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# Durbin watson test in python

### Python for Finance - Second Edition by Yuxing Yan

Use the Durbin-Watson statistic to test for the presence of autocorrelation in the errors of a regression model. Autocorrelation means that the errors of adjacent observations are correlated. If the errors are correlated, then least-squares regression can underestimate the standard error of the coefficients. Underestimated standard errors can make your predictors seem to be significant when they are not. For example, the errors from a regression model on daily stock price data might depend on the preceding observation because one day's stock price affects the next day's price.

The Durbin-Watson statistic D is conditioned on the order of the observations rows. Minitab assumes that the observations are in a meaningful order, such as time order. The Durbin-Watson statistic determines whether or not the correlation between adjacent error terms is zero. To get a conclusion from the test, you can compare the displayed value for the Durbin-Watson statistic with the correct lower and upper bounds in the following table from Savin and White 1.

The table provides values to test for first-order, positive autocorrelation. The significance level for the test is 0. The table is for models with an intercept. You can also use this table to test for first-order, negative autocorrelation. The test statistic is 4 — D.

Computing Durbin-Watson Statistic in Excel

Test for autocorrelation by using the Durbin-Watson statistic Learn more about Minitab To install Python and these dependencies, we recommend that you download Anaconda Python or Enthought Canopyor preferably use the package manager if you are under Ubuntu or other linux. R is a language dedicated to statistics. Python is a general-purpose language with statistics modules.

R has more statistical analysis features than Python, and specialized syntaxes. However, when it comes to building complex analysis pipelines that mix statistics with e.

Some of the examples of this tutorial are chosen around gender questions. The reason is that on such questions controlling the truth of a claim actually matters to many people. The setting that we consider for statistical analysis is that of multiple observations or samples described by a set of different attributes or features.

The data can than be seen as a 2D table, or matrix, with columns giving the different attributes of the data, and rows the observations. We will store and manipulate this data in a pandas. DataFramefrom the pandas module. It is the Python equivalent of the spreadsheet table.

## Durbin–Watson statistic

It is different from a 2D numpy array as it has named columns, can contain a mixture of different data types by column, and has elaborate selection and pivotal mechanisms. The weight of the second individual is missing in the CSV file. Creating from arrays : A pandas. If we have 3 numpy arrays:.

We can expose them as a pandas. DataFrame :. Other inputs : pandas can input data from SQL, excel files, or other formats. See the pandas documentation. For a quick view on a large dataframe, use its describe method: pandas. Other common grouping functions are median, count useful for checking to see the amount of missing values in different subsets or sum.

Groupby evaluation is lazy, no work is done until an aggregation function is applied. What is the average value of MRI counts expressed in log units, for males and females? Pandas comes with some plotting tools pandas. Plot the scatter matrix for males only, and for females only.

Do you think that the 2 sub-populations correspond to gender? For simple statistical testswe will use the scipy.The observations in the model are ordered by the size of z. If set to NULL the default the observations are assumed to be ordered e. Eigenvalues computed have to be greater than tol to be treated as non-zero. By default the variables are taken from the environment which dwtest is called from.

The Durbin-Watson test has the null hypothesis that the autocorrelation of the disturbances is 0. It is possible to test against the alternative that it is greater than, not equal to, or less than 0, respectively.

This can be specified by the alternative argument. Under the assumption of normally distributed disturbances, the null distribution of the Durbin-Watson statistic is the distribution of a linear combination of chi-squared variables. This algorithm is called "pan" or "gradsol". For large sample sizes the algorithm might fail to compute the p value; in that case a warning is printed and an approximate p value will be given; this p value is computed using a normal approximation with mean and variance of the Durbin-Watson test statistic.

Examples can not only be found on this page, but also on the help pages of the data sets bondyieldcurrencysubstitutiongrowthofmoneymoneydemandunemploymentwages. Biometrika 37 Biometrika 38 Biometrika 58 Applied Statistics 29 Heidelberg: Physica. Created by DataCamp. Community examples Looks like there are no examples yet. Post a new example: Submit your example.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. The Durbin-Watson test tests the autocorrelation of residuals at lag 1.

