Abstract
We first provide comprehensive, but simple, access to essential regression knowledge by discussing how regression analysis works, the requirements and assumptions on which it relies, and how you can specify a regression analysis model that allows you to make critical decisions for your business, clients, or project. Each step involved in regression analysis is linked to its execution in Stata (using menus and code). We show how to use a range of Stata’s easy-to-learn statistical procedures that underlie regression analysis, which will allow you to analyze, chart, and validate regression analysis results and to assess your analysis’s robustness. Interpretation of Stata output can be difficult, but we make this easier by means of an annotated case study. We conclude with suggestions for further readings on the use, application, and interpretation of regression analysis.
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Notes
- 1.
Strictly speaking, the difference between the predicted and the observed y-values is \( \widehat{e} \).
- 2.
This only applies to the standardized βs.
- 3.
This is only a requirement if you are interested in the regression coefficients, which is the dominant use of regression. If you are only interested in prediction, collinearity is not important.
- 4.
The VIF is calculated using a completely separate regression analysis. In this regression analysis, the variable for which the VIF is calculated is regarded as a dependent variable and all other independent variables are regarded as independents. The R2 that this model provides is deducted from 1 and the reciprocal value of this sum (i.e., 1/(1 − R2)) is the VIF. The VIF is therefore an indication of how much the regression model explains one independent variable. If the other variables explain much of the variance (the VIF is larger than 10), collinearity is likely a problem.
- 5.
This term can be calculated manually, but also by using the function mmult in Microsoft Excel where x T x is calculated. Once this matrix has been calculated, you can use the minverse function to arrive at (x T x)−1 .
- 6.
In Stata this can be done by using the, robust option.
- 7.
The test also includes the predicted values squared and to the power of three.
- 8.
Specifically, in the mentioned regression model y = α + β 1 x 1 + β 2 x 2 + β 3 x 3 + e, the Breusch-Pagan test determines whether \( \widehat{e^2}=\alpha +{\beta}_{BP1}{x}_1+{\beta}_{BP2}{x}_2+{\beta}_{BP3}{x}_3+{e}_{BP} \).
- 9.
This hypothesis can also be read as that a model with only an intercept is sufficient.
- 10.
The AIC is specifically calculated as AIC = n·ln(SS E /n) + 2·k, where n is the number of observations and k the number of independent variables, while the BIC is calculated as BIC = n·ln(SS E /n) + k·ln(n).
- 11.
Cohen’s (1994) classical article “The Earth is Round (p < 0.05)” offers an interesting perspective on significance and effect sizes.
- 12.
Using the Stata command egen commitment=rowmean(com1 com2 com3)
- 13.
Note that a p-value is never exactly zero, but has values different from zero in later decimal places.
- 14.
Note that it is possible to show all categories for regression tables by typing set showbaselevels on. This can be made permanent by typing set showbaselevels on, permanent.
- 15.
Note that while the constant has the highest value (1.19), this is not a coefficient and should not be interpreted as an effect size.
- 16.
Please note that only Stata 13 or above feature built-in routines to calculate η 2.
- 17.
The seed specifies the initial value of the random-number generating process such that it can be replicated later.
- 18.
We would like to thank Dr. D.I. Gilliland and AgriPro for making the data and case study available.
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Mooi, E., Sarstedt, M., Mooi-Reci, I. (2018). Regression Analysis. In: Market Research. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-5218-7_7
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