Machine Learning vs. Econometric Models to Forecast Inflation Rate in Romania? The Role of Sentiment Analysis

Abstract

Given the high inflationary pressure in Romania, the aim of this paper is to demonstrate the potential of autoregressive distributed lag (ARDL) models incorporating sentiment analysis to provide better inflation forecasts compared to machine learning (ML) techniques. Sentiment analysis based on National Bank of Romania reports on quarterly inflation may provide valuable inputs for econometric models. The ARDL model, utilizing inflation and sentiment index data from the previous period, outperformed the proposed seasonal autoregressive integrated moving average (SARIMA) model and the ML techniques (support vector machine and artificial neural networks). The forecasts based on the ARDL model predicted correctly all the changes in inflation, while accuracy measures (mean error, mean absolute error, root squared mean error) in the short-run 2023: Q1–2024: Q3 indicated the most accurate predictions. The more accurate forecasts are essential for national banks, companies, policymakers, and households.

1. Introduction

Accurate inflation forecasts are essential for Romania because its national bank uses a direct inflation targeting strategy since 2005. Moreover, implementing this strategy has become more challenging because of the high inflation in the last few years caused by the global energy crisis triggered by the Russia–Ukraine conflict, supply chain disruptions generated by the recent health crisis and the war, wage growth.

Accurate forecasts are necessary in making the necessary adjustments to keep the economy on track and maintain public trust in the bank’s effectiveness. First, the National Bank of Romania (NBR) sets interest rates based on the inflation forecasts. An accurate forecast allows banks to adjust rates effectively to maintain price stability. If inflation is predicted to be higher than desired, the bank can raise rates to slow down the economy and temper inflation. In addition, if inflation is forecasted to be too low, the interest rates might be reduced to boost the economy and bring inflation to a higher level [1]. Second, businessmen and consumers rely on inflation forecasts to make their economic decisions. Accurate forecasts create a more stable environment for investment, borrowing, and pricing. Companies can make good plans on future costs and profits, and households can make informed decisions about saving and spending in the near future. Third, when the national bank achieves its inflation target consistently, public confidence in its ability to manage the economy consolidates [2]. This confidence is essential for a healthy financial system.

The objective to provide accurate forecasts depends on many factors: data quality and availability, forecasting method, inclusion of external factors, model updating and refinement, expertise and judgment of forecasters, communication, and transparency, etc. [3]. This paper focuses on selecting the forecasting method that yields the most accurate forecasts.

From this point of view, the recent empirical inflation forecasting can be systematized in a few research directions: the use of dynamic stochastic general equilibrium (known as DSGE) models due to the development of many Bayesian methods that could be employed in estimations, the use of alternative methods to DSGE models (random walk, Bayesian vector autoregressions (VAR), mixed models based on DSGE and various models based on VARs, other types of econometric models, machine learning methods), and the use of predictions provided by professional forecasters. In this framework, two debates were advanced [4]. First, some studies argue that DSGE models outperform all the other inflation forecasting models, while others show the superiority of hybrid models [5]. Second, the ability of one forecasting method to outperform the others depends on the data nature (the use of real-time data that provide better results or revised data) and on the capacity of the method to discern long-term trends presented in data [6].

The recent empirical inflation forecasting is significantly influenced by new theoretical foundations due to fast technological progress. The advent of powerful computing resources, particularly graphical processing units (GPUs), has revolutionized machine learning used in forecasting. GPUs’ parallel processing capabilities have enabled the training of complex models on massive datasets, significantly reducing computational time and enabling the development of more data-intensive algorithms [7]. Therefore, empirical inflation forecasting based on machine learning and other methods should be subordinated to the objective of getting the most accurate predictions. In this paper, the research question is related to the capacity of machine learning to provide better inflation forecasts for Romania compared to econometric models. Moreover, the development of computing resources aids us in extracting sentiments from large amounts of texts and including these sentiment measures in econometric models to improve forecasts. The sentiment analysis is made easily by using natural language processing (NLP), which combines the benefits brought by linguistics, computer science, and artificial intelligence. It aims to bridge the gap between human language and machine understanding. The main objective of this paper is to make inflation rate predictions for Romania on a short-term horizon (2023: Q1–2024: Q3) and show if ARDL models incorporating sentiment analysis provide better inflation forecasts compared to machine learning (ML) techniques.

While both econometrics and machine learning aim to build predictive models using explanatory variables, their approaches differ significantly. Econometrics focuses on statistically rigorous models that explain economic phenomena, often relying on well-defined theories. Machine learning, on the other hand, prioritizes powerful algorithms that can learn complex patterns directly from data, excelling in handling large and diverse datasets [8].

Recent research suggests that machine learning models can outperform traditional econometric methods in terms of prediction accuracy [9]. The main limitation is associated with the cost generated by the reduced interpretability. Econometricians can benefit from understanding these differences and exploring how machine learning tools can be integrated into their models to leverage the strengths of both approaches. This potential for collaboration could lead to more comprehensive and powerful economic analysis.

Given the context of the Romanian economy facing high inflation, the main aim of this paper is to empirically test if inflation rate forecasts based on econometric models that incorporate sentiment analysis outperform machine learning techniques. This target is achieved by introducing the theoretical background supported by previous studies in the next section and developing our own research based on quantitative methods.

2. Literature Review

Accurately forecasting inflation rates is crucial for understanding economic trends and making informed decisions. While traditional econometric models have long been used for this purpose, recent advancements in machine learning (ML) have opened up new possibilities. This section presents the evidence brought by inflation forecasting based on ARDL and SARIMA models and then introduces ML techniques by comparisons with the econometric approach in terms of forecast accuracy.

