We present a time series regression model in which the regressors are past values of the dependent variable, namely the Export data. We use 92 observations of variable EXP, quarterly data from 1991 to 2013 expressed in Euros. Figure 1 shows the related time series.

Looking at Fig. 1, we can see that the Verona export shows relatively smooth growth, although this decreases during the years 2008–2011. Decline in exports is likely caused by economic crisis broken out in Italy in those years. Although the curve may seem apparently growing, it is also possible to notice that there are periodic trends during the years under consideration. In fact, in the fourth quarter of 1992, the curve has a significant growth, then increases fairly linearly until about the second quarter of 1994, in which one can recognize a new increasing period that slightly more obvious than the previous one. This periodicity of 18 months can also be seen in other parts of the curve, but not after the beginning of the current economic crisis, where very likely there will be a structural break. In order to test the goodness of our qualitative analysis based on historical data, we used a software called GRETL, which is particularly useful to perform statical analysis of time series. The mean and standard deviation related to the quarter of this variable EXP are respectively $\mathit{Mean}=1\hspace{0.1667em}579\hspace{0.1667em}900\hspace{0.1667em}000\hspace{2.5pt}\text{€}$ and $\mathit{StandardDeviation}=499\hspace{0.1667em}880\hspace{0.1667em}000\hspace{2.5pt}\text{€}$, whereas the annual mean for EXP is $1\hspace{0.1667em}579\hspace{0.1667em}900\hspace{0.1667em}000\times 4=6\hspace{0.1667em}319\hspace{0.1667em}600\hspace{0.1667em}000\hspace{2.5pt}\text{€}$. The first seven autocorrelations of EXP are $\rho _{1}=\mathit{corr}(\mathit{EXP}_{t},\mathit{EXP}_{t-1})=0.9718$, $\rho _{2}=0.9755$, $\rho _{3}=0.9450$, $\rho _{4}=0.9523$, $\rho _{5}=0.9165$, $\rho _{6}=0.9242$, $\rho _{7}=0.8931$. Previous entries show that inflation is strongly positively autocorrelated; in fact, the first autocorrelation is 0.97. The autocorrelation remains large even at a lag of six quarters. This means that an increase in export in one quarter tends to be associated with an increase in the next quarter. Autocorrelation starts to decrease from the lag of seventh quarters. In what follows, we report the output obtained testing for autoregressive models according to an increasing number of delays, from 1 to 6 delays, on the variable EXP, namely:

the AR(1) case: $\mathit{EXP}=65\hspace{0.1667em}090\hspace{0.1667em}000+0.971606\mathit{EXP}_{t-1}$ the AR(2) case: $\mathit{EXP}=57\hspace{0.1667em}965\hspace{0.1667em}600+0.409313\mathit{EXP}_{t-1}+0.573763\mathit{EXP}_{t-2}$ the AR(3) case: $\mathit{EXP}=54\hspace{0.1667em}025\hspace{0.1667em}100+0.618705\mathit{EXP}_{t-1}+0.726958\mathit{EXP}_{t-2}$-$0.366510\mathit{EXP}_{t-3}$

the AR(4) case: $\mathit{EXP}=54\hspace{0.1667em}498\hspace{0.1667em}000+0.748057\mathit{EXP}_{t-1}+0.466614\mathit{EXP}_{t-2}-0.592869\mathit{EXP}_{t-3}$

the AR(5) case: $\mathit{EXP}=56\hspace{0.1667em}242\hspace{0.1667em}200+0.870848\mathit{EXP}_{t-1}+0.247032\mathit{EXP}_{t-2}-0.417031\mathit{EXP}_{t-3}+0.648298\mathit{EXP}_{t-4}-0.372917\mathit{EXP}_{t-5}$

and the AR(6) case:

We estimate the AR order of our autoregression related to obtained numerical results using both BIC and AIC information criteria (see Table 1).

