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I am working on consumption expenditure data for 5-6 different years but there are 5 years of missing values in the in between those years. For example, I have data for 2004-05 and 2009-10, with 5 years data missing, and so on. And I am forecasting for 2019-20. Can I use moving average for forecasting? Can I impute the missing value or just forecast with the missing values present?

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You can forecast if you have missing values in the data, but the forecast might not be a good one. You can read this link to understand time series.

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