Solutions to the Review Questions at the End of Chapter 5
1. Autoregressive models specify the current value of a series yt as a function of
its previous p values and the current value an error term, ut, while moving
average models specify the current value of a series yt as a function of the
current and previous q values of an error term, ut. AR and MA models have
different characteristics in terms of the length of their “memories”, which has
implications for the time it takes shocks to yt to die away, and for the shapes of
their autocorrelation and partial autocorrelation functions.
2. ARMA models are of particular use for financial series due to their
flexibility. They are fairly simple to estimate, can often produce reasonable
forecasts, and most importantly, they require no knowledge of any structural
variables that might be required for more “traditional” econometric analysis.
When the data are available at high frequencies, we can still use ARMA models
while exogenous “explanatory” variables (e.g. macroeconomic variables,
accounting ratios) may be unobservable at any more than monthly intervals at
best.
3.
yt = yt-1 + ut
yt = 0.5 yt-1 + ut
yt = 0.8 ut-1 + ut
(1)
(2)
(3)
(a) 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.
(b) 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.
(c) 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,
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and hence today’s price should only differ from yesterdays 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.
(d) 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.
4. (a) Box and Jenkins were the first to consider ARMA modelling in this
logical and coherent fashion. Their methodology consists of 3 steps:
Identification - determining the appropriate order of the model using
graphical procedures (e.g. plots of autocorrelation functions).
Estimation - of the parameters of the model of size given in the first stage. This
can be done using least squares or maximum likelihood, depending on the
model.
Diagnostic checking - this step is to ensure that the model actually estimated is
“adequate”. B & J suggest two methods for achieving this:
- Overfitting, which involves deliberately fitting a model larger than
that suggested in step 1 and testing the hypothesis that all the additional
coefficients can jointly be set to zero.
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- Residual diagnostics. If the model estimated is a good description of
the data, there should be no further linear dependence in the residuals of the
estimated model. Therefore, we could calculate the residuals from the
estimated model, and use the Ljung-Box test on them, or calculate their acf. If
either of these reveal evidence of additional structure, then we assume that the
estimated model is not an adequate description of the data.
If the model appears to be adequate, then it can be used for policy analysis and
for constructing forecasts. If it is not adequate, then we must go back to stage 1
and start again!
(b) The main problem with the B & J methodology is the inexactness of the
identification stage. Autocorrelation functions and partial autocorrelations for
actual data are very difficult to interpret accurately, rendering the whole
procedure often little more than educated guesswork. A further problem
concerns the diagnostic checking stage, which will only indicate when the
proposed model is “too small” and would not inform on when the model
proposed is “too large”.
(c) We could use Akaike’s or Schwarz’s Bayesian information criteria. Our
objective would then be to fit the model order that minimises these.
We can calculate the value of Akaike’s (AIC) and Schwarz’s (SBIC) Bayesian
information criteria using the following respective formulae
AIC = ln (  2 ) + 2k/T
SBIC = ln (  2 ) + k ln(T)/T
The information criteria trade off an increase in the number of parameters and
therefore an increase in the penalty term against a fall in the RSS, implying a
closer fit of the model to the data.
5. The best way to check for stationarity is to express the model as a lag
polynomial in yt.
y t  0.803 y t 1  0.682 y t 2  ut
Rewrite this as
yt (1  0.803 L  0.682 L2 )  ut
We want to find the roots of the lag polynomial (1  0.803 L  0.682 L2 )  0 and
determine whether they are greater than one in absolute value. It is easier (in
my opinion) to rewrite this formula (by multiplying through by -1/0.682,
using z for the characteristic equation and rearranging) as
z2 + 1.177 z - 1.466 = 0
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Using the standard formula for obtaining the roots of a quadratic equation,
z
 1177
.
 1177
. 2  4 * 1 * 1466
.
= 0.758 or 1.934
2
Since ALL the roots must be greater than one for the model to be stationary,
we conclude that the estimated model is not stationary in this case.
6. Using the formulae above, we end up with the following values for each
criterion and for each model order (with an asterisk denoting the smallest
value of the information criterion in each case).
