Flexible specification of dynamic factor models in JDemetra+

We present here a common approach to link GDP growth, which is the quarterly variable that we use as a target, with the unobserved factors, which are specified at a monthly frequency and determine the joint dynamics of the whole system. The same modeling approach can be used for variables others than GDP.

Measurement Equation

We can specify more than one factor, but for the sake of simplicity, the following expression links the monthly growth rates of all the variables included in $y_{t}$ to only one monthly factor:

\[\begin{eqnarray} y_{t}&=&\bar{y}+\Lambda_{y} f_{1,t} + \psi_{t} ~~~~~~~~~~~ \text{Measurement equation for monthly series} \label{mbe} \end{eqnarray}\]

where $\psi_{t}$ represents the measurement error, which is assumed to be uncorrelated with the factor at all leads and lags. It is also assumed to be independent and identically distributed (iid) and following a normal distribution: $\psi_{t}\sim N(0,R_{\psi})$. The covariance matrix is assumed to be diagonal, which implies that the factor will account for 100 $\%$ of the comovements implicit in the model.

Variables expressed in terms of monthly growth rates can be linked to a factor representing the underlying monthly growth rate of the economy if “M” is selected in the graphical interface of MODEL tab:

Because the model has been designed for short-term analysis, it makes sense to represent all these series, including GDP, in terms of monthly growth rates or differences. However, the monthly growth rates of official GDP figures are not published, hence, equation \ref{mbe} will need to be modified. Thus, GDP growth rates published by the statistical agencies (i.e. $y^{Q}_{t}$ for the euro area, for example) are linked to the quarterly growth rate of the underlying factor, which can be approximated as a moving average of the monthly growth rates:

\[\begin{eqnarray} y^{Q}_{t}&=&\bar{y}^{Q}+\Lambda^{Q}_{y}f^{Q}_{1,t}+ \psi^{Q}_{t},~~~~~~~~~~~~~t=3,6,9,\ldots ~~~~~ \text{Measurement equation for quarterly series} \label{mbe2} \end{eqnarray}\]

where $f^{Q}_{1,t}={\frac{1}{3} f_{1,t}} +\frac{2}{3}{f_{1,t-1}} + {f_{1,t-2}} +\frac{2}{3}{f_{1,t-3}} +\frac{1}{3}{f_{1,t-4}}$

Monthly or quarterly variables that are correlated with the the underlying quarterly growth rate of the economy can be linked to a weighted average of the factors representing the underlying monthly growth rate of the economy. Such a weighted average is meant to represent quarterly growth rates, and it is implemented by selecting “Q” in the graphical interface of MODEL tab:

Approximation: Let $F_{t}$ be the monthly level of the economy and let $f_{t}=ln F_{t}-ln F_{t-1}$ be its monthly growth rate. Now, define $F^{Q}_{t}$ as the geometric mean of the last three levels. This implies that $ln F^{Q}_{t} = \frac{1}{3}(ln F_{t}+ln F_{t-1}+ln F_{t-2})$. The resulting quarterly growth rate of the factors, which we denote as $f^{Q}_{t}$, can be expressed as $ln F^{Q}_{t}-ln F^{Q}_{t-3}$. By substituting both terms by the geometric mean approximation we obtain $f^{Q}_{t}=\frac{1}{3}(ln F_{t}-ln F_{t-3})+\frac{1}{3}(ln F_{t-1}-ln F_{t-4})+\frac{1}{3}(ln F_{t-2}-ln F_{t-5})$. Finally, a simple expression for the quarterly growth rate of the factors in terms of their monthly growth rates can be obtained as follows: $f^{Q}_{t}=\frac{1}{3}(f_{t}+f_{t-1}+f_{t-2})+\frac{1}{3}(f_{t-1}+f_{t-2}+f_{t-3})+\frac{1}{3}(f_{t-2}+f_{t-3}+f_{t-4})$. Rearranging terms yields the expression $f^{Q}_{t}={\frac{1}{3} f_{t}} +\frac{2}{3}{f_{t-1}} + {f_{t-2}} +\frac{2}{3}{f_{t-3}} +\frac{1}{3}{f_{t-4}}$ presented above.

