- Ironically, however, investment has been lackluster in the period India’s ease of doing business ranking (...?...) improved.
and
- According to World Bank, approximately two-third of the data (...?...) embedded in doing business indicators are based on a reading of the law.
Which Auxiliary Verb should I use in the brackets?
Is it OK if we omit auxiliary?
In which cases can we omit the auxiliary? Is there any specific grammar rule available in English?
In general the auxiliary verb you will use depends on your precise sense of time frame and mood in the sentence. Since you do not say what it is, it is difficult to know which is right for you. And, there are some usage problems in the second sentence beside the missing auxiliary verb.
Here are some suggestions:
According to the World Bank, approximately two-thirds of the data that have been used to calculate business indicators are based on a specific reading of the law.
1) "World Bank" requires an article 'the' (this is just because of the way they style their own name)
2) "two-thirds" is correct. In general, think of "ratio" words as literally being a number of denominator shares grammatically: "one-fifth", but "three-eighths".
3) "embedded in doing" is incorrect use. "embedded in" is used to indicate that one object was contained in another object. The "container" object must immediately follow "in". "Business indicators" are "calculated", and the data is "used to calculate it": in some part of the process of calculation it is utilized.
4) The main idea of the sentence is that the business indicators can come out one way or another, depending on which reading of law you use; they are not universal. To signal this, we should say that they are calculated using a "specific" reading of the law. Now a reader will now that a different reading of the law might give a different value of the business indicator.
5) have been here indicates that the situation described in the sentence happend during some time in the past, continued through the present, and is possibly or likely ongoing. had been or was would indicate that the situation is possibly or likely no longer occurring. might have been or could have been or may have been if we suspect but are not sure that the situation occurred.
Also note that "data" is a strange word in that it is the same plural and singular, in most speech. That wasn't always true, so if you are writing for someone else, or have a style guide, you need to check if it recommends using the singular or plural form or the older use of datum for the singular.
Ironically, however, investment has been lackluster in the period that India’s ease of doing business ranking has improved.
This one is easier. The tense needs to agree since the time frame does not change. Continue using present perfect with has. The addition of that is not required, but makes the sentence flow better. If the improvement continues to the present time and is ongoing, you could also use the present perfect progressive tense:
Ironically, however, investment has been lackluster in the period that India’s ease of doing business ranking has been improving.
I'm not sure that you need any in the brackets.
Yes, in these examples - although there may be other things wrong with the sentences.
Auxiliary verbs are included to form the various tenses, moods, and voices of other verbs. If the verbs themselves are in the correct tense/voice, then auxilaries are not required, so they are omitted.
Just the above - that the tense of any verb has to match the context.
Let's look at your examples:
Ironically, however, investment has been lacklustre in the period India’s ease of doing business ranking () improved.
I don't see a need for an auxiliary verb where the brackets are. "Improved" is already past tense, and this evidently relates to a past period. The thing that seems wrong with this sentence is that "has been" is present perfect tense, and it really should be simple past (ie "was"). There is no auxiliary verb you can place before a word already in the past tense (ie "improved") to change its tense, so this seems the only way to correct the sentence:
Ironically, however, investment was lacklustre in the period India's ease of doing business ranking improved.
According to World Bank, approximately two-third of the data ( ) embedded in doing business indicators are based on a reading of the law.
Again, I don't see how an auxiliary verb in the brackets will fix the sentence. There are other things glaringly wrong with this sentence. It seems to me that it should read:
According to the World Bank, approximately two-thirds of the data embedded in doing business indicators was based on a reading of the law.
In these constructions (using write as an example) ...
is written
is writing
have written
does write
while the verb be, have, or do is sometimes called the auxillary verb, it's still the main verb of the sentence it appears in.
The order is written on the paper - Verb of this sentence: is
The clerk is writing my order down - Verb of this sentence: is
So, this sentence
..., approximately two-third of the data ( ) embedded in doing business indicators are based on a reading of the law.
already has a verb (are). So it forms a complete sentence and nothing else is needed.
Adding a second verb without taking other action to split it into two phrases or clauses will make it ungrammatical. You'd have to do something like this for it to make sense:
According to World Bank, approximately two-third of the data that is embedded in doing business indicators , are based on a reading of the law.
"Auxillaries" are used to communicate:
passive voice
progressive/continuous
perfect
emphatic or do-support for questions
If you aren't doing one of the above, you don't need the auxillary.
The past (-ed or other irregular form like driven, given) participles can work like adjectives, and both the present (-ing) and past participle forms are used in participial phrases. Both of these don't satisfy a sentence's requirement to have a verb, so in those cases, a verb needs to be added if it's not present to be grammatical.
The destroyed building at the end of the block. (incomplete)
John walked past the destroyed building at the end of the block. (complete and OK)
