed testsata 5% levelofsignificance. One concernisthe possible problemofautocorrelation, and Seerveld makesanassessment based uponthefirst-order

autocorrelation coefficientoftheresidualsthatislisted ineach setofoutput. Another concernisthestationarityofthe data.

Finally, theanalyst composesaforecast based oneach equationforthe quarterfollowing theend ofthesample.

Areeitheroftheslope coefficientsstatisticallysignificant?

The simple trend regression is not,but the log-linear trend regression is.

Thesimpletrend regressionis, butnotthelog-lineartrend regression. Yes, both aresignificant.

Explanation

t t t

2

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Question #32 of 106

QuestionID:485704

ᅞ A)

ᅚ B) ᅞ C)

Question #33 of 106

QuestionID:485705

Question #34 of 106

QuestionID:485706

Therespectivet-statisticsare6.7400 / 0.6803 = 9.9074and 0.1371 / 0.0140 = 9.7929. For10 degreesoffreedom, the critical

t-valueforatwo-tailed testata 5% levelofsignificanceis2.228, so both slope coefficientsarestatisticallysignificant. (Study Session3, LOS11.a)

Based upontheoutput, which equationexplainsthe causefor variationof Very Vegan'ssalesoverthesample period? Both the simple linear trend and the log-linear trendhave equal explanatory

power.

The cause cannot be determined using the giveninformation. Thesimplelineartrend.

Explanation

Toactually determinetheexplanatory powerforsalesitself, fitted valuesforthelog-lineartrend would haveto be determined and then compared totheoriginal data. The giveninformation doesnotallow forsuch a comparison. (StudySession3, LOS

11.b)

With respecttothe possible problemsofautocorrelationand nonstationarity, using thelog-lineartransformationappearsto have:

improved the results for nonstationarity but not autocorrelation.

improved theresultsforautocorrelation butnotnonstationarity. notimproved theresultsforeither possible problems.

Explanation

Thefactthatthereisasignificanttrend for both equationsindicatesthatthe dataisnotstationaryineither case. Asfor autocorrelation, theanalystreally cannottestitusing theDurbin-Watsontest becausetherearefewerthan15 observations, which isthelowerlimitoftheDW table. Looking atthefirst-orderautocorrelation coefficient, however, weseethatitincreased (inabsolute valueterms)forthelog-linearequation. Ifanything, therefore, the problem becamemoresevere. (StudySession

3, LOS11.b)

The primarylimitationof both modelsisthat:

each uses only one explanatory variable.

regressionisnotappropriateforestimating therelationship.

theresultsare difficulttointerpret.

Explanation

Themain problem with atrend modelisthatitusesonlyone variablesotheunderlying dynamicsarereallynotadequately

addressed. Astrength ofthemodelsisthattheresultsareeasytointerpret. Thelevelsofmanyeconomic variablessuch as

thesalesofafirm, prices, and gross domestic product (GDP) haveasignificanttimetrend, and aregressionisanappropriate

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Question #3

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Question #36 of 106

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Question #37 of 106

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ᅞ A)

Using thesimplelineartrend model, theforecastofsalesfor Very Veganforthefirstout-of-sample period is: $97.6 million.

$123.0million. $113.0million.

Explanation

Theforecastis10.0015 + (13 × 6.7400) = 97.62. (StudySession3, LOS11.a)

Using thelog-lineartrend model, theforecastofsalesfor Very Veganforthefirstout-of-sample period is: $117.0 million.

$109.4million. $121.2million.

Explanation

Theforecastise = 117.01. (StudySession3, LOS11.a)

AlexisPopov, CFA, wantstoestimate how sales have grownfromone quartertothenextonaverage. Themost direct wayfor

Popov toestimatethis would be:

an AR(1) model with a seasonal lag.

anAR(1)model.

alineartrend model.

Explanation

Ifthe goalistosimplyestimatethe dollar changefromone period tothenext, themost direct wayistoestimate x = b + b × (Trend) + e, whereTrend issimply1, 2, 3, ....T. Themodel predictsa change bythe value b fromone period tothenext.

