Hypothesis Testing and CL in Multiple Regression
7 important questions on Hypothesis Testing and CL in Multiple Regression
What about a hypothesis test on a single coefficient?
Confidence intervals work exactly the same B +/- SE*critical value
Single restriction but not zero while checking for other variables?
1) Directly by software
2) Transform regressors into one by changing the formula by subsequently adding and subtracting.
What about confidence sets for multiple coefficients?
IF the economic correlation is negative, the betas will be positively correlated in the ellipse. For example if X1 increase, X2 would decrease, this means that the beta is positive.
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What about model specifications?
Omitted variable:
- At least one of the regressors is correlated with an ommited variable
- The omitted variable is a determinant of Y
This implies that the OLS estimators are inconsistent.
Note: this problem persists even in large samples.
How to check?
- Base specification: formula based on expert judgement, economic theory, also regarding control variables
- Alternative specification: alternative regressors due to mismeasurement or availability.
IF pretty much the same you can say your base is reliable.
What about R2 and adjusted R2 in practice?
- An increase does not necessarily mean that an added variable is statistically significant (remember: sample uncertainty)
- A high number does not mean that the regressors are a true CAUSE of the dependent variable (coincidence exists and correlation does not imply a causal relation)
- High/low number is not related to omitted variable
- High number does not necessarily mean that you have the most appropriate set of regressors, nor does a low tell you that you have an inappropriate set (economic theory remains most important)
What about analyzing the data?
Best way to present is to put all results in a table to get easy comparison. Using asteriks to highlight statistically significant results and putting the SE in brackets so everyone would be able to construct a t-statistic. Intercept can be labelled constant.
What about a discussion?
- Are they important contributors?
- No need fro them to be statistically significant
- If adding another doesn't make a difference, you can say that it is redundant for this analysis.
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