Revealing Japan’s CPI Fluctuation Mechanisms via a Time-Varying Loading Factor Model

Abstract

In this article, we examine the dynamic interdependencies among components of Japan’s consumer price index (CPI) using a two-lag time-varying loading factor (TLTVLF) model. Whereas previous studies have typically decomposed CPI series into long-term trends, seasonal patterns, and cyclical fluctuations, such approaches mainly describe structural features without fully uncovering the latent mechanisms that drive price dynamics. The proposed TLTVFL modeling framework addresses this limitation by allowing both factor loadings and their lagged effects to evolve over time, thereby capturing gradual structural changes and the time-varying propagation of shocks across CPI categories. Using monthly data for ten major CPI categories from January 1970 to December 2024, we identify evolving common factors, category-specific sensitivities, and dynamic transmission patterns associated with major macroeconomic events. The findings reveal substantial temporal variation in inter-category linkages, offering fresh insights into sectoral contributions to inflationary pressures and providing policy-relevant implications for more effective monetary and fiscal interventions. Methodologically, this study extends the frontier of dynamic factor modeling, while empirically, it deepens the understanding of the mechanisms underlying price fluctuations over a long historical horizon.

1. Introduction

Understanding the detailed mechanisms underlying economy-wide price fluctuations is essential for monitoring inflationary pressures, assessing business-cycle phases, and formulating effective price-stabilization policies. Boskin et al. (1998) emphasize that accurately measuring prices and their rate of change—namely inflation—is central to almost every economic issue; there is virtually no other issue so endemic to every field of economics. Although several indicators are used to measure price levels, the consumer price index (CPI) remains the most widely employed gauge of inflation (Ito & Hoshi, 2020, Chapter 6). Accordingly, this study focuses on the dynamics of the CPI.

Specifically, we investigate the mechanisms driving fluctuations in Japan’s category-level CPIs using the two-lag time-varying loading factor (TLTVLF) model proposed by Kyo and Noda (2025a). While the aggregate CPI is commonly used in empirical and policy analysis, it often masks substantial heterogeneity across sectors, making it difficult to capture category-specific dynamics and the interactions that shape overall price movements (Blinder, 1982; Bryan & Cecchetti, 1993; Hale & Jordà, 2007). This underscores the need for analytical approaches that move beyond aggregate indicators and explicitly account for interdependencies among disaggregated CPI components.

Previous research has attempted to address this issue by decomposing CPI series into structural components such as long-term trends, seasonal effects, and cyclical variations (Baxter & King, 1999; Harvey, 1990). Kyo and Noda (2025b) decompose the CPIs of ten major Japanese categories into trend, seasonal, and cyclical elements to examine the characteristics of each type of fluctuation. Their analysis relies on the moving linear (ML) modeling framework introduced by Kyo and Kitagawa (2023) and its extended version developed by Kyo et al. (2024). Among these components, the trend and cyclical terms are particularly informative. The trend captures long-run movements in the CPI and is relatively straightforward to interpret, whereas the cyclical component reflects short-term, category-specific fluctuations as well as common cyclical movements across categories. These cyclical dynamics contain valuable information about sectoral policy effects and shared business-cycle behavior.

Building on this perspective, the present study focuses on the cyclical components of category-level CPIs and applies the TLTVLF model to examine their dynamic interactions. The TLTVLF model is a multivariate factor-analysis framework in which each variable is associated with a common factor through two lagged terms, each with time-varying loadings. Unlike static factor models or standard state-space representations, the TLTVLF model allows factor loadings to evolve over time, thereby capturing gradual structural changes and the dynamic propagation of shocks. This feature makes it especially suitable for long historical series subject to structural breaks-conditions frequently observed in Japan’s price system. Compared with Bayesian Gaussian-process approaches (e.g., Stock & Watson, 2016) and traditional dynamic-factor (DF) models (e.g., Stock & Watson, 1989, 1991), the TLTVLF modeling framework provides a flexible yet computationally tractable means of representing time-varying interdependencies.

Our empirical analysis employs monthly data for ten major CPI categories from January 1970 to December 2024, encompassing major macroeconomic episodes such as the oil crises, the asset-price bubble, prolonged deflation, and the COVID-19 pandemic. Using the TLTVLF model, we address the following questions: What common fluctuations emerge among the cyclical components of different categories? How have common factors and category-specific sensitivities evolved over time? What patterns characterize the transmission of shocks across categories? And how are these dynamic structures linked to broader business cycles and policy regimes in the Japanese economy?

This study makes four principal contributions. First, it develops an indicator that captures price-fluctuation patterns diverging from those of the aggregate CPI. Second, it advances methodological practice by introducing a framework for modeling dynamic interdependencies in high-dimensional macroeconomic data. Third, it provides novel empirical evidence on the evolution of sectoral price linkages in the Japanese economy over the past five decades. Finally, it offers policy-relevant insights into the sources of inflationary pressures and the propagation of shocks, thereby informing the design of more effective stabilization policies. To the best of our knowledge, no previous research has examined CPI fluctuations from a comparable perspective.

The main findings can be summarized as follows. The TLTVLF model reveals dynamic linkages among disaggregated CPI categories that are obscured in the aggregate index. It identifies differences in leading, coincident, and lagging relationships across categories and traces structural changes in sectoral sensitivities to the common factor over the past five decades. These results suggest that sector-level price dynamics can provide early warnings of inflationary pressures and offer new empirical evidence for developing more timely and targeted policy interventions.

The remainder of this article is organized as follows. Section 2 outlines the methodology, focusing on the TLTVLF model and the estimation procedures. Section 3 presents the empirical findings, emphasizing the dynamic factor structure and its policy implications. Section 4 concludes this paper.