FrankBatchelderand Miriam YenkinareanalystsforBishop Econometrics. Batchelderand Yenkinare discussing themodels

theyusetoforecast changesinChina'sGDPand how they can comparetheforecasting accuracyofeach model. Batchelder

states, "Therootmeansquared error (RMSE) criterionistypicallyused toevaluatethein-sampleforecastaccuracyof autoregressivemodels." Yenkinreplies, "If weusethe RMSE criterion, themodel with thelargest RMSE istheone weshould judgeasthemostaccurate."

With regard totheirstatementsaboutusing the RMSE criterion:

Batchelder is correct;Yenkin is incorrect.

2.9803 + (13 × 0.1371)

t 0 1

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ᅞ B) ᅚ C)

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Question #40 of 106

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ᅞ A)

Batchelderisincorrect; Yenkinis correct. Batchelderisincorrect; Yenkinisincorrect.

Explanation

Therootmeansquared error (RMSE) criterionisused to comparetheaccuracyofautoregressivemodelsinforecasting out

-of-sample values (notin-sample values). Batchelderisincorrect. Out-of-sampleforecastaccuracyisimportant becausethe

futureisalwaysoutofsample, and thereforeout-of-sample performanceofamodelis criticalforevaluating real world performance.

Yenkinisalsoincorrect. The RMSE criteriontakesthesquarerootoftheaveragesquared errorsfromeach model. Themodel with thesmallest RMSE is judged themostaccurate.

Considerthefollowing estimated model:

(Sales-Sales )= 100-1.5 (Sales -Sales ) + 1.2 (Sales -Sales )t=1,2,.. T

and Salesforthe periods1999.1through 2000.2:

t Period Sales

T 2000.2 $1,000 T-1 2000.1 $900 T-2 1999.4 $1,200 T-3 1999.3 $1,400 T-4 1999.2 $1,000 T-5 1999.1 $800

Theforecasted Salesamountfor2000.3isclosestto:

$1,730.

$730. $1,430.

Explanation

Changeinsales = $100-1.5 ($1,000-900) + 1.2 ($1,400-1,000)

Changeinsales = $100-150 + 480 =$430

Sales = $1,000 + 430 = $1,430

Which ofthefollowing is NOTarequirementforaseriesto be covariancestationary? The:

covariance of the time series with itself (lead or lag) must be constant.

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Question #46 of 106

QuestionID:461790

Question #47 of 106

QuestionID:461791

Question #4

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QuestionID:461782

ᅞ A) ᅞ B)

ᅚ C)

testing forserial correlationoftheerrortermsinanautoregressivemodel. Constantand finiteunconditional varianceisnotan

indicatorofserial correlation butratherisoneoftherequirementsof covariancestationarity. (StudySession3, LOS13.e)

Whenusing therootmeansquared error (RMSE) criteriontoevaluatethe predictive powerofthemodel, which ofthefollowing

isthemostappropriatestatement ?

Use the model with the lowest RMSEcalculated using the out-of-sample data.

Usethemodel with the highest RMSE calculated using thein-sample data. Usethemodel with thelowest RMSE calculated using thein-sample data.

Explanation

RMSE isameasureoferror hencethelowerthe better. Itshould be calculated ontheout-of-sample datai.e. the datanot

directlyused inthe developmentofthemodel. Thismeasurethusindicatesthe predictive powerofourmodel. (StudySession3, LOS13.g)

IfCranwellsuspectsthatseasonalitymay be presentin hisAR model, he would mostcorrectly:

use the Durbin Watson statistic.

testforthesignificanceoftheslope coefficients.

examinethet-statisticsoftheresiduallag autocorrelations.

Explanation

Seasonalityinmonthlyand quarterly dataisapparentinthe high (statisticallysignificant)t-statisticsoftheresiduallag autocorrelationsforLag 12and Lag 4respectively. To correctforthat, theanalystshould incorporatetheappropriatelag in his/herAR model.

