ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Sample Final Exam
Part A: Multiple Choice Questions
Answer all questions on the Multiple Choice Answer Sheet. There are 10 questions in this part. Each question is worth 2 points for a total of 20 points. A formula sheet is at the end of the exam paper.
1. (2 points) Let {yt} be a time series. Which one of the following statements is not true?
(a) If {yt} is stationary, the expectation E(yt) is a constant. (b) If {yt} is stationary, the variance Var(yt) is a constant.
(c) If {yt } is stationary, the autocovariance Cov(yt , yt−k ) is time invariant for all k ≥ 1. (d) If {yt} is stationary, {yt} is a white noise (WN) process.
(e) The Ljung-Box test can be used to test if {yt} is WN.
2. (2 points) Let {yt} be a time series. Which one of the following statements is not true?
(a) Ifyt ∼MA(2),theACFof{yt}cutoffatk=2.
(b) Ifyt ∼AR(3),theACFof{yt}cutoffatk=3.
(c) Ifyt ∼WN,theACFof{yt}arezerosforallk≥1.
(d) If yt ∼ ARMA(1, 1) and is stationary, the ACF of {yt} decay at an exponential rate.
(e) If the ACF of {yt} decay slowly, it is necessary to first test the stationarity of {yt} before estimating an ARMA model.
3. (2 points) Let {yt} denote a time series. Which one of the following statements is true?
(a) Ifyt ∼MA(2),thePACFφkk of{yt}cutoffatk=2.
(b) If yt ∼ ARMA(3,1), the PACF φkk of {yt} cut off at k = 3.
(c) Ifyt ∼WN,thePACFφkk of{yt}arezerosforallk≥1. (d) ThePACFφkk =0ifandonlyiftheACFρk =0.
(e) If |φ11| ≈ 1 but |φkk| ≈ 0, ∀k > 1, the most suitable ARMA model for {yt} is AR(1).
4. (2 points) Consider the following time series model
yt =a0 +δt+a1yt−1 +εt
where εt ∼ WN. Which one of the following statements is not true?
(a) If|a1|<1andδ̸=0,theprocesszt =yt−δt/(1−a1)isstationary. (b) Ifa1 =1,a0 ̸=0andδ=0,{yt}isarandomwalkprocesswithdrift.
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(c) Ifa1 =1,a0 ̸=0andδ=0,E(yt|y0)dependsont.
(d) If |a1| < 1, the first differences process ∆yt = yt − yt−1 is stationary.
(e) Ifa0 =a1 =1andδ=0,yt ∼I(2).
5. (2 points) Consider the following ARMA model:
pq
yt =a0 +ajyt−j +blεt−l +εt j=1 l=1
where εt ∼ WN. Which one of the following statements is not true?
(a) One can estimate (a0 , a1 , ..., ap , b1 , ..., bq ) using MLE provided that {yt } is stationary.
(b) If{yt}isstationary,thena1 +···+ap ̸=1musthold.
(c) If a1 = ··· = ap = 0, {yt} is stationary.
(d) The Ljung-Box statistics for residuals have (approximately) χ2k−p−q distribution.
(e) ThebestpredictionforyT+1 givenyt,t=1,...,T isa0 +a1yT +···+apyT−p+1.
6. (2 points) Consider the following test equation for the Dickey-Fuller test:
∆yt =α0 +δt+γyt−1 +εt Which one of the following statements is not true?
(a) Depending on whether α0 = 0 or δ = 0, the ADF test uses different critical values. (b) Ifγ<0andδ=0,{yt}isstationary.
(c) If γ < 0 and δ ̸= 0, {yt} is trend stationary.
(d) Ifγ=0andδ̸=0,{yt}hasunitrootswithadrift.
(e) If γ = 0 and δ ̸= 0, {yt} has unit roots with a quadratic deterministic trend.
7. (2 points) Let xt and yt be independent random walk processes. Suppose we run the
following regression
yt = β0 + β1xt + εt
and test H0: β1 = 0 vs. H1: β1 ̸= 0. Which one of the following statements is not true?
(a) The OLS estimator is not consistent.
(b) We would expect a significant t-statistic and high R2.
(c) The t-test will tend to reject H0 incorrectly.
(d) When the sample size becomes large, the problem with the t-test can be alleviated.
(e) The Durbin-Watson statistic tends to be small.
