<p><strong>Nerlove and Wallis (1966) result</strong></p>
<p><a href="https://www.jstor.org/stable/1909870" rel="nofollow noreferrer">Nerlove and Wallis (1966)</a> have discussed this issue. Their Equation (3) derives the probability limit of the Durbin-Watson statistic as:
<span class="math-container">$$\mathrm{plim}\, d^* = 2 \left[ 1 - \frac{\rho \beta (\beta + \rho)}{1+\beta \rho} \right].$$</span>
(Their notation for <span class="math-container">$\beta$</span> is <span class="math-container">$\alpha$</span> in the paper.) Nerlove and Wallis's (1966) derivation is based on <a href="https://www.jstor.org/stable/1401655" rel="nofollow noreferrer">Malinvaud (1961)</a>, whose beautiful work I can't read, unfortunately.</p>
<p><strong>Derivation</strong></p>
<p>Nerlove and Wallis's (1966) result can be derived. Let me try it here. We need <span class="math-container">$y_t$</span> to be covariance stationary, which we just assume.</p>
<p>Let <span class="math-container">$\gamma_j = E(y_t y_{t-j})$</span>. Let <span class="math-container">$b$</span> be the probability limit of <span class="math-container">$\hat\beta_{ols}$</span>, which is <span class="math-container">$b=(\beta+\rho)/(1+\beta\rho)$</span> according to OP's derivation. Let <span class="math-container">$e_t = y_t - b y_{t-1}$</span>. Then
<span class="math-container">\begin{align}
E(e_t^2) &amp;= (1+b^2) \gamma_0 - 2b \gamma_1\\
E(e_t e_{t-1}) &amp;= \gamma_1 -b\gamma_2 - b\gamma_0 + b^2 \gamma_1 = (1+b^2) \gamma_1 - b(\gamma_0+\gamma_2),
\end{align}</span>
and the probability limit of DW (<span class="math-container">$d^*$</span>) is <span class="math-container">$2[1-E(e_te_{t-1})/E(e_t^2)]$</span>.</p>
<p>We need <span class="math-container">$\gamma_0, \gamma_1$</span> and <span class="math-container">$\gamma_2$</span>. For this, note that <span class="math-container">$\epsilon_t - \rho \epsilon_{t-1} = (y_t - \beta y_{t-1}) - \rho(y_{t-1} - \beta y_{t-2}) = v_t$</span>, that is, <span class="math-container">$y_t = (\beta + \rho) y_{t-1} - \beta \rho y_{t-2} + v_t$</span>. Due to covariance stationarity, we have (because <span class="math-container">$E y_{t-1} v_t = 0$</span> and <span class="math-container">$E y_{t-2} v_t = 0$</span>)
<span class="math-container">\begin{align*}
\gamma_1 &amp;= (\beta + \rho) \gamma_0 - \beta\rho \gamma_1,\\
\gamma_2 &amp;= (\beta + \rho) \gamma_1 - \beta\rho \gamma_0,
\end{align*}</span>
the <a href="https://en.wikipedia.org/wiki/Autoregressive_model#Yule%E2%80%93Walker_equations" rel="nofollow noreferrer">Yule-Walker equations</a>.
The first identity implies <span class="math-container">$\gamma_1/\gamma_0 = (\beta+\rho)/(1+\beta\rho) = b$</span>, which is natural because <span class="math-container">$\hat\beta_{ols}$</span> converges in probability to <span class="math-container">$\gamma_1/\gamma_0$</span>. The second identity implies <span class="math-container">$\gamma_2/\gamma_0 = (\beta+\rho) (\gamma_1/\gamma_0) - \beta\rho = (\beta+\rho) b - \beta\rho$</span>. Thus,
<span class="math-container">\begin{align}
\frac{E(e_te_{t-1})}{E(e_t^2)} &amp;= \frac{(1+b^2) \gamma_1 - b(\gamma_0+\gamma_2)}{(1+b^2)\gamma_0 - 2b\gamma_1}
= \frac{(1+b^2)b - b[1+(\beta+\rho)b -\beta\rho]}{1+b^2 - 2b^2}\\
&amp;= \frac{b^3-(\beta+\rho)b^2+\beta\rho b}{1-b^2} = b \left[ \frac{b^2 - (\beta + \rho) b + \beta\rho}{1-b^2} \right].
\end{align}</span>
Plug in <span class="math-container">$b = (\beta+\rho)/(1+\beta\rho)$</span> to show that the above equals
<span class="math-container">$$
b \left[ \frac{(\beta+\rho)^2 - (\beta+\rho)^2 (1+\beta\rho) + \beta\rho (1+\beta\rho)^2}{(1+\beta\rho)^2 - (\beta+\rho)^2} \right] = b\beta\rho = \frac{\beta\rho(\beta+\rho)}{1+\beta\rho},
$$</span>
which straightforwardly implies Nerlove and Wallis's (1966) result because <span class="math-container">$d^* \to_p 2[1- E(e_te_{t-1})/E(e_t^2)]$</span>.</p>
<p><strong>The Durbin h statistic</strong></p>
<p><a href="https://www.jstor.org/stable/1909547" rel="nofollow noreferrer">Durbin (1970)</a> acknowledges this problem and proposes the so-called <em>Durbin-h statistic</em>, whose formula is given in his Equation (12) on p. 419 as:
<span class="math-container">$$h = a\sqrt{\frac{n}{1-n\hat{V}(b_1)}}, \quad a = 1-\tfrac12 d,$$</span>
asymptotically standard normal under the null. (See also <a href="https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic" rel="nofollow noreferrer">Wikipedia</a>.) You can guess the reason it is called the Durbin &quot;h&quot; statistic.</p>
<p><strong>Meaning of DW biased</strong></p>
<p>Nerlove and Wallis (1966) further explain why their derivation of the probability limit of DW <span class="math-container">$d^*$</span> leads to invalid testing by DW. Yet, under the null hypothesis that <span class="math-container">$\rho=0$</span>, DW <span class="math-container">$d^*$</span> converges in probability to 2 (which is &quot;correct&quot;), so we cannot simply jump to the conclusion that the DW test consistently rejects the correct null. The problem, I would say, is in the standard error of <span class="math-container">$d^*$</span> (or of the estimated serial correlation in <span class="math-container">$y_t - \hat\beta_{ols} y_{t-1}$</span>). And Durbin (1970) gives a solution.</p>
<p><strong>References</strong></p>
<p>Durbin, J. (1970). <a href="https://www.jstor.org/stable/1909547" rel="nofollow noreferrer">Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors are Lagged Dependent Variables</a>, <em>Econometrica</em> 38(3), 410-421.</p>
<p>Malinvaud, E. (1961). <a href="https://www.jstor.org/stable/1401655" rel="nofollow noreferrer">Estimation et Prevision dans les Modeles Economiques Autoregressifs</a>, <em>Review of the International Institute of Statistics</em>, 29(2), 1-32.</p>
<p>Nerlove, M., and K. F. Wallis (1966). <a href="https://www.jstor.org/stable/1909870" rel="nofollow noreferrer">Use of the Durbin-Watson Statistic in Inappropriate Situations</a>, <em>Econometrica</em> 34(1), 235-238.</p>

