ECMT6006 Applied Financial Econometrics 2023S1 Midterm Questions
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ECMT6006 Applied Financial Econometrics
2023S1
Midterm Questions
Note: As required for the online exams, the numbers in the numerical questions and the ordering of the answers in the multiple-answer questions are randomized in the actual exams.
1 Main exam
1.1 Multiple-answer questions
1. Select the incorrect statement(s).
(a) Both t distribution and Cauchy distribution can be used to model the fat tail in the return
distribution.
(b) The kurtosis of normal distributions is zero.
(c) If X and Y are standard normal random variables, then X2 + Y2 must follow chi-squared distribution with degrees of freedom 2.
(d) The probability density function of a normal random variable can take value greater than 1.
(e) The cumulative distribution function of a discrete random variable must be discontinuous. Answer: (b) and (c).
2. Select the correct statement(s).
(a) A strictly stationary time series may not be weakly stationary. This is because the third
or higher order moments may be time-varying.
(b) If the marginal distribution of a strictly stationary time series is Gaussian (normal), then
this time series is weakly stationary.
(c) An i.i.d. time series must be weakly stationary.
(d) A time series with time-varying sknewness can still be weakly stationary.
(e) If the random variables in a time series have too strong mutual dependence, then the time
series cannot be weakly stationary.
Answer: (b) and (d).
3. Select the correct statement(s) about asset returns.
(a) Log return is the logarithm of gross return, and it is a gross return concept. (b) Log return is also called the continuously compound return.
(c) The multi-period log return is the product of the single-period log returns.
(d) The portfolio log return can be written as a value-weighted average of the log returns of individual assets in the portfolio.
(e) For the same amount of deposit at the beginning of the year and the same per annum
interest rate, if the bank pays interest rate more often over the year, then the more return you will earn from the deposit at the end of the year.
Answer: (b) and (e).
4. For all the sequences mentioned below, we assume they have finite first two moments. Select the correct statement(s).
(a) A martingale difference sequence is necessarily a zero mean white noise process.
(b) A martingale difference sequence is necessarily a white noise process but does not necessarily
have zero mean.
(c) An i.i.d. sequence is not necessarily a white noise sequence.
(d) A zero mean i.i.d. sequence is necessarily a martingale difference sequence.
(e) The variables in a martingale difference sequence is necessarily mutually independent. Answer: (a) and (d).
5. Suppose that we run the following regression model:
Yt+1 = β0 + β1 Yt + β2 Xt + β3 Xt −1 + et+1
and would like to test H0 : β 1 = 1,β2 = β3 . We want to formulate this null hypothesis in the general matrix form:
lβ(β)1(0)」
「β(β)3(2) .
Please select correct pair(s) of R and r .
(a) R = 「(l) , r = 「(l) .
(b) R = 「(l) , r = 「(l) .
(c) R = ] , r = [ ]0(1) .
(d) R = ] , r = [ 1].
(e) R = ] , r = [ ]1(0) .
(f) R = ] , r = [ ]1(0) .
(g) R = ] , r = [ ]0(2) .
Answer: (d) and (f).
6. Consider an ARMA model for asset returns:
Rt = µt + εt , µt = Et −1 (Rt ) = c + ϕRt −1 + θεt −1 ,
where c,ϕ,θ are constant parameters. Select the correct statement(s).
(a) If ϕ and θ are nonzero, then the model cannot be weakly stationary because the conditional
mean process is time-varying.
(b) If ϕ = θ = 0, then the current return Rt is uncorrelated with any past returns.
(c) {εt } is a martingale difference sequence.
(d) If we assume that {εt } follows an autoregressive conditional heteroskedastic process, then it must have time-varying variance.
(e) We have evidence to say that {εt } is an i.i.d. white noise.
Answer: (b) and (c).
7. Consider an MA(1) process
Yt = µ + εt + θεt −1 .