ARMA (p,q) model order
(0,0)
(1,0)
(0,1)
(1,1)
(2,1)
(1,2)
(2,2)
(3,2)
0.842*
(2,3)
(3,3)
log (  2 )
0.932
0.864
0.902
0.836
0.801
0.821
0.789
0.773
AIC
0.942
0.884
0.922
0.866
0.841
0.861
0.839
0.833*
SBIC
0.944
0.887
0.925
0.870
0.847
0.867
0.846
0.782
0.764
0.842
0.834
0.851
0.844
The result is pretty clear: both SBIC and AIC say that the appropriate model is
an ARMA(3,2).
7. We could still perform the Ljung-Box test on the residuals of the estimated
models to see if there was any linear dependence left unaccounted for by our
postulated models.
Another test of the models’ adequacy that we could use is to leave out some of
the observations at the identification and estimation stage, and attempt to
construct out of sample forecasts for these. For example, if we have 2000
observations, we may use only 1800 of them to identify and estimate the
models, and leave the remaining 200 for construction of forecasts. We would
then prefer the model that gave the most accurate forecasts.
8. This is not true in general. Yes, we do want to form a model which “fits” the
data as well as possible. But in most financial series, there is a substantial
amount of “noise”. This can be interpreted as a number of random events that
are unlikely to be repeated in any forecastable way. We want to fit a model to
the data which will be able to “generalise”. In other words, we want a model
which fits to features of the data which will be replicated in future; we do not
want to fit to sample-specific noise.
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This is why we need the concept of “parsimony” - fitting the smallest possible
model to the data. Otherwise we may get a great fit to the data in sample, but
any use of the model for forecasts could yield terrible results.
Another important point is that the larger the number of estimated
parameters (i.e. the more variables we have), then the smaller will be the
number of degrees of freedom, and this will imply that coefficient standard
errors will be larger than they would otherwise have been. This could lead to a
loss of power in hypothesis tests, and variables that would otherwise have
been significant are now insignificant.
9. (a) We class an autocorrelation coefficient or partial autocorrelation
1
coefficient as significant if it exceeds  1.96
=  0.196. Under this rule, the
T
sample autocorrelation functions (sacfs) at lag 1 and 4 are significant, and the
spacfs at lag 1, 2, 3, 4 and 5 are all significant.
This clearly looks like the data are consistent with a first order moving average
process since all but the first acfs are not significant (the significant lag 4 acf is
a typical wrinkle that one might expect with real data and should probably be
ignored), and the pacf has a slowly declining structure.
(b) The formula for the Ljung-Box Q* test is given by
m
Q*  T (T  2)
k 1
 k2
T k

m2
using the standard notation.
In this case, T=100, and m=3. The null hypothesis is H0: 1 = 0 and 2 = 0 and
3 = 0. The test statistic is calculated as
 0.420 2 0.104 2 0.032 2 
Q*  100  102  


  19.41.
100  1 100  2 100  3 
The 5% and 1% critical values for a 2 distribution with 3 degrees of freedom
are 7.81 and 11.3 respectively. Clearly, then, we would reject the null
hypothesis that the first three autocorrelation coefficients are jointly not
significantly different from zero.
10. (a) To solve this, we need the concept of a conditional expectation,
i.e. Et 1 ( yt yt 2 , yt 3 ,...)
For example, in the context of an AR(1) model such as , yt  a0  a1 yt 1  ut
If we are now at time t-1, and dropping the t-1 subscript on the expectations
operator
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E ( yt )  a0  a1 yt 1
E ( yt 1 )  a0  a1 E ( yt )
= a0  a1 yt 1 (a0  a1 yt 1 )
a0  a0a1  a12 yt 1
E ( yt 2 )  a0  a1 E ( yt 1 )
= a0  a1 (a0  a1 E ( yt ))
=
etc.
=
a0  a0a1  a12 E ( yt )
=
a0  a0a1  a12 E ( yt )
=
a0  a0a1  a12 (a0  a1 yt 1 )
=
a0  a0a1  a12 a0  a13 yt 1
f t 1,1  a 0  a1 yt 1
f t 1,2  a 0  a1 f t 1,1
f t 1,3  a0  a1 f t 1,2
To forecast an MA model, consider, e.g.
yt  ut  b1ut 1
E ( yt yt 1 , yt 2 ,...)
So
ft-1,1
=
=
E (u t  b1u t 1 )
b1u t 1
=
=
E (u t 1  b1u t )
0
b1u t 1
But
E ( yt 1 yt 1 , yt 2 ,...)
