Method and device for constructing time-varying frequency mixing dynamic factor model, electronic equipment and medium

Abstract

The application discloses a kind of construction method, device, electronic equipment and medium of time-varying mixing dynamic factor model, method includes: based on logarithmic comparison high-frequency time series vector construction model basic structure, model basic structure includes signal component and noise component, signal is the product of time-varying factor load coefficient matrix and common factor vector;Based on model basic structure respectively establishes time-varying factor load updating rule and common factor updating rule;Based on model basic structure and time-varying factor load updating rule and common factor updating rule, construct equivalent linear Gaussian state space model;Based on the observation data of economic variable, equivalent linear Gaussian state space model is solved to obtain model parameter estimation result;Based on model parameter estimation result, construct time-varying parameter confidence interval.The present application can capture the time-varying characteristics of economic structure and the high-frequency information of mixed frequency data in time, accurately decompose signal and noise, ensure the reliability of parameter estimation and the accuracy of economic trend description.

Description

Technical Field

[0001] This invention belongs to the field of economic monitoring technology, specifically relating to the construction method, apparatus, electronic equipment, and medium of time-varying mixed-frequency dynamic factor model. Background Technology

[0002] Economic monitoring can effectively predict economic fluctuations. In order to more accurately capture short-term changes in economic indicators and provide real-time decision support for macroeconomic regulation, a few studies have begun to explore real-time monitoring methods with higher frequencies. For example, a mixed-frequency dynamic factor model has been constructed based on the month-on-month growth rate series of price indicators at three frequencies: daily, weekly, and monthly, to track, monitor, and predict changes in inflation in real time.

[0003] However, in time-varying parameter mixing dynamic factor models (TVP-MFDFM), factor loadings and common factors change simultaneously over time. This characteristic significantly increases the complexity of model specification and thus poses challenges to model estimation. The main reason is that both time-varying factor loadings and time-varying common factors are unobservable processes, and their interaction makes model estimation particularly complex. Estimation methods relying on numerical simulations incur extremely heavy computational burdens, as they require drawing tens of thousands of samples from the conditional posterior to obtain the empirical posterior distribution of the unknown model parameters. Although dynamic factor models themselves represent dimensionality reduction, Bayesian methods still face computational bottlenecks when the number of variables or sample size is extremely large. Especially when considering mixed data, the rapidly increasing state space dimensionality often makes such sampling methods difficult to implement.

[0004] Therefore, it is evident that the existing model specification and estimation method for TVP-MFDFM both need further improvement. Summary of the Invention

[0005] The purpose of this invention is to overcome the limitations of existing time-varying parameter mixing dynamic factor models in terms of model setting and computational complexity, and to provide a method, apparatus, electronic device and medium for constructing time-varying mixing dynamic factor models with more timely and accurate monitoring results.

[0006] To achieve the above objectives, this invention proposes a method for constructing a time-varying mixed-frequency dynamic factor model, comprising: constructing a basic model structure based on a logarithmic year-on-year high-frequency time series vector, wherein the basic model structure includes a signal component and a noise component, and the signal is the product of a time-varying factor loading coefficient matrix and a common factor vector; establishing time-varying factor loading update rules and common factor update rules based on the basic model structure; constructing an equivalent linear Gaussian state-space model based on the basic model structure and the time-varying factor loading update rules and common factor update rules; solving the equivalent linear Gaussian state-space model based on observed data of economic variables to obtain model parameter estimation results; and constructing time-varying parameter confidence intervals based on the model parameter estimation results.

[0007] In one optional implementation, the expression for the basic structure of the model is: ; In the formula, , is an N-dimensional logarithmic year-on-year high-frequency time series vector; for A common factor vector of dimensions; for A time-varying factor loading coefficient matrix of dimension; As signal components, This is a noise component.

[0008] In one optional implementation, time-varying factor loading update rules and common factor update rules are established based on the basic structure of the model, specifically including: assuming that the common factor follows a p-order vector autoregressive process: , In the formula, for A common factor vector of dimensions; for An autoregressive coefficient matrix of dimension 1; random disturbance term Follow the mean The covariance matrix is ​​the identity matrix. The time-varying factor loading coefficients are defined to follow a normal distribution; the generalized conditional score update process is defined as follows: , In the formula, for A time-varying factor loading coefficient matrix of dimension; It is a diagonal matrix; for Maintain residual terms; As a weighted term; For degrees of freedom; This is a noise component.

[0009] In one alternative implementation, the autoregressive coefficient matrix is ​​a diagonal matrix.

[0010] In one optional implementation, the equivalent linear Gaussian state-space model includes observation equations and state equations. Constructing an equivalent linear Gaussian state-space model based on the basic model structure and the time-varying factor load update rule and common factor update rule specifically includes: constructing observation equations based on the basic model structure and the time-varying factor load update rule. : ; In the formula, The observation matrix; Extend the state vector; The observed perturbation term is used; a state equation is constructed based on the basic model structure and the common factor update rule. : ; In the formula, This is the state transition matrix; This is the state noise term; Let be the covariance matrix of the state noise.

[0011] In one optional implementation, the model parameter estimation results are obtained by solving the equivalent linear Gaussian state-space model based on the observed data of economic variables. Specifically, this includes: constructing a log-likelihood function for the equivalent linear Gaussian state-space model based on the observed data of economic variables, wherein the log-likelihood function is used to quantify the goodness of fit between the model parameter vector and the observed data; maximizing the log-likelihood function using the maximum likelihood estimation method to obtain the maximum likelihood estimate of the model parameters, wherein the maximum likelihood estimate of the model parameters follows an asymptotically normal distribution; and calculating the standard error and sum of the maximum likelihood estimates of the model parameters based on the Hessian matrix of the log-likelihood function with respect to the model parameters. The standard error is used to measure the sampling variation of the estimator. The value is used to test the statistical significance of the parameter estimate; the maximum likelihood estimator, standard error, and... The values ​​are integrated into a set that includes parameter values ​​and statistical reliability indicators to obtain the model parameter estimation results.