Let Ft be the information set generated by Yt ,Yt −1 , . . .. Select the correct statement(s).
(a) To ensure the process to be weakly stationary, we need |θ| < 1.
(b) To ensure the process to be weakly stationary, we need |θ| > 1.
(c) The autocorrelation function of {Yt } at lag 3 is zero.
(d) The conditional mean of Yt given Ft −1 is µ + εt −1 .
(e) We can perform maximum likelihood estimation of the model parameters if the conditional
distribution of εt is Cauchy distribution.
Answer: (c) and (e).
1.2 Numerical-answer questions
1. Suppose we use a simple MA(2) model to forecast the index stock returns {Rt }: Rt+1 = c + θ1 εt + θ2 εt −1 + εt+1 ,
where ε −1 = ε0 = 0 and E(εt+1|Ft ) = 0 for all t.
At time t = 1, we have observed R1 = a and R2 = b. What is the conditional mean point forecast of R3 at time t = 2?
Answer: The conditional mean point forecast of R3 at time t = 2 is
Rˆ3 := E(R3 |F2 ) = E(c + θ1 ε2 + θ2 ε 1 + ε3 |F2 ) (1)
= c + θ1 ε2 + θ2 ε 1 + E(ε3 |F2 ) = c + θ1 ε2 + θ2 ε 1 . (2)
Although ε 1 and ε2 are not directly observable, they can be deduced from the observed R1 and R2 because the model implies
R2 = c + θ1 ε 1 + θ2 ε0 + ε2 ⇒ ε2 = R2 − c − θ 1 ε 1 − θ2 ε0 = b − c − θ1 (a − c), (4)
Combining (6), (7) and (4) yields
Rˆ3 = c + θ1 (b − c − θ1 (a − c)) + θ2 (a − c).
2. Consider the following weakly stationary AR(1)-ARCH(1) model for stock return:
εt = σt νt
σt(2) = ω + αεt(2)−1 ,
νt |Ft −1 ∼ F(0, 1),
where F(0, 1) denotes some distribution with mean 0 and variance 1. What is the unconditional variance of Rt ?
Answer: Taking variance on both sides of (5) and imposing stationarity condition yields Var(Rt ) = ϕ2Var(Rt −1) + Var(εt ) = ϕ2Var(Rt ) + Var(εt ).
Therefore, we have
Var(Rt ) = 1 − ϕ2 = 1 − ϕ2 = (1 − ϕ2 )(1 − α) .
3. Consider a simple AR(1) model for asset return Rt :
Rt = ϕRt −1 + εt ,
starting from R0 . Suppose the shocks εt are i.i.d. random variables with probability distribution P(εt = a) = p, P(εt = −b) = 1 − p,
where a,b > 0. What is the conditional mean of R2 given that the realized shock ε 1 in the first period is positive?
Answer: If ε 1 > 0 then
R1 = ϕR0 + a.
Then conditional on this, the return distribution in the second period is
R2 = ϕR1 + ε2 =
So, the conditional mean is
p(ϕ2 R0 + ϕa + a) + (1 − p)(ϕ2 R0 + ϕa − b).
1.3 Short-answer questions
1. Suppose we would like to implement a regression-baed test for testing the serial correlation in an asset return time series.
(1) How the regression should be run [1pt] and what the null hypothesis should be [0.5pt], if you would like to test if there is any significant autocorrelation in the first 5 lags?
(2) Please explain what the “robustness” concern in the serial correlation test is. [1pt] Why do we say that this concern can be address by using a regression-based test? [0.5pt]
Answer : (1) The regression should be
rt = β0 + β1 rt −1 + β2 rt −2 + β3 rt −3 + β4 rt −4 + β5 rt −5 + et
where (rt ) denotes the asset return time series. The null hypothesis should be H0 : β 1 = β2 = β3 = β4 = β5 = 0.
