This work is the first part of a project dealing with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. In particular, we develop an approach mainly based on vector autoregressions, where lagged values of two or more variables are considered, Granger causality, and the stochastic trend approach useful to work with the cointegration phenomenon. Latter techniques constitute the core of the present paper, whereas in the second part of the project, we present how these approaches can be applied to economic data at our disposal in order to obtain concrete analysis of import–export behavior for the considered productive area of Verona.

The analysis of time series data constitutes a key ingredient in econometric studies. Last years have been characterized by an increasing interest toward the study of econometric time series. Although various types of regression analysis and related forecast methods are rather old, the worldwide financial crisis experienced by markets starting from last months of 2007, and which is not yet finished, has put more attention on the subject. Moreover, analysis and forecast problems have become of great momentum even for medium and small enterprizes since their economic sustainability is strictly related to the propensity of a bank to give credits at reasonable conditions.

In particular, great efforts have been made to read economic data not as monads, but rather as constituting pieces of a whole. Namely, new techniques have been developed to study interconnections and dependencies between different factors characterizing the economic history of a certain market, a given firm, a specified industrial area, and so on. From this point of view, methods such as the vector autoregression, the cointegration approach, and the copula techniques have been benefitted by new research impulses.

A challenging problem is then to apply such instruments in concrete situations and the problem becomes even harder if we take into account the economies are hardly hit by the aforementioned crisis. A particularly important case study is constituted by a close analysis of import–export time series. In fact, such an information, spanning from countries to small firms, has the characteristic to provide highly interesting hints for people, for example, politicians or CEOs, to depict future economic scenarios and related investment plans for the markets in which they are involved.

Exploiting precious economic data that the Commerce Chamber of Verona Province has put at our disposal, we successfully applied some of the relevant approaches already cited to find dependencies between economic factor characterizing the Province economy and then to make effective forecasts, very close to the real behavior of studied markets.

For completeness, we have split our project into two parts, namely the present one, which aims at giving a self-contained introduction to the statistical techniques of interest, and the second one, where the Verona import–export case study have been treated in detail.

In what follows, we first recall univariate time series models, paying particular attention to the AR model, which relates a time series to its past values. We will explain how to make predictions, by using these models, how to choose the delays, for example, using the Akaike and Bayesian information crtiteria (AIC, resp. BIC), and how to behave in the presence of trends or structural breaks. Then we move to the vector autoregression (VAR) model, in which lagged values of two or more variables are used to forecast future values of these variables. Moreover, we present the Granger causality, and, in the last part, we return to the topic of stochastic trend introducing the phenomenon of cointegration.

Univariate models have been widely used for short-run forecast (see, e.g., [

The observation on the time-series variable

Formally, the process with stochastic initial conditions results from (

All the variables have nonzero finite fourth moments.

There is no perfect multicollinearity, namely it is not true that, given a certain regressor, it is a perfect linear function of the variables.

In this section, we show how the previously introduced class of models can be used to predict the future behavior of a certain quantity of interest. If

The forecast error is the mistake made by the forecast; this is the difference between the value of

The root mean squared forecast error RMSFE is a measure of the size of the forecast error

Let us recall relevant statistical methods used to optimally choose the number of lags in an autoregression model; in particular, we focus our attention on the

A further relevant topic in econometric analysis is constituted by nonstationarities that are due to trends and breaks. A trend is a persistent long-term movement of a variable over time. A time-series variable fluctuates around its trend. There are two types of trends, deterministic and stochastic. A

A second type of nonstationarity arises when the regression function changes over the course of the sample. In economics, this can occur for a variety of reasons, such as changes in economic policy, changes in the structure of the economy, or an invention that changes a specific industry. These breaks cannot be neglected by the regression model. A problem caused by breaks is that the OLS regression estimates over the full sample will estimate a relationship that holds “on average,” in the sense that the estimate combines two different periods, and this leads to poor forecast. There are two types of testing for breaks: testing for a break at a known date and for a break at an unknown break date. We consider the first option for an

In the following, we consider finite-order moving-average (MA) processes (see, e.g., [

In what follows, we focus our study on the so-called vector autoregression (VAR) econometric model, also using some remarks on the relation between the univariate time series models described in the first part, and the set of simultaneous equations systems of traditional econometrics characterizing the VAR approach (see, e.g., [

We have so far considered forecasting a single variable. However, it is often necessary to allow for a multidimensional statistical analysis if we want to forecast more than one-parameter dynamics. This section introduces a model for forecasting multiple variables, namely the vector autoregression (VAR) model, in which lagged values of two or more variables are used to forecast their future values. We start with the autoregressive representation in a VAR model of order

Since the autocovariance matrix entries link a variable with both its delays and the remaining model variables, we have that if the autocovariance between

An appropriate method for the lag length selection of VAR is fundamental to determine properties of VAR and related estimates. There are two main approaches used for selecting or testing lag length in VAR models. The first consists of rules of thumb based on the periodicity of the data and past experience, and the second is based on formal information criteria. VAR models typically include enough lags to capture the full cycle of the data; for monthly data, this means that there is a minimum of 12 lags, but we will also expect that there is some seasonality that is carried over from year to year, so often lag lengths of 13–15 months are used (see, e.g., [

Iterated multivariate forecasts are computed using a VAR in much the same way as univariate forecasts are computed using an autoregression. The main new feature of a multivariate forecast is that the forecast of one variable depends on the forecast of all variables in the VAR. To compute multiperiod VAR forecasts

An important question in multiple time series is to assign the value of individual variables to explain the remaining ones in the considered system of equations. An example is the value of a variable

(unrestricted)

(restricted)

In the first model, we have

In Section

Critical values for the EG-ADF statistic

Numbers of regressors | 10% | 5% | 1% |

1 | −3,12 | −3,41 | −3,96 |

2 | −3,52 | −3,80 | −4,36 |

3 | −3,84 | −4,16 | −4,73 |

4 | −4,20 | −4,49 | −5,07 |

A different estimator of the cointegrating coefficient is the dynamic OLS (DOLS) estimator, which is based on the equation

In this first part of our ambitious project to use multivariate statistical techniques to study critic econometric data of one of the most influential economy in Italy, namely the Verona import–export time series, we have focused ourselves on a self-contained introduction to techniques of estimating OLS-type regressions, analysis of the correlations obtained between the different variables and various types of

The author would like to acknowledge the excellent support that Dr. Chiara Segala gave him. Her help has been fundamental to develop the whole project, particularly, for the realization of the applied sections, which constitute the core of the whole work.