For the graphical evidence, it will be used the equivalent to the proposed variables used by Buckles et al (2018) regarding the selection of variables in section 2c: IGAE⁴ (Economic Activity General Indicator), CCI4 (Consumer Confidence Index), and IPC Housing4 (Underlying Real Estate Index of Prices to Customer). These variables will be compared to the number of births.

The number of births was used instead of the number of conceptions (or total fertility) due to in Mexico does not exist an official record of the gestation in weeks at the moment of the birth as on the US. Therefore, it is expected that in the graphical evidence, the falls in the indicators could vary from the showed by Buckles et al (2018) and the declines in the number of births could appear at the same time of a fall in the IGAE or the other used indicators.

The number of births as CCI and IPC Housing have a strong seasonal component, therefore it will be used the Census X-12 (a U.S. Census Bureau’s software package included in EViews 9. For more information please check http://www.eviews.com/help/helpintro.html#page/content/series-Seasonal_Adjustment.html), for seasonal adjustment in order to eliminate the seasonal component of the series and to be able to identify the trend and the real behavior of the series before and after the crisis, which in this case is the crisis of 2007-2009. IGAE was got seasonally adjusted from the original source.

As first evidence of the relation between the number of births and the proposed variables, it will be elaborated a graphic that compares the behavior of the time series.

It will be used a very popular econometric approach to state relationships among the used variables, suggested by Toda & Yamamoto (1995), which uses a Vector Autoregressive (VAR) model. A VAR model is accurate for stationary time series, but cointegration relationships (or long-term relationships) could not be shown in these kinds of model. Therefore, this methodology is capable to adapt the Granger causality test to cointegrated series, avoiding the problem originated by the order of integration of the variables and the co-integrant nature of the time series (Calderón-Villarreal, et al, 2017).

It will be also used the Vector Error Correction, (VEC, a VAR family kind of model) which is appropriated to identify linear relationships between non-stationary time series. This model will help to settle long-term relationships and will also show short-term relationships previously captured by the VAR model.

The preparation for the Granger causality test proposed by Toda & Yamamoto (1995) is to analyze each time series individually performing a KPSS test for stationarity in order to determinate each one’s order of integration, being the maximum found order m=1, therefore, the order of integration for the set of variables is 1 (for each country). It is important for the calibration of the model to determine the order of integration because we will add the value of the order of integration of the set to the lags for the exogenous variables with a negative sign, following the Toda & Yamamoto (1995) methodology.

The VAR modeling process was the same for each country: to run a Wald test to find which variables are not significant for the model, to find the appropriated number or lags (avoiding the presence of unit root and looking for serial correlation in the residuals) and after that, to perform a Granger causality test. Then, to estimate a VEC model to show the long-term relationships among the variables and ran a second Granger causality test.

The specifications of the model for each country are presented in the result section with the appropriate number of lags and which variables are endogenous, which are exogenous, and which are not significant for each of the cases.

To contrast the results given by the Granger causality analysis, it was used the phase synchronization, a non-parametric technique to evaluate two non-deterministic systems and their similarities, (Cruz-Aké, 2017), which has its origins in the studies made by Christian Huygens in 1665. He discovered that two pendulums staying in the same surface, will synchronize their movements but this synchronization ends if one of them is moved from the surface (Cruz-Aké et al, 2012, García-Ruiz, 2014 and Calderón-Villarreal et al 2017).

The synchronization is a phenomenon that makes reference to two dynamic systems that tend to adjust their movement and trajectory due to attractors. The most common methodologies to find out synchronization are by differential equations and by time series, identifying a master system and a slave one. These two systems will present identical oscillations after a period of time (García-Ruíz, 2014). The “controller” system (named master) is defined by the state vector x n = x 1 , x 2 , … ,     x n and an “observer” (named slave) which is an output from the master system (Cruz-Aké, et. al. 2012 and García-Ruiz, 2014).

There are four forms of synchronization: complete synchronization, which presents perfect union in the trajectories of two systems; generalized synchronization, in which the output of a system is in function of the input of a first one; phase synchronization, which present two non identical oscillators that present a weak correlation between their amplitudes; delay synchronization, which belongs to the intermediate point between perfect synchronization and phase synchronization (García Ruíz, 2014).