(2) The “robustness” concern is about the presence of heterosekedasticity and autocorrelation in the asset return series. Autocorrelation tests such as Ljung-Box test and Portmanteau test assume that the data are i.i.d. and hence are not robust to the potential heterosekedasticity and autocorrelation in the data. This concern can be addressed easily in a regression-based test because there are standard ways (such as Newey-West standard errors) to construct “het- erosekedasticity and autocorrelation robust” standard errors of the regresstion coefficients, and these standard errors can be used to construct the χ2 test statistic for testing the above null hypothesis.
2. (1) Please explain what a random walk model is. [1.5pt] (2) How is a random walk model related to the Efficient Market Hypothesis? [1.5pt]
Answer: Please refer to Page 87 in Patton (2019) textbook. Students can assume the innovation process εt as either i.i.d., or martingale difference sequence (m.d.s. meaning E(εt |Ft −1) = 0), or white noise, i.e., they can state either version of the random walk model. For part (2) of this question, points are rewarded as long as the answer makes sense. Students may comment on how the implication of a random walk model (assuming the innovation process is i.i.d. or m.d.s) – E(Pt+1|Ft ) = Pt – relates to the EMH saying that at each point of time the market price has already incorporated all the available information. They may also address the issue about how random walk model may not be implied by the EMH (see the three points made on Page 87).
3. (1) Please explain what “expanding window” [0.5pt], “rolling window” [0.5pt], and “fixed win- dow” [0.5pt] methods are in an out-of-sample analysis.
(2) Suppose there is concern that our data is subject to certain changes over time (e.g., structure breaks) , which method shall we use [0.5pt], and why [0.5pt]?
Answer: (1) Please refer to Pages 118– 119 in Patton (2019) textbook. (2) One should use “rolling window” method – see the first paragraph on Page 119.
2 Replacement exam
The total points of this exam is 38. Please attempt all questions.
2.1 Multiple-answer questions (10pt)
The following 5 questions are worth 2 points each. There may be multiple correct answers to each question, please select all correct answers to receive full points.
1. Let P0 ,P1 ,P2 ,P3 be the stock prices in four periods. Suppose there are no dividend payments. For t = 0, 1, 2 and k ≥ 2 being a positive integer, let
• Rt+1 be the single-period gross return from t to t + 1, and Rt →t+k be the multi-period gross return from t to t + k .
• R be the single-period log return from t to t + 1, and Rt+k be the multi-period log return from t to t + k;
• R be the single-period arithmetic net return from t to t + 1, and Rt+k be the multi- period arithmetic net return from t to t + k .
Select the correct statement(s).
(a) R0 →3 = 对t(3)=1 Rt .
(b) R0 →3 = ut(3)=1 Rt(A) .
(c) R0 →3 = (1 + R3(A))R0 →2 .
(d) R0(L)→3 = 对t(3)=1 lnRt(A) .
(e) R0(L)→3 = R3(L)R0(L)→2 .
(f) R0(L)→3 = R 1(L) + R 1(L)→3 .
Answer: (c) and (f).
2. Let {Xt } be a time series and Ft be the information set generated by Xt ,Xt −1 , . . .. Let Et −1 (Xt ) and Vart −1 (Xt ), denote, respectively, the conditional mean and conditional variance of Xt given Ft −1 for all t. Select the correct statements.
(a) E[Vart −1 (Xt )] = Var(Xt ).
(b) Var(Et −1 (Xt )) = Var(Xt ).
(c) Et −2 [Et −1 (Xt )] = Et −1 (Xt ).
(d) Et −1 [Et −2 (Xt )] = Et −2 (Xt ).
(e) Vart −1 (Xt ) = Et −1 (Xt(2)) − [Et −1 (Xt )]2 .
Answer: (d) and (e).
3. Let {Yt } be a time series and Ft be the information set generated by Yt ,Yt −1 , . . .. Construct a new time series {Xt } by
Xt = Yt3 − E[Yt3 |Ft −1].
