# Bridging Dynamic Factor Models and Neural Controlled Differential Equations for Nowcasting GDP ## Reviewer kQMY (2024-03-18) **Paper category:** 3 ### Review In the context of GDP nowcasting, the authors proposed a new framework named NCDENow. This framework first interpolate within the irregular datasets, followed by a NCDE neural networks to predict the coefficient of hidden states. Then, the time series data were sent to Kalman filter to extract the hidden variables. Combining the coefficients and hidden variables gives the desired predicted value. The author then compared the NCDENow with existing models on South Korea and UK GDP datasets. The results showed a significant decreases in MSE and MAPE in most of the studied cases. The NCDENow also showed robustness against data missing and high efficiency with further experiments. The main part of this article is well-organized, the results demonstrated could answer the RQ proposed by the authors. The proposed outperformed other existing model. However, the design motivation/philosophy behind the NCDENow architecture is unclear. The model training should also be complemented. More experimental results is required to prove the advantages of the model. **Influence:** 3 **Novelty:** 3 **Technical soundness:** 2 **Impact:** 4 **Presentation:** 3 **Reproducibility:** 5 ### Summary In the context of GDP nowcasting, the authors proposed a new framework named NCDENow. This framework first interpolate within the irregular datasets, followed by a NCDE neural networks to predict the coefficient of hidden states. Then, the time series data were sent to Kalman filter to extract the hidden variables. Combining the coefficients and hidden variables gives the desired predicted value. The author then compared the NCDENow with existing models on South Korea and UK GDP datasets. The results showed a significant decreases in MSE and MAPE in most of the studied cases. The NCDENow also showed robustness against data missing and high efficiency with further experiments. ### Paper strength 1) The proposed framework of NCDENow is novel. It combines the advantages of DFM and NCDE in one framework to reach better results. 2) The result of NCDENow showed significant improvement of MSE and MAPE compared with existing models on South Korea and UK GDP dataset. 3) The NCDENow also showed higher efficiency and less parameters, while reached a better nowcasting performance. ### Paper weakness 1) The design motivation/philosophy behind the NCDENow architecture is unclear. Is this architecture a pure empirically designed one? 2) The training process of NCDENow is not clear. For instance, how to update the parameters of the neural networks? Would yt (t<=N) be used as labels to train models? 3) The sizes of the dataset seemed to be insufficient. Only two small datasets were tested. I suppose that this is why only one layer of neural networks is enough to reach the best MSE and MAPE. More cases are desired. ### Detailed evaluation and suggestions for authors 1) A Question on the NCDENow framework. As described by the author, the "regression step" does not train directly the coefficients in this step, so it may cause some confusion if this step is termed "regression". 2) The sizes of the dataset seemed to be insufficient. Only two small datasets were tested. I suppose that this is why only one layer of neural networks is enough to reach the best MSE and MAPE. More cases are desired. ### Questions for authors for discussion 1) How to update the parameters of the neural networks? Would yt (t<=N) be used as labels to train models? 2) A question considering the results in table 8: the MAPE seems to be large (57%) even for NCDENow for the sudden drop. How to comment on this result? 3) What is the experiment setting for RQ3? **Overall evaluation:** 3 **Confidence level:** 1 ## Reviewer tRk4 (2024-03-21) **Paper category:** 4 ### Review The paper is technically dense but well-structured, presenting a clear methodology and substantial experimental validation. The originality of combining DFMs with NCDEs for nowcasting GDP represents a significant contribution to the literature, addressing limitations in capturing economic uncertainties and irregular dynamics. Nonetheless, the paper could have better contextualized its contributions within the existing body of work. While the proposed NCDENow framework shows potential in improving nowcasting accuracy and handling irregular time-series data, its significance is somewhat overshadowed by the lack of demonstrated real-world deployment or impact. **Pros** - Innovative integration of methodologies. - Demonstrated real-world applicability and superior performance. - Extensive experimental evaluation and comparison. **Cons** - Limited theoretical depth and discussion on assumptions. - Focus on specific datasets limits generalizability. - Complexity of the approach may hinder broader accessibility. **Influence:** 3 **Novelty:** 4 **Technical soundness:** 5 **Impact:** 5 **Presentation:** 3 **Reproducibility:** 5 ### Summary The paper presents NCDENow, a promising framework that leverages