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Part Two
Relaxing the Assumptions of the Classical Model
What are some of the tests that one can use to detect outliers and leverage points? There are
several tests discussed in the literature, but we will not discuss them here because that will
take us far afield.
45
Software packages such as SHAZAM and MICROFIT have routines to
detect outliers, leverage, and influential points.
Recursive Least Squares
In Chapter 8 we examined the question of the structural stability of a regression model
involving time series data and showed how the
Chow test
can be used for this purpose.
Specifically, you may recall that in that chapter we discussed a simple savings function (sav-
ings as a function of income) for the United States for the period 1970–2005. There we saw
that the savings income relationship probably changed around 1982. Knowing the point of
the structural break we were able to confirm it with the Chow test.
But what happens if we do not know the point of the structural break (or breaks)? This
is where one can use
recursive least squares (RELS).
The basic idea behind RELS is very
simple and can be explained with the savings–income regression.
Y
t
=
β
1
+
β
2
X
t
+
u
t
where
Y
=
savings and
X
=
income and where the sample is for the period 1970–2005.
(See the data in Table 8.11.)
Suppose we first use the data for 1970–1974 and estimate the savings function, obtain-
ing the estimates of
β
1
and
β
2
. Then we use the data for 1970–1975 and again estimate the
savings function and obtain the estimates of the two parameters. Then we use the data for
1970–1976 and re-estimate the savings model. In this fashion we go on adding an addi-
tional data point on
Y
and
X
until we exhaust the entire sample. As you can imagine, each
regression run will give you a new set of estimates of
β
1
and
β
2
. If you plot the estimated
values of these parameters against each iteration, you will see how the values of estimated
parameters change. If the model under consideration is structurally stable, the changes in
the estimated values of the two parameters will be small and essentially random. However,
if the estimated values of the parameters change significantly, it would indicate a structural
break. RELS is thus a useful routine with time series data since time is ordered chronolog-
ically. It is also a useful diagnostic tool in cross-sectional data where the data are ordered
by some “size” or “scale” variable, such as the employment or asset size of the firm. In
Exercise 13.30 you are asked to apply RELS to the savings data given in Table 8.11.
Software packages such as SHAZAM,
EViews,
and MICROFIT now do recursive least-
squares estimates routinely. RELS also generates
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