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Edit: The question that was linked as an answer speaks of heteroskedasticity and multicollinearity, but this data set passed the Breusch-Pagan test and VIF, respectively. And in any case, I still have no idea what use the coefficient is for atr as its insignificant. How am I supposed to create a regression equation using a coefficient that isn't significant? I mean, I could technically do it, but what meaning would it have? Or is it significant within the whole context of the regression equation because the f-test is significant? If the other question actually answered my question, I am not seeing how or am completely lost.

I am trying to analyze the impact of median wages ($1,000s) and average tax rates on jobs per thousand residents among a random sample of the states. But I am having trouble making sense of the results. If one of the independent variables (IVs) is insignificant, shouldn't that render the rest of the results meaningless? I.e., if the p-value for taxes is .54, does it matter what he p-value for median wages or the F-stat are? I want to say that it doesn't and the conclusion should be that the results are insignificant/meaningless regardless of the F-stat, but I cannot find anything that explains what to do in this situation.

The F-stat indicates that the regression is significant, but the regression statistics indicate that 29.9% of the variation in jobs are explained by the variation in median wages and average tax rates. So my inital conclusion would normally be that taxes and median income are weakly correlated with jobs and normally I would just go with that, but the tax rate variable's p-value shows no significance. When I run univariate regression, the wages remain significant yet weakly correlated (p=0.001517, coefficient = 2.572, R^2 = .0288) while tax rates are insignificant (P=.934, coefficient = 43.138, R^2 = 0.002).

For what relevance it may be, the median-wage, jobs per thousand, and residuals are normally distributed per the Shapiro-Wilks test. Average tax rates are not-normally distributed (P=0.004502), though my understanding is that IVs do not necessarily need to be so long as the residuals are, but even with top tax rates instead of average they simply don't correlate. So how do I summarize this?

Regression Results

Residuals

JewJitsu11B
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  • If one of the IVs is insignificant, shouldn't that render the rest of the results meaningless? No. Did you read this somewhere? In fact, it can be that each individual variable lacks significance, yet the overall model significance is enormous. $//$ Could you please say what you mean by an “IV”? I’m pretty sure this is just a regular independent variable, but IV is common to use for instrumental variables, which are not the same. Could you please clarify? – Dave Jul 25 '23 at 02:26
  • My apologies. Yes, IV means Independent variable. In this particular case, the insignificant IV in question is atr (average tax rate). And no, I didn't read it one way or another, that is why I came here. I couldn't find an answer of what it means when one of the independent variables are not significant. So am I to understand that it is conceivable that the model as a whole could be significant while the individual variables are not? – JewJitsu11B Jul 25 '23 at 03:24
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    I notice that medinc/10 is significant, so it is not the case that "the individual variables are not." It is the case that one of the individual variables is not, and the other one is significant. Do you really want to say that "tax rate isn't significant. Therefore, nothing else can be significant no matter what the regression results are telling me?" That seems rather extreme. – jbowman Jul 25 '23 at 03:56
  • @jbowman Correct, one variable is significant but the other is not. But yes, that is what is really throwing me off. That and the fact that it passes the tests for Breusch-pagan/Cook-Weisberg test for heteroskedasticity and VIF. And my confusion is how to summarize this. Also, what use is the coefficient if it's not significant? – JewJitsu11B Jul 25 '23 at 04:32
  • Let's assume both coefficients were significant, and you added another term to the regression: my weight on sufficient randomly-selected days so that you had one observation of my weight for every data point in your sample. Let's further assume that the estimated coefficient of my weight was not significant. What would you conclude? – jbowman Jul 25 '23 at 13:55
  • I would conclude that the variable is irrelevant to my research question so it shouldn’t be significant and would be spurious. (I would also conclude that the variable isn’t at the appropriate level of analysis.) my apologies, I’m assuming that there’s a legitimate point to your response, it may be my own ignorance and limited experience, but I’m not seeing it. – JewJitsu11B Aug 10 '23 at 21:21

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