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I want to test if the slope of a regression line is -1. I used the t test as such:

t = β-(-1)/SE(β)

where β is the slope coefficient and the SE(β) is the standard error of the slope coefficient. I defined α as 0.05 and calculated the critical value for t-statistic as 1.96 since the number of samples in my dataset is 10000. If I understand correctly, if the |t-statistic| is smaller than the critical vale then we can assume that the hypothesis H0 : β=-1 cannot be rejected.

When I actually tested this on my dataset, I got the following results:

β = 2002.39

SE(β) = 1564.45

|t-statistic| = 1.28 which is smaller than critical value so Ho is not rejected.

But the β value of 2002 is far away from -1.

Also, there is another dataset which gives the following

β=-16

SE(β) = 0.39

|t-statistic|= 38.46 which is greater than critical value so Ho is rejected

But one could say the β value of -16 is a lot closer to -1 than 2002 so the t-statistic should be closer to the critical value in the case of -16.

Am I wrong in this assumption ? Is the formula for t-test of β=-1 correct ?

Is there any other way to test if β is equal to -1? I also want a "measure of evidence" statistic that shows how close β is to -1.

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2 Answers 2

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In your first example, the predicted value of the true parameter $\beta$ is:

$\hat{\beta}=2002.39$

Assuming you have one explanatory variable ($x$), the standard error of $\hat{\beta}$ is given by:

$\mathbf{se}(\hat{\beta}) = \sqrt{\frac{\frac{1}{n-2} \times \sum_{i=1}^{i=n}(y_{i}-\hat{y_{i}})^2}{\sum_{i=1}^{i=n}(x_{i}-\bar{x})^2}}=1564.45$

Because the standard error is so large, we are not able to easily pin down the true value of $\beta$. Your t-stat is indicating that the hypothesized value $\beta=-1$ is only $t=1.28$ standard deviations away from your predicted value. Hence, it is quite possible that your null hypothesis $\mathbf{H}_{0}:\beta=-1$ is true.

We can use the same reasoning in your second example. Only here, the standard error is much smaller $\mathbf{se}(\hat{\beta})=0.39$. To go from your predicted value of $\hat{\beta}=-16$ to the hypothesized value of $\beta=-1$ we would have to move $|(-16)-(-1)|/0.39=38.46$ standard deviations to get there. Which is statistically unlikely.

It all has to do with the relative size of $\hat{\beta}$ and $\mathbf{se}(\hat{\beta})$.

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  • $\begingroup$ so, t-statistic cannot be used as a measure of how close estimated β is to -1 ? I had thought that I got it wrong with the formula or the hypothesis. Thank you for the answer. Also, do you have any opinions on how to measure how close β is to -1 ? I would prefer to take SE(β) into account since it defines stability of the estimated β $\endgroup$
    – Kanmani
    Oct 11, 2018 at 1:43
  • $\begingroup$ Hey @Kanmani, The t-stat tells you how many standard errors away the predicted value is from the hypothesized value. In that sense, it is a good metric for measuring how close beta is to a given value. It might be easier to think of it in terms of confidence intervals. In the first example, you will see that -1 falls within your confidence interval. While in the second it clearly does not. $\endgroup$
    – TensorFlow
    Oct 11, 2018 at 2:11
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In the first case the standard error of the estimate is telling you that you really have no clue what the value is. You cannot be reasonably sure it isn't -1, or -100 or -1000, or 4000 (!). Population values between about -1200 and 5000 (roughly) will be more or less consistent with the data. By contrast, with the second one you have a very good sense of its value; only population values between about -15 and -17 could reasonably be consistent with the data, so (if the assumptions are reasonable), you can rule out a population slope of -1 as a remotely reasonable explanation for the data you observed.

I also want a "measure of evidence" statistic that shows how close β is to -1.

I don't know what you mean by "measure of evidence" here. I suggest you consider using confidence intervals as a way to see what population slopes might be reasonable to see as more-or-less consistent with the data.

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