Ordinary Least Squares Regression
The OLS command will estimate the parameters of a
linear regression equation by the method of ordinary least squares.
The general command format is:
OLS depvar indeps / options

where depvar is the dependent variable, indeps is a list of the
explanatory variables and options is a list of desired options.
There are many useful options on the
OLS command and some of these will be illustrated
in this guide.
Examples
Appendixes
[SHAZAM Guide home]
2variable Regression Analysis
This example uses the Griffiths, Hill and Judge
data set on household expenditure for food.
Consider a simple linear regression with FOOD as the dependent
variable and INCOME as the explanatory variable. The following
SHAZAM program reads the data from the file GHJ.txt ,
assigns variable names and runs the regression.
Note that the READ command assumes that the data file
is in the current directory (or folder).
SAMPLE 1 40
READ (GHJ.txt) FOOD INCOME
OLS FOOD INCOME
STOP

The output file of results follows.
_SAMPLE 1 40
_READ (GHJ.txt) FOOD INCOME
UNIT 88 IS NOW ASSIGNED TO: GHJ.txt
2 VARIABLES AND 40 OBSERVATIONS STARTING AT OBS 1
_OLS FOOD INCOME
OLS ESTIMATION
40 OBSERVATIONS DEPENDENT VARIABLE = FOOD
...NOTE..SAMPLE RANGE SET TO: 1, 40
RSQUARE = .3171 RSQUARE ADJUSTED = .2991
VARIANCE OF THE ESTIMATESIGMA**2 = 46.853
STANDARD ERROR OF THE ESTIMATESIGMA = 6.8449
SUM OF SQUARED ERRORSSSE= 1780.4
MEAN OF DEPENDENT VARIABLE = 23.595
LOG OF THE LIKELIHOOD FUNCTION = 132.672
VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 38 DF PVALUE CORR. COEFFICIENT AT MEANS
INCOME .23225 .5529E01 4.200 .000 .563 .5631 .6871
CONSTANT 7.3832 4.008 1.842 .073 .286 .0000 .3129
_STOP
 
