Actors, football players, models are millionaires and receive salaries that sometimes can exceed some developing countries’ annual budgets, while doctors, researchers, and employees in related fields receive in many countries around the world the minimum wage only to keep them and their families alive.

In such times, when the entire human-kind is at threat and faces a global crisis, everyone stands at the researchers’ and doctors’ doorsteps waiting for them to develop a cure or a vaccine to save the world. In such times, the true value of science and researching emerges and the significance of investing in knowledge and education becomes evident and nonnegligible.

Our vision at MSR HUB to “Bridge Knowledge from those who have it to those who need” derives our team and defines our mission, accordingly and as a part of our contribution in alleviating and supporting the world in the forthcoming global recession.

MS Research Hub institute in Germany will administrate and fund the first research project that will be moderated by selected team members to empirically investigate and predict how economies behave – and should behave- in times of the Corona virus using historical data of similar epidemics that have hit the humankind, starting from the Spanish flu at the beginning of the 19 century, passing by MARS and MERS. Our objective will be to develop a prescription that the world economies can use against the forthcoming crises, especially in times where borders are closed, imports and exports are restricted, and the globalization model is simply frozen.

This research project will the official start of our institute “Research Grant Scheme” that aims to fund independent researchers from the least developed countries to carry on their planned human-related research projects in all scientific fields.

We believe first and always in mighty Allah, human-kind and the power of knowledge and science in facing the current crises.

Dr. Sherif Hassan CEO & Academic Division director at MSR HUB- Germany

We declare first some usual assumptions, like closed economy *XN=0*, net investment equals *I=K-δK* where *δ* is a common depreciation rate of the economy for all kinds of capital. There’s no government spending in the model so *G=*0. And finally, we’re setting a function which is going to capture the individual utility *u(c)* given by:

This one is referred to as the constant intertemporal elasticity function of the consumption* c *over time *t*. The behavior of this function can be established as:

This is a utility function with a concave behavior, basically, as consumption in per capita terms is increasing, the utility also is increasing, however, the variation relative to the utility and the consumption is decreasing until it gets to a semi-constant state, where the slope of the points c1 and c2 is going to be decreasing.

We can establish some results of the function here, like

And that

That implies that the utility at a higher consumption point is bigger than on a low consumption point, but the variation of the points is decreasing every time.

The overall utility function for the whole economy evaluated at a certain time can be written as:

Where *U* is the aggregated utility of the economy at a certain time (t=0), *e* is the exponential function, *ρ* is the intergenerational discount rate of the consumption (this one refers to how much the individuals discount their present consumption related to the next generations) *n* is the growth rate of the population, *t* is the time, and *u(c)* is our individual utility function, *dt *is just the differential which indicates what are we integrating.

Think of this integral as a
sum. You can aggregate the proportion of individual utilities at a respective
time considering the population size, but also you need to bring back to the
present the utility of other generations which are far away from our time
period, this is where *ρ* enters and its
very familiar to the role of the interest rate in the present value in
countability.

This function is basically considering all time periods for the sum of individuals’ utility functions, in order to aggregate the utility of the economy U (generally evaluated at t=0).

This is our target function because we’re maximizing the utility, but we need to restrict the utility to the income of the families. So, in order to do this, the Ramsey model considers the property of the financial assets of the Ricardian families. This means that neoclassical families can have a role in the financial market, having assets, obtaining returns or debts.

The aggregated equation to the evolution of financial assets and bonuses *B* is giving by:

Where the left-side term is the evolution of all of the financial assets of the economy over time, w refers to the real rate of the wages, *L* is the aggregate amount of labor, *r* is the interest rate of return of the whole assets in the economy *B*, and finally, *C* is the aggregated consumption.

The equation is telling us that the overall evolution of the total financial assets of the economy is giving by the total income (related to the amount of wages multiplied the hours worked, and the revenues of the total stock of financial assets) minus the total consumption of the economy.

We need to find this in per capita terms, so we divide everything by L

And get to this result.

Where *b=B/L* and c is the consumption in per capita terms. Now we need to find the term with a dot on B/L, and to do this, we use the definition of financial assets in per capita terms given by:

And now we difference respect to time. So, we got.

We solve the derivate in general terms as:

And changing the notation with dots (which indicate the variation over time):

We have

We separate fractions and we got:

Finally, we have:

Where we going to clear the term to complete our equation derived from the restriction of the families.

And we replace equation (2) into equation (1). And we have

To obtain.