Both BIC and AIC are the smallest in the AR(6) model (from the seventh delay onwards the criteria begin to increase); we conclude that the best estimate of the lag length is 6, hence supporting our qualitative analysis. Previous data from Table 1 indicate that as the number of lags increases, the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ increases, and the SER decreases. ${R}^{2}$, $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$, and $\mathit{SER}$ measure how well the OLS estimate of the multiple regression line describes the data. The standard error of the regression (SER) estimates the standard deviation of the error term, and thus, it is a measure of spread of the distribution of a variable Y around the regression line. The regression ${R}^{2}$ is the fraction of the sample variance of Y explained by (or predicted by) the regressors, the ${R}^{2}$ increases whenever a regressor is added, unless the estimated coefficient on the added regressor is exactly zero. An increase in the ${R}^{2}$ does not mean that adding a variable actually improves the fit of the model, so the ${R}^{2}$ gives an inflated estimate of how well the regression fits the data. One way to correct this is to deflate or reduce the ${R}^{2}$ by some factor, and this is what the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ does, which is a modified version of ${R}^{2}$ that does not necessarily increase when a new regressor is added. As seen by numerical output in Table 1, the increase in $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is large from one to two lags, smaller from two to three, and quite small from three to four and in the next lags. Exploiting the results obtained for the AIC/BIC analysis, we can determine how large the increase in the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ must be to justify including the additional lag. In the AR(6) model of Eq. (1), the coefficients of $\mathit{EXP}_{t-1}$, $\mathit{EXP}_{t-4}$, $\mathit{EXP}_{t-5}$, and $\mathit{EXP}_{t-6}$ are statically significant at the $1\% $ significance level because their p-value is less than 0.01, and the t-statistic exceeds the critical value. The constant, however, is statically significant at the $5\% $ significance. The coefficient of $\mathit{EXP}_{t-3}$ is statically significant at the $10\% $ significance, and the coefficient of $\mathit{EXP}_{t-2}$ is not statically significant. In particular, the $95\% $ confidence intervals for these coefficient are as follows:

In order to check whether the EXP variable has a trend component or not, we test the null hypothesis that such a trend actually exists against the alternative EXP being stationary, by performing the ADF test for a unit autoregressive root. Large-sample critical values of the augmented Dickey–Fuller statistic yield the following ADF regression with six lags of $\mathit{EXP}_{t}$, where the subscript t indicates a particular quarter considered: The ADF t-statistic is the t-statistic testing the hypothesis that the coefficient on $\mathit{EXP}_{t-1}$ is zero; this is $t=-1.23$. From Table 2, the 5% critical value is $-2.86$. Because the ADF statistic of $-1.23$ is less negative than $-2.86$, the test does not reject the null hypothesis at the 5% significance level. Based on the regression in Eq. (2), we therefore cannot reject the null hypothesis that export has a unit autoregressive root, that is, that export contains a stochastic trend, against the alternative that it is stationary. If instead the alternative hypothesis is that $Y_{t}$ is stationary around a deterministic linear trend, then the ADF t-statistic results in $t=-4.07$, which is less than $-3.41$ (from Table 2). Hence, we can reject the null hypothesis that export has a unit autoregressive root. We proceed with a test QLR, which provides a way to check whether the export curve has been stable in the period from 1993 to 2010. Specifically, we focus on whether there have been changes in the coefficients of the lagged values of export and of the intercept in the AR(6) model specification in Eq. (1) containing six lags of $\mathit{EXP}_{t}$. The Chow F-statistics (see, e.g., [7, Sect. 5.3.3]) tests the hypothesis that the intercept and the coefficients of $\mathit{EXP}_{t-1},\dots ,\mathit{EXP}_{t-6}$ in Eq. (1) are constant against the alternative that they break at a given date for breaks in the central 70% of the sample. The F-statistic is computed for break dates in the central 70% of the sample because for the large-sample approximation to the distribution of the QLR statistic to be a good one, the subsample endpoints cannot be too close to the beginning or to the end of the sample, so we decide to use 15% trimming, that is, to set $\tau _{0}=0.15T$ and $\tau _{1}=0.85T$ (rounded to the nearest integer). Each F-statistic tests seven restrictions. Restrictions on the coefficients equaled to zero under the null hypothesis (see [5, Sect. 2.4]), and since in our case we have the coefficients of the six delays and the intercept, we get seven restrictions. The largest of these F-statistics is 13.96, which occurs in 2010:I (the first quarter of 2010); this is the QLR statistic. The critical value for seven restrictions is presented in Table 3.

