<p>To be precise, a model is said to be a <strong>linear regression model</strong> when it is <strong>linear in its parameters</strong>. For extension, a model is said to be non-linear when it is <strong>non-linear in its parameter</strong> (Wooldridge, 2010, p.397).</p>

<p>As such, a linear model can have non-linear variables. A standard example is the <a href="https://en.wikipedia.org/wiki/Mincer_earnings_function" rel="noreferrer"><strong>Mincer equation</strong></a>, where wage is a linear function of education, experience and experience squared.</p>

<p>In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because <strong>its parameters can still be recovered using linear methods</strong>. In fact, you do not need non-linear methods at all. <strong>Your model is over-identified</strong> as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.</p>

CC BY-SA 4.0
2019-06-15T11:15:47.090Z
Lucho, sorry for the delay of 3 years... =D Could you tell me how you can recover the parameters using a linear methods?
An old man in the sea.
CC BY-SA 3.0
2016-08-31T10:20:24.937Z
2016-08-31T15:40:07.673Z
2016-08-31T15:40:07.673Z
luchonacho
<p>In response to a comment-request by @Anoldmaninthesea, a linear method to estimate a linear regression with 1st-order auto correlated residuals,</p>
<p><span class="math-container">$$y_t = \alpha + x_t\beta + u_t, \qquad u_t = \rho u_{t-1} + e_t,\;\;\; e_t \sim {\rm WN}$$</span></p>
<p>is the <a href="https://en.wikipedia.org/wiki/Cochrane%E2%80%93Orcutt_estimation" rel="noreferrer"><strong>Cochrane-Orcutt one.</strong></a></p>
<p>Note that the above can be re-written as</p>
<p><span class="math-container">$$y^*_t = \delta + \beta x_t^* + e_t,\qquad y^*_t=y_t - \rho y_{t-1},\;\;\;x^*_t=y_t - \rho x_{t-1},\;\;\; \delta = \alpha(1-\rho).$$</span></p>
<p>The method is iterative. Summarily:</p>
<p><strong>Step 1:</strong> Estimate by OLS <span class="math-container">$y_t = \alpha + x_t\beta$</span>, and obtain the residuals.</p>
<p><strong>Step 2:</strong> Regress the residuals on their lag, and obtain an estimate of <span class="math-container">$\rho$</span>.</p>
<p><strong>Step 3:</strong> Form the series <span class="math-container">$\hat y^*_t = y_t - \hat \rho y_{t-1}$</span> and <span class="math-container">$\hat x^*_t = x_t - \hat \rho x_{t-1}$</span>.</p>
<p><strong>Step 4:</strong> Run the OLS regression <span class="math-container">$\hat y^*_t = \delta + \beta \hat x_t^*$</span> an dobtain the residuals</p>
<p>Continue with Step 2 etc, until the estimate for <span class="math-container">$\hat \rho$</span> is stabilized.</p>
<p>Many econometric software include this algorithm as a ready-made command.</p>

CC BY-SA 4.0
2020-11-21T13:34:00.083Z
2020-11-21T13:34:00.083Z
Alecos Papadopoulos
<p>Our model: $y_t=X_t\beta+u_t$</p>

<p>Our error terms: $u_t=\rho u_{t-1}+\epsilon_t$ with $\epsilon_t\sim IID(0,\sigma^2)$, and $|\rho|&lt;1$.</p>

<p>This results in $y_t=\rho y_{t-1}+X_t\beta-\rho X_{t-1}\beta+\epsilon_t$.
Why is this last model not linear in $\beta$ and $\rho$?</p>

<p>Is it because the parameters are multiplied with each other in the 3rd term? or is there something else?</p>

<p>Any help would be appreciated.</p>

CC BY-SA 3.0
2016-04-06T03:39:21.083Z
I believe you're right. As described on the Wikipedia page (https://en.wikipedia.org/wiki/Linear_model), you need to be
able to write it as a linear model. However, the coefficient on $X_{t-1}$
has a restriction on it that will always depend on $\rho$ or $\beta$.
jmbejara
CC BY-SA 3.0
2016-04-10T14:22:19.797Z
@jmbejara want to post an answer? I would like to put this question in the answered section. ;)
An old man in the sea.
<econometrics>
Why is a 'linear' regression model, with AR(1) error terms, a non-linear model?
An old man in the sea.
# Why is a 'linear' regression model, with AR(1) error terms, a non-linear model?

(Asked by: username_0 on 2016-04-05)

Our model: $y\_t=X\_t\beta+u\_t$

Our error terms: $u\_t=\rho u\_{t-1}+\epsilon\_t$ with $\epsilon\_t\sim IID(0,\sigma^2)$, and $|\rho|<1$.

This results in $y\_t=\rho y\_{t-1}+X\_t\beta-\rho X\_{t-1}\beta+\epsilon\_t$.
Why is this last model not linear in $\beta$ and $\rho$?

Is it because the parameters are multiplied with each other in the 3rd term? or is there something else?

Any help would be appreciated.

**Tags:** econometrics

> **Comments:**
> **username_1**: I believe you're right. As described on the Wikipedia page (https://en.wikipedia.org/wiki/Linear_model), you need to be able to write it as a linear model. However, the coefficient on $X_{t-1}$ has a restriction on it that will always depend on $\rho$ or $\beta$.  
> **username_0**: @username_1 want to post an answer? I would like to put this question in the answered section. ;)  

---

## 2 answers

### Answer by username_2 on 2016-08-31. Score: 8

To be precise, a model is said to be a **linear regression model** when it is **linear in its parameters**. For extension, a model is said to be non-linear when it is **non-linear in its parameter** (Wooldridge, 2010, p.397).

As such, a linear model can have non-linear variables. A standard example is the [**Mincer equation**](https://en.wikipedia.org/wiki/Mincer_earnings_function), where wage is a linear function of education, experience and experience squared.

In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because **its parameters can still be recovered using linear methods**. In fact, you do not need non-linear methods at all. **Your model is over-identified** as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.

> **Comments:**
> **username_0**: Lucho, sorry for the delay of 3 years... =D Could you tell me how you can recover the parameters using a linear methods?  

---

### Answer by username_3 on 2020-11-21. Score: 5

In response to a comment-request by @Anoldmaninthesea, a linear method to estimate a linear regression with 1st-order auto correlated residuals,

$$y\_t = \alpha + x\_t\beta + u\_t, \qquad u\_t = \rho u\_{t-1} + e\_t,\;\;\; e\_t \sim {\rm WN}$$

is the [**Cochrane-Orcutt one.**](https://en.wikipedia.org/wiki/Cochrane%E2%80%93Orcutt_estimation)

Note that the above can be re-written as

$$y^\*\_t = \delta + \beta x\_t^\* + e\_t,\qquad y^\*\_t=y\_t - \rho y\_{t-1},\;\;\;x^\*\_t=y\_t - \rho x\_{t-1},\;\;\; \delta = \alpha(1-\rho).$$

The method is iterative. Summarily:

**Step 1:** Estimate by OLS $y\_t = \alpha + x\_t\beta$, and obtain the residuals.

**Step 2:** Regress the residuals on their lag, and obtain an estimate of $\rho$.

**Step 3:** Form the series $\hat y^\*\_t = y\_t - \hat \rho y\_{t-1}$ and $\hat x^\*\_t = x\_t - \hat \rho x\_{t-1}$.

**Step 4:** Run the OLS regression $\hat y^\*\_t = \delta + \beta \hat x\_t^\*$ an dobtain the residuals

Continue with Step 2 etc, until the estimate for $\hat \rho$ is stabilized.

Many econometric software include this algorithm as a ready-made command.


---