(StudySession3, LOS13.l)

Dianne Hart, CFA, is considering the purchaseofanequity positioninBook World, Inc, aleading sellerof booksinthe United

States. Hart hasobtained monthlysales dataforthe pastsevenyears, and has plotted the data pointsona graph. Which of thefollowing statementsregarding Hart'sanalysisofthe datatimeseriesofBook World'ssalesismostaccurate? Hartshould

utilizea:

linear model to analyze the data because the mean appears to be constant.

mean-reverting modeltoanalyzethe data becausethetimeseries patternis covariancestationary.

log-linearmodeltoanalyzethe data becauseitislikelytoexhibita compound growth

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Explanation

Alog-linearmodelismoreappropriate whenanalyzing datathatis growing ata compound rate. Salesarea classic example ofatypeof dataseriesthatnormallyexhibits compound growth.

The regressionresults fromfitting anAR(1)model to the first-differences in enrollment growth rates at a large university includes aDurbin

-Watson statistic of1.58. The number of quarterly observations in the time series is 60. At 5% significance, the critical values for the

Durbin-Watson statistic are d = 1.55 and d = 1.62. Which of the following is the mostaccurate interpretation of the DW statistic for the

model?

The Durbin-Watson statisticcannot be usedwith AR(1) models.

Since DW > d , the null hypothesis ofno serial correlation is rejected.

Since d < DW < d the results of the DW test are inconclusive.

Explanation

The Durbin-Watson statistic is not useful when testing for serial correlation inanautoregressive model where one of the independent

variables is a lagged value of the dependent variable. The existence of serial correlation inanAR model is determined by examining the

autocorrelations of the residuals.

David Brice, CFA, hasused anAR(1)modeltoforecastthenext period'sinterestrateto be0.08. TheAR(1) hasa positive slope coefficient. Iftheinterestrateisameanreverting process with anunconditionalmean, a.k.a., meanreverting level, equalto0.09, then which ofthefollowing could be hisforecastfortwo periodsahead?

0.081.

0.113. 0.072.

Explanation

AsBricemakesmore distantforecasts, each forecast will be closertotheunconditionalmean. So, thetwo period forecast

would be between0.08 and 0.09, and 0.081istheonly possibleanswer.

TroyDillard, CFA, hasestimated thefollowing equationusing quarterly data: x = 93-0.5× x + 0.1× x + e. Giventhe data

inthetable below, whatisDillard's bestestimateofthefirst quarterof2007?

Time

V

a

l

u

e

2005:I 62 2005:II 62

l u

l

l u,

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Question #

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QuestionID:485690

Question #

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ᅚ A)

ᅞ B)

ᅞ C)

imports willnot grow inthelong runand tend toreverttoalong-runmean, but hesaystheactuallong-runmeanis 54.545. Briarsthen computestheforecastofimportsthreeyearsintothefuture.

With respecttothestatementsmade by Holmesand Briars concerning serial correlationand theimportanceoftheDurbin

-Watsonstatistic:

Holmes was correct and Briars was incorrect.

they were both incorrect.

Briars was correctand Holmes wasincorrect.

Explanation

Briars wasincorrect becausetheDW statistic isnotappropriatefortesting serial correlationinanautoregressivemodelofthis sort. Holmes wasincorrect becauseserial correlation can certainly bea probleminsuch amodel. Theyneed toanalyzethe

residualsand computeautocorrelation coefficientsoftheresidualsto better determineifserial correlationisa problem. (Study Session3, LOS10.k)

With respecttothestatementthatthe company'sstatisticianmade concerning the consequencesofserial correlation, assuming the company'sstatisticianis competent, we would mostlikely deducethat Holmesand Briars did nottellthe statistician:

the model's specification.

the valueoftheDurbin-Watsonstatistic.

thesamplesize.

Explanation

Serial correlation will biasthestandard errors. It canalso biasthe coefficientestimatesinanautoregressivemodelofthistype.

Thus, Briarsand Holmes probably did nottellthestatisticianthemodelisanAR(1)specification. (StudySession3, LOS10.m)

Thestatistician'sstatement concerning the benefitsofthe Hansenmethod is:

correct,because the Hansen method adjusts for problems associatedwithboth serial correlation andheteroskedasticity.

not correct, becausethe Hansenmethod onlyadjustsfor problemsassociated with serial correlation butnot heteroskedasticity.

not correct, becausethe Hansenmethod onlyadjustsfor problemsassociated with

heteroskedasticity butnotserial correlation.

