**4. Econometric model and data specification**

To look at the relationship between population aging and savings in an unemployment context in Tunisia over the period 1970–2019, we apply a structural VAR model, as defined by Sims [25]. This enables us to approach a multivariate causal setting allowing the coexistence of both short and long-term forces derived from the aging influences on saving decisions. Finally, we deepen our dynamic analysis by application the techniques of impulse response functions (IRFs) of different shocks for all variable's fluctuations and of the variance decomposition (VDC).

#### **4.1 Data specification**

To undertake the aggregate saving model estimating, we use as an independent variable the national savings rate unlike previous studies, which generally referred to the household savings rate. National saving is important as it is a source of investment and one of the major determinants of macroeconomic growth. Also, as it is closely related to the demand for financial and real assets and it may affect asset price formation. In addition, we seek to avoid narrowing the aging impact as its takes into account companies and public sector saving (related to social sector, health, education and pensions). On another side, the household savings refers to survey measure which undervalues personal income as it provides information related to expenditure than to income sources. Likewise, it does not capture the same share of total saving for persons at different ages, so the estimate of relationship between savings and age may be fallacious.

As a definition, between the two known alternative measures of savings (S) we adopt that of national account (as income minus consumption expenditure) given data availability.<sup>5</sup> Explicitly, we use the savings rate with respect to the gross national income disposable income<sup>6</sup> .

For independent variables, we refer to the main population aging indicators. We consider the mortality rate (MR) and fertility rate (FR) to capture demographic changes and its impact on the age structure composition, and likewise on the dependency ratio. We consider the old-age dependency ratio (EDR) to accurately look at the effect of aging besides the broadly used the total dependency ratio (TDR). Then, we could deduce if the aging impact is due to the fertility decline or to the longevity increase.

Concerning economic variables, we include three macroeconomic variables. (1) Basing to the neoclassical approach we introduce the interest rate (MMR) in particular the money market rate as a driver of the real interest rate (credit and debit). (2) As a one quantitative measure of aggregate income uncertainty we consider the aggregate unemployment rate (U). (3) In order to check the economic effect of saving, we examine the economic growth (G) measured by the GDP per capita at constant domestic prices. It is computed by dividing GDP per capita at current domestic prices by the consumption price index (base 1990). Hence, the inflation rate is indirectly considered.

<sup>5</sup> The second defines savings as the changes in net wealth. Net wealth accumulation includes capital gains and losses, adjusted for general inflation, and is more relevant for purposes of measuring changes individual's economic well-being.

<sup>6</sup> Gross national income equal to the gross national income minus the current transfers (current taxes on income and wealth, social security contributions, social security benefits) paid to non-residents units plus the current transfers received from the rest of the world by the residents.

The main statistical characteristics of these variables used are summarized in **Table 1** (in the Appendix). Data are drawn from the Central Bank of Tunisia (CBT), the National Institution of Statistics (NIS) and Tunisian Institute of competitively and quantitative study (ITCQS).

## **4.2 Econometric models**

Our analysis is based on the identification and estimation of structural vector autoregressive (SVAR). The SVAR model is used in macroeconomic analysis in order to check the effect of exogenous shocks (of the demographic change, for instance) on macroeconomic variables.

Our basic model VAR is the following:

$$\mathbf{Y\_t = \Gamma(L)} \ \mathbf{Y\_t + \nu\_t} \tag{1}$$

where Yt is a column vector of stationary variables considered in the estimate.

The selection and order of independent variables are essential in the SVAR estimate. Thus, the independent demographic variables are introduced with caution following the demographic transition theory. As mortality decline brings that of fertility, so we first introduce the mortality rate (MR) followed by the fertility rate (FR). Then, we integrate the dependency ratio as an indicator of the population aging and the age structure change following the demographic transition.

After what, we consider the economic variable exogenous effect on saving. So, we insert the interest rate (MMR) as a saving determinant and the aggregate unemployment rate (U) as a measure of aggregate income uncertainty. Lastly, we introduce the national savings rate (S) followed by the economic growth (G) to check the aging impact on economic growth through the savings evolution.

As we use two dependency ratios, we estimate two distinct vectors autoregression. A vector includes the total dependency ratio which reflects the effect of both the mortality and fertility evolution as a result of the demographic policy as follows (MRt, FRt, TDRt, Ut, MMRt, St, Gt).

