Durbin Watson Test for an AR(1) process

$$(1) y_t =\beta y_{t-1} +\epsilon_t$$

$$(2) \epsilon_t =\rho \epsilon_{t-1} +v_t$$

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 $$Cov[y_{t-1},\epsilon_t]=\frac{\rho \sigma^2_\epsilon}{(1-\beta \rho)}$$.

• See 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). Jul 23, 2021 at 14:32
• The above paper is written by Durbin himself. Also this one: 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). Jul 23, 2021 at 14:39
• @chan1142 please consider expanding it into an answer so it can be accepted
– 1muflon1
Jul 24, 2021 at 13:38

Nerlove and Wallis (1966) result

Nerlove and Wallis (1966) 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), 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. 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) 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.) 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, Econometrica 38(3), 410-421.

Malinvaud, E. (1961). Estimation et Prevision dans les Modeles Economiques Autoregressifs, 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, Econometrica 34(1), 235-238.

• Incredible answer, just one minor typo. After "For this, note that", you wrote $e_t -e_{t-1}$ where I think you meant $e_t-\rho e_{t-1}$ Jul 25, 2021 at 12:46
• Thanks! Fixed it. Jul 25, 2021 at 12:50
• Also added discussions on the "bias" of DW. Jul 25, 2021 at 13:09