Currently, I am studying the association between the frequency of extreme temperature and mortality rate. Before I add the Sulfur dioxide emissions as a control variable, the coefficient of my key variable (The frequency of extreme temperature) is insignificant but after adding this control variable, the coefficient becomes significant. So how to explain this kind of situation.
Dear Jianhong Xu ,
There could be at least two reasons:
(1) Even if the variabla A is not correlated with variable B, if variable B is relevant in the regression you are trying to estimate, the estimated coefficient of variable B is not efficient, that is, is less precise than if you are able to control for B. Therefore, it can happen that, adding additional controls, even if those controls are not correlated with variable A, the standard error of A decreases and the estimated coefficient of variable becomes different from 0.
(2) (1) The regressor (variable A) is correlated with the variables you did not include earlier (let's call it variable B). In that case, in the regression without variable B, the estimated coefficient of variable A is biased. This is the omitted variable problem. The estimated coefficient is a biased estimated of true parameter and naturally there is the possibility that the estimated coefficient in the regression is not significantly different from 0 and the true parameter is indeed different from 0.
Of course, it is possible that both things happen.
You can find this in any introductory book such as Wooldridge's introductory Econometrics.
Mark the question as solved if it is the case.
I hope it helps.
Best regards,
José-Ignacio
Hi, I disagree with the first point in the previous answer: by adding extra variables, the standard errors of the coefficient estimates should increase and therefore make less significant the variables already included. This line of argument cannot explain what is going on here.
I agree with the second point in the previous answer.
A third possible explanation: Suppressor variables. Sulfur diox could be a suppressor variable, that is a variable independent of mortality rate, but correlated with "extreme temperatures". Suppressor variables suppress noise that is overshadowing the true relationship between a variable x and y. By adding this suppressor variable to the regression, suddently, the true relationship between X and Y emerges.
You can google the term "suppressor variable". For a proof, you can take a class with me!! :)
Cheers!
See, e.g., Wooldridge's Introductory Econometrics 3.4, the case of b2 != 0 and X2 and X1 are uncorrelated.
What you state only applies if X2 is irrelevant (b2 = 0).
If b2 != 0, overall, we can expect that the there is correlation between X2 and the regressors already in the equation. This would lead to bias in the estimated b1, a far more severe and serious than the efficiency loss.
Overall, as we are not ready to assume that X2 and X1 are not correlated (particularly, if X1 includes many variables), then, the most conservative explanation (in terms of being cautious) is the second one, but it is obvious than omitting relevant variables inflates standard errors if the omitted variable is uncorrelated with the rest of regressors.
This is also the reason for which some authors control for covariates in experimental designs (the covariates are uncorrelated with the treatment by design, but they can reduce standard errors if relevant).
As you add explanatory variables, what that first one does is explain the residual variation in the dependent variable. This variation could look quite different than the total variation in the dependent variable. Thus, it could be that the first right side variable does not appear to be correlated with the dependent variable when it enters alone but appears to be significant when included with other explanatory variables.
Hi Jose Ignacio,
I see your point regarding Wooldridge. If X2 and X1 are uncorrelated and you add to the regression X2 and compute standard errors in the usual homoskedastic way, then you are right. My initial disagreement was based on the book by Green. Yet in that book, it is assumed that "sigma", the population standard error of the error term is either known or always correctly estimated, regardless of the ommission of relevant or irrelevant variables. This assumption is unlikely in practice, yet interesting in theory.
Good point, sorry about my drastic initial disagreement.
As to help Jianhong, I guess you should evaluate the correlation between all three variables: mortality, Sulfur dioxide and extreme temperatures to see which one of all these plausible explanations might apply to your case.
Cheers!
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