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K. REGRESSION EXTENSIONS
1. Residual Analysis
a. Presence of Outliers
b. Patterns of Residuals
(i) Nonlinear Equations
(ii) Heteroskedasticity
(iii) Missing Variables
(iv) Autocorrelation
2. Dummy Variables
a. Limits on numbers of Dummy Variables (Overdetermination)
b. Use in ANOVA
3. Nonlinear regression
Semilog, Double log, Reciprocal, Polynomial and Cyclical Forms.
4. Runs test
Consider the sequence + + + - - + + - - or AAABBAABB The Runs Test is a test to see if the
sequences of two items, plusses and minuses in the first case and ‘A’s and ‘B’s in the second case is
random. If the sequence is not random, the alternation of the two kinds of item tends to follow a predictable
pattern. Our null hypothesis is randomness. In the sequences above, let n be the total number of items, n1
be the total number of items of the first kind (plusses or ‘A’s), n 2 be the total number of items of the
second kind (minuses or ‘B’s) and r be the total number of runs, that is the number of sequences of one
kind. In the sequences above there are two sequences of plusses or ‘A’s and two of minuses or ‘B’s, so that
r  4. (Note that the sequence ABA is three runs in itself). In the sequences above, we can also see that
n1  5, n 2  4 and n  n1  n 2  9 .
To test the null hypothesis of randomness for a small sample, assume that the significance level is 5% and
use the table entitled 'Critical values of r in the Runs Test.’ For n1  5 and n 2  4 , the top part of the table
gives a critical value of 2 and the bottom part of the table gives a critical value of 9. This means that we
reject the null hypothesis if r  2 or r  9. In this case, since r  4 we do not reject the null hypothesis.
For a larger problem (if n1 and n 2 are too large for the table), r follows the normal distribution with
2n1 n 2
  1  2 . Example: Assume that the significance level is 5% and that we
 1 and  2 
n
n 1
2n n
225 30 
 1  28.27273
find that r  5 , n1  25, n 2  30 and n  n1  n 2  55 . Then   1 2  1 
n
55
  1  2  27 .27273 26 .27273   13 .26905 . So z  r    5  28 .27273  6.39 . Since
and  2 
n 1
54

13 .26905

this value of z is not between  z  1.960 , we reject H 0 : Randomness.
2
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5. Durbin-Watson test
This is a test for randomness of regression residuals . The alternative hypothesis is (first-order)
 e
autocorrelation. The computer will print d 
t
 et 1 2
, and compute the Durbin-Watson statistic,
et
DW . Use a Durbin-Watson table to fill in the diagram below.
0
+
 0
dL
+
?
dU
+
 0
2  0
+
4  dU
+
?
4 dL
+
 0 4
+
For example, if for a significance level of   .05 and we want a 2-sided test for a regression with a sample
size of n  20 observations with k  4 independent variables and DW  1.01 ,we go to a Durbin-Watson
table for  2  .025 . There we find for these values of n and k that d L  0.79 and d U  1.70 , so that
4  dU  4  1.70  2.30 and 4  d L  4  0.79  3.21 . Our diagram is now
0
+
 0
0.79
+
?
1.70   0
+
2  0
+
2.30
+
?
3.21
+
 0
4
+
Because DW  1.01 is between 0.79 and 1.70, it is in an area marked with a question mark, and we cannot
be sure whether to reject H 0 : randomness. If, however, DW were between 0 and 0.79 or 3.21 and 4 we
could reject the null hypothesis, while if it were between 1.70 and 2.30 we could say that we cannot reject
the null hypothesis.
We can also do a one-sided test for an alternative hypothesis of positive autocorrelation by looking up
critical values for   .05 and saying that we reject the null hypothesis if DW is below d L , that we do not
reject it if DW is above d U and that we are not sure if DW is between d L and d U . If the alternative
hypothesis is negative autocorrelation, we use the same critical values but we reject the null hypothesis if
DW is above 4  d L , we do not reject it if DW is above 4  d U and we are not sure if DW is between
4  d L and 4  d U . For example, if DW  0.85 and H 1 is   0 , and we again have a regression with a
sample size of n  20 observations with k  4 independent variable, a one sided 5% table gives us
d L  0.90 and dU  1.83. Thus, since our computed DW is below d L , we reject the null hypothesis and
conclude that there is positive autocorrelation. The most accessible D-W table can be found through D-W
table.