This is the equation to find the evolution of financial assets in per capita terms, where we can see it depends positively on the rate of wages of the economy and the interest rate of returns of the financial assets, meanwhile it depends negatively on the consumption per capita and the growth rate of the population.

The maximization problem of the families is giving then as

Where we assume that b(0)>0 which indicates that at the beginning of the time, there was at least one existing financial asset.

We need to impose that utility function is limited, so we state:

Where in the long run, the limit of utility is going to equal 0.

Now here’s the tricky thing, the use of dynamical techniques of optimization. Without going into the theory behind optimal control. We can use the Hamiltonian approach to find a solution to this maximization problem, the basic structure of the Hamiltonian is the following:

*H(.) = Target Function + v (Restriction)*

We first need to identify two types of variables before implementing it in our exercise, the control variable, and the state variable. The control variable is the one that focuses on the agent which is a decision-maker, (in this case, the consumption is decided by the individual, and the state variable is the one relegated in the restriction). The state variable is the financial assets or bonus b. Now the term v is the dynamic multiplier of Lagrange, consider it, as the shadow price of the financial assets in per capita terms, and it represents an optimal change in the individual utility given by one extra unit of the assets.

We’re setting what is inside of our integral as our objective, and our restriction remains the same and the Hamiltonian is finally written as:

The first-order conditions are giving by:

One could ask why we’re setting the partial derivates as this? Well, it’s part of the optimum control theory, but in sum, the control variable is set to be maximized (that’s why it’s equally to 0) but our financial bonus (the state variable) needs to be set negatively to the shadow prices of the evolution of the bonus because we need to find a relationship where for any extra financial asset in time we’ll decrease our utility.

The solution of the first-order condition *dH/dc *is giving by:

To make easier the derivate we can re-express:

To have now:

Solving the first part we got:

To finally get:

Now solving the term of the first-order condition we obtain:

thus the first-order condition is:

Now let’s handle the second equation of first-order condition in *dH/db*.

Which is a little bit easier since:

So, it remains:

Thus we got.

And that’s it folks, the result of the optimization problem related to the first-order conditions are giving by

Let’s examine these conditions: the first one is telling us that the shadow price of the financial assets in per capita terms it’s equal to the consumption and the discount factor of the generations within the population grate, some better interpretation can be done by using logarithms. Lets applied them.

let’s differentiate respect to time and we get:

Remember that the difference of logarithms it’s equivalent approximately to a growth rate, so we can write this another notation.

Where

In equation (4) we can identify that the growth rate of the shadow prices of the financial assets is negatively related to the discount rate ρ, and the growth rate of consumption. (in the same way, if you clear the consumption from this equation you can find out that is negatively associated with the growth rate of the shadow prices of the financial assets). Something interesting is that the growth rate of the population is associated positively with the growth in the shadow prices, meaning that if the population is increasing, some kind of pull inflation is going to rise up the shadow prices for the economy.

If we multiply (4) by -1, like this

and replace it in the second equation of the first order which is

Multiplied by both sides by v, we get

Replacing above equations drive into:

Getting n out of the equation would result in:

Which is the Euler equation of consumption!

Mankiw, N. G.,
Romer, D., & Weil, N. D. (1992). A CONTRIBUTION TO THE EMPIRICS OF ECONOMIC
GROWTH. *Quarterly Journal of Economics*, 407- 440.

Ramsey, F. P.
(1928). A mathematical theory of saving. *Economic Journal, vol. 38, no. 152,*,
543–559.

Solow, R. (1956). A
Contribution to the Theory of Economic Growth. *The Quarterly Journal of Economics,
Vol. 70, No. 1 (Feb., 1956),*, 65-94.

“The Holy Grail of applied econometrics is a tight theoretical model, which fits the data well. Like the Holy Grail, such models are hard to find.” (Dunne, Smith, & Willenbockel, 2005)

When one, as a researcher meditate this, one really knows that matching theoretical models with regression equations it’s indeed hard. Even when econometrics can be defined as the measure and validation part of the economic science, the relationships which are addressed to study are not exactly as accurate as the theory states.

I want to put an example, let’s see the conclusions of the Solow Swan (1956) model with technology. which are compiled in the next equation.

Where *Y/L* is the gross domestic product -GDP- of the economy measured in per capita units, *A* is a level of technology, α is the elasticity of the aggregate stock of capital of the economy, *s* is an exogenous saving rate, δ is the depreciation rate, *x* is the growth rate of the technology, and n is the growth rate of the population.

