Let's say I'm trying to tell whether my company's sales have had a downward trend over the last 10 years. Let's also say I have available to me the last 10 years of quarterly sales. If I run a bivariate regression with time as the independent variable and sales as the dependent variable, and if the regression coefficient for time is negative, do I need to examine the regression coefficient's t-statistic? I mean, aren't these data based on a census of my population of interest? If they're a census, why would I need statistical testing?

Basically I'm wondering how often my knee-jerk reaction to apply statistical testing has been wrong-headed.


I think the correct answer depends on the question. If your question is, "On average have the observed sales of my company decreased over time?" there is no reason for statistical inference. Suppose on the other hand you think that the data generating process is:

observed_sales = constant + fundamental_trend * year + white_noise

You are interested in testing the hypothesis "fundamental_trend = 0". In this case, you can think of the observed sales of your company as a finite sample of an infinite data generating process, and employing statistical inference makes sense.

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  • $\begingroup$ I tend to follow this approach in practice. While there is no sampling error, there is still the possibility that some other noise ("luck" or "random chance") could have caused the downturn, and that that "luck" may change in the future. Of course it may not be truley a random stochastic process, but with no explanatory variables it might behave like one. $\endgroup$ – david25272 Mar 20 '14 at 4:35
  • $\begingroup$ What you wrote is helpful. However, I'm having trouble choosing between the two options you provided. My core question is the first one you wrote -- I seek to describe my sales over time. But you bring up a good point about random noise. From the narrow perspective of the model you wrote, the "white noise" is from exogenous factors, right? Therefore, even if I have the simple descriptive question, maybe I should use statistical inference to test whether the linear effect of time overcomes the noise of exogenous factors? $\endgroup$ – user42226 Mar 20 '14 at 5:32
  • $\begingroup$ It seems to me that observed sales are just a set of numbers. For example, suppose we have a number x and a number y. I don't need statistics to check if x > y. On the other hand, if I think that x = a + noise and y = b + noise, and I am interested in testing whether a > b, then I will need statistics. Since you are talking about exogenous factors, I guess you have something more like the second case in mind. $\endgroup$ – veryshuai Mar 20 '14 at 13:50
  • $\begingroup$ I think it's perfectly acceptable to treat time as a possible exogenous variable and test whether its coefficient is non-zero. What makes less sense is to apply confidence intervals to point estimates (since they aren't really estimates). $\endgroup$ – david25272 Mar 20 '14 at 21:21

I believe you're correct. Confidence bounds are needed when we are sampling from a random process. If you assume that the data you have exactly represents the companies true financials and there is no measurement error then you can accurately say the regression coefficient is the true coefficient.

However, if the data might be subject to measurement error (the data may not reflect true financials) or there is huge 'jitter' (like your companies financials change on a daily basis) then I think you would need confidence bounds...

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    $\begingroup$ The confidence bounds computed by standard procedures assume that uncertainty comes from sampling a finite sample from a population, not from measurement error. So, even if you need it, the standard approach won't give the bounds you are looking for. $\endgroup$ – Maarten Buis Mar 20 '14 at 8:33

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