Finally, we present the aggregate efect of the above socioeconomic factors. In brief, we found that an increase of cigarettes smoked, of fat and alcohol consumption and the number of passenger cars will result to a relevant increase regarding mortality. The latter one is also affected by the changes in unemployment rate. On the other hand, an increase of personal disposable income may negatively affect mortality, by almost the same portion.

Given that till now, there is no any detailed quantitative study for the case of Greece on this particular field, we decided to present this model which gives a clear picture regarding the extend that each of the main factors considered, influences the above mortality rate. The methodology of model testing, integrates this attempt to correctly assess the significance of specific etiological factors on the population health status, as far as the ishaemic disease is concerned.

On theoretical grounds, the choice of the particular factors considered here is based upon relevant research works discussing the effect of fat and alcohol consumption, cigarette smoking, quality of nutrition, physical exercise and frustration, on various heart diseases^1–5.

The model specification is based on similar works^6–12, where the great majority of risk factors biochemical, behavioral stress or physical and health care utilization, are directly or indirectly related to economic growth factors and socioeconomic status. There is an indirect effect of many factors of this kind on the mortality changes. Which factors are to be considered in model specification, depends on their importance as well as the available data. In our case, this combination produced the following structural form, which is presented after estimation by OLS (Ordinary Least Squares) from annual data obtained from various sources as indicated in table 1.

Table 1. Main characteristics of the variables used

• where MORTALITY_i = Ischemic mortality rate at year i.
• ALCOCHOL_i = Per capita average annual consumption of alcohol at year i.
• CARS_i-2 = Total number of passenger cars (national level), two years earlier.
• CIGARS_i-3 = Per capita average annual consumption of cigarettes, three years earlier.
• ΔUNR_i = UNR_i – UNR_i-1 (UNR_i = unemployment rate at year i).
• FATS_i-3 = Per capita fat and land animal consumption, three years earlier.
• PerCaGDP_i-2 = Per capita Gross Domestic Product (in real terms), with two years time lag.

It is recalled that in (1) the numbers in brackets are standard errors of the corresponding estimated coefficients, the adjusted coefficient of determination, d is the Durbin-Watson statistic, PC is the Amemiya criterion and BIC is the Bayesian Information Criterion, known also as Schwarz Bayesian Criterion. Note also that p values refer to the lowest level of significance (α), for the corresponding coefficient.

Although the d statistic reveals the possible existence of a slight negative first order autocorrelation, which will be discussed later, we may conclude from (1), since all coefficients seem to be significant at an acceptable α, that at national (aggregate) level, the influence of the major socio-economic factors considered here on mortality rate, can be summarized in what follows.

• We observe a positive effect of the quantity of alcohol consumed during the current year.
• There is a cumulative effect of the cigarettes smoked and the fat consumed during the last three year.
• Also there is a cumulative effect of the increase in the number of passenger cars, during the last two years, since the more intensive use of private cars, restrains physical exercise and increases stress.
• The current change in unemployment rate seems to have a positive effect too, through increasing uncertainty in many individuals, who in turn feel more frustrated.
• The increase of per capita GDP in the last two years seems to have a negative effect, mainly due to the improvements in nutrition quality, the better exercise and health care too.

More details regarding the model variables and further calculations to facilitate the computation of individual elasticities, are presented in table 1.

The majority of the empirical works on this field suffer from the lack of further tests^13–16, which establish the grounds to conclude that the estimates we obtained have the desired statistical properties and the specified model is not going to lead us towards misleading inference. With this in mind, we suggest the following tests.

It has been pointed out, that the number of explanatory variables selected, heavily depends on available data. This implies that we may exclude some other variables, which play an important role in explaining the variation of the dependent variable. In this case we have a specification error, resulting to biased end inconsistent estimates. We have tested this kind of specification error, with the following two ways.

1. By computing the recursive residuals, denoted by û_j⁰(j = m+1,.....T), with mean ū⁰.It is recalled, that m denotes the number of coefficients in (1), and T the number of observations actually used in the estimation process. From these residuals we calculate the statistic ψ from

2. By applying the RESET (Regression Specification Error Test) of Ramsey¹⁹. For simplicity, denote the estimated values of the dependent variable ŷ_i. Then, consider ŷ_i as the dependent variable in new (auxiliary) regressions, with explanatory variables ŷ_i², or (ŷ_i²,ŷ_i³), or (ŷ_i²,ŷ_i³,ŷ_i⁴) and all the independent variables in (1) plus a constant term. From the ESS^* of these regressions we consider the largest F^* which is computed from

It is recalled that linearity is one of the basic assumptions, in order to apply OLS for model estimation. This test presupposes the estimation of the regression

Given that some of the explanatory variables have a very small total variation (0.12E-2 and 0.86F-2) the use of condition number (which is the largest condition index) will produce misleading results²⁰. In such cases it is preferable to employ the Variance Inflation Factor (VIF), for testing multicollinearity. VIF has to be calculated for each explanatory variable i (i = 1, 2,...., m-1) taken as dependent and the remaining as independent, in order to get the corresponding R_i². For variable i, the VIF_i is computed from

The results obtained are presented in table 2.

Table 2. Estimated Variance Inflation Factors (VIFi) and the corresponding R2.