But so does testing the autocorrelation at lag 1 directly. Plus, you can test the autocorrelation at lag 2,3,4 and there are good portmanteau tests for autocorrelation at multiple lags, and get nice, easily interpretable graphs [e. Durbin-Watson is not intuitive to understand, and often produces inconclusive results. So why ever use it? This was inspired by this question on the inconclusiveness of some Durbin-Watson tests, but is clearly separate from it. As pointed out before in this and other threads: 1 The Durbin-Watson test is not inconclusive.

Only the boundaries suggested initially by Durbin and Watson were because the precise distribution depends on the observed regressor matrix. So neither inconclusiveness nor limitation of lags is an argument against the Durbin-Watson test. In comparison to the Wald test of the lagged dependent variable, the Durbin-Watson test can have higher power in certain models. Specifically, if the model contains deterministic trends or seasonal patterns, it can be better to test for autocorrelation in the residuals as the Durbin-Watson test does compared to including the lagged response which isn't yet adjusted for the deterministic patterns.

I include a small R simulation below. One important drawback of the Durbin-Watson test is that it must not be applied to models that already contain autoregressive effects. Thus, you cannot test for remaining residual autocorrelation after partially capturing it in an autoregressive model. In that scenario the power of the Durbin-Watson test can break down completely while for the Breusch-Godfrey test, for example, it does not.

For a data set with trend plus autocorrelated errors the power of the Durbin-Watson test is higher than for the Breusch-Godfrey test, though, and also higher than for the Wald test of autoregressive effect. I illustrate this for a simple small scenario in R. I draw 50 observations from such a model and compute p-values for all three tests:.

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The Durbin-Watson test is how you test for autocorrelation. Being able to eyeball a Q-Q plot to test for normality is useful, but a Kolmogorov-Smirnov or Levene test supplements what you see in the plot because a hypothesis test for normality is more conclusive.

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With regard to multiple lags, you could use a generalized Durbin-Watson statistic, run a few hypothesis tests, and do a Bonferroni correction to correct for multiple testing.

You could also run a Breusch-Godfrey testwhich tests for the presence of a correlation of any order. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Why ever use Durbin-Watson instead of testing autocorrelation? Ask Question. Asked 4 years, 10 months ago. Active 4 years, 10 months ago. Viewed 13k times. Look up Generalized Durbin-Watson statistics. Active Oldest Votes.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

It only takes a minute to sign up. From my data it's clear that residuals shows a strong autocorrelation. The null hypothesis is that there is no autocorrelation.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 4 years, 11 months ago. Active 4 years, 11 months ago. Viewed 16k times. Giacomo Rosaspina Giacomo Rosaspina 33 1 1 gold badge 1 1 silver badge 6 6 bronze badges. My impression is that residui is a matrix of residuals and their lags. Then the results are non-sensical.

The input for the function should be the regression itself and not its residuals. And the regression must not contain lagged responses because the Durbin-Watson test is not consistent for these and other autocorrelation tests should be used e.

Maybe you can post a reproducible example that illustrates your analysis approach. So the input of the function is directly the variable object of the fitting and the explanatory variables. I don't have to first compute the residuals? This is a procedure made automatically by the function? So if I understood well for this test is implied that the lag is 1. The Durbin-Watson test assesses the autocorrelation of residuals of a linear regression fit. The function dwtest expects you to either supply a fitted lm object or equivalently the corresponding formula plus data. The implementation in dwtest only allows to test lag 1.In this post I will use Python to explore more measures of fit for linear regression. This will be an expansion of a previous post where I discussed how to assess linear models in R, via the IPython notebook, by looking at the residual, and several measures involving the leverage.

