I would like to know if the coefficient of an independent variable is still relevant if the R-squared is low (assuming the p-value for the independent variable is less than 0.05).

For example, assume I have ice cream sales as my dependent variable and GDP as my independent variable. If I run a regression analysis the R-squared will be very low (say 0.001), because clearly there are other factors that will explain ice cream sales, but the GDP p value is significant (i.e. <0.05) and the coefficient is 0.0001. Can I still conclude from this regression that for every $1 increase in GDP, ice cream sales will increase by 0.0001?

For the avoidance of doubt, I am not interested in predicting future ice cream sales I just want to know if there is a relationship between ice cream sales and gdp and to what extent.

Many thanks in advance.


If you do not include all relevant independent variables, then the coeffcients of the variables that you keep in your regression will be biased (unless they are independent of the variables that you left out), so whether your $R^2$ is high or low does not matter, you have a biased estimate of your coefficient and your hypothesis test can lead to wrong conclusions because of the bias.

This problem is known as the 'omitted variable bias', you find it via google.


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