Explanation

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Question #

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Question #60 of 106

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Question #61 of 106

QuestionID:461852

Using themodel'sresults, Briar'sforecastforthreeyearsintothefutureis: $47.151 million.

$54.543million. $54.108 million.

Explanation

Briars' forecastsforthenextthreeyears would be: yearone:3.8836 + 0.9288 × 54 = 54.0388

yeartwo:3.8836 + 0.9288 × (54.0388) = 54.0748

yearthree:3.8836 + 0.9288 × (54.0748) = 54.1083

(

S

t

u

d

y

S

essio

n

3 ,

L

O

S

11 .

a

)

With respecttothe commentsof Holmesand Briars concerning themeanreversionoftheimport data, thelong-runmean valuethat:

Briars computes is not correct, andhis conclusion is probably not accurate.

Briars computesisnot correct, but his conclusionis probablyaccurate. Briars computesis correct.

Explanation

Briars has computed a valuethat would be correctiftheresultsofthemodel werereliable. Thelong-runmean would be

3.8836 / (1 − 0.9288)= 54.5450. (StudySession3, LOS11.a)

Giventhenatureoftheiranalysis, themostlikely potential problemthatBriarsand Holmesneed toinvestigateis:

multicollinearity. unitroot.

autocorrelation.

Explanation

Multicollinearity cannot bea problem becausethereisonlyoneindependent variable. ForatimeseriesAR model,

autocorrelationisa bigger worry. Themodelmay have beenmisspecified leading tostatisticallysignificantautocorrelations. Unitroot doesnotseemto bea problem giventhe valueof b1<1. (StudySession3, LOS11.e)

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Question #62 of 106

QuestionID:461860

following isthemostlikely conclusionabouttheappropriatenessofthemodel? Thetimeseries:

L

agged

A

ut

ocorrela

t

ions

of

t

he

L

og

of

Qu

ar

t

erl

y

Thea

t

er

Tic

k

e

t

S

ales

L

ag

A

ut

ocorrela

t

ion

St

andard

Error

t-St

a

t

is

t

ic

1 −

0 .

0

73

8

0 .

1

667 −

0 .

44

2

7

1

2 −

0 .

10

47

0 .

1

667 −

0 .

6

2

8

0

7

3 −

0 .

02

5

2

0 .

1

667 −

0 .

1

5

11

7

4

0 .55

2

8

0 .

1

667

3 .

3

1

6

1

4

contains ARCH(1) errors.

contains seasonality.

would be more appropriately described with anMA(4)model.

Explanation

The time series contains seasonalityas indicated by the strong and significant autocorrelation of the lag-4residual.

Considerthefollowing estimated model:

(Sales-Sales ) = 30 + 1.25 (Sales -Sales ) + 1.1 (Sales -Sales )t=1,2,.. T

and Salesforthe periods1999.1through 2000.2:

t Period Sales

T 2000.2 $2,000 T-1 2000.1 $1,800 T-2 1999.4 $1,500 T-3 1999.3 $1,400 T-4 1999.2 $1,900 T-5 1999.1 $1,700

Theforecasted Salesamountfor2000.3isclosestto:

$2,625.

$1,730. $2,270.

Explanation

Notethatsince weareforecasting 2000.3, thenumbering ofthe "t" column has changed.

Changeinsales = $30 + 1.25 ($2,000-1,800) + 1.1 ($1,400-1,900)

Changeinsales = $30 + 250- 550 = -$270

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Question #6

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QuestionID:461834

ᅞ A)

ᅞ B) ᅚ C)

y

2002

.

1

10

3 2002

.

2

5

2 -

5

1 2002

.

3

2 -

20 -

5

1 2002

.

4

6

8 +

36 -

20

200

3 .

1

9

1 +

2

3 +

36

200

3 .

2

44 -47

+

2

3 -

5

1

200

3 .

3

0 -

1

4 -47

-

20

200

3 .

4

6

0 +

3

0 -

1

4 +

36

200

4 .

1

77 +

1

7 +

3

0 +

2

3

200

4 .

2

3

8 -39

+

1

7 -47

200

4 .

3

2

9 -9

-39

-

1

4

200

4 .