The previously reported values indicate that the hypothesis of stable coefficients is rejected at the 1% significance level. Thus, there is an evidence that at least one of these seven coefficients changed over the sample. These results also confirm the assumptions that we made earlier since the year 2010 coincides with an increasing import of the financial crisis before arriving at a partial economic recovery. A forecast of Verona export in 2014:I using data through 2013:IV can be then based on our established AR(6) model of export, which gives Therefore, substituting the values of export into each of the four quarters of 2013, plus the two last quarters of 2012, we have

so that, for 2014:II, we obtain and forecasts for all 2014 quarters are as follows:

It is worth mentioning that the forecast error increases as the number of considered quarters increases. Figure 2 shows, through a graph, forecasts since 2002 in sample and forecasts for 2014, highlighting the confidence intervals.

It is also useful to analyze the time series of the growth rate in exports that we denoted by $\Delta \mathit{EXP}$. Economic time series are often analyzed after computing their logarithms or the changes in their logarithms. One reason for this is that many economic series exhibit growth that is approximately exponential, that is, over the long run, the series tends to grow by a certain percentage per year on average, and hence the logarithm of the series grows approximately linearly. Another reason is that the standard deviation of many economic time series is approximately proportional to its level, that is, the standard deviation is well expressed as a percentage of the level of the series; hence, if this is the case, the standard deviation of the logarithm of the series is approximately constant. It follows that it turns to be convenient to work with the variable $\Delta \mathit{EXP}_{t}=\ln (\mathit{EXP}_{t})-\ln (\mathit{EXP}_{t-1})$. Taking into account the data shared in Fig. 3, we retrieve the following information: The first four autocorrelations of $\Delta \mathit{EXP}$ are $\rho _{1}=-0.6133$, $\rho _{2}=0.5698$, $\rho _{3}=-0.6100$, $\rho _{4}=0.7029$.

Even if it might seem contradictory that the level of export is strongly positively correlated but its change is negatively correlated, we have to consider that such values measure different things. The strong positive autocorrelation in export reflects the long-term trends in export; in contrast, the negative autocorrelation of the change of export means that, on average, an increase in export in one quarter is associated with a decrease in export in the next one. Analogously to what we have seen in Section 2.1, we perform an AIC/BIC analysis for $\Delta \mathit{EXP}$ obtaining that the best choice for the lag lay is 4, so that we have

In our AR(4) model, the coefficients of $\Delta \mathit{EXP}_{t-4}$ are statically significant at the $1\% $ significance level because their p-value is less than 0.01 and the t-statistic exceeds the critical value. The coefficient of $\Delta \mathit{EXP}_{t-3}$ is statically significant at the $10\% $ significance. The constant and the other coefficients are not statically significant. Even when the information criteria are very low, this is not a good model because ${R}^{2}$ and $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ are relatively small. So this AR(4) model turns out to be not very useful to predict the growth rate in exports. Figure 3 shows that the frequency in this case is annual; moreover, an increase in $\Delta \mathit{EXP}$ in one quarter is associated with a decrease in the next one. In this case, the results of ADF test allow us to reject the null hypothesis that rate of growth in export has a unit autoregressive root both with the alternative hypothesis of stationarity and of stationarity around a deterministic linear trend. It follows that the QLR statistic is 5.02, which occurs in 2009:I, and hence the hypothesis that the coefficients are stable is rejected at the 1% significance level. Again, the results of the software GRETL confirm that the crisis of recent years has greatly affected the exports from Verona. Consequently, by the results obtained we have that the forecast of $\Delta \mathit{EXP}$ for 2014, given in the table and also sketched in Fig. 4, is not very accurate, and the predictions do not perceive the lower peaks of the variable, which is confirmed by the low value of ${R}^{2}$.

We now turn to the empirical problem to predict Verona import by analyzing its historical series. We present an autoregressive model that uses the history of Verona import to forecast its future. We use 92 observations of variable import, quarterly data from 1991 to 2013 expressed in Euros. Figure 5 shows the time series.