[0012] In one optional implementation, constructing a time-varying parameter confidence interval based on the model parameter estimation results specifically includes: repeatedly extracting multiple model parameter vectors based on the maximum likelihood estimate of the model parameters; calculating the corresponding time-varying factor loading estimate based on each model parameter vector; and taking the corresponding quantile of the time-varying factor loading estimate to construct the time-varying parameter confidence interval.

[0013] On the other hand, the present invention also proposes a device for constructing a time-varying mixed-frequency dynamic factor model, characterized by comprising: a basic structure construction module, used to construct a basic structure of the model based on a logarithmic year-on-year high-frequency time series vector, wherein the basic structure of the model includes a signal component and a noise component, and the signal component is the product of a time-varying factor loading coefficient matrix and a common factor vector; an update rule establishment module, used to establish time-varying factor loading update rules and common factor update rules based on the basic structure of the model respectively; an equivalent model transformation module, used to construct an equivalent linear Gaussian state-space model based on the basic structure of the model and the time-varying factor loading update rules and common factor update rules; a parameter estimation module, used to solve the equivalent linear Gaussian state-space model based on observed data of economic variables to obtain model parameter estimation results; and a confidence interval construction module, used to construct time-varying parameter confidence intervals based on the model parameter estimation results.

[0014] On the other hand, the present invention also proposes an electronic device, comprising: at least one processor; a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, the instructions being executed by the at least one processor to enable the at least one processor to perform the method for constructing the time-varying mixing dynamic factor model as described in any one of the present invention.

[0015] On the other hand, the present invention also proposes a medium storing a computer program, which, when executed by a processor, implements the method for constructing the time-varying mixing dynamic factor model as described in any one of the claims.

[0016] The beneficial effects of this invention are as follows: By deeply embedding time-varying factor loads and mixed-frequency data characteristics into the basic structure of the dynamic factor model, and combining customized factor update rules to transform it into an equivalent linear Gaussian state-space model, it not only breaks through the limitations of the traditional dynamic factor model of "static loads + same-frequency data" and achieves seamless integration of multi-frequency economic data, but also solves the computational complexity problem of high-dimensional time-varying models by using the state-space model estimation framework; at the same time, the time-varying load update mechanism adapted to mixed-frequency characteristics makes the dynamic correlation between variables and common factors more in line with the structural changes of the real economic environment, ultimately enabling the model to have higher stability, accuracy and interpretability in economic monitoring, and significantly improving computational efficiency and outlier robustness, while also conveniently introducing high-frequency data to expand the monitoring dimensions. Attached Figure Description

[0017] Figure 1 A flowchart illustrating the method for constructing a time-varying mixing dynamic factor model provided in one embodiment of the present invention. Detailed Implementation

[0018] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0019] like Figure 1 As shown, according to an embodiment of the present invention, in one aspect, a method for constructing a time-varying mixing dynamic factor model is provided, comprising the following steps:

[0020] Step S101: Construct the basic structure of the model based on the logarithmic year-on-year high-frequency time series vector. The basic structure of the model includes signal components and noise components. The signal components are the product of the time-varying factor loading coefficient matrix and the common factor vector.

[0021] Step S103: Establish time-varying factor load update rules and common factor update rules based on the basic structure of the model.

[0022] Step S105: Construct an equivalent linear Gaussian state-space model based on the basic structure of the model and the time-varying factor load update rule and the common factor update rule.

[0023] Step S107: Solve the equivalent linear Gaussian state-space model based on the observed data of economic variables to obtain the model parameter estimation results.

[0024] Step S109: Construct confidence intervals for time-varying parameters based on the model parameter estimation results.

[0025] The model is built on the basis of logarithmic year-on-year high-frequency time series vectors, which naturally fits the fluctuation patterns and statistical characteristics of high-frequency economic data (such as daily interest rates and weekly consumption indices), avoiding the information loss caused by the "down-frequency processing" of high-frequency data in traditional models.

[0026] The dynamic factor model assumes that each observable variable is influenced by two factors: firstly, one or more common factors affecting all variables, which, together with factor loadings, constitute the signal component of the economic variable; secondly, the specific components (or random disturbances) of each observable variable, which are important factors distinguishing each observed variable and constitute the noise component of the variable system. The entire variable system can be viewed as the sum of the signal and noise components. The essence of economic monitoring is to accurately extract the (high-frequency) signal at every moment.

[0027] The system clearly decomposes economic variables into signal components ("time-varying factor loading coefficient matrix × common factor vector") and noise components (variable-specific components). Through structured decomposition, it accurately removes noise interference such as market fluctuations and short-term disturbances, ensuring that the high-frequency signals extracted at each point in time (such as economic growth momentum and inflationary pressure) are closer to the actual operating trend of the variables, and providing reliable data support for real-time macroeconomic monitoring (such as monthly economic prosperity assessment).

[0028] By independently constructing time-varying factor loading update rules and common factor update rules, dynamic adjustment of the model's core parameters can be achieved. Among them, the time-varying factor loading can capture the time-varying characteristics of the correlation between economic variables and common factors (such as aggregate demand and aggregate supply) (such as the change in the contribution of a certain industry to economic growth as policy adjustments change), and the common factor update rules can track the dynamic evolution of the core driving factors of the macroeconomy (such as the phased changes in the pace of demand recovery after the pandemic), breaking through the limitation of the traditional static dynamic factor model with "fixed parameters" that cannot adapt to the transformation of economic structure.