The term ε is just added as the stochastic error in the equation to proceed with the regression analysis, which theoretically is defined as independent of the variables in the regression and represents external shocks in the per capita product. However, if this doesn’t happen in the time series context, it could be possible that this term contains all the variables not included in the regression, therefore violating the exogeneity assumption and inducing an omitted variable bias with misspecification.

Basically, the model is telling us that the growth of the economy is positively given by the technology and the rate of saving of the economy which is invested in physical capital.

Now the Augmented Solow-Model proposed by Mankiw, Romer & Weil (1992) also known as the MRW model, concludes the following:

Where we got some new terms denoted with β as the elasticity of the aggregate stock of human capital in the production function, and separated terms of the savings, denoted by s_k as the saving rate dedicated to the accumulation of physical capital and s_h which is the saving rate dedicated to the accumulation of human capital.

The Augmented Model proposed by Mankiw, Romer & Weil has more variables in the specification of the growth of the economy.

Which one is correct? The answer relies on the regressions they have performed with both models, in general, the augmented model explains better the economic growth and the convergence of the economies than the simple Solow-Swan model.

The simple Solow-Swan model has a problem in the specification and an omitted variable problem, the augmented Solow-Swan model correct this by introducing the measure and importance of human capital accumulation. Both are theoretical constructions, but the augmented model fits better in reality than the original model.

Going further, one could ask if it would be wrong to consider all variables as endogenous? In the last two models, we have seen that the savings of physical or human capital are exogenous along with the growth rate of technology, but more theoretical considerations, like the Ramsey (1928) model could determinate the savings as endogenous, even the depreciation and the technology can be endogenized, so regressing the above equation with two-stage or three-stage least squares would be the best approach.

Considering this set of ideas, econometricians then will have to face a difficult situation when the theoretical approach might not be suitable for the reality of the sample, and I say this because this is a complex world, where a single explanation for all the situations is not plausible.

We need to remember also, that the whole objective of the theory is to explain reality, and if this theory fails to succeed in this objective, even the most logical explanation would be useless. Something completely out of sense is to modify reality to match with the theory.

The holy grail then would be the adequacy of the theory with the reality, and in econometrics, this means that we need to find a strong theoretical framework that matches our data generating process. But the validation techniques should have some logical approaches considering the assumptions of the regression.

Going backward, before theory and empirical methods, we are interested in finding the truth, and this truth goes from discovery existing or non-existing relationships and causality, in order to explain reality. Such findings, even when they start from a deviated or wrong approach are useful to build the knowledge.

A great example of this is the Phillip’s Curve (Phillips, 1958), it started as an empirical fact which correlated positive rates of inflation with employment, and then it began to be strongly study on Phelps (1967) and Friedman (1977) with more theoretical concepts as rational expectations over the phenomenon of inflation.

Econometricians should then do research with logical economic sense when they are heading to estimate relationships, but have to be aware that samples and individuals are not the same in the space (they change according to locations and the time itself). However, the theoretical framework is the main basis we need to always consider during the economic research, but also remember we can propose a new theoretical framework, to explain the reality on the basis of facts and past theories.

Dunne, J., Smith,
R. P., & Willenbockel, D. (2005). MODELS OF MILITARY EXPENDITURE AND
GROWTH: A CRITICAL REVIEW. *Defence and Peace Economics, Volume 16, 2005 –
Issue 6*, 449-461.

Friedman. (1977). Nobel
Lecture Inflation and Unemployment. *Journal of Political Economy, Vol. 85,
No. 3 (Jun., 1977)*, 451-472.

Kwat, N. (2018). *The
Circular Flow of Economic Activity.* Obtenido de Economics Discussion:
http://www.economicsdiscussion.net/circular-flow/the-circular-flow-of-economic-activity/18159

Mankiw, N. G.,
Romer, D., & Weil, N. D. (1992). A CONTRIBUTION TO THE EMPIRICS OF
ECONOMIC GROWTH. *Quarterly Journal of Economics*, 407- 440.

Marmolejo, I.
(2012). *Indifference Curve Confusion and Possible Critique.* Obtenido de Radical
Subjectivist:
https://radicalsubjectivist.wordpress.com/2012/02/10/indifference-curve-confusion-and-possible-critique/

Nicholson, W.
(2002). *Microeconomic Theory.* México D.F.: Thompson Learning.

Phelps, E. (1967).
Phillips Curves, Expectations of Inflation and Optimal Unemployment over Time
. *Economica, New Series, Vol. 34, No. 135 (Aug., 1967)*, 254-281.

Phillips, A. W.
(1958). The Relation between Unemployment and the Rate of Change of Money Wage
Rates in the United Kingdom, 1861-1957. *Economica, New Series, Vol. 25, No.
100. (Nov., 1958),*, 283-299.