CC BY-SA 4.0
2021-07-25T12:46:44.157Z
Incredible answer, just one minor typo. After "For this, note that", you wrote $e_t -e_{t-1}$ where I think you meant $e_t-\rho e_{t-1}$
Michael Gmeiner
CC BY-SA 4.0
2021-07-25T13:09:42.300Z
Also added discussions on the "bias" of DW.
chan1142
CC BY-SA 4.0
2021-07-24T16:27:00.990Z
2021-08-10T01:13:45.607Z
2021-08-10T01:13:45.607Z
chan1142
<p><span class="math-container">$(1)  y_t =\beta y_{t-1} +\epsilon_t$</span></p>
<p><span class="math-container">$(2)  \epsilon_t =\rho \epsilon_{t-1} +v_t$</span></p>
<p>Where <span class="math-container">$v_t$</span> is i.i.d white noise. I know that OLS estimates of (1) are biased. It would then follow that estimates of <span class="math-container">$\epsilon_t$</span> are biased, which should imply a Durbin-Watson Test using those residual estimates is biased. I have been attempting to prove this, but have struggled, and likely I am making algebra mistakes. Is there a straightforward proof that the Durbin-Watson test is biased?</p>
<p>I have that <span class="math-container">$plim \hat{\beta}_{OLS}=\beta +\frac{\rho (1-\beta^2)}{1+\beta\rho}$</span> which implies <span class="math-container">$\hat{\epsilon_t}=\epsilon_t-y_{t-1}\frac{\rho (1-\beta^2)}{1+\beta\rho}$</span>
and <span class="math-container">$Cov[y_{t-1},\epsilon_t]=\frac{\rho \sigma^2_\epsilon}{(1-\beta \rho)}$</span>.</p>

