definition of Wikipedia
Advertizing ▼
In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation (a relationship between values separated from each other by a given time lag) in the residuals (prediction errors) from a regression analysis. It is named after James Durbin and Geoffrey Watson. However, the small sample distribution of this ratio was derived in a path-breaking article by John von Neumann (von Neumann, 1941). Durbin and Watson (1950, 1951) applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially independent (not autocorrelated) against the alternative that they follow a first order autoregressive process. Later, John Denis Sargan and Alok Bhargava developed several von Neumann–Durbin–Watson type test statistics for the null hypothesis that the errors on a regression model follow a process with a unit root against the alternative hypothesis that the errors follow a stationary first order autoregression (Sargan and Bhargava, 1983).
Contents |
If e_{t} is the residual associated with the observation at time t, then the test statistic is
where T is the number of observations. Since d is approximately equal to 2(1 − r), where r is the sample autocorrelation of the residuals,^{[1]} d = 2 indicates no autocorrelation. The value of d always lies between 0 and 4. If the Durbin–Watson statistic is substantially less than 2, there is evidence of positive serial correlation. As a rough rule of thumb, if Durbin–Watson is less than 1.0, there may be cause for alarm. Small values of d indicate successive error terms are, on average, close in value to one another, or positively correlated. If d > 2 successive error terms are, on average, much different in value to one another, i.e., negatively correlated. In regressions, this can imply an underestimation of the level of statistical significance.
To test for positive autocorrelation at significance α, the test statistic d is compared to lower and upper critical values (d_{L,α} and d_{U,α}):
Positive serial correlation is serial correlation in which a positive error for one observation increases the chances of a positive error for another observation.
Although positive serial correlation does not affect the consistency of the estimated regression coefficients, it does affect our ability to conduct valid statistical tests. First, the F-statistic to test for overall significance of the regression may be inflated because the mean squared error (MSE) will tend to underestimate the population error variance. Second, positive serial correlation typically causes the ordinary least squares (OLS) standard errors for the regression coefficients to underestimate the true standard errors. As a consequence, if positive serial correlation is present in the regression, standard linear regression analysis will typically lead us to compute artificially small standard errors for the regression coefficient. These small standard errors will cause the estimated t-statistic to be inflated, suggesting significance where perhaps there is none. The inflated t-statistic, may in turn, lead us to incorrectly reject null hypotheses, about population values of the parameters of the regression model more often than we would if the standard errors were correctly estimated. This Type I error could lead to improper investment recommendations.
To test for negative autocorrelation at significance α, the test statistic (4 − d) is compared to lower and upper critical values (d_{L,α} and d_{U,α}):
Negative serial correlation implies that a positive error for one observation increases the chance of a negative error for another observation and a negative error for one observation increases the chances of a positive error for another.
The critical values, d_{L,α} and d_{U,α}, vary by level of significance (α), the number of observations, and the number of predictors in the regression equation. Their derivation is complex—statisticians typically obtain them from the appendices of statistical texts.
An important note is that the Durbin–Watson statistic, while displayed by many regression analysis programs, is not relevant in many situations. For instance, if the error distribution is not normal, if there is higher-order autocorrelation, or if the dependent variable is in a lagged form as an independent variable, this is not an appropriate test for autocorrelation. A suggested test that does not have these limitations is the Breusch–Godfrey (serial correlation LM) Test.
The Durbin–Watson statistic is biased for autoregressive moving average models, so that autocorrelation is underestimated. But for large samples one can easily compute the unbiased normally distributed h-statistic:
using the Durbin–Watson statistic d and the estimated variance
of the regression coefficient of the lagged dependent variable, provided
For panel data this statistic was generalized as follows by Alok Bhargava et al. (1982):
This statistic can be compared with tabulated rejection values [see Alok Bhargava et al. (1982), page 537]. These values are calculated dependent on T (length of the balanced panel—time periods the individuals were surveyed), K (number of regressors) and N (number of individuals in the panel). This test statistic can also be used for testing the null hypothesis of a unit root against stationary alternatives in fixed effects models using another set of bounds (Tables V and VI) tabulated by Alok Bhargava et al. (1982).
sensagent's content
Dictionary and translator for handheld
New : sensagent is now available on your handheld
Advertising ▼
Webmaster Solution
Alexandria
A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !
SensagentBox
With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.
Business solution
Improve your site content
Add new content to your site from Sensagent by XML.
Crawl products or adds
Get XML access to reach the best products.
Index images and define metadata
Get XML access to fix the meaning of your metadata.
Please, email us to describe your idea.
Lettris
Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
boggle
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
English dictionary
Main references
Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).
Copyrights
The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.
Translation
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
last searches on the dictionary :
computed in 0.047s