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First we will look at a small data set from DASL library, regarding the correlation between tobacco and alcohol purchases in different regions of the United Kingdom. The interesting feature of this data set is that Northern Ireland is reported as an outlier.

Notwithstanding, we will use this data set to describe two tools for calculating a linear regression. We will alternatively use the statsmodels and sklearn modules for caluclating the linear regression, while using pandas for data management, and matplotlib for plotting. To begin, we will import the modules. I copied the data from here and pasted it between a pair of triple quotes in the IPython Notebook, as so. Each line ends in a newline, and each datum is delimited by a tab, so we first split the string over the newlines, and then split each new datum using the tabs, like this. Next, we make sure any numbers register as numbers, while leaving the strings for the regions alone.

Finally, we wrap this data in a pandas DataFrame. The neat thing about a DataFrame, is that it lets you access whole variables by keyword, like a dictionary or hash, individual elements by position, as in an array, or through SQL-like logical expressions, like a database.

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Furthermore, it has great support for dates, missing values, and plotting. We give the DataFrame two arguments, the data, and then labels for the columns, taken from the first row of our list, d.

This will allow us to refer to the column containing alcohol data as df. This convention makes it easier to read the code at a later date. We notice that there seems to be a linear trend, and one outlier, which corresponds to North Ireland. To perform ordinary least squares regression on the alcohol consumption as a function of tobacco consumption, we enter the following code.

Note that we are excluding the last datum, which refers to the outlying North Ireland data.

## Durbin–Watson statistic

Since we want a linear model that looks likewe need to add an extra array or vector of ones to our independent variable, df. Tobacco because the statsmodels OLS function does not assume that we would like a constant or intercept intercept term. This is not so uncommon as it would seem; several regression packages make this requirement. The summary method produces the following human-readable output.

And now we have a very nice table of mostly meaningless numbers. The left column of the first table is mostly self explanatory. The degrees of freedom of the model are the number of predictor, or explanatory variables. The degrees of freedom of the residuals is the number of observations minus the degrees of freedom of the model, minus one. Most of the values listed in the summary are available via the result object.

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For instance, the R 2 value is obtained by result. If you are using IPython, you may type results. The term is the coefficient of determination and it usually reflects how well the model fits the observed data. The coefficient of determination is usually given by. Where is an observed response, is the mean of the observed responses, is a prediction of the response made by the linear model, and is the residual, i. Also, is called the sum of the squared error, or the sum of the squared residuals, and is called the total sum of squares.

This is known as the Curse of Dimensionality. The adjusted takes into account the number of predictor variables the degrees of freedom and number of observations. Let be the number of observations, and be the number of predictors, then the adjusted is given by. In the following discussion of -tests and -tests, please bear in mind that squinting over a p-values at significance levels is silly, because your model is built upon simplifying and inaccurate assumptions.

Hypothesis testing should guide your decision making, not dictate it. That being said, the null hypothesis of the -test is that the data can be modeled accurately by setting the regression coefficients to zero.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I'm experimenting to decide if a time-series as in, one list of floats is correlated with itself. Perhaps this is supposed to be passed in the exog parameter to OLS? Side-note: I'm not sure what a "nobs x k" array means. Maybe an array with is x by k? So what should I be doing here? Am I expected to pass the data twice, or to lag it manually myself, or? OLS is a regression that needs y and x or endog and exog.

Durbin Watson is a test statistic for serial correlation. It is included in the OLS summary output. There are other tests for no autocorrelation included in statsmodels. This small program computes the durbin-watson correlation for a linear range which should be highly correlated, thus giving a value close to 0 and then for random values which should not be correlated, thus giving a value close to 2 :. Learn more. Durbin—Watson statistic for one dimensional time series data Ask Question.

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It seems like this kind of thing should work: from statsmodels. Warren Weckesser Edd Barrett Edd Barrett 2, 1 1 gold badge 20 20 silver badges 32 32 bronze badges. Active Oldest Votes.