2. Methodology

2.1. Major Categories of the CPI

In Japan, the CPI is classified into ten major categories, as shown in

Table 1. Major Categories of the CPI in Japan.

No.	Category
1	Food
2	Housing
3	Fuel, Light and Water Charges
4	Furniture and Household Utensils
5	Clothes and Footwear
6	Medical Care
7	Transportation and Communication
8	Education
9	Culture and Recreation
10	Miscellaneous

.

In this study, the cyclical component of each category is extracted and treated as a separate variable. Accordingly, the analysis is conducted with ten variables in total, corresponding to the ten CPI categories.

2.2. The TLTVLF Model

2.2.1. The Model

Let $\{x_{ti};i=1,2,\dots ,10\}$ denote the dataset of variables within the context of our framework. The model is specified as

$$x_{ti}=a_{ti}{f}_{t-L_i^{\left(1\right)}}+b_{ti}{f}_{t-L_i^{\left(2\right)}}+v_{ti}(i=1,2,\dots ,10),$$

(1)

where $x_{ti}$ is the i-th variable and $f_t$ is the common factor. $f_{t-L_i^{\left(1\right)}}$ and $f_{t-L_i^{\left(2\right)}}$ are its values at lags $L_i^{\left(1\right)}$ and $L_i^{\left(2\right)}$, respectively. The coefficients $a_{ti}$ and $b_{ti}$ are time-varying factor loadings (TVFLs). $v_{ti}$ is the idiosyncratic component. Throughout this article, $t=1,2,\dots ,N$ is assumed unless stated otherwise, where N denotes the sample size. The specification in Equation (1) is referred to as the two-lag time-varying loading factor (TLTVLF) model, originally proposed by Kyo and Noda (2025a) for analyzing business cycle mechanisms in Japan.

This framework allows for the examination of lead-lag relationships between the variables and the common factor. For simplicity, we fix $L_i^{\left(2\right)}$ and focus on $L_i^{\left(1\right)}$. If $L_i^{\left(1\right)}<0$, the i-th variable leads the common factor, suggesting a causal role in price fluctuations. When $L_i^{\left(1\right)}>0$, the variable lags behind, reflecting the consequences of fluctuations. Moreover, if $L_i^{\left(1\right)}\approx 0$, the variable is nearly coincident. Combinations of $L_i^{\left(1\right)}$ and $L_i^{\left(2\right)}$ introduce more complex dynamics, which are discussed later. In addition, the sign and magnitude of the TVFLs indicate the direction and strength of the common factor’s influence, and their evolution over time reveals changes in interdependencies.

The TLTVLF model has two defining features: (1) Each variable is linked to the common factor through two lagged terms, facilitating the analysis of causal relationships. (2) The factor loadings vary over time, enabling the model to capture structural changes in the dependence between the common factor and individual variables. Additionally, it is worth noting that these features enhance monitoring of CPI dynamics and provide a more detailed understanding of fluctuation mechanisms.

Relative to Stock–Watson-type dynamic factor (DF) models, the framework based on the TLTVLF model differs in two key respects as follows: (1) The DF models difference variables to remove trends, whereas the TLTVLF model uses decomposed cyclical components, avoiding information loss. (2) The DF models typically apply autoregressive structures to both common and idiosyncratic factors, while the TLTVLF model introduces lags into the common factor and employs time-varying loadings, thereby simplifying structural analysis. Although the TLTVLF modeling approach requires specialized estimation techniques, this challenge can be addressed within the distribution-free dynamic linear modeling framework (Kyo & Noda, 2025c).

Stock and Watson (2009) and Su and Wang (2017) have also introduced time-varying factor loadings. The former emphasized sudden shocks, whereas the latter focused on continuous structural change. While similar in principle, these approaches remain within general factor analysis frameworks and face estimation difficulties, whereas the TLTVLF model is specifically designed for business cycle analysis.

In the TLTVLF model, the two factor loadings and their associated lags must be estimated simultaneously, which makes the estimation procedure and algorithm complex. Once these parameters are estimated, the common factor can be derived as a byproduct. For further details, see Kyo and Noda (2025a).

2.2.2. Estimation Procedure and Convergence

In the TLTVLF model used in this study, both the TVFLs and the common factor are unknown, which poses challenges for identification and estimation. To address this, ordinary principal component analysis (PCA) is first applied to each variable, and the first principal component, after normalization, is taken as the initial common factor. The TVFLs are then estimated by extending the ML model approach, allowing the relationship between the factors and the variables to vary over time. Subsequently, PCA is again applied to the products of the estimated TVFLs and the common factor for each variable to update the common factor. By repeating this process, the TVFLs and the common factor are iteratively refined. The iteration is terminated when the variance of the first principal component reaches its maximum. Through this procedure, both time-varying structural changes and lag structures can be effectively captured, enabling the stable and simultaneous estimation of factors and loadings. For further details, see Kyo and Noda (2025a).

Concretely, the proposed estimation scheme iteratively updates the common factor and the TVFLs. The procedure begins by estimating the TVFLs at the first lag. Let $f_t^{\left(0\right)}$ be the initial setting of the common factor, which is determined using the first principal component of the data $\{x_{t,1},x_{t,2},\dots ,x_{t,10}\}$. Then, for $k=1,2,\dots ,K$ with K being an arbitrarily chosen integer, the procedure proceeds as follows:

