Paper Code: ECON920 Page 1 of 7 Paper Code: ECON920 Department: ULMS Examiner: Chi Wan Cheang Tel. No: 51771 Email: C.Cheang@liverpool.ac.uk Office: 261 Central FINAL EXAMINATIONS 2022 Applied Macroeconometrics TIME ALLOWED: 24 Hours INSTRUCTIONS TO CANDIDATES You must answer all four questions Show necessary working for numerical questions All statistical tests should be performed at the 5% significance level unless otherwise stated in the questions The 97.5th percentile of the standard normal distribution is 1.96 FINAL EXAMINATIONS 2022 Paper Code: ECON920 Page 2 of 7 Applied Macroeconometrics QUESTION 1 a) Describe three key features that you may observe in time series data. For each feature, provide one practical example in economics or finance, and write a data generating process that can potentially capture the specific feature. (6 Marks) b) Let ! denote an (0, “) and ! = #!”” + “!$%. Is ! weakly stationary Is ! ergodic Is ! an IID Show all working and explain why or why not. (6 Marks) c) For each of the following, derive the mean and autocovariance functions, and state if it is a weakly stationary process. Here !~(0, “). (i) ! = 2 + ! + 2.5!$% 1.5!$” (ii) ! = 0.5!$% + ! (iii) ! = 0.8!$” + ! (9 Marks) d) Consider a Gaussian white noise process !~(0,1). If the process ! = !”, derive the mean and autocovariance function of ! . Is ! weakly stationary (4 marks) TOTAL [25 Marks] – End of Question 1 – Paper Code: ECON920 Page 3 of 7 QUESTION 2 Consider the following GARCH(1,1) process ! = !, !|!$%~(0, !”) !” = + %!$%” + %!$%” where !$% is the information available up to and including time 1, is constant. Answer part a) to d) for the above model. a) Use the conditional variance process !”, explain briefly how does the GARCH(1,1) model capture the volatility clustering feature in time series !. (2 Marks) b) Obtain the ARMA representation of the GARCH(1,1) process. Show all working and state the stationarity condition of the conditional variance process. (4 Marks) c) Obtain the ARCH(∞) representation of the GARCH(1,1) process. Show all working and state the condition when the ARCH(∞) representation exist. (4 Marks) d) Obtain the unconditional expectation of !”, i.e. [!”]. (2 Marks) Consider modelling a series of daily exchange rates between US dollar and British Pound using an ARMA-GARCH-type model. Answer part e) to h) for this case study. e) A plot of the USD/GBP daily exchange rates is shown below. What feature you may observe from this time series plot How to systematically verify the presence of this feature What kind of data transformation may be needed in order to estimate the model (3 Marks) Paper Code: ECON920 Page 4 of 7 f) The transformed time series has the following correlogram with ACF, PACF and Q-Stat. What ARMA(p,q) process, or more specifically, what values for p and q you would suggest for the mean equation of the model Explain your answer. (2 Marks) g) After estimating the mean equation by an ARMA(p,q) process, the presence of ARCH/GARCH effect can be detected by some diagnostic checks. Suggest at least two diagnostic checks to identify the presence of ARCH/GARCH effect. (4 Marks) h) Suppose a GARCH(1,1) process is chosen according to some ARCH/GARCH effect diagnostic checks, the following is the estimated variance equation of the model. Is the conditional variance process stationary What kind of post-estimation diagnostic checks you would suggest in order to check the model adequacy (4 Marks) TOTAL [25 Marks] – End of Question 2 – Paper Code: ECON920 Page 5 of 7 QUESTION 3 a) Let ! be a white noise process. Consider the processes ! = ! and ! = ( 1)!!. Show that (i) ! is covariance-stationary. (ii) ! + ! is nonstationary. (6 Marks) b) Consider an autoregressive model of order p as ! = + %!$% + “!$” + + &!$& + ! for = 1,2, … , and !~(0, “). Define the characteristic polynomial for this AR(p) model. What is the stationarity condition of the AR(p) model For what parameter restriction the AR(p) model have a unit root Is an AR(3) model with % = 0.5, ” = 0.3 and ‘ = 0.2 a unit root process (4 Marks) c) Suppose we want to test the presence of a unit root in an AR(p) model ! = + %!$% + “!$” + + &!$& + ! with !~(0, “). Write down the testable equation you will estimate in order to perform Augmented Dickey- Fuller (ADF) test and state clearly its lag order. Explain the details of the ADF test by clearly stating the null and alternative hypotheses and test statistic, and commenting on the limiting distribution of the test statistic under the null hypothesis and the rejection rule. (5 Marks) d) Explain the importance of including different deterministic trend (no intercept, with intercept, intercept and trend) in the ADF test. Describe which type of deterministic trend may be suitable for which type of trend pattern in nonstationary time series. Provide practical examples from macroeconomics or finance. (6 Marks) e) Consider the following estimated test regression model Δ! = K + M!$% + M + ! . Describe the dynamic properties of time series ! in terms of the potential existence of trend(s) assuming the following scenarios. (4 Marks) (i) None of K, M and M is statistically significant. (ii) Only K is statistically significant. (iii) Only M is statistically significant. (iv) All of K, M and M are statistically significant. TOTAL [25 Marks] – End of Question 3 – Paper Code: ECON920 Page 6 of 7 QUESTION 4 a) Consider a bivariate I(1) processes ! = (%! , “!)′. Suppose %! and “! are two independent random walks, discuss the consequences of the OLS regression %! = + “! + !. Specifically, comment on the properties of the OLS residuals M!, the coefficient estimates of the OLS regression, and the behaviours of the usual OLS test statistics for hypothesis testing. (6 Marks) b) Let a bivariate I(1) system ! = (! , !)( where ! denotes the log of spot prices and ! denotes the log of futures prices in the aluminium commodity market. Suppose the cost-of-carry model implies that spot and futures prices of a commodity are cointegrated with = (1, 1)′. (i) Write down the long-run equilibrium of the cost-of-carry relationship of !. (ii) Assume that there is no lagged term dynamics of the differenced series, write the error correction representation of the system !. (iii) Suppose the adjustment coefficients of !and ! are 0 and -0.4 respectively. Comment respectively when the disequilibrium error at time t-1 is positive and negative, what are the reactions of ! and ! at time t (6 Marks) c) Let !%)* be the 1-year market yield on US treasury securities, and !%+)* be the 10-year market yield on US treasury securities. The theoretical relationship between the short term and long term interest rates, known as the term structure of interest rates, is often modelled by their term spread, that is ! = + !%+)* !%)*. (i) See the following figure for the time series plot of the 1-year and 10-year market yield from 1962 to 2000. Comment on the pattern of the relationship between short term and the long term interest rates. (3 Marks) Paper Code: ECON920 Page 7 of 7 (ii) Describe the Engle-Granger approach to test the validity of this term spread relationship: ! = + !%+)* !%)*. (6 Marks) (iii) The ADF test for a unit root on the term spread between 1-year and the 10-year market yield has the following result, comment on the support of the theoretical term spread relationship. (4 Marks) TOTAL [25 Marks] – End of Question 4 –