Since none of the VIF values exceeds 10, we conclude that there is no any severe collinearity problem, regarding the columns of data matrix X of the explanatory variables, which implies that X has full column rank.

To apply this test, we computed the Spearmans's rank correlation coefficient r_s, considering each explanatory variable and the (absolute values of) residuals. The obtained coefficients, together with the computed t^* values (given that the standard error is 0.2357), are presented in table 3.

Table 3. The computed rank correlation coefficients rs and the corresponding t* values

Since the last t^* statistic is greater than the tables t, for (T-m) degrees of freedom at α=0.05, we see that there is a problem of heteroscedastisity of the form:

In such a case we have to transform the initial model, dividing throughout by PerCaGDP_i and to re-estimate it. The results are presented bellow.

where

• YSTAR_i =MORTALITY_i/PerCaGDP_i
• ONEovGDP_i = 1/ PerCaGDP_i
• ALCovGDP_i = ALCOHOL_i/PerCaGDP_i
• CARovGDP_i = CARS_i-2/PerCaGDP_i
• CIGARovGDP_i = CIGARS_i-3/ PerCaGDP_i
• ΔUNRovGDP_i = ΔUNR_i/ PerCaGDP_i
• FATovGDP_i = FATS_i-3/ PerCaGDP_i

It should be noted that we avoided applying the general test of White²¹, to test for heteroscedasticity, since there are too many explanatory variables in the model.

Before going back to the original form of the model seeing in (1), by multiplying (3) by PerCaGDP_i, it is advisable to perform one more test, regarding autocorrelation.

Given the value of d statistic in (3), one may suspect that there is a problem of first order (negative) autocorrelation. Applying ordinary Durbin-Watson test, we found d_L<d<d_U, so that the test was inconclusive. However, according to other tests (Brausch-Godfrey), it seems that the problem of autocorrelation is present. At this point it is useful to recall that we may have some indications for autocorrelation, although the real problem is an ARCH (AutoRegressive Conditional Heteroscedasticity) effect. To be sure that we have to face the problem as being a pure autocorrelation problem, we should trace any ARCH effect, by performing the proper test. With this in mind, and considering that the order h (maximum lag) is 3, we get the following estimates.

We further compute LM = (T-h)xR², which is much smaller than the tables X², for h =3 degrees of freedom, so that we may conclude that the error terms do not follow an ARCH scheme, and thus we are facing a pure autocorrelation problem. The application of Hildreth-Lu iterative procedure to get the minimum |d-2|, produces the following results.

To test the coefficients stability, we consider the recursive residuals and compute the CUSUM (cumulative sums) and CUSUMSQ (cumulative sums of squared residuals) statistics, presented in 1 and 2. From these graphs one may conclude that no problem of coefficients stability exists.

To transform model (4) to its original form, which is model (1), we multiply (4) throughout by PerCaGDP_i to obtain

With the new estimates we get 3, which refers to the observed and estimated values of the dependent variable.

In this paper a relevant model has been specified, estimated and properly tested, in order to investigate the quantitative effect of specific socioeconomic factors on the ischemic mortality rate. We presented step by step all the tests required in order to obtain correct and unbiased results which reflect the real world in the best possible way. We also discussed the effect of each factor on the dependent variable in a rather general manner. Further we may obtain some more detailed results, if we combine the ratios of means of the explanatory variables over the mean value of the dependent variable presented in table 1, together with the estimated coefficients in (5), in order to get estimates of the elasticities. Thus we may determine the extend that each explanatory variable affects mortality, which is the dependent variable in our model. These elasticities are presented in table 4, bellow.

Table 4. Estimated elasticities

It is noted, that all positive elasticities are less than one and only the (absolute) value of the last one is slightly greater than unity. According to the last column of table 4, we can say that a 10% increase of cigarettes smoked (i.e. 100 cigarettes), will result to a 1.77% increase in mortality. This harmful effect of smoking will be realized with a delay of three years. Also an increase of fat consumption by 10%, will gradually increase mortality by 1%, in the following three years An increase in alcohol consumption by 10% (i.e. 100 kgr) will increase mortality rate by 1.2% without any delay, i.e. in the same time period. A similar interpretation holds for the other socioeconomic factors considered, which constitute the set of the explanatory variables. In other words, a 1% increase of the number of passenger cars, will result to a gradual increase in mortality by 0.48%, which is to be realized two years later. Changes in unemployment rate by 1%, will affect mortality by 0.1%, at the same time period, through increasing frustration. On the other hand, an increase of personal disposable income by 1%, may contribute towards reducing mortality, by almost the same portion. This effect is due to the expected improvements in nutrition quality, the better exercise and health care too.

Finally it is worthy to mention that we haven't met in the literature an equivalent quantitative analysis on this particular field.

• From the mathematical point of view, a model is a mathematical expression to describe a real phenomenon. In this context, the flight of an air plain can be simulated by a single differential equation. On the other hand, to model a national economy one may need more than a hundred of difference equations. A model is used for simulation purposes, and mainly to investigate the effect of some predetermined factors on the characteristic we are interested in.
• The right specification is the first step in model building. This implies that we must know in advance the factors (usually called explanatory variables), which actually affect the evolution of the characteristic under consideration (usually called dependent variable).
• A model should comply with some specific tests, in order to be efficient and reliable. All these tests are described in details and properly applied in this paper, given that there is a lack of such an analysis in the relevant literature.