The second vector includes the old-age dependency ratio and takes the mortality choc as the main cause of the elderly proportion evolution as follows (MRt, EDRt, Ut, MMRt, St, Gt).

Otherwise, Γ(L) = Γ1L1 + Γ2L2 + … + Γp. Lp is a lag operator in the form of polynomial matrix and ν<sup>t</sup> is a vector of idiosyncratic errors, where νt = (μ<sup>1</sup> t, … ,μ<sup>5</sup> t) **'** . These errors are not auto correlated and are homoscedastic. Then, the representation (1) can be written in the form of a moving average of infinite order VMA (∞) (representation theorem of Wald):

$$\mathbf{Y\_t = C(L) \ v\_t} \tag{2}$$

where C(L) = [I - Γ(L)]�<sup>1</sup> .

The structural form (SF) of the model (1) can be written as follows:

$$\mathbf{Y}\_{\mathbf{t}} = \mathbf{A}(\mathbf{L})\,\mathbf{e}\_{\mathbf{t}} \tag{3}$$

where A(L) = C(L) H is the coefficient matrix (aij) of (7 � 7 or 6 � 6 for the two vectors respectively) size, and more precisely it represents the impulse response functions of the elements of Yt following the various shocks. Moreover, H is the transition matrix and ε is the vector of structural shocks where E (εtεt')=IN.

However, the identification of these shocks requires the Cholesky decomposition in the order to identify the structure of the shocks. As a result, the

*The Life Cycle Hypothesis and Uncertainty: Analyzing Aging Savings Relationship in Tunisia DOI: http://dx.doi.org/10.5772/intechopen.100459*

decomposition of the variance covariance matrix of the reduced form residuals is written in a lower triangular matrix A(L). The number of constraints imposed on A (L) is equal to 21 i.e. n � (n-1) / 2 with n = 7 variables and where some of the structural shocks do not have contemporaneous impact on other variables.

Additionally, the Cholesky decomposition assumes that series listed earlier in the VAR order impact the others variables contemporaneously. But series listed later in the VAR order impact those listed earlier only with lag. Therefore, the variables listed early in the VAR order are considered more exogenous. As mentioned above, the order of endogenous variables is central to the identification of structural shocks, i.e. it determines the structure of the shocks. More precisely, the first variable has impacts on all the variables that are below it, but it does not receive any impacts from these variables. This rule applies to all subsequent variables. For instance, the triangular matrix A(L), for the case of n = 7 variables, is as follows.

$$\mathbf{A(L)} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf{C21} & 1 & 0 & 0 & 0 & 0 & 0 \\ \mathbf{C31} & \mathbf{C32} & 1 & 0 & 0 & 0 & 0 \\ \mathbf{C41} & \mathbf{C42} & \mathbf{C43} & 1 & 0 & 0 & 0 \\ \mathbf{C51} & \mathbf{C52} & \mathbf{C53} & \mathbf{C54} & 1 & 0 & 0 \\ \mathbf{C61} & \mathbf{C62} & \mathbf{C63} & \mathbf{C64} & \mathbf{C65} & 1 & 0 \\ \mathbf{C71} & \mathbf{C72} & \mathbf{C73} & \mathbf{C74} & \mathbf{C75} & \mathbf{C76} & 1 \end{pmatrix} \tag{4}$$

Henceforth, we have to estimate four matrixes: a matrix (Alt1) for the total dependency ratio and one for the old-age dependency ratio (Alt2). Two others matrixes are also estimates as a robustness test by omitting the unemployment rate, respectively the matrixes (Alt3) and (Alt4).

To undertake the SVAR estimate model, we first study the stationarity of all variables using the Phillips-Perron test [26]. As reported in **Table 2** (in the Appendix) all the considered variables are I(0) suggesting that a long-run (cointegration) relationship could exist between the considered variables.

Then we determine the order p of the VAR process to remember. To do this, we consider various processes for VAR lag orders p ranging from 1 to 4. For each model, we calculate the Akaike information criteria (AIC) and Schwarz (SC), and the log-likelihood (LV) to hold the p lag (=4) that minimizes these criteria as indicated in **Table 3** (in the Appendix).

Accordingly, four alternatives are estimated, respectively with the identification of cointegration relationships by using the cointegration test of Johansen [27] as well as the structural factorization (with 500 iterations).