Modeling yearly growth rate of a flow variable like GDP. In this case, the measurent equation requires a larger number of lagged factors. The new measurement equation for these variables takes the following form: $\begin{equation} y^{Y}_{t}=\bar{y}^{Y}+\Lambda^{Y}_{y}f^{Y}_{t}+ \psi^{Y}_{t},~~~~~~~~~~~~~t=12,24,36,\ldots ~~~~~ \text{Measurement equation for yearly growth rates} \label{mbeY} \end{equation}$

where $\tiny f^{Y}_{t}=\frac{1}{12} f_{t} + \frac{2}{12}f_{t-1} +\frac{3}{12}f_{t-2} + \frac{4}{12}f_{t-3} +\frac{5}{12}f_{t-4} +\frac{6}{12}f_{t-5} +\frac{7}{12}f_{t-6}+\frac{8}{12}f_{t-7} +\frac{9}{12}f_{t-8} +\frac{10}{12}f_{t-9} +\frac{11}{12}f_{t-10} +f_{t-11} + \frac{11}{12}f_{t-12}+\frac{10}{12}f_{t-13} +\frac{9}{12}f_{t-14} +\frac{8}{12}f_{t-15} +\frac{7}{12}f_{t-16} + \frac{6}{12}f_{t-17} +\frac{5}{12}f_{t-18} +\frac{4}{12}f_{t-19} +\frac{3}{12}f_{t-20} +\frac{2}{12}f_{t-21} +\frac{1}{12}f_{t-22}$ The icon needed to implement this concept is

Approximation:Let $F_{t}$ be the monthly level of the economy and let $f_{t}=ln F_{t}-ln F_{t-1}$ be its monthly growth rate. Now, define $F^{Y}_{t}$ as the geometric mean of the last twelve levels. This implies that $ln F^{Y}_{t} = \frac{1}{12}(ln F_{t}+ln F_{t-1} ... + ln F_{t-11})$. The resulting yearly growth rate of the factors, which we denote as $f^{Y}_{t}$, can be expressed as $ln F^{Y}_{t}-ln F^{Y}_{t-12}$. By substituting both terms by the geometric mean approximation and using the same method as in the approximation of quarterly growth we obtain $\tiny f^{Y}_{t}=\frac{1}{12} f_{t} + \frac{2}{12}f_{t-1} +\frac{3}{12}f_{t-2} + \frac{4}{12}f_{t-3} +\frac{5}{12}f_{t-4} +\frac{6}{12}f_{t-5} +\frac{7}{12}f_{t-6}+\frac{8}{12}f_{t-7} +\frac{9}{12}f_{t-8} +\frac{10}{12}f_{t-9} +\frac{11}{12}f_{t-10} +f_{t-11} + \frac{11}{12}f_{t-12}+\frac{10}{12}f_{t-13} +\frac{9}{12}f_{t-14} +\frac{8}{12}f_{t-15} +\frac{7}{12}f_{t-16} + \frac{6}{12}f_{t-17} +\frac{5}{12}f_{t-18} +\frac{4}{12}f_{t-19} +\frac{3}{12}f_{t-20} +\frac{2}{12}f_{t-21} +\frac{1}{12}f_{t-22}$ .

Measurement Equations for Surveys

The link between the time series and the factors can be more sophisticated. Apart from the two options mentioned above, the measurement equation for some variables (typically survey data) may require an additional effort.

