Quality of the Time Series Model

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 ) ${\displaystyle s_{ZRM}={\sqrt {{\frac {1}{T}}\sum \limits _{i=1}^{P}\sum \limits _{j=1}^{k}(x_{i,j}-{\widehat {x}}_{i,j}^{ZRM})^{2}}}}$ the coefficient of variation ${\displaystyle v={\frac {S_{ZRM}}{\bar {x}}}}$ coefficient of determination (applicable only if the trend was calculated with the least square method.) ${\displaystyle R^{2}=1-{\frac {s_{ZRM}^{2}}{s_{x}^{2}}}}$ ${\displaystyle s_{x}^{2}={\frac {1}{T}}\sum \limits _{i=1}^{P}\sum \limits _{j=1}^{k}(x_{i,j}-{\bar {x}})^{2}\,\qquad 0\leq {\frac {s_{ZRM}^{2}}{s_{x}^{2}}}\leq 1}$ 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.