For example, the study of Osman et al. delves into the factors influencing inflation within the Saudi Arabian economy between 1980 and 2018. Utilizing the ARDL framework, the research examines the dynamic interconnections between inflation and key macroeconomic variables, including broad money supply, stock prices, real GDP, oil prices, global inflation, and the exchange rate. The analysis confirms the presence of a long-term equilibrium relationship among these variables. Empirical findings indicate that domestic inflation exhibits positive correlations with broad money supply, oil prices, and real GDP across both short-term and long-term horizons. On the other hand, stock prices exert a negative influence. While global inflation exerts a significant positive impact on domestic inflation in the long run, its short-term effect is insignificant. Notably, the exchange rate demonstrates no statistically significant impact on inflation in either the short or long term. Further analysis, employing impulse response functions and variance decomposition, reveals that broad money supply and global inflation are the primary drivers of long-term inflationary pressures in Saudi Arabia [10].

Another study made by Pahlavani and Rahimi investigates the key factors influencing inflation in Iran using annual time series data spanning 1971 to 2006. Employing the ARDL model, the analysis incorporates the unique characteristics of the Iranian economy and draws upon recent empirical research on inflation. The resulting model emphasizes the influence of liquidity, exchange rate fluctuations, real GDP growth, expected inflation, imported inflation, and the impact of the Iran–Iraq war. Empirical findings reveal that liquidity, exchange rates, expected inflation, and imported inflation are significant determinants of inflation in the long run. These variables also exhibit significant short-run effects on the inflation rate. Furthermore, the study demonstrates a positive correlation between the Iran–Iraq war and the inflation rate. Finally, the presence of a negative and statistically significant error correction term (−0.3995) suggests a relatively rapid adjustment process towards long-run equilibrium [11].

The research of Reda and Nourhan examines the dynamic causal relationships between inflation, the foreign exchange rate, money supply, and GDP within the Egyptian economy from the first quarter of 2005 to the second quarter of 2018. Employing the ARDL model, the study investigates the presence of a long-run equilibrium relationship between inflation and these key determinants. The findings suggest that both the exchange rate and growth in money supply exert a significant influence on inflation in Egypt, while real GDP exhibits no statistically significant impact [12].

The SARIMA models are also frequently used in practice for forecasting inflation rates. For Turkey, Saz proposed an optimal SARIMA model that is identified through a systematic modeling approach, employing the innovative Hyndman–Khandakar (HK) algorithm for stepwise model selection. The findings suggest a single, parsimonious SARIMA model that accurately captures the dynamics of the Turkish inflation process during 2003–2009 [13]. Lidiema compared the forecasting performance of two models: the SARIMA model and the Holt-Winters triple exponential smoothing method. The analysis utilizes published consumer price index data from the Kenya National Bureau of Statistics (KNBS) covering the period from November 2011 to October 2016. Based on the obtained results, the SARIMA model demonstrates superior forecasting accuracy compared to the Holt–Winters triple exponential smoothing method [14].

Utilizing the identified optimal SARIMA model, inflation rates in the Philippines were forecasted from April 2023 to March 2024 by Corpin et al. [15]. The projections suggest a gradual decline in the inflation rate over the next 12 months, from 6.93% in April 2023 to an estimated 4.85% in March 2024. These findings offer valuable insights that can inform government decision making regarding monetary policies, potentially contributing to a more stable and improved economic outlook for the Philippines. Hassan et al. indicated that a SARIMA (2,0,0) × (1,0,1)12 model for Bangladesh exhibits the best fit for the observed data. Utilizing this model, inflation forecasts were generated from February 2023 to December 2024. A comparative analysis between the predicted values and actual inflation data reveals a strong correlation. While minor discrepancies were observed, they did not significantly impact the predictive accuracy of the model, as the estimated model parameters fall within the 95% confidence interval [16].

Besides the econometric approach, a growing body of research has explored the application of ML techniques to economic forecasting, particularly in the context of inflation prediction [17,18,19,20]. Pérez-Pons et al. conducted a systematic literature review to compare the performance of ML and traditional econometric models. Their findings suggest that, in many cases, ML models outperform traditional methods. However, hybrid models that combine the strengths of both approaches may yield the best results [21].

It is important to recognize that the field of ML is rapidly evolving. While ML currently offers promising results for inflation forecasting, it is uncertain whether this advantage will persist over time. Therefore, a careful evaluation of both ML and traditional methods is essential for making informed decisions about forecasting techniques.

Ülke et al. compared machine learning (ML) models and time series models (univariate and multivariate) to forecast US inflation over 16 different horizons. Their results indicated that ML models outperformed time series models in seven cases, while time series models were more accurate in nine [22]. Additionally, multivariate time series models generally produce more accurate forecasts than univariate models [23]. While ML models, such as artificial neural networks (ANN), support vector machines (SVM), and k-nearest neighbors (k-NN), are less commonly used for inflation forecasting [22], they have shown promise in predicting other macroeconomic variables like exchange rates, income, and stock market return volatility [24,25,26,27]. According to Ülke et al., there is no universally superior model for inflation forecasting. The optimal approach depends on the specific research objectives and context [22].

Building on previous research that explored ML methods for inflation forecasting [28,29], studies by Aras and Lisboa, Pavlov, Baybuza, Choudhary and Haider, and Marcek and Marcek aim to demonstrate the viability of ML models compared to traditional methods and provide valuable solutions [30,31,32,33].

The findings of these studies suggest that ML models offer a powerful approach to inflation forecasting, particularly for short- and medium-term horizons. By considering non-linear relationships, ML models can often produce more accurate predictions than traditional methods. However, when compared to conventional benchmarks, ML models generally perform similarly, without a clear advantage. Non-linear ML models, in particular, may struggle to provide accurate forecasts.

Baybuza applied neural networks and support vector machines to forecast Russian inflation and employed Shapley value decomposition to interpret the neural network’s predictions. The results showed that both ML models performed comparably to or better than traditional methods, such as autoregression and ridge regression. Moreover, Shapley decomposition proved to be a useful tool for understanding the economic factors influencing the neural network’s forecasts [31].

Chakraborty and Joseph presented three case studies illustrating the potential applications of ML in central banking. These studies covered alert detection in financial institution balance sheets, UK CPI inflation forecasting, and identifying disruptive innovators in financial technology. While ML models often excel at prediction tasks, their ability to draw causal inferences remains an ongoing area of research [32].

