Skeleton Solutions to the Exercises

Skeleton Solutions to the Exercises

(These are only skeleton solutions, you would need to add more detail if
these were in a test or exam) Also asymptotic t-statistics are exactly the
same as t-statistics in terms of their interpretation.

Lecture 1

1) The main reason for using panel data is to expand the degrees of
freedom and avoid collinearity. However it also has some advantages
over a cross sectional or times series approach. The advantage over a
cross sectional regression is that it can be used to account for any
unobserved heterogeneity. In addition it can be used to introduce a
dynamic structure into a cross sectional regression. Panel data can
also overcome some of the problems of aggregation bias as well as pick
up effects that cross section and time series regressions miss out.

2) Unobserved heterogeneity refers to the unobserved effects on the
individual or firm, which can not be directly measured. For
individuals these can include ambition, parental influence etc. If
this effect is ignored it induces inefficiency in the estimator.

3) Fixed effects overcomes the problem of unobserved heterogeneity, given
the standard panel data model of the following form:

[pic]
Where:
Y is the dependent variable
Xj are the observed explanatory variables
Zp are unobserved explanatory variables

If we assume the unobserved effect does not vary over time and given that
it is unobserved and difficult to measure, the model can be rewritten as:

[pic]

Where the ?i is referred to as an individual unobserved specific effect
which can be remedied in a number of ways although the most common is to
include individual dummy variables for each cross section microunit.

4) The three ways of introducing fixed effects include the above dummy
variable approach as well as using differenced variables in the model:

- Within-groups fixed effects, where the variation is explained about
the mean of the dependent variable in terms of the variations about
the means of the explanatory variables. However this method has
potential problems such as the loss of the x variables that remain
constant for an individual.

[pic]
The unobserved effect disappears from this model and is known as
the within-groups regression model as it explains the variations
about the mean of the dependent variable in terms of the variations
about the means of the explanatory variables for the group of
observations relating to a given individual.

- Taking first-differences of the variables. Again the problem with
the x variable remains, but a potential advantage is that it could
remove any problem of first-order autocorrelation.

5) If the unobserved effects are distributed randomly, we can treat the ?i
as random variables, drawn from a given distribution. This involves
subsuming the unobserved effects into the disturbance term to give:

[pic]
This is a random effects type of model is in general better than the fixed
effects model as characteristics that remain constant for each individual
remain in the model but have to be removed for fixed effects models.

6) This involves a general set of answers on the model and results, why
panel data were used and how the model could be improved. In this case the
panel data is used as they only have bi-annual data, which is not enough to
give a time series based regression.

Lecture 2

1) A time series is stationary if it has a constant mean, variance and
constant structure to the co-variance, i.e. the covariance between
lags 1 and 4 is the same as 10 and 14. This is the definition of a
weakly stationary series, a strictly stationary series has a
distribution of its values that remain the same as time progresses.

2) An autocorrelation coefficient measures the correlation between lags
of a specific variable, whereas the partial autocorrelation
coefficient measures the correlation between lag t and t + k, whilst
removing the effects of the intervening lags. Both coefficients are
used to determine the order of an ARIMA model, however the ACF is used
to determine the lags in the MA process and the PACF is used to
determine the lags in the AR part of the model.

3) If ?k is plotted against k, the population correlogram is obtained. To
produce a correlogram:

a) Compute the sample covariance and variance at lag k for series y.

[pic]


Where n is the sample size and [pic]is the sample mean.


b) Then [pic]plotted from k = 1 onwards.

If a time series is stationary, the correlogram will indicate that all
[pic]values would be zero, non-stationary series usually have values
significantly above zero, then declining for higher values of k. The
statistical significance of [pic]can be judged either by its standard error
or the Q-statistic.

4)
[pic]
The Q statistic suggests the series is stationary, the Ljung-Box that it
is non-stationary, the different results are due to the small sample size..

4) An AR(3) model has 3 lags on the dependent variable.

[pic]


6) [pic]

This assumes that as the E(u)t = 0, then E(u)t-I = 0. Also that the
variance of the error term is constant and the cross product terms of the
lags of the error term all equal zero.





Exercise 3

1) To determine if the AR model is stationary, the characteristic
equation needs to be defined and the roots of the equation examined:

[pic]
As both roots are more than one, they therefore lie outside the unit
circle and yt is stationary. (I think I incorrectly said it was the
other way round in the lecture!)


2) To determine the variance of the random walk, we first need to state
that for any AR(p) process, according to Wold's decomposition, it can
be expressed as an MA([pic]) process. (You don't need to know the
details of this, just the result)

[pic]




Given that the variance of a random variable is:


[pic]
As the error term is assumed to be Gaussian we can ignore the cross
product terms.


[pic]
3) There are two main criticisms of the Box-Jenkins methodology, firstly
that it lacks theory and secondly that it tends only to reveal if the
model is under parameterised rather than over parameterised. In
general we prefer models to be as small or parsimonious as possible.
The choice of lag length tends to be determined by the ACF and PACF or
information criteria such as the Akaike criteria, without reference to
a set theory determining the ARIMA lags. If the diagnostic tests are
failed, the process then involves respecifying the model with more
lags, however it can be argued that this is ad hoc and may not result
in the best model. For this reason it is often referred to as more art
than science, despite its ability to forecast well.

4) An out-of-sample forecast tends to be a better measure of how well a
model forecasts than the in-sample forecast. This is because in the in-
sample forecast the model used to produce the forecasts includes the
observations that are used for the forecast. The MSE is:

[pic]
In the above case, the smaller the value of the MSE, the better the
forecast. However the statistic in isolation is not very informative
as its value depends on the units of the variables being forecast, so
it needs to be used as a comparison with the MSE from a competing
model such as the random walk.


5) The formula for measuring how well a model forecasts the correct signs
of a variable is:

[pic]


This means that when the correct sign is predicted, zt+s takes the
value of 1, if it gets the sign wrong it takes the value of 0, these
are then added to produce the number of correct predictions. In
finance in particular this is important, as a profit is often made
when the correct direction of movement an asset takes can be
predicted, whereas the magnitude of the movement is less important.


6) This answer would need to refer to Harvey's two main criticisms of the
ARIMA models, particularly the difficulty in obtaining the best ARIMA
model and the lack of theory behind these models compared to a
structural model, as he suggests this can have serious implications
for forecasting over the long-run as it fails to pick up the cyclical
nature of some time series. On the other hand other practitioners,
such as Granger suggest they forecast better than far more complex
structural models.


Exercise 4


1) A stationary process has a constant mean, variance and covariance
structure, whereas a trend stationary process is stationary around a
time trend. This involves including a time trend in the regression:

[pic]


An I(1) variable needs to be differenced once to ensure it is
stationary, whereas an I(2) variable needs to be differenced twice.
This has implications for the cointegration tests.


2) The Dickey-Fuller and Augmented Dickey-Fuller (ADF) tests are
basically a test for whether a series follows the random walk, which
is a non-stationary I(1) process. The test itself is for the null
hypothesis