Going back to the example above,
yt  0.036  0.69 yt 1  0.42u t 1  ut
Suppose that we know t-1, t-2,... and we are trying to forecast yt.
Our forecast for t is given by
E ( yt yt 1 , yt 2 ,...) = f t 1,1  0.036  0.69 y t 1  0.42u t 1  u t
= 0.036 +0.693.4+0.42(-1.3)
= 1.836
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ft-1,2 = E( yt 1 yt 1 , yt 2 ,...)  0.036  0.69 yt  0.42ut  ut 1
But we do not know yt or ut at time t-1.
Replace yt with our forecast of yt which is ft-1,1.
ft-1,2
= 0.036 +0.69 ft-1,1
= 0.036 + 0.69*1.836
= 1.302
ft-1,3
= 0.036 +0.69 ft-1,2
= 0.036 + 0.69*1.302
= 0.935
etc.
(b) Given the forecasts and the actual value, it is very easy to calculate the
MSE by plugging the numbers in to the relevant formula, which in this case is
MSE 
1
N
N

n 1
( x t 1 n  f t 1, n ) 2
if we are making N forecasts which are numbered 1,2,3.
Then the MSE is given by
1

(1.836  0.032) 2  (1.302  0.961) 2  (0.935  0.203) 2 
3
1
 (3.489  0.116  0.536)  1.380
3
MSE 
Notice also that 84% of the total MSE is coming from the error in the first
forecast. Thus error measures can be driven by one or two times when the
model fits very badly. For example, if the forecast period includes a stock
market crash, this can lead the mean squared error to be 100 times bigger than
it would have been if the crash observations were not included. This point
needs to be considered whenever forecasting models are evaluated. An idea of
whether this is a problem in a given situation can be gained by plotting the
forecast errors over time.
(c) This question is much simpler to answer than it looks! In fact, the inclusion
of the smoothing coefficient is a “red herring” - i.e. a piece of misleading and
useless information. The correct approach is to say that if we believe that the
exponential smoothing model is appropriate, then all useful information will
have already been used in the calculation of the current smoothed value
(which will of course have used the smoothing coefficient in its calculation).
Thus the three forecasts are all 0.0305.
(d) The solution is to work out the mean squared error for the exponential
smoothing model. The calculation is
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
1
(0.0305  0.032) 2  (0.0305  0.961) 2  (0.0305  0.203) 2
3
MSE 


1
0.0039  0.8658  0.0298  0.2998
3
Therefore, we conclude that since the mean squared error is smaller for the
exponential smoothing model than the Box Jenkins model, the former
produces the more accurate forecasts. We should, however, bear in mind that
the question of accuracy was determined using only 3 forecasts, which would
be insufficient in a real application.
11. (a) The shapes of the acf and pacf are perhaps best summarised in a table:
Process
White
noise
AR(2)
MA(1)
ARMA(2
,1)
acf
No significant coefficients
pacf
No significant coefficients
Geometrically declining or
damped sinusoid acf
First 2 pacf coefficients
significant, all others
insignificant
Geometrically declining or
damped sinusoid pacf
Geometrically declining or
damped sinusoid pacf
First acf coefficient significant,
all others insignificant
Geometrically declining or
damped sinusoid acf
A couple of further points are worth noting. First, it is not possible to tell what
the signs of the coefficients for the acf or pacf would be for the last three
processes, since that would depend on the signs of the coefficients of the
processes. Second, for mixed processes, the AR part dominates from the point
of view of acf calculation, while the MA part dominates for pacf calculation.
(b) The important point here is to focus on the MA part of the model and to
ignore the AR dynamics. The characteristic equation would be
(1+0.42z) = 0
The root of this equation is -1/0.42 = -2.38, which lies outside the unit circle,
and therefore the MA part of the model is invertible.