In recent years, artificial neural networks (ANNs) have gained significant attention as a powerful tool for forecasting. Haider and Hanif, Estiko and Wahyuddin, Yusif et al., and Akhter have demonstrated that ANN models often outperform traditional univariate time series models like AR(1) and ARIMA in terms of forecasting accuracy [34,35,36,37].

The results of a few recent studies provide important insights on inflation forecasts. Araujo and Gaglianone investigated the application of ML algorithms to enhance inflation forecasting in the Brazilian context. The authors conducted a comprehensive out-of-sample forecasting analysis, encompassing multiple forecast horizons, a vast dataset comprising 501 time series, and a diverse set of 50 forecasting models [38]. This ensemble includes novel ML techniques developed for this research, established econometric models, and various forecast combination methodologies. To enhance interpretability and identify key predictive variables, the authors introduced tools to partially “open the black box” of ML models. While these findings do not reveal a single universally superior model, they demonstrate that ML approaches can significantly outperform conventional econometric models in terms of mean-squared error across numerous instances. Furthermore, the results provide compelling evidence of non-linearities within the inflation dynamics, highlighting their critical role in accurate inflation forecasting. The set of consistently top-performing forecasts frequently includes model combinations, tree-based methods such as Random Forest and XGBoost, breakeven inflation rates, and survey-based inflation expectations. Mirza et al. explored the predictive power of both traditional econometric models and ML algorithms in forecasting inflation within an emerging market economy, Pakistan. ML models, particularly random forest and gradient boosting algorithms, consistently demonstrate superior predictive performance compared to traditional econometric models [39]. Furthermore, the inclusion of foreign exchange reserves as a predictor variable significantly enhances the predictive accuracy of both traditional and ML-based inflation forecasting models. The SVR model in conjunction with walk-forward validation exhibits the highest predictive accuracy in Sri Lanka based on the data from the period 1988–2021, as Bandara and De Mel showed [40].

Another recent study of Malladi investigates the predictive power of various ML models and ordinary least squares regression (OLS) in forecasting annual U.S. inflation rates up to one year ahead. The analysis utilizes the FRED-MD dataset, encompassing 134 economic and financial variables spanning 1959 to April 2022. A comprehensive evaluation of 25 distinct ML algorithms alongside OLS is conducted. The findings demonstrate that the ML models consistently outperform OLS in predicting annual inflation. Furthermore, OLS fails to rank among the top 10 performers across all forecasting horizons. The study underscores the robust classification of the labor market as the primary driver of inflation forecasts by the ML models. The analysis reveals a significantly stronger influence of labor market indicators on inflation compared to housing or stock market factors [41].

Another study for inflation forecasting in the US investigates the efficacy of various ML algorithms, emphasizing a deeper understanding of the underlying mechanisms rather than solely replicating existing models. The results obtained by Kanaparthi suggest that ML models, particularly non-linear algorithms such as Random Forests and AdaBoost, demonstrate superior predictive accuracy, especially during periods of high inflation. While traditional economic indicators like price trends and the Phillips Curve maintain some explanatory power, the analysis highlights the amplified influence of inflation itself on price dynamics during these periods, suggesting unique and complex feedback loops at play [42].

Overall, it is difficult to definitively conclude which models, machine learning or traditional econometric, consistently outperform the other in inflation forecasting. Both approaches have their strengths and limitations, and the optimal choice often depends on specific economic conditions and the desired level of accuracy. As Takur et al. emphasized, considering country-specific economic factors is crucial for accurate inflation forecasting. A one-size-fits-all approach may not yield the best results. Therefore, a combination of both machine learning and traditional econometric models, along with a deep understanding of the underlying economic factors, may be the most effective strategy for predicting inflation [43].

3. Materials and Methods

The quarterly inflation rate is predicted on the horizon 2023: Q1–2024: Q3 using econometric models and machine learning techniques that are described in this section of the paper. Moreover, the quarterly unemployment rate is also used among the variables in machine learning methods.

The reports on inflation rate released by the National Bank of Romania are used to compute sentiment indexes based on the abstract of the section Inflation outlook. IntelliDockers, version 1 (https://zettacloud.ai/sentiment-analysis/ (accessed on 20 December 2024)) was used to calculate automatically the values of the sentiment index in each quarter. The text analytics platform, designed for businesses seeking to automate cognitive tasks, leverages natural language processing (NLP) to extract insights from unstructured data. The sentiment analysis, which categorizes text as negative, neutral, or positive. The underlying technology utilizes advanced AI techniques, including deep learning and neural networks, to power the platform’s sophisticated processing capabilities.

Economic sentiment is a secondary factor that reflects perceptions, not the direct cause of inflation. Therefore, in this paper, the connection is empirically supported. The relevance of sentiment analysis depends on the quality and specificity of the analyzed texts. For instance, reports may introduce noise into the data. To address this issue, several techniques were employed to clean and preprocess the text data before conducting sentiment analysis: removing irrelevant characters and punctuation and tokenization (this involves splitting the text into individual words or phrases, which can be used to identify patterns and trends).

As an example, we selected more sentences from the inflation report released in August 2023 by the National Bank of Romania: “According to the baseline scenario, the annual CPI inflation rate will decline steadily until the projection horizon, albeit at a slower pace than that recorded in the first half of this year. The projected values will remain above the upper bound of the variation band of the target. Specifically, the indicator is expected to decelerate in December 2023”.

For this text, IntelliDockers reported a positive sentiment with a score of 98% since there are key words supporting the optimistic perspective on future inflation: it “will decline steadily”, “will remain above the upper bound”, and “is expected to decelerate”. The positive sentiment on the future evolution of inflation is confirmed by the registered values of this indicator that decreased. In the next quarter, Romania’s annual CPI inflation rate continued its downward trajectory. This disinflationary trend was primarily driven by a decline in food prices, including both processed and unprocessed items. Romania also experienced a slowdown in economic growth in 2023, with the rate dropping to 2.1% (from 4.1% in 2022). Nevertheless, this pace remains one of the highest regionally and even in Europe. At the same time, the annual rate of CPI inflation has declined from the extremely high double-digit levels recorded in 2022.