4

5

3 +

2

4 -9

+

3

0

Winstonsubmitsthefollowing resultsto hissupervisor. Thefirstistheestimationofatrend modelforthe period 2002:1to

2004:4. Themodelis below. Thestandard errorsarein parentheses. (Warrantyexpense) = 74.1-2.7* t + e

R-squared = 16.2% (14.37) (1.97)

Winstonalsosubmitsthefollowing resultsforanautoregressivemodelonthe differencesintheexpenseoverthe period

2004:2to2004:4. Themodelis below where "y" representsthe changeinexpenseas defined inthetableabove. The standard errorsarein parentheses.

y

=

-

0 .

7 -

0 .

0

7 *

y

+

0 .8

3 *

y

+ e

R

-

sq

u

ar

ed =

99 .

9 8%

(

0 .

643)

(

0 .

0222

)

(

0 .

01

8 6)

Afterreceiving theoutput, Collier'ssupervisorasks himto computemoving averagesofthesales data.

Collier'ssupervisors would probablynot wanttousetheresultsfromthetrend modelforallofthefollowing reasons EXCEPT:

it does not give insights into the underlyingdynamics of the movement of the dependent variable.

theslope coefficientisnotsignificant.

themodelisalineartrend modeland log-linearmodelsarealwayssuperior.

Explanation

t t-1 t-4

t t

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Question #7

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ᅞ A)

ᅞ B) ᅚ C)

Question #76 of 106

QuestionID:461850

Question #77 of 106

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Question #7

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ᅚ A)

Which ofthefollowing statementsregarding theinstabilityoftime-seriesmodelsismostaccurate? Modelsestimated with:

a greater number of independent variables are usually more stable than those

with a smaller number.

longertimeseriesareusuallymorestablethanthose with shortertimeseries. shortertimeseriesareusuallymorestablethanthose with longertimeseries.

Explanation

Thosemodels with ashortertimeseriesareusuallymorestable becausethereislessopportunityfor varianceinthe estimated regression coefficients betweenthe differenttime periods.

Which ofthefollowing isaseasonallyadjusted model?

(Sales - Sales )=b +b (Sales - Sales ) +b (Sales - Sales ) +ε . Sales = b + b Sales + b Sales + ε .

Sales = b Sales + ε .

Explanation

ThismodelisaseasonalAR with first differencing.

Which of the following is leastlikelya consequence ofamodel containing ARCH(1) errors? The:

regression parameters will be incorrect.

model's specification can be corrected byadding anadditional lag variable.

variance of the errors can be predicted.

Explanation

The presence ofautoregressive conditional heteroskedasticity (ARCH) indicates that the variance of the error terms is not constant. This

is a violation of the regressionassumptions upon which time series models are based. The addition ofanother lag variable to amodel is

not ameans for correcting forARCH (1) errors.

Atimeseriesthat hasaunitroot can betransformed intoatimeseries withoutaunitrootthrough:

first differencing.

t t-1 0 1 t-1 t-2 2 t-4 t-5 t

t 0 1 t-1 2 t-2 t

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ᅞ B)

ᅞ C)

Question #

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causethetransformed seriesto belinear.

Ananalystmodeled thetimeseriesofannualearnings pershareinthespecialty departmentstoreindustryasanAR(3)

process. Uponexaminationoftheresidualsfromthismodel, shefound thatthereisasignificantautocorrelationforthe

residualsofthismodel. Thisindicatesthatsheneedsto:

alter the model to an ARCH model.

switch modelstoamoving averagemodel.

revisethemodeltoincludeatleastanotherlag ofthe dependent variable.

Explanation

Sheshould estimateanAR(4)model, and thenre-examinetheautocorrelationsoftheresiduals.

The procedurefor determining thestructureofanautoregressivemodelis:

estimate an autoregressive model (e.g., an AR(1) model),calculate the autocorrelations for the model's residuals, test whether the autocorrelations are different from zero, and revise the model if there are significant

autocorrelations.

testautocorrelationsoftheresidualsforasimpletrend model, and specifythenumber

ofsignificantlags.

estimateanautoregressivemodel (forexample, anAR(1)model), calculatethe

autocorrelationsforthemodel'sresiduals, test whethertheautocorrelationsare differentfrom zero, and add anAR lag foreach significantautocorrelation.

