Chapter 4:
BASIC ESTIMATION TECHNIQUES
Essential Concepts
1. A simple linear regression model relates a dependent variable Y to a single independent (or
explanatory) variable X in a linear fashion
2. Because the variation in Y is affected not only by variation in X but also by various random effects as
well, the actual value of Y cannot be predicted exactly. The regression equation is correctly
interpreted as providing the average value, or the expected value, of Y for a given value of X.
3. Parameter estimates are obtained by choosing values of a and b that minimize the sum of the squared
residuals. The residual is the difference between the actual value of Y and the fitted value of Y,
i i
ˆ
Y Y–
.
4. The estimates
ˆ
a
and
ˆ
b
do not, in general, equal the true values of a and b. Since
ˆ
a
and
ˆ
b
are
computed using data from a random sample, the estimates themselves are random variables—the
5. It is the randomness of the parameter estimates that necessitates testing for statistical significance.
Just because the estimate
ˆ
b
ˆ
b
is not zero does not mean the true value of b is not zero. Even when b does
6. There are two ways to determine whether an estimated parameter is statistically significant. Either a t–
test can be performed or the p-value for the parameter estimate can be examined.
7. To perform a t-test for significance, a researcher must first determine the level of significance for the
test. The significance level of a test is the probability of finding a parameter estimate to be
significantly different from zero when, in fact, b is zero. This mistake is called a Type I error. Lower
levels of significance, other things equal, are more desirable. One minus the level of significance is
8. An alternative method of assessing the statistical significance of parameter estimates is to treat as
9. The coefficient of determination R2 measures the percentage of the total variation in the dependent
variable that is explained by the regression equation. The value of R2 ranges from 0 to 1. A high R2
indicates Y and X are highly correlated and the scatter diagram tightly fits the sample regression line.
10. The F-test is used to test for significance of the overall regression equation. The F-statistic from the
computer printout is compared to the critical F-value obtained from the F-table at the end of your
11. Multiple regression uses more than one explanatory variable to explain the variation in the dependent
12. Two types of nonlinear models can be easily transformed into linear models that can be estimated
using linear regression analysis. These are quadratic regression models and log-linear regression
models.
(a) Quadratic regression models are appropriate when the curve fitting the scatter plot is either –
Chapter 4: Basic Estimation Techniques
Answers to Applied Problems
1. a. The intercept a is expected to be positive because even if no advertising is undertaken, some sales
are expected to occur. b is expected to have a positive sign since Vanguard’s sales are positively
related to its level of advertising expenditures. Vanguard’s sales should be inversely related to its
rivals’ expenditures on advertising, so c is expected to be negative.
S
2. a. At the 95% level of confidence, the critical F-value is Fk–1,n–k = F1,15 = 4.54. Since the computed
F-ratio 42.674 is greater than 4.54, the regression equation provides evidence of a statistically
V = 53.682 – 0.528(0) = 53.682,
or 53.7% are expected to favor Proposition 103.
c. For
ˆ
b
: t = |–6.519| > 2.131; statistically significant
Remember that both V and P are measured as percentages. Thus, a 1% increase in P is estimated
Chapter 4: Basic Estimation Techniques
3. a. The F-statistic provides evidence that the regression equation as a whole is statistically
significant. The p-value for the F-statistic is significant at less than 0.01%. The R2 indicates the
regression equation explains 83% of the variation in E. The p-values for the individual
4. a. Estimate the model
Q‘ = a‘ + bH‘ + cS‘,
where Q‘ = ln Q, a’ = ln a, H‘ = ln H, and S‘ = ln S.
d. 54.52% of the variation in Q is explained by this model. The R2 could be increased by adding
some additional explanatory variables such as the sales experience of the salespersons employed.
Whether the sales day is a weekday or a Saturday/Sunday, and the level of advertising in
newspapers the previous week.
e. The critical t–ratio is tn–k = t50 ≈ 2.