Time Series Analysis and Stock Market Prices
Univariate Time Series Models
(M.Sc. Finance - Exercise 4)
Walter Distaso
Imperial College Business School w.distaso@imperial.ac.uk
Question 1
Consider the following three models that a researcher suggests might be reasonable models of stock market prices. yt yt yt = yt−1 + ut = 0.5yt−1 + ut = 0.8yt−1 + ut
(a) What classes of models are these examples of? (b) What would the autocorrelation function for each of these processes look like? (not exactly, just the shape) (c) Which model is more likely to represent stock market prices from a theoretical perspective, and why? If any of the three models truly represented the way stock market prices move, which could potentially be used to make money? (d) Consider the extent of persistence of shocks in the series in each case.
Question 2
You obtain the following estimates for an AR(2) model of some returns data. yt = 0.803yt−1 + 0.682yt−2 + ut , where ut is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity.
Question 3
A researcher is trying to determine the appropriate order of an ARMA model to describe some data, with 200 observations available. She has the following figures for the log of estimated residual variance (log(ˆ 2 )) for various candidate models. She has σ assumed that an order greater than (3,3) should not be necessary to model the dynamics of the data. What is the “optimal” model order? ARMA(p, q) model order log(ˆ 2 ) σ (0,0) 0.932 (1,0) 0.864 (0,1) 0.902 (1,1) 0.836 (2,1) 0.801 (1,2) 0.821 (2,2) 0.789 (3,2) 0.773 (2,3) 0.782 (3,3) 0.764
Question 4
“Given that the objective of any econometric modeling exercise is to find the model that most closely ‘fits’ the data, then adding more lags to an ARMA model will almost invariably lead to a better fit. Therefore, a large model is best because it will fit the data more closely.” Comment on the validity (or otherwise) of this statement.
Question 5
(a) You obtain the
(M.Sc. Finance - Exercise 4)
Walter Distaso
Imperial College Business School w.distaso@imperial.ac.uk
Question 1
Consider the following three models that a researcher suggests might be reasonable models of stock market prices. yt yt yt = yt−1 + ut = 0.5yt−1 + ut = 0.8yt−1 + ut
(a) What classes of models are these examples of? (b) What would the autocorrelation function for each of these processes look like? (not exactly, just the shape) (c) Which model is more likely to represent stock market prices from a theoretical perspective, and why? If any of the three models truly represented the way stock market prices move, which could potentially be used to make money? (d) Consider the extent of persistence of shocks in the series in each case.
Question 2
You obtain the following estimates for an AR(2) model of some returns data. yt = 0.803yt−1 + 0.682yt−2 + ut , where ut is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity.
Question 3
A researcher is trying to determine the appropriate order of an ARMA model to describe some data, with 200 observations available. She has the following figures for the log of estimated residual variance (log(ˆ 2 )) for various candidate models. She has σ assumed that an order greater than (3,3) should not be necessary to model the dynamics of the data. What is the “optimal” model order? ARMA(p, q) model order log(ˆ 2 ) σ (0,0) 0.932 (1,0) 0.864 (0,1) 0.902 (1,1) 0.836 (2,1) 0.801 (1,2) 0.821 (2,2) 0.789 (3,2) 0.773 (2,3) 0.782 (3,3) 0.764
Question 4
“Given that the objective of any econometric modeling exercise is to find the model that most closely ‘fits’ the data, then adding more lags to an ARMA model will almost invariably lead to a better fit. Therefore, a large model is best because it will fit the data more closely.” Comment on the validity (or otherwise) of this statement.
Question 5
(a) You obtain the