CC BY-SA 4.0
2021-07-23T14:32:19.330Z
See https://www.jstor.org/stable/1909547 (Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors are Lagged Dependent Variables, Econometrica, Vol 38, No 3, 1970, 410-421).
chan1142
CC BY-SA 4.0
2021-07-23T14:39:24.640Z
The above paper is written by Durbin himself. Also this one: https://www.jstor.org/stable/1909870 (Nerlove and Wallis, Use of the Durbin-Watson Statistic in Inappropriate Situations, Econometrica, Vol 34, No 1, 1966, 235-238). Their equations (1) and (2) are identical to your (1) and (2).
chan1142
CC BY-SA 4.0
2021-07-24T13:38:28.763Z
@chan1142 please consider expanding it into an answer so it can be accepted
1muflon1
<econometrics><time-series><autoregressive>
Durbin Watson Test for an AR(1) process
Michael Gmeiner
# Durbin Watson Test for an AR(1) process

(Asked by: username_0 on 2021-07-22)

$(1) y\_t =\beta y\_{t-1} +\epsilon\_t$

$(2) \epsilon\_t =\rho \epsilon\_{t-1} +v\_t$

Where $v\_t$ is i.i.d white noise. I know that OLS estimates of (1) are biased. It would then follow that estimates of $\epsilon\_t$ are biased, which should imply a Durbin-Watson Test using those residual estimates is biased. I have been attempting to prove this, but have struggled, and likely I am making algebra mistakes. Is there a straightforward proof that the Durbin-Watson test is biased?

I have that $plim \hat{\beta}\_{OLS}=\beta +\frac{\rho (1-\beta^2)}{1+\beta\rho}$ which implies $\hat{\epsilon\_t}=\epsilon\_t-y\_{t-1}\frac{\rho (1-\beta^2)}{1+\beta\rho}$
and $Cov[y\_{t-1},\epsilon\_t]=\frac{\rho \sigma^2\_\epsilon}{(1-\beta \rho)}$.

**Tags:** econometrics, time-series, autoregressive

> **Comments:**
> **username_1**: See https://www.jstor.org/stable/1909547 (Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors are Lagged Dependent Variables, Econometrica, Vol 38, No 3, 1970, 410-421).  
> **username_1**: The above paper is written by Durbin himself. Also this one: https://www.jstor.org/stable/1909870 (Nerlove and Wallis, Use of the Durbin-Watson Statistic in Inappropriate Situations, Econometrica, Vol 34, No 1, 1966, 235-238). Their equations (1) and (2) are identical to your (1) and (2).  
> **username_2**: @username_1 please consider expanding it into an answer so it can be accepted  

---

## 1 answers

### Answer by username_1 on 2021-07-24. Score: 5 | Accepted answer

**Nerlove and Wallis (1966) result**

[Nerlove and Wallis (1966)](https://www.jstor.org/stable/1909870) have discussed this issue. Their Equation (3) derives the probability limit of the Durbin-Watson statistic as:
$$\mathrm{plim}\, d^\* = 2 \left[ 1 - \frac{\rho \beta (\beta + \rho)}{1+\beta \rho} \right].$$
(Their notation for $\beta$ is $\alpha$ in the paper.) Nerlove and Wallis's (1966) derivation is based on [Malinvaud (1961)](https://www.jstor.org/stable/1401655), whose beautiful work I can't read, unfortunately.

**Derivation**

Nerlove and Wallis's (1966) result can be derived. Let me try it here. We need $y\_t$ to be covariance stationary, which we just assume.

Let $\gamma\_j = E(y\_t y\_{t-j})$. Let $b$ be the probability limit of $\hat\beta\_{ols}$, which is $b=(\beta+\rho)/(1+\beta\rho)$ according to OP's derivation. Let $e\_t = y\_t - b y\_{t-1}$. Then
\begin{align}
E(e\_t^2) &= (1+b^2) \gamma\_0 - 2b \gamma\_1\\
E(e\_t e\_{t-1}) &= \gamma\_1 -b\gamma\_2 - b\gamma\_0 + b^2 \gamma\_1 = (1+b^2) \gamma\_1 - b(\gamma\_0+\gamma\_2),
\end{align}
and the probability limit of DW ($d^\*$) is $2[1-E(e\_te\_{t-1})/E(e\_t^2)]$.