• First TVFL estimation: For $i=1,2,\dots ,10$, the TVFLs $\left\{a_{ti}^{\left(k\right)}\right\}$ are estimated based on the values of $f_t^{(k-1)}$ and $x_{ti}$ with the model in Equation (1), by temporarily setting $b_{ti}=0$.
• Factor estimation: The common factor is updated as $f_t^{\left(k\right)}$ using the first principal component of the data $\{a_{t,1}^{\left(k\right)}{x}_{t,1},a_{t,2}^{\left(k\right)}{x}_{t,2},\dots ,a_{t,10}^{\left(k\right)}{x}_{t,10}\}$, with the first eigenvalue denoted by $V^{\left(k\right)}$, which is a function of the first lags $L_i^{\left(1\right)}$.
• First lag estimation: The first lags $L_i^{\left(1\right)}$ are estimated by maximizing the eigenvalue $V^{\left(k\right)}$ within an interval $[-L,L]$ for an arbitrarily chosen integer L.
• Second TVFL estimation: Given ${\widehat{a}}_{ti}$ and $f_t^{\left(k\right)}$, estimate $\left\{b_{ti}\right\}$ conditionally from

  $$x_{ti}-{\widehat{a}}_{ti}{f}_{t-L_i^{\left(1\right)}}^{\left(k\right)}=b_{ti}{f}_{t-L_i^{\left(2\right)}}^{\left(k\right)}+e_{ti},(i=1,\dots ,10).$$
• Second lag estimation: Determine $L_i^{\left(2\right)}$ by minimizing the variance of residuals $e_{ti}$ within $[-L,L]$.

This alternating process is iterated until the change in the first eigenvalue between two successive iterations becomes sufficiently small:

$$|V^{\left(k\right)}-V^{(k-1)}|<\epsilon ,$$

where $\epsilon$ is a small positive threshold.

To ensure identification, we impose the standard normalization constraint $\mathrm{Var}(f_t^{\left(k\right)})=1$ and fix the sign of one representative loading throughout the iterations. These constraints eliminate the rotational indeterminacy common in factor models and make the estimates of $\left\{f_t\right\}$ and $\{a_{ti},b_{ti}\}$ comparable across iterations.

Empirically, the iterative process converged within fewer than ten iterations in all experiments. Although several approaches can be considered for the initial setting of the common factor, using the principal component of each variable’s data has been empirically found to ensure convergence and is logically consistent. The results were nearly identical across all settings, suggesting that the likelihood surface is well-behaved and that the final estimates are practically unique and numerically stable.

2.3. Application of the TLTVLF Model

2.3.1. Constructing the Cyclical Price Factor Index

The common factor estimated using the TLTVLF model captures the co-movements embedded in the cyclical components of each CPI category. Its estimates therefore provide a basis for analyzing short-term CPI variations, such as the direction of price movements and the dynamics of business cycles.

Because the common factor is standardized with mean zero and variance one, its numerical values serve only as a relative indicator of economic conditions. To enhance interpretability, we adopt a standardized score transformation and define the cyclical price factor index (CPFI) as follows:

$${\mathrm{CPFI}}_t=10f_t+50(t=1,2,\dots ,N),$$

(2)

so that the series $\{{\mathrm{CPFI}}_t;t=1,2,\dots ,N\}$ has an approximate mean of 50 and a standard deviation of 10, typically ranging from 0 to 100. This transformation makes the index suitable for evaluating short-term CPI fluctuations.

While the cyclical component derived from the aggregate CPI provides one measure of short-term price variation, the CPFI offers several distinct advantages as follows:

• Capturing Structural Heterogeneity The cyclical component of the aggregate CPI is obtained as a weighted average of category-specific indices and therefore reflects only the aggregated outcome. It obscures interdependencies and asynchronous movements across categories, leading to the omission of structural heterogeneity and propagation mechanisms. In contrast, the CPFI is based on the common factor extracted from the cyclical components of all categories using the TLTVLF model. Rather than relying on simple averaging, it emphasizes shared cyclical dynamics while filtering out idiosyncratic noise, yielding a clearer measure of business cycle-related price fluctuations.
• Preservation of Lead–Lag Information Because the aggregate CPI collapses information into a single index, its cyclical component retains little or no lead-lag information. By incorporating the lag structure $(L_i^{\left(1\right)},L_i^{\left(2\right)})$ estimated through the TLTVLF model, the CPFI preserves this dimension, making it particularly valuable for detecting turning points and analyzing the transmission of shocks across sectors.
• Revealing Distinctive Fluctuation Patterns Opposing category movements often offset each other in the aggregate CPI, masking meaningful dynamics. For example, if two categories move in opposite directions—one upward and the other downward—their effects may cancel out in the aggregate. In contrast, the CPFI incorporates the positive and negative factor loadings of each category, allowing both to influence the common factor. This enables researchers to assess both the magnitude and direction of each category’s contribution.
• Noise Reduction and Policy Relevance Temporary shocks in specific categories (e.g., energy or fresh food) can distort the aggregate CPI, leading to biased assessments of cyclical movements. By extracting the common factor, the CPFI reduces such distortions and provides a cleaner signal of underlying business cycle dynamics, making it a more reliable benchmark for monetary policy and macroeconomic evaluation.
• Facilitating International and Historical Comparisons Because the CPFI is standardized (mean $=50$, standard deviation $=10$), it supports meaningful comparisons across countries and historical periods. By contrast, the aggregate CPI varies substantially in level and volatility across contexts, complicating direct comparison. The CPFI therefore provides a robust basis for cross-country and long-term time series analyses.

2.3.2. Analyzing Lead–Lag Relationships Across Categories

As discussed in Section 2.2, when a category’s CPI is strongly correlated with the common factor at a negative lag, it precedes overall price fluctuations and can be regarded as a causal category influencing broader price dynamics. By contrast, a positive lag indicates that the category responds to overall price movements, reflecting outcomes rather than driving them. A zero lag implies that the category moves synchronously with the common factor and is therefore a coincident category.

It is worth noting that a category may be associated with multiple lags, functioning as a causal category at one lag and as an outcome or coincident category at another. The TLTVLF model in Equation (1) represents the simplest case of such multi-lag relationships.

