<p>The reasoning behind it is simple:</p>
<ol>
<li><p>Multicolinearity distorts t-test and consequently p-values because it inflates standard errors of the <span class="math-container">$\beta$</span> coefficients. Since in multivariate regression the variance of estimated coefficients depends on correlation between independent variables.</p>
</li>
<li><p>F-test is not affected by multicolinearity.</p>
</li>
<li><p>the t-test tests whether coefficients are individually significant (with null <span class="math-container">$\beta_i=0$</span> and alternative hypothesis <span class="math-container">$\beta_i\neq0$</span></p>
</li>
<li><p>The F-test tests the null of joint insignificance <span class="math-container">$\beta_1 = \beta_2 = … = \beta_k=0$</span> against alternative that at least one of the coefficients is not equal to zero.</p>
</li>
</ol>
<p>Given 1-4 it is very strange to observe situation where using F-test we cannot reject the hypothesis that at least one coefficient is non-zero, while individual t-test we cannot reject the hypothesis that both coefficients are individually equal to zero.</p>
<p>In that case results become a bit suspicious which warrants further investigation. One likely explanation is that there is a multicolinearity present because as mentioned above multicolinearity distorts t-tests but not F-test.</p>
<p>However, this itself is not a test for multicolinearity just common symptom. If you want to make sure that multicolinearity is present you should make further investigation by for example calculating variance inflation factors. VIF&gt;5 (some sources say VIF&gt;10 these values are just rule of thumbs) tell you that multicolinearity is indeed present.</p>

CC BY-SA 4.0
2021-10-26T20:38:44.960Z
Thanks @1muflon1, from the calculation of p-value based on std.err (https://www.bmj.com/content/343/bmj.d2304), I know that the higher the std.err, the lower p-value. But I don't understand why multicollinearity cause the std.err inflation
Phil Nguyen
CC BY-SA 4.0
2021-10-26T20:47:37.100Z
and can I ask why small N is also an indicator of multicollinearity in this case if it is convenient to you
Phil Nguyen
CC BY-SA 4.0
2021-10-27T18:19:34.533Z
@Louise 1. Small N is not an indicator of multicolinearity. 2. It is because in formula for se in a *multivariate* regression there is also term $(1/(1-r_{x1x_2}^2$
1muflon1
CC BY-SA 4.0
2021-10-26T20:26:31.563Z
2021-10-26T20:26:31.563Z
1muflon1
<p>Today I read a document about multicollinearity <a href="https://www3.nd.edu/%7Erwilliam/stats2/l11.pdf" rel="nofollow noreferrer">here</a>.</p>
<p>In the first page, the author runs a simple regression that</p>
<p><span class="math-container">$y = \beta_1x1+ \beta_2x2$</span></p>
<p>the p-value( or t-statistic) of coefficients of <span class="math-container">$x1$</span> and <span class="math-container">$x2$</span> are <strong>insignificant</strong> but the overall <strong>F-test is significant</strong>. And he said that it is an indicator of multicollinearity.</p>
<p>I do not understand what is the reason for this conclusion, would you please help me to understand it intuitively?</p>
<p>Much appreciated.</p>

CC BY-SA 4.0
2021-10-27T08:25:35.480Z
If $y = x_1+ x_2$ there is no need for a regression.
Bertrand
CC BY-SA 4.0
2021-10-27T08:40:23.860Z
@Betrand sorry, I forget to add the coefficient, it is a regression
Phil Nguyen
<econometrics>
Why p-value of coefficients of each variable are insignificant but the overall F is significant is a indicator of multicollinearity?
Phil Nguyen
# Why p-value of coefficients of each variable are insignificant but the overall F is significant is a indicator of multicollinearity?

(Asked by: username_0 on 2021-10-26)

Today I read a document about multicollinearity [here](https://www3.nd.edu/%7Erwilliam/stats2/l11.pdf).

In the first page, the author runs a simple regression that

$y = \beta\_1x1+ \beta\_2x2$

the p-value( or t-statistic) of coefficients of $x1$ and $x2$ are **insignificant** but the overall **F-test is significant**. And he said that it is an indicator of multicollinearity.

I do not understand what is the reason for this conclusion, would you please help me to understand it intuitively?

Much appreciated.

**Tags:** econometrics
> **Comments:**
> **username_1**: If $y = x_1+ x_2$ there is no need for a regression.  
> **username_0**: @Betrand sorry, I forget to add the coefficient, it is a regression  

---

## 1 answers

### Answer by username_2 on 2021-10-26. Score: 1 | Accepted answer

The reasoning behind it is simple:

1. Multicolinearity distorts t-test and consequently p-values because it inflates standard errors of the $\beta$ coefficients. Since in multivariate regression the variance of estimated coefficients depends on correlation between independent variables.
2. F-test is not affected by multicolinearity.
3. the t-test tests whether coefficients are individually significant (with null $\beta\_i=0$ and alternative hypothesis $\beta\_i\neq0$
4. The F-test tests the null of joint insignificance $\beta\_1 = \beta\_2 = … = \beta\_k=0$ against alternative that at least one of the coefficients is not equal to zero.

Given 1-4 it is very strange to observe situation where using F-test we cannot reject the hypothesis that at least one coefficient is non-zero, while individual t-test we cannot reject the hypothesis that both coefficients are individually equal to zero.

In that case results become a bit suspicious which warrants further investigation. One likely explanation is that there is a multicolinearity present because as mentioned above multicolinearity distorts t-tests but not F-test.

However, this itself is not a test for multicolinearity just common symptom. If you want to make sure that multicolinearity is present you should make further investigation by for example calculating variance inflation factors. VIF>5 (some sources say VIF>10 these values are just rule of thumbs) tell you that multicolinearity is indeed present.

> **Comments:**
> **username_0**: Thanks @1muflon1, from the calculation of p-value based on std.err (https://www.bmj.com/content/343/bmj.d2304), I know that the higher the std.err, the lower p-value. But I don't understand why multicollinearity cause the std.err inflation  
> **username_0**: and can I ask why small N is also an indicator of multicollinearity in this case if it is convenient to you  
> **username_2**: @Louise 1. Small N is not an indicator of multicolinearity. 2. It is because in formula for se in a *multivariate* regression there is also term $(1/(1-r_{x1x_2}^2$  

---

