13 American Academic Scientific Research Journal for Engineering, Technology, and Sciences ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 https://asrjetsjournal.org/index.php/American_Scientific_Journal/index Could Econometric Models Predict Higher Inflation? Time Series Modelling and Inflation Rate Forecasting Mir Md. Moheuddin a*, Saddam Hossainb aDept. of Economics and Management, University of Brescia, Brescia-25122, Italy bBasic Science Division, World University of Bangladesh (WUB), Dhaka-1205, Bangladesh aEmail: mirmoheuddin2305@gmail.com bEmail: s.hossain656@gmail.com Abstract This research utilizes annual time series data on HICP to identify the most suitable econometric model for forecasting the inflation rate in Poland. In this study, we have compared and applied the Autoregressive Moving Average (ARMA) model and the Vector Autoregressive (VAR) model to predict the annual inflation rate using data from January 2002 to December 2020. Various methods are employed to determine the optimal model specifications, followed by the generation of forecasts within a rolling estimation window. The results consistently indicate that the ARMA (2, 0) model outperforms other specifications in forecasting Polish inflation. These findings suggest that the inflation rate is expected to continue its downward trend in the coming years. Consequently, this study offers valuable insights for guiding future actions and policymaking in response to prospective inflation scenarios. Keywords: Time series data; HCIP; ARMA; VAR; residual analysis; inflation modelling and forecasting. 1. Introduction Inflation is one of the most crucial economic indicators, employed by governments, stakeholders, politicians, and economists to understand the state of the economy. It is defined as the rate of change in the Consumer Price Index (CPI), a key term that reveals the overall price level of goods and services in a country's economy. We use the Harmonized Consumer Price Index (HICP), which is measured by generally accepted price levels, and it comprises a weighted average of various indexes, including unprocessed food, processed food, non-energy industrial goods, energy, and services, among others. Inflation determines the relative changes in the cost of living as the general price level of goods and services increases. ------------------------------------------------------------------------ Received: 8/16/2024 Accepted: 10/16/2024 Published: 10/26/2024 ------------------------------------------------------------------------ * Corresponding author. https://asrjetsjournal.org/index.php/American_Scientific_Journal/index American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 14 Inflation is a vital variable that directly impacts a country's economic situation. Understanding inflation forecasting techniques is crucial for making both fiscal and monetary policies to maintain stability and sustainable economic progress. Controlling inflation at a manageable level is necessary for the balanced development of the economy. Higher inflation can signal economic instability and excessive expenditure, potentially hindering economic development. Forecasting is a primary objective of econometric modeling, with two main methods: qualitative and quantitative. For central bankers, forecasting is crucial in the decision-making process, as it helps control inflation rates based on economic uncertainty. Making the right decisions regarding forecast values plays a pivotal role in navigating the complex economic system. This research aims to address central questions at the core of our forecasting analysis: which forecasting method should be utilized? We will compare two classes of methods, namely Univariate (ARMA) models and multivariate (VAR) models, to forecast Polish HICP inflation. The primary objective of this thesis is to model and forecast inflation rates in Poland as accurately as possible. We will compare two forecasting methods: the Autoregressive Moving Average (ARMA) model and the Vector Autoregressive (VAR) model, for predicting Polish HICP inflation as a rate of change. VAR is a standard tool used for long-term forecasting, while ARMA is relatively robust for short-term forecasting. We will use these models to forecast the yearly inflation rate and determine the highest forecast accuracy using a range of criteria to identify optimal forecasts in our study. 2. Literature review In the current era, statistical and economic modelling are widely utilized in empirical and theoretical research. Quality prediction is essential for helping investors understand market behaviors and improve their profitability based on applied modeling techniques. Policymakers want to know about the pattern of inflation rates and modelling is also helpful for them to apply appropriate policies to control inflation. Nowadays, these types of research play a significant role in developing economic sectors as well as leading the transparent relationship between variables and external factors.Khan, S., & Alghulaiakh, H. [1] applied and compared auto ARIMA (Auto Regressive Integrated Moving Average model). They showed two customized ARIMA (p, D, q) to get an accurate stock forecasting model by using Netflix stock historical data for five years. Szafranek K. [2] utilized a thick modelling approach to investigate the quality of the out-of-sample short-term headline inflation forecasts generated by a combination of bagged single hidden-layer feed-forward artificial neural networks. Facebook’s Prophet Forecasting Model and ARIMA Forecasting Model were utilized to compare their performance and accuracy on a dataset containing the confirmed cases, deaths, and recovered numbers, obtained from the Kaggle website (Hernandez-Matamoros, A., Fujita, H., Hayashi, T., & Perez-Meana, H.,). In their paper, the forecast models were then compared to the last 2 weeks of the actual data to measure their performance against each other, and the result showed that Prophet generally outperforms ARIMA, despite it being further from the actual data the more days it forecasts [3]. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 15 Abonazel, M. R. and his colleagues, [4] used the Box-Jenkins approach to build the appropriate Autoregressive- Integrated Moving-Average (ARIMA) model for the Egyptian GDP data. Egypt’s annual GDP data was obtained from the World Bank for the years 1965 to 2016. Khan, Firdos, Alia Saeed, and Shaukat Ali [5] demonstrated modelling and forecasting of new cases, deaths and recovery cases of COVID-19 by using the Vector Autoregressive model and their forecasted model results showed a maximum of 5,363/day new cases with a 95% confidence interval of 3,013–8,385 on 3rd of July, 167/day deaths with 95% confidence interval of 112–233 and maximum recoveries 4,016/day with 95% confidence interval of 2,182–6,405 in the next 10 days. Ramyar, S. and his colleagues, [6] tried to develop a multilayer perceptron neural network and trained historical data from 1980 to 2014 and using mean square error for testing data, the optimal number of hidden layer neurons was determined and the designed MLP neural network was used for estimation of the forecasting