The objective of using this technique is to separate the time series in parts, getting the cyclical component which will allow to get the number of cycles present in each time series and to appreciate which of the time series is the master one and which series are slaves. It also lets us observe if the cycles occur at the same time which would give us a “perfect phase synchronization” between the studied variables.

It is needed to consider that a cycle has the form of a sinus function, and two changes in the signs represent a complete cycle when the line that describes this kind of function “touches” the zero twice: when it presents a fall and becomes negative, and when it raises getting a positive value again.

Now, the process for phase synchronization is described. time series will be compared in pairs: the number of births time series against each one of the other time series. The reason for doing this is that it is wanted to know how fertility is affected by the macroeconomic variables or vice versa. The steps will be the same for each studied country.

As a first step to prepare the time series for the phase synchronization, the seasonal component from the number of births and the number of marriages will be retired by using the Census Bureau’s X12-ARIMA package within EViews 9.

Since the time series present different magnitudes (percentages, millions of dollars, etc.) the next step is to normalize the series, which consists on divide each value of the series between its maximum value, maintaining their movement properties.

The process for phase synchronization is described by the following function as in Cantú-Esquivel (2018):

X t = ( ε × X t - 1 ) + ( 1 - ε ) K t (1)

Where X t represents the smoothed series, ε is the smoothing level, X t − 1 represents the smoothed series lagged one period and K t is the original time series.

After the smoothing process, calculate the phase differential for each pair of compared series using the formula

Φ X ( t ) = 2 π ( t - t k ) / ( t k + 1 - t k ) + 2 π k (2)

Where k represents the number of the cycle for the variable and t k + 1 − t k corresponds to the duration of each period (Calderón-Villarreal et al 2017). The process will be repeated for each time series in order to get the differential of analyzed pair of series. When the differential is the same for each observation in the series, it means that the series are synchronized in phase. The series could be synchronized in phase, but then lose that synchronization to recover it again or not. When the differential is the same for a period of time it represents strong synchronization, whereas if the difference between the differentials stays close to zero (but not zero) the series presents weak synchronization. After finding the phase differential it is possible to identify a master system and a slave one and replicates its behavior at the same time or with a delay.

The first variables were selected trying to represent lifelong desitions: having a baby (number of births); the number of marriages; and the changes of prices of the real estate market, represented by the IPC Housing. This last variable is also relevant to this study because of the nature of the Great Recession of 2007-2009 and it also involves a life-long desition due to the fact that most of the people need a long-term loan to purchase their houses.

Some macroeconomic variables are widely accepted as representatives of the performance of an economy: interest rates, foreign exchange rates, and industrial production (Muradoglu et al, 2000). Interest rate is vastly accepted as a variable to consider for the kind of studies that focus on crisis literature (Frankel & Saravelos, 2010).

The other variables were selected because of their very known relevance in macroeconomics literature: GDP (Gross Domestic Product, for the pool of studied countries), unemployment rate, interest rate, CCI and exchange rate. IGAE is a previous estimation of the GDP. It reflects the monthly evolution of the real economy, therefore, it is a short-term indicator.

The importance of the CCI and unemployment is supported by the fact that “the available evidence indicates that a change in unemployment or in consumer confidence matters more for fertility changes than the levels of these indicators”. (Sobotka et al 2011, p. 291). CCI affects negatively fertility because of its properties to capture uncertainty (Shneider, 2015)

Buckles et. al (2018) proposed that fertility could be a forward-looking economic indicator and that the factor behind it have a rapid effect on the family decisions, specifically on the decision of having a baby or not. Their study compared fertility rates to other important indicators: the consumer confidence index, purchases of personal durable goods and growth in housing prices. They presented graphical evidence that fertility rates decreased rapidly before these other important indicators.

Prior authors have highlighted the importance of fertility and its behavior from an economic perspective. Fertility is procyclical and it responds to negative shocks (Jones & Schoonbroodt, 2016, and Sobotka et al, 2011, Buckles et al, 2018, Bellido & Marcén, 2019, Seltzer, 2019) and to unemployment with a decline (Currie & Schwandt, 2014). For men and women, unemployment has a negative effect on conceptions and negative changes in the income could have long-term effects on fertility (Andersen & Ozcan, 2011).