SHAZAM automatically includes an intercept coefficient in the regression
and this is given the name CONSTANT .
On the SHAZAM output, the intercept estimate is listed as the
final coefficient estimate.
The results show that the estimated coefficient on INCOME (the
slope coefficient) is 0.23225 and the intercept estimate
is 7.3832 . The estimated equation can be written as:
FOOD = 7.38 + 0.232 INCOME + ê
where ê is the estimated residual. The figure below shows a
scatterplot of the observations and the estimated regression line.
(This figure corresponds to Figure 5.9 of Griffiths, Hill and
Judge [1993, p. 187]).
The LIST option on the OLS command
will give more extensive output that includes a listing of the
estimated residuals and the predicted values for the dependent variable.
The use of the LIST option is shown with the SHAZAM
command:
The interested reader can look at the
SHAZAM output generated with the LIST option.
Interpreting tratios
The OLS estimation results report the ESTIMATED COEFFICIENT
and the estimated STANDARD ERROR .
With the assumption that the errors
are normally distributed these estimates can be used for hypothesis
testing purposes. In the above example, a useful question to ask is:
Is the estimated coefficient on INCOME significantly different
from zero ? That is, does household income have an effect on the level of
household expenditure for food ? To help answer this question the SHAZAM
output reports the test statistic:
TRATIO = ESTIMATED COEFFICIENT / STANDARD ERROR
The estimated coefficient is significantly different from zero
(that is, the null hypothesis of a zero coefficient is rejected) if the
tratio is "relatively large". The critical value is obtained
from tables for the tdistribution with NK degrees of
freedom (N is the number of observations and
K is the number of estimated
coefficients). These tables are usually printed in the appendix to
econometrics textbooks.
For the household food expenditure example the reported tratio for the
coefficient on INCOME is 4.20 .
The number of observations is 40 and the number of estimated coefficients
is 2 and so the degrees of freedom (DF ) is 38. By choosing a
signficance level of 5% and considering a twosided test
(so that the critical region in each tail is 2.5%) the critical
value obtained from printed tables is 2.024 .
(Note that this critical value was approximated using the tabulated values
for 30 and 40 degrees of freedom that are reported in the tables.)
In absolute value, the tratio exceeds this critical value.
Therefore, there is strong evidence to conclude that the estimated coefficient
on INCOME is significantly different from zero.
Interpreting pvalues
When interpreting tratios it can be inconvenient to consult
statistical tables. To assist the user, SHAZAM
reports the PVALUE on the OLS estimation output.
This value is computed as the tail probability for a twotail
test of the null hypothesis that the coefficient is 0.
This is the probability of a Type I error  the probability of
rejecting a true hypothesis.
The null hypothesis is rejected if the pvalue is "small" (say smaller than
0.10, 0.05 or 0.01). For example, if the pvalue is 0.078, this means
that the null hypothesis cannot be rejected at a
5% significance level but can be rejected at a 10% significance level.
Note: SHAZAM only reports three decimal places for the
pvalue. So a value that is reported as .000 actually means
a value less than .0005 .
This can be interpreted as meaning that the null hypothesis of a
zero coefficient is rejected at any reasonable significance level.
It is possible to use SHAZAM commands to
compute pvalues for test statistics.
Interpreting elasticities
For the household food expenditure relationship the estimated coefficient
on INCOME measures the marginal effect. This gives the amount
by which FOOD changes in response to a one unit
change in INCOME .
Another measure of interest to economists is elasticity. This
gives the percentage change in the dependent variable that results
from a 1% change in the explanatory variable.
The final column on the SHAZAM OLS estimation output reports the
ELASTICITY AT MEANS .
For the example illustrated here,
let B1 be the estimated coefficient on INCOME
and let CM and PM be the sample means of
FOOD and INCOME respectively.
The income elasticity evaluated at the sample means is computed as:
B1 (PM/CM) = 0.6871
When interpreting the meaning of the estimated coefficients and the
elasticities users should take careful note of the units of measurement
of the variables in the regression equation.
The LIST option
The SHAZAM output that follows shows the use of the
LIST option on the OLS command.
_OLS FOOD INCOME / LIST
OLS ESTIMATION
40 OBSERVATIONS DEPENDENT VARIABLE = FOOD
...NOTE..SAMPLE RANGE SET TO: 1, 40
RSQUARE = .3171 RSQUARE ADJUSTED = .2991
VARIANCE OF THE ESTIMATESIGMA**2 = 46.853
STANDARD ERROR OF THE ESTIMATESIGMA = 6.8449
SUM OF SQUARED ERRORSSSE= 1780.4
MEAN OF DEPENDENT VARIABLE = 23.595
LOG OF THE LIKELIHOOD FUNCTION = 132.672
VARIABLE ESTIMATED STANDARD TRATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 38 DF PVALUE CORR. COEFFICIENT AT MEANS
INCOME .23225 .5529E01 4.200 .000 .563 .5631 .6871
CONSTANT 7.3832 4.008 1.842 .073 .286 .0000 .3129
OBS. OBSERVED PREDICTED CALCULATED
NO. VALUE VALUE RESIDUAL
1 9.4600 13.382 3.9223 * I
2 10.560 15.352 4.7918 * I
3 14.810 17.254 2.4440 * I
4 21.710 18.241 3.4689 I *
5 22.790 18.599 4.1913 I *
6 18.190 18.710 .52021 *
7 22.000 18.915 3.0854 I *
8 18.120 19.446 1.3265 *I
9 23.130 20.002 3.1285 I *
10 19.000 20.127 1.1270 *I
11 19.460 20.496 1.0362 *I
12 17.830 21.047 3.2167 * I
13 32.810 21.116 11.694 I *
14 22.130 21.488 .64204 *
15 23.460 21.579 1.8815 I*
16 16.810 22.038 5.2284 * I
17 21.350 22.703 1.3526 *I
18 14.870 22.805 7.9348 * I
19 33.000 23.738 9.2615 I *
20 25.190 23.752 1.4376 I*
21 17.770 24.101 6.3308 * I
22 22.440 24.105 1.6655 *I
23 22.870 24.159 1.2889 *I
24 26.520 24.159 2.3611 I *
25 21.000 24.440 3.4399 * I
26 37.520 24.628 12.892 I *
27 21.690 24.749 3.0588 * I
28 27.400 25.111 2.2889 I *
29 30.690 26.200 4.4896 I *
30 19.560 26.393 6.8332 * I
31 30.580 26.558 4.0219 I *
32 41.120 26.737 14.383 I *
33 15.380 26.753 11.373 * I
34 17.870 28.706 10.836 * I
35 25.540 28.706 3.1664 * I
36 39.000 28.973 10.027 I *
37 20.440 29.487 9.0468 * I
38 30.100 30.934 .83371 *I
39 20.900 33.890 12.990 * I
40 48.710 34.199 14.511 I *
DURBINWATSON = 2.3703 VON NEUMANN RATIO = 2.4310 RHO = .28193
RESIDUAL SUM = .36060E12 RESIDUAL VARIANCE = 46.853
SUM OF ABSOLUTE ERRORS= 207.53
RSQUARE BETWEEN OBSERVED AND PREDICTED = .3171
RUNS TEST: 22 RUNS, 17 POS, 0 ZERO, 23 NEG NORMAL STATISTIC = .4755
_STOP
 
The LIST option displays a table of results that
contains the following:
OBSERVED VALUE 
The observed value of the dependent variable.

PREDICTED VALUE 
The predicted value (also called
estimated value or fitted value) of the dependent variable.

CALCULATED RESIDUAL 
The difference between the observed and predicted values.

The right hand side of the output displays a rough plot of the residuals.
A property of ordinary least squares regression (when an intercept
is included) is that the sum of the estimated residuals (and hence the
mean of the estimated residuals) is 0. Note that the final part of the
SHAZAM output reports:
RESIDUAL SUM = .36060E12
That is, SHAZAM computes the sum of residuals as
.00000000000036060 . This shows that computer calculations
can have some imprecision. Different computers may have numerical differences
in the reporting of this result.
