Notes

1 It should be noted that dummy variables are usually subject to some kind of constraints that

require treatments different from other explanatory variables. For example, if a full set of

dummies is mutually exclusive and exhaustive, such as the case of foreign and domestic firm

dummies, the sum of the full set of dummies should equal to a vector of one. As we illustrate

in the following sections, misinterpretations committed by researchers often originate from

the failure to recognize the implicit restriction among the full set of dummies. Owing to this

implicit restriction, some of the discussions of multiplicative terms formed by non-dummy

independent variables (Aiken and West, 1991; Friedrich, 1982; Irwin and McClelland,

2001; Stone and Hollenbeck, 1989) may not be applicable to the case of multiplicative

terms with dummy variables.

2 If the dependent variable is a categorical variable, the regression will be a limited dependent

variable estimation (see Maddala, 1983 for more details).

3 Vermeulen and Barkema’s (2001) study is one of the few examples that use the partition

approach. Under the column ‘Survival Analysis 2’ in their Table 2, the multiplicative terms

between Zit (i.e. number of preceding greenfields or number of preceding acquisitions) and

two dummy variables, namely greenfield and acquisition, together partition the effect of Zit

for greenfield subsidiaries and acquired subsidiaries respectively. Another example is Kim

et al.’s (2004) study, which uses two dummy variables to distinguish between keiretsu member

firms with strong power and those with weak power.

4 If the regression contains Zi and Zi

2 as explanatory variables, one can also expand them into

partition terms of Zi and partition terms of Zi

2 . Nevertheless, researchers should always check

whether their polynomial equation is a good approximation of the true relationship.

5 In some other disciplines, the coefficients are sometimes rightly interpreted as the difference

of differences in the slopes. Nevertheless, as we show later, the partition specification still has

obvious advantages over the base specification in terms of its flexibility in formulating and

testing hypotheses among various combinations of coefficients.

6 For simplicity of presentation, we have compressed all the other explanatory variables in Xit

~

of equations 3# and 7#. Both equations are in fact well-specified models under the base

approach.

7 There seems to be a typo here. It should be hypothesis 4 rather than hypothesis 5.

8 The purpose of our survey was to demonstrate that this problem of misinterpretation of

interaction effects is more widespread than many readers might suppose; but this purpose

did not include criticizing particular researchers or lines of research. Hence, the specific

papers in our survey are not listed here, but they are available upon request.

9 As the PIMS dataset used in the study only has three categories of entrants, we do not have

any strong objection to the use of dummy variables. Nevertheless, the use of dummies

instead of continuous time entry variables could result in loss of information. Whenever

possible, researchers are advised to code their data in continuous measures instead of categories.

10 The two equations in the second panel of their Table 3 (and Table 4) are jointly estimated.

Although we only discuss the first equation here for simplicity of presentation, similar comments

apply to the second equation.

11 The proof is available from us upon request.

12 With the help of the partition specification and equations 9a–h, the researcher can use the

regression result of the base specification in equation 9 to test any combination of coefficients

without the need of re-doing the estimation. For example, if he/she wants to test whether

c7  c8, he/she can conduct an equivalent test on whether

Thank you very much for your share Falah Khalaf Ali Alrubaie!

Falah Khalaf Ali Alrubaie Many thanks for this well explained and insightful answer!!

Thank you dear Ngo Thai Hung and Imen Mbarki