Prep ISYE 6402 Midterm Prep– Complete Solutions
(2025/2026)
1. Getting a 3 variableVAR model from summary(model) output ofVAR(1)
a model:first matrix:first row
arecoefficientsfor Xt1, secondrowarecoefficientsfor Xt2, etc...
secondmatrixis Xt-1, i b/cthis is a VAR(1) model last matrix are the
constants
eta_tis covariancematrix,directcopy
2. (c) Basedon thefittedmodel,is therecontemporaneous cross-correlation? Istherelaggedcross-correlation?
Is therelaggedauto-correlation?Explain.-
: contemporaneous cross-correlationis NOT presentif thevariance-covariance matrix is a diagonal matrix
thereis laggedcorrelationif theorderp of theVAR(p) model> 0
3. T/F - Differencingthedatamight not maketheseriesstationaryin the presence of cointegration.:
True
4. Cointegrationandlong-run equilibrium:Seeimage
5. Doescov(x,x)= var(x)?:You betcha
6. AutocovarianceT/F: see image
7. T/F -The AR(1) processis causalif andonlyif theautoregressive parameter phi is between 0 and 1. Howe
it is always invertible.:
FALSE! the absolute value of phi must lie b/w -1 and 1
8. T/F - A linearprocessis a specialcaseof themovingaveragemodel.:FALSE
- themovingaverageis a specialcaseof a linearprocess.
9. T/F - A guassiantimeseriesis alwaysstationary:false- guassianprocesses can have varying means
10. T/F 'In autoregressive models the current value of dependent variable is influenced by past valu
both dependent and independent variables.': - FALSE - thereareno analogiesof dependent/independent
variablesw/AR models, as there are w/ regression models
11. in AR models the current value of the dependent variable is affectedpast by the
valuesof bothdependent
andindependentvariables:False- Wedon't have dependent and independent variables in
AR models like we
do in regressionmodels
12. howdo ACF andPACF differ?:TBD
13. whatin an ACF plotwouldshownon-stationarity?:slowlydecreasinglags
1/7
,Prep ISYE 6402 Midterm Prep– Complete Solutions
(2025/2026)
14. whatin an ACF andPACF plotwouldshowstationary?:fewlagsoutsideof confidence bands, quickly
decreasing
15. can confidenceintervalsbe used forsignificance?:youbet- shouldall be same sign for significance
16. in a VAR model w/ seasonalityfor twelvemonths,how many seasonality dummy variables will you
have?:just 11- # of categories- 1
17. T/F 1. Time series processesgenerallycan be decomposedinto a compo- nent modeling systemat
variation (trend and seasonality) and a component modeling stochastic stationary variation.:TRUE!
stochastic, like white noise
In mathematicsand statistics, astationary process (a.k.a.a strict/strictlystation- ary processor
strong/stronglystationary process) is a stochasticprocesswhose unconditional joint probability distributi
does not change when shifted in time. Consequently, parameters such as mean variance
and alsodo not
change overtime.
18. T/F Consecutive observations in time series data are independent and identically distributed.:FALSE -
otherwise,youwouldn'thavetoconsiderauto- correlation
19. T/F - Var(a+bY) = b * Var(Y): FALSE - Var(a+bY) = b^2 * Var(Y)
20. T/F - Onemodelforthetrendcomponentofa timeseriesis thesimplelinearregression model in which tim
is used as an explanatory variable.:
TRUE
21. T/F - If Cov(X,Y)=0 then X and Y are independent:FALSE - If X and Y are independentvariables,
thentheircovarianceis 0: Cov(X, Y ) = E(XY ) µXµY = E(X)E(Y ) µXµY = 0
The converse,however,is notalwaystrue.Cov(X, Y ) canbe0 for variablesthatare not independent
22. T/F If =ÁC orr(X ,Y )=0, thenX andYareindependent: FALSE - However,if X andY are uncorrelated, then
they can still be dependent
23. T/F If X andY areindependentrandomvariables,thenwehave that
Var(X+Y )V̀ar(X)+Var(Y ).: FALSE
24. T/F If X andY and areindependentrandomvariables,thenwe Cov(a+bX, c+DY) = have that
bdCov(X,Y).: FALSE
25. T/F The 0th lag order autocovariance of a variableYt is equalto the varianceof that variable.:TRUE -
Cov(X,X) = Var(X)
26. T/FThe conditionthatthecovariancebetween Yi andYi-j dependsonlyonj is sufficientfor theprocesstobe
stationary:FALSE - Stationaryif it hasconstantmean for all time points, t, it hasfinite
a variance, or more
specifically, has a finite second
moment,andthecovariancefunctiondoesnotchangewhenshiftedin time.
