<p>Case 1: $$\mathbb{E}(x'\epsilon) = \mathbb{E}(\mathbb{E}(x'\epsilon|x)) = \mathbb{E}(x'\mathbb{E}(\epsilon|x))$$</p>

<p>Therefore, 
$$\mathbb{E}(\epsilon|x) = 0 \implies \mathbb{E}(x'\epsilon) = 0 $$
Equivalently,
$$\mathbb{E}(x'\epsilon) \neq 0 \implies \mathbb{E}(\epsilon|x) \neq 0 $$</p>

<p>Thus, violating assumption (2).</p>

<p>Case 2:
If two random variables are independent, then they are uncorrelated i.e. their covariance is 0. Equivalently, if two random variables are correlated, then they are not independent. Therefore, $Cov(x_i, \epsilon)\neq 0$ implies that $x$ is not independent of $\epsilon$ violating assumption (4).</p>

CC BY-SA 4.0
2018-06-06T03:56:16.883Z
2018-06-06T03:56:16.883Z
Amit
<p>I  assume that x is a scalar.( but argument is similar if it isn't).  </p>

<p>They both violate condition 4) because they both say that x, the regressor, is not independent of $\epsilon$. Case 1 a theoretical statement about the regressor and the noise term. Case 2 is an empirical statement about some of the  actual regressor data and the noise term.   If you want a nice intuitive explanation of econometrics ( not a standard textbook exposition ), I highly recommend Peter Kennedy's text. I think it's called "econometrics"but I'm pretty sure he only wrote one book so you can google for him on amazon.</p>

CC BY-SA 4.0
2018-06-06T05:56:49.500Z
@amit: nice on case 1. but I think case 1 also violates 4) because $E(x^\prime \epsilon)$ is the definition of the covariance of $x$ and $\epsilon$. ( assuming mean of $\epsilon$ = 0 ). thanks for fix on case 1.
mark leeds
CC BY-SA 4.0
2018-06-06T02:20:47.363Z
2018-06-06T02:20:47.363Z
mark leeds
<p>For the given model $y = x\beta + \varepsilon$, the four assumptions would be</p>

<ol>
<li>The model is linear in $x$ and $\beta$, also additive in $\varepsilon$</li>
<li>The conditional mean of the error terms are zero (i.e. $E[\varepsilon|x] = 0)$</li>
<li>$Var[\varepsilon|x] = \sigma^2I_n$</li>
<li>Exogeneity of $x$ (i.e. either $x$ is fixed or independent of $\varepsilon$)</li>
</ol>

<p>My questions is that in the case of the following situations which assumptions have been violated.</p>

<p>Case 1. $E[x'\varepsilon] \neq 0$ [I am not sure whether this implies the second or fourth assumption has been violated]</p>

<p>Case 2. $Cov(x_i,\varepsilon_i) \neq 0$ [I suppose the third one has been violated but could it imply the violation of other assumptions?]</p>

<p>Any help will be greatly appreciated.</p>

<econometrics>
Regarding the assumption of Classical Linear Regression Model
James
# Regarding the assumption of Classical Linear Regression Model

(Asked by: username_0 on 2018-06-05)

For the given model $y = x\beta + \varepsilon$, the four assumptions would be

1. The model is linear in $x$ and $\beta$, also additive in $\varepsilon$
2. The conditional mean of the error terms are zero (i.e. $E[\varepsilon|x] = 0)$
3. $Var[\varepsilon|x] = \sigma^2I\_n$
4. Exogeneity of $x$ (i.e. either $x$ is fixed or independent of $\varepsilon$)

My questions is that in the case of the following situations which assumptions have been violated.

Case 1. $E[x'\varepsilon] \neq 0$ [I am not sure whether this implies the second or fourth assumption has been violated]

Case 2. $Cov(x\_i,\varepsilon\_i) \neq 0$ [I suppose the third one has been violated but could it imply the violation of other assumptions?]

Any help will be greatly appreciated.

**Tags:** econometrics

---

## 2 answers

### Answer by username_1 on 2018-06-06. Score: 1 | Accepted answer

Case 1: $$\mathbb{E}(x'\epsilon) = \mathbb{E}(\mathbb{E}(x'\epsilon|x)) = \mathbb{E}(x'\mathbb{E}(\epsilon|x))$$

Therefore,
$$\mathbb{E}(\epsilon|x) = 0 \implies \mathbb{E}(x'\epsilon) = 0 $$
Equivalently,
$$\mathbb{E}(x'\epsilon) \neq 0 \implies \mathbb{E}(\epsilon|x) \neq 0 $$

Thus, violating assumption (2).

Case 2:
If two random variables are independent, then they are uncorrelated i.e. their covariance is 0. Equivalently, if two random variables are correlated, then they are not independent. Therefore, $Cov(x\_i, \epsilon)\neq 0$ implies that $x$ is not independent of $\epsilon$ violating assumption (4).


---

### Answer by username_2 on 2018-06-06. Score: 1

I assume that x is a scalar.( but argument is similar if it isn't).

They both violate condition 4) because they both say that x, the regressor, is not independent of $\epsilon$. Case 1 a theoretical statement about the regressor and the noise term. Case 2 is an empirical statement about some of the actual regressor data and the noise term. If you want a nice intuitive explanation of econometrics ( not a standard textbook exposition ), I highly recommend Peter Kennedy's text. I think it's called "econometrics"but I'm pretty sure he only wrote one book so you can google for him on amazon.

> **Comments:**
> **username_2**: @amit: nice on case 1. but I think case 1 also violates 4) because $E(x^\prime \epsilon)$ is the definition of the covariance of $x$ and $\epsilon$. ( assuming mean of $\epsilon$ = 0 ). thanks for fix on case 1.  

---