the strengths of DFMs and NCDEs for GDP nowcasting, aiming to address challenges related to economic uncertainties and data irregularities. While it demonstrates methodological innovation and improved forecasting accuracy through experimental validation, it falls short in evidencing real-world deployment and impact, limiting its significance within the applied data science context. ### Paper strength - The paper presents a novel approach by combining NCDEs with traditional DFM, enhancing the model's capability to handle irregular time series data and capture economic uncertainties. - Extensive experiments conducted on real-world datasets from South Korea and the United Kingdom, demonstrating the model's superior performance over existing benchmarks. - By addressing the specific challenge of GDP nowcasting, the paper contributes valuable insights that can assist policymakers and analysts in making informed decisions based on timely and accurate economic indicators. ### Paper weakness - The paper could provide a deeper theoretical foundation for the choice of integrating NCDEs with DFM, explaining why this combination is particularly suited for addressing the challenges of nowcasting GDP. - While the model shows promising results on the datasets used, the paper could benefit from a discussion on its applicability to other types of economic data or different geographic regions. - More detailed information on model architecture, hyperparameters, and training details could enhance the reproducibility of the results. Additionally, discussing how the model performs with varying amounts of available data could provide insights into its robustness. ### Detailed evaluation and suggestions for authors - Authors are encouraged to provide more theoretical insights into the integration of NCDEs with DFM, expand on the model's applicability to other economic indicators or regions. - Conducting additional experiments across more diverse economic contexts. - A comparative analysis with existing models across a broader range of economic conditions would strengthen the paper's contributions. - Enhance the paper's reproducibility through more detailed model documentation and analysis on data variability. ### Questions for authors for discussion - Can the authors provide additional theoretical background on the integration of NCDEs with DFM? - Could you elaborate on the robustness of the model in scenarios with varying data availability or quality? - Could the authors detail the preprocessing steps for handling irregular time series data, particularly any data imputation or transformation techniques employed before applying the NCDE-DFM framework? - Has any sensitivity analysis been conducted to assess the impact of key parameters on the forecasting performance of NCDENow? How sensitive is the model to changes in its architecture or hyperparameters? - Are there limitations or challenges in applying NCDENow that were not discussed, and how might these affect its broader applicability? **Overall evaluation:** 3 **Confidence level:** 4 ## Reviewer 5y5M (2024-03-22) **Paper category:** 1 ### Review The research is interesting and meaningful. The quality is good and the manuscript has some innovations. The proposed model is based on existing related researches, and the innovations should be further illsturated to demonistrate the contribution of the research in a clearer way. There are many other researches that can detect abnormal fluctuations in large-scale time series, the research should discuss the unique advantages of the proposed model between thoses researches. The illustration of the research could be further improved. For example, the mechanism of NCDE to detect idiosyncratic disturbances; The domain expert knowledge data for factor group information data and etc. **Influence:** 2 **Novelty:** 3 **Technical soundness:** 4 **Impact:** 3 **Presentation:** 3 **Reproducibility:** 4 ### Summary DFM suffers from two issues: i) the lack of capturing economic uncertainties such as sudden recessions or booms; and ii) the limitation of capturing irregular dynamics from data of mixed frequency. To overcome the above two challenges, the research introduce a new GDP Nowcasting framework called NCDENow , which combines neural controlled differential equations (NCDEs) to DFM, which effectively handles the dynamics of irregular time series. NCDENow consists of two main parts: i) one extracts factors reflecting the economic conditions based on DFM, and ii) the other captures temporal dependencies of the impact of these factors on GDP growth based on NCDE. The research evaluates NCDENow against 6 baselines on 2 real-world GDP datasets from South Korea and United Kingdom, demonstrating its enhanced predictive capability. The empirical results favor our method, highlighting the significant potential of integrating NCDE into nowcasting models. ### Paper strength The research is meaningful and interesting. The method of combining DFM with NCDEs is novel. The experiment is solid and can better support the effectiveness of the proposed model. ### Paper weakness The