# Portmanteau Test Statistic: Test the Adequacy of a Fitted Model

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## What Is Portmanteau?

The dictionary meaning of ‘Portmanteau’ is a new word created by blending the two others and their meanings. Few words can be considered as an example here like motel, brunch, Brexit, smog etc. The word ‘motel’ is a portmanteau in the sense that it comprises two words motor and hotel which provides a person with the convenient parking space along with the typical facilities of a hotel. In the same way, the word ‘brunch’ comprises two words namely breakfast and lunch which is usually used to denote the food served in the late morning. There might be a myriad of such examples of portmanteau. Portmanteau Test Statistic is something that is very much pertinent to the nature and meaning of the word ‘Portmanteau’. The concept of portmanteau test has been prevalent in the literature of time series modelling and forecasting for several years and has also attracted the attention of the researchers and mathematicians in several countries.

A fitted model may not be the best one, but a best model is always a fitted one.

— Amit Sharma

## What Is Portmanteau Test Statistic?

Portmanteau Test Statistic is a criterion to check the adequacy, appropriateness of a fitted time series stochastic model by using the autocorrelation values of the residuals which are calculated at different lag points. It had primarily been propounded and developed by George E.P. Box and David A. Pierce in 1978. Therefore, it was also known by Box-Pierce Test in those days. Afterwards, this test had been modified due to the weakness of it arisen from its extreme simplicity. Modified version has been put forward by Greta M. Ljung and George E.P. Box in the same year. Since then, this test is also known by Ljung-Box Q Test. This test doesn’t take into consideration the randomness of the residuals at each individual lag, rather, it tests the overall randomness by taking into consideration the combined number of autocorrelations of residuals at a time. In simple words, Ljung-Box Q test considers the autocorrelations of the residuals and takes into account the combined number of autocorrelations of the residuals calculated at different lag points to determine the value of Q-statistic. Due to these characteristics, this test is popularly known as Portmanteau Test Statistic.

## Where to Use?

In the Box-Jenkins Methodology, technically known as ARIMA (Autoregressive Integrated Moving Average), especially at the time of diagnostic checking when the residuals are considered to diagnose the fitness of the different stochastic models to a given time series, the portmanteau test plays very significant role. One can encounter a number of fitted models on the basis of the identical white noise properties of their residuals. These fitted models are called the competing models. It becomes a challenge for the forecasters to select the best model among these competing models. The portmanteau test is one of the best model selection criteria that can be deployed along with others like Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) which is also known as Schwarz Information Criterion (SIC), adjusted R square etc. On the basis of testing the randomness of the residuals by taking into consideration the group of autocorrelations, It checks whether a stochastic model departs from the underlying data generation stochastic process or it is adequate and appropriate as far as the prognosticative values are considered.

In any time series, residuals are always random, but significant relationships between the predictive values or in the residuals at the adjacent lag points kill the randomness.

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## What Is the Hypothesis for the Test?

Portmanteau Test is a statistical hypothesis test in nature. It is deployed to test a null hypothesis regarding the adequacy of the model as shown in Fig. 1.

### What to Do?

To test the above-mentioned hypothesis, one has to calculate the value of the Q-statistic (Box-Pierce, 1978), which takes into consideration the sum of the square of a group of residual autocorrelations of a model calculated at different lags as depicted by Fig. 2. If the model is appropriate, the average values of the Q-statistic would approximately follow or resemble the Chi-square distribution otherwise they will be tending to increase in case the model is inappropriate which shows the increment in the autocorrelation values of the residuals. But it’s apparently clear that a well-fitted model would give the residuals having the autocorrelation values not significantly different from zero which are calculated by taking into consideration different lag points. Ljung and Box (1978) argued that the Chi-square is not the accurate approximation of the Q-statistic and revealed that the values of Q-statistic are less than the expected values of the Chi-square. Therefore, they modified the existing Q-statistic and put forward a new one known as the modified Q-statistic depicted by Fig. 3. Furthermore, they also argued that the modified test approximately and accurately follows the Chi-square distribution.

## How to Test a Model for Its Adequacy and Appropriateness?

To determine the appropriateness and adequacy of the model or to test the null hypothesis, one has to take the reference of the table values of the Chi-square distribution after calculating the values of modified Q-statistic. From the table, one has to select the value corresponding to the provided degree of freedom and required level of significance. After selecting the appropriate value, a comparison has to be made between calculated value of the modified Q-statistic and selected table value of the Chi-square distribution. If the calculated value of the modified Q-statistic is greater than the table value of Chi-square distribution, model is inadequate and inappropriate as it leads to the rejection of the null hypothesis. In the simple words, it indicates that there is significant autocorrelation in the values of the residuals of the model or the autocorrelation values in the residuals at several lag points are significantly different from zero. On the contrary, if the modified Q-statistic is less than the table value, model is adequate and appropriate as it leads to the acceptance of the null hypothesis as shown in Fig. 4.