[0029] Whether it's a sudden shock (such as natural disasters or a surge in international commodity prices), a policy shift (such as adjustments to monetary policy), or a long-term trend (such as changes in population structure or technological progress), the model can respond quickly through real-time updates of time-varying parameters, ensuring good explanatory and fitting capabilities in different scenarios such as economic stability, volatility, and transformation.

[0030] The original time-varying mixing dynamic factor model is transformed into a linear Gaussian state-space model, directly adapting to mature state-space model solving algorithms such as Kalman filtering and particle filtering. These algorithms have undergone long-term theoretical verification and practical application, possessing stable convergence and efficient computational logic. They eliminate the need to design complex custom solver programs for the original model, significantly lowering the technical threshold for model development and application.

[0031] The solution framework of the linear Gaussian state-space model can rely on efficient computational logic such as matrix operations and recursive iterations to process massive observations of high-frequency data. Even when facing scenarios with multiple variables (such as covering dozens of economic indicators) and long time series (such as high-frequency data over many years), it can still maintain a fast computation speed. At the same time, the mature algorithm has stronger fault tolerance and robustness, which can reduce the fluctuations in the solution caused by outliers in the data and deviations in the model settings, and ensure the stability of the parameter estimation results.

[0032] Model parameter estimation is based entirely on actual observed data of economic variables. By maximizing the likelihood function of the observed data or minimizing the prediction error, the estimation results strictly match the actual economic operation data, avoiding the bias caused by subjective parameter setting, and ensuring that the parameters (such as factor loading coefficients and common factor mean) can truly reflect the intrinsic relationship between economic variables.

[0033] Building upon parameter estimation, time-varying parameter confidence intervals can be further constructed to clearly quantify the statistical uncertainty of parameter estimation at each point in time (such as the 95% confidence interval range of the inflation factor estimate for a certain period). This design not only helps users judge the reliability of parameter estimation results (e.g., the smaller the confidence interval width, the higher the estimation accuracy), but also provides a risk boundary reference for model-based economic decision-making (such as policy formulation and investment judgment), reducing decision-making errors caused by parameter uncertainty.

[0034] Furthermore, set This is an N-dimensional logarithmic year-on-year high-frequency time series vector. Assume that at each time point... Since all can be observed, the basic structure of the model can be expressed as follows:

[0035] ;

[0036] (1)

[0037] In the formula, , is an N-dimensional logarithmic year-on-year high-frequency time series vector; for A common factor vector of dimensions, which characterizes the common change features of various variables in the system; for A time-varying factor loading coefficient matrix of dimension 3, which characterizes the degree of response of each variable to changes in factors; As signal components, This is a noise component.

[0038] The basic structure of the model is a dynamic factor model with time-varying parameters. The setting of time-varying factor loadings allows for structural changes in the connections between variables and common factors at different times, thus making the model more closely reflect the current complex and volatile economic environment and serving as a source of robustness. Equation (1) assumes a random disturbance term. Follow the mean The covariance matrix is The normal distribution of time-varying parameters is a fundamental assumption of the dynamic factor model.

[0039] The model is based on an N-dimensional logarithmic year-on-year high-frequency time series vector, through... (Signal components) and The explicit decomposition of (noise components) can accurately extract the common systemic change characteristics depicted by common factors from high-frequency economic data such as daily and weekly data, avoiding interference from short-term disturbances and market noise on the core economic trend, and providing clear signal support for real-time macroeconomic monitoring (such as daily liquidity monitoring and weekly consumption prosperity judgment).

[0040] when When it is mixed data, for example, when it is ,in , , and These refer to daily, weekly, monthly, and quarterly variables, respectively. Based on the approximate average relationship of logarithmic year-on-year growth rate under frequency mismatch:

[0041] , ; (2)

[0042] in, For time-frequency observable variables With potential high-frequency variables The relationship between them. For daily mixed data, there are generally... , , and .

[0043] By applying an approximate average relationship to the logarithmic year-on-year growth rates of different frequency data (daily D, weekly W, monthly M, and quarterly Q) using Equation (2), unified modeling of mixed-frequency data is achieved. For example, multiple frequency data such as daily interest rates, weekly retail data, monthly industrial output, and quarterly GDP can be incorporated simultaneously, solving the problem that traditional models cannot integrate and analyze data due to inconsistent data frequencies, and significantly improving the integration of multi-dimensional information in the economic system.

[0044] Time-varying factor loading coefficient matrix The design allows for structural changes in the connections between economic variables and common factors over time. In scenarios involving economic restructuring (such as industrial upgrading), drastic policy adjustments (such as a shift in monetary policy), and external shocks (such as sudden changes in international oil prices), the model can dynamically capture changes in the correlation between variables (e.g., the sensitivity of a particular industry to aggregate demand changes with the strength of policy support). This overcomes the limitations of traditional static dynamic factor models with their "fixed parameters," making the model more closely reflect the complex and ever-changing real-world economic environment and improving the accuracy of economic trend predictions and the model's robustness.

[0045] Assume random disturbance term This conforms to the statistical characteristics of the disturbance term of economic variables, constructs a rigorous probabilistic framework for the model, enables parameter estimates (such as factor loadings and common factors) to have clear statistical significance, and provides a theoretical basis for subsequent statistical inferences such as confidence interval construction and hypothesis testing.

[0046] The model can be equivalently transformed into a linear Gaussian state-space model, relying on mature algorithms such as Kalman filtering for parameter estimation and signal extraction. These algorithms have been validated through long-term practice, exhibiting high computational efficiency and stable convergence. Even when facing complex data scenarios with high frequency and multiple variables, they can still ensure the efficiency and accuracy of parameter estimation, reducing the technical threshold and computational cost of model application.

[0047] To ensure that the TVP-MFDFM model is complete in its specification, the update process for the common factor and time-varying factor loadings needs to be provided.