On the other hand, the inflation report released in August 2022 provided pessimistic opinions related to the future evolution of this indicator in the third quarter of 2022: “the annual CPI inflation rate will remain high”, “the projected values being revised higher”. Indeed, the economy was affected by inflation pressure with a significant increase in prices, mainly because of the depreciation of national currency (RON) against the US dollar.

Sentiment analysis, when effectively integrated into econometric models, can potentially improve inflation forecasts by capturing subjective expectations and early warning signals, improving model fit. First, central bank reports often reflect the prevailing economic sentiment and the subjective expectations of policymakers and market participants regarding inflation. Sentiment analysis can quantify these subjective expectations, providing valuable insights into the likely direction of future inflation that may not be fully captured by traditional economic indicators. Second, changes in sentiment can sometimes precede actual changes in economic activity and inflation. By incorporating sentiment data, models can potentially identify emerging inflationary pressures or easing trends earlier, allowing for more timely policy responses. Third, sentiment data can potentially enhance the explanatory power of econometric models by capturing factors that are not fully captured by traditional economic variables. This can lead to improved model fit and more accurate forecasts.

3.1. Econometric Models

Autoregressive distributed lag (ARDL) models are a type of linear regression model employed to show dynamic relationships in time series data. These models incorporate both the dependent variable and exogenous variables with a specific lag (delay) among their regressors. ARDL models offer several advantages over traditional regression techniques, making them a valuable tool for economic and financial research. Some key advantages of ARDL models are:

• Unveiling short-run and long-term connections: ARDL models effectively capture both short and long-term relations between variables, even when the variables exhibit different orders of integration. This flexibility allows researchers to investigate the immediate and long-lasting impacts of different factors on the dependent variable.

• Handling mixed orders of integration: Unlike traditional cointegration techniques that require both series to be integrated of order I(1), ARDL models can accommodate series with mixed orders of integration, including I(0) and I(1). This versatility makes ARDL models applicable to a wider range of empirical scenarios.

• Small sample efficiency: ARDL models provide reliable and efficient estimations even with limited sample sizes. This robustness is essential when data availability is constrained [44].

While ARDL models offer numerous advantages, they also present certain methodological challenges that need to be addressed to ensure the validity of the results.

• Autocorrelation: The presence of the endogenous variable among the regressors can introduce autocorrelation in the error terms. To address this issue, robust estimation techniques such as seemingly unrelated regressions (SUR) can be employed.

• Multicollinearity: The inclusion of lagged values of the dependent and explanatory variables can lead to multicollinearity. Careful variable selection and appropriate data transformation methods can help mitigate this problem [45].

To effectively utilize ARDL models, certain assumptions and conditions need to be fulfilled:

• Dependent variable integration: The dependent variable is assumed to be integrated of order one (I(1)).

• Maximum integration order: The time series of the variables in the model cannot have an order of integration higher than one.

• Exogeneity assumption: The explanatory variables are assumed to be weakly exogenous, meaning they are not caused by the endogenous variable.

Given the endogenous variable Y and X₁, X₂, …, X_k as k predictors, the form of ARDL (p, q₁, q₂, …, q_k) model is represented as:

(1)$Y_t={\alpha }_0+{\alpha }_{1}t+\sum _{i=1}^p{\beta }_i{Y}_{t-i}+\sum _{j=1}^k\sum _{l_j=0}^{q_j}{\gamma }_{j,l_j}{X}_{j,t-l_j}+{\epsilon }_t$

${\alpha }_0$—constant;

${\alpha }_1$—trend coefficient;

${\beta }_i$—coefficients for lagged values of predictor;

${\gamma }_{j,l_j}$—coefficients for lagged values of the k regressors for j = 1, 2,…, k;

${\epsilon }_t$—errors.

Considering L as lag operator, the polynomials in L denoted by $\beta \left(L\right)$ and ${\gamma }_j(L)$ can be written as:

$\beta \left(L\right)=1-\sum _{i=1}^p{\beta }_i{L}^i$

${\gamma }_j\left(L\right)=1-\sum _{l_j=1}^{q_j}{\gamma }_{j,l_j}{L}^{l_j}$

Given the polynomials in L, the equation in (1) becomes:

(2)${\beta \left(L\right)Y}_t={\alpha }_0+{\alpha }_{1}t+{\sum }_{j=1}^k{\gamma }_j\left(L\right)X_{j,t}+{\epsilon }_t$

There are three alternative representations of the ARDL (p, q₁, q₂, …, q_k) model. The first representation is employed for estimating intertemporal dynamics, the second for deriving the post-estimation long-run equilibrium relationship, and the third is a reduction to the error-correction model representation conditional on the bounds test for cointegration, known as the Pesaran, Shin, and Smith test, or the PSS test [46]. All of these alternative representations are grounded in the Beveridge–Nelson decomposition, a procedure for decomposing a non-stationary time series into stochastic components (a permanent one following a random walk with drift and a transitory one that is stationary around a null average) [47].

Sentiment analysis, when integrated into econometric models like ARDL, can enhance inflation forecast accuracy for more reasons. First, sentiment analysis of reports from the National Bank of Romania can capture shifts in economic sentiment, such as growing concerns about inflation or changing consumer confidence. These shifts often precede actual changes in inflation, providing valuable early warning signals. Second, sentiment can act like a leading indicator for inflation. For example, if reports consistently express concerns about rising prices, it might signal future inflationary pressures even before they are reflected in official data on inflation. Third, traditional econometric models primarily rely on macroeconomic indicators like GDP, unemployment, and interest rates. Sentiment analysis incorporates subjective factors like experts’ expectations, which can significantly influence economic behavior and, consequently, inflation.