To classify categories systematically, we define six groups based on the lag values $L_i^{\left(1\right)}$ and $L_i^{\left(2\right)}$ for each variable $i=1,2,\dots ,10$.

• Co-varying category: $L_i^{\left(1\right)}=0$ and $L_i^{\left(2\right)}=0$; moves in sync with the overall cycle.
• Causal category: $L_i^{\left(1\right)}<0$ and $L_i^{\left(2\right)}<0$; consistently leads overall price fluctuations.
• Outcome category: $L_i^{\left(1\right)}>0$ and $L_i^{\left(2\right)}>0$; consistently lags overall price movements.
• Bipolar category: $L_i^{\left(1\right)}<0,L_i^{\left(2\right)}>0$ or $L_i^{\left(1\right)}>0,L_i^{\left(2\right)}<0$; exhibits both leading and lagging characteristics.
• Co-varying and causal category: $L_i^{\left(1\right)}=0,L_i^{\left(2\right)}<0$ or $L_i^{\left(1\right)}<0,L_i^{\left(2\right)}=0$.
• Co-varying and outcome category: $L_i^{\left(1\right)}=0,L_i^{\left(2\right)}>0$ or $L_i^{\left(1\right)}>0,L_i^{\left(2\right)}=0$.

These classification rules provide a consistent framework for analyzing lead–lag relationships across categories and highlight the diversity of sectoral contributions to aggregate price dynamics.

2.3.3. Analyzing the Dynamics of Category Dependence on the CPFI

The first and second TVFLs capture each category’s interaction with the CPFI at the first and second lags, respectively. The sign of a loading indicates the direction of co-movement, while its magnitude reflects the strength of interdependence. Because these loadings evolve over time, they trace the changing dependence of each category on the common factor.

Combining the time-varying loadings with the estimated lag structure makes it possible to identify causal effects, feedback loops, and bidirectional linkages between individual categories and the CPFI. This, in turn, provides deeper insights into the mechanisms underlying business cycle dynamics.

In special cases, the two lags may coincide. When $L_i^{\left(2\right)}=L_i^{\left(1\right)}$ for a given variable i, the TLTVLF model simplifies by merging the two loadings into a single measure:

$$x_{(t+L_i^{\left(1\right)})i}={\widehat{a}}_{ti}^{\ast }{f}_t+v_{(t+L_i^{\left(1\right)})i}.$$

(3)

In Equation (3), the combined factor loading (CFL) is defined as ${\widehat{a}}_{ti}^{\ast }={\widehat{a}}_{ti}+{\widehat{b}}_{ti}$. The CFL offers a concise representation of the category’s overall dependence on the CPFI when the two lagged effects align.

3. Results and Analysis

In this section, we present the estimation results obtained from applying the TLTVFL model to the cyclical components of ten major CPI categories in Japan. We then discuss the implications of these results, highlighting the dynamic interactions among categories and their relationships with the common factor.

3.1. Dataset

As outlined in Section 2.1, the cyclical component of each CPI category in Table 1 is treated as a distinct variable. Data of all variables can be obtained from the website of the Statistics Bureau of Japan (https://www.stat.go.jp/english/data/cpi/1581-z.html, accessed on 1 September 2025), with historical series linked to maintain continuity across changes in 2020 base CPI. The dataset consists of N = 660 monthly observations covering the period from January 1970 to December 2024, as reported in Kyo and Noda (2025b) A visual summary of the time series for all ten variables is provided in Figure A1 in Appendix A.

3.2. Analyzing the Common Factor

The maximum lag length is set to 12, so that for each variable the lag parameters are estimated within the range of $-12$ to 12. The resulting estimates will be examined in detail in the next subsection.

Figure 1a displays the estimated common factor, while Figure 1b shows the standardized cyclical component of the overall CPI for comparison.

From Figure 1a,b, we find that both series exhibit pronounced fluctuations until the mid-1970s, after which the amplitude of variation gradually decreases. Moreover, the cyclical component of the overall CPI features numerous irregular peaks and troughs, whereas the estimated common factor remains comparatively stable. Furthermore, the common factor tends to decline during recession periods, reflecting the impact of macroeconomic downturns on category-level price dynamics.

Using Equation (2), the CPFI can be directly constructed from the estimated common factor. As this involves only a linear scale transformation, the corresponding figure is omitted.

3.3. Lead–Lag Relationships Between CPI Categories and the Common Factor

Table 2. Estimated Lags $(L_i^{\left(1\right)},L_i^{\left(2\right)})$ for Each CPI Category.

Variable No. $\left(\mathit{i}\right)$	1	2	3	4	5	6	7	8	9	10
Estimate for $L_i^{\left(1\right)}$	0	$-1$	0	0	$-2$	11	0	12	0	1
Estimate for $L_i^{\left(2\right)}$	4	1	6	2	$-2$	1	3	0	$-12$	$-12$

reports the estimated lags $(L_i^{\left(1\right)},L_i^{\left(2\right)})$ for the ten CPI categories.

These estimates allow each category to be classified into one of the six groups defined in Section 2.3.2, clarifying their roles in leading or responding to common cyclical fluctuations. The raw estimates reveal diverse lead–lag patterns. For example, Food (No. 1) with the lags $(0,4)$ and Fuel, Light and Water Charges (No. 3) with the lags $(0,6)$, Furniture and Household Utensils (No. 4) with the lags $(0,2)$, Transportation and Communication (No. 7) with the lags $(0,3)$, and Education (No. 8) with the lags $(12,0)$ belong to the co-varying and outcome group, indicating that these essential goods generally move with the common cycle but respond with delays due to factors such as prices of imported products, energy shocks, or supply constraints. Clothes and Footwear (No. 5) with the lags $(-2,-2)$ serves as a clear causal driver, leading overall CPI fluctuations. Housing (No. 2) with the lags $(-1,1)$ and Miscellaneous (No. 10) with the lags $(1,-12)$ fall into the bipolar category, exhibiting both leading and lagging behavior depending on historical context. This feature reflects structural heterogeneity and context-dependent responses to macroeconomic shocks. Medical Care (No. 6) with the lags $(11,1)$ corresponds to outcome category with long lags, highlighting structural rigidity and regulatory constraints. Culture and Recreation (No. 9) with the lags $(0,-12)$ is classified into the co-varying and causal group, because it relates to the timing of consumer behavior in response to prevailing trends in recreational goods and services.