proposed model is based on existing related researches, and the innovations should be further illsturated to demonistrate the contribution of the research in a clearer way. There are many other researches that can detect abnormal fluctuations in large-scale time series, the research should discuss the unique advantages of the proposed model between thoses researches. The illustration of the research could be further improved. For example, the mechanism of NCDE to detect idiosyncratic disturbances; The domain expert knowledge data for factor group information data and etc. The research is suggested to add new content to evaluate the effectivenss of the proposed model in real applications. **Detailed evaluation and suggestions for authors:** Please see the content in paper weakness. **Questions for authors for discussion:** Please see the content in paper weakness. **Overall evaluation:** 3 **Confidence level:** 3 ## Reviewer bngv (2024-04-03) **Paper category:** 4 ### Review The paper proposes a novel framework called NCDENow, which integrates dynamic factor models (DFM) with neural controlled differential equations (NCDE) for GDP nowcasting. It is well-written and easy to understand, and the problem of nowcasting GDP is important. However, my major concern is that the novelty of the proposed method is insufficient. **Influence:** 2 **Novelty:** 2 **Technical soundness:** 4 **Impact:** 3 **Presentation:** 3 **Reproducibility:** 4 ### Summary The paper proposes a novel framework called NCDENow, which integrates dynamic factor models (DFM) with neural controlled differential equations (NCDE) for GDP nowcasting. While the methodology demonstrates improved performance in handling irregular data and capturing economic uncertainties compared to baselines, it lacks novelty in its extension of DFM and relies heavily on NCDE for addressing irregular dynamics. Further, the paper could benefit from presenting more innovative designs built upon NCDE. ### Paper strength 1. This paper is well-written and easy to understand. 2. The problem of nowcasting GDP is important. 3. Experiments are condudcted by comparing with six baselines. ### Paper weakness 1. The noverty of the proposed method is limited. 2. The proposed method does not significantly distinguish itself from NCDE. 3. Discussion on design Innovations is not sufficient. ### Detailed evaluation and suggestions for authors 1. The proposed method seemly simply integrate NCDE with DFM. The extension seems relatively straightforward, lacking significant innovation beyond leveraging NCDE's capabilities. 2. The paper heavily relies on NCDE for handling irregular dynamics, without introducing substantial enhancements or innovations within the NCDE framework itself. 3. The paper lacks in-depth discussions on innovative designs or adaptations within the NCDE framework that could further distinguish NCDENow from existing approaches, potentially limiting its novelty and significance. ### Questions for authors for discussion 1. Can you elaborate on any novel designs or adaptations within the NCDE framework that differentiate NCDENow from existing approaches? How does NCDENow leverage the unique capabilities of NCDE beyond standard applications? 2. Given the comment about the reliance on NCDE for addressing irregular dynamics, are there additional enhancements or innovations within the NCDE framework that could further strengthen the proposed method's performance and novelty? 3. How does NCDENow address potential limitations or challenges associated with the integration of DFM and NCDE? Are there specific considerations or strategies employed to ensure the synergy and effectiveness of these two components in the nowcasting process? **Overall evaluation:** 2 **Confidence level:** 2 ## Discussion, meta-review and decision (in time order) ### Rebuttal -- Authors (2024-04-12) ### Rebuttal We sincerely thank you for the review and feedback. Below, we address each of your concerns and questions: --- **Q1. Can you elaborate on any novel designs or adaptations within the NCDE framework that differentiate NCDENow from existing approaches? How does NCDENow leverage the unique capabilities of NCDE beyond standard applications?** Our novelty is the integration of the DFM with the NCDE framework. NCDENow can be seen as a unification of CDEs that generalize recurrent models. It can also be linked to linear CDEs, which can theoretically generalize state-space models[1] (e.g., S4[2], Mamba[3]), which have recently gained prominence. Thus, NCDENow encompasses traditional recurrent models and, according to [1], is more expressive than traditional recurrent models, while also leveraging the strengths of DFMs. As you know, our NCDENow is based on the NCDE framework, and we had the design goal of integrating it with DFM as a loose dependency. We plan to integrate the Kalman filter of a DFM into NCDE in our future work. The advantage of our NCDENow is that it is a framework that can use not only the DFM we used, but also other kinds of DFM methods, and our method is extensible in this respect. > [1] Cirone, Nicola Muca, et al. "Theoretical Foundations of Deep Selective State-Space Models." arXiv preprint arXiv:2402.19047 (2024). > > [2] Gu, Albert, Karan Goel, and Christopher Ré. "Efficiently modeling long sequences with structured state spaces." arXiv preprint arXiv:2111.00396 (2021). > > [3] Gu, Albert, and Tri Dao. "Mamba: Linear-time sequence modeling with selective state spaces." arXiv preprint arXiv:2312.00752 (2023). **Q2. Given the comment about the reliance on NCDE for addressing irregular dynamics, are there additional enhancements or innovations within the NCDE framework that could further strengthen the proposed method's performance and novelty?