2/7
(2025/2026)
1. Getting a 3 variableVAR model from summary(model) output ofVAR(1)
a model:first matrix:first row
arecoefficientsfor Xt1, secondrowarecoefficientsfor Xt2, etc...
secondmatrixis Xt-1, i b/cthis is a VAR(1) model last matrix are the
constants
eta_tis covariancematrix,directcopy
2. (c) Basedon thefittedmodel,is therecontemporaneous cross-correlation? Istherelaggedcross-correlation?
Is therelaggedauto-correlation?Explain.-
: contemporaneous cross-correlationis NOT presentif thevariance-covariance matrix is a diagonal matrix
thereis laggedcorrelationif theorderp of theVAR(p) model> 0
3. T/F - Differencingthedatamight not maketheseriesstationaryin the presence of cointegration.:
True
4. Cointegrationandlong-run equilibrium:Seeimage
5. Doescov(x,x)= var(x)?:You betcha
6. AutocovarianceT/F: see image
7. T/F -The AR(1) processis causalif andonlyif theautoregressive parameter phi is between 0 and 1. Howe
it is always invertible.:
FALSE! the absolute value of phi must lie b/w -1 and 1
8. T/F - A linearprocessis a specialcaseof themovingaveragemodel.:FALSE
- themovingaverageis a specialcaseof a linearprocess.
9. T/F - A guassiantimeseriesis alwaysstationary:false- guassianprocesses can have varying means
10. T/F 'In autoregressive models the current value of dependent variable is influenced by past valu
both dependent and independent variables.': - FALSE - thereareno analogiesof dependent/independent
variablesw/AR models, as there are w/ regression models
11. in AR models the current value of the dependent variable is affectedpast by the
valuesof bothdependent
andindependentvariables:False- Wedon't have dependent and independent variables in
AR models like we
do in regressionmodels
12. howdo ACF andPACF differ?:TBD
13. whatin an ACF plotwouldshownon-stationarity?:slowlydecreasinglags
1/7
,Prep ISYE 6402 Midterm Prep– Complete Solutions
(2025/2026)
14. whatin an ACF andPACF plotwouldshowstationary?:fewlagsoutsideof confidence bands, quickly
decreasing
15. can confidenceintervalsbe used forsignificance?:youbet- shouldall be same sign for significance
16. in a VAR model w/ seasonalityfor twelvemonths,how many seasonality dummy variables will you
have?:just 11- # of categories- 1
17. T/F 1. Time series processesgenerallycan be decomposedinto a compo- nent modeling systemat
variation (trend and seasonality) and a component modeling stochastic stationary variation.:TRUE!
stochastic, like white noise
In mathematicsand statistics, astationary process (a.k.a.a strict/strictlystation- ary processor
strong/stronglystationary process) is a stochasticprocesswhose unconditional joint probability distributi
does not change when shifted in time. Consequently, parameters such as mean variance
and alsodo not
change overtime.
18. T/F Consecutive observations in time series data are independent and identically distributed.:FALSE -
otherwise,youwouldn'thavetoconsiderauto- correlation
19. T/F - Var(a+bY) = b * Var(Y): FALSE - Var(a+bY) = b^2 * Var(Y)
20. T/F - Onemodelforthetrendcomponentofa timeseriesis thesimplelinearregression model in which tim
is used as an explanatory variable.:
TRUE
21. T/F - If Cov(X,Y)=0 then X and Y are independent:FALSE - If X and Y are independentvariables,
thentheircovarianceis 0: Cov(X, Y ) = E(XY ) µXµY = E(X)E(Y ) µXµY = 0
The converse,however,is notalwaystrue.Cov(X, Y ) canbe0 for variablesthatare not independent
22. T/F If =ÁC orr(X ,Y )=0, thenX andYareindependent: FALSE - However,if X andY are uncorrelated, then
they can still be dependent
23. T/F If X andY areindependentrandomvariables,thenwehave that
Var(X+Y )V̀ar(X)+Var(Y ).: FALSE
24. T/F If X andY and areindependentrandomvariables,thenwe Cov(a+bX, c+DY) = have that
bdCov(X,Y).: FALSE
25. T/F The 0th lag order autocovariance of a variableYt is equalto the varianceof that variable.:TRUE -
Cov(X,X) = Var(X)
26. T/FThe conditionthatthecovariancebetween Yi andYi-j dependsonlyonj is sufficientfor theprocesstobe
stationary:FALSE - Stationaryif it hasconstantmean for all time points, t, it hasfinite
a variance, or more
specifically, has a finite second
moment,andthecovariancefunctiondoesnotchangewhenshiftedin time.
2/7
