Durbin Watson Table

Decoding the Durbin-Watson Table: A Guide to Autocorrelation Testing

The Durbin-Watson (DW) test is a cornerstone of regression analysis, crucial for assessing the presence of autocorrelation in the residuals of a time series model. Autocorrelation, the correlation between a variable and its lagged values, violates the assumption of independent errors, potentially leading to inefficient and biased estimates. Understanding the Durbin-Watson test, particularly how to interpret its statistic using the DW table, is therefore paramount for the reliability of your regression results. This article aims to address common challenges and questions associated with using the Durbin-Watson table effectively.

Understanding the Durbin-Watson Statistic

The DW statistic ranges from 0 to 4. A value of 2 indicates no autocorrelation. Values significantly below 2 suggest positive autocorrelation (consecutive residuals tend to have the same sign), while values significantly above 2 suggest negative autocorrelation (consecutive residuals tend to have opposite signs). However, the "significance" isn't determined by a simple comparison to 2; this is where the DW table comes into play.

The DW table provides critical values (dL and dU) for a given sample size (n) and number of independent variables (k). These critical values depend on the significance level (α, typically 0.05). The interpretation follows these rules:

dL < DW < dU: The test is inconclusive. Further investigation might be necessary using other autocorrelation tests.
DW ≤ dL: Indicates positive autocorrelation.
DW ≥ 4 - dL: Indicates negative autocorrelation.
4 - dU < DW < 4 - dL: The test is inconclusive.

Locating and Using the Durbin-Watson Table

Durbin-Watson tables are readily available in statistical software packages (like SPSS, R, Stata) and statistical textbooks. They typically present dL and dU values for various combinations of n and k, and for common significance levels (e.g., α = 0.01, 0.05, 0.10). It's crucial to use the correct table for your chosen significance level and to identify the correct values for 'n' (the number of observations) and 'k' (the number of independent variables, excluding the constant).

Example: Let's assume we have a regression model with 30 observations (n=30) and 2 independent variables (k=2). We calculate a DW statistic of 1.2. Looking up the table for α = 0.05, we might find dL = 1.1 and dU = 1.5. Since 1.2 < 1.5 (dU), and 1.2 ≤ 1.1 (dL), we conclude there's evidence of positive autocorrelation.

Dealing with Inconclusive Results

The inconclusive region (dL < DW < dU) is a common challenge. Several strategies can help:

1. Increase Sample Size: A larger sample size generally reduces the inconclusive region and increases the power of the test.
2. Alternative Tests: If the DW test is inconclusive, consider using alternative autocorrelation tests, such as the Breusch-Godfrey test, which is more powerful and can handle more complex autocorrelation structures.
3. Visual Inspection: Plotting the residuals against time can reveal patterns indicative of autocorrelation, even if the DW test is inconclusive.
4. Consider Model Specification: Incorrect model specification can lead to spurious autocorrelation. Re-evaluate the model's variables and functional form.

Handling Different Autocorrelation Structures

The Durbin-Watson test primarily detects first-order autocorrelation (correlation between consecutive residuals). Higher-order autocorrelation might require more sophisticated tests. If you suspect higher-order autocorrelation, the Breusch-Godfrey test is a more suitable alternative.

Interpreting Results and Taking Action

Once the presence and type of autocorrelation are identified, appropriate corrective measures should be taken. These might include:

Autoregressive (AR) models: Incorporating lagged dependent variables or lagged residuals into the model can often mitigate autocorrelation.
Generalized Least Squares (GLS): This method explicitly accounts for the autocorrelation structure, leading to more efficient estimates.
Transformations: Transforming the data (e.g., taking logarithms or differencing) might reduce autocorrelation.

Conclusion

The Durbin-Watson test, in conjunction with its accompanying table, provides a valuable tool for detecting autocorrelation in regression models. However, the inconclusive region and limitations regarding higher-order autocorrelation necessitate careful interpretation and, sometimes, the use of alternative techniques. Understanding the table's usage and the limitations of the DW test are critical for ensuring the reliability and validity of your regression analysis.

FAQs:

1. What if my Durbin-Watson statistic is exactly 2? A DW statistic of 2 suggests no autocorrelation. However, this is a point estimate, and statistical noise could lead to a value of 2 even if slight autocorrelation exists.