8. (2 points) Consider the following AR(1)-ARCH(1) model
yt = a0 + a1yt−1 + εt ε = v α + α ε2
t t 0 1t−1
iid
where |a1| < 1, vt ∼ N(0,1) and independent of εt−s for all t,s. Which one of the
following statements is not true?
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(a) The ARCH coefficients (α0, α1) should be nonnegative. (b) If α1 ≤ 1, the process {ε2t } is stationary.
(c) E(εtεt−k)=0forallk≥1. (d) E(εt|εt−1,εt−2,...)=0.
(e) E(ε2t |εt−1, εt−2, ...) = α0 + α1ε2t−1.
9. (2 points) Which one of the following statements on volatility models is not true?
(a) A GARCH(1, 1) model captures the idea that when a large positive shock εt hap- pens, εt+1 is more likely to have large positive mean.
(b) GARCH(p, q) models (with small p and q) are often used as parsimonious alterna- tives to high-order ARCH models.
(c) An ARCH-M model can be used to estimate risk premium.
(d) An EGARCH model can be used to test leverage effects.
(e) Stochastic volatility (SV) models can be used to capture volatility clustering.
10. (2 points) Consider the following VAR(p) model for wt = (yt, xt)′:
p
wt =a0 +Ajwt−j +et, et =(ey,t,ex,t)′
j=1
where E(et) = 0 and Var(et) = Ω. Which one of the following statements is not true?
(a) If Ω is known, the number of unknown paremeters in this model is 2(1 + 2p).
(b) The VAR(p) model has a companion VAR(1) representation.
(c) For the case with p = 1, wt is stationary if all eigenvalues of A1 are greater than 1 in absolute value.
(d) For the case with p = 1, wt is stationary if all roots of det(I2 − A1z) = 0 are greater than 1 in absolute value.
(e) For the case with p = 1, the impulse response function (IRF) for xt to a shock in ey,t−k is the lower left element of Ak1.
Solution:
1 2 3 4 5 6 7 8 9 10
(d) (b) (c) (e) (e) (d) (d) (b) (a) (c)
Part B: Short Answer and Problem Solving Questions B.1 Unit Roots and Cointegration
Let {xt} and {yt} be two time series.
(a) (2points)Supposeyouknow{yt}doesnothaveanydeterministictrendsbutwanttotest if {yt} has unit root(s). Which test(s) can be used for this purpose? Write out a proper test equation.
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(b) (3 points) You want to apply the Engle-Granger method to test if {xt} and {yt} are coin- tegrated. What is the null hypothesis of the Engle-Granger test? Describe the procedure of performing the Engle-Granger test.
(c) (1 point) The computed Engle-Granger statistic and critical values are presented in Table 1. Do you conclude {xt} and {yt} are cointegrated? Explain your answer.
(d) (1 point) You fit an ARDL(p, q) models to the data. The regression outputs are summa- rized in Table 2 (page 7). Write out the estimated ARDL model.
(e) (1 point) Which test can be used to test if the residuals of the ARDL model is WN? What is the (asymptotic) distribution of the test statistic for an ARDL(p, q) model?
(f) (2 points) Write out the ECM representation of the ARDL model estimated in Part (d). Compute the long-run multiplier and adjustment parameter.
Table 1: Engle-Granger Test
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Table 2: ARDL(p, q) Estimation
Solution:
(a) The Dickey-Fuller test with the test equation not containing deterministic trend, i.e.,
∆yt = a0 + γyt−1 + εt
p
∆yt =a0 +γyt−1 +βj∆yt−j+1 +εt
j=2
(b) H0: xt and yt are not cointegrated, i.e., there is no β such that yt − βxt ∼ I(0). To
conduct the Engle-Granger test, we should
(1) pretest if {xt} and {yt} are of the same order of integration.
(2) run regression yt = β0 + β1xt + εt and save the residuals {εt}.
(3) conduct the Dickey-Fuller (or ADF) test using test equation ∆εt = a1εt−1 + ut (or ∆εt = a1εt−1 + pj=1 aj+1∆εt−j + ut).
(c) The test statistic is −10.312 < −3.941, the critical value for the 1% significance level. Hence, we can reject the null hypothesis at any conventional significance levels. This provides strong evidence for xt and yt being cointegrated.