** Our NCDENow focuses on integrating the respective strengths of NCDE and DFM. Although we mention it as future work in the conclusion section, our goal of NCDENow in this paper is not our intention to replace the functions of NCDE with the algorithms of DFM. Instead, our NCDENow has the advantage of being able to use some kind of DFM with NCDE, which can generalize recurrent models. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **Q3. How does NCDENow address potential limitations or challenges associated with the integration of DFM and NCDE? Are there specific considerations or strategies employed to ensure the synergy and effectiveness of these two components in the nowcasting process?** NCDE has the advantage of effectively modeling non-linear and irregular time series data, but it may not sufficiently reflect economic theory. On the other hand, DFM has the advantage of capturing correlations between economic indicators well, but is limited by linearity and normality assumptions. Utilizing the strengths and weaknesses of each of these models to complement each other is an effective approach to improving overall prediction performance. As a strategy for this, the NCDENow framework clearly distinguishes between the roles of DFM and NCDE. DFM extracts latent factors by taking advantage of the ability to capture correlations between economic indicators, and NCDE models non-linear and irregular characteristics well to derive idiosyncratic disturbances and factor loadings. By integrating the results derived from these two modules in the regression module, the advantages of DFM and NCDE can be effectively combined to improve overall prediction performance. This design is expected to maximize the synergy effect between NCDE and DFM and enable effective GDP nowcasts that reflect both economic theory and non-linear dynamics. **W1. The novelty of the proposed method is limited.** Please refer to our response for Q1. **W2. The proposed method does not significantly distinguish itself from NCDE.** Please refer to our response for Q2. **W3. Discussion on design Innovations is not sufficient.** Please refer to our response for Q3. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal We sincerely thank you for the review and positive feedback. Below, we address each of your concerns and questions: **W1. The proposed model is based on existing related researches, and the innovations should be further illsturated to demonistrate the contribution of the research in a clearer way. There are many other researches that can detect abnormal fluctuations in large-scale time series, the research should discuss the unique advantages of the proposed model between thoses researches. The illustration of the research could be further improved. For example, the mechanism of NCDE to detect idiosyncratic disturbances; The domain expert knowledge data for factor group information data and etc. The research is suggested to add new content to evaluate the effectivenss of the proposed model in real applications.** (1) The contribution of the research GDP nowcasting has been studied based on DFM models. Our novelty is that to the best of our knowledge, no research has been done to integrate Neural CDEs, which can generalize recurrent neural networks, with DFMs, which have strengths in economic indicator data. Our contribution is that our NCDENow framework can be used in conjunction with many other variants of DFMs. This implies a promising extensibility of NCDENow, which is our strength. We would like to explain that time series abnormal detection models and NCDENow have different objectives as you mentioned. The main difference between NCDENow and time series abnormal detection models is their respective goals and applications. While time series abnormal detection models are designed to identify unusual patterns or outliers in the data that may indicate errors or new events, NCDENow was developed specifically for GDP nowcasting. Our model aims to accurately nowcast current economic conditions using a combination of DFM and NCDE, and focuses on capturing and interpreting the underlying economic dynamics rather than detecting anomalies. In addition, unlike anomaly detection models that focus on pattern recognition and identifying deviations, NCDENow leverages NCDE to model the ongoing dynamics of economic indicators, providing a deeper understanding of the temporal dependence and the impact of macroeconomic factors on GDP growth. And when we say that we are good at capturing abrupt fluctuations in **Figure 1**, we mean that DFM is a linear state-space model, whereas NCDE has the advantage of compensating for the non-linearity of these fluctuations. