ECO2061 Financial Time Series Analysis Final Exam Answer Key Fall 2021 PDF

ECO2061: Financial Time Series Analysis Final Exam Answer Key Prof. Hyojin Han Fall 2021 You have 75 minutes to complete this exam. All questions are equally weighted. Total points = 100. Good luck!...

ECO2061: Financial Time Series Analysis Final Exam Answer Key Prof. Hyojin Han Fall 2021 You have 75 minutes to complete this exam. All questions are equally weighted. Total points = 100. Good luck! 1. Consider the following ARMA(1,1) model: yt = α + a1 yt−1 + ut + θ1 ut−1 , where a1 and θ1 are some nonzero constants. Choose the correct statement. (a) The regressors in this model are strictly exogenous (b) The regressors in this model are predetermined but not strictly exogenous (c) The regressors in this model are contemporaneously uncorrelated but not pre- determined (d) The regressors in this model are endogenous (d) (Questions 2-3) A forecaster have a monthly data {yt } from January 1970 (1970:1) to December 2017 (2017:12). Using the sample over this period, the forecaster estimates the following AR(3) model for yt : ŷt = 1.002 + 0.2 yt−1 + 0.1 yt−2 + 0.005 yt−3. (0.05) (0.021) (0.015) (0.0078) The values of yt for October 2017 through December 2017 are as follows: y2017:10 = 10, y2017:11 = 5, y2017:12 = 20. 2. Use this AR(3) and the values of yt for October 2017 through December 2017 given above to forecast the values of yt in January 2018 (Choose the closest one). (i) 5 (ii) 5.55 (iii) 6) (iv) 6.55 (ii) 3. Use this AR(3) and the values of yt for October 2017 through December 2017 given above to forecast the values of yt in February 2018 (Choose the closest one). (i) 3.14 (ii) 4.14 (iii) 5.14 (iv) 6.14 (ii) 1 4. Suppose you have two sets of time series data {yt } and {xt }. The simple regression of yt on xt gives ŷt = −0.521 + 0.967 xt (0.018) (0.014) with T = 1417, R2 = 0.982. Evaluate the following claims. Claim 1: The t statistic testing for the statistical significance of the coefficient on xt is 140. Claim 2: From the t statistic and R2 , we can always conclude that there is a significant causal effect of xt on yt. (a) Both Claim 1 and Claim 2 are correct (b) Claim 1 is correct and Claim 2 is incorrect (c) Claim 2 is correct and Claim 1 is incorrect (d) Both Claim 1 and Claim 2 are incorrect (d) 5. Suppose that we have the following partial-adjustments model: yt = δ + ρyt−1 + zt0 α + ut where ut is independent white noise and |ρ| < 1. What is the long-run multiplier of z on y? (a) α/(1 − ρ) (b) δ/(1 − ρ) (c) δ (d) α (a) (For questions 6-8) Consider the following partial adjustment model: yt∗ = γ0 + γ1 xt + et (1) yt − yt−1 = λ(yt∗ − yt−1 ) + at , (2) where 0 < λ < 1, yt∗ is the desired or optimal level of y and yt is the actual (observed) level. From the equations (1) and (2), we can write yt = β0 + β1 yt−1 + β2 xt + ut. Assume that Et [et |xt , yt−1 , xt−1 ,...] = Et [at |xt , yt−1 , xt−1 ,...] = 0 and all series are weakly dependent and weakly stationary. 2 6. What is β2 in terms of γ0 , γ1 , and λ? (find the βj ’s in terms of the γj ’s and λ and find ut in terms of et and at.) (a) λγ0 (b) (1 − λ) (c) λγ1 (d) γ1 (c) 7. If β̂1 =.7 and β̂2 =.2, what is the estimate of λ? (a) 0.1 (b) 0.2 (c) 0.3 (d) 0.4 (c) 8. If β̂1 =.7 and β̂2 =.2, what is the estimate of γ1 ? (a) 0.52 (b) 0.67 (c) 0.75 (d) 1 (b) (Questions 9-10) Consider the following AR(1) model: yt = α + ρyt−1 + ut , ut ∼ i.i.d.(0, σ 2 ). 9. For what value(s) of ρ does {yt } has a unit root? (a) ρ = 0 (b) ρ = 1 (c) ρ = −1 (d) |ρ| > 1 (b) 3 10. For this question, assume ρ = 1. Let ρ̂ denote the OLS estimator of ρ. Choose a correct statement. (a) ρ̂ is consistent because {yt } is weakly stationary and the contemporaneously uncorrelation assumption is satisfied. (b) ρ̂ is NOT consistent because {yt } is neither weakly stationary nor weakly de- pendent. √ (c) T (ρ̂−ρ) is asymptotically normally distributed even though {yt } is not weakly stationary. √ (d) T (ρ̂ − ρ) converges in probability to 0. (d) 11. Consider the AR(4) process: yt = α0 + α1 yt−3 + α2 yt−4 + t where t ∼ i.i.d.