We need $\gamma\_0, \gamma\_1$ and $\gamma\_2$. For this, note that $\epsilon\_t - \rho \epsilon\_{t-1} = (y\_t - \beta y\_{t-1}) - \rho(y\_{t-1} - \beta y\_{t-2}) = v\_t$, that is, $y\_t = (\beta + \rho) y\_{t-1} - \beta \rho y\_{t-2} + v\_t$. Due to covariance stationarity, we have (because $E y\_{t-1} v\_t = 0$ and $E y\_{t-2} v\_t = 0$)
\begin{align\*}
\gamma\_1 &= (\beta + \rho) \gamma\_0 - \beta\rho \gamma\_1,\\
\gamma\_2 &= (\beta + \rho) \gamma\_1 - \beta\rho \gamma\_0,
\end{align\*}
the [Yule-Walker equations](https://en.wikipedia.org/wiki/Autoregressive_model#Yule%E2%80%93Walker_equations).
The first identity implies $\gamma\_1/\gamma\_0 = (\beta+\rho)/(1+\beta\rho) = b$, which is natural because $\hat\beta\_{ols}$ converges in probability to $\gamma\_1/\gamma\_0$. The second identity implies $\gamma\_2/\gamma\_0 = (\beta+\rho) (\gamma\_1/\gamma\_0) - \beta\rho = (\beta+\rho) b - \beta\rho$. Thus,
\begin{align}
\frac{E(e\_te\_{t-1})}{E(e\_t^2)} &= \frac{(1+b^2) \gamma\_1 - b(\gamma\_0+\gamma\_2)}{(1+b^2)\gamma\_0 - 2b\gamma\_1}
= \frac{(1+b^2)b - b[1+(\beta+\rho)b -\beta\rho]}{1+b^2 - 2b^2}\\
&= \frac{b^3-(\beta+\rho)b^2+\beta\rho b}{1-b^2} = b \left[ \frac{b^2 - (\beta + \rho) b + \beta\rho}{1-b^2} \right].
\end{align}
Plug in $b = (\beta+\rho)/(1+\beta\rho)$ to show that the above equals
$$
b \left[ \frac{(\beta+\rho)^2 - (\beta+\rho)^2 (1+\beta\rho) + \beta\rho (1+\beta\rho)^2}{(1+\beta\rho)^2 - (\beta+\rho)^2} \right] = b\beta\rho = \frac{\beta\rho(\beta+\rho)}{1+\beta\rho},
$$
which straightforwardly implies Nerlove and Wallis's (1966) result because $d^\* \to\_p 2[1- E(e\_te\_{t-1})/E(e\_t^2)]$.

**The Durbin h statistic**

[Durbin (1970)](https://www.jstor.org/stable/1909547) acknowledges this problem and proposes the so-called *Durbin-h statistic*, whose formula is given in his Equation (12) on p. 419 as:
$$h = a\sqrt{\frac{n}{1-n\hat{V}(b\_1)}}, \quad a = 1-\tfrac12 d,$$
asymptotically standard normal under the null. (See also [Wikipedia](https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic).) You can guess the reason it is called the Durbin "h" statistic.

**Meaning of DW biased**

Nerlove and Wallis (1966) further explain why their derivation of the probability limit of DW $d^\*$ leads to invalid testing by DW. Yet, under the null hypothesis that $\rho=0$, DW $d^\*$ converges in probability to 2 (which is "correct"), so we cannot simply jump to the conclusion that the DW test consistently rejects the correct null. The problem, I would say, is in the standard error of $d^\*$ (or of the estimated serial correlation in $y\_t - \hat\beta\_{ols} y\_{t-1}$). And Durbin (1970) gives a solution.

**References**

Durbin, J. (1970). [Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors are Lagged Dependent Variables](https://www.jstor.org/stable/1909547), *Econometrica* 38(3), 410-421.

Malinvaud, E. (1961). [Estimation et Prevision dans les Modeles Economiques Autoregressifs](https://www.jstor.org/stable/1401655), *Review of the International Institute of Statistics*, 29(2), 1-32.

Nerlove, M., and K. F. Wallis (1966). [Use of the Durbin-Watson Statistic in Inappropriate Situations](https://www.jstor.org/stable/1909870), *Econometrica* 34(1), 235-238.

> **Comments:**
> **username_0**: Incredible answer, just one minor typo. After "For this, note that", you wrote $e_t -e_{t-1}$ where I think you meant $e_t-\rho e_{t-1}$  
> **username_1**: Also added discussions on the "bias" of DW.  

---

