CS代考 ECMT2130 - 2021 semester 1 final exam 2 solutions

1 ECMT2130 – 2021 semester 1 final exam 2 solutions
1. (15 points) ARMA model over-parameterisation
Lucy wants to fit the following model to a univariate time-series:
Equation 1: xt = α + φ1xt−1 + εt + θ1εt−1

where the shocks are independently and identically distributed through time, εt ∼ 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 + φ1xt−1 + φ2xt−2 + εt + θ1εt−1 + θ2εt−2
where the shocks are independently and identically distributed through time, εt ∼ 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 -1.46
Standard error 1.09
The maximised log-likelihood function value for the ARMA(1,1) model is -1555.96.
Coefficient
α φ1 φ2 θ1 θ2
Point estimate -1.27
Standard error 0.85
The maximised log-likelihood function value for the ARMA(2,2) model is -1466.39.

(a) To be invertible, the roots of (1 − θ1L) = 0 to lie outside the unit circle. The single root is L = −1/θ1. In other words, we require:
(b) To be I(1), requires the roots of (1 − φ1L) = 0 to lie on the unit circle. The single root is
L = 1/φ1. In other words, we require:
(c) To be causal, the roots of (1 − φ1L) = 0 to lie outside the unit circle. The single root is
L = 1/φ1. In other words, we require:
(d) If Lucy works with an ARMA(1,1) model instead of an ARMA(2,2) model, then the residuals will continue to have autocorrelation structure that is not captured by the ARMA(1,1) struc- ture. This will show up in the ACF as significant autocorrelations at various lags, depending upon the coefficients of the ARMA(2,2) model. For many coefficient values, the significant autocorrelations in the ACF will taper off slowly because of autoregressive structure in the residuals of the estimated ARMA(1,1) model.
(e) Likelihood ratio test:
1. Test the hypothesis: H0: φ2 = θ2 = 0 against H1: φ2 ̸= 0 and/or θ2 ̸= 0 at the 5% level of significance.
2. The test statistic, LR, is computed as -2 times the difference between the restricted and unrestricted maximised log likelihood function values.
3. Under the null hypothesis, and assuming that the shocks are normally distributed, the sample is large enough to use the χ2 distribution (with 2 degrees of freedom because we are testing 2 restrictions jointly) for the LR test statistic.
4. The test is a one-sided upper-tail test. The critical value is χ2(0.95,2) = 5.9915. The decision rule is reject the null hypothesis if the test statistic lies above 5.9915. Otherwise, fail to reject the null hypothesis.
5. The LR test statistic is computed as LR = −2(Lr − Lu) = −2(−1555.96 − −1466.39) = 179.14.
6. The computed test statistic lies in the rejection region (179.14 > 5.9915). Thus, we reject the null hypothesis at the 5% level of significance. There is sufficient evidence to warrant including the second lag of the dependent variable and the seccond lag of the shock in the ARMA model.

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;
• rft: 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 −rft =αi +βi(rmt −rft)+γiSMBt +δiHMLt +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
αi βi γi δi
Point estimate 0.37
0.55 -0.33 -0.00
Standard error 0.21
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 0.33
Standard error 0.21
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.

(a) CAPM and APT assumptions.
The main assumptions underlying the version of the CAPM that allows investors access to a risk-free asset are:
• Investors are rational, mean-variance optimizers (or asset returns have a Joint Normal distribution).
• Investors share a single period planning horizon.
• Investors have homogeneous expectations. They share the same knowledge of the distri-
butions of returns on all risky assets and all relevant information is publicly available.
• All assets are infinitely divisible, publicly held, and trade on public exchanges.
• Investors can borrow or lend at a common risk-free rate without default risk.
• Investors can take both long and short positions on traded securities.
• There are no taxes or transaction costs (frictionless markets).
• Investors are price takers (no market power).
The main assumptions underlying the APT are:
• Arbitrage: Well-functioning security markets do not allow for the persistence of arbitrage opportunities.
• Factor models: Security returns can be described by a factor model – some factor model is “truth”.
• Diversification: There are sufficient securities to diversify away idiosyncratic risk associated with individual securities.
The APT assumptions are considerably more innocuous than those for the CAPM. While the assumption that there is a true factor model is somewhat challenging in the APT, it is not as hard to accept as the assumptions required to derive the CAPM. The CAPM assumptions are clearly unrealistic. While many of them can be somewhat relaxed in alternative CAPM formulations, the assumptions about tradeable assets, information, planning horizons, market frictions etc. are all clearly not reflections of the financial markets that we live with.
(b) The excess return on the market portfolio proxy and the SMB factor are both individually signficant at the 5% level, suggesting that the is sufficient evidence to include them in the asset pricing model. The HMB factor is not supported as an explanatory variable, at any reasonable level of significance.
(c) Exclusion test:
1. Testthehypothesis: H0: γi =δi =0againstH1: γi ̸=0and/orδi ̸=0atthe5%levelof significance.
2. The test statistic, F , is computed as:
F∗ = (Ru2 −Rr2)/2 = 0.04/2 =13.35
(1 − Ru2 )/(231 − 4) 0.34/227
3. Under the null hypothesis, and with the large sample, the test statistic can be assumed to have an F distribution with 2 numerator degrees of freedom and 227 denominator degrees of freedom.