model. The exchange rate played a dominant role as a policy instrument in the 1990s (Gottschalk, J. and Moore, D.). In their papers, linkages between the short-term interest rate and inflation had been weak [7]. Moser, G. and his colleagues, [8] applied factor models proposed by Stock and Watson and VAR and ARIMA models to generate 12-month out-of-sample forecasts of Austrian HICP inflation and its subindices processed food, unprocessed food, energy, industrial goods, and services price inflation. Rumler, F., Moser, G. and Scharler, J. Reference [9] evaluated the performance of VAR and ARIMA models to forecast Austrian HICP inflation. Additionally, they investigated whether disaggregate modelling of five subcomponents of inflation is superior to specifications of headline HICP inflation. Uko, A. K and Nkoro, E [10] examined the relative predictive power of ARIMA, VAR and ECM models in forecasting inflation in Nigeria. Comparatively, they examined the performance of the forecasting ability of the models, and how well the simulated series track the actual data. Poulos and his colleagues, [11] described an automated forecasting system that encompasses an objective ARIMA method with the Holt-Winters procedure in a weighted averaging scheme. Arratibel, O. and his colleagues, [12] provided stylized facts on monetary versus non-monetary (economic and fiscal) determinants of inflation in these countries as well as formal econometric evidence on the forecast performance of a large set of monetary and non-monetary indicators. Hubrich K. [13] analyzed whether the accuracy of forecasts of aggregate euro area inflation can be improved by aggregating forecasts of subindices of the Harmonized Index of Consumer Prices (HICP) as opposed to forecasting the aggregate HICP directly. The analysis included univariate and multivariate linear time series models and distinguished between different forecast horizons, HICP components and inflation measures. Pufnik A and Kunovac D. [14] attempted to forecast changes in the index’s components to obtain a more detailed insight into the sources of future inflationary or deflationary pressures and to determine whether a forecast of developments in the total consumer price index obtained by aggregating forecasted values of the index’s components is more precise than a direct forecast. Junttila J. [15] forecasted the rate of future inflation in Finland for the time period of unregulated financial markets since the beginning of 1987. Wigati, Y., Rais, R., & Utami, I. T. [16] developed a model with the best time series Autoregressive Integrated Moving Average (ARIMA) to predict the movement of data Consumer Price Index (CPI) in Palu - Central Sulawesi. Mohamed J. [17] compared Autoregressive Integrated Moving Average (ARIMA) and regression with ARIMA errors, where the covariate is the time, to forecast Somaliland Consumer Price Index using monthly time series data from 2013 – 2020. Tandon H. and his colleagues. [18] developed for forecasting future COVID-19 cases in American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 16 India. In their papers, the time series analysis indicated that the cases would keep on increasing in India in the coming month as the peak. Djawoto, D. [19] researched to forecast the inflation rate in November 2010 with the Consumer Price Index (CPI) by using ARIMA. He also mentioned in his analysis that the ARIMA method is sufficient for short-term forecasting, whereas if used for long-term forecasting, the resulting value will tend to be constant. Kumar M and Anand M. [20] had been used a time series modelling approach (Box-Jenkins’ ARIMA model) in their study to forecast sugarcane production in India. 3. Data description The dataset used in this study is secondary data based on the Harmonised Consumer Price Index (HICP). This quantitative dataset covers the period from January 2002 to December 2020 and is published by the European Statistical System (ESS) online on the official website of the European Union (https://ec.europa.eu/eurostat). For the purpose of our research, we analyzed monthly HICP data to forecast Polish HICP using R software (version 3.6.3). We applied both the Autoregressive Moving Average (ARMA) model and the Vector Autoregressive (VAR) model using the R programming language in R Studio. 3.1 Autoregressive Moving Average (ARMA) model 3.1.1 Introduction of ARMA The econometric models include several popular time series models and model testing processes like simple autoregressive (AR) models, simple moving average (MA) models, mixed autoregressive moving-average (ARMA) models, unit-root nonstationary, regression models with time series errors, etc. The ARMA model comprises two processes: the Autoregressive (AR) model, which establishes a relationship between past and present values, and the Moving Average (MA) model, which indicates that the present value depends on past errors. For practical representation, it is desired to obtain models which drive parameters parsimoniously. The parsimony has been gained through a linear process with a small number of parameters for the autoregressive- moving average (ARMA) model. So, the ARMA model takes into consideration a parsimonious explanation of a (weakly) stationary stochastic process for two polynomials, one for the AR model and another for the MA model. In our study, we used the Autoregressive Moving Average model (ARMA) for analyzing univariate time series and forecasting. With the ARMA model, researchers can easily understand equilibrium and how inflation reverts to equilibrium. It also helps in analyzing the impact of shocks on our series and how inflation reacts to such shocks. 3.1.2 Autoregressive (AR) model The autoregressive model is the response variable as a function of past values (i.e., lag of the series). For a series 𝑦𝑡 , an autoregressive process of order p, AR(p) can be written as: 𝑦𝑡 = 𝛼0 + 𝛼1𝑦𝑡−1 + 𝛼2𝑦𝑡−2 + ⋯ … … + 𝛼𝑝𝑦𝑡−𝑝 + 𝑢𝑡; 𝑡 = 1,2, … … 𝑇 (1) https://ec.europa.eu/eurostat American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 17 Alternatively, we can write 𝑦𝑡 = 𝛼0 + ∑ 𝛼𝑖𝑦𝑡−𝑖 𝑝 𝑖=1 + 𝑢𝑡 . Where, 𝑦𝑡 is the response variable at time t, 𝛼𝑖,…., 𝛼𝑝 are the parameters of the model, 𝑦𝑡−𝑖 is the dependent variable at time t - i (i =1, 2, …., p), the random variable 𝑢𝑡 is the error term of white noise process at time t assumed independently and identically distributed (iid) normal random variables with 𝐸(𝑢𝑡) = 0 and 𝑣𝑎𝑟(𝑢𝑡) = 𝜎2; i.e. 𝑢𝑡 ~ 𝑖𝑖𝑑 𝑁(0, 𝜎2). Using the backshift operator B, the autoregressive (AR) model can be defined in the equivalent form, (1 − 𝛼1𝐵 − 𝛼1𝐵2 − ⋯ . . − 𝛼𝑝𝐵𝑝)𝑦𝑡 = 𝑢𝑡 Alternatively, we can define, 𝛼(𝐵)𝑦𝑡 = 𝑢𝑡 Besides, all preceding values of this model may have an additive impact on the series of 𝑦𝑡 and so on. So, this model is also called a long-term memory model. 3.1.3 Moving-average (MA) model This model is a linear form of a series model that is dependent on past errors. For a time-series 𝑦𝑡 , a moving average process of order q, MA(q) can be expressed as follows. 