Finally, to examine the dynamic of the model, we refer to the impulse response function (IRF). It helps us to judge and to appreciate the channel(s) of population age structure change transmission. It allows to see if there is really a robust, stable and predictable relationship between aging and savings. In this respect, we will identify the different responses of all the variables in the model to various shocks. It should be noted that we focused on the effects of the shock on 10 periods and that errors are generated by Monte Carlo with 100 repetitions. Such analysis is strengthened with the variance decomposition analysis (VDC), which however, indicates the proportion of the variable changes due to own shocks versus shocks on the other variables. Namely, the variance of the forecast error of the change in savings rate is partitioned among the contributions of the innovations in each variable of the system.

### **5. Results interpretation**

Interestingly, results put forward that the LCH prediction is not automatically confirmed, but it is up to the economic uncertainty extent. Indeed, the LCH is not validated in the unemployment setting, which is considered in our study as an indirect measure of income uncertainty.

#### **5.1 The SVAR results interpretation**

The SVAR's estimate results (reported in **Table 4** in the Appendix) points out that uncertainty encourages saving. Indeed, for the two aging indicators used (TDR and EDR) the LCH prediction is not validated once unemployment is introduced unlike Ahmedova's [28] findings. As we note from matrixes Alt1 and Alt2, the two indicators present a significant positive effect on savings rate in the context of enduring unemployment without allowance, like in Frini' work [29]. However, this effect is found more significant for old-age dependency ratio, unlike in Wong and Tang [30] and Loumrhari's [31] findings.

In contrast, the LCH prediction is validated by the estimate omitting the unemployment rate, however, for only the total dependency ratio. Indeed, a significant negative effect is found in the matrix Alt3.

Such results confirm that the demographic changes impact on saving depends on the perception of the economic context and the confidence towards future. Furthermore, the LCH's validation appears to be, as well, empirically related to the aging indicator used in the estimate. This confirms our proposal that the old-age dependency ratio indicator seems more efficient to explicitly check the effect of aging.

The unemployment which hints uncertainty about future income pushes the savers to keep up savings. It reduces confidence and intensifies incentives for precautionary saving so that, it prevents savings from decline. Indeed, the unemployment rate displays a significant and positive coefficient. The weight of the future uncertainty boosts the employed population to form a precautionary savings. This precautionary saving is important to offset the small amount of wealth accumulated after an enduring unemployment without any allocation benefit. When enduring a long unemployment period and facing great difficulties for credit liquidity at old age, elderly try to continue to save to keep up a certain level of consumption. Such behavior is very pronounced in Tunisia since retired do not benefit from a sufficient pension in a distressed pay-as-you-go system. The insufficiency of pension and medical care benefits entails the elderly saving's behavior adjustment by continuing to work and to save at the beginning of the retirement period (as long as they remain in better health). As pointed out by Frini [32, 33] the new retirees or the youngest elderly (which share, generally, weighs more than that of the old retirees or the old elderly) maintain their savings mainly for precautionary motives in high uncertainty economic environment.

This Tunisian elderly saving behavior may in part be strengthened by the intentional transfers motivation of the old generation towards the young one. Indeed, Tunisian families, as stressed by Mahfoud [34] and Frini [35, 36], are strongly linked and directed by an intergenerational altruistic motive. So, the old generation do not seek to cut savings so as to help the young generation to face uncertain environment and hard economic conditions.

As expected, mortality drop induces a fertility decline, putting on show the demographic transition theory. This fertility decline increases the savings rate. It seems that the youth share decrease outweighs the small increase in the elderly share since the aggregate savings rate increases. Household with fewer children are likely to incur less expenditure in respect to their income for looking after them and

#### *The Life Cycle Hypothesis and Uncertainty: Analyzing Aging Savings Relationship in Tunisia DOI: http://dx.doi.org/10.5772/intechopen.100459*

then would save more. In addition, a reduced family size leads to a competition between children as a mean of transferring income from present to future and as a financial asset.<sup>7</sup> Henceforth, by the fall of fertility rate, the demand of financial and capital market as a substitute of youth assurance service will increase and thus savings. Additionally, the decline of government expenditures for youth (given their share decline), seems to make up or even more the government expenditures increase for elderly (due to their share increase) to not lead savings decrease.