2. Can I use the Durbin-Watson test with non-linear regression models? The DW test is primarily designed for linear regression models. For non-linear models, other autocorrelation tests are generally more appropriate.

3. How do I choose the correct significance level (α)? The choice of α depends on the context and the risk tolerance for making a Type I error (falsely concluding autocorrelation exists). A common choice is 0.05, but 0.01 is used when a stricter significance level is preferred.

4. What are the assumptions of the Durbin-Watson test? The DW test assumes that the errors are normally distributed and homoscedastic (constant variance). Severe deviations from these assumptions can affect the test's validity.

5. My software doesn't provide a DW table; what should I do? Most statistical software packages (R, Stata, SPSS, Python's Statsmodels) calculate the DW statistic and often provide p-values directly, eliminating the need to consult a table. If a p-value is available, it provides a more direct measure of significance. If your software only provides the DW statistic, you can find DW tables readily online.

Search Results:

Critical Values for the Durbin-Watson Test: 5% Significance Level Critical Values for the Durbin-Watson Test: 5% Significance Level K includes intercept T K dL dU

Durbin-Watson test - National Sun Yat-sen University Critical Values of the Durbin-Watson Statistic. The fundamental assumptions in linear regression are that the error terms εi have mean zero and constant variance and uncorrelated [E(εi) = 0, Var(εi) = σ2, and E(εiεj) = 0].

Table A-1 Models with an intercept (from Savin and White) Table A-1 Models with an intercept (from Savin and White) Durbin-Watson Statistic: 1 Per Cent Significance Points of dL and dU k’ *=1 *k’ is the number of regressors excluding the intercept k’=2 k’=3 k’=4 k’=5 k’=6 k’=7 k’=8 k’=9 k’=10 n dL dU dL dU …

Tabel Durbin-Watson (DW), α = 5% - WordPress.com Tabel Durbin-Watson (DW), α = 5% sumber: http://www.standford.edu Catatan-Catatan Reproduksi dan Cara Membaca Tabel: 1. Tabel DW ini direproduksi dengan merubah format …

TABLE de DURBIN-WATSON : Test unilatéral de ρ = 0 contre ρ TABLE de DURBIN-WATSON : Test unilatéral de ρ = 0 contre ρ > 0, au seuil de 5% (test bilatéral : seuil = 10%) n d L du d L du d L d L d L d L d L d L d L d L d.

Lampiran Tabel DW - UNJ Lampiran Tabel DW.

Critical Values for the Durbin-Watson Test: 5% Significance Level Critical Values for the Durbin-Watson Test: 5% Significance Level =100 to 200, K=2 to 21 K includes intercept N K dL dU 100. 2. 1.65404 1.69439 100. 3. 1.63369 1.71517 100. 4. 1.61306 …

Durbin-Watson Significance Tables - هيئة التدريس جامعة ... In this appendix, we have reproduced two sets of tables. Savin and White (1977) present tables for sample sizes ranging from 6 to 200 and for 1 to 20 regressors for models in which an intercept is included.

Durbin-Watson Significance Tables - University of Notre Dame Because of the dependence of any computed Durbin-Watson value on the associated data matrix, exact critical values of the Durbin-Watson statistic are not tabulated for all possible cases.

k = 1 k = 2 k = 3 k = 4 k = 5 - qualitytechno.com Critical values for the Durbin-Watson d Statistic n d L d U d L d U d L d U d L d U d L d U 15 1.08 1.36 0.95 1.54 0.82 1.75 0.69 1.97 0.56 2.21 16 1.10 1.37 0.98 1.54 0.86 1.73 0.74 1.93 0.62 2.15 17 1.13 1.38 1.02 1.54 0.90 1.71 0.78 1.90 0.67 2.10 …

Lampiran 7 Tabel Durbin-Watson (DW), α = 5% - UMG Tabel Durbin-Watson (DW), α = 5% n k=1 k=2 k=3 k=4 k=5 dL dU 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 29 30 31 32 33 34 35 36 37 38 1 39 40 1 41 1 42 43 44 45 46 47 48 49 50 51 1 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 0.6102 0.6996 1 0.7 629 0.8243 0.8791 0.9273 0.9708 1.0097 1.0450 1.0770

