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I have panel data and for the I have a following equation

$$ logY = \beta_1 + \beta_2 logX + \beta_3 m logW $$

Problem is with $\beta_3$ coefficient. Since m is outside the log and it is a very small value, the overall $mlogW$ is very small and the resulting coefficient is very large (in thousands). One alternative would be to use standardization but my results change drastically if I standardize the variables. A solution I thought of was to multiply the variable with a constant such as 10000. So the new equation will become:

$$ logY = \beta_1 + \beta_2 logX + \beta_3 m logW*10000 $$

My question is, is this a right thing to do? It should only change the magnitude of the coefficients just for the sake of reporting the coefficients but actually it is changing how all the other coefficients behave when I multiply it say by 1000 or 10000.

Or is there a way to standardize the coefficients after estimation? since I have panel data, the division and multiplication with standard deviations seem complex.

Any idea how to make it work?

Thank you

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    $\begingroup$ Your results will not change at all by standardizing the variables. The fact that they do implies you are not actually standardizing them. I suspect--assuming that "$logX$" means $\log(X)$ and "$mlogW$" means $m\times\log(W)$--that you might have standardized $X$ and $W$. (It is unclear whether $m$ is a variable or a constant.) That drastically changes the entire model. The variables you might wish to standardize (although it will make no difference) are $\log(X)$ and $\log(W)$. What's the problem with large coefficients, anyway? Just use appropriate units of measurement to report them. $\endgroup$
    – whuber ♦
    Commented Jul 8, 2014 at 14:20
  • $\begingroup$ thank you for the quick response. $m$ is a share of imports to GDP so it is not a constant. Unfortunately, standardization IS changing my results alot. if I standardize $log(W)$ the very small number is $m$ will again make the variable very small and problem will persist. Problem with large coefficients is that the table looks very ugly. two coefficients around the value "1" and this one ranging from 4000 to 20000 in different models. Is multiplying a variable with a constant a serious crime? $\endgroup$
    – Ali-Jena
    Commented Jul 8, 2014 at 14:25
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    $\begingroup$ You need to understand "results" in terms of the estimated relationship between the independent and dependent variables. Standardizing will not change this. (In fact, almost all regression software will automatically standardize the variables in order to do its calculations.) Because you changed how you express the variables' values, of course the estimated coefficients will change! Multiplying a variable by a constant merely changes the units by which you measure it. There's absolutely nothing wrong with that. $\endgroup$
    – whuber ♦
    Commented Jul 8, 2014 at 14:34
  • $\begingroup$ "Multiplying a variable by a constant merely changes the units by which you measure it. There's absolutely nothing wrong with that" exactly as I thought but significance levels are changing when I do that. which is very weird. $\endgroup$
    – Ali-Jena
    Commented Jul 8, 2014 at 14:36
  • $\begingroup$ It sounds like you might need to share more details about what you are doing, then. But please note that if $m$ is one of your independent variables then the standardization also changes the interpretation of the coefficients: there's a lot of discussion about that on this site. Perhaps that answers your question? $\endgroup$
    – whuber ♦
    Commented Jul 8, 2014 at 16:13

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Your question is, is this (multiplying by a scalar) a right thing to do?

My answer is: Yes.

Multiplying by 1000 should not be changing the other coefficients, its a totally normal processes to multiply your predictors by any coefficient. Actually if you multiply a coefficient by 1000 you should see your coefficient decrease by a factor of .001 exactly.

An example is if I measured weight in kg and wanted to convert it to pounds I would multiply it by 2.2 and include that in the model. This would not change the R-squared or the fits of the other parameters at all. By multiplying by 2.2 we are not actually changing the correlations or relationship to the other variables in the model, they would still have the exact same correlation. If the other coefficients are changing you may have accidentally changed something else?

I am not aware of a way to scale parameters after fitting.

You could always try exploring the scale() function in R: http://stat.ethz.ch/R-manual/R-patched/library/base/html/scale.html

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  • $\begingroup$ thank you for your comment. The effect on other variables after this rescaling is not too big but the significance and sign of the coefficient itself changes dramatically if I multiply it by 10000 instead of 100000. Just to add one more thing (if that might help) I am using fully modified OLS for estimation of long run estimates of cointegrated equations. $\endgroup$
    – Ali-Jena
    Commented Jul 10, 2014 at 14:47

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