Considering uncertainty, mortality evolution positively influences savings rate when considering the economic and social facts, but negatively when they are neglected. The increase of mortality risk and health problem intensifies precautionary behavior to face health care expenditure at old age.

The uncertainty related to interest rate affects positively the savings rate. An increase in interest rate will make saving more attractive. Finally, like in AbuAl-Foul's [37] work results show that no long-run relationship exists between saving and GDP growth. This in part due to that saving is, generally, done in real estate, which is known as a small creator of wealth with a small ripple effect.

#### **5.2 The IRF's and VDC's results interpretation**

Likewise, the IRF's and VDC's results underline that population aging on savings evolution changes respect to the economic uncertainty context. Savings positively respond to age structure changes once unemployment is taken into account. The different graphs of impulse responses (**Figure 2** in the Appendix) show that savings respond quickly to demographic changes (mortality rate, fertility rate and dependency ratio jointly), but weakly to the shock of the money market rate. The response due to unemployment rate shock can be judged as significant with a return to equilibrium in the long-term. The saving response to economic growth innovations is, however, slow and limited. This analysis is corroborated out by the variance decomposition as displayed in **Table 5** (in the Appendix).<sup>8</sup> In detail, a relatively constant proportion of the change in savings rate variance is recorded for both ratios. The total dependency ratio shock is by about 3.75 percent for Alt1 and by 4.26 percent for Alt3 after three years. The old-age dependency ratio shocks are, however, of a less proportion by 1.42 percent for Alt2 and 0.16 percent for Alt4 over ten years. The noteworthy result is that savings evolution follow-up a shock of the total dependency ratio is more significant (by three times more) than of the elderly one. This fact is also proved by the dynamic response path. Fewer children lower the dependent population and consumption without contribution to income. The decrease of the youth dependent proportion out weights the increase of the elderly dependent in the proportion, which limits saving rate depression. This brings up the role of relative weight of the youth share to the elderly share on savings evolution. Further, the increase of elderly proportion appears not to cut savings rate. Thus, savings rises when fertility declines and longevity increases, but less intensively. In the contrary, to the LCH prediction, the old-age dependency ratio shock instantaneously and positively affects saving rate, however, more weakly than the total dependency ratio.

<sup>7</sup> Children is treated as pure capital goods and a kind of safety assets which returns are "elderly assurance".

<sup>8</sup> The VDC indicates the proportion of the variable changes due to own shocks versus shocks on the other variables. The Cholesky decomposition method is used in orthogonalizing the innovations across equation. Percentage of forecast variances is explained by innovations.

Remarkably, once we ignore the labour market unbalance (or uncertainty) of the estimates the relationship between aging and savings becomes consistent with the LCH prediction. The total dependency ratio shocks present a negative shortterm impact on saving to disappear at long-run (after eight years). However, no impact is found for the old-age dependency ratio. This discrepancy in estimated magnitude through the two dependency ratios used refers back to our assumption that aging impact may be sensitive to the measurement used to describe it.

Moreover, demographic indicators shocks trace the variance of savings innovations. Mortality rate explains saving variance by almost the same small proportion (by about 2.30 percent) for all alternatives in the variance decomposition, but relatively less without the unemployment rate. In the impulse function graph, a negligible positive impact is found of mortality shock. Hence, with the rise of longevity and elderly proportion savings may not decline. Fertility decline significantly contributes in the savings change variance (by about 5.16 percent in Alt1) and even much more when forsaking the unemployment rate (by about 17.79 percent in Alt3). The corresponding impulse function displays a negative influence over six years to reverse positively after.

However, saving is less sensitive to interest rate shock. Money market rate contribution is more pronounced for the total dependency ratio than the elderly one. The same evidence is observed through the impulse function graph shown a very small positive influence which disappears in the long-term. This small impact of the real interest rate on saving may hide the offsetting of its two effects (of income and substitution). In other hand, it may be related to the Tunisian household's behavior which seems to comply more with the Keynesian approach.

Finally, in the long-term, savings shocks seem to produce an effect on economic growth, but weakly when the imbalance labour market is considered (as reported in **Table 6** in the Appendix). As mapped out by the response functions this dynamic is non-instantaneous. In contrast, a very small 'feed-back'seems to be produced of economic growth over three years on saving.
