Durbin-Watson Test

Basic Concepts

A key assumption in regression is that the error terms are independent of each other. On this webpage, we present a simple test to determine whether there is autocorrelation (aka serial correlation), i.e. where there is a (linear) correlation between the error term for one observation and the next. This is especially relevant with time series data where the data are sequenced by time.

The Durbin-Watson test uses the following statistic:

where the e_i = y_i – ŷ_i are the residuals, n = the number of elements in the sample, and k = the number of independent variables.

d takes on values between 0 and 4. A value of d = 2 means there is no autocorrelation. A value substantially below 2 (and especially a value less than 1) means that the data is positively autocorrelated, i.e. on average, a data element is close to the subsequent data element. A value of d substantially above 2 means that the data is negatively autocorrelated, i.e. on average a data element is far from the subsequent data element.

Example

Example 1: Find the Durbin-Watson statistic for the data in Figure 1.

Figure 1 – Durbin-Watson Test

The d statistic (cell J3) is 0.725951, but what does this tell us about the autocorrelation?

Hypothesis Testing

The Durbin-Watson statistic can also be tested for significance using the Durbin-Watson Table. For each value of alpha (.01 or .05) and each value of the sample size n (from 6 to 2000) and each value of the number of independent variables k (from 1 to 20), the table contains a lower and upper critical value (d_L and d_U).

Since most regression problems involving time-series data show a positive autocorrelation, we usually test the null hypothesis H₀: the autocorrelation ρ ≤ 0 (which we believe is ρ = 0) versus the alternative hypothesis H₁: ρ >  0, using the following criteria:

If d < d_L reject H₀ : ρ ≤ 0 (and so accept H₁ : ρ > 0)

If d > d_U do not reject H₀ : ρ ≤ 0 (presumably ρ = 0)

If d_L < d < d_U test is inconclusive

Note that if d > 2 then we should test for negative autocorrelation instead of positive autocorrelation. To do this, simply test 4–d for positive autocorrelation as described above.

For Example 1, with α = .05, we know that n = 11 and k = 2. From the Durbin-Watson Table, we see that d_L = .75798 and d_U = 1.60439. Since d = 0.72595 < .75798 = d_L, we reject the null hypothesis, and conclude that there is a significant positive autocorrelation.

Worksheet Functions

Real Statistics Function: The following two versions of the DURBIN function are available in the Real Statistics Resource Pack.

DURBIN(R1) = the Durbin-Watson statistic d where R1 is a column vector containing residuals

DURBIN(R1, R2) = the Durbin-Watson statistic d where R1 is a m × n range containing X data and R2 is an m × 1 column vector containing Y data.

DLowerCRIT(n, k, α, h) = lower critical value of the Durbin-Watson statistic  for samples of size n (6 to 2,000) based on k independent variables (1 to 20) for α = .01, .025 or .05 (default). If h = TRUE (default) harmonic interpolation is used; otherwise linear interpolation is used.

DUpperCRIT(n, k, α, h) = upper critical value of the Durbin-Watson statistic  for samples of size n (6 to 2,000) based on k independent variables (1 to 20) for α = .01, .025 or .05 (default). If h = TRUE (default) harmonic interpolation is used; otherwise linear interpolation is used.

Actually, the DURBIN function is an array function, described as follows:

DURBIN(R1, R2, lab, α): returns a column range with the values d, d_L, d_U and sig where R1 is a m × n range containing X data and R2 is an m × 1 column vector containing Y data,

DURBIN (R1, k, lab, α): returns a column range with the values d, d_L, d_U and sig where R1 is a column vector containing residuals and k = the # of independent variables (default = 2)

Here α = .01, .025 or .05 (default). If lab = TRUE (default = FALSE) then an extra column of labels is added to the output.

Note that the functions DLowerCRIT and DUpperCRIT support a much larger range of values of n than the Durbin-Watson Table. Also these functions support α = .01, .025 and .05 (actually any value between .01 and .05 by using interpolation), while the table only provides values for α = .01 and .05.

Observation: Referring to Figure 1, we can calculate the statistic d = 0.72595 using either one of the formulas: = DURBIN(G4:G14) or =DURBIN(B4:C14,D4:D14). In fact, if we highlight the range I3:J6 and enter either of these formulas and then press Ctrl-Shft-Enter the result will be the same as shown in range I3:J6 of Figure 1.

Data Analysis Tool

Real Statistics Data Analysis Tool: The Linear Regression data analysis tool provided by the Real Statistics Resource Pack also supports the Durbin-Watson Test as described next.

To conduct the test in Example 1, press Ctrl-m and double click on the Linear Regression data analysis tool. Now fill in the dialog box that appears as shown in Figure 2.

Figure 2 – Durbin-Watson data analysis

The output is similar to that generated by the formula

=DURBIN(B4:C14,D4:D14,TRUE,O24)

Test for Large Samples

For n > 200, you can test the null hypothesis that there is no autocorrelation by noting that based on this null hypothesis the following test statistic has a standard normal distribution

where df = n – k – 1 (n = sample size and k = # of independent variables). Thus, if ABS(z) > NORM.S.INV(α/2), then you reject the null hypothesis and conclude that there is autocorrelation.

For sample sizes 200 ≥ n ≥ 50, you can use the following test statistic instead

where c = 0.0026991244575048159 + 3.11878179323157 / df

References

Wikipedia (2018) Durbin-Watson statistic
https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic

Lee, M-Y. (2016) On the Durbin Watson statistic based on a Z-test in large samples, Int. J. Computational Economics and Econometrics, Vol. 6, No. 1, pp.114-121.
https://ideas.repec.org/a/ids/ijcome/v6y2016i1p114-121.html