Ramsey, F. P.
(1928). A mathematical theory of saving. *Economic Journal, vol. 38, no.
152,*, 543–559.

Solow, R. (1956).
A Contribution to the Theory of Economic Growth. *The Quarterly Journal of Economics,
Vol. 70, No. 1 (Feb., 1956),*, 65-94.

Governments and public communities should pay close attention to improving the widespread, availability and affordability of education for girls and women nationwide regardless of the women’s residence area and levels of income. Subsidizing and income transfer programs should make sure that girls continue their education and do not leave schools due to income constraints. Availability and reachability of schools especially for girls living in slums or refugee camps that are located outside the peripheral areas of public services should be improved and families need to be constantly advised and guided about the importance of education to their children.

Within the MENA region that has the lowest global share of female literacy, Palestinian women are classified as the best-educated (The Royal Academy of Science International Trust [RASIT], 2017). Our analysis suggests that the better educational attainment of Palestinian women explains the low prevalence of child marriage and having relatively lower health deprivations relative to their counterparts in the other countries. Better educated women are not only capable of better caring about their health and the health of their children but also they are better wives, citizens and a catalyst for the development of their countries. As narrated by Hafez Ibrahim the Nile poet in his poem about knowledge and morals (Ibrahim, 1937): “A mother is a school, whenever you equipped her well, you prepared a nation with a fine race”.

Reference: Hassan, S.M. (2019). **H**ealth Repercussions of Child Marriage on Middle-Eastern Mothers and Their Children. preprint

In this article, I want to start with an introduction of the basic concept of budget constrain related to the income in microeconomics, and that’s the linear constraint given a set of quantities and prices of the goods which determine the utility for the consumer, this is closely related to the Cobb-Douglas utility function (and overall utility functions) since it is one of the main aspects of the microeconomic theory.

Keeping the utility function as the traditional Cobb-Douglas function:

We know that the utility is sensitive to the elasticity **α**and *B*. With **α**and *B* lesser or equal to one. And since resources are not infinite, we can establish that the amount of goods that the consumer can pay is not infinite. In markets, the only way to get goods and services is with money, and according to the circular flow of the economy, the factor market can revenue two special productive factors: labor and capital, we can say that consumers have a level of income derived from his productive activities.

Inside the microeconomic theory in general, utility U is restricted to the income of the consumer within a maximization process with a linear constraint containing the goods and prices which are consumed. The budget constraint for the two good model looks as it follows:

Where *I* is the income of the individual, *Px* is the price of the good X and *Py* is the price of the good Y. One might wonder if the income of the customer is the sum of prices times goods, which doesn’t seem as close to what the circular flows states in a first glance. Income could be defined as the sum of the salary and overall returns of the productive activities (like returns on assets) of the consumer, and there’s no such thing as that in the budget equation.

However, if you look at the equation as a reflection of all the spending on goods (assuming the consumer will spend everything) this will equally match all that he has earned from his productive activities.

The maximization problem of the consumer is established as:

And typical maximization solution is done by using the Lagrange operator where the whole expression of the Lagrange function can be stated as:

A useful trick to remember how to write this function is to remember that if λ is positive then the income is positive and the prices and goods are negative (we’re moving everything to the left from the constraint equation). And the first-order conditions are given by:

By simply dividing the first two differential equations you’ll get the solution to the consumer’s problem which satisfies the relation as the next ratio:

Each good then is primarily sensitive to his own price and the weight (elasticity) in the utility function, seconded by the prices and quantities of the other good Y. If we replace one of the solutions in the last differential equation, say X, we’ll get:

Taking as a general factor the Py*Y will result in:

The quantities of the good Y are a ratio of the Income times the elasticity* B* and this is divided by the price of the same good Y given the sum of the elasticities. Before we stated that α+ *B = 1* so we got that B*=1- *α and the optimum quantities of the goods can be defined now as:

This optimal place its graphically displayed ahead, and it represents the point where the utility is the maximum given a certain level of income and a set of prices for two goods, if you want to expand this analysis please refer to Nicholson (2002).

**The budget constrains: An econometric appreciation**

Suppose we got a sample of *n* individuals which only consumes a finite number of goods. The income is given for each individual and also the quantities for each good. How we would be able to estimate the average price that each good has? If we start by assuming that the income is a relation of prices and quantities from the next expression:

Where X_1 is the good number one associated with the price of the good P_1, the income would be the sum of all quantities multiplicated by their prices or simply, the sum of all expenses. That’s the approach on demand-based income. In this case we got m goods consumed.

