Durbin–Watson statistic
In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis. It is named after James Durbin and Geoffrey Watson. The small sample distribution of this ratio was derived 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 uncorrelated against the alternative that they follow a first order autoregressive process. Note that the  distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors. A similar assessment can be also carried out with the Breusch–Godfrey test and the Ljung–Box test. Computing and interpreting the Durbin–Watson statistic. If "e" is the residual given by formula_1 the Durbin-Watson statistic states that null hypothesis: formula_2 , alternative hypothesis formula_3, then the test statistic is where "T" is the number of observations. If one has a lengthy sample, then this can be linearly mapped to the Pearson correlation of the time-series data with its lags . Since "d" is approximately equal to 2(1 − formula_5), where formula_5 is the sample autocorrelation of the residuals, "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 positively correlated. If "d" > 2, successive error terms are 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" and "d"): Positive serial correlation is serial correlation in which a positive error for one observation increases the chances of a positive error for another observation. To test for negative autocorrelation at significance "α", the test statistic (4 − "d") is compared to lower and upper critical values ("d" and "d"): 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" and "d", vary by level of significance ("α") and the degrees of freedom in the regression equation. Their derivation is complex—statisticians typically obtain them from the appendices of statistical texts. If the design matrix formula_7 of the regression is known, exact critical values for the distribution of formula_8 under the null hypothesis of no serial correlation can be calculated. Under the null hypothesis formula_8 is distributed as where "n" are the number of observations and "k" the number of regression variables; the  formula_11 are independent standard normal random variables; and the formula_12 are the nonzero eigenvalues of formula_13 where formula_14 is the matrix that transforms the residuals into the formula_8 statistic, i.e. formula_16 . A number of computational algorithms for finding percentiles of this distribution are available. Although 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 under positive serial correlation 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. If the Durbin–Watson statistic indicates the presence of serial correlation of the residuals, this can be remedied by using the Cochrane–Orcutt procedure. The Durbin–Watson statistic, while displayed by many regression analysis programs, is not applicable in certain situations. For instance, when lagged dependent variables are included in the explanatory variables, then it is inappropriate to use this test. Durbin's h-test (see below) or likelihood ratio tests, that are valid in large samples, should be used. Durbin h-statistic. 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