Explanation

The procedureisiterative: continuallytestforautocorrelationsintheresidualsand stop adding lags whentheautocorrelations oftheresidualsareeliminated. Evenifseveraloftheresidualsexhibitautocorrelation, thelagsshould beadded oneatatime.

Inthetimeseriesmodel:y=b + b t + ε , t=1,2,...,T, the:

disturbance term is mean-reverting.

changeinthe dependent variable pertime period is b . disturbancetermsareautocorrelated.

Explanation

Theslopeisthe changeinthe dependent variable perunitoftime. Theinterceptistheestimateofthe valueofthe dependent

t 0 1 t

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variable beforethetimeseries begins. The disturbancetermshould beindependentand identically distributed. Thereisno

reasontoexpectthe disturbancetermto bemean-reverting, and iftheresidualsareautocorrelated, theresearch should correctforthat problem.

Considertheestimated AR(2)model, x = 2.5 + 3.0 x + 1.5 x + ε t=1,2,...50. Making a predictionfor valuesof x for1 ≤ t ≤ 50isreferred toas:

requires more information to answer the question.

anin-sampleforecast. anout-of-sampleforecast.

Explanation

Anin-sample (a.k.a. within-sample)forecastismade withinthe boundsofthe dataused toestimatethemodel. Anout-of

-sampleforecastisfor valuesoftheindependent variablethatareoutsideofthoseused toestimatethemodel.

AlexisPopov, CFA, isanalyzing monthly data. Popov hasestimated themodel x = b + b × x + b × x + e. Theresearcher findsthattheresiduals haveasignificantARCH process. The bestsolutiontothisisto:

re-estimate the model using a seasonal lag.

re-estimatethemodel with generalized leastsquares. re-estimatethemodelusing onlyanAR(1)specification.

Explanation

Iftheresiduals haveanARCH process, thenthe correctremedyis generalized leastsquares which willallow Popov to better

interprettheresults.

Suppose that the following time-series model is found to have aunit root:

Sales = b + b Sales + ε

What is the specification of the model iffirst differences are used?

(Sales - Sales )=b +b (Sales - Sales ) +ε . Sales = b Sales + ε .

Sales = b + b Sales + b Sales + ε .

Explanation

t t-1 t-2 t

t 0 1 t-1 2 t-2 t

t 0 1 t-1 t

t t-1 0 1 t-1 t-2 t

t 1 t-1 t

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ᅞ A)

Estimation with first differences requires calculating the change in the variable from period to period.

William Zox, ananalystfor OpalMountainCapitalManagement, usestwo differentmodelstoforecast changesintheinflation rateinthe United Kingdom. Both models were constructed using U.K. inflation datafrom1988-2002. Inorderto comparethe

forecasting accuracyofthemodels, Zox collected actual U.K. inflation datafrom2004-2005, and compared theactual datato whateach model predicted. ThefirstmodelisanAR(1)modelthat wasfound to haveanaveragesquared errorof10.429

overthe12month period. Thesecond modelisanAR(2)modelthat wasfound to haveanaveragesquared errorof11.642

overthe12month period. Zox then computed therootmeansquared errorforeach modeltouseasa basisof comparison. Based ontheresultsof hisanalysis, which modelshould Zox concludeisthemostaccurate?

Model 1 because it has an RMSE of 5.21. Model1 becauseit hasan RMSE of3.23.

Model2 becauseit hasan RMSE of3.41.

Explanation

Therootmeansquared error (RMSE) criterionisused to comparetheaccuracyofautoregressivemodelsinforecasting out

-of-sample values. To determine which model willmoreaccuratelyforecastfuture values, we calculatethesquarerootofthe

meansquared error. Themodel with thesmallest RMSE isthe preferred model. The RMSE forModel1is √10.429 = 3.23, whilethe RMSE forModel2is √11.642 = 3.41. SinceModel1 hasthelowest RMSE, thatistheone Zox should concludeisthe

mostaccurate.

