ECMT2130 – Final exam

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ECMT2130 – Final exam

Author: Geo Shuetrim

1. ( points) Amanda’s ARMA model

Amanda is wanting to model a univariate time-series using an autoregressive moving average model.

(a) (4 points) Show that the following model, where ∈t is white noise, is over-parameterised and write out the appropriately simpli ed model.

北t = 0.5北t 1 + ∈t _ 0.4∈t 1 _ 0.05∈t 2

(b) (2 points) Given a large enough set of data on 北t , what features would you expect to observe in the autocorrelation function

(c) (2 points) Given a large enough set of data on 北t , what features would you expect to observe in the partial autocorrelation function

(d) (2 points) Assuming you did not know the true model, suggest a diagnostic test that would help in assessing whether the estimated model had enough lags

2. ( points) Cherie’s unit root testing

Cherie is an analyst who believes that productivity shocks in uence asset returns and so she wants to include US real industrial production in a multi-factor asset pricing model, based upon the Arbitrage Pricing Theory. Her research assistant has downloaded an Industrial Production Index that measures real output for all facilities located in the United States manufacturing, mining, and electric, and gas utilities (excluding those in U.S. territories). The data is shown in the graph below:

(a) (4 points) From visual inspection, is real industrial production in the US weakly stationary Ex- plain your response in terms of the de nition of weakly stationary time series.

(b) (4 points) The analyst takes the natural log of the data before beginning further data analysis. She then tests for a unit root using the logged data from January 2000 through to the December 2019, using an Augmented Dickey Fuller test, allowing for a drift and trend. The analyst includes 4 lagged di erences of the logged data as regressors. The test statistic is -2.54. The critical value de ning the rejection region for the Augmented Dickey-Fuller test with the number of observations that have been used is as -4.00 at the 1% level of signi cance. Perform the Augmented Dickey-Fuller test at the 1% level of signi cance, stating both the null hypothesis and the alternative hypothesis, the decision rule, and the conclusion of the test.

(c) (2 points) Given the test result, should the analyst introduce a factor into the multifactor asset pricing model that is based on the log-level of real industrial production or the rst di erence in the log-level of real industrial production Explain your reasoning.

3. ( points) EWMA for Q-Group

Fund manager, Q-Group, wants to be able to make return volatility predictions. They use the Exponen- tially Weighted Moving Average (EWMA) technique to model the daily volatility of the rate of return on a security as:

σt(2) = λrt(2) + (1 _ λ)σt(2) 1

where rt is the centred daily rate of return on the security (the daily rate of return after subtracting the average daily rate of return over the sample).

λ = 0.5, and given rt == 5% and σt(2) 1 = 0.003.

(a) (3 points) Showing your working, what is the estimate of σt(2)

(b) (3 points) Showing your working, what is the estimate of σt(2)+1

(c) (2 points) If the fund manager increased the value of λ from 0.5 to 0.1, what would that imply about the impact, on volatility estimates, of a daily return that was a lot larger than average

(d) (2 points) The fund manager wants to extend the model of σt(2) to include an additive seasonal component with a cycle through the 5 working days of the week. Write out the Holt-Winters model that they should use.

4. ( points) GARCH question

(a) (2 points) What stylised facts characterise daily data on nancial rates of return (b) (2 points) Consider the Normal GARCH(1,1) model:

rt _ μ = ∈t = ut σt

σt(2) = a0 + a1 ∈t(2) + b1 σt(2) 1

where:

. ut ~ N (0, 1) is independently and identically distributed over time;

. a0 and a1 are both strictly greater than 0;

. b1 is greater than or equal to 0; and

. a1 + b1 < 1 and b1 < 1. Which of the stylised facts about nancial returns can be characterised by this model (c) (2 points) What would motivate you to use the GARCH model in part B instead of an ARCH model (d) (2 points) After tting the GARCH(1,1) model to daily returns on the broad stock-market index a researcher found that the autocorrelation function (ACF) of the residuals, ∈t , indicated a series of positive autocorrelations outside the 95% con dence interval and tapering o to zero at longer lags. How might the researcher adjust the GARCH(1,1) model in response (e) (2 points) What would your suggested change to the model speci cation (in part D) imply in relation to the weak form of the e cient markets hypothesis