To emphasize broader patterns, categories are aggregated by group as summarized in

Table 3. CPI Categories Classified by Lead–Lag Group.

Group	Categories	$({\mathit{L}}^{\left(1\right)},{\mathit{L}}^{\left(2\right)})$	Interpretation
Causal	Clothes and Footwear (No. 5)	($-2$,$-2$)	This variable leads overall price fluctuations. Therefore, it is expected to provide a useful roadmap for predicting broad price movements driven by seasonal cycles and import costs.
Outcome	Medical Care (No. 6)	(11,1)	These variables respond slowly. This is due to regulatory and structural rigidity. The long delays suggest the presence of limitations in short-term responsiveness.
Co-varying and Outcome	Food (No. 1), Fuel, Light and Water Charges (No. 3), Furniture and Household Utensils (No. 4), Transportation and Communication (No. 7), Education (No. 8)	(0,4), (0,6), (0,2), (0,3), (12,0)	These variables move contemporaneously with the common cycle, but exhibit delayed responses reflecting cost-push effects, energy price shocks, or exchange rate fluctuations.
Co-varying and Causal	Culture & Recreation (No. 9)	(0,$-12$)	This variable exhibits characteristics that lead and coincide with overall price fluctuations. This is mainly due to households’ purchasing behavior in response to current trends in recreational goods and services.
Bipolar	Housing (No. 2), Miscellaneous (No. 10)	($-1$,1), (1,$-12$)	Depending on the historical context, these variables exhibit both leading and lagging behaviour. This implies that lead–lag relationships vary across events, reflecting the structural and institutional features of the Japanese economy.

.

As Table 3 demonstrates, Japan’s CPI categories exhibit substantial heterogeneity in their interactions with the common factor. As a result, our analysis shows that the ten major variables listed in Table 1 fall into one of the following categories: causal, outcome, bipolar, co-varying and causal, or co-varying and outcome. In other words, no variables belong to the co-varying category in Japan’s CPI data.

It should be noted that five of the ten variables were found to belong to the co-varying/outcome category. That is, within our grouping, the largest number of variables exhibit characteristics of the co-varying/outcome category. Using sectoral price data for the 588 items included as components of Japan’s CPI, K. Watanabe and Watanabe (2018) and T. Watanabe (2024) investigated the distribution of item-level price changes. The results of their empirical analysis can be summarized as follows: The proportion of items with inflation rates near zero was approximately 20% from the 1970s through the early 1990s. However, in the late 1990s, the proportion of items with inflation rates near zero rose to about 50%, and this level has persisted through the early 2020s. This suggests that price fluctuation patterns within the categories of items covered by Japan’s CPI differ between the period from the 1970s to the early 1990s and the period from the mid-1990s onward. Therefore, their results are consistent with our findings described above.

K. Watanabe and Watanabe (2018) argued that there are two possible reasons why prices in Japan have become stickier since the mid-1990s. The first possibility is a structural change in the Japanese economy, such as a change in the competitive environment in which firms operate, which leads firms to alter their pricing behaviour, resulting in increased price stickiness. Another explanation is that the increase in price stickiness since the mid-1990s may have occurred endogenously rather than exogenously. For example, based on a menu cost model, Ball and Mankiw (1994) point out that price stickiness can change depending on the level of trend inflation. In other words, when trend inflation is high, the profits foregone by a firm that does not adjust its prices will be considerable. Such a firm would fall behind if it alone does not raise prices while its rivals do. Since the profits foregone by not adjusting prices are substantial, firms will choose to adjust prices despite incurring menu costs. Accordingly, prices are flexible and the Phillips curve is steeper. Conversely, when trend inflation is close to zero, as has been the case in Japan since the mid-1990s, the profit foregone by not adjusting prices is smaller than the menu costs. Consequently, firms will delay adjusting prices. Prices become stickier and the slope of the Phillips curve becomes smaller. Thus, changes in trend inflation lead to endogenous changes in price stickiness.

The following is an additional note regarding the interpretation of our analytical results described in Table 3. A primary goal of monetary policy is usually considered to be price stability (Ito & Hoshi, 2020, Chapter 6). From a policy perspective, monitoring sector-specific dynamics can enhance the timeliness and effectiveness of stabilization measures. In particular, causal variables may function as early-warning indicators of inflationary pressures or a transition to a deflationary phase. Additionally, outcome variables are useful for guiding central bank policymakers when making ex post adjustments aimed at stabilizing the macroeconomy. Furthermore, it should be noted that co-varying and outcome variables, co-varying and causal variables possess two distinct characteristics. This is because the characteristics of each variable can change due to factors such as changes in the phases of the domestic business cycles or global shocks like the COVID-19 pandemic. Therefore, policymakers should closely monitor the macroeconomic situation in a timely manner and accurately recognize the characteristics of the variables during the relevant period.

3.4. Analyzing the Dynamics of Relationships Between the Variables and the Common Factor

In this subsection, we present the estimated TVFLs and examine the dynamic relationships between each variable and the common factor. Both the variables and the common factor are centered and standardized, so that each TVFL represents the time-varying correlation coefficient between a variable and the common factor at the corresponding lag.