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal ***The following continues from the previous rebuttal for W1.*** (2) The mechanism of NCDE to detect idiosyncratic disturbances As shown in **Eq. (4)**, the exposure estimation module in NCDENow uses an augmented NCDE to estimate factor exposures and idiosyncratic disturbances. The module uses the cubic spline method to generate continuous paths from economic indicators. From these paths, the NCDE continuously learns how economic indicators change over time and identifies and captures idiosyncratic disturbances, which are unexpected fluctuations. This mechanism effectively detects idiosyncratic disturbances in complex time series, enabling more precise GDP nowcasting. (3) The economic domain expert knowledge The dataset of economic indicators for South Korea was selected, taking into account its economic structure. In South Korea, manufacturing and imports and exports account for a large share of the economy, and we included economic indicators such as manufacturing production and shipments, and the size of exports and imports, as in previous work [1]. We have also classified the economic indicators into four blocks (global, real, soft, and labor) as in the previous work [1]: the global group includes all economic indicators; the real group includes indicators related to the real economy; the soft group includes survey data, considering the importance of soft indicators in GDP nowcasting [2]; the labor group consists of indicators related to the labor market. In our project, we are constantly analyzing the characteristics of the economic indicators in order to update the dataset. For example, we have recently been working with economic experts to correct measurement errors in the ‘Number of job openings per unemployed person’ indicator. We hope our response will address your concerns. > [1] Hyun Chang Yi, Dongkyu Choi, and Yonggun Kim. 2022. Dynamic Factor Model and Deep Learning Algorithm for GDP Nowcasting. Issue Note 28 (2022). > > [2] KURZ‐KIM, Jeong‐Ryeol. A note on the predictive power of survey data in nowcasting euro area GDP. Journal of Forecasting, 2019, 38.6: 489-503. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal We sincerely thank you for the review and positive feedback. Below, we address each of your concerns and questions: --- **Q1. Can the authors provide additional theoretical background on the integration of NCDEs with DFM?** Below, we describe the theoretical background of the integration of NCDE and DFM. The DFM is expressed in state-space form and can be estimated using Kalman filters and multiple solution algorithms. While DFMs have been well studied on data using economic indicators, they may be insufficient to model complex patterns or rapid changes in data due to the lack of non-linear activation. A possible complement to this is the use of Neural CDEs, which can theoretically generalize not only RNNs but also state-space models [1], which have recently gained attention. Thus, our NCDENow is a framework that utilizes both the advantages of NCDEs, which are more expressive than traditional RNN-based models [1], and the advantages of DFMs, which are complementary to each other. Also, our NCDENow has the advantage of being an extensible framework that can be extended to other kinds of DFM methods in addition to the DFM we used. > [1] Cirone, Nicola Muca, et al. "Theoretical Foundations of Deep Selective State-Space Models." arXiv preprint arXiv:2402.19047 (2024). ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **Q2. Could you elaborate on the robustness of the model in scenarios with varying data availability or quality?** (1) Robustness of the model with respect to data quality To check the robustness of NCDENow with respect to data quality, we performed GDP nowcasting after arbitrarily removing a portion of the economic indicators (10% and 20%). The results are shown in **Table 7**. Even with poor data quality, NCDENow outperformed DFM, the leading GDP nowcasting model used by many central banks around the world. (2) Robustness of the model with respect to data availability We used economic indicators from South Korea and the UK to nowcast the GDP growth rate of each country. The South Korean economic indicators were available through a web service provided by the Bank of Korea [1]. However, some of the UK economic indicators used in previous work [2], such as the PMI construction and the UK focused equity index, were unavailable due to their proprietary nature. Nevertheless, in both scenarios, NCDENow outperforms all baseline models (including DFM) in terms of MSE and MAPE, demonstrating the robustness of NCDENow with respect to data availability. In the future, we will consult with government agencies in South Korea that are participating in this project to find ways to use high-frequency data collected by financial authorities, which will allow us to conduct various experiments on quantitative data. In addition, our project is considering applying NCDENow to exchange rate forecasting to enhance its usefulness. We will provide more details on this in our response to question 5. > [1] [Bank of Korea: Economic Statistics System](https://ecos.bok.or.kr) > > [2] Anesti, Nikoleta, Ana Beatriz Galvão, and Silvia Miranda-Agrippino. "Uncertain Kingdom: nowcasting GDP and its revisions." (2018). ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **Q3. Could the authors detail the preprocessing steps for handling irregular time series data, particularly any data imputation or transformation techniques employed before applying the NCDE-DFM framework?** (1) Data imputation The exposure estimation module uses the cubic spline method. This interpolation method imputes missing values or outliers by connecting data points with a curve between them. It has the advantage of preserving the natural dynamics of time series. (2) Data transformation The factor extraction module uses two different types of transformation methods (differencing or log differencing), depending on the nature of the economic indicator. For some indicators, such as GDP growth, we use the raw values. In our paper, **Table 2** shows the transformation method for each economic indicator. - Differencing: $\Delta D_n = D_n - D_{n-1}$ - Log Differencing: $\Delta \log(D_n) = \log(D_n) - \log(D_{n-1})$ ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **Q4. Has any sensitivity analysis been conducted to assess the impact of key parameters on the forecasting performance of NCDENow? How sensitive is the model to changes in its architecture or hyperparameters?** (1) ODE Solver In our paper, **Table 9** shows the GDP nowcasting results when using the Euler method and the 4th order Runge-Kutta (RK4) method. For all datasets, the RK4 method performs better than the Euler method because it performs more precise operations. For example, the Euler method performs only one computation at each time step, while the RK4 method performs four computations and then uses the average value. The RK4 method is known to be more accurate for solving nonlinear differential equations [1]. Because it is specifically designed for various types of differential equations, it provides better accuracy than the simplest method, the Euler method. > [1] Islam, Md Amirul. "A comparative study on numerical solutions of initial value problems (IVP) for ordinary differential equations (ODE) with Euler and Runge Kutta Methods." American Journal of computational mathematics 5.03 (2015): 393-404. (2) Number of hidden layers In our paper, **Table 10** shows how the performance of NCDENow varies depending on the number of hidden layers in the exposure estimation module. For the South Korea dataset, the lowest MSE was achieved when using a single layer, and for the UK dataset, the lowest MSE was obtained when using three layers. (3) Dimension of hidden layer Although we could not include it due to the page limit, we also carried out a sensitivity analysis on the dimension of the hidden layer. The results are as follows: | | Korean GDP | | UK GDP | | |:---:|:----------:|:------:|:------:|:------:| | D | MSE | MAPE | MSE | MAPE | | 4 | 0.6390 | **0.4763** | 0.0186 | 0.2312 | | 8 | 0.6398 | 0.5803 | 0.0121 | 0.2304 | | 16 | 0.5987 | 0.6582 | 0.0168 | 0.2024 | | 32 | 0.5620 | 0.5096 | 0.0145 | 0.2359 | | 64 | 0.6164 | 0.6193 | 0.0146 | 0.2171 | | 128 | **0.5104** | 0.7049 | 0.0142 | 0.1988 | | 256 | 0.5191 | 0.5808 | **0.0070** | **0.1385** | | 512 | 0.5552 | 0.8736 | 0.0146 | 0.1979 | The South Korea dataset had the lowest MSE when the size of the hidden dimension was 128, and the UK dataset had the lowest MSE and MAPE when the size of the hidden dimension was 256. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **Q5. Are there limitations or challenges in applying NCDENow that were not discussed, and how might these affect its broader applicability?** (1) Nowcasting of other economic indicators We are currently considering expanding the use of the NCDENow model to nowcast the KRW-USD exchange rate. Economic experts are involved in collecting various daily or monthly economic indicators that affect the exchange rate. For example, the economic indicators relevant to the KRW-USD exchange rate are shown in the following table. | Group | Economic indicators | |:-------------------:|:--------------------------------------------------------------------:| | U.S. Interest Rates | 1-year / 2-year / 10-year treasury bond yield | | South Korea Interest Rates | 1-year / 2-year / 10-year treasury bond yield, 3-year corporate bond | | South Korea Stock Market | Cumulative net purchase amount by foreigners in KOSPI | | Commodities | WTI (West Texas Intermediate crude oil price) | | Exchange Rate | Dollar index, KRW-USD exchange rate | We are also working on structuring appropriate factor groups considering the characteristics of each economic indicator. Once this work is completed, we expect NCDENow to be used to nowcast the KRW-USD exchange