Surveys such as the euro area economic sentiment indicator produced by the European Commission, which are correlated with the year-on-year growth rate of GDP. Those variables may be linked to the cumulative sum of the last 12 monthly factors. If the model is designed in such a way that the monthly factors represent monthly growth rates, the resulting cumulative sum boils down to the year-on-year growth rate. Thus, variables expressed in terms of year-on-year growth rates or surveys that are correlated with the year-on-year growth rates of the reference series should be linked to the factors using this approach:
Variables representing expected changes over a precise horizon may be added to the measurement equation in a model-consistent way. For example, think of a quantitative or qualitative survey $S_t$ where respondents have been asked to report how much they expect prices to change over the next three months. That is, $S_{t}=P_{t+h | t}-P_{t}$, where $P_{t+h | t}$ stands for the price level expected three months ahead when $h=3$. Variables that contains expectations about the future are likely to be informative about the current state of the economy, $f_t$, so in principle they could be linked to the factors in the usual way: $\begin{eqnarray} S_{t} &=& \Lambda_{s} f_{t} + \psi^{s}_{t} \label{naive} \end{eqnarray}$ However, we can be more precise and introduce a model-consistent link between the surveys and the factors: $\begin{eqnarray} S_{t} &=& \Lambda_{s}(f_{t+3| t} +f_{t+2| t} + f_{t+1| t} ) + \psi^{s}_{t} \\ S_{t} &=& \Lambda_{s}(A^3 + A^2 + A ) f_{t} + \psi^{s}_{t} ,~~~~~ \text{Measurement equation for surveys regarding t+3 } \label{modelConsistentM} \end{eqnarray}$ Equation (\ref{modelConsistentM}) is obtained, without loss of generality, by assuming that the factors follow a VAR(1), i.e. $f_{t}=A f_{t-1} + u_{t}$, so that $f_{t+h| t} = A^{h} f_{t}$. Notice that while the factor loadings in measurement equation (\ref{naive}) are equal to $\Lambda_{s}$ , the factor that enters the more sophisticated measurement equation (\ref{modelConsistentM}) loads on $\Lambda_{s}(A^3 + A^2 + A )$.
Variables represening forecasts for a given horizon. The solution is independent on the frequency of the variable and what it represents. In practice it is equivalent to shifting the time series forward by $h$ periods. For example, one can model quarterly growth expectations produced by professional forecasters for a given quarter in the future. You may think it is a good idea to simply introduce quantitative surveys representing expectations exactly as you would do with the remaining indicators. However, you may want to achieve efficiency gains by making those forecasts consistent with your model. How to do it? A GDP growth forecast for, say $h$ quarters ahead is represented as $GDP_{t+h\times 3|t}$ because $t$ refers to the montly frequency and a quarter has 3 months. This is introduced in a model as follows: $\begin{eqnarray} GDP_{t+h \times 3|t} &=& \Lambda_{GDP} f^{Q}_{t+h\times 3 | t} + \psi^{Q}_{t+ h \times 3} \nonumber \\ GDP_{t+h \times 3|t} &=& \Lambda_{GDP} (\frac{1}{3}f_{t+h \times 3| t} +\frac{2}{3}f_{t+h \times 3-1 | t} + f_{t+h \times 3-2 | t} +\frac{2}{3}f_{t+h \times 3-3 | t} +\frac{1}{3}f_{t+h \times 3-4| t} ) + \psi^{Q}_{t+h \times 3} \label{modelConsistentQ} \end{eqnarray}$ If $h=0$, we are back to the case where we simply model the quarterly growth rate of a variable, which uses a moving average of the factors $f_t$, $\ldots$, $f_{t-4}$. If instead we want to model one quarter ahead forecasts of GPD, i.e. $h=1$, then equation (\ref{modelConsistentQ}) is shifting the time indices 3 months ahead. This is implemented by using

In this case, equation (\ref{modelConsistentQ}) does not need so many lags of the factors, but only $f_t$ and $f_{t-1}$: $\begin{eqnarray} GDP_{t+h \times 3|t} &=& \Lambda_{GDP} f^{Q}_{t+h\times 3 | t} + \psi^{Q}_{t+ h \times 3} \nonumber \\ GDP_{t+h \times 3|t} &=& \Lambda_{GDP} (\frac{1}{3}f_{t+h \times 3| t} +\frac{2}{3}f_{t+h \times 3-1 | t} + f_{t+h \times 3-2 | t} +\frac{2}{3}f_{t+h \times 3-3 | t} +\frac{1}{3}f_{t+h \times 3-4| t} ) + \psi^{Q}_{t+h \times 3} \nonumber \\ GDP_{t+h \times 3|t} &=& \Lambda_{GDP} (\frac{1}{3}A^{h \times 3}f_{t} +\frac{2}{3}A^{h \times 3-1} f_{t} + A^{h \times 3-2}f_{t} +\frac{2}{3} f_{t } + \frac{1}{3} f_{t-1} ) + \psi^{Q}_{t+h \times 3} \nonumber \\ GDP_{t+h \times 3|t} &=& \Lambda_{GDP} (\frac{1}{3}A^{h \times 3} +\frac{2}{3}A^{h \times 3-1} + A^{h \times 3-2} +\frac{2}{3} A^{h \times 3-3}) f_{t} + \Lambda_{GDP} \frac{1}{3}f_{t-1} +\psi^{Q}_{t+h \times 3} \label{modelConsistentQ2} \end{eqnarray}$ This has a more complex loading structure than equations (\ref{naive}) and (\ref{modelConsistentM}). As before, it is obtained, without loss of generality, by assuming that the factors follow a VAR(1), i.e. $f_{t}=A f_{t-1} + u_{t}$, so that $f_{t+h| t} = A^{h} f_{t}$. The icon you need to select to implement this idea is different than the previous one. After using the Q icon to to link quarterly growth rates with the moving average of the factors, we need to tell the sofware we are dealing with a forcast using the Shift(h) icon.
Further options. Once the difference between Shift(h) and X(h) has been understood, they can be used in combination with Q, Y or YoY in order to exploit alternative modeling options:
- Modeling yearly growth forecasts for given year in the future. The procedure is equivalent to the one followed in the previous point. After using the Y icon to link yearly growth rates with the moving average of the monthly factors, we need to tell the sofware we are dealing with a forcast using the Shift(h), where $h$ may be set to 1 if our variable, which has annual frequency, represents a forecast for the next year.
- Modeling expectations regarding cummulative growth over $h$ quarters. Instead of using icon Shift(h), you will select the icon X(h), where $h=4$ quarters for instance. In this case, you will be modeling the expected growth rate over the next 4 quarters (12 months, since the base frequency in our current set up is monthly): $\begin{eqnarray} GDP_{t+1 \times 3|t}+GDP_{t+2 \times 3|t}+GDP_{t+3 \times 3|t}+GDP_{t+4 \times 3|t} &=& \Lambda_{GDP} (f^{Q}_{t+1\times 3 | t}+f^{Q}_{t+2\times 3 | t}+f^{Q}_{t+3\times 3 | t}+f^{Q}_{t+h\times 3 | t}) + \psi^{Q}_{t+ h \times 3} \nonumber \\ \label{modelConsistentY} \end{eqnarray}$ which is different from GDP growth expected for the last quarter of the year, i.e. Shift(h) with $h=4$ and also different from yearly growth rates that are typically reported (they compare the GDP flow of a year with that of the previous year), so be careful with this. This expression can be expanded by replacing $f^{Q}_{t+h| t}$ by the corresponding weighted average, and of course, by dividing both sides of this expressioin by 4, we get the average growth rate for all quarters of the year.