Looking at Fig. 5, we can see that Verona import shows relatively smooth growth, although this decreases during the years 2008–2011; the curve is very similar to the time series of export, and hence it is reasonable to deduce that decline in import is likely caused by economic crisis broken out in Italy in those years. Although the curve may seem apparently growing, periodic trends appear during years under consideration. This curve has an annual periodicity. Looking at a minimum of the curve, exactly one year later, another minimum exists. The mean and standard deviation on a quarterly basis for IMP are $\mathit{Mean}=2\hspace{0.1667em}177\hspace{0.1667em}300\hspace{0.1667em}000\hspace{2.5pt}\text{€}$ and $\mathit{StandardDeviation}=697\hspace{0.1667em}420\hspace{0.1667em}000\hspace{2.5pt}\text{€}$, whereas the annual mean export is $2\hspace{0.1667em}177\hspace{0.1667em}300\hspace{0.1667em}000\times 4=8\hspace{0.1667em}709\hspace{0.1667em}200\hspace{0.1667em}000\hspace{2.5pt}\text{€}$. The first five IMP autocorrelation values are $\rho _{1}=0.9424$, $\rho _{2}=0.9280$, $\rho _{3}=0.9060$, $\rho _{4}=0.9260$, $\rho _{5}=0.8750$. These entries show that inflation is strongly positively autocorrelated; in fact, the first autocorrelation is 0.94. The autocorrelation remains large even at the lag of four quarters. This means that an increase in import in one quarter tends to be associated with an increase in the next quarter. Autocorrelation, as expected, starts to decrease starting from the lag of five quarters. As with the variable EXP, we estimated the AR order of an autoregression in IMP using both the AIC and BIC information criteria, finally obtaining that the optimal lag length is 4.

Therefore, we have

We check now if the model has a trend. The null hypothesis that Verona import has a stochastic trend can be tested against the alternative that it is stationary by performing the ADF test for a unit autoregressive root. The ADF regression with four delays of IMP gives The ADF t-statistic is the t-statistic testing the hypothesis that the coefficient on $\mathit{IMP}_{t-1}$ is zero, and it turns to be $t=-1.78$. From Table 2, the 5% critical value is $-2.86$. Because the ADF statistic of $-1.78$ is less negative than $-2.86$, the test does not reject the null hypothesis at the 5% significance level. We therefore cannot reject the null hypothesis that import has a unit autoregressive root, that is, that import contains a stochastic trend, against the alternative that it is stationary. If the alternative hypothesis is that $Y_{t}$ is stationary around a deterministic linear trend, then the ADF t-statistic results in $t=-2.6$, which is less negative than $-3.41$. So, in this case, we also cannot reject the null hypothesis that export has a unit autoregressive root.

We proceed with a QLR test, which provides a way to check whether the import curve has been stable during the years sparing from 1993 to 2010. The Chow F-statistic tests the hypothesis that the intercept and the coefficients at $\mathit{IMP}_{t-1},\dots ,\mathit{IMP}_{t-4}$ in Eq. (3) are constant against the alternative that they break at a given date for breaks in the central 70% of the sample. Each F-statistic tests five restrictions. The largest of these F-statistics is 10.26, which occurs in 1995:III; the critical values for the five-restriction model at different levels of significance are given in Table 4. These values indicate that the hypothesis that the coefficients are stable is rejected at the 1% significance level. Thus, there is an evidence that at least one of these five coefficients changed over the sample; namely, we have a structural break, which might be caused by the devaluation that the Lira currency experienced during the period 1992–1995. According to the previous analysis, the predictions of import of Verona for the year 2014 are as follows: They result in a slight increase for the next year, as shown by Fig. 6.

The fourth variable of interest is represented by the logarithm of the ratio between consecutive values of IMP, that is, The first six autocorrelations values of $\Delta \mathit{IMP}$ are presented in Table 5.