(0, σ 2 ). See that this process has a unit root if α1 + α2 = 1. This process may be rewritten in the form ∆yt = β0 + β1 yt−1 + β2 ∆yt−1 + β3 ∆yt−2 + β4 ∆yt−3 + t. When does this process (yt ) have a unit root? (Find the relationship between α and β parameters.) (a) β1 = 0 (b) β2 = 0 (c) β3 = 0 (d) β4 = 0 (a) 12. You estimate the process ∆yt given in question 11 by OLS using a sample of 120 observations and obtain the following results (standard errors in parentheses) ˆ t = 0.72 − 0.47 yt−1 − 0.62 ∆yt−1 − 0.74 ∆yt−2 − 0.21∆yt−3. ∆y (0.94) (0.13) (0.51) (0.43) 0.34 You test for the unit root using augmented Dickey-Fuller (DF) test. The 5% critical value of the DF distribution is -2.86. Choose a correct statement. (a) The unit root null is rejected because the absolute value of the t statistic is larger than | − 2.86| 4  (b) The unit root null is not rejected because the absolute value of the t statistic is larger than | − 2.86| (c) The unit root null is rejected because the t statistic is smaller than -2.86 (d) The unit root null is not rejected because the t statistic is smaller than -2.86 (c) 13. Let xt denote series that we are interested. Suppose that we consider a seasonal model with an AR component for xt such that xt = ρ12 xt−12 + ut , ut ∼ i.i.d.(0, σ 2 ). Evaluate the following claims. Claim1: The correlation between xt and xt−12 = ρ12. Claim2: xt has an autocorrelation function with a spike at lag 12 with zero auto- correlation everywhere else. (a) Both Claim 1 and Claim 2 are correct (b) Claim 1 is correct and Claim 2 is incorrect (c) Claim 2 is correct and Claim 1 is incorrect (d) Both Claim 1 and Claim 2 are incorrect (b) Figure1: ACF of monthly returns of S&P500 1.0 0.8 0.6 ACF 0.4 0.2 0.0 0 10 20 30 40 Lag(h) 5 14. Figure1 shows the ACF of monthly returns data of the S&P500 index (yt ). Assume that the ACFs at h > 40 are all zero. Based on your answer in question 10 above, which seasonal ARMA model would be most appropriate for the monthly returns of the S&P500? (a) yt = ut + θ1 ut−1 + θ2 ut−2 + θ3 ut−3 , where ut ∼ i.i.d.(0, σ 2 ). (b) yt = ut + θ1 ut−12 + θ2 ut−24 + θ3 ut−36 where ut ∼ i.i.d.(0, σ 2 ). (c) yt = a1 yt−1 + a2 yt−2 + a3 yt−3 + ut , where ut ∼ i.i.d.(0, σ 2 ). (d) yt = a1 yt−12 + a2 yt−24 + a3 yt−36 + ut , where ut ∼ i.i.d.(0, σ 2 ). (b) 15. Recall the Akaike and Bayesian Information Criteria for model selection: AICi = T ln(SSRi ) + 2ni BICi = T ln(SSRi ) + ni ln(T ), where SSRi is the sum of squared residuals from model i, ni is the number of parameters in model i and T is the sample size. Another popular model selection technique is to minimize the Hannan-Quinn Information Criterion, given by HQCi = T ln(SSRi ) + 2ni ln[ln(T )]. Rank AICi , BICi and HQCi in terms of how parsimonious a model they will tend to choose (for large enough samples). (a) AICi - BICi - HQCi (b) HQCi - BICi - AICi (c) BICi - HQCi - AICi (d) BICi - AICi - HQCi (c) (For questions 16-19) Global warming is a topic of considerable importance and has attracted much attention in recent years. If the rise in global temperature con- tinues, it will have a major impact on the global economy, and thus we are interested in analyzing global temperature anomalies. Using a monthly data of global tem- perature anomalies from January 1880 to August 2010 for 1568 observations, we fit three models as follows. Model 1: Based on the time series graph and the ACF, we apply a first differencing to the series {Gt } and fit the first differencing (∆Gt ≡ xt ) by a model with a multiplicative ARMA model using AR(1) to model the nonseasonal component and 6  an AR and an MA component where s = 12 since we are dealing with monthly data. The fitted model for xt is then (1 − 0.2L)xt = 0.9xt−12 − 0.18xt−13 + ut − 0.9ut−12 , ut ∼ i.i.d.