Now assume we can replace each price for another variable.

Looks familiar, isn’t it? It’s a regression structure for the equation, so in theory, we are able to estimate each price with ordinary least squares. Assuming as the prices, the estimators associated with each good with *B*-coefficients. And that all the income is referred to as the other side of the coin for the spending process.

**The simulation exercises**

Assume we got a process which correlates the following variables (interpret it as the Data Generating Process):

Where *I* is the total income, Px, Py, Pz are the given prices for the goods X, Y, and Z and we got *s* which refers to a certain amount of savings, all of this of the individual *i*. This population according to the DGP not only uses the income for buying the goods X, Y, and Z, but also deposits an amount of savings in *s*. The prices used in the Monte Carlo approach are Px=10, Py=15, and Pz=20.

If we regress the income and the demanded quantities of each good, we’ll have:

The coefficients don’t match our DGP and that is because our model is suffering from a bias problem related to omitted variables. In this case, we’re not taking into account that the income is not only the sum of expenses in goods but also the income is distributed in savings. Regressing the expression with the *s* variable we have:

The coefficients for the prices of each good (X, Y, Z) match our DGP almost accurate, R squared has gotten a significant increase from 51.45% to 99.98%. And the overall variance of the model has been reduced. The interesting thing to note here is that the savings of the individuals tend to be associated with an increase in the income with an increase of one monetary unit in the savings.

Remember that this is not an exercise of causality, this is more an exercise of correlation. In this case, we’re just using the information of the goods for the individuals of our sample to estimate the average price for the case of two goods. If we have a misspecification problem, such an approach cannot be performed.

This is one way to estimate the prices that the consumers pay for each good, however, keep in mind that the underlying assumptions are that 1) the prices are given for everyone, they do not vary across individuals, 2) The quantities of X, Y and the amount of savings must be known for each individual and it must be assumed that the spending (including money deposited in savings) should be equivalent to the income. 3) The spending of each individual must be assumed to be distributed among the goods and other variables and those have to be included in the regression, otherwise omitted variable bias can inflict problems in the estimators of the goods we’re analyzing.

**References**

Kwat, N. (2018). *The Circular Flow of Economic Activity.* Economics Discussion. Recuperated from: http://www.economicsdiscussion.net/circular-flow/the-circular-flow-of-economic-activity/18159

Marmolejo, I. (2012). *Indifference Curve Confusion and Possible Critique.* Radical Subjectivist. Recuperated from: https://radicalsubjectivist.wordpress.com/2012/02/10/indifference-curve-confusion-and-possible-critique/

Nicholson, W.
(2002). *Microeconomic Theory.* México D.F.: Thompson Learning.

In this article, I want to start with an introduction of a basic concept in microeconomics, which is the Cobb-Douglas utility function and its estimation with Stata. So we’re reviewing the basic utility function, some mathematical forms to estimate it and finally, we’ll see an application using Stata.

Let’s start with the traditional Cobb-Douglas function:

Depending on the elasticity *α* and β for goods X and Y, we’ll have a respective preference of the consumer given by the utility function just above. In basic terms, we restrict *α* + β =1 in order to have an appropriate utility function which reflects a rate of substitution between the two goods X and Y. If we assume a constant value of the utility given by U* for the consumer, we could graph the curve by solving the equation for *Y*, in this order of ideas.

And the behavior of the utility function will be given by the number of quantities of the good Y explained by X and the respective elasticities *α* and β. We can graph the behavior of the indifference curve given a constant utility level according to the quantities of X and Y, also for a start, we will assume that *α* =0.5 and β=0.5 where the function has the following pattern for the same U* level of utility (example U=10), this reflects the substitution between the goods.

If you might wonder what happens when we alter the elasticity of each good, like for example, *α* =0.7 and β=0.3 the result would be a fast decaying curve instead of the pattern of the utility before.

Estimating the utility function of the Cobb-Douglas type will require data of a set of goods (X and Y in this case) and the utility.

Also, it would imply that you somehow measured the utility (that is, selecting a unit or a measure for the utility), sometimes this can be in monetary units or more complex ideas deriving from subjective utility measures.

Applying logarithms to the equation of the Cobb-Douglas function would result in:

Which with properties of logarithms can be expressed as:

This allows a linearization of the function as well, and we can see that the only thing we don’t know regarding the original function is the elasticities of *α* and β. The above equation fits perfectly in terms of a bivariate regression model. But remember to add the stochastic part when you’re modeling the function (that is, including the residual in the expression). With this, we can start to do a regressing exercise of the logarithm of the utility for the consumers taking into account the amount of the demanded goods X and Y. The result would allow us to estimate the behavior of the curve.