(c) Since no values for the series y or the lagged residuals are given, the
answers should be stated in terms of y and of u. Assuming that information is
available up to and including time t, the 1-step ahead forecast would be for
time t+1, the 2-step ahead for time t+2 and so on. A useful first step would be
to write the model out for y at times t+1, t+2, t+3, t+4:
y t 1  0.036  0.69 y t  0.42u t  u t 1
y t  2  0.036  0.69 y t 1  0.42u t 1  u t  2
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y t 3  0.036  0.69 y t  2  0.42u t  2  u t 3
y t  4  0.036  0.69 y t 3  0.42u t 3  u t  4
The 1-step ahead forecast would simply be the conditional expectation of y for
time t+1 made at time t. Denoting the 1-step ahead forecast made at time t as
ft,1, the 2-step ahead forecast made at time t as ft,2 and so on:
E( yt 1 yt , yt 1 ,...)  f t ,1  Et [ yt 1 ]  Et [0.036  0.69 yt  0.42ut  ut 1 ]  0.036  0.69 yt  0.42ut
since Et[ut+1]=0. The 2-step ahead forecast would be given by
E( yt  2 yt , yt 1,...)  ft , 2  Et [ yt  2 ]  Et [0.036  0.69 yt 1  0.42ut 1  ut  2 ]  0.036  0.69 f t ,1
since Et[ut+1]=0 and Et[ut+2]=0. Thus, beyond 1-step ahead, the MA(1) part of
the model disappears from the forecast and only the autoregressive part
remains. Although we do not know yt+1, its expected value is the 1-step ahead
forecast that was made at the first stage, ft,1.
The 3-step ahead forecast would be given by
E( yt 3 yt , yt 1,...)  ft ,3  Et [ yt 3 ]  Et [0.036  0.69 yt  2  0.42ut  2  ut 3 ]  0.036  0.69 f t , 2
and the 4-step ahead by
E( yt  4 yt , yt 1,...)  ft , 4  Et [ yt  4 ]  Et [0.036  0.69 yt 3  0.42ut 3  ut  4 ]  0.036  0.69 f t ,3
(d) A number of methods for aggregating the forecast errors to produce a
single forecast evaluation measure were suggested in the paper by Makridakis
and Hibon (1995) and some discussion is presented in the book. Any of the
methods suggested there could be discussed. A good answer would present an
expression for the evaluation measures, with any notation introduced being
carefully defined, together with a discussion of why the measure takes the
form that it does and what the advantages and disadvantages of its use are
compared with other methods.
(e) Moving average and ARMA models cannot be estimated using OLS – they
are usually estimated by maximum likelihood. Autoregressive models can be
estimated using OLS or maximum likelihood. Pure autoregressive models
contain only lagged values of observed quantities on the RHS, and therefore,
the lags of the dependent variable can be used just like any other regressors.
However, in the context of MA and mixed models, the lagged values of the
error term that occur on the RHS are not known a priori. Hence, these
quantities are replaced by the residuals, which are not available until after the
model has been estimated. But equally, these residuals are required in order to
be able to estimate the model parameters. Maximum likelihood essentially
works around this by calculating the values of the coefficients and the
residuals at the same time. Maximum likelihood involves selecting the most
likely values of the parameters given the actual data sample, and given an
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assumed statistical distribution for the errors. This technique will be discussed
in greater detail in the section on volatility modelling in Chapter 8.
12. (a) Some of the stylised differences between the typical characteristics of
macroeconomic and financial data were presented in Chapter 1. In particular,
one important difference is the frequency with which financial asset return
time series and other quantities in finance can be recorded. This is of
particular relevance for the models discussed in Chapter 5, since it is usually a
requirement that all of the time-series data series used in estimating a given
model must be of the same frequency. Thus, if, for example, we wanted to
build a model for forecasting hourly changes in exchange rates, it would be
difficult to set up a structural model containing macroeconomic explanatory
variables since the macroeconomic variables are likely to be measured on a
quarterly or at best monthly basis. This gives a motivation for using pure timeseries approaches (e.g. ARMA models), rather than structural formulations
with separate explanatory variables.
It is also often of particular interest to produce forecasts of financial variables
in real time. Producing forecasts from pure time-series models is usually
simply an exercise in iterating with conditional expectations. But producing
forecasts from structural models is considerably more difficult, and would
usually require the production of forecasts for the structural variables as well.
(b) A simple “rule of thumb” for determining whether autocorrelation
coefficients and partial autocorrelation coefficients are statistically significant
is to classify them as significant at the 5% level if they lie outside of
1
, where T is the sample size. In this case, T = 500, so a particular
 1.96 *
T
coefficient would be deemed significant if it is larger than 0.088 or smaller
than –0.088. On this basis, the autocorrelation coefficients at lags 1 and 5 and
the partial autocorrelation coefficients at lags 1, 2, and 3 would be classed as
significant. The formulae for the Box-Pierce and the Ljung-Box test statistics
are respectively
m
Q  T 
k 1
2
k
and
 k2
m
Q*  T (T  2)
k 1
T k
.