(d) The estimated model is
y =0.171 + 0.002t − 0.214y
t−1
+ 0.170y (0.052)
t−2
+ 0.469x + 0.037x
t t−1
(0.051) (0.066)
t
(0.198) (0.004) (0.062)
R2 =0.968
Give students partial credits if their answers are not entirely correct (e.g., 1/2 point for esti- mated coefficients, 1/4 point for SE and 1/4 point for R2).
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(e) The Ljung-Box test. The corresponding Q-statistic converges to χ2k−5 for k ≥ 6. (f) The ECM representation is
∆yt =0.171 + 0.002t − 1.044(yt−1 − 0.485xt−1) − 0.170∆yt−1 + 0.469∆xt The LRM is 0.485 and the adjustment parameter is −1.044.
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B.2 ARCH/GARCH Models
In this question, you analyze the daily share prices of the Goldman Sachs Group, Inc. (GS hereafter) for the period January 3, 2011 - November 30, 2018. Let {Yt} denote the time series of the GS share prices.
(a) (3 points) It is well known that {Yt} is not likely to be stationary, and hence you decide to work with the process {yt} of log returns, i.e., yt = ln(Yt/Yt−1). You fit the following ARMA(1, 1) model to {yt} and obtain regression outputs presented in Table 3 (page 9),
yt =a0 +a1yt−1 +b1εt−1 +εt
where εt ∼ WN. Report the estimated model. How many observations for Yt do you
have in the raw data?
(b) (3 points) From the data, you know yT = −0.0216 and YT = 189.9230. From the estima- tionresults,youknowεT =−0.0241.UsetheseinformationtopredictthevaluesofyT+1 and YT +1. Here, T = 30/11/2018 and so T + 1 means the first trading day in December, 2018.
(c) (4 points) You doubt the validity of the homoskedasticity assumption of the ARMA model. As a result, you try fitting three ARMA(1, 1)-GARCH(p, q) models of the form below to the data:
yt =a0 +a1yt−1 +b1εt−1 +εt εt = vtht
qp
h t = α 0 + α j ε 2t − 1 + β l h t − l
j=1 l=1
iid
where vt ∼ N (0, 1) and independent of εt−s for all t, s. The estimation results are sum-
marized in Tables 4-6 (pp. 10-12). In terms of BIC, which model do you think is the best for analyzing the conditional mean and variance of yt? Justify your answer and report the estimated model.
(d) (3 points) Suppose you know yT = −0.0216, YT = 189.9230, εT = −0.0235, εT−1 = −0.0125, hT = 0.0004, and hT−1 = 0.0005. Based on your selected ARMA-GARCH model in Part (c), forecast yT +1, YT +1, and hT +1.
(e) (2 points) To capture the potential leverage effects, you estimate the following ARMA- TGARCH model:
yt =a0 +a1yt−1 +b1εt−1 +εt εt = vtht
ht = α0 + α1ε2t−1 + λdt−1ε2t−1 + β1ht−1 iid
where dt−1 = 1 if εt−1 > 0 and 0 otherwise, vt ∼ N (0, 1) and independent of εt−s for all t, s. The regression outputs are presented in Table 7 (page 13). Do you think there exist significant leverage effects? Explain your answer.
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Table 3: ARMA(1, 1) Estimation
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Table 4: ARMA(1, 1)-GARCH Estimation (1)
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Table 5: ARMA(1, 1)-GARCH Estimation (2)
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Table 6: ARMA(1, 1)-GARCH Estimation (3)
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Table 7: ARMA(1, 1)-TGARCH Estimation
Solution:
(a) The estimated model is
y =0.000 − 0.883y
t t−1
(0.000) (0.069)
+ 0.855ε
T = 1993.