However, some assumptions must be noted: 1) We’re assuming that our sample (or subsample) containing the set of individuals *i* tend to have a similar utility function, 2) the estimation of the elasticity for each good, would also be a generalization of the individual behavior as an aggregate. One could argue that each individual *i* has a different utility function to maximize, and also that the elasticities for each good are different across individuals. But we can argue also that if the individuals *i* are somewhat homogenous (regarding income, tastes, and priorities, for example, the people of the same socioeconomic stratum) we might be able to proceed with the estimation of the function to model the consumer behavior toward the goods.

**The Stata application**

As a first step would be to inspect the data in graphical terms, scatter command, in this case, would be useful since it displays the behavior and correlation of the utility (U) and the goods (X and Y), adding some simple fitting lines the result would be displayed like this:

twoway scatter U x || lfit U x twoway scatter U y || lfit U y

Up to this point, we can detect a higher dispersion regarding good Y. Also, the fitted line pattern relative to the slope is different for each good. This might lead to assume for now that the overall preference of the consumer for the n individuals is higher on average for the X good than it is for the Y good. The slope, in fact, is telling us that by an increase of one unit in the X good, there’s a serious increase in the utility (U) meanwhile, the fitted line on the good Y regarding to its slope is telling us comparatively speaking, that it doesn’t increase the utility as much as the X good. For this cross-sectional study, it also would become more useful to calculate Pearson’s correlation coefficient. This can be done with:

correlate U y x

The coefficient is indicating us that exists somewhat of a linear association between the utility (U) and the good Y, meanwhile, it exists a stronger linear relationship relative to the X good and the utility. As a final point, there’s an inverse, but not significant or important linear relationship between goods X and Y. So the sign is indicating us that they’re substitutes of each other.

Now instead of regressing U with X and Y, we need to convert it into logarithms, because we want to do a linearization of the Cobb-Douglas utility function.

gen ln_U=ln(U)

gen ln_X=ln(x)

gen ln_Y=ln(y)

reg ln_U ln_X ln_Y

And now performing the regression without the constant.

Both regressions (with and without the constant) tends to establish the parameters in *α* =0.6 and β=0.4 which matches the Data Generating Process of the Montecarlo simulation. It appears that the model with the constant term has a lesser variance, so we shall select these parameters for further analysis.

How would it look then our estimation of this utility function for our sample? well, we can start using the mean value of the utility using descriptive statistics and then use a graphical function with the parameters associate. Remember that we got:

And we know already the parameters and also we can assume that the expected utility would be the mean utility in our sample. From this, we can use the command:

sum U y x

And with this, the estimated function for the utility level U=67.89 with approximated elasticities of 0.6 and 0.4 would look like this:

In this order of ideas, we just estimated the indifference curve for a certain population which consists of a set of *i* individuals. The expected utility from both goods was assumed as the mean value of the utility for the sample and with this, we can identify the different sets of points related to the goods X and Y which represents the expected utility. This is where it ends our brief example of the modeling related to the Cobb-Douglas utility function within a sample with two goods and defined utilities.

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]]>This term spurious correlation was invented on or before 1897 i.e.
in less than 15 years after invention of regression analysis. In 1897, Karl
Pearson wrote a paper entitled, ‘*Mathematical Contributions to the Theory of Evolution: On a
Form of Spurious Correlation Which May Arise When Indices Are Used in the
Measurement of Organs’.* The title indicates the terms spurious regression
was known at least as early as 1897, and it was observed in the data related to
measurement of organs. The reason for this spurious correlation was use of
indices. In next 20 years, many reasons for spurious correlation were unveiled
with the most popular being missing third variable. This means if X is a cause
of Y and X is also a cause of Z, but Y and Z are not directly associated. If
you regress Y on Z, you will find spurious regression.

In 1974, Granger and Newbold (Granger won noble prize later) found that two non-stationary series may also yield spurious results even if there is no missing variable. This finding only added another reason to the possible reasons of spurious regression. Neither this finding can be used to argue that the non-stationarity is one and only reason of spurious regression nor this can be used to argue that the spurious regression is time series phenomenon. However, unfortunately, the economists adapted the two misperception. First, they thought that spurious regression is time series phenomenon and secondly, although not explicitly stated, it appears that the economists assume that the non-stationarity is the only cause of spurious regression. Therefore, although not explicitly stated, most of books and articles discussing the spurious regression, discuss the phenomenon in the context of non-stationary time series.