In this instance, the statistics would be calculated respectively as
Q  500  [0.307 2  (0.013 2 )  0.086 2  0.0312  (0.197 2 )]  70.79
and
 0.307 2 (0.013 2 ) 0.086 2
0.0312 (0.197 2 ) 
Q*  500  502  




  71.39
500  2
500  3 500  4
500  5 
 500  1
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The test statistics will both follow a 2 distribution with 5 degrees of freedom
(the number of autocorrelation coefficients being used in the test). The critical
values are 11.07 and 15.09 at 5% and 1% respectively. Clearly, the null
hypothesis that the first 5 autocorrelation coefficients are jointly zero is
resoundingly rejected.
(c) Setting aside the lag 5 autocorrelation coefficient, the pattern in the table is
for the autocorrelation coefficient to only be significant at lag 1 and then to fall
rapidly to values close to zero, while the partial autocorrelation coefficients
appear to fall much more slowly as the lag length increases. These
characteristics would lead us to think that an appropriate model for this series
is an MA(1). Of course, the autocorrelation coefficient at lag 5 is an anomaly
that does not fit in with the pattern of the rest of the coefficients. But such a
result would be typical of a real data series (as opposed to a simulated data
series that would have a much cleaner structure). This serves to illustrate that
when econometrics is used for the analysis of real data, the data generating
process was almost certainly not any of the models in the ARMA family. So all
we are trying to do is to find a model that best describes the features of the
data to hand. As one econometrician put it, all models are wrong, but some are
useful!
(d) Forecasts from this ARMA model would be produced in the usual way.
Using the same notation as above, and letting fz,1 denote the forecast for time
z+1 made for x at time z, etc:
Model A: MA(1)
f z ,1  0.38  0.10u t 1
f z , 2  0.38  0.10  0.02  0.378
f z , 2  f z ,3  0.38
Note that the MA(1) model only has a memory of one period, so all forecasts
further than one step ahead will be equal to the intercept.
Model B: AR(2)
xˆ t  0.63  0.17 xt 1  0.09 xt 2
f z ,1  0.63  0.17  0.31  0.09  0.02  0.681
f z , 2  0.63  0.17  0.681  0.09  0.31  0.718
f z ,3  0.63  0.17  0.718  0.09  0.681  0.690
f z , 4  0.63  0.17  0.690  0.09  0.716  0.683
(e) The methods are overfitting and residual diagnostics. Overfitting involves
selecting a deliberately larger model than the proposed one, and examining
the statistical significances of the additional parameters. If the additional
parameters are statistically insignificant, then the originally postulated model
is deemed acceptable. The larger model would usually involve the addition of
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one extra MA term and one extra AR term. Thus it would be sensible to try an
ARMA(1,2) in the context of Model A, and an ARMA(3,1) in the context of
Model B. Residual diagnostics would involve examining the acf and pacf of the
residuals from the estimated model. If the residuals showed any “action”, that
is, if any of the acf or pacf coefficients showed statistical significance, this
would suggest that the original model was inadequate. “Residual diagnostics”
in the Box-Jenkins sense of the term involved only examining the acf and pacf,
rather than the array of diagnostics considered in Chapter 4.
It is worth noting that these two model evaluation procedures would only
indicate a model that was too small. If the model were too large, i.e. it had
superfluous terms, these procedures would deem the model adequate.
(f) There are obviously several forecast accuracy measures that could be
employed, including MSE, MAE, and the percentage of correct sign
predictions. Assuming that MSE is used, the MSE for each model is
MSE ( Model A) 
MSE ( Model B) 


1
(0.378  0.62) 2  (0.38  0.19) 2  (0.38  0.32) 2  (0.38  0.72) 2  0.175
4


1
(0.681  0.62) 2  (0.718  0.19) 2  (0.690  0.32) 2  (0.683  0.72) 2  0.326
4
Therefore, since the mean squared error for Model A is smaller, it would be
concluded that the moving average model is the more accurate of the two in
this case.
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“Introductory Econometrics for Finance” © Chris Brooks 2008