Figure 1: Given a virtual object with specified elasticity material parameters (blue=soft, red=stiff), our method computes an assemblage of small-scale structures that approximates the desired elastic behavior and requires only a single material for fabrication. AbstractWe propose a method for fabricating deformable objects with spatially varying elasticity using 3D printing. Using a single, relatively stiff printer material, our method designs an assembly of smallscale microstructures that have the effect of a softer material at the object scale, with properties depending on the microstructure used in each part of the object. We build on work in the area of metamaterials, using numerical optimization to design tiled microstructures with desired properties, but with the key difference that our method designs families of related structures that can be interpolated to smoothly vary the material properties over a wide range. To create an object with spatially varying elastic properties, we tile the object's interior with microstructures drawn from these families, generating a different microstructure for each cell using an efficient algorithm to select compatible structures for neighboring cells. We show results computed for both 2D and 3D objects, validating several 2D and 3D printed structures using standard material tests as well as demonstrating various example applications.

Mixed data sampling (MIDAS) regressions allow us to estimate dynamic equations that explain a low frequency variable by high frequency variables and their lags. When the difference in sampling frequencies between the regressand and the regressors is large, distributed lag functions are typically employed to model dynamics avoiding parameter proliferation. In macroeconomic applications, however, differences in sampling frequencies are often small. In such a case, it might not be necessary to employ distributed lag functions. We discuss the pros and cons of unrestricted lag polynomials in MIDAS regressions. We derive unrestricted-MIDAS (U-MIDAS) regressions from linear high frequency models, discuss identification issues and show that their parameters can be estimated by ordinary least squares. In Monte Carlo experiments, we compare U-MIDAS with MIDAS with functional distributed lags estimated by non-linear least squares. We show that U-MIDAS performs better than MIDAS for small differences in sampling frequencies. However, with large differing sampling frequencies, distributed lag functions outperform unrestricted polynomials. The good performance of U-MIDAS for small differences in frequency is confirmed in empirical applications on nowcasting and short-term forecasting euro area and US gross domestic product growth by using monthly indicators.

In this article, we merge two strands from the recent econometric literature. First, factor models based on large sets of macroeconomic variables for forecasting, which have generally proven useful for forecasting. However, there is some disagreement in the literature as to the appropriate method. Second, forecast methods based on mixed-frequency data sampling (MIDAS). This regression technique can take into account unbalanced datasets that emerge from publication lags of high-and lowfrequency indicators, a problem practitioner have to cope with in real time. In this article, we introduce Factor MIDAS, an approach for nowcasting and forecasting low-frequency variables like gross domestic product (GDP) exploiting information in a large set of higher-frequency indicators. We consider three alternative MIDAS approaches (basic, smoothed and unrestricted) that provide harmonized projection Å The authors are grateful for helpful comments and discussions to three anonymous referees, Riccardo and Gerhard Rünstler. JEL Classification numbers: E37, C53.Factor MIDAS 519 methods that allow for a comparison of the alternative factor estimation methods with respect to nowcasting and forecasting. Common to all the factor estimation methods employed here is that they can handle unbalanced datasets, as typically faced in real-time forecast applications owing to publication lags. In particular, we focus on variants of static and dynamic principal components as well as Kalman filter estimates in state-space factor models. As an empirical illustration of the technique, we use a large monthly dataset of the German economy to nowcast and forecast quarterly GDP growth. We find that the factor estimation methods do not differ substantially, whereas the most parsimonious MIDAS projection performs best overall. Finally, quarterly models are in general outperformed by the Factor MIDAS models, which confirms the usefulness of the mixed-frequency techniques that can exploit timely information from business cycle indicators.

This paper compares the mixed-data sampling (MIDAS) and mixed-frequency VAR (MF-VAR) approaches to model speci…cation in the presence of mixed-frequency data, e.g., monthly and quarterly series. MIDAS leads to parsimonious models based on exponential lag polynomials for the coe¢ cients, whereas MF-VAR does not restrict the dynamics and therefore can su¤er from the curse of dimensionality. But if the restrictions imposed by MIDAS are too stringent, the MF-VAR can perform better. Hence, it is di¢ cult to rank MIDAS and MF-VAR a priori, and their relative ranking is better evaluated empirically. In this paper, we compare their performance in a relevant case for policy making, i.e., nowcasting and forecasting quarterly GDP growth in the euro area, on a monthly basis and using a set of 20 monthly indicators. It turns out that the two approaches are more complementary than substitutes, since MF-VAR tends to perform better for longer horizons, whereas MIDAS for shorter horizons.