rate. (2) GDP Nowcasting for other countries As mentioned in **Section 5**, a limitation of our study is that we focused only on South Korea and the United Kingdom. Therefore, in the future, we will extend our experiments to nowcast GDP growth for a variety of countries. In particular, we will consider datasets from countries whose economic indicators tend to be volatile. (3) Model Architecture In NCDENow, the factor extraction module based on the Kalman filter and smoother extracts factors by economic indicator group, and the exposure estimation module based on NCDE estimates factor exposures and idiosyncratic disturbances. As mentioned in **Section 5**, we will attempt to use the mechanisms of NCDE to model the continuous modelling of the factor estimation process within the Kalman filter and smoother. Through this work, we hope to explore the potential of the NCDENow model for GDP nowcasting. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **W1. The paper could provide a deeper theoretical foundation for the choice of integrating NCDEs with DFM, explaining why this combination is particularly suited for addressing the challenges of nowcasting GDP.** Please refer to our respones for Q1. **W2. While the model shows promising results on the datasets used, the paper could benefit from a discussion on its applicability to other types of economic data or different geographic regions.** Please refer to our respones for Q5. **W3. More detailed information on model architecture, hyperparameters, and training details could enhance the reproducibility of the results. Additionally, discussing how the model performs with varying amounts of available data could provide insights into its robustness.** (1) Model architecture Please refer to our respones for Q1. (2) Hyperparameter settings Please refer to our respones for Q4. (3) Amounts of data As the data collected in this project are monthly/quarterly economic indicators, the amount of time series data is not abundant. Therefore, we will consult with Korean government agencies participating in this project to find ways to use high-frequency data, and we will conduct various experiments regarding the amount of data in the future. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal We sincerely thank you for the review and positive feedback. Below, we address each of your concerns and questions: --- **Q1. How to update the parameters of the neural networks? Would yt (t<=N) be used as labels to train models?** (1) Updating neural network parameters For training the neural network, please refer to **Algorithm 1** in **Section 3.5**, where line 11 describes updating the parameters. (2) Labels to train models If the GDP growth rate we want to nowcast is at time $N+1$, as shown in **Figure 3**, we use the GDP growth rate up to time $N$ and related economic indicators up to time $N+1$ for training. The label (ground truth) you mentioned is the GDP growth rate at time $N+1$. This learning method is about the concept of nowcasting; please refer to **Figure 2**. **Q2. A question considering the results in table 8: the MAPE seems to be large (57%) even for NCDENow for the sudden drop. How to comment on this result?** We apologize for the confusion. There was an error in the calculation of the MSE and MAPE for the COVID-19 period (2020Q1, 2020Q2, 2020Q3) that caused the 2020Q3 forecast to be missing. As shown in the table below, the corrected MAPE for NCDENow decreases to 53.87%, while the MAPE for DFM increases to 99.55%. The hyperparameters are the best settings based on the MSE, which are the same as those described in **Table 3**. | COVID-19 | Ground Truth | NCDENow | DFM | |------|:-----:|:------:|:-------:| | 2020Q1 | -1.2615067 | 0.1034569 | 0.10668433 | | 2020Q2 | -3.1507382 | -3.3880901 | -1.14140557 | | 2020Q3 | 2.2334244 | 1.2085018 | -0.5897278 | | MSE | - | **0.9900** | 4.6265 | | MAPE | - | **0.5387** | 0.9954 | The best setting based on MAPE (learning rate: 0.01, hidden dimension: 512, # layers: 3) results in a performance of 44.53%, which you could also consider poor. However, we can see from this study that the performance can be significantly improved compared to DFM, which is used by several central banks for GDP nowcasting. **Q3. What is the experiment setting for RQ3?** The experimental setting for RQ3 is identical to that shown in **Table 5**. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **W1. The design motivation/philosophy behind the NCDENow architecture is unclear. Is this architecture a pure empirically designed one?** Our architecture is not an empirical construct but a fusion of DFM and NCDE, drawing from a theoretical background[1,2]. The novelty of NCDENow lies in integrating DFM with NCDE's advanced handling of complex, non-linear dynamics. This integration is motivated by the recognition that traditional DFMs, while powerful for linear analysis, struggle to capture complex patterns and rapid economic changes. NCDEs, generalizing over RNNs, offer a more expressive framework. Our strategic design aims to synergize the interpretability of DFMs with the temporal modeling capabilities of NCDEs. This is not an ad-hoc combination but a thoughtful response to the limitations of existing nowcasting models, with a clear goal to enhance predictive accuracy and adaptability. Also, our NCDENow can incorporate diverse DFM methods, highlighting the extensibility of NCDENow. > [1] Cirone, Nicola Muca, et al. "Theoretical Foundations of Deep Selective State-Space Models." arXiv preprint arXiv:2402.19047 (2024). > > [2] Kidger, Patrick, et al. “Neural controlled differential equations for irregular time series.” NeurIPS. 