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I know that GDP can be calculated from the national accounts using data which says how much of each industry's output was bought by each other industry.

But how is this kind of supply/use data gathered? Is it via questionnaire?

If so, is a random sample of businesses quizzed, or does every business have to submit data on what they bought and from whom?

Where is this data-gathering process documented? (For, e.g., the UK.)

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In the UK, which uses what is know as "the output approach", the Office for National Statistics (ONS) sends out around 45,000 surveys to business, called the Monthly Business Survey (MBS).

The ONS claims that this covers "around 55% of the economy.".

In this survey, questions about turnover and purchases are asked which then allow the ONS to assign total outputs and intermediate expenditures to industry classifications. In the case of the ONS, they use Standard Industrial Classification (UK SIC 2007).

There is a separate survey for construction called the Construction Output Survey, and there are additional forecasts and models to fill in the gaps.

These output and expenditure estimates are then deflated by things like the Consumer Price Index (CPI) to produce an inflation-adjusted GDP figure.

More detail on this process along with many more links to data sources is available in the GDP quality and measurement information document.

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Answering to the who: the GDP is usually calculated by a statistics agency of the government. In the US it is the America’s Bureau of Economic Analysis; in Britain, it is the Office for National Statistics.

About how it is done in practice, take a look at this question: https://economics.stackexchange.com/questions/12311/how-to-measure-gdp-in-practice/17812

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The System of National Accounts (SNA) is the international standard regarding measurement and dissemination of National Accounts. It is produced by the United Nations. It's latest edition is 2008.

Chapter 14 in this manual provides a very detailed account of who and how these statistics are obtained. A summary is shown in the image below:

enter image description here

As the document states:

Data for national accounts come from three main sources:
a) Administrative records;
b) Statistical methods: data obtained by extrapolating survey results on the basis of a benchmark census, a complete enumeration of the entire population;
c) Estimation methods in the national accounts departments.

Regarding administrative records, these come from:

a) Government revenue and expenditure statistics (traditionally compiled by the ministry of finance on government budget);
b) Foreign trade statistics (i.e., exports and imports of goods through customs);
c) Money and banking statistics (traditionally compiled by the central bank);
d) Report on insurance companies by the insurance regulatory authority;
e) Tax records, with limited information on sales, cost of sales and income (processed by tax authority);
f) Business accounts of publicly traded corporations submitted to the stock exchange regulation commission.

Data from statistical methods are based on census and surveys. So in effect, National Accounts do not include a sample of every single firm/household. As the document states:

Surveys are based on a scientifically selected random sample from a population. Data for the population is obtained by blowing up the sample data by extrapolating the sample size to the
population size.

Finally, estimation is also used to produce national accounts:

There are three methods of estimation in national accounts for data that are not available in administrative records, censuses or surveys:
a) Balancing item method;
b) Commodity flow method;
c) Benchmark ratio method.

You can read more about these issues on the aforementioned chapter.

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