# Durbin Watson Contrast

The Durbin-Watson (DW) test is used to perform an AR autocorrelation test on a data set. This contrast focuses on the study of Ordinary Least Squares (OLS) residuals.

DW is a statistical test that contrasts the presence of autocorrelation in the residuals of a regression. The main characteristic of a data series with autocorrelated residuals is the defined trend of the data.

Autocorrelation occurs when the independent variables have a temporal structure that is repeated on certain occasions over time. Then, today's residuals (t = 2) will depend on the past residuals (t = 1) and the assumption of independence of the classical linear model will not be fulfilled.

## Durbin Watson in financial series

We can find this autocorrelation problem in data series with a clearly defined trend. For example, the Japanese NIKKEI 225 index with the number of *ski passes* issued in the ski resort of Aspen, USA. Both series have the same growing trend although they do not share, at first, any relationship. The most common case of autocorrelation occurs in financial series, where the trend of the data is very well defined.

A practical solution to reduce autocorrelation and heteroscedasticity in financial series would be to apply the natural logarithm (*ln*). Through the first difference, lnPt - lnPt-1, we isolate the series from its trend. In this case, it represents the prices in time *t*.

The result is the conditional DW distribution in Xi that meets the assumptions of the classical linear model, with special importance the assumption of normality in the residuals.

This contrast is known by the upper and lower limits for critical values that depend on the significance level of the confidence interval. These general levels are:

- dU: Upper limit.
- dL: Lower limit.

Although we do not have an exact distribution, dU and dL are defined in the DW tables. The limits are a function of the number of variables (*n*) and the number of explanatory variables (*k*).

## Procedure

1. We arrange the residuals in temporal order such that

2. We define H0 and H1.

3. Contrast statistic *t*.

4. Rejection rule.

In large samples, DW is approximately equal to 2 (1-r) where *r* is the first-order estimate on the residuals.

The approximate range for DW is [0.4]

- If 0 ≤ DW <dL → We reject H0
- If dL <DW <dU → Inconclusive test
- If dU <DW <If 4 - dU → There is no first order autocorrelation
- If 4 - dU <DW <If 4 - dL → Inconclusive test
- If 4 - dL <DW ≤ 4 → We do not have enough significant evidence to reject H0

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