1 ECMT2130 – Final exam solutions
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 εt is white noise, is over-parameterised and write out the appropriately simplified model.
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xt = 0.5xt−1 + εt − 0.4εt−1 − 0.05εt−2
(b) (2 points) Given a large enough set of data on xt, what features would you expect to observe in
the autocorrelation function?
(c) (2 points) Given a large enough set of data on xt, 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?
(a) The model can be factored as:
xt − 0.5xt−1 = εt − 0.4εt−1 − 0.05εt−2 (1 − 0.5L)xt = (1 − 0.4L − 0.05L2)εt (1 − 0.5L)xt = (1 − 0.5L)(1 + 0.1L)εt
There is a common factor, (1 − 0.5L), so the ARMA(1,2) model is over parameterised. Cancelling the common factor, (1 − 0.5L), we get:
xt = (1 + 0.1L)εt
which is an invertible MA(1) model.
(b) The ACF would cut off abruptly after the autocorrelation at lag 1 because the model is a pure MA(1).
(c) The PACF would taper off to insignificant PACF estimates as the lag length increased. There would be no abrupt cut-off because the model is a pure MA(1).
(d) Any test for serial correlation is fine. We covered the Ljung-Box text but the Durbin-Watson or Breusch-Godfrey would also be acceptable. 1 point for naming the test. 2 points for identifying that there was a need to choose the number of lags to include in the test if using the Ljung-Box test.
It is not enough to say that there should be diagnostic testing of residuals to see if they are white noise. The answer must suggest a test to use.
1 mark off if the answer names multiple tests, some relevant and some not relevant.
No marks for suggesting a likelihood-ratio test applied to a model that may or may not be over-fitted. The likelihood-ratio test is a test of a set of joint restrictions on a model and in this context it is examining whether there are too many lags. The question asked for a diagnostic test exploring whether there are enough lags in whatever model had been estimated.
2. (?? points) Cherie’s unit root testing
Cherie is an analyst who believes that productivity shocks influence 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 definition 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 test, allowing for a drift and trend. The analyst includes 4 lagged differences of the logged data as regressors. The test statistic is -2.54. The critical value defining 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 significance. Perform the Augmented Dickey-Fuller test at the 1% level of significance, 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 first difference in the log-level of real industrial production? Explain your reasoning.
(a) Real industrial production grows over time according to an approximately exponential trend. Thus the mean of the process is a function of time. Weak stationarity requires that the mean and variance and autocovariances for the stochastic process are invariant to a time-shift. This is clearly not the case for real industrial production where the mean increases with time. Adding more detail that is not needed to get the points for the question, the trend appears to be stochastic because big shocks to the process, especially negative ones, do not appear to unwind over time. That suggests that the process is not stationary around a deterministic trend.
(b) There are 20 years of 4 observations per year so there are 80 observations. This is enough data to rely on the asymptotic properties of the ADF test.
let yt be the log of real industrial production in period t. The regression used for the ADF test is:
∆yt =α+βt+πyt−1 +γi∆yt−i +ut
(c) The null hypothesis is H0 : π = 0 implying at least one unit root in the log of real industrial production. The alternative hypothesis is H0 : π < 0 implying that there are no unit roots in the log of real industrial production. It is weakly stationary around a deterministic trend. Under the null hypothesis of at least one unit root, the test statistic has a distribution with lower tail critical values described in the distribution tables. The decision rule, at the 1% level of significance is, reject the null hypothesis if the test statistic, τ is less than the critical value, -4.00 and otherwise fail to reject the null hypothesis because there is insufficient evidence to do so. The test statistic estimate is -2.54. This does not lie in the rejection region so we do not have enough evidence at the 1% level of significance to reject the null hypothesis that the log of real industrial production has one or more unit roots. (d) I would use the first difference of logged real industrial production as an anomaly regressor in a CAPM model of excess financial returns (over the risk-free rate of return) because excess returns over the risk-free rate of return typically have a mean that does not trend upwards. The regression will be miss-specified if we try to explain variation in a non-trending variable with a stochastically trending variable. 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: σ2 = λr2 + (1 − λ)σ2 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 σt2−1 = 0.003. (a) (3 points) Showing your working, what is the estimate of σt2? (b) (3 points) Showing your working, what is the estimate of σ2 ? t+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 σt2 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. (a) Finding σt2: σt2 = 0.5×0.052 +0.5∗0.003 = 0.5 × 0.0025 + 0.0015 = 0.00125 + 0.0015 , note that our best guess of the unobserved squared return value in the forecast period t + 1 is the variance in period t + 1. Substituting and rearranging, we have: (1−λ)σ2 = (1−λ)σ2 (b) Finding σ2 t+1 (c) Decreasing λ puts less weight on volatility of the most recent rate of return. This would make the volatility estimation less sensitive to new data. A daily return that was a lot larger than average would drive up the volatility estimates much less with λ = 0.1. (d) Adding a working week additive seasonal, with a period equal to 5, the model would become: σ2 =λ(r2 −s )+(1−λ)σ2 Using this forecasting equation, we obtain: σ2 = 1 × σ2 t+1 t = 0.00275 ttt−5 t−1 st = γ(rt2 − σt2−1) + (1 − γ)st−5 4. (?? points) GARCH question (a) (2 points) What stylised facts characterise daily data on financial rates of return? (b) (2 points) Consider the Normal GARCH(1,1) model: rt − μ = εt = utσt σ2=a +aε2+bσ2 t 0 1t 1t−1 • 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 financial 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 fitting 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, εt, indicated a series of positive autocorrelations outside the 95% confidence interval and tapering off 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 specification (in part D) imply in relation to the weak form of the efficient markets hypothesis? (a) Daily financial returns are characterised by: Clustered sets of observations with high volatility (volatility clustering) Very little, if any, serial correlation (supporting the efficient markets hypothesis) Leptokurtosis (fat-tails in the distribution) suggesting deviation from normality Negative skew, with more negative extreme values than positive extreme values Leverage effects, whereby a large negative shock tends to cause volatility to rise more than a large positive shock. clustering and the fat tails of the distribution. It is crucial that it is recognised that the GARCH(1,1) model will not capture negative skew or leverage effects. For full points the answer should also indicate that the model will have zero serial correlation in daily financial returns (consistent with the weak form of the EMH). It is fine to also note that the model will be consistent with the weak stationarity of financial return because financial returns are I(0) in the model. (c) A GARCH model is more parsimonious than an ARCH model that allows for the same amount of volatility clustering so it avoids overfitting problems, Being more parsimonious, a GARCH model is less likely to breach the non-negativity constraints on the parameters. (d) The researcher could add auto-regressive or moving average structure to the equation for the level of daily returns. For example, it could be expressed as an ARMA(1,1) model: rt =α+φrt−1 +εt +θεt−1 crucial to state that the GARCH(1,1) model will be able to capture both the volatility (e) It would imply a violation of the EMH because it would mean that the level of returns are pre- dictable in a way that investors could profit from. Specifically, if an ARMA(1,1)-GARCH(1,1) model were true, investors could profit from the model predictions for future values of expected daily returns because those predictions would be better than predictions just based on the long-run mean daily return. 1 point if they know the weak form of the EMH but are relating it to a model adjustment with no implications for EMH. Full points if they got part D wrong but know the weak form of the EMH and explicitly state that the change they made has no implications for the EMH. 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com