Granger and his coauthors in 1998 wrote a paper entitled “Spurious regressions with
stationary series”, in which they show that spurious regression can occur in
the stationary data. Therefore, they clear one of the common misconception that
the spurious regression is only due to non-stationarity, but they were
themselves caught in the second misconception that the spurious regression is
time series phenomenon. They define spurious
regression as *“A spurious regression occurs when a pair of independent
series but with strong temporal properties, are found apparently to be related
according to standard inference in an OLS regression”*. The use of term *temporal
properties* implies that they assume the spurious regression to be time
series related phenomenon. But a 100 years ago, Pearson has shown the spurious
regression a cross-sectional data.

The unit root and cointegration analysis were developed to cope with the problem of spurious regression. The literature argues that spurious regression can be avoided if there is cointegration. But unfortunately, cointegration can be defined only for non-stationary data. What is the way to avoid spurious regression if the underlying are stationary? The literature is silent to answer this question.

Pesaran et al (1998) developed a new technique ‘ARDL Bound Test’ to test the existence of level relationship between variables. People often confuse the level relationship with cointegration and the common term used for ARDL Bound test is ARDL cointegration, but the in reality, this does not necessarily imply cointegration. The findings of Bound test are more general and imply cointegration only under certain conditions. The ARDL is capable of testing long run relationship between pair of stationary time series as well as between pair of non-stationary time series. However, the long run relationship between stationary time series cannot be termed as cointegration because by definition cointegration is the long run relationship between stationary time series.

In fact, ARDL bound test is a better way to deal with the spurious regression in stationary time series, but several misunderstandings about the test has restricted the usefulness of the test. We will discuss the use and features of ARDL in a future blog.

]]>In this article, we’re going to explore the utility of the logarithm transformation to reduce the presence of structural breaks in the time series context. First, we’ll review what’s a structural break, what are the implications of regressing data with structural breaks and finally, we’re going to perform a short empirical analysis with the Gross Domestic Product -GDP- of Colombia in Stata.

**The structural break**

We can define a structural break as a situation where a sudden, unexpected change occurs in a time series variable, or a sudden change in the relationship between two-time series (Casini & Perron, 2018). In this order of ideas, a structural change might look like this:

The basic idea is to identify abrupt changes in time series variables but we’re not restricting such identification to the domain of time, it can be detected also when we scatter X and Y variables that not necessarily consider the dependent variable as the time. We can distinguish different types of breaks in this context, according to Hansen (2012) we can encounter breaks in 1) Mean, 2) Variance, 3) Relationships, and also we can face single breaks, multiple breaks, and continuous breaks.

**Basic problems of the structural breaks**

Without going into complex mathematical definitions of the structural breaks, we can establish some of the problems when our data has this situation. The first problem was identified by Andrews (1993) regarding to the parameter’s stability related to structural changes, in simple terms, in the presence of a break, the estimators of the least square regression tend to vary over time, which is of course something not desirable, the ideal situation is that the estimators would be time invariant to consolidate the Best Linear Unbiased Estimator -BLUE-.

The second problem of structural breaks (or changes) not taken in account during the regression analysis is the fact that the estimator turns to be inefficient since the estimated parameters are going to have a significant increase in the variance, so we’re not getting a statistical unbiased estimator and our exact inferences or forecasting analysis wouldn’t be according to reality.

A third problem might appear if the structural break influences the unit root identification process, this is not a wide explored topic but Tai-Leung Chong (2001) makes excellent appoints related to this. Any time series analysis should always consider the existence of unit roots in the variables, in order to provide further tools to handle a phenomenon, that includes the cointegration field and the forecasting techniques.

**An empirical approximation **

Suppose we want to model the tendency of the GDP of the Colombian economy, naturally this kind of analysis explicitly takes the GDP as the dependent variable, and the time as the independent variable, following the next form:

In this case, we know that the GDP expressed in *Y* is going to be a function of the time *t*. We can assume for a start that the function *f(t)* follows a linear approximation.

With this expression in (1), the gross domestic
production would have an independent autonomous value independent of time
defined in *a*, and we’ll get
the slope coefficient in α
which has the usual interpretation that by an increase of one-time unit, the
GDP will have an increase of α.

The linear approximation sounds ideal to model the GDP against the changes over time, assuming that t has a periodicity of years, meaning that we have annual data (so we’re excluding stational phenomena); however, we shall always inspect the data with some graphics.

