if we accept or reject the Null hypothesis.

The null hypothesis of the Dickey-Fuller test is a random walk, possibly with drift. The differenced process is not serially correlated under the null of I(1). There is a great need for the generalization of this specification. The augmented Dickey-Fuller (ADF) test, originally proposed in , adjusts for the serial correlation in the time series by adding lagged first differences to the autoregressive model,

Lecture 7 - Using Minitab to Calculate Hypothesis .

When the null hypothesis  holds, the quadratic form  has the characteristic function

If t* critical value, == reject null hypothesis, i.

In the Durbin-Watson test, the marginal probability indicates positive autocorrelation () if it is less than the level of significance (), while you can conclude that a negative autocorrelation () exists if the marginal probability based on the computed Durbin-Watson statistic is greater than . presented tables for bounds tests of fourth-order autocorrelation, and has given tables for a 5% significance level for orders two to four. Using the AUTOREG procedure, you can calculate the exact p-values for the general order of Durbin-Watson test statistics. Tests for the absence of autocorrelation of order p can be performed sequentially; at the th step, test given against . However, the size of the sequential test is not known.

we will accept null hypothesis of no autocorrelation, ..

In the Durbin-Watson test, the marginal probability indicates positive autocorrelation () if it is less than the level of significance (), while you can conclude that a negative autocorrelation () exists if the marginal probability based on the computed Durbin-Watson statistic is greater than . presented tables for bounds tests of fourth-order autocorrelation, and has given tables for a 5% significance level for orders two to four. Using the AUTOREG procedure, you can calculate the exact p-values for the general order of Durbin-Watson test statistics. Tests for the absence of autocorrelation of order p can be performed sequentially; at the th step, test given against . However, the size of the sequential test is not known.


we can perform a hypothesis test in which the null hypothesis ..

While the prospect of having an inconclusive test result is less than desirable, there are some programs which use exact and approximate procedures for calculating a p-value.

and the null hypothesis is H0: m £ 100

Notice that under the null hypothesis, can be estimated by ordinary least squares regression of on an intercept and the time trend. Following the original work of Kwiatkowski, Phillips, Schmidt, and Shin, under the null (), statistic converges asymptotically to three different distributions depending on whether the model is trend-stationary, level-stationary (), or zero-mean stationary (, ). The trend-stationary model is denoted by subscript and the level-stationary model is denoted by subscript . The case when there is no trend and zero intercept is denoted as 0. The last case, although rarely used in practice, is considered in .

Durbin–Watson statistic - Wikipedia

where iid , and an intercept (in the original paper, the authors use instead of , here we assume .) The null hypothesis of trend stationary is specified by , while the null of level stationary is the same as above with the model restriction . Under the alternative that , there is a random walk component in the observed series .

and the null hypothesis is p=0 which means ..

where is an data matrix, is a coefficient vector, and is a disturbance vector. The error term is assumed to be generated by the th-order autoregressive process where , is a sequence of independent normal error terms with mean 0 and variance . Usually, the Durbin-Watson statistic is used to test the null hypothesis against . generalized the Durbin-Watson statistic:

test statistics for the null hypothesis that the errors on a ..

where is the rank of th observation in the sequence of observations. For large sample, the statistic follows the standard normal distribution under the null hypothesis of independence. For small samples of size between 11 and 100, the critical values through simulation would be more precise; for samples of size no more than 10, the exact CDF is applied.

The Null hypothesis of the Durbin-Watson test ..

The statistics of both the runs test and the turning point test have the standard normal distribution under the null hypothesis of independence.