From an economic perspective, a son could be seen as a consumption good and a factor of production that could provide an income to their families (Becker, 1960) and (Del Río et al, 2010).

Income is considered as a factor that affects demographic behavior (D’Addio & d’Ercole, 2005), therefore the absence of it or a low one could be a determinant in childbearing decisions. Shocks in the level of income have long term effects on fertility (Andersen & Ozcan, 2011).

All selected macroeconomic variables for the model satisfy the NBER (National Bureau of Economic Research) criteria which consider the economic significance, the statistical reliability, a uniform behavior through the time, large coverage of the economy, smoothness of the series, timely availability and high frequency (Del Río et al, 2010). Macroeconomic variables are useful to observe the general behavior of an economy.

The Great Recession of 2007-2009 was declared by the NBER began in December 2007 (Buckles et al, 2007). Therefore, and according to Buckles et al 2018, it is expected to see a decline in the number of births months before this date.

Graph 1 shows both time series (number of births and IGAE) and it is possible to see that two months before December 2007, the number of births had a decline. It changed the trend of the series and even though it presented a little recover for February 2008, this downward trend continued and increased its fall in November 2008 almost at the same time that IGAE presented an abrupt fall. As we said before, in the case of Mexico, the most dramatic falls in the number of births and IGAE is present at the same time (October 2008) due to the fact that a birth is the result of nine months of gestation, therefore we can assume that families perceived an adverse economic environment before the crisis was declared. The previous support the results in Buckles et al 2018, (with its proper considerations about the differences between total fertility and number of births) in which it is possible to see a fall in the fertility rates, quarters before the beginning of the crisis.

The CCI is an indicator that shows how optimistic are the consumers in relation to the general performance of the economy and how they feel about their financial perspectives for the future. This indicator is important for our study because the consumers could perceive negative conditions that could affect their life-long decisions, related to their aims of having a baby.

Graphic 1 shows the behavior of both time series (number of births and ICC) and it is possible to appreciate that the CCI presents a decline in August 2007 (four months before the official beginning of the crisis) and the number of births shows its fall in October 2007, but again, a birth it is the result of a period of nine months of gestation. Therefore, we can say that the abrupt fall in the number of births is the result of the perception of problems for a period of at least nine months. We could have seen the decline in the birth series a couple of months before as in Buckles et al (2018).

These results support the Malthus theory that stated that an increase in the income would cause an increase in fertility due to the families (Becker, 1960) have a more positive perspective about their future income, therefore, they will not postpone the decision of having a baby.

The nature of the Great Recession of 2007-2009 is strongly related to the real estate sector. The expansionist policies (specifically low-interest rates) of the U.S. the years before the crisis, eased a great number of consumers to purchase a house despite their low credit quality. The interest rates let them buy a property regardless of their capacity to pay for it. A great number of loans were granted at high-interest rates because of the risk they represented. These credits were known as subprime. These conditions created a bubble that began to be untenable in the middle of 2006, due to an increase in the prices of some commodities and had a fatal end with the bankruptcy of some funds linked to the risky mortgages (Cruz-Aké et al, 2011). The growth in the offer of properties caused that the prices of the real estate sector fell and the promise of can be able to sell their house at a higher price than they paid (if they were not capable to pay for it), caused a decline in the prices and a great number of credit holders lost their heritage.

Graphic 1 shows an increasing trend in the IPC Housing series but we are still able to identify a fall in the trend of the time series in December 2008. For the time series number of births, is clear that the respective fall happened at the same time, but again, is clear that birth is a result of nine months of gestation and the previous months to take the decision.

For these variables, it is clear that when the IPC Housing had a recovery a year after the fall, the number of births continued falling during the crisis and it did not recover its past positive trend until January 2010, supporting the hypothesis in Buckles et al (2018) that stated that the number of conceptions has a slow recovery after a crisis.

Graph 1 Source: with data from INEGI

The Granger causality test for the VAR model showed a causal relationship between unemployment and the number of births in both ways. It supports conclusions from Currie & Schwandt (2014), Andersen & Ozcan (2011), Buckles et al, (2018), Bellido & Marcén (2019) and Seltzer (2019). The ICC did not reveal a causal relationship to the number of births.

