How to Measure the Importance of a Variable in a Regression Equation

[From the working paper “On the economic importance of the determinants of long-term growth” by Olivier Sterck, posted at the Centre for the Study of African Economies working papers website]
“The long-run growth literature abounds with statements about the“economic significance” or “economic importance” of variables of interest. To justify these statements, authors usually report the standardized beta coefficient associated with the variable of interest, or, more rarely, the partial R2 or the Shapley value.”
“I show that these statistics are inadequate to measure economic importance.”
“I …propose two complementary methods to measure the economic importance of variables and residuals in regressions. … Both methods aim at measuring the percentage contribution of each explanatory variable to variation in the dependent variable. They differ in how they handle variation induced by explanatory variables that are correlated.”
“According to the ceteris paribus approach, the importance of a variable xi is measured as:”
TRN1(20190330)
“The measure qi2 captures the variation generated by the explanatory variable xi ceteris paribus and expresses it in percentage terms.”
“The second method focuses on non-ceteris paribus variation: the importance of an explanatory variable not only depends on the variation it generates alone, but also on the variation co-generated with other explanatory variables. …. According to the non-ceteris paribus approach, … the importance of a variable xi is measured as:”
TRN2(20190330)
“…I show that it can be interpreted as the elasticity of Var(y) with respect to Var(βixi). The measure is therefore positive if a marginal increase in βi increases Var(y), and negative if a marginal increase in βi reduces Var(y).”
“The two measures are complementary. They bring different information, especially when they sharply differ. For example, if the ceteris paribus importance of a variable is much larger than its non-ceteris paribus importance, it means that the effect of that variable is going against the effects of other variables included in the estimated model. On the contrary, if the ceteris paribus importance of the variable is much smaller than its non-ceteris paribus importance, it means that the effect of that variable is reinforcing the effect of other important variables.”
“As research analyzing large datasets is more and more frequent, evaluating economic importance is becoming as – if not more – important than assessing statistical significance. This research proposed two intuitive methods to measure importance that usefully complement standard measures of statistical significance.”
To read the article, click here.

 

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