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Suppose I have a no intercept regression model such as $Y = \beta x + \epsilon$ where $x$ is univariate and so is $y$.

I want to test the null hypothesis that $\beta = -1.0$.

So, I calculate the sse of the model assuming that $\beta = -1.0$ and call that $sseA$.

Then I estimate the model and call that modelB and get all the sums of squares of that modelB.

Then I calculate the usual F-test of $(sseA - seeB)/1$ divided by $sseB/(n-1)$ where $n$ is the number of observations.

I'm not sure if this F-test is valid for 2 reasons.

  1. the intercept is not in the model

  2. can modelA be viewed as nested in modelB? F-tests are only valid for nested models ( assuming normality) and I'm not clear if this case can be thought of as nested. Usually, the null being true results in a model that is a subset of the full model but here things are a little different.

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    $\begingroup$ Why is the intercept not in the model? $\endgroup$
    – Mark White
    May 22, 2017 at 11:50
  • $\begingroup$ Hi Michael: It's not always the case that an intercept be in the model. Suppose one wanted to build a model his car's miles per gallon and he used gas from different gas stations. the person filled his car up 20 times over some time period and calculated the miles driven each time he filled it. then, he-she wanted to estimate the miles per gallon coefficient for his-her car. in this case, the person would have 20 observations but having an intercept in the model: $ miles driven_i = miles/galion coefficient \times (gallons in tank)$ doesn't make sense. what would it represent ? $\endgroup$
    – mlofton
    May 24, 2017 at 15:12

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An alternative approach is to use an offset (a covariate with known coefficient). So change your model to

$y = \beta x +$ offset($-x$) + $\epsilon$

and then test whether your estimated $\beta = 0$ or establish a confidence interval about it.

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I see no reason not to estimate the regression coefficient (beta) for the model (zero intercept), then test whether the estimated value is equal to -1 using a t-test.

You would also be interested in how much different your estimate is from the hypothesized value of -1 whatever the test shows. A large but insignificant difference might still be suspicious. It also would suggest your sample size is insufficient to get a good estimate of beta.

If your estimated beta is much different from -1 then you should examine the residuals from both the estimated model and from your hypothesized model (beta = -1, intercept = 0) carefully for the usual stuff but also for differences between the two models. I suspect that either model might work poorly near the ends of the range of x you have observed.

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  • $\begingroup$ Hi: mdewey: Yes, the offset approach turns it into a test of beta = 0 which makes it nested. I figured that out after I sent this post. Thanks. david smith: the t-test will give the same answer as the F-test, if one uses offset. my problem actually turns out to be the opposite of what you described. I have so many observations that the power of the F-test is too great so I get rejection of H_o even for very small differences in the estimate and -1. To me, this is where statistics really doesn't work. If you have tons of observations, then, statistical tests can be problematic for sure. thx $\endgroup$
    – mlofton
    May 22, 2017 at 18:10
  • $\begingroup$ Statistical tests were designed for small samples, not large ones. An estimate and one or more intervals should serve your needs. How important is a small difference in practice? And look at the pattern of residuals. $\endgroup$ May 22, 2017 at 18:20
  • $\begingroup$ Thanks David: Definitely statistical tests can fall apart when one has a large number of observations. $\endgroup$
    – mlofton
    May 24, 2017 at 15:19

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