[0048] Further, in step S103, based on the basic structure of the model, time-varying factor load update rules and common factor update rules are established respectively, specifically including the following steps:

[0049] Step S1031: Define the dynamic update process of the common factors: Assume that the common factors follow a p-order vector autoregressive process:

[0050] , (3)

[0051] In the formula, for A common factor vector of dimensions; for An autoregressive coefficient matrix of dimension 1; random disturbance term Follow the mean The covariance matrix is ​​the identity matrix. The normal distribution is assumed, with the identity matrix assumption primarily for the identifiability of the factor model.

[0052] The common factors follow a p-order vector autoregression process (Equation (3)), through the autoregression coefficient matrix and random disturbance term This dynamically depicts the evolutionary patterns of core driving factors in the economic system (such as aggregate demand and aggregate supply factors). For example, during the economic recovery phase, the autoregressive process of demand factors can capture its dynamic rhythm of "continuous recovery - accelerated growth - gradual stabilization," breaking through the limitation of "fixed factor trends" in traditional static factor models and enabling common factors to reflect the time-varying characteristics of economic driving forces in real time.

[0053] The autoregressive coefficient matrix is ​​a diagonal matrix. When the matrix is ​​diagonal, the autoregressive process of each common factor can be characterized independently. That is, the dynamic evolution of the i-th common factor is only affected by its own lagged terms and is unrelated to the lagged terms of other common factors. This structure makes the dynamic change logic of the economic meaning of each common factor (such as inflation factor and growth factor) clearer. For example, the autoregressive process of the inflation factor can explain the evolution of "inflation inertia" on its own, and the autoregressive process of the growth factor can reflect the "sustainability of growth momentum" on its own. This avoids the problem of "ambiguity of meaning" caused by the dynamic correlation between factors and makes economic analysis based on common factors (such as factor trend interpretation and shock source identification) have a clearer logical chain.

[0054] Common factor random disturbance term It follows that the covariance matrix is ​​the identity matrix. The assumption of a normal distribution provides crucial support for the identifiability of factor models. In complex scenarios with multiple factors and variables, the identity matrix constraint eliminates redundant correlations between factors, ensuring that the economic meaning of each common factor (such as inflation factor and growth factor) can be uniquely identified. This avoids the problems of "ambiguous meaning and overlapping" of factors, giving factor-based economic analysis (such as factor contribution decomposition) a clear logical foundation.

[0055] The time-varying factor loading coefficients are defined to follow a generalized conditional score update process:

[0056] , ; (4)

[0057] In the formula, It is a diagonal matrix, and its first... diagonal elements Corresponding to the The frequency difference between individual variables and daily variables; for The residual term is defined if and only if the corresponding element is... It is not zero when observable; As a weighted term; For degrees of freedom. In other words, The time-varying frequency of is the same as the time-varying frequency of the corresponding variable.

[0058] Through the diagonal matrix The i-th diagonal element This design precisely corresponds to the frequency multiple difference between the i-th variable and the daily variable (e.g., weekly variable m(W)=7, monthly variable m(M)=30, quarterly variable m(Q)=90), achieving quantitative adaptation of different frequency data (daily, weekly, monthly, and quarterly) in the time dimension. This design solves the problem of "information distortion caused by frequency mismatch" in traditional mixed-frequency models, ensuring that high-frequency data such as daily and weekly data and low-frequency data such as monthly and quarterly data can maintain their respective time granularity characteristics during modeling, maximizing the preservation of the original information value of multi-frequency data.

[0059] residual terms Only The mechanism that ensures the corresponding element is not zero when observable enables precise matching of the "partially observed" characteristics of mixed-frequency data. For example, quarterly GDP data is only observable at the end of the quarter. The model automatically identifies and adapts to this type of low-frequency data with "discontinuous observations" through the observability constraint of the residual term, without the need for forced interpolation or down-frequency processing of the data. This avoids information distortion during data preprocessing and improves the accuracy of mixed-frequency data modeling.

[0060] The time-varying factor loading coefficients are dynamically adjusted through the generalized conditional score update process (Equation (4)), and the update amplitude is... With residuals Common factors Frequency adaptation matrix They are closely related. This mechanism allows factor loadings to be adjusted in real time according to the strength of the association between economic variables and common factors. When the contribution of a certain industry to the economic growth factor increases due to policy support, the corresponding factor loadings will respond quickly through this update rule, ensuring that the model's capture of changes in economic structure (such as adjustments in industry weights and changes in policy transmission paths) is timely and accurate.

[0061] The update rules for both time-varying factor loadings and common factors are based on a data-driven dynamic adjustment logic, rather than subjectively set fixed rules. This design enables the model to offset interference through adaptive parameter updates when facing economic shocks (such as the impact of the pandemic or fluctuations in international oil prices) and data noise, maintaining its ability to depict core economic trends. Simultaneously, the introduction of the degree of freedom v provides the model with a "flexible adjustment" space, allowing for optimization of the update magnitude based on data distribution characteristics (such as heavy tails), further enhancing the model's robustness in non-ideal data scenarios.

[0062] Based on equations (1)-(4), the model can be written as an equivalent linear Gaussian state-space expression.

[0063] Step S105: The equivalent linear Gaussian state-space model includes the observation equation and the state equation. Based on the basic model structure and the time-varying factor load update rule and the common factor update rule, an equivalent linear Gaussian state-space model is constructed, specifically including the following steps:

[0064] Step S1051: Construct observation equations based on the basic model structure and time-varying factor loading update rules. :

[0065] , (5)

[0066] In the formula, The observation matrix; Extend the state vector; For the observed perturbation term;

[0067] Step S1053: Construct the state equation based on the basic model structure and common factor update rules. :

[0068] , (6)

[0069] In the formula, This is the state transition matrix; This is the state noise term; Let be the covariance matrix of the state noise.

[0070] in, , This is formed by stacking time-varying factor loads according to the structures of equations (1)-(2). Similarly, and It is a first-order VAR expression formed by stacking the structures of equations (2)-(3).