Figure 2, Figure 3 and Figure 4 display the TVFLs for each variable, excluding variable No. 5. Regarding variable No. 5, because the two estimated lags are equal, the corresponding TVFLs are combined into a single CFL, shown in Figure 5. This CFL provides a concise representation of the overall dynamic relationship between variable No. 5 and the common factor.

3.4.1. Food (No. 1)

The first-lag TVFL (Figure 2a) remains nearly constant at 0.6 in the early 1970s, indicating a weak but stable correlation. It gradually increases from 1973, reaching around 1.0 by the late 1970s, reflecting a strengthening association after the first oil crisis (1973–1974). During the 1980s to early 1990s, the loading peaks above 1.2 before stabilizing between 0.8 and 1.0. From the 2000s, the TVFL rises sharply, exceeding 2.0 in the mid-2000s and stabilizing at 2.26 toward the end of the sample, suggesting a persistently strong link with the common factor.

The second-lag TVFL (Figure 2b) exhibits more volatile dynamics. It starts at $-0.13$ in the early 1970s, rises toward zero by the late 1970s, and alternates between positive and negative values throughout the 1980s–2000s. Toward the end of the sample, it stabilizes around 0.07, indicating a weak but positive delayed correlation. Overall, the first-lag effect dominates over time, gradually loses explanatory power.

3.4.2. Housing (No. 2)

The first-lag TVFL (Figure 2c) begins near 0.6 and remains stable in the early 1970s. A gradual increase begins in the mid-1970s, reaching above 2.0 by 1990, followed by corrections and eventual stabilization around 1.3 by 2008. These dynamics reflect increasing sensitivity to the common factor over time, possibly due to structural changes or policy shifts.

The second-lag TVFL (Figure 2d) starts near $-0.06$, indicating a weak negative correlation. It rises steadily from the late 1970s, peaks around 2.0 in early 1993, and then declines to stabilize around 0.10. This trajectory illustrates a transition from weak negative association to strong positive alignment and finally to a moderate correlation, highlighting structural changes in the relationship.

3.4.3. Fuel, Light and Water Charges (No. 3)

The first-lag TVFL (Figure 2e) shows how short-term energy price shocks are transmitted to the common factor, indicating that this category primarily co-varies with, but slightly lags overall CPI fluctuations. The second-lag TVFL (Figure 2f) captures more persistent cost-push effects stemming from imported fuel and utilities, highlighting delayed yet sustained contributions to cyclical price movements.

3.4.4. Furniture and Household Utensils (No. 4)

The first-lag TVFL (Figure 3a) reflects immediate adjustments to household spending cycles, while the second-lag TVFL (Figure 3b) reveals slower inventory and replacement effects. Together, they suggest that this category follows the common cycle with modest delays but can amplify turning points during periods of rapid demand change.

3.4.5. Medical Care (No. 6)

The first-lag TVFL (Figure 3c) begins at 0.74 and gradually decreases, turning negative around 1974–1976, reflecting a temporary reversal. It peaks at approximately 2.7 between 1986 and 1994 and later stabilizes around 0.55–0.8. The second-lag TVFL (Figure 3d) starts near 0.05, rises gradually to 0.5, peaks above 1.0 in the period from 1983 to 2001, and eventually declines toward zero. These patterns indicate that the first-lag effect captures the dominant long-term alignment, while the second-lag effect reflects transient structural deviations.

3.4.6. Transportation and Communication (No. 7)

The first-lag TVFL (Figure 3e) indicates that transportation and communication prices tend to move contemporaneously with the common factor, reflecting their sensitivity to fuel costs and general demand conditions. The second-lag TVFL (Figure 3f) points to lingering effects from regulatory adjustments and contract renewals, implying a gradual transmission of shocks across subcomponents.

3.4.7. Education (No. 8)

The first-lag TVFL (Figure 4a) remains stable near 0.35 until early 1973, then rises steadily, reaching about 1.16 by the late 1970s. It fluctuates moderately during 1981–1989, declines to near zero by 1994, and exhibits a sharp rise between 1999 and 2009, surpassing 3.7. After a gradual decline, a second peak occurs around 2019, followed by stabilization at 1.74. The second-lag TVFL (Figure 4b) starts at 0.14, gradually rises to 0.6 in the mid-1970s, and fluctuates until the mid-1990s. It then experiences a period, from 2001 to 2004, of strong influence, followed by a temporary negative phase ($-0.55$) around 2006–2009 and final stabilization around 0.38. These dynamics show that the first-lag effect dominates long-term behavior, while the second-lag effect captures temporary and occasionally inverse relationships.

3.4.8. Culture and Recreation (No. 9)

The first-lag TVFL (Figure 4c) reveals episodes when discretionary spending in culture and recreation leads or coincides with the common factor, acting as an early signal of changing household sentiment. The second-lag TVFL (Figure 4d), in contrast, captures delayed responses during downturns when households postpone non-essential consumption, producing oscillatory patterns.

3.4.9. Miscellaneous (No. 10)

The first-lag TVFL (Figure 4e) shows a weak but variable contemporaneous link to the common factor, reflecting the diverse nature of this category. The second-lag TVFL (Figure 4f) uncovers protracted and sometimes opposing effects from heterogeneous subitems, underscoring the importance of disaggregated analysis for accurate policy interpretation.

3.4.10. Combined Factor Loading (Variable No. 5)

For Clothes and Footwear (No. 5), the two lags coincide, yielding a combined factor loading (CFL, Figure 5). The CFL captures the overall dependence on the common factor, exhibiting long-term stability with moderate fluctuations, thereby highlighting consistent influence on the CPFI.

