Durbin-Watson Test

Durbin-Watson Test is conducted to gauge the autocorrelation from the residual in the regression analysis. The residual here refers to the errors present in the analysis. Analysts and stock traders use the test to predict stock price movement based on historical data,. Autocorrelation is also known as serial correlation.

A positive autocorrelation today means yesterday's price positively correlates with today's price. In simple terms, by calculation, if a particular stock declined in price yesterday, it may also drop today. In contrast, a negative autocorrelation means if the price fell yesterday, it will increase today.

• The Durbin-Watson test for autocorrelation was introduced in 1950 and is employed to detect autocorrelation from a regression analysis residual.
• The test value always varies between 0 to 4, and a value equal to 2 signifies no autocorrelation in the residual.
• James Durbin and Geoffrey Watson invented it at the London School of Economics.
• One of the key limitations of the test is that it can only be applied to a single time lag, which means if it is not for first-order serial correlation, the analyst will have to shift to another method.

The Durbin-Watson test is employed to find the autocorrelation from the errors in the regression analysis. James Durbin, a British mathematician, and Geoffrey Watson, an Australian statistician, introduced this method in 1950, and both came together to invent the test at the London School of Economics. When the Durbin-Watson test in r programming is used, it assumes no correlation between the independent residuals. The test is applied with a null hypothesis indicating the assumption is correct and an alternative hypothesis indicating that the errors are autocorrelated.

The test consists of three important components. Autocorrelation refers to the degree of correlated variables between two or more data sets or sample sizes. It is generally sighted in time series, in which observations are taken at different points in time. The second component is the residual, which means the errors depicting the gap between the observed and mean values. It defines the variation. Thirdly, a regression analysis is performed to identify the impacting group.

The Durbin-Watson test for autocorrelation is important because, if not detected, it can cause problems in the least squares regression. It mostly happens when an incorrect model is taken for analysis. Detecting autocorrelation is important for a successful regression analysis without any errors, and this test is one of the simplest ways to check for any autocorrelation.

The formula for the Durbin-Watson test is as follows:

Here,

e_t = residual or error value

T = number of observations

Check out these examples to get a better idea:

Example #1

Suppose Jeffrey, a long-term stock trader and investor, follows a stock in the stock market for over two months. He collects the stock's historical data and performs a regression analysis; Jeffrey's objective is to predict the stock's future price movement. He implies the Durbin-Watson test to find the autocorrelation from the residual.

Jeffrey reaches the value of 4 as the outcome of the test, which signifies a negative correlation. In simple terms, the securities price has declined in the past, and there is a high chance of it increasing. With calculated risk, Jeffrey invests in the stock and, with precision, generates high returns and short-term profit from the stock.

It is a simple example, but statistical calculation and Durbin-Watson test interpretation are complex in the real world. They require practice, understanding of values, and knowledge of deducing sense from table values. If the test value had come to 0, it would mean that the stock price was declining in the past and will continue to move in the same direction.

Example #2

Another good example of the Durbin-Watson test in r comes from studying the autocorrelation of CO2 and temperature time series. The temperature change was assumed to be linear with the CO2 concentration. The least squares method assumes no residual in the CO2 concentration measurement.

The Durbin-Watson tests that the residual and null hypotheses are not autocorrelated; the test value is less than 2, indicating a positive correlation. The calculation was performed on a linear regression with two variables. As per the data, the autocorrelation can impact the regression statistics when the CO2 and temperature are regressed against each other.

When further computed for 2nd order polynomial regression, the test value for one year lag came to 0.9 and indicated the autocorrelation. Yet, the remaining correlation was positive, signifying that CO2 has a small influence on temperature. For each year, 76% of the global temperature and 90% of CO2 measurement is based on the previous year’s value.

The two most popular tests conducted to figure out the autocorrelation in statistics are - the Durbin-Watson and Breusch-Godfrey tests. Though the objective of both these tests is the same, they still differ from each other in various aspects. Let us check the differences between them below:

• The Durbin-Watson test seeks autocorrelation for lag 1. In contrast, the Breusch-Godfrey test looks at all autocorrelation.
• If an analyst can recognize and point out the autocorrelation beyond lag 1, then the Durbin-Watson test is sufficient; otherwise, the analyst must employ the Breusch-Godfrey test.
• It was introduced in 1950; therefore, it is an old method, but Breusch Godfrey came into existence in 1978.

This has been a guide to what is Durbin-Watson Test. Here, we explain the concept with its formula, examples, and comparison with the Breusch-Godfrey test. You may also have a look at the following articles –

• Serial Correlation
• Correlation Coefficient
• Spearman Rank Correlation Coefficient