[0071] Based on the predetermined structure of the time-varying factor load in equation (4) based on the generalized conditional scoring method, and the linear Gaussian state-space model in (5)-(6), the classical Kalman filtering method can be used to obtain... The conditional mean.

[0072] Expand the state vector Defined as a form containing common factors from multiple periods ( ), and through Matching the lag length of data at different frequencies is a customized state vector construction method for the characteristics of mixed-frequency data, which solves the problem that conventional state-space models are difficult to directly accommodate multi-frequency data.

[0073] The time-varying factor loading is transformed into a measurement matrix through a "stacked structure". Furthermore, by combining the generalized conditional scoring method with the pre-defined structure of time-varying factor loads, a seamless embedding of time-varying features and state-space framework is achieved, simplifying the estimation process of time-varying parameters.

[0074] The original time-varying mixed-frequency dynamic factor model is equivalently transformed into a linear Gaussian state-space model (including observation equation (5) and state equation (6)), which is directly adapted to the mature Kalman filter solution algorithm. Kalman filtering has stable recursive iterative logic and efficient computational performance. It does not require the development of a custom solution program for the original model, which greatly reduces the technical threshold of the model in parameter estimation, signal extraction and other aspects, enabling researchers or practitioners to quickly master and apply the model to carry out economic analysis.

[0075] The state-space model's observation and state equations are both linear in structure. Combined with the matrix operations and recursive characteristics of Kalman filtering, it can efficiently process high-frequency, multi-variable economic data (such as macroeconomic monitoring systems containing dozens of daily and weekly indicators). Compared to the custom solution logic of traditional dynamic factor models, its computational time complexity is significantly reduced, and it can maintain a fast computation speed even when dealing with long-term, high-dimensional data.

[0076] Kalman filtering under the linear Gaussian assumption exhibits good robustness and can effectively handle noise, outliers, and small deviations in model specifications. (State transition matrix) and observation matrix The "stacked" construction logic (based on the structure of equations (1)-(3)) further ensures the structural stability of the model in mixed and time-varying scenarios, and avoids convergence failure or result fluctuation caused by structural complexity during the solution process.

[0077] The "generalized conditional score update" structure of time-varying factor loadings is highly compatible with the pre-defined characteristics of the linear Gaussian state-space model, enabling Kalman filtering to naturally adapt to the dynamic adjustment logic of time-varying parameters during the solution process. This compatibility ensures that while the model can characterize time-varying economic structures (such as abrupt changes in the correlation between variables and factors under policy shocks), it can still obtain reliable statistics such as the conditional mean of state variables through classical algorithms, providing a solid quantitative foundation for subsequent analyses such as constructing confidence intervals for time-varying parameters and predicting economic trends.

[0078] Step S107 involves solving an equivalent linear Gaussian state-space model based on observed data of economic variables to obtain model parameter estimation results. This includes the following steps:

[0079] Step S1071: Based on the observed data of economic variables, construct a log-likelihood function for the equivalent linear Gaussian state-space model. The log-likelihood function is used to quantify the fit between the model parameter vector and the observed data.

[0080] Step S1073: Maximize the log-likelihood function using the maximum likelihood estimation method to obtain the maximum likelihood estimate of the model parameters, wherein the maximum likelihood estimate of the model parameters follows an asymptotic normal distribution.

[0081] Step S1075: Based on the Hessian matrix of the log-likelihood function with respect to the model parameters, calculate the standard error of the maximum likelihood estimator of the model parameters. The standard error is used to measure the sampling variation of the estimator. The value is used to test the statistical significance of the parameter estimate.

[0082] Step S1077: Combine the maximum likelihood estimator, standard error, and... The values ​​are integrated into a set that includes parameter values ​​and statistical reliability indicators to obtain the model parameter estimation results.

[0083] In this embodiment, let This is an unknown parameter vector containing all model parameters. Given the initial values ​​of the time-varying factor loading coefficients and... Given the prior distribution, the log-likelihood function of the model can be written as:

[0084] (7)

[0085] Given any point in time , The conditional log-likelihood value is given by equation (7). Given a set of parameters It can be recursively obtained from the score-driven update equation (4) and the Kalman filtering method. and , in Therefore, the log-likelihood function can be calculated. Then, the maximum likelihood estimation method is used to obtain... The maximum likelihood estimator.

[0086] By maximizing the log-likelihood function using the maximum likelihood estimation method (Equation (7)), under the condition of regularity, a consistent estimate of the model parameters (such as time-varying factor loadings, common factor autoregressive coefficients, etc.) can be obtained. That is, when the sample size tends to infinity, the estimate converges to the true parameter value in probability. This characteristic ensures that the parameter estimation results can truly reflect the intrinsic relationship between economic variables, providing a reliable quantitative basis for subsequent economic analysis.

[0087] Based on observed economic variable data, a log-likelihood function is constructed using the probability structure of a linear Gaussian state-space model. This step is fundamental to the entire parameter estimation process—the core function of the log-likelihood function is to quantify the degree of fit between the model parameters and the observed data, providing a clear objective function for subsequent parameter solving. Building upon this, the maximum likelihood estimation method is used to maximize the log-likelihood function, thereby obtaining the maximum likelihood estimator of the model parameters. Simultaneously, it is clarified that this estimator follows an asymptotically normal distribution. This distribution characteristic is not an isolated conclusion but rather serves as a basis for subsequent calculations of the standard error of the parameters. The value provides a key theoretical basis, building a bridge from "parameter point estimation" to "statistical significance test".

[0088] The construction of the log-likelihood function fully integrates the score-driven update rule of time-varying factor loading (Equation (4)) with the recursive logic of Kalman filtering, and can adapt to the complex model structure of "time-varying + mixing + dynamic factors". Even if the model contains multi-dimensional time-varying parameters and multi-frequency data associations, maximum likelihood estimation can still accurately capture the matching relationship between parameters and data by maximizing the likelihood contribution of the data, thus ensuring the effectiveness of the estimation.