(0, σ 2 ), σ̂ 2 = 270.6, AIC = 13, 234.4, RM SF E = 14.526, where RMSFE means “root mean squared forecast error” defined in class. Model 2: Some researchers use time trend to model the global temperature anoma- lies: Gt = β0 + β1 t + zt. Then, in order to capture seasonality and time dependence in the nonseasonal com- ponent of zt = Gt − β0 − β1 t, the model is refined as a multiplicative model. The fitted model is now then given as: (1 − 1.2L)(Gt + 38.72 − 0.53t) = (1 − 0.75L)(1 − 0.09L24 )ut , ut ∼ i.i.d.(0, σ 2 ), σ̂ 2 = 272.8, AIC = 13, 247.5, RM SF E = 15.341. Model 3: We fit a model with a time trend and deterministic seasonality using monthly dummy variables rather than stochastic one. Let F ebt denotes a dummy variable that is equal to 1 if t is an observation in February and 0 otherwise. Other dummy variables in the equation below are all defined in the same way with other months. The fitted model is Gt = −36.24 − 0.21F ebt + 0.02M art + · · · − 0.42Dect + ut , σ̂ 2 = 274.8, AIC = 13, 240.3, RM SF E = 15.792. 16. From Model 1, which lower order ARMA process is {xt } equivalent to? (Hint: Rewirte the process using the lag operator.) (a) White noise (b) AR(1) (c) MA(1) (d) ARMA(1,1) (b) 17. Rank the models in terms of their in-sample performance (the order of best - second best - third best). (a) M odel1 - M odel2 - M odel3 (b) M odel1 - M odel3 - M odel2 (c) M odel2 - M odel1 - M odel3 7  (d) M odel2 - M odel3 - M odel1 (b) 18. Rank the models in terms of their out-of-sample performance (the order of best - second best - third best). (a) M odel1 - M odel2 - M odel3 (b) M odel1 - M odel3 - M odel2 (c) M odel2 - M odel1 - M odel3 (d) M odel2 - M odel3 - M odel1 (a) 19. The time plot of global temperature anomalies in Figure? shows an increase in time slope around 1980. We now consider a regression model with trend-shift. The fitted model is given as: Gt = β0 + β1 t + β2 xt + t , where t is white noise, and xt is defined as a multiplication of t and an indicator function that is equal to 0 if t < 1212 and equal to 1 if t ≥ 1212 where t = 1212 corresponds to December 1980. The fitted model is then Ĝt = 28.92 + 0.0313t + 0.0214xt. (1.2) (0.002) (0.02) Evaluate the claims. Claim1: A positive β2 implies that the time slope increases from January 1981. Claim2: According to the above fitted model, there was no statistically significant trend-shift in early 1980s in the global temperature. (a) Both Claim 1 and Claim 2 are correct (b) Claim 1 is correct and Claim 2 is incorrect (c) Claim 2 is correct and Claim 1 is incorrect (d) Both Claim 1 and Claim 2 are incorrect (a) 8  15 0.8 10 ACF r1 0.4 5 0.0 0 1970 1980 1990 2000 2010 0 5 10 15 20 year Lag 15 time plot of 3 year maturity rates ACF of 3 year maturity rates 0.8 10 ACF r2 0.4 5 0.0 1970 1980 1990 2000 2010 0 5 10 15 20 year Lag time plot of residuals ACF of residuals 0.8 0.5 residuals ACF -0.5 0.4 -1.5 0.0 1970 1980 1990 2000 2010 0 5 10 15 20 year Lag 20. We consider the relationship between two US weekly interest rate series: 1. r1t = the 1-year treasury constant maturity rate 2. r3t = the 3-year treasury constant maturity rate Those interest rate series are commonly considered as integrated of order 1 processes. They appear to move together and look highly correlated. In order to describe the relationship between those two series, we run a simple regression of r3t on r1t using 2398 observations from January 1965. The fitted model is r3t = 0.832 + 0.93r1t + et. Figure 3 presents time series plots of r1t and r3t , and the ACF of the residual et. Choose a correct statement. (a) The 1-year maturity interest rates appear to be I(0) series (b) The 3-year maturity interest rates appear to be I(0) series (c) The 1-year and 3-year maturity interest rates appear to be cointegrated with the coefficient b = 0.93, i.e. r3t − 0.93r1t looks like I(0) series (d) The 1-year and 3-year maturity interest rates appear not to be cointegrated with the coefficient b = 0.93, i.e. r3t − 0.93r1t does not appear to be I(0) series (d) 9 10