# Quality of the Time Series Model

### From MM*Stat International

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In the preceding paragraph it likely became clear that, a priori, there is no best time series model. In particular, there are different methods for the estimation of the trend which do not differ in the parameters only, but follow different methodologies.
In order to select one model from the variety of possible models one needs a criteria to justify a decision. How well a model describes (fits) the available data, can be seen from the structure and the fluctuation of the residuals. The following measures, which offer information about the fluctuation of the residuals, have already been studied.
**Mean square dispersion (estimated )**
the coefficient of variation coefficient of determination (applicable only if the trend was calculated with the least square method.)
**Explanation**
Like in the preceding paragraph you can select a , which will be decomposed into estimated trend , seasonal component and residuals. Consider with each selection the time frequency in which the data were observed (e.g. monthly).
In the result window you now also find measurements of the quality of the fit of your model to the data (standard deviation and coefficient of variation). The coefficient of determination is not shown here, because the trend is estimated with the moving average method, and r-square does not make sense here.
**Suggestion**
Your first step is to select a filter which seams reasonable to you.
Repeat the calculations and pay attention to the consequences of the application of different filters to the fit of the model (i.e. its ability to fit or ”explain” the observed data). How do the estimated standard deviation and the variance coefficient change, if you compare an appropriate model with one, which uses an unsuitable filters for smoothing out the actual seasonal fluctuations? What is the effect of selecting a shorter/longer filter?
Which measure for the examination of the quality of the model fit do you consider more suitable? What does a good description of the data by the model mean (thus a good adjustment)? Should this differ if we consider the fit of our model for forecasts?
**Data**
You can select one of the following time series.

**Circulation of money**Circulation of money in Germany Time period: 1968:1 - 1998:3Periodicity (frequency): Monthly data**M3**Money supply M3: change in % relative to the preliminary periodTime period: 1956:1 - 1998 :3Periodicity (frequency): Monthly data**Balance of payments**Balance of payments of Germany Time period: 1977 – 1995 Periodicity (frequency): Yearly data**New car registrations**Number of newly registered cars in Berlin Time period: 1977:1 – 1998 :4Periodicity (frequency): Quarterly data**Rain**Precipitation in Potsdam, GermanyTime period: 1984:1 - 1995:1Periodicity (frequency): Monthly data**DAX stock market index**Yearly changes of the DAX in % Time period: December 1960 - December 1997 Periodicity (frequency): Yearly data Note: Until 1987 the index of the Deutsche Börsenzeitung is reported.**Public income**Changes in the incomes of all public households in Germany in %. Time period: 1951 – 1991 Periodicity (frequency): Yearly data**National debt**Indebtedness of all public households in Germany (change in % to the last year)Time period: 1967:4 – 1997:3Periodicity (frequency): Quarterly data