𝑦𝑡 = 𝑢𝑡 − 𝛽1𝑢𝑡−1 − 𝛽2𝑢𝑡−2 − ⋯ … . . − 𝛽𝑞𝑢𝑡−𝑞; 𝑡 = 1, 2, … … 𝑇 (2) Or we can write 𝑦𝑡 = 𝑢𝑡 − ∑ 𝛽𝑖𝑢𝑡−𝑖 𝑞 𝑖=1 . Whereas 𝛽1, 𝛽2, … … , 𝛽𝑞 are model parameters and q is the lags of this MA model. The moving-average model is considered as the past errors of explanatory variables. Hence, only q errors can affect the dependent variable (𝑦𝑡), but larger order errors do not affect on 𝑦𝑡 . That means it is a short-term memory model. Applying the backshift operator B𝑢𝑡= 𝑢𝑡−1, the moving-average (MA) model can define in the equivalent form as 𝑦𝑡 = (1 − 𝛽1𝐵 − 𝛽1𝐵2 − ⋯ . . − 𝛽𝑞𝐵𝑞) 𝑢𝑡 Or more succinctly as 𝑦𝑡 = 𝛽(𝐵)𝑢𝑡 Now, adding equations (1) and (2), we can get ARMA (p, q) i.e., the combination of the AR(p) and MA (q) model which constructs the ARMA (p, q) model is written below, 𝑦𝑡 = 𝛼0 + 𝛼1𝑦𝑡−1 + 𝛼2𝑦𝑡−2 + ⋯ … … + 𝛼𝑝𝑦𝑡−𝑝 + 𝑢𝑡 − 𝛽1𝑢𝑡−1 − 𝛽2𝑢𝑡−2 − ⋯ … . . − 𝛽𝑞𝑢𝑡−𝑞 Alternatively, 𝑦𝑡 = 𝛼0 + ∑ 𝛼𝑖𝑦𝑡−𝑖 𝑝 𝑖=1 + 𝑢𝑡 − ∑ 𝛽𝑖𝑢𝑡−𝑖 𝑞 𝑖=1 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 18 Where, 𝑦𝑡 and 𝑢𝑡 are the actual value and random error at time t; 𝛼𝑚(𝑚 = 1,2,3, … . . , 𝑝) and 𝛽𝑛(𝑛 = 1,2,3, … … , 𝑞) are model’ parameters, p and q are considered as integers for the order of AR (p) and MA (q), the error term 𝑢𝑡 is assumed to be independent and identically distributed (i.i.d) with a mean value of zero and constant variance 𝜎2 . Again, using the backward shift operator, the ARMA (p, q) can be expressed in the following form, 𝛼(𝐵)𝑦𝑡 = 𝛽(𝐵)𝑢𝑡 Where, 𝛼(𝐵) = 1 − 𝛼1𝐵 − 𝛼1𝐵2 − ⋯ . . − 𝛼𝑝𝐵𝑝 and 𝛽(𝐵) = 1 − 𝛽1𝐵 − 𝛽1𝐵2 − ⋯ . . − 𝛽𝑞𝐵𝑞 3.1.4 Identification of correlograms To assess graphical stationary and determining the appropriate values for the order p in AR(p) and the order q in MA(q), the formal test procedure is applied using the autocorrelation function (ACF) and partial autocorrelation function (PACF). The ACF measures the correlation between observations in a time series for a set of lags. It identifies which lags have significant correlations, helping us understand the patterns and properties of the time series and enabling us to determine the order of an MA model. The basic definition of ACF is given below, The autocorrelation function (ACF) at lag k, denoted as 𝑠𝑘, of a stationary stochastic process is defined as 𝑠𝑘 = 𝜌𝑘 𝜌0 Where, 𝜌𝑘 = 𝑐𝑜𝑣(𝑦𝑡 , 𝑦𝑡+𝑘) for all t and 𝜌0 is the variance of the stochastic process. Mathematically, the mean of a series 𝑦1, … . , 𝑦𝑇 is in the following, �̅� = 1 𝑇 ∑ 𝑦𝑖 𝑇 𝑡=1 The auto covariance function at lag k, for k ≥ 0, of the time series is defined by 𝜌𝑘 = 1 𝑇 ∑(𝑦𝑡 𝑇−𝑘 𝑡=1 − �̅�)(𝑦𝑡+𝑘 − �̅�) = 1 𝑇 ∑ (𝑦𝑡 𝑇 𝑡=𝑘+1 − �̅�)(𝑦𝑡−𝑘 − �̅�) Now, the autocorrelation function (ACF) at lag k, for k ≥ 0, of the time series is defined by, 𝑠𝑘 = 𝜌𝑘 𝜌0 = ∑ (𝑦𝑡 𝑇−𝑘 𝑡=1 − �̅�)(𝑦𝑡+𝑘 − �̅�) ∑ (𝑦𝑡 − �̅�)2𝑇 𝑡=1 Where, 𝜌0 is the variance of the time series and the plot of 𝑆𝑘 with lag, k is called a correlogram. The PACF tells us about the partial correlation between the series and its lag values. It shows the correlation American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 19 between the observation at the current time and at the preceding time spots and finds out the order of an AR model. The theoretical PACF is in the following form, 𝑟𝑘,𝑘 = 𝑠𝑘 − ∑ 𝑟𝑘−1,𝑖 𝑠(𝑘 − 𝑖)𝑘−1 𝑖=1 1 − ∑ 𝑟𝑘−1,𝑖 𝑠(𝑖)𝑘−1 𝑖=1 Where, 𝑟𝑘,𝑖 = 𝑟𝑘−1,𝑖 − 𝑟𝑘,𝑘𝑟𝑘−1,𝑘−𝑖 for i = 1, 2, …., k-1 and 𝑠𝑘 is the autocorrelation function. 3.1.5 Unit root (UR) test This test is conducted to check whether the time series data is stationary or not. The Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test were utilized for the UR test. The test is considered on the assumption that a time series data 𝑦𝑡 follows a random walk: 𝑦𝑡 = 𝜎𝑦𝑡−1 + 𝑢𝑡 Where, 𝜎 = 1 and 𝑦𝑡−1 will be subtracted from both sides. We get, ∆𝑦𝑡 = 𝛿𝑦𝑡−1 + 𝑢𝑡 and 𝛿 = 𝜎 − 1. The null hypothesis is H0: 𝛿 = 0, i.e., the series is non-stationary and therefore the alternative hypothesis is H1: 𝛿 < 0, i.e., the series is stationary and 𝜎 < 1. 3.1.6 Model specification We use information criteria to select the best models in terms of the in-sample fit. AIC: Akaike information criterion (Akaike, 1974), AIC = −2 𝑙 𝑇 + 2 𝐾 𝑇 SIC: Schwarz information criterion is known as Bayesian information criteria (Schwarz, 1997), BIC = −2 𝑙 𝑇 + 2 𝐾 𝑇 ln (𝑇) HQIC: Hannan-Quinn information criterion (Hanna and Quinn, 1979), HQIC = −2 𝑙 𝑇 + 2 𝐾 𝑇 ln (𝑙𝑛𝑇) Where K is the number of estimated parameters and 𝑙 = ln (𝜓) is the log-likelihood. Generally, the BIC penalizes free parameters stronger than the AIC and good models are gained by decreasing the value of AIC, HQIC, and BIC and optimizing the log-likelihood. We choose the best model with the lowest information criteria (IC). Note that: K(BIC) ≤ K(HQIC) ≤ K(AIC), IC depends on the fit (log-likelihood) and penalty on the number of parameters. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 20 3.1.7 Likelihood ratio (LR) test In this test, we can compare the model fit. H0: the fit of the big ARMA (m parameters more) is the same as the fit of the small ARMA model. LR = −2(𝑙𝑟 − 𝑙𝑢)~𝜒2(𝑚) Where m is the number of additional parameters, and l is the log-likelihood for restricted (small) and unrestricted (big) models. 3.1.8 Model validation and forecasting using the ARMA model The estimated model will be considered a good model if it simulates the historical behavior properly. The quality of the residual term is assessed through diagnostic tests based on residuals. The Ljung-Box test is utilized to verify the overall adequacy of the estimated model. One notable feature of the ARMA model is its ability to make accurate forecasts. To predict the future values of the time series, we use the given point forecast equation (calculated recursively), 𝑦𝑇+ℎ 𝑓 = 𝛼0 + ∑ 𝛼𝑖𝑦𝑇+ℎ−𝑖 𝑓 𝑝 𝑖=1 + 𝑢𝑡 − ∑ 𝛽𝑖𝑢𝑇+ℎ−𝑖 𝑓 𝑞 𝑖=1 Where, 𝑢𝑇+ℎ 𝑓 = 0 for h > 0 and 𝑦𝑇+ℎ 𝑓 = 𝑦𝑇+ℎ for h ≤ 0. The forecast error is: 𝑦𝑇+ℎ − 𝑦𝑇+ℎ 𝑓 = ∑ 𝜃ℎ 𝐻−1 ℎ=0 𝑢𝑇+𝐻−ℎ Also, the forecast variance (only due to stochastic term) is: Var (𝑦𝑇+ℎ) = ∑ 𝜃ℎ 2𝐻−1 ℎ=0 3.2 The Vector Autoregressive (VAR) model 3.2.1 Introduction of VAR VAR is one of the most successful, flexible models applied for multivariate time series data. The VAR model establishes dynamic relationships within the structural model without limitations, where all variables are jointly considered endogenous. The VAR model provides a way to the structural form of higher-scale simultaneous equations for which it is differed from the system of simultaneous equations. In the context of the VAR model, the availability of lagged values of dependent variables enables it to make robust predictions about future American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 21 economic developments. This model's purpose is not solely to assess one-way relationships among variables but also to reveal linkages between variables through lagged effects. It is a valuable tool extensively used to express the dynamic behavior of macro econometric modeling, financial time series analysis, and forecasting. So, the model is an n-variables, n-equations model, which expresses each variable as a linear function of its past values, the past values of all other variables being treated, and a serially uncorrelated error term. VAR models maintain a coherent and credible approach to describing data, inferring structural relationships, and conducting policy analysis. 