(b) Based on the estimated model in Part (a), we have
yT +1 = 0.000 − 0.883yT + 0.855εT
= 0.000 + 0.883 × 0.0216 − 0.855 × 0.0241 = −0.0015327
and by definition,
Y = Y × eyT +1 = 189.9230 × e−0.0015327 = 189.632 T+1 T
t−1 (0.076)
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(c) We choose ARMA(1,1)-GARCH(1,1) model as it has the smallest BIC. The estimated model is
yt =0.000 − 0.817yt−1 + 0.794εt−1 + εt (0.000) (0.221) (0.232)
εt =vtht
ht =0.000 + 0.068ε2t−1 + 0.908ht−1
(0.000) (0.001) (0.010) (d) Based on the estimated model in Part (c), we have
yT +1 =0.000 − 0.817yT + 0.794εT
=0.000 + 0.817 × 0.0216 − 0.794 × 0.0235 = − 0.0010118
hT +1 =0.000 + 0.068ε2T + 0.908hT
=0.000 + 0.068 × 0.02352 + 0.908 × 0.0004 =0.00040075
Y = Y × eyT +1 = 189.9230 × e−0.0010118 = 189.731 T+1 T
(e) λ = −0.0398 and the p-value is 0.001. We can reject H0 : λ = 0 at any conven- tional significance level, which can be regarded as strong evidence for the existence of leverage effects.
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B.3 VAR Models
You have daily prices data of SPDR S&P500 and S&P 400 ETF, denoted as {yt} and {xt}, re- spectively. You employ VAR model(s) to study the dynamic interrelationship between these two assets.
(a) (4 points) Two (reduced form) VAR(p) models are estimated for the system (lryt,lrxt), where lryt = log(yt/yt−1) and lrxt = log(xt/xt−1) denote log returns of holding these two assets, respectively. The estimation results are summarized in Tables 8-9 (pp. 15-16). In terms of BIC, which model do you think fits the data best? Justify your answer and write out the estimated model.
(b) (2 points) Name an alternative (other than information criteria) method that can be used to select p for a VAR model. Briefly explain how it works.
(c) (2 points) Is it possible to identify a unique structural VAR (SVAR) model from the VAR model estimated in Part (a)? Briefly explain your answer.
(d) (3 points) Write out the companion VAR(1) form of the VAR(p) model estimated in Part (a). Briefly discuss the usefulness of working with the companion expression.
(e) (4 points) Explain the meaning of Granger causality. Suppose you test if lryt and lrxt Granger cause each other and obtain test results summarized in Table 10 (page 17). What are the H0 for these two tests? What are your rejection decisions?
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Table 8: VAR Estimation (1)
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Table 9: VAR Estimation (2)
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Table 10: Testing Granger Causality
Solution:
(a) We choose VAR(2) as it has smaller BIC. The estimated model is
lryt =0.000 − 0.257lryt−1 − 0.228lryt−2 + 0.160lrxt−1 + 0.133lrxt−2 + eyt (0.000) (0.054) (0.054) (0.047) (0.047)
lrxt =0.000 − 0.136lryt−1 − 0.198lryt−2 + 0.056lrxt−1 + 0.118lrxt−2 + ext (0.000) (0.062) (0.062) (0.054) (0.054)
(b) We can perform likelihood ratio tests, i.e., compute a likelihood ratio which tests the number of lags against a shorter alternative under the null hypothesis. Rejecting the null implies the shorter alternative is not sufficient to capture the dynamics of the model and the residuals will still show autocorrelations.
(c) Let wt denote (lryt, lrxt)′. In general, it is impossible to identify a unique structural VAR model like
Bwt = γ0 + Γ1wt−1 + Γ2wt−2 + εt, Var(εt) = Σ without imposing additional identification restrictions.
(d) From Part (a), we have
where
wt =a0 + A1wt−1 + A2wt−2 + et
0.000 −0.257 0.160 −0.228 0.133
a0 = 0.000 , A1 = −0.136 0.056 , A2 = −0.198 0.118 The companion representation is
wt a0 A1 A2wt−1 et w=0+I0w+0
t−1 2 t−2
The companion (VAR(1)) representation is useful in analyzing VAR(p) models as the stationarity of the dynamic system can be assessed by check the eigenvalues of the
matrix
A1 A2 I2 0
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Moreover, it is much easier to work with the companion VAR(1) form to derive the VMA(1) representation of a VAR(p) model. Based on the VMA(1) representation, we can easily compute IRFs and FEVDs of interest.
(e) For two stochastic processes {yt} and {zt}, if E[yt+1|yt,…,y0,zt,…,z0] ̸= E[yt+1|yt,…,y0]
(=)
then zt (does not) Granger causes yt. The null hypotheses of these two tests are respectively H0 : a1,12 = a2,12 = 0 and H0 : a1,21 = a2,21 = 0, where aj,lm is the (l, m)-element of Aj . The p-values for these two tests are both less than 0.01. We can reject H0 at any conventional significance level.
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