I'm regressing a response versus a ratio between two measurements as an independent variable. I'm getting a significant positive effect and I'd like to test whether the contribution of the increase in the numerator measurement is lower than that of the decrease in the denominator measurement. Is there a better way to get to this other than comparing the slopes of the response fitted to each of them separately?
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2 Answers
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Your starting model is:
$Y = \beta_0 + \beta_1\frac{A}{B}$
But, you are really interested in:
$Y = \beta_0 + \beta_1 \cdot \Delta \text A + \beta_2\cdot \Delta \text B$
In your statement I think that is what you are doing. And, if that is it, that's fine. That would be the best way to figure out what are the separate influences of changes in A ($\Delta \text A$) and changes in B ($\Delta \text B$) on your dependent variable Y.
If you do that, you may also want to detrend your dependent variable so that it also reflects a change. By doing so, you will avoid unit root issues in both your dependent and independent variable. Your model will also probably test better in terms of residuals structure (heteroskedasticity, autocorrelation, Normality).
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$\begingroup$ Thanks. By change in A and change in B do you A-min(A) and B-min(B)? $\endgroup$– danCommented Jun 16, 2016 at 0:02
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$\begingroup$ Depending on the shape of your variable there are different ways of capturing change. Let's say your variable is a nominal value like GDP. And, the GDP of the country you are looking at is $600 billion. In that case, the best detrended transformation is % growth in GDP over the previous period or: (GDP/GDP t-1) -1. And, you will get some value, let's say 1.5%. But, if your variable is the Unemployment rate, then you have to use a First Difference. So, if Unemployment decreased from one period to the next from 7% down to 6% your First Difference would be: 6% - 7% = -1%. $\endgroup$– SympaCommented Jun 16, 2016 at 3:26
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You should be fitting the interaction model: $$ Y = \beta_0 + \beta_A A + \beta_B B + \beta_{AB} (A \times B) $$ See Kronmal (1993), sections 4 & 5 for the argument.
answered Mar 16, 2018 at 16:52