Seasonal autoregressive integrated moving average (SARIMA) models expand upon the capabilities of traditional ARIMA models by explicitly incorporating seasonality into the analysis. Seasonality considers repetitive patterns in series that occur at predictable intervals, denoted by “s” in this context. SARIMA models capture the seasonal component through three key elements:

• Autoregressive Seasonal Part (Order P): This part considers the impact of past seasonal values (lags of multiples of “s”) on the current value. The number of autoregressive seasonal terms is denoted by “P”.

• Moving Average Seasonal Part (Order Q): This component incorporates the impact of past seasonal errors (residuals) on the current value. The number of moving average seasonal terms is denoted by “Q”.

• Seasonal Differencing (D): Similar to differencing in ARIMA models, seasonal differencing removes non-stationarity caused by seasonal patterns. The number of seasonal differences imposed to achieve stationarity is denoted by “D”.

To determine the presence of seasonality and select appropriate values for P, Q, and D, it is crucial to describe the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the time series data. Significant spikes in these functions at lags that are multiples of “s” indicate the presence of seasonal components.

A stationary process SARIMA (p, q) × (P, Q)_s with D = 0 can be represented in its basic form as: $\varphi \left(L\right)\phi \left(L^s\right)\left(y_t-\mu \right)=\theta \left(L\right)\Theta \left(L^s\right){\epsilon }_t$.

This formula incorporates the non-seasonal ARMA components (p, q) alongside the seasonal components (P, Q)_s and the number of seasonal differences (D).

The seasonal components are written as: $\phi \left(L^s\right)=1-{\phi }_1{L}^s-{\phi }_2{L}^{2s}-\dots -{\phi }_P{L}^{P_s}$ and SMA(Q): $\Theta \left(L^s\right)=1-{\Theta }_1{L}^s-{\Theta }_2{L}^{2s}-\dots -{\Theta }_Q{L}^{Q_s}$.

The non-seasonal components are given by: AR(p): $\varphi \left(L\right)=1-{\varphi }_{1}L-{\varphi }_2{L}^2-\dots -{\varphi }_p{L}^p$ and MA(q): $\varphi \left(L\right)=1-{\theta }_{1}L-{\theta }_2{L}^2-\dots -{\theta }_q{L}^q$.

The non-seasonal components control for short-term dynamics, while p < 0.5 s and q < 0.5 s.

In the case of different periodicity for autoregressive and moving average seasonal component, (p, q) × (P, Q)_s,s^′ is determined by: $\varphi \left(L\right)\phi \left(L^s\right)\left(y_t-\mu \right)=\theta (L)\Theta (L^{s^{\prime }}){\epsilon }_t$.

S represents the autoregressive component seasonality, while; s’ is the moving average component seasonality.

The seasonal components are: SAR(P): $\phi \left(L^s\right)=1-{\phi }_1{L}^s-{\phi }_2{L}^{2s}-\dots -{\phi }_P{L}^{P_s}$ and SMA(Q): $\Theta \left(L^{s^{\prime }}\right)=1-{\Theta }_1{L}^{s^{\prime }}-{\Theta }_2{L}^{2s^{\prime }}-\dots -{\Theta }_Q{L}^{Q_{s^{\prime }}}$. The non-seasonal components are: AR(p): $\varphi \left(L\right)=1-{\varphi }_{1}L-{\varphi }_2{L}^2-\dots -{\varphi }_p{L}^p$ and MA(q): $\varphi \left(L\right)=1-{\theta }_{1}L-{\theta }_2{L}^2-\dots -{\theta }_q{L}^q$.

3.2. Machine Learning Methods

Random forests (RF) are a powerful algorithm employed for classification and regression. They are based on creating a collection of decision trees, like a forest, and then combining the predictions of each tree to obtain a final, more accurate prediction. Regression trees used in RF are nonparametric models with recursive partitioning of covariate space. Given K terminal nodes, the minimization of the sum of squared error is made to select the splits:

(3)${\pi }_{t+h}={\sum }_{k=1}^K{c}_k{I}_k(x_t;{\theta }_k)$

${\theta }_k$—parameters.

If $R_k\left({\theta }_k\right)$ is the k-th region, then:

$I_k\left(x_t;{\theta }_k\right)=\left\{\begin{array}{l}1,\mathrm{for}{x}_t\in R_k\left({\theta }_k\right)\\ 0,\mathrm{else}\end{array}\right.$

Bootstrapping is employed to aggregate the regression trees. A number of 1000 bootstrap samples is considered. In the case of one bootstrap sample, one may find a tree including K_b regions. The forecasts are computed as:

(4)${\widehat{\pi }}_{t+h}={\displaystyle \frac{1}{B}}{\sum }_{b=1}^B\left[{\sum }_{k=1}^{K_b}{\widehat{c}}_{k,b}{I}_{k,b}(x_t;{\widehat{\theta }}_{k,b})\right]$

SVMs (support vector machines) are also used in predicting future values based on historical data due to a few key reasons. First, real-world time series data often exhibit non-linear connections. SVMs can effectively capture these non-linearities by employing kernel functions, which project the data into a space of higher dimension where a linear relationship is represented. Second, time series data can have many features, incorporating past values and other relevant factors. SVMs are comfortable dealing with such high-dimensional data, reducing the need for feature selection (which can sometimes remove valuable information).

