ECMT3150: The Econometrics of Financial Markets (Semester 1, 2022) Tutorial 1a
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ECMT3150: The Econometrics of Financial Markets
(Semester 1, 2022)
Tutorial 1a
1. Consider the AR(2) model:
yt = μ + ut ;
ut = o1 ut — 1 + o2 ut —2 + "t :
(a) Show that the autocovariance function is
y0 = 
(1 _ o2 );
y 1 = 
o1 ;
yj = o1yj — 1 + o2yj —2 for j 2 2;
y —j = yj ;
where D 三 (1 + o2 )(1 + o1 _ o2 )(1 _ o1 _ o2 ).
(b) Show that the autocorrelation function (ACF) is
y 1 o1
y0 1 _ o2 ;
pj = o1 pj — 1 + o2 pj —2 for j 2 2;
p —j = pj :
2. [Assignment 1, 2019s1] Consider the AR(2) process
yt = 0:3 _ 0:5yt — 1 + 0:1yt —2 + "t :
where "t ~ wn(0; 0:01).
(a) Is the process stationary? Explain.
(b) Is the process invertible? Explain.
(c) Compute the unconditional mean of yt .
(d) Compute the unconditional variance of yt .
(e) Compute the autocorrelations of yt at lags 1, 2 and 3.
Suppose the forecast origin is at time t, and that yt = 0:3 and yt — 1 = 0:2.
(f) What is the 1-step ahead forecast estimate? What is its standard error? (g) What is the 2-step ahead forecast estimate? What is its standard error?
3. [Assignment 1, 2020s2] Suppose {yt } is a covariance stationary time series generated from the process:
yt = 1 + ayt —5 + "t ;
where "t ~ wn(0; 0:1) and |a| < 1.
(a) Compute the unconditional mean of yt in terms of a.
(b) Compute the unconditional variance of yt in terms of a.
(c) Derive the autocorrelation function of yt in terms of a.
(d) Suppose a = 0:5. Plot the autocorrelogram of yt .
(e) Suppose a = 0:5 and y100 = 1.
i. Compute the j-step ahead forecast y^100 (j) for j = 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 and the standard deviations of their forecast errors.
ii. What is the o-step ahead forecast and the standard deviation of its forecast error?
4. [Assignment 1, 2020s2] Suppose {yi } is generated from the process deÖned by yi = zi + zi — 1 , where zi ~ iid N(0; 1).
(a) Derive the moment generating function (mgf) of yi . [Hint: use the result of Q1(a) in Tutorial 0.]
(b) Derive the mgf of E
Ai yi (A1 ;A2 ;:::;An are Önite real numbers).
(c) Derive the mgf of E
Ai yi+k .
(d) Is {yi } strictly stationary? Explain. [Hint: use (b) and (c), and that the mgf of E
Ai yi fully characterises the joint distribution of (y1 ;y2 ;:::;yn ).]
(e) Is {yt } covariance stationary? Explain.
5. [R corner] We want to get a feeling of how the sample paths of an autoregressive model look like. To keep things simple, let us focus on an AR(1) model yt = oyt — 1 +ut , where ut are iid N(0; 1) and o = 0:5.
(a) Simulate Öve sample trajectories, each of length 100, from the AR(1) model. Plot the sample trajectories.
(b) Plot the sample ACF and PACF based on the Örst sample trajectory. Do they make sense based on your understanding on the AR(1) model?
(c) Staring at the sample trajectories in (a), you wonder whether they are stationary or not. Conduct a hypothesis testing by applying the ADF test on the Örst sample trajectory.
(d) Repeat (a) for other values of o (e.g., o = 0, o = 1). Describe how the sample trajec- tories behave di§erently as you change the value of o.
(e) Repeat (b) for o = 1. Does the sample ACF look di§erently from what you would expect? If so, what is a possible reason for the di§erence?
(f) Repeat (c) for o = 1. Does your conclusion change or not? If so, how?
6. [RT Ex: 2.3] Consider the monthly U.S. unemployment rate from January 1948 to March
2009 in the Öle m-unrate .txt. The data are seasonally adjusted and obtained from the Federal Reserve Bank of St Louis.
Build a time series model for the series and use the model to forecast the unemployment rate for the April, May, June, and July of 2009. In addition, does the Ötted model imply the existence of business cycles? Why? (Note that there may be multiple models that Öt the data well. You only need an adequate model.)
2023-04-05