With Stata once we already tsset the database, we can watch the graphical behavior with the command “scatter y t”.

In sum, the linear approximation might not be a good idea with this behavior of the real GDP of the Colombian economy for the period of analysis (1950-2014). And it appears to be some structural changes judging by the tendency which changes the slope of the curve drastically around the year 2000.

If we regress the expression in (1), we’ll get the next results.

The linear explanation of the time (in years) related to the GDP is pretty good, around 93% of the independent variable given by the time, explains the GDP of the Colombian economy, and the parameter is significant with a level of 5%.

Now I want you to focus in two basic things, the variance of the model which is 1.7446e+09 and the confidence intervals, which positions the estimator between 7613.081 and 8743.697. Without having other values to compare these two things, we should just keep them in mind.

Now, we can proceed with a test to identify structural breaks in the regression we have just performed. So, we just type “estat sbsingle” in order to test for a structural break with an unknown date.

The interesting thing here is that the structural break test identifies one important change over the full sample period of 1950 to 2014, the whole sample test is called “supremum Wald test” and it is said to have less power than average or exponential tests. However, the test is useful in terms of simply identify structural terms which also tend to match with the graphical analysis. According to the test, we have a structural break in the year 2002, so it would be useful to graph the behavior before and after this year in order to conclude the possible changes. We can do this with the command “scatter y t” and include some if conditions like it follows ahead.

twoway (scatter Y t if t<=2002)(lfit Y t if t<=2002)(scatter Y t if t>=2002)(lfit Y t if t>=2002)

We can observe that tendency is actually changing if we adjust the line for partial periods of time, given by t<2002 and t>2002, meaning that the slope change is a sign of structural break detected by the program. You can attend this issue including a dummy variable that would equal 0 in the time before 2002 and equal 1 after 2002. However, let’s graph now the logarithm transformation of GDP. The mathematical model would be:

Applying natural logarithms, we got:

α now becomes the average growth rate per year of the GDP of the Colombian economy, to implement this transformation use the command “gen ln_y=ln(Y)” and the graphical behavior would look like this:

gen ln_Y=ln(Y) scatter ln_Y t

The power of the monotonic transformation is now visible, there’s a straight line among the variable which can be fitted using a linear regression, in fact, let’s regress the expression in Stata.

Remember that I told you to keep in mind the variance and the confidence intervals of the first regression? well now we can compare it since we got two models, the variance of the last regression is 0.0067 and the intervals are indeed close to the coefficient (around 0.002 of difference between the upper and lower interval for the parameter). So, this model fits even greatly than the first.

If we perform again the “estat sbsingle” test again, it’s highly likely that another structural break might appear. But we should not worry a lot if this happens, because we rely on the graphical analysis to proceed with the inferences, in other words, we shall be parsimonious with our models, with little, explain the most.

The main conclusion of this article is that the logarithms used with its property of monotonic transformation constitutes a quick, powerful tool that can help us to reduce (or even delete) the influences of structural breaks in our regression analysis. Structural changes are also, for example, signs of exogenous transformation of the economy, as a mention to apply this idea for the Colombian economy, we see it’s growing speed changing from 2002 until the recent years, but we need to consider that in 2002, Colombia faced a government change which was focused on the implementation of public policies related to eliminating terrorist groups, which probably had an impact related to the investment process in the economy and might explain the growth since then.

Andrews, D. W.
(1993). Tests for Parameter Instability and Structural Change With Unknown
Change Point. *Journal of the Econometric Society Vol. 61, No. 4 (Jul., 1993)*,
821-856.

Casini, A., &
Perron, P. (2018). *Structural Breaks in Time Series.* Retrieved from
Economics Department, Boston University: https://arxiv.org/pdf/1805.03807.pdf

Hansen, B. E.
(2012). *Advanced Time Series and Forecasting.* Retrieved from Lecture 5
tructural Breaks. University of Wisconsin Madison:
https://www.ssc.wisc.edu/~bhansen/crete/crete5.pdf

Nau, R. (2019). *The
logarithm transformation.* Retrieved from Data concepts The logarithm
transformation: https://people.duke.edu/~rnau/411log.htm

Shresta, M., & Bhatta, G. (2018). *Selecting
appropriate methodological framework for time series data analysis.* Retrieved from The
Journal of Finance and Data Science: https://www.sciencedirect.com/science/article/pii/S2405918817300405

Tai-Leung Chong, T.
(2001). *Structural Change In Ar(1) Models.* Retrieved from Econometric
Theory,17. Printed in the United States of America: 87–155