ECMT2130 - 2021 semester 1 final exam 2
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ECMT2130 - 2021 semester 1 final exam 2
Author: Geoff Shuetrim
1. (15 points) ARMA model over-parameterisation
Lucy wants to fit the following model to a univariate time-series:
Equation 1: xt = α + φ 1 xt _ 1 + ∈t + θ1 ∈t _ 1
where the shocks are independently and identically distributed through time, ∈t ~ N ╱0, σ2 、 (a) (2 points) What conditions on the model coefficients must be satisfied for the model to be invertible? (b) (2 points) What conditions on the model coefficients must be satisfied for the model to be I(1)?
(c) (2 points) What conditions on the model coefficients must be satisfied for the model to be causal?
(d) (3 points) What should Lucy expect the autocorrelation function of the residuals to look like if the true stochastic process is given by equation 2 but she fits equation 1? Explain your answer. Equation 2: xt = α + φ 1 xt _ 1 + φ2 xt _2 + ∈t + θ1 ∈t _ 1 + θ2 ∈t _2
where the shocks are independently and identically distributed through time, ∈t ~ N ╱0, σ2 、 (e) (6 points) Estimates of equation 1 and equation 2 are provided in the tables below. Use those
estimates to conduct a Likelihood ratio test of the null hypothesis:
H0 : θ2 = φ2 = 0
against a suitable alternative at the 5% level of significance. Write up all steps in the likelihood ratio test.
Equation 1: ARMA(1,1) model estimates
Coefficient |
Point estimate |
Standard error |
α φ 1 θ 1 |
-1.46 0.97 0.04 |
1.09 0.01 0.03 |
The maximised log-likelihood function value for the ARMA(1,1) model is -1555.96.
Coefficient |
Point estimate |
Standard error |
α φ 1 φ2 θ 1 θ2 |
-1.27 0.65 0.28 0.42 0.49 |
0.85 0.06 0.06 0.05 0.03 |
The maximised log-likelihood function value for the ARMA(2,2) model is -1466.39.
2. (15 points) Agnew’s Empirical Asset Pricing Model tests
Agnew wants to test whether the model for financial returns based upon the Arbitrage Pricing Theory (APT) can be simplified to a model for financial returns based upon the Capital Asset Pricing Model (CAPM).
(a) (5 points) Compare the assumptions underlying the risk-free asset version of the CAPM and the
APT. Which set of assumptions are more unlikely to hold. Explain why.
He has access to 231 monthly observations on the following variables:
● rit : the simple monthly rates of return for asset i;
● rmt : the simple monthly rates of return for the proxy of the market portfolio;
● rf t : the simple monthly rates of return for the risk-free rate of return;
● SMBt : the difference between the simple monthly rates of return for a portfolio of small market capitalisation companies and a portfolio of large market capitalisation companies; and
● HMLt : the difference between the simple monthly rates of return for a portfolio of high book- to-market value companies and a portfolio of low book-to-market value companies.
He estimates the following 3 factor model for excess returns on asset i as a function of excess returns on the market portfolio, the SMB factor and the HML factor.
rit - rf t = αi + βi (rmt - rf t ) + γi SMBt + δi HMLt + uit
where uit is the error term.
The OLS estimates of the model coefficients are shown in the table below.
3 factor model OLS estimates
Coefficient |
Point estimate |
Standard error |
αi βi γi δi |
0.37 0.55 -0.33 -0.00 |
0.21 0.05 0.09 0.07 |
The R-squared for the 3 factor model is 0.34.
(b) (3 points) Indicate which factors have sufficient evidence to warrant including them in the factor model, assessed at the 5% level of significance. You do not need to write up the full details of the hypothesis test for each exclusion restriction.
(c) (5 points) He also estimates a one factor model for excess returns on asset A as a function of excess returns on the proxy of the market portfolio. The results are shown below. Use these results and those reported in part B to test the joint exclusion restrictions on the SMB and HML factors, at the 5% level of significance. Include all steps in this hypothesis test.
1 factor model OLS estimates
Coefficient |
Point estimate |
Standard error |
αi βi |
0.33 0.48 |
0.21 0.05 |
The R-squared for the 3 factor model is 0.30.
(d) (1 point) Explain the Asset Pricing Model implications of the findings in parts B and part C. (e) (1 point) What is one way that he could assess the robustness of the findings in parts B and C.
3. (15 points) Angus’ CAPM+GARCH model
Angus is concerned about ignoring clustered volatility when estimating the CAPM beta for a financial asset using daily data on rates of return for that asset, a proxy of the market portfolio, and a risk-free asset.
He has access to 500 observations (for time periods t = 1, 2, 3, ..., 500) on:
● the daily rate of return on the asset of interest, rit
● the daily rate of return on the proxy for the market portfolio, rmt
● the daily rate of return on the risk-free asset, rf t
He uses the available data to estimate a CAPM model while ignoring clustered volatility. His results are shown in the table below.
Coefficient |
Point estimate |
Standard error |
Intercept CAPM β |
-0.02 0.93 |
0.03 0.08 |
(a) (5 points) Ignoring clustered volatility, test the null hypothesis that the CAPM Beta of the financial asset is greater than or equal to 1 at the 5% level of significance.
(b) (3 points) In what circumstances would clustered volatility be expected to impact upon inference using OLS estimates of the CAPM Beta?
To assess whether clustered volatility is an issue, he uses the Engle LM test for ARCH effects in the residuals from the regression estimated for part A. The test is conducted with 10 lags. The test statistic is 19.354.
(c) (5 points) Report the hypothesis test, conducted at the 5% level of significance, including all steps.
(d) (2 points) What are the implications of your results in part C for Angus’ concerns about ignoring clustered volatility when estimating the CAPM Beta? [2 points]
2022-06-17