[0089] Steps S1071-S1077 establish a standardized process of "log-likelihood function construction → maximum likelihood estimation → standard error calculation," providing clear logical steps for parameter estimation. Researchers can quickly reproduce the estimation process using this process, reducing the technical barriers to model application.

[0090] By combining the recursive nature of Kalman filtering with a score-driven time-varying update rule, maximum likelihood estimation can handle high-frequency, long-term time-series data through a "period-by-period recursive calculation of likelihood contribution". This computational logic avoids the high complexity of traditional global optimization methods, significantly improves the computational efficiency of parameter estimation, and can still complete the solution in a reasonable time even when facing large-scale models containing dozens of economic variables.

[0091] By calculating the standard error of the model parameters using the Hessian matrix, the statistical uncertainty of each model parameter (such as an element of the time-varying factor loading or the autoregressive coefficient of the common factor) can be quantified. This quantification provides a crucial basis for subsequently constructing confidence intervals for time-varying parameters (step S109) and conducting parameter significance tests (such as determining whether the time-varying nature of a factor loading has economic significance). This ensures that model-based economic decisions (such as whether the policy effect is significant) have rigorous statistical support and reduces the decision-making risk caused by parameter uncertainty.

[0092] The maximum likelihood estimators of the model parameters follow an asymptotic normal distribution.

[0093] Based on certain regularization conditions The maximum likelihood estimator follows the following asymptotic normal distribution:

[0094] (8)

[0095] In the formula, For sample size; For the log-likelihood function with respect to The Hessian matrix. In the empirical study, this paper maximizes equation (7) based on a nonlinear optimization algorithm of the BFGS type, and then obtains... The maximum likelihood estimator is obtained, and the standard error and t-value of the model parameters are calculated based on the inverse Hessian estimator of the log-likelihood at the optimal point.

[0096] Based on the fact that the maximum likelihood estimator follows an asymptotic normal distribution under regularity (Equation (8)), a solid theoretical support is provided for the statistical inference of model parameters. Researchers can rely on this distributional characteristic to carry out analyses such as significance tests of parameters (e.g., t-tests) and confidence interval construction (step S109). For example, by judging whether the t-value of the model parameter is greater than the critical value, it can be determined whether the parameter (e.g., a time-varying factor loading) is statistically significantly non-zero, thereby distinguishing between "substantive association" and "random disturbance association" between economic variables and improving the rigor of economic analysis.

[0097] Hessian matrix using log-likelihood function The standard error and t-value of the model parameters are calculated using (or their inverse matrix estimators), ensuring the consistency and unbiasedness of these statistics. In empirical research, the standard error quantifies the dispersion of parameter estimates, and the t-value reflects the statistical significance of the deviation between the parameters and the true values. The combination of the two provides a quantitative basis for judging whether the parameters have economic significance (e.g., when the absolute value of the t-value is greater than 2, the parameter is usually considered statistically significant), making parameter-based economic conclusions (e.g., whether a policy has a significant impact on economic factors) more convincing.

[0098] Based on the Hessian matrix of the log-likelihood function with respect to the parameters, the standard error and t-value of the estimated parameters are calculated. The former is used to measure the degree of fluctuation of the parameter estimate, and the latter is used to test the statistical significance of the parameter estimate. These two indicators are integrated with the previously obtained parameter point estimates to finally form a complete model parameter estimation result containing parameter point estimates, standard error, and t-value.

[0099] A nonlinear optimization algorithm of the BFGS type is employed to maximize the log-likelihood function, offering advantages such as fast convergence and strong stability. This type of algorithm iteratively updates the approximate value of the Hessian matrix using a quasi-Newton method, eliminating the need for complex calculations of the true Hessian matrix. This allows for efficient handling of the "high-dimensional, nonlinear" likelihood optimization problem in time-varying mixed-frequency dynamic factor models. Even with complex models containing a large number of time-varying parameters, it can quickly converge to the maximum likelihood estimator, significantly improving the computational efficiency of parameter estimation and creating conditions for the model's application in practical scenarios (such as real-time economic monitoring).

[0100] Step S109 involves constructing confidence intervals for time-varying parameters based on the model parameter estimation results, specifically including the following steps:

[0101] Step S1091: Repeatedly extract multiple model parameter vectors based on the maximum likelihood estimator of the model parameters.

[0102] Step S1093: Calculate the corresponding time-varying factor loading estimate based on each model parameter vector.

[0103] Step S1095: Take the corresponding quantiles for the time-varying factor loading estimates and construct the confidence intervals for the time-varying parameters.

[0104] For the estimator The confidence interval can be estimated by referring to the method of calculating the confidence interval of time-varying parameters based on numerical simulation scores. as well as Then from M model parameter vectors are repeatedly extracted. Based on this, the time-varying factor loading estimate is calculated. Thus, by constructing the corresponding quantiles from M samplings. The confidence interval.

[0105] In this embodiment, by constructing confidence intervals for time-varying parameters, the statistical uncertainty of model parameters (such as time-varying factor loadings) at each time point can be quantified. For example, the width and location of the 95% confidence interval for a certain time-varying factor loading can intuitively reflect the accuracy (the narrower the interval, the higher the accuracy) and deviation of the parameter estimation (whether the interval contains 0 can determine whether the parameter is statistically significant).

[0106] Confidence intervals for time-varying parameters can be used to verify the statistical significance of time-varying economic structures. If the confidence interval of factor loadings for a certain period is consistently far from 0 and the interval is narrow, it indicates that the change in the correlation between the variable and common factors is statistically robust (e.g., the increased sensitivity of an industry to growth factors is a real structural change, not an estimation error). This characteristic provides a crucial basis for policy making (e.g., determining whether industrial policies trigger substantial structural adjustments) and market analysis (e.g., identifying the real time-varying correlations of asset factors), avoiding decision-making errors caused by parameter uncertainty.