The support vector regression (SVR) represents the SVM utilization for regression and function approximation. The training dataset for features and output is represented as: $Z={\{X}_j,Y_j|j=1,2,\dots ,n\}$, $X_j\in {\mathfrak{R}}^q$ represent features values for training example j, $Y_j\in \mathfrak{R}$ shows target value for training example j, q represents the number of features in input vector, n represents the dimension of training dataset. The form of the regression model is shown below:

(5)$Y={\omega }^T·\theta \left(X\right)+b$

b—bias; $\theta \left(X\right)$—non-linear mapping function; $\omega$—weight vector that is computed by minimizing the regularized function in Equation (6) using the restriction in Equations (7)–(9):

(6)$\mathrm{m}\mathrm{i}\mathrm{n}\{0.5·{\omega }^2+C{\sum }_{j=1}^N({\epsilon }_j+{\epsilon }_j^{\left(*\right)})\}$

(7)$Y_j-\left\{\left({\omega }^T·\theta \left(X_j\right)\right)+b\right\}\le \psi +{\epsilon }_j,j=1,2,\dots ,N$

(8) ${\epsilon }_j,{\epsilon }_j^{\left(*\right)}\ge 0,j=1,2,\dots ,N$

(9) $f\left(X\right)={\sum }_{j=1}^N\left({\delta }_j-{\delta }_j^{*}\right)K\left(X,X_j\right)+b$

C—penalization coefficient corresponding to error; ${\epsilon }_j,{\epsilon }_j^{\left(*\right)}$—positive slack variables; $\psi$—function approximation accuracy; ${\delta }_j,{\delta }_j^{*}$—Lagrange multipliers.

Artificial neural networks (ANNs) are also powerful tools for time series forecasting. They are able to reflect non-linear complex relationships, allowing for scenarios where traditional statistical methods might struggle. First, unlike some statistical methods, ANNs do not require assumptions about linear relationships in the data. Second, the network architecture (number of layers, neurons) can be adapted to the specific time series. Third, ANNs can identify complex patterns and dependencies within the data that cannot be shown by simple models.

ANNs contain interconnected sample of neurons that are linked between them. For the j-th neuron, let us consider the n-dimension input vector $x=(x_1,\dots ,x_n)$ that corresponds to vector weights $w=(w_{j1},\dots ,w_{jn})$ and the bias $b_j.$ The output neuron ($y_j$) is computed based on activation function f as: $y_j=f\left(\sum _{t=1}^n{w}_{jt}{x}_t+b_j\right)$.

The inflation forecasts based on SVR and ANN use as inputs inflation in the previous period and the unemployment rate in the actual period. The versions including the sentiment index did not provide acceptable results in terms of forecast accuracy. Forecasts based on hybrid methods based on the combination between ML techniques and econometric models were constructed.

Walk forward validation (WFV) is employed as a cross-validation method. It is a time series evaluation technique that simulates empirical forecasting. It divides the data into sequential intervals, iteratively trains the model on past data, and tests it on the next interval. This process, repeated with an expanding training set, provides a more realistic assessment of model performance by accounting for time dependence and adapting to evolving patterns within the data. The optimal model configuration is determined based on the average performance across all iterations, typically measured by metrics like RMSE.

3.3. Forecast Accuracy

Forecast accuracy is assessed to determine the method that generated the best predictions. The usual accuracy measures are used: mean absolute error (MAE), mean error (ME), and root mean square error (RMSE).

(10)$ME={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n(y_i-{\widehat{y}}_i)$

(11) $MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|y_i-{\widehat{y}}_i|$

(12) $MSE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2$

(13) $RMSE=\sqrt{MSE}$

However, these measures do not have the capacity to detect the correct direction of change for the predicted inflation. Therefore, ${∆}_{t+h}^f$ and ${∆}_{t+h}$ defined below are computed to identify the predicted and actual direction of inflation change in the next quarter.

(14)${∆}_{t+j}^f=\left\{\begin{array}{c}1if{forecast}_{t+1}>{inflation}_t\\ -1otherwise\end{array}\right.$

(15) ${∆}_{t+j}=\left\{\begin{array}{c}1if{inflation}_{t+1}>{inflation}_t\\ -1otherwise\end{array}\right.$

The correct directional change is given by the percent of correct anticipated changes. Moreover, the Diebold–Mariano (DM) test is employed to compare forecasts in terms of accuracy. In this case, the comparison is made with a naïve forecast that considers as unchanged the value in the previous value.

4. Results and Discussion

Romania’s inflation rate surged in late 2022, reaching a peak of 16.37% as Figure 1 shows. This likely resulted from a hyperinflationary episode stemming from the global price increase of energy and food commodities due to the war in Ukraine. Romania, along with many other countries, relies on imports for these essential goods, making them more expensive domestically. Additionally, lingering supply chain disruptions caused by the recent pandemic limited the availability of various goods in 2022. This confluence of factors, restricted supply meeting high global demand, pushed prices in Romania upwards. In contrast, Romania experienced its highest disinflation in the third quarter of 2019, likely due to a decrease in excess consumer spending.

More tests for verifying the stationarity of series were applied: Augmented Dickey–Fuller (ADF) test, Phillips–Perron (PP) test, KPSS test, and Elliott, Rothenberg, and Stock (ERS) unit root test. The results in Table 1 suggest that all the data series are stationary in their first difference. This indicates that the series in level are integrated of order one (I(1)).

Cointegration is not supported between these series. The best model selected is ARDL(1,0,1), which is presented in Table 2. All the parameters of the model are statistically significant at the 10% level. The model is well specified according to the RESET test, while the errors are independent, homoscedastic, and normally distributed at the 1% significance level. Another ARDL model was proposed to check for the robustness of the results. Potential non-linearity was considered, but the interaction between inflation and the sentiment index was not relevant.

The best SARIMA model selected is SARIMA (1,0,0) (2,0,0)4 that is described in Table 3. All the coefficients are statistically significant at the 5% level. The errors are independent, homoscedastic, and normally distributed at the 1% significance level.

A grid search procedure is employed to identify the optimal parameters for SVM. In this case, the optimal parameters have the values: C = 8, gamma = 0.7, Coef0 = 3, epsilon = 0.1, kernel = poly, and degree = 1.

TimeSeriesSplit class is a part of the scikit-learn library in Python, version 3.13. that splits time series data into training and testing sets, ensuring that the testing data always come after the training data in chronological order. The description of the parameters is presented below:

C: This parameter controls the regularization strength. It determines the trade-off between fitting the training data closely and avoiding overfitting. A higher C value prioritizes fitting the training data, which can lead to overfitting, while a lower C value prioritizes avoiding overfitting, which might result in underfitting.

