TF-MIDAS: a transfer function based mixed-frequency model

ABSTRACT

This paper tackles the mixed-frequency modelling problem from a new perspective. Instead of drawing upon the common distributed lag polynomial model, we use a transfer function representation to develop a new type of models, named TF-MIDAS. We derive the theoretical TF-MIDAS implied by the high-frequency VARMA family models for two common aggregation schemes, flow and stock. This exact correspondence leads to potential gains in terms of nowcasting and forecasting performance against the current alternatives. The estimation of the model proposed is also addressed via its state space equivalent form. A Monte Carlo simulation exercise confirms that TF-MIDAS beats U-MIDAS models (its natural competitor) in terms of out-of-sample nowcasting performance for several data generating high-frequency processes.

Keywords:

• Mixed-frequency models
• U-MIDAS
• nowcasting
• forecasting
• TF-MIDAS

JEL:

• C18
• C51
• C53

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

In bold rows where median MSFE of TF-MIDAS is at least 5% better than U-MIDAS.

In bold rows where median MSFE of TF-MIDAS is at least 5% better than U-MIDAS.

In bold rows where median MSFE of TF-MIDAS is at least 5% better than U-MIDAS.

1 In the case of USA, the Bureau of Economic Analysis (BEA) releases a first estimate of GDP 28 days after the end of quarter, followed by a second and a third estimate 60 and 90 days after the end of quarter, respectively. In Latin American countries the schedule is similar. In Uruguay, for example, GDP value is released by the Central Bank 75 days after the end of quarter.

2 In the case of a stock variable, it would be ${\omega }_0=1,{\omega }_1=\cdots ={\omega }_{k-1}=0$. In the case of a flow variable, it would be ${\omega }_0={\omega }_1=\cdots ={\omega }_{k-1}=1$, if the values are aggregated by addition, or ${\omega }_0={\omega }_1=\cdots ={\omega }_{k-1}=1/k$, if the values are aggregated by average.

3 The Exponential Almon weighting function was proposed in Ghysels et al. [20] and it has the following expression, with Q shape parameters: (33) $b(j;\theta )={\displaystyle \frac{\exp ({\theta }_{1}j+\cdots +{\theta }_Q{j}^Q)}{\sum _{j=0}^K\exp ({\theta }_{1}j+\cdots +{\theta }_Q{j}^Q)}}$(33) Beta weighting function, proposed for the first time in Ghysels et al. [4], includes only two shape parameters: (34) $b(j;{\theta }_1,{\theta }_2)={\displaystyle \frac{f({\displaystyle \frac{j}{K}};{\theta }_1,{\theta }_2)}{\sum _{j=1}^{K}f({\displaystyle \frac{j}{K}};{\theta }_1,{\theta }_2)}}$(34) where f is the Beta probability density function.

4 We also develop the theoretical TF-MIDAS representations corresponding to additional VAR$(p)$, VMA$(q)$ and VARMA$(p,q)$ HF processes. The developments are available from the authors upon request.

5 The value of ψ in (17(17) ${\eta }_{t_q}=(1+\psi Z)e_{y^A,t_q},$(17) ) can be calculated straightforwardly by using the ACF for a MA(1) model: ${\rho }_1=\psi /(1+{\psi }^2)$ and ${\rho }_k=0$ for $k=2,3,\dots$ Solving the ${\rho }_1$ equation for ψ will return two solutions, from where we consider the one satisfying the invertibility condition: $|\psi |<1$.

6 Note that if $\varphi (Z)=b_i(Z)$ for some i, then $a_i^{\ast }(Z)=a_i(Z)$ and this polynomial will be of finite order.

7 In order to keep focused on the TF-MIDAS discussion, we do not present in this paper the ML function and the Kalman filter equations, as this is standard in the state space models literature. However, for readers who are not familiar with this type of models, all the necessary equations needed to compute the ML can be found in Casals et al. [18, Section 5.3.2], where expression (5.50) specifically shows the corresponding log-likelihood function used.

8 Notice that the polynomial $\hat{\theta }(Z)$ does not include the unit term as ${\hat{a}}_{k{T}_q+es}$ is not known at period $k{T}_q+es$.

9 However, some alternative simulations have been performed using the AIC criterion to select the best forecasting model. As this criterion overweight the goodness-of-fit against the parsimony relative to BIC, the criterion does choose the TF-MIDAS with an MA term most of the times, leading to the expected nowcasting performance improvement with respect to U-MIDAS models. The results of these simulations are available from the authors upon request.

10 A detailed analysis is available from the authors upon request.

Additional information

Funding

This work was supported by UCM-Santander [PR75/18-21570].