The findings obtained through detailed investigations from Figure 2, Figure 3, Figure 4 and Figure 5 can be summarized as follows. Recall that the first and second TVFLs capture the interaction between each CPI category and the CPFI at the first and second lags, respectively. Thus, each TVFL represents the time-varying correlation coefficient between the relevant variable and the common factor at the corresponding lag. First, it is confirmed that the magnitude of the correlation between all variables and their common factors changes over time. Second, each variable exhibits distinct patterns of variation in correlation coefficients; that is, the fluctuation patterns of correlation coefficients differ by variable. Third, throughout the analysis period, the sign of the correlation coefficient for most variables is not consistently the same; it alternates between positive and negative values depending on the period. Therefore, the relationship between price movements in each category and the common factor varies by period. Fourth, across the examined variables, first-lag TVFLs generally reflect the dominant, long-term alignment with the common factor, while second-lag TVFLs capture transient, oscillatory, or occasionally opposing effects. Fifth, these dynamics reveal that the contemporaneous impact of the common factor often strengthens over time, whereas delayed effects are less consistent. Finally, the observed temporal evolution of these relationships is influenced by changes in the phases of the business cycle, various policy shifts, domestic and international regime changes, changes in consumer sentiment, and exogenous shocks (e.g., the first oil crisis of 1973–1974, the global financial crisis triggered by the collapse of Lehman Brothers in 2008, and the recent COVID-19 pandemic).

These findings provide the following insight: understanding prices requires elucidating the distinct fluctuation mechanisms of various categories with differing characteristics, and modeling that accounts for structural changes is indispensable for this purpose.

3.5. Analyzing the Interaction Between the Variables and the Common Factor

Even within the same variable, its relationship with the common factor can be intricate and time-varying. For instance, one lag may load negatively, indicating that the variable leads fluctuations in the common factor, while another lag may load positively, reflecting a delayed response. Variable No. 2 exemplifies this pattern, with the first lag negative and the second lag positive. These contrasting signs and magnitudes illustrate how the two TVFLs capture complex and multi-layered interactions. This feature distinguishes the TLTVLF model from static factor-loading approaches.

Figure 6 depicts a scatter plot with line graphs showing the simultaneous evolution of the first- and second-lag TVFLs for variable No. 2.

To emphasize underlying patterns, the TVFL time series are smoothed using the ML model approach. In Figure 6, the first-lag TVFL is plotted on the horizontal axis, representing the variable’s leading influence on the common factor, while the second-lag TVFL is plotted on the vertical axis, representing its delayed response. These TVFLs reflect the link between the variable and the common factor through the latter, which acts as a latent intermediary summarizing overall price dynamics.

Several observations can be made from Figure 6.

• Early stage (before 1975): Both TVFLs are small and negative (around −0.05 to −0.06). The joint movement indicates that variable No. 2 was initially weakly but stably negatively aligned with the common factor, with nearly equal contributions from both lags.
• Transition phase (end of 1974 to mid-1986): The second-lag TVFL rises and turns positive, while the first-lag TVFL also trends upward but with smaller amplitude. This divergence implies that the second lag increasingly dominates the explanatory power.
• Peak synchronization phase (mid-1986 to early 1993): The second-lag TVFL surges to nearly 2.0, whereas the first-lag TVFL rises modestly. Variable No. 2 becomes strongly synchronized with the common factor, dominated by the second-lag effect.
• Late stage (after early 1993): Both TVFLs decline and stabilize at weak positive levels (around 0.1), suggesting that variable No. 2 gradually decouples from the common factor.

Overall, the joint dynamics of the two TVFLs reveal regime shifts in the role of variable No. 2. Initially balanced at weak negative levels, subsequently dominated by the second lag, and finally reconverging to weak, stable positive values. These patterns reflect structural changes in the underlying system, highlighting the TLTVLF model’s ability to uncover hidden asymmetries and time-varying dependencies not detectable by static factor loadings. In practical terms, this implies that housing prices not only drive overall price fluctuations but also tend to move broadly in the same direction as the general price level.

Regarding variable No. 2 (Housing) in Figure 6, several stages can be distinguished. In the early stage (before 1975), both first- and second-lag TVFLs are small and negative, indicating that housing prices were weakly but stably negatively aligned with the common factor, with nearly equal contributions from both lags. During the transition phase (end of 1974 to mid-1986), the second-lag TVFL rises and turns positive, while the first-lag TVFL increases more moderately, suggesting that the delayed effect increasingly dominates. In the peak synchronization phase (mid-1986 to early 1993), the second-lag TVFL surges to nearly 2.0, whereas the first-lag TVFL rises only modestly, indicating strong synchronization with the common factor, primarily via the second-lag effect. Finally, in the late stage (after early 1993), both TVFLs decline and stabilize at weak positive levels, showing that housing prices gradually decouple from the common factor. These dynamics highlight structural shifts and the TLTVLF model’s ability to reveal time-varying dependencies not captured by static loadings.

From a policy perspective, these findings highlight the importance of considering the heterogeneous and time-varying contributions of individual CPI components. In particular, the evolving influence of housing prices on the common factor suggests that policies aimed at stabilizing overall price levels should take into account the lagged and dynamic effects of key components. By monitoring such component-specific dynamics, policymakers can better anticipate potential sources of inflationary or deflationary pressures and design targeted interventions that reflect the differential roles of each sector in driving overall price movements. This emphasizes that aggregate measures alone may obscure critical temporal asymmetries, and component-level analysis is essential for effective monetary and fiscal policy formulation.

4. Concluding Remarks

This study applied the TLTVLF model to the cyclical components of ten major CPI categories in Japan, covering the period from January 1970 to December 2024. The primary objective was to elucidate the dynamic interactions between individual CPI categories and the common factor driving overall price fluctuations, while explicitly accounting for time-varying relationships. By integrating estimated lag structures with TVFLs, our proposed approach identified contemporaneous and delayed effects, leading and lagging roles, and complex interdependencies across categories.