In the case of the growth rate of export, the negative autocorrelation of the change of import means that, on average, an increase in import in one quarter is associated with a decrease in the next one. From the fifth lag, autocorrelation starts to be less significant. So, it can be easily seen from Fig. 7 and the autocorrelations in Table 5 that the right estimate of the lag length is 4. The consequent AR(4) model reads as follows: and the following Fig. 7 shows the time series of $\Delta \mathit{IMP}$, and we can see how an increase in import in one quarter is associated with a decrease in the next one.

The QLR statistic for AR(4) model in Eq. (5) is 22.58, which occurs in 1995:II. This value indicates that the hypothesis that the coefficients are stable is rejected at the 1% significance level. As for imports, we can associate this structural break to the last crisis of Lira occurred in that period. We observe the dynamics of the real effective exchange rate in Fig. 8.

As shown in Fig. 9, the devaluation of the Lira has produced some benefits for the growth of Italian exports (goods and services), especially looking at analogous economical data for Germany and France.

As shown in Fig. 10, the devaluation of the Lira did not stop the value of imports, but you can still easily perceive the rupture of 1995.

We would like also to briefly analyze the variable “Active Enterprises” ($\mathit{ACTE}_{t}$), namely the time series with quarterly data from 1995 to 2013, where each observation is the number of firms operating in a given quarter in the province of Verona. With the software GRETL we obtain the AR(4) model The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ of this regression is 0.94, and the QLR statistic is 37.52, which occurs in 2011:I. This value indicates that the hypothesis that the coefficients are stable is rejected at the 1% significance level. Also, for the variable $\mathit{ACTE}_{t}$, we can conclude that the number of active businesses were affected by the crisis of those years. However, the ADF t-statistic for this variable does not reject the null hypothesis, so we cannot reject the fact that the time series of the numbers of active enterprises has a unit autoregressive root, that is, that $\mathit{ACTE}_{t}$ contains a stochastic trend, against the alternative that it is stationary. From Fig. 11 we can see that the curve has a quite regular annual pattern and that active enterprises tend to decline in the first quarter of each year and then return generally to grow. It is worth to mention the drastic rise of the curve during the first period of the time interval under consideration. Such an increase has been caused by a particular type of bureaucratic constraints, namely by a sort of forced registration imposed to a rather large set of farms companies previously not obliged to be part of the companies register. Such a norm has been introduced in two steps, first by a simple communication (1993), and later in the form of legal disclosure (2001).

As we saw in Chapter 2, the import end export of Verona are subject to a stochastic trend, so that it is appropriate to transform it by computing its logarithmic first differences in order to obtain stationary variables. Figure 12 shows a multiple graph for the time series of $\Delta \mathit{EXP}_{t}$, $\Delta \mathit{IMP}_{t}$, and $\Delta \mathit{ACTE}_{t}$.

The VAR for $\Delta \mathit{EXP}_{t}$, $\Delta \mathit{IMP}_{t}$, and $\Delta \mathit{ACTE}_{t}$ consists of three equations, each of which is characterized by a dependent variable, namely by $\Delta \mathit{EXP}_{t}$, $\Delta \mathit{IMP}_{t}$, and $\Delta \mathit{ACTE}_{t}$, respectively. Because of the apparent breaks in considered time series for the years 1995 and 2010, the VAR is estimated using data from 1996:I to 2008:IV. The number of lags of this model are obtained through information criteria BIC and AIC using the software GRETL, which gives the results in Table 6, where the asterisks indicate the best (or minimized) of the respective information criteria.

The smallest AIC has been obtained considering six lags; indeed, the BIC estimation of the lag length is $\hat{p}=2$. We decide to choose two delays because, for $\hat{p}=6$, we have a VAR with three variables and six lags, so we will have 19 coefficients (eight lags with three variables each, plus the intercept) in each of the three equations, with a total of 57 coefficients, and we saw in [5, Sect. 4.2] that estimation of all these coefficients increases the amount of the forecast estimation error, resulting in a deterioration of the accuracy of the forecast itself. We also prefer consider the BIC estimation for its consistency; however, the AIC overestimate p (see [5, Sect. 2.2]. Estimating the VAR model with GRETL produces the following results:

In the first equation ($\Delta \mathit{EXP}_{t}$) of VAR system (7), we have the coefficients of $\Delta \mathit{EXP}_{t-1}$, $\Delta \mathit{IMP}_{t-2}$, and $\Delta \mathit{ACTE}_{t-2}$, which are statically significant at the $1\% $ significance level because their p-value is less than 0.01 and the t-statistic exceeds the critical value. The constant and the coefficients of $\Delta \mathit{IMP}_{t-1}$, however, are statically significant at the $5\% $ significance, and the other coefficients are not statically significant. The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is 0.53. In the second equation ($\Delta \mathit{IMP}_{t}$) of VAR system (7), we have the coefficients of $\Delta \mathit{EXP}_{t-1}$, $\Delta \mathit{EXP}_{t-2}$, $\Delta \mathit{IMP}_{t-1}$, and $\Delta \mathit{IMP}_{t-2}$, which are statically significant at the $1\% $ significance level. The constant, however, is statically significant at the $10\% $ significance, and the other coefficients are not statically significant. The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is 0.45. In the last equation of (7), we have only the constant statically significant, at the $5\% $ level. The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is −0.04. These VAR equations can be used to perform Granger causality tests. The results of this test for the first equation of (7) are as follows:

The F-statistic testing the null hypothesis that the coefficients of $\Delta \mathit{IMP}_{t-1}$ and $\Delta \mathit{IMP}_{t-2}$ are zero in the first equation is 12.46 with p-value 0.0001, which is less than 0.01. Thus, the null hypothesis is rejected at the level of 1%, so we can conclude that the growth rate in Verona import is a useful predictor for the growth rate in export, namely $\Delta \mathit{IMP}_{t}$ Granger-causes $\Delta \mathit{EXP}_{t}$. Also, $\Delta \mathit{ACTE}_{t}$ Granger-causes the change in export at the 1% significance level. The results for the second equation of (7) are as follows:

For the $\Delta \mathit{IMP}_{t}$ equation, we can also conclude that the growth rate in Verona export is a useful predictor for the growth rate in import, but the change in the number of active enterprises is not. The results for the last equation of (7) are as follows:

The F-statistic testing the null hypothesis that the coefficients of $\Delta \mathit{EXP}_{t-1}$ and $\Delta \mathit{EXP}_{t-2}$ are zero in the first equation is 1.09 with p-value 0.34, which is greater than 0.10. Thus, the null hypothesis is not rejected, so we can conclude that the growth rate in Verona import is not a useful predictor for the growth rate in active enterprises, namely, $\Delta \mathit{IMP}_{t}$ does not Granger-cause $\Delta \mathit{ACTE}_{t}$. The F-statistic testing the hypothesis that the coefficients of the two lags of $\Delta \mathit{EXP}_{t}$ are zero is 1.64 with p-value of 0.2; thus, $\Delta \mathit{EXP}_{t}$ also does not Granger-cause $\Delta \mathit{ACTE}_{t}$ at the 10% significance level. Forecasts of the three variables in system (7) are obtained exactly as discussed in the univariate time series models, but in this case, the forecast of $\Delta \mathit{EXP}_{t}$, we also consider past values of $\Delta \mathit{IMP}_{t}$ and $\Delta \mathit{ACTE}_{t}$.

By means of the forecasts from 2009 to 2013, we can establish a comparison with the real data, noting that the predictions with this VAR model are not very reliable since the error is quite high and it increases in recent years. The lack of accuracy was confirmed previously by low values of the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$. Figures 13, 14, and 15 show the real time series of the three variables with a red line and the prediction made with the estimated models with a blue line. It can be seen from these graphs that the confidence intervals (green area in the figures) are very high.

In this section, we analyze the three variable ($\mathit{EXP}_{t}$, $\mathit{IMP}_{t}$, and $\mathit{ACTE}_{t}$), considering quarterly Verona data from 1995 to 2013. We analyze these time series without avoiding structural breaks and without considering the first differences, and we check if the analysis produces different results with respect to the previous ones. Figure 16 shows a multiple graph for the time series respectively of $\mathit{EXP}_{t}$, $\mathit{IMP}_{t}$, and $\mathit{ACTE}_{t}$.