Gamma: This parameter influences the radius of influence of support vectors for specific kernels like “rbf”, “poly”, and “sigmoid”. It controls how much impact a single training point has on the decision boundary. A higher gamma value leads to a more localized influence, while a lower gamma value gives training points a broader influence.

Coef0: This parameter applies only to “poly” and “sigmoid” kernels. It defines the constant term in the kernel function, affecting how data points are transformed before similarity is measured.

Epsilon: It defines an epsilon-tube around the actual target values. Errors within this range are not penalized during training. It essentially controls the width of the margin of tolerance for errors in regression.

Kernel: This parameter specifies the similarity measure used between data points. It determines how data points are mapped into a higher-dimensional space for some kernel types (like “rbf” or “poly”). Common choices include “linear”, “poly”, “rbf”, and “sigmoid”.

Degree: This parameter is only applicable to the “poly” kernel. It defines the degree of the polynomial used in the kernel function, which can influence the model’s complexity.

Popular Python libraries like TensorFlow, PyTorch, and Keras (built on top of TensorFlow) provide high-level abstractions for building and training ANNs. We used the Sequential API from Keras to define the network architecture, specifying the number of layers, neurons, and activation functions for each layer.

According to Table 4, the ARDL model performed better in terms of forecast accuracy on the horizon 2023: Q1–2024: Q3. All the predictions based on the ARDL model correctly anticipated the direction of change in the inflation rate. Machine learning techniques performed better than the SARIMA model based on ME, MAE, and RMSE. Moreover, all the forecasts are more accurate than naïve forecasts, as the DM test indicated. Various combined models were proposed, but they performed worse than the ARDL model and worse than each of the methods in the combination.

SHapley Additive exPlanations (SHAP) assigns a contribution value to each feature in a model’s prediction. It helps to understand how much each feature influenced the outcome, providing a global understanding of the model’s behavior. Shapley values quantify the impact of each feature on the difference between a model’s prediction for a specific instance and the average prediction across all instances. When interpreting Shapley values, it is crucial to focus on the overall trend of their values rather than solely on their sign. A negative Shapley value does not inherently signify a negative contribution to the prediction; it merely indicates that the feature’s contribution deviates from the average prediction in a negative direction. However, the absolute magnitude of the Shapley value is essential for comparing the relative importance of different features in influencing the model’s output [48].

Averaged SHAP values are computed for SVR and ANN for the inputs: inflation in the previous period (averaged SHAP value is 2.345 for SVR and 3.267 for ANN) and unemployment in the actual period (averaged SHAP value is 1.556 for SVR and 1.332 for ANN). It is clear that inflation in the previous period is the most important variable in explaining actual inflation, while the unemployment rate is less important.

All in all, the ARDL model that included the sentiment index among exogenous variables determined the best quarterly inflation forecasts on the horizon 2023: Q1–2024: Q3. This type of model outperformed machine learning techniques. The results are contrary to those obtained by Malladi for US inflation when ML techniques outperformed OLS regressions [41]. The inclusion of the sentiment index in the ARDL models might explain the superiority of the econometric approach.

Accurate inflation forecasts have significant economic implications for various decision-makers. First, accurate inflation forecasts are crucial for policymakers to make informed decisions regarding monetary policy, fiscal policy, and other economic interventions. Models like ARDL and machine learning techniques can provide valuable insights into future inflation trends, enabling policymakers to adjust policies proactively to stabilize prices and promote economic growth. Second, businesses rely on accurate inflation forecasts to make informed decisions about pricing, investment, and inventory management. By understanding future inflation trends, businesses can adjust their strategies to mitigate risks and capitalize on opportunities. Third, investors and financial institutions use inflation forecasts to assess the risks and returns of various assets. Accurate forecasts can help them make informed investment decisions and manage their portfolios effectively. Fourth, consumers’ purchasing decisions are influenced by inflation expectations. Accurate forecasts can help consumers make informed decisions about spending and saving, thereby influencing aggregate demand and economic activity.

Sentiment indices can serve as early warning signals for potential inflationary pressures, allowing policymakers to take preemptive action. Sentiment index reflects subjective views of experts but is based on objective knowledge of the previous evolution of inflation. Our approach, based on ARDL models and sentiment index, is in line with the traditional paradigm of foresight: McNees and Donihue suggested that the predictions based on econometric models should be subjectively adjusted [49,50]. In this study, the adjustment was not made after getting the forecasts based on ARDL models but from the very beginning by including the subjective factor given by the sentiment index.

Few possible explanations are provided to explain why ML techniques performed worse. If the training data are biased, the model will learn and perpetuate those biases. This can lead to inaccurate predictions on data outside the training set’s limitations. On the other hand, empirical data often contain noise, inconsistencies that affect forecast accuracy. Models can overfit the training data, meaning they perform well on the training set but poorly on unseen data. This happens when the model has learned the training data too well, including its noise and idiosyncrasies. ML model performance could be improved by using large, diverse, and unbiased datasets. It is recommended to clean and preprocess the data to remove noise and handle missing values. The diversity of the training data could be increased by using back testing or cross-validation to evaluate the stability of sentiment’s contribution. Back-translation, synonym replacement, and creation of variations of existing sentences are practical solutions in this particular application. Incorrect forecasts based on ML can lead to inappropriate monetary policy decisions, potentially exacerbating inflation or slowing down economic growth unnecessarily. Inaccurate predictions can lead to incorrect pricing decisions, inventory management issues, and difficulties in financial planning. Unexpected inflation can erode purchasing power, impact consumption decisions, and increase uncertainty.

5. Conclusions

The technological progress allows us to use sophisticated statistical models to identify patterns and connections in economic series and predict inflation. Moreover, ML and artificial intelligence (AI) techniques are increasingly used to build predictive models that can show non-linear and complex connections and adapt to dynamic economic conditions.

Both econometric models and machine learning have their strengths and weaknesses in economic analysis. The choice of approach depends on the research objective. Econometrics is better suited for understanding causal relationships and interpreting economic phenomena when a strong theoretical foundation exists. Machine learning offers powerful prediction capabilities and can handle large, complex datasets, though interpretability can be a challenge. A growing trend is to combine these approaches to leverage the strengths of both for a more comprehensive analysis [51].