Our analytical results revealed substantial heterogeneity among CPI categories. Specifically, Clothes and Footwear (No. 5) consistently acted as a causal driver, leading overall CPI movements. In contrast, Medical Care (No. 6) primarily served as an outcome indicator, responding with long lags due to regulatory and structural constraints. Essential goods and services such as Food (No. 1) and Fuel, Light and Water Charges (No. 3), Furniture and Household Utensils (No. 4), Transportation and Communication (No. 7), and Education (No. 8) co-varied with the common factor but exhibited delayed responses indicative of cost-push influences. Housing (No. 2) and Miscellaneous (No. 10) exhibited bipolar behaviour, displaying both leading and lagging tendencies, thereby highlighting the structural heterogeneity and context-dependent nature of their cyclical responses. Culture and Recreation (No. 9) was classified as a co-varying/causal category, moving contemporaneously with overall price fluctuations and, at times, leading them, depending on the timing of consumer behaviour in response to prevailing trends in recreational goods and services.

This study applied the TLTVLF model to the cyclical components of ten major CPI categories in Japan over the period from January 1970 to December 2024. The primary objective was to elucidate the dynamic interactions between individual CPI categories and the common factor underlying overall price fluctuations, while explicitly accounting for time-varying relationships. By integrating estimated lag structures with time-varying factor loadings (TVFLs), the proposed approach identified contemporaneous and delayed effects, as well as leading and lagging roles and complex interdependencies across categories. Our empirical results revealed substantial heterogeneity among CPI categories. Clothes and Footwear (No. 5) consistently acted as a causal driver, leading movements in the overall CPI. In contrast, Medical Care (No. 6) primarily functioned as an outcome indicator, responding with long lags owing to regulatory and structural constraints. Essential goods and services—including Food (No. 1), Fuel, Light and Water Charges (No. 3), Furniture and Household Utensils (No. 4), Transportation and Communication (No. 7), and Education (No. 8)—co-moved with the common factor but exhibited delayed responses, suggesting the presence of cost-push mechanisms. Housing (No. 2) and Miscellaneous (No. 10) displayed bipolar behavior, alternating between leading and lagging roles, thereby highlighting structural heterogeneity and the context-dependent nature of their cyclical responses. Culture and Recreation (No. 9) was classified as a co-varying and causal category, moving contemporaneously with overall price fluctuations while, at certain times, also leading them, depending on the timing of consumer responses to prevailing trends in recreational goods and services. Time-varying factor loadings provided a detailed depiction of how each category’s relationship with the common factor evolved over time. First-lag TVFLs generally exhibited persistent trends, reflecting the gradual strengthening of contemporaneous relationships, whereas second-lag TVFLs showed oscillatory behavior, capturing delayed and occasionally opposing effects. For instance, Food (No. 1) displayed a steadily increasing first-lag TVFL, indicating a growing alignment with the common factor over the decades, while its second-lag TVFL fluctuated substantially, reflecting unstable delayed responses. Housing (No. 2) exhibited distinct phases—an initial period of weak negative alignment, an intermediate phase dominated by the second lag, and eventual stabilization at weak positive levels. These patterns likely reflect structural changes and regime shifts in the Japanese economy.

The TLTVLF model provides distinct methodological advantages relative to conventional static factor models. By permitting factor loadings to evolve over time and explicitly incorporating multiple lag structures, it is capable of capturing asymmetric, non-stationary, and regime-dependent relationships that are inherently inaccessible to static approaches. In addition, the introduction of combined factor loadings (CFLs) for equal lags offers a compact yet informative measure of overall dependence on the common factor, thereby improving interpretability without sacrificing analytical richness. The systematic classification of CPI categories into causal, outcome, co-varying, and bipolar groups further facilitates a structured understanding of sectoral dynamics within a unified modeling framework.

From a policy perspective, the results underscore the importance of closely monitoring sector-specific price behavior rather than relying solely on aggregate indicators. Categories identified as causal can function as early-warning signals of emerging inflationary pressures, whereas outcome categories reflect delayed transmission mechanisms, highlighting the need for forward-looking and pre-emptive policy actions. The presence of bipolar categories indicates that the effects of macroeconomic shocks are neither uniform across sectors nor stable over time, reinforcing the necessity for flexible and context-sensitive policy design. Moreover, continuous tracking of time-varying factor loadings can help detect emerging structural changes or shifts in cyclical sensitivities, thereby supporting more timely and effective policy responses.

Based on the above discussion, the main contributions of this study can be summarized as follows. Existing CPI-related research has largely focused on inflation forecasting (e.g., Joseph et al., 2024; Nason & Palasciano, 2026; Sengupta et al., 2025) or inflation-targeting frameworks (e.g., Fukuda & Soma, 2019; Hattori et al., 2021; Kinugawa, 2019). In contrast, the present study adopts a distinct analytical perspective by explicitly examining the dynamic and heterogeneous interrelationships among CPI components through a time-varying factor structure. By addressing an aspect that has received limited attention in prior empirical studies, this research offers a novel viewpoint and identifies an important dimension for future CPI analysis. The proposed TLTVLF framework thus contributes both a new methodological tool and a deeper conceptual understanding of inflation dynamics, with direct relevance for evidence-based price stabilization policies.

Finally, the TLTVLF modeling framework provides a foundation for future research and a range of potential extensions. Its flexible structure makes it readily applicable to other price indices, such as the corporate goods price index or the services producer price index. Furthermore, given the well-established interdependence between price and wage dynamics in the context of wage-price spiral theory (see, e.g., Blanchard, 1986; Lorenzoni & Werning, 2023), extending the framework to jointly model prices and wages represents a promising avenue for further investigation. These issues are left for future research.