Bill Johnson, CFA, has prepared data concerning revenuesfromsalesof winter clothing made byPolarCorporation. This data

is presented (in $ millions)inthefollowing table:

ChangeInSales LaggedChange

InSales

Seasonal Lagged ChangeInSales

Quarter Sales Y Y+(−1) Y+(−4) 2013.1 182

2013.2 74 −108

2013.3 78 4 −108

2013.4 242 164 4

2014.1 194 −48 164

2014.2 79 −115 −48 −108

2014.3 90 11 −115 4

2014.4 260 170 11 w

The preceding table will beused by Johnsontoforecast valuesusing:

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ᅞ B) ᅚ C)

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aserially correlated model with aseasonallag. anautoregressivemodel with aseasonallag.

Explanation

Johnson willusethetabletoforecast valuesusing anautoregressivemodelfor periodsinsuccessionsinceeach successive

forecastreliesontheforecastforthe preceding period. Theseasonallag isintroduced toaccountforseasonal variationsin theobserved data.

(LOS13.a,l)

The valuethat Johnsonshould enterinthetablein placeof "w" is:

164.

−115. −48.

Explanation

Theseasonallagged changeinsalesshowsthe changeinsalesfromthe period 4 quarters beforethe current period. Salesin theyear2013 quarter4increased $164millionoverthe prior period.

(LOS13.l)

Imaginethat Johnson preparesa change-in-salesregressionanalysismodel with seasonality, which includesthefollowing:

Coefficien

t

s

I

n

t

e

r

cep

t

−

6 .

0

3

2 L

a

g

1

0 .

01

7 L

a

g

4

0 .

9

8

3

Based onthemodel, expected salesinthefirst quarterof2015 will beclosestto:

155.

210. 190.

Explanation

Substituting the1-period lagged datafrom2014.4and the4-period lagged datafrom2014.1intothemodelformula, changein

salesis predicted to be −6.032 + (0.017 × 170) + (0.983 × −48) = −50.326. Expected salesare260 + (−50.326) = 209.674. (LOS13.l)

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ᅞ A)

ᅞ B) ᅚ C)

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ᅞ A)

heteroskedasticity of model residuals.

nonstationarityintimeseries data. cointegrationinthetimeseries.

Explanation

Johnson'smodeltransformsraw sales data byfirst differencing itand thenmodeling changeinsales. Thisismostlikelyan adjustmenttomakethe datastationaryforuseinanAR model.

(LOS13.k)

Totestfor covariance-stationarityinthe data, Johnson would mostlikelyusea:

Dickey-Fuller test.

Durbin-Watsontest.

t-test.

Explanation

TheDickey-Fullertestforunitroots could beused totest whetherthe datais covariancenon-stationarity. TheDurbin-Watson testisused for detecting serial correlationintheresidualsoftrend models but cannot beused inAR models. At-testisused to

testforresidualautocorrelationinAR models (LOS13.k)

The presenceof conditional heteroskedasticityofresidualsin Johnson'smodelis would mostlikelytolead to:

invalid standard errors of regression coefficients,but statistical tests will still

be valid.

invalid estimatesofregression coefficients, butthestandard errors willstill be valid. invalid standard errorsofregression coefficientsand invalid statisticaltests.

Explanation

The presenceof conditional heteroskedasticitymayleadstoincorrectestimatesofstandard errorsofregression coefficients

and henceinvalid testsofsignificanceofthe coefficients. (LOS13.j)

Which ofthefollowing statementsregarding ameanreverting timeseriesisleastaccurate? If the current value of the time series is above the mean reverting level, the

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ᅞ B) ᅚ C)

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Ifthetime-series variableis x, then x = b + b x .

Ifthe current valueofthetimeseriesisabovethemeanreverting level, the prediction

isthatthetimeseries willincrease.

Explanation

Ifthe current valueofthetimeseriesisabovethemeanreverting level, the predictionisthatthetimeseries will decrease; if the current valueofthetimeseriesis below themeanreverting level, the predictionisthatthetimeseries willincrease.