If Pérez-Pons et al. showed that in most of the comparative studies ML performs better than econometric models [21], this study shows that the incorporation of the sentiment index in econometric models might outperform the ML forecasts.

This paper presents few limitations. For example, only a few types of econometric models and machine learning methods were employed. The forecasts are made on a short horizon, and only a few forecast accuracy measures were calculated. In future studies, more forecasting methods and accuracy measures will be tackled. Moreover, combining these forecasts, each with its strengths and weaknesses, can be considered in a next study to check if overall forecast accuracy is improved. A clear limitation of this research is that the horizon is short. However, the actual economic context, characterized by a high degree of uncertainty with complex international challenges (the war in Ukraine, close to the Romanian border, post-pandemic recovery), does not recommend the elaboration of forecasts for long-term horizons.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request. Data were derived from the following resources available in the public domain: https://www.bnr.ro/en/12063-inflation-reports (accessed on 10 December 2024), http://statistici.insse.ro:8077/tempo-online/#/pages/tables/insse-table (accessed on 28 November 2024).

Conflicts of Interest

The author declares no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ARDL	Autoregressive distributed lag
SARIMA	Autoregressive integrated moving average
SVM	Support vector machine
ANNs	Artificial neural networks

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figure and Tables

Figure 1. The evolution of quarterly inflation rate in Romania (2023: Q1–2024: Q3).

Table 1

The results of tests for checking the stationarity of series.

Variable	Unit Root Test/Stationarity Test	Data in the Second Difference	Data in the First Difference	Data in Level
Inflation rate	ADF test	−7.3246 (<0.01)	−7.7983 (<0.01)	−0.5793 (0.4558)
	PP test	−27.8988(<0.01)	−7.9467(<0.01)	−0.6223 (0.4267)
	KPSS test	0.0229	0.2926	0.2005
	ERS unit root test	266.8365	0.9582	5.4253
Sentiment index	ADF test	−6.4463 (<0.01)	−14.7735 (<0.01)	−0.4997 (0.4987)
	PP test	−53.8365 (<0.01)	−41.3385 (<0.01)	−0.4566 (0.5211)
	KPSS test	0.2294	0.2345	0.1308
	ERS unit root test	5798.9322	1.6538	2.4329
Unemployment rate	ADF test	−8.3382 (<0.01)	−6.9963 (<0.01)	−0.4228 (0.5087)
	PP test	−31.1926 (<0.01)	−6.9654 (<0.01)	−0.4509 (0.5095)
	KPSS test	0.1833	0.1587	0.2265
	ERS unit root test	8.2393	0.0364	1.9196 1

¹ Source: own calculations. Note: reported statistics and p-values in brackets. For KPSS test, the critical value at 5% significance level is 0.463. Critical values for ERS test at 5% significance level: 3.02 for data in level, 3.01 for data in the first difference and 3.017 for data in the second difference.

Table 2

The estimations for ARDL(1,0,1) model.

Variable	Coefficient	p-Value	VIF	Coefficient	p-Value
Intercept	5.9238	0.0112	-	4.335	0.0564
Inflation (t − 1)	0.8345	<0.01	2.037	0.8558	<0.01
Sentiment index (t)	−5.2983	0.0302	1.967	−4.8867	0.0532
Sentiment index (t − 1)	−2.8785	0.0906	1.897	−2.9853	0.0866
inflation (t − 1) × sentiment index (t)	-	-	-	−0.0378	0.4338 1
Adjusted R-squared: 0.6792	RESET test: RESET = 2.1985, p-value = 0.1255	-	-	-	-
studentized Breusch–Pagan test: BP = 1.5933, p-value = 0.6544	Ljung–Box test: X-squared = 0.08337, p-value = 0.7566	Shapiro–Wilk normality testW = 0.9223, p-value = 0.159	-	-	-

¹ Source: own calculations in R.

Table 3

The results of estimation for SARMA (1,0,0) (2,0,0)4 model with non-zero mean.

Variable	Coefficient	p-Value
Mean	6.3233	0.0128
AR(1)	0.9122	<0.01
SAR(1)	−0.2326	0.045
SAR(2)	0.3113	0.030
Sigma squared	3.8760	-
R-squared: 0.786	Durbin–Watson statistic: 1.9623	Jarque–Bera stat. (p-value in brackets): 1.293 (0.552) 1

¹ Source: own calculations in EViews 9.

Table 4

Forecast accuracy evaluation for one-step-ahead inflation predictions (horizon: 2023: Q1–2023: Q4).

Method	ME	MAE	RMSE	DM Stat. Compared to Naive Forecasts	Percentage of Quarters with Correct Direction of Change for Forecasts
ARDL(1,0,1)	−1.511	1.544	1.919	−5.93 (<0.01)	100%
SARMA (1,0,0) (2,0,0)4	2.019	2.031	2.445	−4.72 (<0.01)	71.43%
Random forest	1.985	1.985	2.246	4.11 (<0.01)	71.43%
SVR for testing dataset	−1.789	1.794	2.051	4.59 (<0.01)	71.43%
ANN for testing dataset	2.009	2.006	2.342	3.93 (<0.01)	57.14%
ARDL(1,01) + random forest	1.993	2.033	2.332	4.557 (<0.01)	57.14%
ARDL(1,01) + SVR	2.003	2.001	2.044	(<0.01)	57.14%
ARDL(1,01) + ANN	2.156	2.998	3.003	(<0.01)	57.14%
SARMA (1,0,0) (2,0,0)4) + random forest	2.115	2.338	2.675	(<0.01)	57.14%
SARMA (1,0,0) (2,0,0)4 + SVR	2.166	2.535	2.996	(<0.01)	57.14%
SARMA (1,0,0) (2,0,0)4 + ANN	2.138	2.343	2.739	(<0.01)	57.14% 1

¹ Source: own calculations.

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