MSc Introductory Financial Econometrics
MGT 5380
Academic Year 2020-21
Practice Degree Exam
Answer Key

Version 1: Initially posted on 2021 July 24
Version 2: A2 added

A1
(1) Trivial t-test.

(2) Trivial t-test.

(3) No.

(4) F-test

(5) See Brooks 3rd Edition Section 5.4.1

A2
(1) - (4) See Brooks 3rd edition Sections 4.6
(5) See Brooks 3rd edition Box 4.1
B1
(1) The Durbin Watson test is a test for first order autocorrelation. The test is calculated as follows.
You would run whatever regression you were interested in, and obtain the residuals. Then calculate
the statistic
( )
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=
=
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=
T
t
t
T
t
tt
u
uu
DW
2
2
2
2
1
?
??
You would then need to look up the two critical values from the Durbin Watson tables, and these
would depend on how many variables and how many observations and how many regressors
(excluding the constant this time) you had in the model. The rejection / non-rejection rule would be
given by selecting the appropriate region from the following diagram:

(2) We have 60 observations, and the number of regressors excluding the constant term is 3. The
appropriate lower and upper limits are 1.48 and 1.69 respectively, so the Durbin Watson is lower than
the lower limit. It is thus clear that we reject the null hypothesis of no autocorrelation. So it looks like
the residuals are positively autocorrelated.

(3) ttttt uxxxy +?+?+?+=? 4433221 ????
The problem with a model entirely in first differences, is that once we calculate the long run
solution, all the first difference terms drop out (as in the long run we assume that the values
of all variables have converged on their own long run values so that yt = yt-1 etc.) Thus when
we try to calculate the long run solution to this model, we cannot do it because there isn’t a
long run solution to this model/

(4) tttttttt vXXxxxxy ++++?+?+?+=? ??? 1471361254433221 ???????
The answer is yes, there is no reason why we cannot use Durbin Watson in this case. You may have
said no here because there are lagged values of the regressors (the x variables) variables in the
regression. In fact this would be wrong since there are no lags of the DEPENDENT (y) variable and
hence DW can still be used.

(5) See Brooks 3rd edition Section 5.5.7 (Dealing with autocorrelation).

B2
Coming Soon.
C1
(1) The first two models are roughly speaking AR(1) models, while the last is an MA(1). Strictly, since
the first model is a random walk, it should be called an ARIMA(0,1,0) model, but it could still be viewed
as a special case of an autoregressive model.

(2) We know that the theoretical acf of an MA(q) process will be zero after q lags, so the acf of the
MA(1) will be zero at all lags after one. For an autoregressive process, the acf dies away gradually. It
will die away fairly quickly for case (2), with each successive autocorrelation coefficient taking on a
value equal to half that of the previous lag. For the first case, however, the acf will never die away,
and in theory will always take on a value of one, whatever the lag.

Turning now to the pacf, the pacf for the first two models would have a large positive spike at lag 1,
and no statistically significant pacf’s at other lags. Again, the unit root process of (1) would have a pacf
the same as that of a stationary AR process. The pacf for (3), the MA(1), will decline geometrically.

(3) Clearly the first equation (the random walk) is more likely to represent stock prices in practice. The
discounted dividend model of share prices states that the current value of a share will be simply the
discounted sum of all expected future dividends. If we assume that investors form their expectations
about dividend payments rationally, then the current share price should embody all information that
is known about the future of dividend payments, and hence today’s price should only differ from
yesterday’s by the amount of unexpected news which influences dividend payments.

Thus stock prices should follow a random walk. Note that we could apply a similar rational
expectations and random walk model to many other kinds of financial series. If the stock market really
followed the process described by equations (2) or (3), then we could potentially make useful forecasts
of the series using our model. In the latter case of the MA(1), we could only make one-step ahead
forecasts since the “memory” of the model is only that length. In the case of equation (2), we could
potentially make a lot of money by forming multiple step ahead forecasts and trading on the basis of
these.

Hence after a period, it is likely that other investors would spot this potential opportunity and hence
the model would no longer be a useful description of the data.

(4) See the book for the algebra. This part of the question is really an extension of the others. Analysing
the simplest case first, the MA(1), the “memory” of the process will only be one period, and therefore
a given shock or “innovation”, ut, will only persist in the series (i.e. be reflected in yt) for one period.
After that, the effect of a given shock would have completely worked through.

For the case of the AR(1) given in equation (2), a given shock, ut, will persist indefinitely and will
therefore influence the properties of yt for ever, but its effect upon yt will diminish exponentially as
time goes on.

In the first case, the series yt could be written as an infinite sum of past shocks, and therefore the
effect of a given shock will persist indefinitely, and its effect will not diminish over time.

(5) See Brooks 3rd edition Section 6.7.1 (Information criteria for ARMA model selection).

C2
Coming Soon.