Transition Equation

Specifying the joint dynamics of all variables requires a second equation representing the factor as a vector autoregressive (VAR) process with a non-diagonal covariance matrix for the error term. Because the measurement equation links GDP growth with a moving average of the monthly factors, the dimension of the state vector must be at least equal to 5. The minimum dimension would be 12 if at least one of the variables is expressed on year-on-year growth rates. To sum up, the representation given by equations \ref{mbenchmark} and \ref{ssbenchmark} conforms to the so-called state-space representation of this model in the case where you observed both quarterly and monthly growth rates:

\[\begin{eqnarray} \label{mbenchmark} \left( \begin{array}{c} y^{Q}_{t}- \bar{y}^{Q}\\ y_{t}-\bar{y} \end{array} \right)&=& \left( \begin{array}{ccccc} \Lambda_{y}^{Q} & 2\Lambda_{y}^{Q} & 3\Lambda_{y}^{Q} & 2\Lambda_{y}^{Q} & \Lambda_{y}^{Q} \\ \Lambda_{y} & 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array}{c} f_{1,t} \\ f_{1,t-1} \\ f_{1,t-2} \\ f_{1,t-3} \\ f_{1,t-4} \\ \end{array} \right)+ \left( \begin{array}{c} \psi^{Q}_{t}\\ \psi_{t} \\ \end{array} \right) \\ \label{ssbenchmark} \left( \begin{array}{c} % f_{1,t} \\ f_{1,t-1} \\ f_{1,t-2} \\ f_{1,t-3} \\ f_{1,t-4} \\ \end{array} \right)&=& \left( \begin{array}{ccccc} A_{11} & A_{12} & A_{13} & A_{14} & 0 \\ I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ \end{array} \right) \left( \begin{array}{c} f_{1,t-1} \\ f_{1,t-2} \\ f_{1,t-3} \\ f_{1,t-4} \\ f_{1,t-5} \\ \end{array} \right) + \left( \begin{array}{c} u^{f}_{t} \\ 0\\ 0\\ 0\\ 0\\ \end{array} \right) \end{eqnarray}\]

This equation represents a VAR of order 4 for the factor, but one could restrict it to a VAR(1) by setting $A_{12} = A_{13} = A_{14} = 0$. The error component $u^{f}_{t}$ is uncorrelated with the measurement error term $\psi_{t}$ , in line with the literature on factor models.

State-Space framework in JDemetra+

The sophisticated measurement and transition equations that have been described below conform a state-space representation (SSF) of the mixed-frequencies dynamic factor model. Such state-space system inherits all properties and algorithms of the general linear gaussian state-space model used throughout JD+ 3.0. The particularities of the design for the dynamic factor models used for nowcasting is described in Section: Design issues in Java. You will observe that the matrices in expressions (\ref{mbenchmark}) and (\ref{ssbenchmark}) are sparse, i.e. they have a lot of zeros. Since multiplying by zeroes and inverting large matrices is inneficient, we exploit the object oriented framework of Java in such a way that the algorithms of the different dynamic factor models that can be constructed have been optimized to a very large extent. This will ensure estimation is feasible even for large models without the need to use the EM algorithm.

As it will be discussed in Section: Design issues in Java, our approach is very different from that of alternative softwares packages built in Matlab, SAS, and OX.