We know that E(Xt(2)) < ∞ . Select the statement(s) that must be true.
(a) {Xt } is a white noise.
(b) {Xt } is a martingale difference sequence.
(c) {Xt } is an i.i.d. sequence.
(d) {Xt } is serially correlated.
(e) {Xt } has nonzero mean.
Answer: (a) and (b).
4. Consider the following regression model
Yt+1 = β0 + β1 Xt + β2 Xt −1 + β3 Xt −2 + et+1 .
We would like to test
H0 : β0 = 0,β1 = 1 and β2 − 2β3 = β 1 .
Which pairs(s) of R and r is/are able to help us formulate this null hypothesis in the general matrix form
lβ(β)1(0)」
「β(β)3(2)
(a) R = 「(l) 2 1 , r = 「(l)2 .
(b) R = 「(l) 0(0) 2(0) 0(1) 2 , r = 「(l)2(1) .
(c) R = ] , r = [ ]0(1) .
(d) R = ] , r = [ 2].
(e) R = 「(l) 2 , r = 「(l) .
(f) R = 「(l) 2 , r = 「(l) .
Answer: (a) and (b).
5. Consider the following regression:
Yt = β0 + β1 Xt −1 + β2 Xt −2 + β3 Xt −3 + et .
Suppose that we want to test the following null hypothesis
H0 : β 1 = β2 = β3 = 0
using a χ2 test at the 10% level. Select the correct statement(s).
(a) Since there are 4 unknown parameters in the model, the test statistic follows χ2 distribution
with degrees of freedom 4 under H0 .
(b) Since 3 parameters are involved in the null hypothesis, the test statistic follows χ2 distri-
bution with degrees of freedom 3 under H0 .
(c) Since there are 3 restrictions in H0 , the test statistic follows χ2 distribution with degrees of freedom 3 under H0 .
(d) Since there are 1 restrictions in H0 , the test statistic follows χ2 distribution with degrees of freedom 1 under H0 .
(e) The critical value of the test is the 0.9 quantile of the χ2 distribution with the correct
degrees of freedom.
(f) The critical value of the test is the 0.1 quantile of the χ2 distribution with the correct
degrees of freedom.
(g) The critical value of the test is the 0.95 quantile of the χ2 distribution with the correct
degrees of freedom.
(h) The critical value of the test is the 0.05 quantile of the χ2 distribution with the correct
degrees of freedom.
Answer: (c) and (e).
6. Select the incorrect statement(s).
(a) The Akaike Information Criterion (AIC) imposes a lighter penalty on extra parameters in
the model than the Schwarz’s Bayesian Information Criterion (BIC).
(b) Unlike the information criteria, the R2 does not impose penalty on extra parameters in the
model.
(c) The out-of-sample analysis does not impose penalty on the estimation error.
(d) The mean squared forecast error in the out-of-sample analysis contains penalty on the estimation error.
(e) If we use MLE to estimate a model, then the maximized likelihood measures goodness-of-fit
of the model.
(f) If we see the BIC of an estimated model produced by different types of statistical software,
then something must have gone wrong.
Answer: (c) and (f).
7. Consider an AR(1) process
Yt = µ + ρYt −1 + εt ,
with information set Ft . The conditional distribution of εt given Ft −1 has mean zero and constant variance σ 2 > 0. Select the incorrect statement(s).
(a) The strict exogeneity condition, E(εt |Y1 , . . . ,YT ) = 0 for all t = 1, . . . ,T, fails in this
model.
(b) If the process is stationary, then the variance of Yt is σ 2 .
(c) The conditional variance of Yt is time-varying.
(d) The process is not stationary if ρ = 1.
(e) If the process is stationary, then the OLS estimator for ρ can be a good estimator for the
autocorrelation function of {Yt } at lag 1.
Answer: (b) and (c).
2.2 Numerical-answer questions (8pt)
The following 4 questions are 2 points each.