The GRETL lag length selection gives the results in Table 8; then, according to the considerations made to determine the number of delays for the model (7), we decide to choose three delays, obtaining the following model: In the first equation ($\mathit{EXP}_{t}$) of VAR system (8), we have the coefficients of $\mathit{EXP}_{t-1}$, $\mathit{EXP}_{t-2}$, $\mathit{IMP}_{t-2}$, and $\mathit{ACTE}_{t-3}$, which are statically significant at the $1\% $ level because their p-value is less than 0.01 and the t-statistic exceeds the critical value. However, the coefficients of $\mathit{EXP}_{t-3}$ and $\mathit{IMP}_{t-1}$ are statically significant at the $5\% $ level, and the other coefficients are not statically significant. The coefficient of $\mathit{IMP}_{t-3}$ is statically significant at the $10\% $ level, and the others are not statically significant. The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is 0.93. In the second equation ($\mathit{IMP}_{t}$) of VAR system (8), we have the coefficients of $\mathit{EXP}_{t-2}$, $\mathit{EXP}_{t-3}$, $\mathit{IMP}_{t-1}$, $\mathit{IMP}_{t-3}$, and $\mathit{ACTE}_{t-1}$, which are statically significant at the $1\% $ significance level. The constant is statically significant at the $5\% $ level, and the other coefficients are not statically significant. The $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is 0.85. In the last equation of (8), we have only $\mathit{EXP}_{t-1}$, $\mathit{ACTE}_{t-1}$, and $\mathit{ACTE}_{t-2}$ statically significant respectively at the $5\% ,1\% $, and $10\% $ levels, whereas the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ is 0.94. If we perform Granger causality tests, then we have that all p-values of the F-statistic of the three equations are less than 0.01; only for the third equation of (8), the Granger causality test for the variable $\mathit{EXP}_{t}$ has the p-value 0.0852, and hence $\mathit{EXP}_{t}$ Granger-causes $\mathit{ACTE}_{t}$, but in this case, the null hypothesis is rejected at the level of 10%. Notice that the model (8) has high values of the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$, so it can be very useful to make prediction of future values of the three variables. The forecasts for $\mathit{EXP}_{t}$ concerning 2014 are given by the table

Forecasts for $\mathit{IMP}_{t}$ are as follows: and prediction for $\mathit{ACTE}_{t}$ reads as follows:

Figures 17, 18, and 19 show the time series of the three variables and their forecasts. The area of confidence interval for $\mathit{EXP}_{t}$ is rather small, which is confirmed by the value 0.93 of the $\mathit{Adjusted}\hspace{2.5pt}{R}^{2}$ of the first equation in system (8). This area is slightly wider for the second graph, and in Fig. 19 we show the confidence interval for $\mathit{ACTE}_{t}$ becoming wider at each quarter.

We saw in Sections 2.1 and 2.3 that the time series for $\mathit{EXP}_{t}$ and $\mathit{IMP}_{t}$ are both integrated of order 1 (I(1)); hence, we perform an EG-ADF test to verify if these two variables are cointegrated. The cointegrating coefficient θ is estimated by the OLS estimate of the regression $\mathit{EXP}_{t}=\alpha +\theta \mathit{IMP}_{t}+z_{t}$; hence, we obtain $\mathit{EXP}_{t}=197\hspace{0.1667em}119\hspace{0.1667em}000+0.641536\mathit{IMP}_{t}+z_{t}$, so that $\theta =0.641536$. Then we use a Dickey–Fuller test to test for a unit root in $z_{t}=\mathit{EXP}_{t}-\theta \mathit{IMP}_{t}$. The statistic test result is −2.77065, which is greater than −3.96 (see [5, Table 1] for critical values); therefore, we cannot refuse the null hypothesis of a unit root for $z_{t}$, concluding that the series $\mathit{EXP}_{t}-\theta \mathit{IMP}_{t}$ is not stationary. Moreover, we have that the variables $\mathit{EXP}_{t}$ and $\mathit{IMP}_{t}$ are not cointegrated.