Estimation Results

Estimation Results

The OUTPUT tab of each model contains all the results. The figure below presents the tree structure that includes all relevant information.

By clicking on each one of the branches of this tree, the corresponding results will be dynamically computed using the data and the estimated parameters:

1. The "Input" branch

It contains information regarding the model specification, estimation options and input data (both original and transformed). This is obviously not part of the estimation output, but it is presented in the Output tab so that it can be easily access and copied by the user.

2. The "Estimation" branch

It contains all the estimaton results.

• MODEL: factor loadings and variance of the idiosyncratic components {Z,R}, transition matrix for the factor an innovations covariance matrix {A1,…,Ap,Q}

• SHOCKS DECOMPOSITION from Choleski factorization of Q (the VAR innovations covariance). Such decomposition is shown in the image above for the variable industrial production in the euro area (demeaned or centered around zero). The difference between the signal extracted by the model and the actual data is an estimator of the idiosyncratic component (Noise) at each point in time. The factor accounting for the largest contribution in the signal is the first one (F1). The graph also shows that other two factors (F2 and F6) have also played a role during the Great Recession. Click on the name of the variable to visualize the same graph for other indicators, and scroll or click on the graph to zoom in (left) and out (right).

• FIT: The in-sample fit of the model can be analysed along many different dimensions.

  • The first sub-branch, "Signals vs Data" displays the actual data together with the signal extracted by the model, i.e. E[y(t) |y(1), ..., y(T)], using the entire sample. Advanced users can think of such signal as the Kalman smoothed estimates of the de-noised data.

    • Tip 1: Models with many factors may yield a signal that resembles the actual data in the sample. However, this is not a guarantee that the model will also perform well at forecasting out-of-sample.
    • **Tip 2:**Note that the plot for Data relative to a quarter is plotted at the middle of the quarter, so it may appear that it leads the monthly signal, which represents a weighted average of the factors. The third month of the signal aims to represent the underlying growth rate for the whole quarter.

  • The "Residuals" sub-branch is also important to assess the in-sample fit of the model. The residuals are defined as y(t) - E[Z f(t) | y(1), ..., y(T) ]. The program displays a table with their variance and their autocorrelation of order one (using red colours when it is large). The cross-correlation patterns are also displayed in matrix form and highlighted. For the analysis of large datasets, we use a tool that is more powerful than the data matrix: the Schema ball. This tool allows you to identify correlation patterns that are common to large groups of variables, which could suggest the omission of relevant factors. You can also click on a given variable, as shown in the video, and identify all the variables that are correlated with it.

• FACTORS: Plot for each one of the factors. The so-called “smoothed” factors E[f(t) | y(1), ..., y(T) ] are displayed together with 95 percent confidence bounds around them. The so-called filtered factors are also displayed E[f(t) | y(1), ..., y(t-1) ]. Note that both definitions coincide for t>T.

• ANALYSIS of impulse response functions and variance decompositions consistent a Choleski factorization of the VAR innovations matrix. Advanced users can easily introduce a triangular pattern of restrictions in the loadings in order to interpret those results from an economic point of view (i.e. structural VAR analysis), but the current version of the plugin does not offer an interface for implementing other decompositions based on sign restrictions or long run restrictions.

3. The "Forecasts" branch

Forecasts based on the information set used in the estimation stage:

• DATA: Grid containing the forecasts for the transformed data, e.g. differenced, seasonally adjusted, and corrected for outliers.

• STDEVS: Grid with the standard deviation of those forecasts.

• UNTRANSFORMED DATA: The data corresponding to the forecasts for the original data, i.e. not differenced or seasonally adjusted, are also included.

• FORECAST UNCERTAINTY plots out-of-sample forecasts for the transformed data, including a forecasting interval that represents the sum of the uncertainty resulting from all news to come and the variance of the measurement errors.

• NEWS UNCERTAINTY plots out-of-sample forecasts for the transformed data, including a forecasting interval that represents only the uncertainty resulting from all news to come. Estimates of past values or missing observations are also displayed. In those cases, the bounds represents the uncertainty of the signal extraction problem.

  

4. The "Forecasts Simulation" branch

This the last branch of the results. It contains data and graphs only if simulations based on recursive or rolling estimation schemes have taken place. This part will be discussed in detail in the Real-Time simulation section.