1. Suppose today’s price of Apple stock is a (dollars). Given some information set, we have formed a conditional mean prediction of b (dollars) and c (squared dollars) for, respectively, the Apple stock price and squared Apple stock price tomorrow. What is the conditional standard deviation of the Apple stock net return from today to tomorrow?
Answer: Let Pt = be the today’s price of Apple stock, and t+1 = Et (Pt+1) be the predicted price of Apple stock tomorrow. Let rt+1 be the net return of the Apple stock from today to tomorrow. The conditional variance of the net return is
Vart (Pt+1) Et (P) − [Et (Pt+1)]2 c − b2
by the answer to Question 1 (b) in Section 1.10.2 of Patton (2019, p. 44) and the definition of conditional variance. Therefore, the conditional standard deviation of the net return is ^Vart (rt+1).
2. Consider the following MA(2) process:
Yt = c + θ1 εt −1 + θ2 εt −2 + εt , εt ∼ WN(0,σ2 ).
What is the second-order autocorrelation of {Yt }?
Answer: We have
Var(Yt ) = Var(εt ) + Var(θ1 εt −1) + Var(θ2 εt −2)
= σ 2 + θ1(2)σ 2 + θ2(2)σ 2 = (1 + θ1(2) + θ2(2))σ2 .
Cov(Yt ,Yt −2) = Cov(c + εt + θ1 εt −1 + θ2 εt −2, c + εt −2 + θ1 εt −3 + θ2 εt −4) = Cov(θ2 εt −2,εt −2) = θ2 σ 2 .
The second-order autocorrelation is
γ2 Cov(Yt ,Yt −2) θ2
γ0 Var(Yt ) 1 + θ1(2) + θ2(2) .
3. Consider a simple AR(1) model for asset return Rt :
Rt = ϕRt −1 + εt ,
starting from R0 . Suppose the shocks εt are i.i.d. random variables with probability distribution P(εt = a) = 0.6, P(εt = −b) = 0.4
What is the conditional mean of R2 given that the realized shock ε 1 in the first period is negative?
Answer: If ε 1 < 0 then
R1 = ϕR0 − b.
Then conditional on this, the return distribution in the second period is
R2 = ϕR1 + ε2 =
So, the conditional mean is
0.6(ϕ2 R0 − ϕb + a) + 0.4(ϕ2 R0 − ϕb − b).
4. Suppose we use an ARMA(1,1) model to forecast the index stock returns {Rt }: Rt+1 = c + ϕRt + θεt + εt+1 ,
where R0 = r , ε0 = 0 and E(εt+1|Ft ) = 0 for all t. At time t = 1, we have observed R1 = α . What is the conditional mean point forecast of R2 at time t = 1?
Answer: The conditional mean point forecast of R2 at time t = 1 is
Rˆ2 := E(R2 |F1 ) = E(c + ϕR1 + θε1 + ε2 |F1 ) = c + ϕR1 + θε1 (6)
under the model assumptions. Although ε 1 is not directly observable, it can be deduced from the observed R1 because the model implies
R1 = c + ϕR0 + θε0 + ε 1 ⇒ ε 1 = R1 − c − ϕR0 − θε0 = R1 − c − ϕR0 = α − c − ϕr. (7)
Rˆ2 = c + ϕα + θ(α − c − ϕr).
2.3 Short-answer questions
The first two questions are answered by entering texts in the text boxes in the Canvas Quiz. The last question is answered by uploading a file via Canvas Assignment.
1. Please explain why we may say that “The Efficient Market Hypothesis is in essence an extension of the zero profit competitive equilibrium condition” in the classical price theory” .
2. Please give an real-world example of “ephemeral forecastatiblity” in the financial market when we talk about Efficient Market Hypothesis.
3. Please explain why we can test the presence of serial correlation in an asset return series by linear regression.
4. Please comment on the following statement: “The ARMA model is a powerful tool for forecast- ing the asset return in the financial market.”
2023-06-01