AlbertMorris, CFA, isevaluating theresultsofanestimationofthenumberof wireless phoneminutesused ona quarterly

basis withintheterritoryofCar-telInternational, Inc. Someoftheinformationis presented below (in billionsofminutes):

Wi

r

eless

P

ho

n

e

M

i

n

u

t

es (W

PM)

= b + b W

PM

+ ε

A

N

OVA

Degrees

of

F

reedom

Su

m

of

Squ

ares

M

ean

Squ

are

Reg

r

essio

n

1

7 ,

212 .

64

1

7 ,

212 .

64

1 E

rr

o

r

2

6

3 ,

102 .

4

10

11

9 .

3

2

4 T

o

t

a

l

2

7

10 ,

3

1

5.

0

5

1 Coefficien

t

s Coefficien

t St

andard

Error

of

t

he

Coefficien

t

I

n

t

e

r

cep

t

-

8.

02

37

2 .

9

02

3 W

PM

1 .

0

9

2

6

0 .

0

673

The varianceoftheresidualsfromonetime period withinthetimeseriesisnot dependentonthe varianceoftheresidualsin anothertime period.

MorrisalsomodelsthemonthlyrevenueofCar-telusing dataover96monthlyobservations. Themodelisshown below: Sales (CAD$ millions) = b + b Sales + ε

Coefficients Coefficient StandardErroroftheCoefficient

Intercept 43.2 12.32

Sales 0.8867 0.4122

The valuefor WPMthis period is 544 billion. Using theresultsofthemodel, theforecast WirelessPhoneMinutesthree periods inthefutureis:

586.35. 691.30.

683.18.

Explanation

Theone-period forecastis −8.023 + (1.0926 × 544) = 586.35. t 0 1 t-1

t o 1 t-1 t

t-1

0 1 t−1 t

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ᅚ B) ᅞ C)

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Thetwo-period forecastisthen −8.023 + (1.0926 × 586.35) = 632.62.

Finally, thethree-period forecastis −8.023 + (1.0926 × 632.62) = 683.18. (LOS11.a)

The R-squared forthe WPMmodelisclosestto:

70%. 97%.

33%.

Explanation

R-squared = SSR/SST = 7,212.641/10,315.051 = 70%.

The WPMmodel wasspecified asa(n):

Moving Average (MA) Model.

Autoregressive (AR)Model with aseasonallag. Autoregressive (AR)Model.

Explanation

Themodelisspecified asanAR Model, butthereisnoseasonallag. Nomoving averagesareemployed intheestimationof themodel.

(LOS11.a, l)

Based upontheinformation provided, Morris would mostlikely getmoremeaningfulstatisticalresults by:

doing nothing.No information provided suggests that any of these will improve the specification.

first differencing the data. adding morelagstothemodel.

Explanation

Sincetheslope coefficientis greaterthanone, the processmaynot be covariancestationary (we would havetotestthisto be definitive). A commontechniqueto correctforthisistofirst differencethe variableto performthefollowing regression:

Δ(WPM) = b + b Δ(WPM) + ε . (LOS11.j)

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ᅚ A)

ᅞ B) ᅞ C)

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Themeanreverting levelofmonthlysalesisclosestto:

43.2 million. 381.29million. 8.83million.

Explanation

(LOS11.f)

Morris concludesthatthe current priceofCar-telstockis consistent with singlestage constant growth model (with g=3%). Based onthisinformation, thesalesmodelismostlikely:

Incorrectly specified and taking the natural log of the data wouldbe an appropriate remedy.

Correctlyspecified.

Incorrectlyspecified and first differencing the data would beanappropriateremedy.

Explanation

If constant growth rateisanappropriatemodelforCar-tel, its dividends (as wellasearningsand revenues) will grow ata constantrate. Insuch a case, thetimeseriesneedsto beadjusted bytaking thenaturallog ofthetimeseries. First

differencing would removethetrending componentofa covariancenon-stationarytimeseries but would not beappropriatefor transforming anexponentially growing timeseries.

(LOS11.b)

Amonthlytimeseriesof changesinmaintenanceexpenses (ΔExp)foranequipmentrental company wasfittoanAR(1)

modelover100months. Theresultsoftheregressionand thefirsttwelvelagged residualautocorrelationsareshowninthe

tables below. Based ontheinformationinthesetables, doesthemodelappearto beappropriatelyspecified? (Assumea 5% levelofsignificance.)