FINAL EXAM ECONOMICS

1. A researcher analyses a series from seasonal quarterly process: yt = ϕyt−4 + ϵt + θϵt−4, ϵt ∼ N(0, σ2 ) (1) Due to limited experience with R, the researcher does not plot the series, but rather decides to run ADF test for stationarity in levels and in first difference. ADF in level and in first difference are tested against the alternative with intercept. 1.1 Assume ϕ = 1, θ = 0. i. [2 marks] Solve for roots of the characteristic equation for (1). ii. [3 marks] Deduce the roots of the characteristic equation for ∆yt . iii. [4 marks] Compute theoretical values of α for Augmented Dickey Fuller (ADF) tests for xt = {yt , ∆yt} in Equation (2). ∆xt = δ + αxt−1 + X p i=1 ϕ ∗ i ∆xt−i + et (2) iv. [2 marks] Clearly specify H0 and HA hypotheses for ADF test and conclude whether ADF H0 is likely to be accepted or rejected for yt and ∆yt . v. [2 marks] Conclude whether ADF test for ∆yt correctly detects non-stationarity. vi. [13 marks] Compute autocorrelation functions for yt , ∆yt for t > 0 assuming yt = 0 for all t ≤ 0 using the following formula for arbitrary xt , xt−l : ρl = E[(xt − E(xt))(xt−l − E(xt−l))] p V ar(xt)V ar(xt−l) vii. [3 marks] Suggest a version of ADF test that allow to test correctly the process in Eq. (1) for non-stationarity. Carefully write down the corresponding test specifications, H0 and Ha. 1.2 Assume |ϕ| < 1, |θ| < 1. 3 of 7 ECON60522 i. [4 marks] The researcher decided to use Akaike Information Criterion to identify the AR order of the model for yt . What would you expect to observe for the optimal lag order for AR approximation of y as the sample size increases? Total: 33 4 of 7 ECON60522 2. VAR question 2.1 [2 marks] State the properties of the k × 1 vector white noise process ϵt . 2.2 Assume that you have a vector moving average process: yt = δ + Ψ0ϵ0 + Ψ1ϵt−1 + ϵt , Cov(ϵ0) = Cov(ϵt) = Σ, ∀t > 0 (3) where ϵt satisfies all vector white noise assumptions. i. [7 marks] Compute E(yt) and Cov(yt) for t = 1, 2, . . .. Conclude whether the process is stationary. ii. [7 marks] Compute the autocovariance matrices of yt: Note: Cov(yt, yt+s) = E(yt − E(yt))(yt+s − E(yt+s))′ for t = 1, 2, . . ., s = 1, 2, . . .. Think carefully about how many different covariances the process has. iii. [2 marks] Write down the definition of impulse response function associated with (2). iv. [4 marks] Using your impulse response function definition in Equation (3). How will yt react on a change in ϵt−s, s = 1, . . . , t for t > 2? v. [4 marks] Explain how to apply Cholesky decomposition to the model above. vi. [7 marks] Construct conditional expectations E(yt+h|It), It = {ϵt , ϵt−1, . . . , ϵ0} t > 1, h = 1, 2, 3, . . . and corresponding errors of forecast. Highlight the differences (if any) between E(yt) and E(yt+h|It) for large h. Total: 33 5 of 7 ECON60522 3. Cointegration 3.1 Assume that there are two independent random walk processes zt ∼ I(1), xt ∼ I(1), i.e. zt = zt−1 + et xt = xt−1 + ut (4) where et and ut are independent white noise processes. A 4 × 1 vector random process yt is formed from {xt}, {zt} of equation 4 as follows: yt =   zt xt xt−2 xt−4   , yt−1 =   zt−2 xt−2 xt−4 xt−6   , yt−2 =   zt−4 xt−4 xt−6 xt−8   i. [3 marks] Show that yt is cointegrated. Hint: check definition of cointegration ii. [3 marks] Represent the process of {yt , yt−1, yt−2, . . .} as VAR(1) and check whether ϵt satisfy the white noise assumption for error term. iii. [3 marks] How many linearly independent cointegrating vectors does the system have? iv. [4 marks] Represent the system in Vector Error Correction form. Explicitly write down a vector of cointegration relationships. Conclude whether the VAR system can be reduced without any loss of predictive power. Briefly explain. 3.2 The probability of the state of the economy of a particular country depends on two variables: inflation growth x and output growth y. All relevant information is presented in the following table: x > 0, y > 0 x ≤ 0, y > 0 x > 0, y ≤ 0 x ≤ 0, y ≤ 0 P(x, y|st = 0) 0 0.1 0.2 0.7 P(x, y|st = 1) 0.6 0.2 0.15 0.05 6 of 7 ECON60522 Transition probabilities between states are defined by P r(st = 1|st−1 = 1) = p = 0.8 P r(st = 0|st−1 = 0) = q = 0.6 Note: This information is relevant for the rest of the question. i. [2 marks] Due to the COVID pandemic, for some period of time only data for GDP growth was collected. Collapse the table so that it depends only on y, i.e. fill y > 0 y ≤ 0 P(y|st = 0) P(y|st = 1) ii. [6 marks] For the last quarter prior to the pandemics, a filtered probability of expansion is P(st−1 = 1|ψt−1) = 0.7, where ψt−1 = {xt−1, xt−2, . . . , yt−1, yt−2, . . .}. Compute a value of P(st = 1|ψt−1) and update this probability observing at time t only yt , yt ≤ 0, i.e. compute P(st = 1|ψt), ψt = {yt ≤ 0, ψt−1}. How different are the two? Briefly explain. iii. [8 marks] Due to quarantine neither data for CPI nor for GDP growth were collected at time t+ 1. The Office of Statistics was back online at time t+2 with the following CPI and GDP growth recordings: xt+2 > 0, yt+2 ≤ 0. Update the probability that the economy is in expansion state (st+2 = 1) given the data ψt+2 = {xt+2 > 0, yt+2 ≤ 0, ψt}. iv. [5 marks] Assume that GDP growth for two regimes follows two different AR(1) processes: yt = ϕ0,0 + ϕ1,0yt−1 + e0,t, e0,t ∼ IID(0, σ2 0 ) st = 0 yt = ϕ0,1 + ϕ1,1yt−1 + e1,t, e1,t ∼ IID(0, σ2 1 ) st = 1 Assume that P(st = 1|ψt) = 1. Compute MSE for one period ahead and two period ahead forecasts. Total: 34 7 of 7 END OF EXAMINATION