The Granger causality test for the VEC model revealed a causal relationship from unemployment to the number of births. In this case, the IPC Housing also showed a causal relationship to the number of births. Similar results were captured in the graphical evidence section in this study and in Buckles et al (2018) and Shneider (2015), who found that fertility declines with the consumer confidence. All the variables together (IPC Housing, unemployment, TIIE, and ICC) also showed a causal relation as a set, a result not captured before for any other study.

The Granger causality test for the VAR model revealed that there is no causal relationship from GDP and unemployment to the number of births individually or as a set.

After the elaboration of a VEC model, we found that that the number of births could be explained by all the other variables as a set, and individually, by the prices of housing and by the number of marriages. This last relation is in both ways. It seems that the decision of having a baby in Germany is more reasonable as people look more interested in economic variables and a stable environment to take the decision.

The Granger causality test for the VAR showed that there is not a causal relation from any of the variables to the number of births, individually or as a set, but the number of births presents a causal relation to the CCI and the unemployment. The causal relation between the number of births and unemployment could be explained by educational factors where a woman prefers to raise her family instead of having a job or due to the deficient maternity leaving policies or the lack of day care centers.

For the case of the VEC model it was found that there is an individual causal relation from the confidence index, the number of marriages and the exchange rate to the number of births. The causal relation is present from all the variables as a set. The number of births has a causal relation to unemployment.

Chile seems to be another country where people are interested in ensuring stable economic conditions before to take the decision of having a baby, being CCI a factor to consider, supporting Shneider (2015). There is no previous evidence of relation from the exchange rate to the number of births.

There was not a CCI reliable source, therefore, for this country, it is not included. Despite the lack of a variable, the elaborated model did not present unit root and the residuals showed serial correlation, avoiding the problem of an omitted variable.

The Granger causality test for the VAR model showed that marriages have a causal relation explaining the number of births. Marriages and housing as a set have a causal relationship to the number of births, and the number of births has a causal relation to marriages. Singapore seems to be a more traditional country where marriage is still important to take the decision of having a baby. Religious reasons could be a factor behind it and the desire of having a heritage for the families result important.

After the elaboration of the VEC model, we found that the number of births could not be explained by any of the variables individually or as a set, but the number of births could explain the number of marriages in a long term. The relation between marriages and the number of births also appears in the long term.

The Granger causality test for the VAR model revealed that exists a causal relationship between the exchange rate and the number of births and from the GDP to the number of births. The whole set of endogenous variables has a causal relation to the number of births, revealing the interest of people of the general economic performance instead of the individual variables.

The VEC model showed a causal relation from the house prices to the number of births. The set of variables also showed a causal relationship to the number of births for the long term, showing that in the long term the whole set of variables are relevant.

Table 2

VAR GRANGER CAUSALITY/BLOCK EXOGENITY WALD TEST
H o : There is not causality
COUNTRY: MEXICO					COUNTRY: GERMANY
Dependent variable: Number of births					Dependent variable: Number of births
Excluded	Chi sq	df	Prob.		Excluded	Chi sq	df	Prob.
CCI	7.813	13	0.8556		GDP	2.9198	2	0.2323
Unemployment	33.9438	13	0.0012			3.8153	2	0.1484
All	55.9132	26	0.0006		All	4.3569	4	0.3599
Dependent variable: Unemployment					COUNTRY: SINGAPORE
Excluded	Chi sq	df	Prob.		Dependent variable: Number of births
Births	35.458	13	0.0007		Excluded	Chi-sq	df	Prob.
CCI	14.2772	13	0.3546		Housing	7.5527	4	0.1094
All	61.3164	26	0.0001		Marriages	18.9668	4	0.0008
					All	40.9397	8	0
COUNTRY: CHILE
Dependent variable: Number of births					Dependent variable: Housing
Excluded	Chi sq	df	Prob.		Excluded	Chi-sq	df	Prob.
CCI	7.39	3	0.0605		Births	4.5268	4	0.3394
Unemployment	4.6535	3	0.199		Marriages	8.5906	4	0.0722
Interest Rate	0.2897	3	0.962		All	13.6909	8	0.0902
GDP	2.8068	3	0.4224
All	20.8633	12	0.0524		Dependent variable: Marriages
					Excluded	Chi-sq	df	Prob.
Dependent variable: CCI					Housing	20.4339	4	0.0004
Excluded	Chi sq	df	Prob.		Births	14.4662	4	0.0059
Unemployment	5.4561	3	0.1413		All	35.5826	8	0
Interest Rate	3.015	3	0.3893
Births	10.572	3	0.0143		COUNTRY: SOUTH AFRICA
GDP	3.1003	3	0.3764		Dependent variable: Number of Births
All	19.9513	12	0.068		Excluded	Chi-sq	df	Prob.
					CCI	2.4928	2	0.2875
Dependent variable: Unemployment					Housing	3.229	2	0.199
Excluded	Chi-sq	df	Prob.		GDP	6.0031	2	0.0497
CCI	8.3996	3	0.0384		Exchange Rate	7.4467	2	0.0242
Interest Rate	3.1762	3	0.3652		All	29.2832	8	0.0003
Births	24.2776	3	0
GDP	7.4915	3	0.0578
All	63.8011	12	0