2020. **W2. The training process of NCDENow is not clear. For instance, how to update the parameters of the neural networks? Would yt (t<=N) be used as labels to train models?** Please refer to our response for Q1. **W3. The sizes of the dataset seemed to be insufficient. Only two small datasets were tested. I suppose that this is why only one layer of neural networks is enough to reach the best MSE and MAPE. More cases are desired.** Since the release frequency of the economic indicators used in our project is monthly or quarterly, the size of the dataset is not sufficient. Therefore, in future work, we will consult with the Korean government agencies involved in this project to find a way to use high-frequency data. ### Rebuttal -- Authors (2024-04-12) ### Rebuttal **S1. A Question on the NCDENow framework. As described by the author, the "regression step" does not train directly the coefficients in this step, so it may cause some confusion if this step is termed "regression".** We apologize for the confusion. We will change the terminology to **"fusion module"** in our updated version. **S2. The sizes of the dataset seemed to be insufficient. Only two small datasets were tested. I suppose that this is why only one layer of neural networks is enough to reach the best MSE and MAPE. More cases are desired.** Please refer to our response for W3. ### Official Comment -- Authors (2024-04-19) ### Comment Dear Reviewer kQMY, as the discussion period comes to a close, we would like to thank you once again for your positive assessment. Your support and comments have been invaluable to us. We remain open and eager to incorporate any further feedback or insights you might offer. ### Official Comment -- Authors (2024-04-19) ### Comment Dear Reviewer tRk4, as the discussion period comes to a close, we would like to thank you once again for your positive assessment. Your support and comments have been invaluable to us. We remain open and eager to incorporate any further feedback or insights you might offer. ### Official Comment -- Authors (2024-04-19) ### Comment Dear Reviewer 5y5M, as the discussion period comes to a close, we would like to thank you once again for your positive assessment. Your support and comments have been invaluable to us. We remain open and eager to incorporate any further feedback or insights you might offer. ### Official Comment -- Authors (2024-04-19) ### Comment Dear Reviewer bngv, as the discussion period comes to a close, we would like to thank you once again for your insightful comments have been invaluable to us. We remain open and eager to incorporate any further questions or suggestions you might offer. ### Meta Review -- Program Chairs (2024-05-14) ### Meta review The reviewers largely agree on the overall merit of the work but also point at weaknesses. Our primary concern is the very limited applicability of the insights from this research outside the very specific application to GDP prediction. For the specific goal of GDP nowcasting for the purpose of policy adjustments, it is unclear whether the approach is in fact used as of today and the value those improvements may have had. For an applied paper, some more color around the utility and impact of a MAPE/MSE reduction by 50% would be important - as well as a justification of either MAPE or MSE as the most relevant metric given that both assume symmetric impact of both over and underprediction. All those application relevant nuances are unfortunately not presented. **Recommendation:** 1 **Confidence:** 3 **Paper category:** 1 ### Meta Review -- Area Chair JPEo (2024-05-14) ### Meta review Reviews are quite heterogeneous (both in terms of reviewer confidence and specific questions/observations). Overall, there is good consensus on the relevance on the problem, and appreciation for the solid experimentation methodology and evaluation against multiple baselines (although in this specific case the available real world datasets are theoretically too small for an accurate evaluation of the proposed algorithm). The paper is well written and all the questions of the reviewers have been answered. As noted by one reviewer, the paper may be slightly out-of-scope for the ADS track, since it shows good quality results validated on real GDP data of two specific countries, but it falls short in evidencing significant real-world deployment and impact, and does not introduce a benchmark dataset. **Recommendation:** 1 **Confidence:** 2 **Paper category:** 3 ### Decision -- Program Chairs (2024-05-16) **Decision:** Revise and Resubmit