3.2.2 The Vector Autoregressive (VAR) model For a vector of time series data 𝑦𝑡 , a general VAR (p) model with n endogenous variables, p lags, and m exogenous variables can be expressed mathematically in the following form 𝑦𝑡 = 𝐴1𝑦𝑡−1 + 𝐴2𝑦𝑡−2 + ⋯ … + 𝐴𝑝𝑦𝑡−𝑝 + 𝐵𝑥𝑡 + 𝑢𝑡 , 𝑢𝑡~𝒩(0, ∑), 𝑡 = 1, 2, … , 𝑇 Where, 𝑦𝑡 = (𝑦1,𝑡 , 𝑦2,𝑡 , … , 𝑦𝑛,𝑡) is an n × 1 vector of endogenous data, 𝐴1, 𝐴2, … . , 𝐴𝑝 are p matrices of parameters in dimension 𝑛 × 𝑛, B is a 𝑛 × 𝑚 matrix, and 𝑥𝑡 is a 𝑚 × 1 vector of exogenous regressors that can be constants terms, time trends, or exogenous data series, 𝑢𝑡 = (𝑢1,𝑡 , 𝑢2,𝑡 , … . , 𝑢𝑛,𝑡) is a vector of residuals (serially uncorrelated or independent) with a time-invariant covariance matrix ∑ . The simplest VAR (p) model with two variables and lag p =1, can be written in matrix form (more compact notation) as [ 𝑦1,𝑡 𝑦2,𝑡 ] = [ 𝑎1,1 𝑎1,2 𝑎2,1 𝑎2,2 ] [ 𝑦1,𝑡−1 𝑦2,𝑡−1 ] + [ 𝑢1,𝑡 𝑢2,𝑡 ] Also, we can write the two systems of equations, 𝑦1,𝑡 = 𝑎1,1𝑦1,𝑡−1 + 𝑎1,2𝑦2,𝑡−1 + 𝑢1,𝑡 𝑦2,𝑡 = 𝑎2,1𝑦1,𝑡−1 + 𝑎2,2𝑦2,𝑡−1 + 𝑢2,𝑡 3.2.3 Estimating VAR model In compact form, the VAR model can be written (Source: Dieppe, 2016), 𝑦𝑡 = 𝐴1𝑦𝑡−1 + 𝐴2𝑦𝑡−2 + ⋯ … + 𝐴𝑝𝑦𝑡−𝑝 + 𝐵𝑥𝑡 + 𝑢𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 2, … , 𝑇 (1) Where, 𝑢𝑡 = (𝑢1,𝑡 , 𝑢2,𝑡 , … . , 𝑢𝑛,𝑡) is a vector of residuals following a multivariate normal distribution. 𝑢𝑡 ~ 𝒩(0, ∑) (2) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 22 For further calculations, a reformulation of (1) consists of a transpose form as 𝑦𝑡′ = 𝑦′ 𝑡−1 𝐴′ 1 + 𝑦′ 𝑡−2 𝐴′ 2 + ⋯ … + 𝑦′ 𝑡−𝑝 𝐴′ 𝑝 + 𝑥′ 𝑡𝐵′ + 𝑢′𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑡 = 1, 2, … , 𝑇 Reformulating the model into a single matrix, we get VAR(p) in matrix notation, 𝑌 = 𝑋𝐶 + 휀 (3) An estimate �̂� (LS estimates) of the parameter C in (3) obtains from: �̂� = (𝑋, 𝑋)−1𝑋, 𝑌 (4) An OLS estimate 휀̂ (residuals) of the residual matrix 휀 can be obtained from the direct application of (3): 휀̂ = 𝑌 − 𝑋�̂� Also, a (degree of freedom-adjusted) estimates ∑̂ of the covariance matrix ∑ in (2) can be gained from: ∑̂ = 1 𝑇−𝑘−1 (휀̂ , 휀̂) Where k = 1+np is the number of parameters in each equation. 3.2.4 Model validation and forecasting When the model's parameters are estimated, it is essential to perform diagnostic checks to assess the quality of the time series model. After investigating and identifying the data-generating process for time series models, it becomes useful for forecasting h steps ahead. Generally, the point forecast for horizon h is given below, 𝑦𝑡+ℎ|𝑡 = 𝐴1𝑦𝑡+ℎ−1|𝑡 + 𝐴2𝑦𝑡+ℎ−2|𝑡 + ⋯ … + 𝐴𝑝𝑦𝑡+ℎ−𝑝|𝑡 The VAR moving average (VMA) representation is given as 𝑦𝑡 = 𝜙0𝑢𝑡 + 𝜙1𝑢𝑡−1 + 𝜙2𝑢𝑡−2 + ⋯ .. The error of forecast for horizon h due to future shocks, 𝑦𝑡+ℎ − 𝑦𝑡+ℎ|𝑡 = 𝜙0𝑢𝑡+ℎ + 𝜙1𝑢𝑡+ℎ−1 + ⋯ . . +𝜙ℎ−1𝑢𝑡+1 The variance of forecast error for horizon h due to future shocks, 𝑉𝑎𝑟𝑡(𝑦𝑡+ℎ) = 𝜙0∑𝜙′ 0 + 𝜙1∑𝜙′ 1 + ⋯ + 𝜙ℎ−1∑𝜙′ ℎ−1 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 23 3.3 Forecasting accuracy measurements The predicting performance of the ARMA and VAR model is compared by measuring forecasting accuracy statistics like Mean Forecasting Error (MFE) and Root Mean Square Forecast Error (RMSFE). The lower the calculated value of MFE, RMSFE, the better the model and hence, the forecasted values are considered accurate. Mathematically, The mean forecast error (MFE) for horizon h is given below: 𝑀𝐹𝐸ℎ = 1 𝑇ℎ ∑ (𝑦𝑡+ℎ − 𝑦𝑡,ℎ 𝑓 𝑇−ℎ 𝑡=𝑇1+1 ) The root mean squared forecast error (RMSFE) for horizon h is: 𝑅𝑀𝑆𝐹𝐸ℎ = √ 1 𝑇ℎ ∑ (𝑦𝑡+ℎ − 𝑦𝑡,ℎ 𝑓 )2 𝑇−ℎ 𝑡=𝑇1+1 Where, 𝑇ℎ = 𝑇 − 𝑇1 − ℎ + 1 Moreover, I applied the Diebold-Mariano test for equal forecast accuracy. Forecast errors from two competing models (ARMA & VAR) are: 𝑒1𝑡,ℎ = 𝑦𝑡+ℎ − 𝑦1𝑡,ℎ 𝑓 and 𝑒2𝑡,ℎ = 𝑦𝑡+ℎ − 𝑦2𝑡,ℎ 𝑓 The quadratic loss differential: 𝑑𝑡,ℎ = 𝑒1𝑡,ℎ 2 − 𝑒2𝑡,ℎ 2 The null of equal forecast accuracy: H0: E(𝑑𝑡,ℎ) = 0. Test statistic of Diebold-Mariano (DM) is: DM = 𝑑𝑡,ℎ̅̅ ̅̅ ̅̅ √ 𝑆 𝑇ℎ ~𝑁(0,1) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 24 4. Results and discussions for the ARMA model 4.1 Time series plot Data preprocessing is important for forecasting time series datasets. We have extracted the higher inflation period from my database to model and decide whether it plays an important role in shaping inflation. We use the vintage database for the study of forecasting inflation. After filtering and clearing my dataset, We figure out that the use of preprocessing data reduces the risk of computational problems as well as makes the training procedure more efficient. Figure 1 indicates the time series plot of the Polish HICP series from January 2002 to December 2020. The series exhibits a fluctuating pattern over time. It is seen that our time series data looks non- stationary. The mean does not seem to be constant over time and the series displays an upward and downward trend in time. In the below figure, the x-axis is the year, and the y-axis represents the Polish HICP inflation rate. Source: Figure 1 was created using R language (version 3.6.3) in R Studio, and the same process was applied to all other figures in this chapter. Figure 1: Time series plot of Polish HICP 4.2 Auto Correlation Function (ACF) Figure 2: ACF plot of the inflation data American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 25 From Figure-2, We can see that the spikes of the ACF cross the cut-off line, suggesting that the current level of HICP is significantly autocorrelated with its lagged values. So, there is autocorrelation in the series. For our data series, it is evident that most of the lags fall outside the 95% confidence interval level, and lags from the first one up to the 23rd are statistically significant. 4.3 Partial Auto Correlation Function (PACF) In Figure-3, we observe that the 1st lag and 2nd lag are statistically significant in the data series, while all other lags either lack statistical significance or show only marginal significance, notably at the 13th and 25th lags. This pattern is visually represented in the PACF plot, which exhibits two prominent spikes at lag 1 and lag 2, effectively indicating the points where the data deviates significantly from the baseline. Figure 3: PACF plot of HICP series 4.4 Unit Root (UR) test Here, we will apply the statistical test called Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test to see whether our data series is stationary or not. By applying the Augmented Dickey-Fuller (ADF) test, we can see the following results. Source: the given table has been made by using Microsoft Office carefully and we follow the same manner to create other tables as well. Table 1: ADF test of original series Level ADF statistic Critical values (5% sig.) Decision Drift (intercept) -2.49 -2.88 Non-Stationary American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 26 From Table 1, it is seen that the value of test-statistic t is: -2.49 and DF critical value is -2.88 at a 5% significant level. Since the calculated statistic is higher than the critical value for the ADF test, the null (H0) about non- stationary cannot be rejected. Hence, the time series is non-stationary in the optics of the ADF test. By applying Phillips-Perron (PP) test, we see the following results of our data. Table 2: PP test of original series Level PP statistic Critical values (5% sig.) Decision Constant (intercept) -2.20 -2.87 Non-Stationary From Table 2, it can be said that the value of test-statistic t is: -2.20 and PP critical value is -2.87 at a 5% significant level. Since the calculated statistic is higher than the critical value for the PP test, the null (H0) about non-stationary cannot be rejected. Hence, the time series is non-stationary in the optics of the PP test. After conducting unit root (UR) testing, we observed that utilizing non-stationary time series data is more suitable for modeling our restricted dataset. Using stationary time series data poses challenges in our cases due to the lower power of UR tests, indicating high persistence in annual inflation and forecasts stabilizing at a high value without returning to the mean. 4.5 ARMA model specification We start by estimating ARMA (2, 2) model. Table 3: Three methods result using ARMA (2, 2) Methods ar1 ar2 ma1 ma2 Constant CSS-ML 0.584 0.353 0.7617 0.263 2.285 CSS 0.257 0.665 1.088 0.353 1.926 ML 0.200 0.729 1.102 0.303 2.316 Referring to Table-03, we prioritize simplicity in model selection, aiming for a minimal number of parameters. We prefer straightforward models; for instance, if a one-lag model adequately describes inflation, we favor it over a two-lag model, even if the second parameter is close to zero or statistically insignificant. My objective is to limit the complexity of the models while ensuring a good fit to the data. However, it's important to note that different methods can yield varying results. Despite this variability, the parameters we obtain are reliable. We initiated the modeling process with the full-maximum likelihood approach, which provides a solid starting point. Subsequently, we employ conditional maximum likelihood (OLS estimator) and then return to the full maximum likelihood estimation (ML). In these models, a constant term holds significance. This constant represents an equilibrium value, which plays a American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 27 pivotal role in forecasting and ARMA modeling. Specifically, in this model, the equilibrium value stands at 2.285 (for instance), indicating that inflation in this context gradually returns towards this equilibrium. It's worth noting that the equilibrium values differ across various cases, signifying distinct patterns of equilibrium return. 4.6 Information criteria Table 4: Bayesian information criteria (BIC) Coefficient ma0 ma1 ma2 ma3 ar0 860.32 595.96 425.19 333.99 ar1 183.57 158.62 160.21 165.61 ar2 154.85 160.27 165.54 170.71 ar3 160.07 163.50 170.07 166.24 Table5: Akaike information criteria (AIC) Table 6: Hannan-Quinn information criteria (HQIC) Coefficient ma0 ma1 ma2 ma3 ar0 853.10 586.59 412.68 318.15 ar1 176.29 150.83 150.22 153.58 ar2 146.78 150.11 152.38 154.93 ar3 150.06 152.23 155.19 157.07 Referring to Table-4(a-c), we determine the appropriate lag structure by selecting a maximum of three lags. We use three different information criteria—Akaike Information Criteria (AIC), Normalized Bayesian Information Criteria (BIC), and Hannan-Quinn Information Criteria (HQIC)—to identify the best model. Based on the information criteria, the results are as follows: BIC suggests that ARMA (2, 0) is the best model with the lowest value of 154.85, AIC favors ARMA (3, 3) with a value of 138.91, and HQIC points to ARMA (2, 0) with a value of 146.78. In summary, ARMA (3, 3) is a larger model with three lags, while ARMA (2, 0) is a more parsimonious model with only two lags. Ultimately, ARMA (3, 3) emerges as a good choice for analyzing and predicting inflation in this context. Coefficient ma0 ma1 ma2 ma3 ar0 853.46 585.67 411.47 316.84 ar1 173.30 144.92 143.08 145.06 ar2 141.17 143.17 145.02 146.77 ar3 142.99 143.01 146.16 138.91 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 28 4.7 Likelihood ratio (LR) test After performing the LR test, we obtained a test statistic of 10.093 with a p-value of 0.038, which is less than the significance level of 0.05. This indicates a significant difference between the larger ARMA model and the smaller ARMA model. The restriction imposed on the model is indeed having a significant impact on the fit to the data. In other words, the larger ARMA model and the smaller ARMA model show notably different fits to the data as a result of this restriction. 4.8 Estimating ARMA (2, 0) & ARMA (3, 3) We am now comparing the ARMA (2, 0) and ARMA (3, 3) models to determine which one is a better fit. Opting for the smaller ARMA (2, 0) model offers simplicity with only two parameters, making them easier to interpret. Conversely, the larger ARMA (3, 3) model introduces more complexity, with parameters like the first auto regression parameter (ar1) taking values such as -0.938 and 0.865, which can be challenging to intuitively understand. Additionally, working with a larger model can be more sophisticated due to the intricacies of its moving average representation. Table 7: Results from two ARMA models Model Log- likelihood AIC AICc BIC ME RMSE MAE MPE MAPE MASE AF1 ARMA(2,0,0) -66.5 141 141 155 - 0.005 0.322 0.242 0.104 21.4 0.176 0.005 ARMA(3,0,3) -61.5 139 140 166 - 0.006 0.314 0.238 0.092 20.1 0.173 0.020 In Table 5, we present the results obtained from ARMA models with varying orders. To identify the best model, we consider several metrics, including log-likelihood, AIC, AICc, BIC, ME, RMSE, MAE, MPE, MAPE, MASE, and AF1. In general, a superior model exhibits the highest log-likelihood and the lowest values for AIC, AICc, BIC, ME, RMSE, MAE, MPE, MAPE, MASE, and AF1. Our aim is to minimize all these metrics when forecasting HICP inflation in Poland.After careful evaluation, the ARMA (2, 0) model emerges as the optimal choice. It not only meets the aforementioned criteria but also adheres to the parsimony principle, favoring the inclusion of the fewest parameters in the model. Therefore, we confidently select the ARMA (2, 0) model for our forecasting purposes. 4.9 Diagnostic checking of residuals Figure-04(a, b) provides a visual diagnostic assessment of the ARMA (2, 0) model's fit to the time series data. Notably, the residuals of this model exhibit clear modeling behavior toward the end of the sample, indicating American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 29 that standard errors remain consistent in both mean and variance over time. Additionally, the ACF and PACF plots of the residuals demonstrate white noise characteristics, with exceptions being significant spikes at lag 12 for ACF and at lag 12 and lag 24 for PACF. These spikes could be attributed to random factors, possibly related to base effects. However, the Q-Q plot reveals that the residuals are not normally distributed, displaying a lack of fit in the tails of the distribution. Figure 4: Correlograms plots of ARMA (2, 0) model American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 30 Figure 5: Normal Q-Q plot of residuals-ARMA (2, 0) We conducted an Autocorrelation (Ljung-Box) test to assess the overall adequacy of the ARMA (2, 0) model. The test examined whether there was any serial autocorrelation in the model residuals. The results, with p- values of 0.6 for horizon h = 3, indicate that there is no statistically