Source: own elaboration with E-Views9

Table 3

VEC Granger Causality/Block Exogeneity Wald Tests
Country: Mexico					Country: Chile
Dependent variable: Number of births					Dependent variable: Number of births
Excluded	Chi-sq	df	Prob.		Excluded	Chi-sq	df	Prob.
Unemployment	24.52463	12	0.0172		CCI	11.18586	3	0.0108
CCI	11.94151	12	0.4504		Unemployment	2.634946	3	0.4514
IGAE	13.50274	12	0.3336		Housing	2.831318	3	0.4184
IPC Housing	14.59866	12	0.2641		Interest Rate	4.7761	3	0.1889
Exchange Rate	14.50703	12	0.2695		Marriages	18.45861	3	0.0004
TIIE	11.31941	12	0.5018		GDP	6.059365	3	0.1088
Marriages	15.80657	12	0.2003		Exchange Rate	9.62919	3	0.022
All	114.2521	84	0.0157		All	35.61293	21	0.0242
Country: Germany					Dependent variable: Unemployment
Dependent variable: Number of births					Excluded	Chi-sq	df	Prob.
Excluded	Chi-sq	df	Prob.		CCI	6.048302	3	0.1093
CCI	3.868807	3	0.276		Housing	7.647812	3	0.0539
Unemployment	4.743866	3	0.1915		Interest Rate	8.954831	3	0.0299
Housing	14.37491	3	0.0024		Marriages	7.607323	3	0.0549
Interest rate	4.646641	3	0.1996		Number of births	22.24062	3	0.0001
Marriages	32.12842	3	0		GDP	9.454478	3	0.0238
GDP	4.671733	3	0.1975		Exchange rate	0.893691	3	0.827
Exchange Rate	0.203616	3	0.977		All	149.8801	21	0
All	120.4529	21	0
					Country: Singapore
Dependent variable: Housing					Dependent variable: Number of births
Excluded	Chi-sq	df	Prob.		Excluded	Chi-sq	df	Prob.
CCI	0.357774	3	0.9488		Unemployment	2.303539	2	0.3161
Unemployment	2.057802	3	0.5605		Housing	0.877581	2	0.6448
Interest Rate	0.892172	3	0.8273		Interest rate	0.19976	2	0.9049
Marriages	15.58692	3	0.0014		Marriages	1.099302	2	0.5772
Number of Births	10.66256	3	0.0137		GDP	2.130256	2	0.3447
GDP	3.679083	3	0.2983		Exchange rate	0.424638	2	0.8087
Exchange Rate	0.055219	3	0.9966		All	7.758269	12	0.8037
All	25.93748	21	0.2088
					Dependent variable: Marriages
Country: South Africa					Excluded	Chi-sq	df	Prob.
Dependent variable: Number of births					Unemployment	0.51853	2	0.7716
Excluded	Chi-sq	df	Prob.		Housing	12.27894	2	0.0022
CCI	3.826776	2	0.1476		Interest rate	0.604732	2	0.7391
Unemployment	4.816439	2	0.09		Number of births	6.767615	2	0.0339
Housing	6.238688	2	0.0442		GDP	2.254135	2	0.324
Interest rate	4.115934	2	0.1277		Exchange rate	3.631438	2	0.1627
Marriages	3.144149	2	0.2076		All	25.0905	12	0.0144
GDP	2.815184	2	0.2447
Exchange rate	3.905352	2	0.1419
All	38.16482	14	0.0005

Source: own elaboration with E-Views9

Unemployment presented four periods of synchronization which confirms the relationship found with the VAR and VEC models and the affirmation in Currie & Schwandt (2014), Andersen & Ozcan (2011) and D’Addio & d’Ercole (2005). The VEC model could find long term relationships which also appeared with the phase synchronization, which lead to establishing that the relationships are non-linear and long term.