[0107] The numerical simulation method employing "repeatedly extracting model parameter vectors → calculating time-varying factor loadings → taking quantiles" (refer to Blasques et al. (2016)) has two major advantages:

[0108] Reliability: By approximating the asymptotic normal distribution Repeated sampling fully considers the distribution characteristics of parameter estimation, making the coverage probability of the confidence interval closer to the theoretical level (e.g., the actual coverage probability of the 95% confidence interval is close to 95%).

[0109] Flexibility: This method does not rely on complex analytical derivations and can be adapted to complex model structures of "time-varying + mixing + dynamic factors". Even if the model contains high-dimensional time-varying parameters, it can still efficiently construct confidence intervals through numerical simulation, thus expanding the application boundaries of the model in complex economic scenarios.

[0110] This invention utilizes a generalized conditional scoring method to provide an update path for time-varying factor loads in TVP-MFDFM. Under this setting, the likelihood function of the model has a closed-form expression, making the model computationally feasible and thus suitable for modeling and estimating large-dimensional systems. By incorporating the latest quasi-conditional scoring setting, the update path of time-varying factor loads can be robust to the influence of outliers, thereby making the model parameter estimation results more stable and resulting in more stable monitoring performance. Furthermore, a conditional likelihood algorithm for the model is designed, thus providing the maximum likelihood estimation process of the model. A simulation-based method for calculating the confidence interval of time-varying parameters is also provided, thus giving a method for measuring the uncertainty of model parameter estimation results and monitoring prediction results.

[0111] The present invention has achieved the following beneficial technical effects.

[0112] 1. Define low-frequency observation variables using equation (2) With high-frequency latent variables The "approximate average relationship" was determined and specifically set. Match daily / weekly / monthly / quarterly frequency differences, and simultaneously convert these frequency differences through a diagonal matrix. The embedded time-varying factor loading update process (Equation 4) solves the problem that traditional dynamic factor models cannot directly accommodate multi-frequency data, and achieves "seamless integration" of mixed-frequency data in the model, rather than simple frequency alignment or missing value filling.

[0113] 2. Equation (4) loads the time-varying factor. The update is combined with the "generalized conditional score" and the residual term is introduced at the same time. The observation state (only when) (When observable, the values ​​are non-zero), ensuring that the update frequency of time-varying factor loads is consistent with the frequency of the corresponding variables. This customized update mechanism, designed for the dual characteristics of "time-varying + mixed frequency," not only guarantees the dynamic nature of factor loads but also adapts to the observation rhythms of variables with different frequencies, breaking through the limitations of the conventional "uniform frequency update" of time-varying factor loads.

[0114] 3. The high-dimensional time-varying mixing factor model is equivalently transformed into a linear Gaussian state-space model (Equation 5-6), and the state vector is... Defined as a form containing common factors from multiple periods ( Simultaneously, the time-varying factor loading is transformed into a measurement matrix through a "stacked structure". This transformation solves the estimation complexity problem of large-dimensional time-varying mixing models. It achieves recursive estimation of high-dimensional parameters by leveraging the mature framework of Kalman filtering, while preserving the time-varying and mixing characteristics of the model. It is an innovative adaptation of "classic tools + scenario customization".

[0115] In summary, this invention fundamentally improves the stability, accuracy, and interpretability of monitoring results. Compared with existing economic variable monitoring models, the model has higher computational efficiency, stronger robustness to outliers, and its factor loadings, due to their time-varying characteristics, offer greater interpretability. Furthermore, this invention enhances monitoring accuracy. Moreover, it exhibits good scalability; based on this model framework, various high-frequency daily data can be easily incorporated and integrated with weekly, monthly, and quarterly data.

[0116] On the other hand, the present invention also proposes a time-varying mixing dynamic factor model construction device, comprising:

[0117] The basic structure construction module is used to construct the basic structure of the model based on the logarithmic year-on-year high-frequency time series vector. The basic structure of the model includes a signal component and a noise component. The signal component is the product of the time-varying factor loading coefficient matrix and the common factor vector.

[0118] The update rule establishment module is used to establish time-varying factor load update rules and common factor update rules based on the basic structure of the model, respectively.

[0119] The equivalent model conversion module is used to construct an equivalent linear Gaussian state-space model based on the basic structure of the model and the time-varying factor load update rule and the common factor update rule.

[0120] The parameter estimation module is used to solve the equivalent linear Gaussian state-space model based on the observed data of economic variables to obtain the model parameter estimation results;

[0121] The confidence interval construction module is used to construct time-varying parameter confidence intervals based on the model parameter estimation results.

[0122] In the basic structure building module, the logarithmic year-on-year high-frequency time series vector is formed by integrating logarithmic year-on-year sequences of multiple economic indicators with different frequencies.

[0123] In the update rule establishment module, both the time-varying factor load update rule and the common factor update rule adopt a recursive dynamic update algorithm.

[0124] The parameter estimation module uses the maximum likelihood estimation method to solve for the parameters of the linear Gaussian state-space model.

[0125] On the other hand, the present invention also proposes an electronic device comprising: at least one processor; a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, the instructions being executed by the at least one processor to enable the at least one processor to perform any one of the time-varying mixing dynamic factor models.

[0126] On the other hand, the present invention also proposes a computer storage medium storing a computer program, wherein the computer program, when executed by a processor, implements any one of the time-varying mixing dynamic factor models.