significant autocorrelation in the residuals. This assessment was extended up to 12 lags, revealing that most lags did not exhibit serial autocorrelation, except for h = 12 (as shown in Table-6). Table 8: Ljung-Box test table Horizon LB Stat. p-value h=3 0.311 0.577 h=4 1.065 0.587 h=5 1.082 0.781 h=6 1.687 0.793 h=7 2.994 0.701 h=8 3.107 0.795 h=9 3.490 0.836 h=10 6.007 0.646 h=10 7.685 0.566 h=12 34.306 0.000 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 31 Furthermore, we apply the two normality tests of the residuals for ARMA (2, 0). By Shapiro-Wilk and Anderson-Darling normality test, we can see that the residuals are not normally distributed (table-7) Table 9: Test of normality-ARMA (2, 0) In figure 5, the histogram of the residuals displays the constant mean and variance over time graphically. Figure 6: Histogram of residuals-ARMA (2, 0) 4.10 Out-of-sample: Forecasting inflation for the next 5 years from ARMA (2, 0) Using the ARMA (2, 0) model, we have illustrated how inflation forecasts revert to equilibrium more rapidly, as depicted in Figure 6. Over the observed time periods, inflation displays fluctuations. In 2021, the predicted inflation stands at 2.952, but it gradually decreases continuously until 2025. For instance, in 2023, it is expected to be 2.421, and in 2024, it is projected to reach 2.334. In the graph, the blue line represents predicted inflation, while the green line represents the mean value of the dataset. The x-axis corresponds to the year, while the y-axis denotes the inflation rate (HICP). Between 2021 and 2025, a notable trend emerges: the predicted inflation converges quickly toward the mean value. Test of normality Statistics p-value Shapiro-wilk 0.98028 0.002857 Anderson-Darling 1.0731 0.007976 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 32 Figure 7: Forecasting graph using ARMA (2, 0) From table-08, we can see that in Poland inflation will be lower and it will return to equilibrium quickly after 2024 years. In the table below, the bold font denotes forecasts dates and rates. Table 8: Forecasting results-ARMA (2, 0) Year HICP (x) Year HICP (x) 2002-12-01 1.958 2014-12-01 0.083 2003-12-01 0.725 2015-12-01 -0.700 2004-12-01 3.633 2016-12-01 -0.192 2005-12-01 2.183 2017-12-01 1.617 2006-12-01 1.258 2018-12-01 1.192 2007-12-01 2.617 2019-12-01 2.117 2008-12-01 4.167 2020-12-01 3.667 2009-12-01 4.025 2021-12-01 2.952 2010-12-01 2.658 2022-12-01 2.595 2011-12-01 3.900 2023-12-01 2.421 2012-12-01 3.700 2024-12-01 2.334 2013-12-01 0.817 2025-12-01 2.291 In conclusion, our analysis demonstrates that ARMA (2, 0) provides a superior model for inflation prediction. Notably, it yields forecasts that exhibit a steady return to the mean, contributing to economic stability. Additionally, our model performs satisfactorily during periods of low inflation. Furthermore, our analysis American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 33 underscores the robustness of our findings across various model specifications considered. These results collectively support the effectiveness and reliability of the ARMA (2, 0) model for inflation forecasting. 5. Results and discussions for the VAR model 5.1 Time series plot between PL and EA Figure 7 illustrates the fluctuating pattern of inflation over time for both Poland (PL) and the Euro Area (EA). Both series exhibit alternating upward and downward trends. The x-axis represents the year, while the y-axis depicts the Polish HICP inflation rate. Source: The figure was generated using R language (version 3.6.3) in R Studio, following the same procedure used for all other figures in this chapter. Figure 8: Time series plot of HICP 5.2 Information criteria From table-9 of the information criteria, we can say that 2 lag is sufficient for a VAR model. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 34 Table 9: information criteria Criteria 1 2 3 4 5 6 AIC(n) -4.939 -5.080 -5.120 -5.150 -5.233 -5.219 HQ(n) -4.902 -5.019 -5.033 -5.039 -5.097 -5.058 SC(n) -4.847 -4.927 -4.905 -4.874 -4.896 -4.820 FPE(n) 0.007 0.006 0.006 0.005 0.005 0.005 5.3 Estimating parameters for the VAR model In Table-10 (a), the equation for Euro Area inflation reveals that one estimated parameter stands out as statistically significant, indicated by a star mark (*) due to p-values approaching zero, which allows us to reject the null hypothesis at a 5% significance level. Specifically, the autoregression coefficients are 1.090 (statistically significant) and -0.120. Additionally, there are coefficients linked to inflation in Poland (PL), with 0.023 indicating a slight positive impact and -0.030 suggesting a minor reduction. Notably, the coefficient for inflation in PL is much lower than the autoregression coefficients. The high multiple R-squared value of 0.932 indicates a strong correlation between the response and the fitted values. Source: The table was meticulously created using Microsoft Office, following the same methodology applied to other tables in this study. Estimation results for equation EA: EA = EA. l1 + Pl. l1 + EA. l2 + PL. l2 + Constant Table 10: the equation of inflation for the Euro Area Variables Estimate Std. Error t value Pr(>|t|) EA. l1 1.090 0.070 15.44 <2 × 10−16 Pl. l1 0.023 0.052 0.45 0.652 EA. l2 -0.120 0.071 -1.68 0.094 PL. l2 -0.030 0.052 -0.57 0.570 Constant 0.050 0.035 1.43 0.153 In Table 10(b), the equation for inflation in Poland reveals that all estimated parameters are statistically significant, with p-values close to zero, allowing us to reject the null hypothesis at both 1% and 5% significance levels. Specifically, the coefficients include 0.200 for the inflation rate in the Euro Area, which is higher than - 0.2046. Additionally, we find statistically significant coefficients of 1.2948 and -0.3258. Each of these coefficients proves to be individually useful for prediction. The high adjusted R-squared value of 0.96 suggests Selection AIC(n) HQ(n) SC(n) FPE(n) 5 5 2 5 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 35 that additional input variables (independent variables) significantly contribute to the VAR model, accounting for 96% of the variance. Estimation results for equation PL: PL = EA. l1 + Pl. l1 + EA. l2 + PL. l2 + Constant Table 11: the equation of inflation for Poland Variables Estimate Std. Error t value Pr(>|t|) EA. l1 0.200 0.087 2.29 0.023 Pl. l1 1.294 0.065 19.83 <2 × 10−16 EA. l2 -0.204 0.089 -2.30 0.022 PL. l2 -0.325 0.065 -4.96 1.4 × 10−6 Constant 0.071 0.043 1.65 0.100 5.4 Model checking Figures 8(a, b) display the graphical diagnosis of the VAR (2) model to assess its fit to the time series data. The residuals of this model reflect the sample modeling. In terms of the residuals' behavior, both the ACF plots for PL and EA exhibit white noise characteristics. However, notable spikes are observed at lag 12 for both PL and EA, likely attributed to base effects. On the other hand, the Q-Q plot reveals that the residuals do not follow a normal distribution, displaying a lack of fit in the distribution's tails. Figure 9: Diagnostic plot of residuals-VAR (2) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 36 Figure 10: Normal Q-Q plot- VAR (2) We conducted an Autocorrelation (Ljung-Box) test with a lag length of 6 to assess serial autocorrelation in the residuals of the VAR (2) model. The p-value (8.878×10^(-07)) is less than 0.05, indicating the presence of autocorrelation. Consequently, finding a VAR model with no autocorrelation is challenging, and options include models with 2 or 5 lags.Additionally, we applied two normality tests (Shapiro-Wilk and Anderson-Darling) to the residuals of the VAR (2) model, both of which indicate that the residuals are not normally distributed (as shown in Table-11). Table 12: Test of normality-VAR (2) Also, in figure 9, the histogram of the residuals shows the distribution of errors over time graphically. Test of normality Statistics p-value Shapiro-wilk 0.98504 0.0001316 Anderson-Darling 1.4449 0.0009781 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 37 Figure 11: Histogram of residuals-VAR (2) 5.5 Out-of-sample: Forecasting inflation for Euro Area and Poland for the next 5 years using VAR (2) model Figure-10 reveals a gradual trend in forecasting inflation for Poland. In 2021, the predicted inflation is 3.082, but it steadily decreases until 2025. Over the next four years, inflation is expected to remain low, with forecasts of 2.643 for 2023 and 2.506 for 2024. In the graph, the x-axis corresponds to the year, while the y-axis represents the inflation rate (HICP). The straight line represents the mean value, and the blue line indicates the inflation rate. Notably, between 2021 and 2025, the predicted inflation gradually converges toward the mean value. Figure12: Forecasting graph using VAR (2) for Poland American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 38 From table-12, we can see that in Poland inflation will be lower and it will return to equilibrium quickly after 2024 years. The bold font indicates forecasts in the given table. Table 13: Forecasting results-VAR (2) for Poland Year HICP (x) Year HICP (x) 2002-12-01 1.958 2014-12-01 0.083 2003-12-01 0.725 2015-12-01 -0.700 2004-12-01 3.633 2016-12-01 -0.192 2005-12-01 2.183 2017-12-01 1.617 2006-12-01 1.258 2018-12-01 1.192 2007-12-01 2.617 2019-12-01 2.117 2008-12-01 4.167 2020-12-01 3.667 2009-12-01 4.025 2021-12-01 3.082 2010-12-01 2.658 2022-12-01 2.827 2011-12-01 3.900 2023-12-01 2.643 2012-12-01 3.700 2024-12-01 2.506 2013-12-01 0.817 2025-12-01 2.407 Figure-11 depicts a gradual trend in forecasting inflation for the Euro Area. In 2021, the predicted inflation is - 0.066, but it steadily increases until 2025. Over the next four years, inflation is expected to remain very low, with forecasts of 0.571 for 2023 and 0.758 for 2024. In the graph, the x-axis corresponds to the year, while the y-axis represents the inflation rate (HICP). The straight line represents the mean value, and the blue line indicates the inflation rate. Notably, between 2021 and 2025, the predicted inflation slowly converges toward the mean value. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 39 Figure 13: Forecasting graph using VAR (2) for Euro Area From table-13, we see that in the euro area inflation will be lower and it will return to equilibrium slowly after 2024 years. The bold font indicates forecasts in the given table. Table 14: Forecasting results-VAR (2) for Euro Area Year HICP (x) Year HICP (x) 2002-12-01 1.958 2014-12-01 0.083 2003-12-01 0.725 2015-12-01 -0.700 2004-12-01 3.633 2016-12-01 -0.192 2005-12-01 2.183 2017-12-01 1.617 2006-12-01 1.258 2018-12-01 1.192 2007-12-01 2.617 2019-12-01 2.117 2008-12-01 4.167 2020-12-01 3.667 2009-12-01 4.025 2021-12-01 -0.066 2010-12-01 2.658 2022-12-01 0.307 2011-12-01 3.900 2023-12-01 0.571 2012-12-01 3.700 2024-12-01 0.758 2013-12-01 0.817 2025-12-01 0.890 In summary, our analysis indicates that inflation prediction using VAR (2) yields favorable results. However, it's noteworthy that the forecast values gradually converge toward the mean value for both Poland (PL) and the Euro Area (EA), a contrast to the faster convergence observed with ARMA (2, 0). We have presented the predicted results comprehensively and accurately throughout our analysis. Ultimately, our findings highlight the robustness and effectiveness of the analysis across various model specifications. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 40 6. The comparison table of ARMA (2, 0) and VAR (2) Table-14 demonstrates that both ARMA and VAR models exhibit a trend of predicted inflation converging toward equilibrium when compared to the years 2021 and 2022. Notably, the ARMA model's forecast results show a faster return to the mean value compared to the VAR model. Table15: Forecast comparison table of the AMRA (2, 0) and VAR (2) models 6.1 Comparison between the ARMA model and VAR model graphically Figure-12 provides a graphical comparison of the proposed models with actual inflation, focusing on their predicted inflation rates. It is evident from the graph that the ARMA model is converging towards equilibrium at a faster pace than the VAR model. This comparison strongly suggests the superior performance of the ARMA (2, 0) model in the coming years. In the figure, the x-axis corresponds to the year in the database, while the y- Year Actual inflation ARMA (2, 0) VAR (2) 2002-12-01 0.8 1.958 1.958 2003-12-01 1.7 0.725 0.725 2004-12-01 4.3 3.633 3.633 2005-12-01 0.8 2.183 2.183 2006-12-01 1.4 1.258 1.258 2007-12-01 4.3 2.617 2.617 2008-12-01 3.3 4.167 4.167 2009-12-01 3.9 4.025 4.025 2010-12-01 2.9 2.658 2.658 2011-12-01 4.6 3.900 3.900 2012-12-01 2.1 3.700 3.700 2013-12-01 0.6 0.817 0.817 2014-12-01 -0.7 0.083 0.083 2015-12-01 -0.4 -0.700 -0.700 2016-12-01 0.9 -0.192 -0.192 2017-12-01 1.7 1.617 1.617 2018-12-01 0.9 1.192 1.192 2019-12-01 3.0 2.117 2.117 2020-12-01 3.4 3.667 3.667 2021-12-01 8.0 2.952 3.082 2022-12-01 15.3 2.595 2.827 2023-12-01 - 2.421 2.643 2024-12-01 - 2.334 2.506 2025-12-01 - 2.291 2.407 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 41 axis represents the inflation rate. Source: The figure was created using R language (version 3.6.3) in R Studio. Figure14: Comparison graph In summary, the ARMA (2, 0) model proposes lower inflation forecasts and exhibits a faster return to the mean value, which is a crucial aspect for accurate forecasting. In contrast, the VAR (2) model takes a slower approach to reach the mean value. The graphical representation with three colors, where red represents actual inflation, green represents the ARMA (2, 0) model, and blue represents the VAR (2) model, vividly illustrates the differences in inflation predictions. 7. Evaluation to achieve the best model using various test 7.1 Mean forecast error (MFE) and Root mean squared forecast error (RMSFE) We have initiated an analysis to assess the accuracy of forecasts using two commonly employed criteria: Mean Forecast Error (MFE) and Root Mean Squared Forecast Error (RMSFE). As seen in Table 15(a, b), when evaluated using the MFE criterion, the ARMA model consistently exhibits lower errors across most forecast horizons. Similarly, in terms of the Root Mean Squared Forecast Error, the ARMA model outperforms the VAR model, particularly at shorter horizons. In summary, the comparative accuracy analysis reveals that the ARMA model provides satisfactory results for forecasting inflation, surpassing the VAR model in terms of accuracy. These findings underscore the advantages of employing the ARMA model over the VAR model. