Microsoft Word – Tutorial 3.docx
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ECON3206/5206 Financial Econometrics
Tutorial 3
Question 1. Consider the AR(1) model
(a) Calculate unconditional and .
(b) What is the (optimal) forecast of on the basis of time t information?
(c) Calculate conditional variance
and form confidence interval for forecast.
(d) Is a white noise process?
(e) When is a covariance stationary process?
(f) Think about an economic example where AR(1) is relevant?
Question 2. Suppose that a researcher estimated the lag 1 autocorrelation coefficient using a series of
T=100 observations, and found it to be equal to 0.15. Is the autocorrelation coefficient significantly
different from 0? Specify the null hypothesis, the alternative, test statistics, null distribution and
decision criterion.
Question 3. Find the least squares estimator of the coefficient in the AR(1) model
*[Show also the under the null hypothesis of the correlation coefficient being zero, the OLS estimator
of , , is asymptotically normally distributed with mean zero and variance 1/T.
You need to use the following elements:
1. Normality of OLS (MLE) estimator
2. Computation of the variance of the OLS estimator for large T.]
Question 4.
Let 𝑓!”#|! be the forecast based on Ω!. Namely, 𝑓!”#|! is a function of elements in Ω!. Which 𝑓!”#|!
minimises the mean square forecast error (MSFE)?
𝑀𝑆𝐹𝐸 = 𝐸[*𝑦!”# − 𝑓!”#|!-
= + +t t ty by
2(0, )t WNe s!
( ), var( )t tE y y corr( , ) for 1,2- =t t iy y i
, for 1,2
var( | )t ty + W
,t t ty by
2(0, )t WNe s!
*[Proof your answer formally
Hint: there are several ways to proof this.
Option 1. You may explicitly write down the definition of the expectation in terms of the integral
(sum for discrete rv, but we typically use continuous rv in time series). Be careful to specify the
correct conditional expectation, 𝑦!”# is the random variable here. Take non-random terms outside of
the expectation and take the derivative.
Option 2. Subtract and add the correct answer in the squared term. Open the squares
(a+b)2.= a2 +2ab+b2 . Show that the term 2ab is equal to zero using the properties of the conditional
expectation. After this, the answer follows automatically as a2 term is not a function of 𝑓!”#|!]
5. Estimating the CAPM and making sense of betas
Open the file CAPM.XLSX which contains the following daily data for 40 years starting on 12
August 1975 and ending on 12 August 2015 (source Datastream):
Gold Bullion LBM U$/ : the price of Gold
S&P 500 COMPOSITE – PRICE INDEX: a proxy for market portfolio
US T-BILL SEC MARKET 3 MONTH: a proxy for risk-free rate (annualized)
GENERAL ELECTRIC: the price of General Electric (GE) shares
Note: GE is one of the oldest companies in the index. It was founded in late 1800s. One of its co-
founders, , is the inventor of a commercially viable light bulb.
(a) Note that the Tbill interest is quoted on annual basis while the other returns are
daily returns. Transform the annual returns to daily returns using compounding
formula: (1+Rd) = (1+Ry)1/360 . Note you may check that the answer is similar to
the one where you simply approximate Rd by Ry/360.
(b) Calculate the (log) returns for gold gold_r, S&P500 sp500_r and ge, ge_r
(c) Calculate the corresponding excess returns gold_re, sp500_re, sp500_re
(d) Plot the excess return of gold “gold_re” against the excess market return
“sp500_re”.
(e) Do the same for ge excess return
(f) Estimate the CAPM models.
(g) Inspect the estimation output table. Is the CAPM supported by Gold and GE data?
Interpret the estimated beta coefficients. Comment on the R-square and the DW
statistics.
(h) Find time series plot for the residuals, actual and fitted. Find the histogram for the
residuals? Is the error term is normally distributed?
(i) Test for heteroskedasticity in the residuals.
(j) Test for autocorrelation in the residuals.
(k) Is the model stable?
These are less routine, but more interesting questions
(l) Construct a portfolio based on the market portfolio (S&P 500) and risk-free T-Bill
which would yield the same expected return as Gold and GE.
(m) Verify that the expected returns of the original assets and the corresponding
constructed portfolios are the same.
(n) Compare the risk (standard deviation) of the original assets and the portfolio
replicating the expected returns of these assets. Decompose the risk (st. dev) into
the systematic and idiosyncratic risk.
(o) Where would you place Gold on the efficiency frontier, capital allocation line
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