The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

12.8
Model Mis-Specification versus Pure Autocorrelation
Let us return to our wages–productivity regression given in Eq. (12.5.2). There we saw that
the 
d
value was 0.2176 and based on the Durbin–Watson 
d
test we concluded that there was
positive correlation in the error term. Could this correlation have arisen because our model
was not correctly specified? Since the data underlying regression (12.5.1) is time series
data, it is quite possible that both wages and productivity exhibit trends. If that is the case,
then we need to include the time or trend, 
t
, variable in the model to see the relationship
between wages and productivity net of the trends in the two variables.
To test this, we included the trend variable in Eq. (12.5.2) and obtained the following
results
ˆ
Y
t
=
0.1209
+
1.0283
X
t
−
0.0075
t
se
=
(0.3070)
(0.0776)
(0.0015)
t
=
(0.3939)
(13.2594)
(
−
4.8903)
(12.8.1)
R
2
=
0.9900;
d
=
0.4497
The interpretation of this model is straightforward: Over time, the index of real wages has
been decreasing by about 0.75 units per year. After allowing for this, if the productivity
index went up by one unit, on average, the real compensation went up by about one unit.
What is interesting to note is that even allowing for the trend variable, the 
d
value is still
very low, suggesting that Eq. (12.8.1) suffers from pure autocorrelation and not necessarily
specification error.
How do we know that Eq. (12.8.1) is the correct specification? To test this, we regress 
Y
on 
X
and 
X
2
to test for the possibility that the real wage index may be nonlinearly related
to the productivity index. The results of this regression are as follows:
ˆ
Y
t
= −
1.7843
+
2.1963
X
t
−
0.1752
X
2
t
t
=
(
−
2.7713)
(7.5040)
(
−
5.2785)
(12.8.2)
R
2
=
0.9906
d
=
0.3561
We leave it to the reader to interpret these results. For the present purposes, look at the
Durbin–Watson, which is still quite low, suggesting that we still have positive serial corre-
lation in the residuals.
It may be safe to conclude from the preceding analysis that our wages–productivity re-
gression probably suffers from pure autocorrelation and not necessarily from specification
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bias. Knowing the consequences of autocorrelation, we may therefore want to take some
corrective action. We will do so shortly.
Incidentally, for all the wages–productivity regressions that we have presented above,
we applied the