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 42 Table 16: Mean forecast error (ME) Horizon 1 2 3 6 9 12 ARMA 0.0223 0.0585 0.0999 0.242 0.381 0.530 VAR 0.0357 0.0844 0.1385 0.319 0.499 0.685 Table 17: Root mean squared forecast error (RMSE) Horizon 1 2 3 6 9 12 ARMA 0.314 0.519 0.682 0.849 0.889 1.07 VAR 0.317 0.524 0.692 0.891 0.955 1.17 7.2 Diebold Mariano test We conducted a Diebold-Mariano test with a horizon of 12, and the results allow us to reject the null hypothesis (H0) of equal forecast accuracy. The DM statistic (-4.45) falls outside the critical z-value range of -1.96 to 1.96, indicating that both the ARMA and VAR models do not exhibit equal forecast accuracy. Furthermore, in Figure-13, we observe that the forecasting errors between the two models exhibit variations over time. The ACF plot suggests the presence of autocorrelation in errors at the beginning of the lag. Figure 15: Errors difference Considering the Diebold-Mariano (DM) test results, which indicate that both models do not have equal forecast accuracy, it is reasonable to infer that one model outperforms the other. The test results provide evidence that one of the models exhibits superior accuracy compared to the other. 7.3 Forecast accuracy measures: efficiency test for AMRA (2, 0) We conducted an Efficiency/Unbiasedness test, involving regression between the actual value and forecast value. The results indicate that our forecast parameter (x level) is statistically significant, with a p-value of 0.001, which is less than 0.05 at a 5% significance level. Consequently, we can reject the null hypothesis (H0). American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 43 Furthermore, when subjected to a linear hypothesis test, the parameters remain statistically significant. These findings suggest that the unbiasedness criterion is not met. 7.4 Efficiency test - graphical illustration - AMRA (2, 0) Figure-14 provides a clear graphical representation of the efficiency test for ARMA (2, 0). The scatterplot highlights a systematic underprediction trend, indicating that the model consistently underpredicts actual values. Figure 16: Scatter plot of efficiency test-ARMA (2, 0) 7.5 Forecast accuracy measures: efficiency test for VAR (2) A relatively good forecast accuracy does not imply that they are satisfactory in the absolute sense. Absolute performance includes ME and efficiency tests. So, we manage the following efficient test. From the Efficiency/unbiasedness test (regression between actual value and forecast value), we can say that our forecast parameter (x level) is statistically significant due to the p-value (0.002) being less than 0.05 at a 5% significant level. So, we can reject the null (H0). By the linear hypothesis test, once again parameters are statistically significant. Hence, we can see that unbiasedness is no met. 7.6 Efficiency test - graphical illustration – VAR (2) In Figure-15, we provide a clear graphical representation of the efficiency test for the VAR (2) model. The scatterplot illustrates a consistent pattern of systematic underprediction, indicating that the VAR (2) model consistently underpredicts actual values. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 44 Figure 17: Scatter plot of efficiency test-VAR (2) 8. Sequential forecasts for ARMA (2, 0) & VAR (2) In Figure-16(a), we present a visual representation of the rolling window forecast analysis using the ARMA (2, 0) model with a restricted dataset spanning from 2010 to 2024. The analysis highlights the model's ability to rapidly revert towards the sample mean when forecasting future events. The x-axis corresponds to the year, while the y-axis represents the Polish inflation rate. It becomes evident that when inflation is high, it tends to return to equilibrium. Notably, when inflation is at 2, the model exhibits a faster convergence towards equilibrium compared to other scenarios. Conversely, when inflation is at 4, the reversion to equilibrium occurs more gradually. Figure 18: Sequential forecast-ARMA (2, 0) In Figure-16(b), we provide a graphical representation of the rolling window forecast analysis using the VAR (2) model. While the VAR (2) model also exhibits reversion towards equilibrium, the dynamics can be more complex compared to the ARMA (2, 0) model. However, it is evident from the figure that there is still a discernible reversion towards the mean (equilibrium). American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 45 The graph illustrates that the VAR (2) model gradually moves towards the mean value over time. Figure 19: Sequential forecast-VAR (2) 9. Conclusion This research aimed to model and forecast Polish inflation using two methodologies: the Autoregressive Moving Average (ARMA) model and the Vector Autoregressive (VAR) model. The study utilized annual data from January 2002 to December 2020 and conducted various diagnostic checks, graphical investigations, and statistical tests to assess the adequacy of the models. The comparison between the ARMA and VAR models provided valuable insights into their respective abilities to accurately predict the inflation rate. The evaluation, based on Mean Forecast Error (MFE) and Root Mean Squared Forecast Error (RMSFE), revealed that the ARMA (2, 0) model outperformed other models in forecasting inflation for the next five years. Additionally, the analysis indicated a downward trend in the Harmonized Index of Consumer Prices (HICP) for Poland over the forecasted period. As a result, this study suggests that fiscal and monetary authorities should consider implementing robust economic policies to maintain the inflation rate at an equilibrium level in the coming years. 9.1 Recommendations for future works In future research, we intend to explore the application of Bayesian inference alongside the methods employed in this thesis for enhanced analysis and forecasting. Additionally, we plan to expand our repertoire by delving deeper into various methods and staying updated with the latest developments in the field. The solution methodology presented in this thesis holds promise for addressing a wide range of real-life challenges in the finance and economic sectors. We believe that further research in this direction will yield valuable insights and contribute to the advancement of predictive modeling and analysis. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 99, No 1, pp 13-47 46 References [1] Khan, S., & Alghulaiakh, H. (2020). ARIMA model for accurate time series stocks forecasting. International Journal of Advanced Computer Science and Applications, 11(7). [2] Szafranek K. Bagged neural networks for forecasting Polish (low) inflation. International Journal of Forecasting. 2019 Jul 1;35(3):1042-59. [3] Hernandez-Matamoros, A., Fujita, H., Hayashi, T., & Perez-Meana, H. (2020). Forecasting of COVID- 19 per region using ARIMA models and polynomial functions. Applied soft computing, 96, 106610. [4] Abonazel, M. R., & Abd-Elftah, A. I. 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