Report di Statistica per la sperimentazione e le previsioni in ambito tecnologico

REPORT

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Index

1. INTRODUCTION .................................................................................................................................. 2

.................................................................................................................................. 2

1.1 D

ATA DESCRIPTION .................................................................................................................... 2

1.2 O

BJECTIVE OF THE ANALYSIS

2. PRELIMINARY ANALYSIS ...................................................................................................................... 3

.................................................................................................... 4

3. ARIMA MODEL AND DIAGNOSTIC

4. OUTLIERS IDENTIFICATION ................................................................................................................. 6

.................................................................................................................................. 9

5. FORECASTING ........................................................................................................................... 9

5.1 E - F

X ANTE ORECASTING

.......................................................................................................................... 10

5.2 E - F

X POST ORECASTING

.................................................................................................................................. 12

6. CONCLUSION

1. Introduction

1.1 Data description

The data analysed in this report were retrieved from the ISTAT website and refer to the Italian Import Price Index (base

year: 2015). The series is monthly (frequency S=12) and spans from January 2005 to December 2023, for a total of 228

observations.

Only the raw (unadjusted) series was selected, excluding both seasonally adjusted and calendar adjusted data, in order to

preserve the original structure of the time series. This choice ensures that any seasonal pattern and potential structural

changes are captured directly by the modelling framework, rather than being partially removed y external adjustment

procedures. The dataset can be accessed via ISTAT at the following link:

https://esploradati.istat.it/databrowser/#/it/dw/categories/IT1,Z0400PRI,1.0/DCSC_PREIMPIND/IT1,143_222_DF_DC

SC_PREIMPIND_1,1.0

1.2 Objective of the analysis

The objective of this analysis is to identify the best statistical model capable of approximating the data-generating process

of the time series and based on this model, to produce reliable forecast for future values.

To achieve this goal, the Box-Jenkins methodology is applied, following its main steps:

1. Preliminary analysis of the time series: graphical inspection, decomposition and ACF/PACF to assess trend,

seasonality and stationarity.

2. Specification of the ARIMA model

(P,

ARIMA(p, d, q)x D, Q)

3. Estimation of !"

4. Diagnostics: residual analysis (ACF and PACF, residual plot, Ljung-Box test, normality checks) to validate

model adequacy.

5. Forecasting: assessment of predictive performance through ex-ante forecasts and ex-post forecast evaluation on

a hold-out sample. 2

2. Preliminary analysis

The preliminary analysis represents the first step of the Box-Jenkins procedure. The time series under investigation is

displayed in Figure 1, which shows the evolution of the import Price Index over the considered time period.

Figure 1: Plot of analysed time series

Figure 1 reports the time series over time (X-axis: months from 2005 to 2023; Y-axis: Import Price Index). Between 2005

and 2010, the index exhibits several oscillations, with a generally moderate growth. From 2010 to 2015, the upward trend

becomes more pronounced, indicating an acceleration in import prices. This phase is followed by a period of slight decline

and fluctuations between 2015 and 2020 onward, a sharp increase can be observed, after which the series enters a phase

characterised by limited variability, suggesting a new and higher level of import prices in the most recent years.

These considerations are consistent with the evidence emerging from the additive and multiplicative decomposition

(Figure 2). Figure 2: decomposition of additive and multiplicative time series 3

Figure 3: ACF and PACF plots

After analysing the time series plot and its decomposition, the next step consists in examining the Autocorrelation

Function (ACF) and the Partial Autocorrelation Function (PACF) of the original series, shown in Figure 3. In both the

ACF and PACF plots, the blue dashed horizontal lines represent the approximate 95% confidence bounds for testing the

statistical significance of each autocorrelation value.

The ACF displays a slow and gradual decay, confirming the presence of non-stationarity in the series. Since the data are

monthly, the seasonal period of the ARIMA model is set to S=12. The smooth exponential decrease of the ACF indicates

persistent correlation among observations over time, suggesting that first-order differencing (d=1) is required to achieve

stationarity in the mean.

The PACF shows a significant spike at lag 1, while all subsequent lags remain within the confidence bounds.

3. Arima model and diagnostic

After a preliminary analysis of the initial data, which led to the selection of d=1, the study proceeded with the identification

of the most suitable ARIMA model for the data under examination. The modelling process began with the baseline model

ARIMA(0, 0, 0) x (0, 0, 0) , progressively adjusting the orders p, d, q, P, D, Q, increasing their values and progressively

!"

evaluating the results, as shown in Table 1.

Model selection was based on a comparison of the Akaike Information Criteria (AIC) and the Bayesian Information

Criteria (BIC), but also on the improvement of residual diagnostics. Specifically, the ACF and PACF plots of the residuals

were inspected to