The absence of synchronization during the period of the 2007-2009 crisis is relevant due to the fact that it is a similar result as in Calderón-Villarreal et al (2017) who find the loss of synchronization during periods of crisis. Unemployment and marriages presented synchronization during that period but just for 2 months for both cases. For the case of unemployment, it also appeared in the VEC model. CCI also presented a period of perfect synchronization from October 2007 to January 2008.

It is important to highlight the absence of synchronization between marriages and the number of births, representing a reality in which marriage is not the only option for raising a family. The prices in properties were the variable with most synchronized periods converting it into a variable relevant for family decisions and confirming the nature of the Great Recession of 2007-2009.

The relationship found with the VEC model between the number of births and marriages is not present with the phase synchronization, but the variable housing presented two periods of synchronization: from 2008Q2 to 2008Q3 and a long one from 2011Q2 to 2013 Q2, representing the aim of the parents for have their own houses before taking a decision about fertility turning it into a rational decision as in Davia & Legazpe (2013).

Once again it is important to highlight the absence of synchronization for the variables during the subprime crisis, even when variables like exchange rate and CCI presented synchronization before the crisis, it seems to lose the synchronization during crisis periods, remarking the importance previously stated by Muradoglu et al (2000) and Schneider (2015).

For Chile, the relationship found with the VAR and VEC models between the CCI and the number of births is also present with the phase synchronization with the period of phase synchronization present from 2007Q2 to 2009Q4 and during the two first quarters of 2011 and confirming the relationship in Schneider (2015).

Marriages also presented a synchronized behavior from 2008Q3 to 2009Q4 and from 2011Q2 to 2011Q4. This relationship is also present in the VEC model. Exchange rate presented a synchronized behavior from 2012Q3 to 2015Q1, confirming the causal relationship found the VEC model. Interest rate presented a period of synchronization from 2009Q1 to 2009Q4 remarking the relation established by Murgadoglu et al (2000).

The results for Chile are interesting because the synchronized period corresponds to the subprime crisis for the variables marriages, interest rate and CCI, which is contrary to the findings for Mexico and Germany, where the variables did not present synchronization during the crisis. It maybe represents that the importance of these variables is only relevant for chilean families during periods of crisis when having a baby requires a more planned decision.

The results for Singapore confirm the relationship found with VAR model between the number of births and marriages, being that these two variables presented a period of synchronization from 2010Q2 to 2014Q1 and a short one during the first two quarters of 2007 confirming that for families in Singapore marriage is still relevant for making decisions about having a baby.

It is interesting that the variables housing, unemployment, interest rate, and exchange rate presented important periods of synchronization, but only housing and interest rate presented two quarters of synchronization during the 2007-2009 crisis. The synchronization for these variables is present again after the crisis for long periods, confirming the loss of synchronization during crisis periods as in Calderón-Villarreal et al (2017). Synchronization for marriages, housing, unemployment, interest rate, and the exchange rate was present basically for the same quarters.

For South Africa, there are periods synchronized in phase for the exchange rate (as with VAR model) and housing (as with VEC model). The relation present with GDP is not present with the phase synchronization technique.

Marriages, housing, and interest rate shared a period of synchronization from 2006Q4 to 2009Q1, but it was lost when the effects of the Great Recession were deeper. Short periods of synchronization are common for housing, interest rate, exchange rate, and unemployment, confirming the importance as in Murgadoglu (2000) and Lovenheim & Mumford (2013).

In general, for all studied economies it is important to highlight the fact that the compared variables lost synchronization during the 2007-2009 crisis., or in its case, presented periods were short. For South Africa, marriages, housing, and interest rate were the only variables with a significant period of synchronization during that crisis. The synchronization present in Chile during the crisis was a weak one. These results are similar to the findings in Claderón-Villarreal (2017) for periods of crisis.

Figure 1

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Figure 3

Figure 4

Figure 5