4. The test is a one-sided test. At the 5% level of significance, with 2 and 227 degrees of freedom, the critical value is 3.036. The decision rule is: reject the null hypothesis if the test statistic lies above 3.036. Otherwise, fail to reject the null hypothesis.
5. The computed test statistic does lies in the rejection region. Thus, we reject the null hypothesis at the 5% level of significance. There is sufficient evidence to warrant including the two factors, SMB and HML, in the regression model.
(d) The findings in B and C suggest that the APT is a better model than the CAPM because factors other than the excess return on the proxy for the market portfolio are able to help explain variation in the excess returns on asset i. That result, may however, be because the proxy for the market portfolio is not on the efficient frontier. The result from part B suggest that the main additional factor is the SMB factor rather than the HML factor. The finding in part C just generalises this result from part B, testing joint exclusion restrictions for all “anomaly” factors that were included in the unrestricted regression.
(e) The robustness of the findings could be explored further by either performing the test across many assets, or performing the test across many time-periods, examining the stability of the coefficients and their significance, for each of the anomaly factors. Other suggestions should be considered on their merits.

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, rft
He uses the available data to estimate a CAPM model while ignoring clustered volatility. His results are shown in the table below.
Coefficient Intercept CAPM β
Point estimate -0.02
Standard error 0.03
(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 M 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]

(a) Coefficient restriction test:
1. Test the hypothesis: H0: β >= 1 against H1: β < 1 at the 5% level of significance. 2. The test statistic, t = βˆ−1 . SE(βˆ) 3. Under the null hypothesis, and assuming that the shocks are normally distributed,and given the the sample is large, it is reasonable to assume that the test statistic has a Student’s t distribution with N − 2 = 498 degrees of freedom. 4. The test is a one-sided lower-tail test. The critical value is t0.05,498 = −1.6479. The deci- sion rule is: reject the null hypothesis if the test statistic lies below −1.6479. Otherwise, fail to reject the null hypothesis. 5. The t test statistic is computed as t∗ = (0.93−1) = −0.88. 0.08 6. The computed test statistic lies outside of the rejection region (−0.88 > −1.6479). Thus, we fail to reject the null hypothesis at the 5% level of significance. There is insufficient evidence to suggest that the CAPM β is less than 1.
(b) Clustered volatility could be expected to impact upon inference using OLS estimates of the CAPM Beta if it caused heteroskedasticity. Using OLS estimates for hypothesis testing in the presence of heteroskedasticity will result in test statistics that do not have the usual Student’s t and F distributions because the estimates of the standard errors will be biased and inconsistent. In the context of this CAPM regression, clustered volatility would lead to heteroskedasticity problems when:
1. the variation and clustered volatility in excess returns on the market did not fully explain the clustering of volatility in the dependent variable, the excess return on the asset of interest so that the error terms in the CAPM regression feature clustering of volatility; and
2. That time-variation in the variance of the CAPM regression equation variance is dependent upon the regressor, the excess returns on the market.
(c) Engle’s LM ARCH test:
1. Test the hypothesis: H0: The error term in the model does not feature ARCH structure using 10 lags against H1: the error term in the CAPM model does feature ARCH structure using 10 lags, at the 5% level of significance.
2. Under the null hypothesis, and assuming that the shocks are normally distributed, and given the the sample is large, the LM test statistic has a Chi-squared distribution with 10 degrees of freedom.
3. The test is a one-sided upper-tail test. The critical value is χ20.05,10 = 18.3070. The decision rule is: reject the null hypothesis if the test statistic lies above 18.3070. Otherwise, fail to reject the null hypothesis.
4. The test statistic, LM = 19.354.
5. The computed test statistic lies inside the rejection region (19.354 > 18.3070). Thus, we reject the null hypothesis at the 5% level of significance. There is sufficient evidence to suggest that the CAPM regression has error terms that feature clustered volatility.
(d) The findings from Engle’s LM ARCH test suggest that there is remaining clustered volatility in the errors of the CAPM regression. Thus, there is potential for clustered volatility to cause heteroskedasticity, if this clustering of high volatility is dependent, in some way, upon the excess returns on the market, the one regressor.

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