[0127] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous DRAM (SDRAM), Dual Data SDRAM (DDRSDRAM), Enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), Rambus Direct RAM (RDRAM), Direct Memory Bus Dynamic RAM (DRDRAM), and Memory Bus Dynamic RAM (RDRAM). The various embodiments described in this specification are presented in a progressive manner, and similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on describing the differences from other embodiments. In particular, for embodiments of apparatus, devices, and non-volatile computer storage media, since they are substantially similar to the method embodiments, the description is relatively simple, and relevant parts can be referred to the description of the method embodiments.

[0128] The above embodiments are merely illustrative examples and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for constructing a time-varying mixing dynamic factor model, characterized in that, include: The basic structure of the model is constructed based on the logarithmic year-on-year high-frequency time series vector. The basic structure of the model includes a signal component and a noise component. The signal component is the product of the time-varying factor loading coefficient matrix and the common factor vector. Based on the basic structure of the model, time-varying factor load update rules and common factor update rules are established respectively. Based on the basic structure of the model and the time-varying factor load update rule and common factor update rule, an equivalent linear Gaussian state-space model is constructed. The model parameter estimation results are obtained by solving the equivalent linear Gaussian state-space model based on the observed data of economic variables. Confidence intervals for time-varying parameters are constructed based on the model parameter estimation results.

2. The method for constructing the time-varying mixing dynamic factor model according to claim 1, characterized in that, The expression for the basic structure of the model is: ； ； In the formula, , is an N-dimensional logarithmic year-on-year high-frequency time series vector; for A common factor vector of dimensions; for A time-varying factor loading coefficient matrix of dimension; As signal components, This is a noise component.

3. The method for constructing the time-varying mixing dynamic factor model according to claim 1, characterized in that, Based on the basic structure of the aforementioned model, time-varying factor loading update rules and common factor update rules are established respectively, specifically including: Assume the common factors follow a p-order vector autoregressive process: ， ； In the formula, for A common factor vector of dimensions; for An autoregressive coefficient matrix of dimension 1; random disturbance term Follow the mean The covariance matrix is ​​the identity matrix. The normal distribution; The time-varying factor loading coefficients are defined to follow a generalized conditional score update process: ； ； In the formula, for A time-varying factor loading coefficient matrix of dimension; It is a diagonal matrix; for Maintain residual terms; As a weighted term; For degrees of freedom; This is a noise component.

4. The method for constructing the time-varying mixing dynamic factor model according to claim 3, characterized in that, The autoregressive coefficient matrix is ​​a diagonal matrix.

5. The method for constructing a time-varying mixing dynamic factor model according to any one of claims 1 to 4, characterized in that, The equivalent linear Gaussian state-space model includes observation equations and state equations. Based on the basic structure of the model and the time-varying factor load update rule and common factor update rule, an equivalent linear Gaussian state-space model is constructed, specifically including: Based on the basic structure of the model and the time-varying factor load update rule, the observation equation is constructed. : ； ； In the formula, The observation matrix; Extend the state vector; For the observed perturbation term; Based on the basic structure of the model and the common factor update rule, a state equation is constructed. : ； ； In the formula, This is the state transition matrix; This is the state noise term; Let be the covariance matrix of the state noise.

6. The method for constructing a time-varying mixing dynamic factor model according to any one of claims 1 to 4, characterized in that, The model parameter estimation results are obtained by solving the equivalent linear Gaussian state-space model based on the observed data of economic variables, specifically including: Based on the observed data of economic variables, a log-likelihood function is constructed for the equivalent linear Gaussian state-space model. The log-likelihood function is used to quantify the fit between the model parameter vector and the observed data. The maximum likelihood estimation method is used to maximize the log-likelihood function to obtain the maximum likelihood estimate of the model parameters, which follows an asymptotic normal distribution. Based on the Hessian matrix of the log-likelihood function with respect to the model parameters, calculate the standard error of the maximum likelihood estimator of the model parameters. The standard error is used to measure the sampling variation of the estimator. The value is used to test the statistical significance of the parameter estimate; The maximum likelihood estimator, standard error, and The values ​​are integrated into a set that includes parameter values ​​and statistical reliability indicators to obtain the model parameter estimation results.

7. The method for constructing the time-varying mixing dynamic factor model according to claim 6, characterized in that... It lies in, Based on the model parameter estimation results, a confidence interval for the time-varying parameters is constructed, specifically including: Based on the model parameter estimation results, multiple model parameter vectors are repeatedly extracted from the asymptotic normal distribution of the maximum likelihood estimator of the model parameters; Calculate the corresponding time-varying factor loading estimate based on each model parameter vector; The time-varying factor loading estimate is used to construct the confidence interval of the time-varying parameter by taking the corresponding quantile.

8. A device for constructing a time-varying mixing dynamic factor model, characterized in that, include: The basic structure construction module is used to construct the basic structure of the model based on the logarithmic year-on-year high-frequency time series vector. The basic structure of the model includes a signal component and a noise component. The signal component is the product of the time-varying factor loading coefficient matrix and the common factor vector. The update rule establishment module is used to establish time-varying factor load update rules and common factor update rules based on the basic structure of the model, respectively. The equivalent model conversion module is used to construct an equivalent linear Gaussian state-space model based on the basic structure of the model and the time-varying factor load update rule and the common factor update rule. The parameter estimation module is used to solve the equivalent linear Gaussian state-space model based on the observed data of economic variables to obtain the model parameter estimation results; The confidence interval construction module is used to construct confidence intervals for time-varying parameters based on the model parameter estimation results.

9. An electronic device, characterized in that, include: At least one processor; A memory that is communicatively connected to the at least one processor; The memory stores instructions that can be executed by the at least one processor, which are executed by the at least one processor to enable the at least one processor to perform the method for constructing the time-varying mixing dynamic factor model as described in any one of claims 1 to 8.

10. A medium, characterized in that, The system contains a computer program that, when executed by a processor, implements the method for constructing a time-varying mixing dynamic factor model as described in any one of claims 1 to 8.

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