Table A-2 Models with an intercept (from Savin and White) Table A-2 Models with an intercept (from Savin and White) Durbin-Watson Statistic: 5 Per Cent Significance Points of dL and dU k’*=1 *k’ is the number of regressors excluding the intercept k’=2 k’=3 k’=4 k’=5 k’=6 k’=7 k’=8 k’=9 k’=10 ndL dUdL dUdL …

Tabel Durbin-Watson (DW), α = 5% - Junaidi Tabel DW ini direproduksi dengan merubah format tabel mengikuti format tabel DW yang umumnya dilampirkan pada buku-buku teks statistik/ekonometrik di Indonesia, agar lebih mudah dibaca dan diperbandingkan 2. Simbol ‘k’ pada tabel menunjukkan banyaknya variabel bebas (penjelas), tidak termasuk variabel terikat. 3.

Lampiran 1 Tabel Durbin Watson - UMG Sumber : J. Durbin and G.S Watson. "Testing for serial correlation in least squares regression, (II), "Biometrika" dalam J. Supranto (1995)

Department of Economics - The University of Warwick Table Contents Page 1 Normal Distribution 1 2 Student-t Distribution 2 3 Chi-Squared Distribution 3 4 F-Distribution ( 001. ) 4-6 5 F-Distribution ( 005. ) 7-9 6 F-Distribution ( 010. ) 10-12 7 Binomial Distribution 13-27 8 Poisson Distribution 28-30 9 Durbin-Watson Critical Values 31

TABLE 2. Critical Values of the Durbin - Watson Statistic TABLE 2. Critical Values of the Durbin - Watson Statistic Probability in Lower Tail k = Number of Regressors (Excluding the Intercept) Sample (Significance 1 2 3 4 5 Size Level= α) dL dU dL dU dL dU dL dU dL dU.01 .81 1.07 .70 1.25 .59 1.46 .49 1.70 .39 1.96 15 .025 .95 1.23 .83 1.40 .71 1.61 .59 1.84 .48 2.09

The Durbin-Watson Test for Serial Correlation with Extreme This paper presents extended tables for the Durbin and Watson [3 and 4] bounds test. The tables can be used for samples with 6 to 200 observations and for as many as 20 regressors.

Durbin Watson test statistic : 5% significance points of and - John … Durbin–Watson test statistic d: 1% significance points of dL and dU. k′=1 k′=2 k′=3 k′=4 k′=5 n dL dU dL dU dL dU dL dU dL dU 15 0.81 1.07 0.70 1.25 0.59 1.46 0.49 1.70 0.39 1.96 16 0.84 1.09 0.74 1.25 0.63 1.44 0.53 1.66 0.44 1.90 17 0.87 1.10 0.77 1.25 0.67 1.43 0.57 1.63 0.48 1.85 18 0.90 1.12 0.80 1.26 0.71 1.42 0.61 1.60 0.52 1.80

DURBIN-WATSON STATISTIC 1% SIGNIFICANCE POINTS OF … durbin-watson statistic 2.5% significance points of ql and qu Λ = 2 Λ = 3 Λ = 4 Λ = 5 Λ = 6 n ql qu ql qu ql qu ql qu ql qu 15 0.949 1.222 0.827 1.405 0.706 1.615 0.588 1.848 0.478 2.099 16 0.980 1.235 0.864 1.403 0.748 1.594 0.636 1.806 0.527 2.035 17 1.009 1.248 0.899 1.403 0.788 1.578 0.680 1.773 0.574 1.983

SPS Home Stats Tables Durbin Watson 0.05 Table Durbin-Watson Critical Values - 95% (d) Page 1 of 4 SPS Home > Stats Tables > Durbin Watson 0.05 Table . Critical Values for the Durbin-Watson Statistic (d) Level of Significance α = .05 . n k = l . k = 2 . k = 3 . k = 4 . k = 5 d. L . d. U . d. L . d. U . d. L . d. U . d. L . d. U . d. L . d. U . 6 . 0.61 1.40 1.90 2.29 2.13 0.30 2.59 . 7 . 0 ...