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I doubted whether I should use the preposition "with", "by" or no preposition at all in the following sentence:

I do not estimate equation (#) by using/with use of/using [name of some econometric method] because [explaining the reasons].

To answer this question myself I did some research. The following web page says that one uses the preposition "by" to indicate a mean or method. As I refer to a particular (econometric) method, it could be an option to use "by". However, I checked how other authors did it in the literature I write about and I came to the conclusion that it should be:

I do not estimate equation (#) using [name of some econometric method] because [explaining the reasons].

However, this sounds a bit odd to a non-native English speaker. Is it the last quote correct anyway?

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There are a number of different ways to negate a sentence, or even whether to negate it at all. Sometimes the problem is more where you put the "not" than whether a particular preposition fits.

In this case you're right, the sentence is already awkward so no matter how you phrase it, it's still going to feel "off". This is a good opportunity to rephrase:

I chose not to estimate [some equation] with [some method] because [some reason].

I was unable to estimate [some equation] with [some method] because [some reason].

It was not possible to estimate [some equation] with [some method] because [some reason].

In general, the verb "to use" something is itself overused. If you can, good style suggests that, at least in formal writing, you insert a more descriptive and targeted verb that fits the context

I used a Venn diagram to describe [some thing].
I described [some thing] with a Venn diagram.
I illustrated [some thing] with a Venn diagram.

If you feel you must use the sentence as written, you do not need any preposition. In some formal, academic, or technical writing, you can omit words that neither add to nor subtract from the meaning of the sentence:

I did not estimate equation using [name of some econometric method] because [explaining the reasons].

Although, again, the plain preposition "with" is better:

I did not estimate equation with [name of some econometric method] because [explaining the reasons].

Also, note I changed your example to the past tense, which I feel is more natural in formal academic / technical writing.

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