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.

]]>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.

In our first YEAR “2019”, we had

**Over 120,000 visits to our institute website

**Over 50 completed interactive econometrics private and group training

**Around 70 researchers, 10 of them successfully completed their Ph.D. degrees, have booked our methodological and statistical guidance services

** 1 annual conference, and around 5 onsite and online workshops

** 5 online webinars

** 6 research ambassadors around the World and 10 esteemed academic council members

** New developed division “Program Design and Impact evaluation” that is managed and operated by two world first-class senior consultants.

— > Nothing could have happened without your support, trust, and faith in our mission

Stay tuned for 2020 new calendar of events and training programs.

]]>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

Applications of DiD are quite diverse, amongst are

- Impact evaluation (public policy analysis)
- Measuring the variations overtime (Time Series) and over individuals (Cross-Sectional Data)
- Focusing on the establishment of effects on a dependent variable derived from the interaction of exogenous variables given a treatment.
- Variants of the DiD method can account to deal with auto-selection bias and endogeneity problems.
- Comparing the differences between observed outcomes from partial and non-randomized samples in groups.

Although this method is highly important, few learning resources are available to instruct researchers and scientists how to properly implement and design it. There may be some resources that discuss the theoretical foundations of this method while listing a few examples of its applications. However a fully-fledged learning material for DiD that covers both theory, and guides researchers to implement this method on statistical software using real and simulated data applications are rather scarce.

At M&S Research Hub we recently launched a video library wherein our team of academic experts record offline training videos for advanced econometrics methods. This material is designed to fit researchers at different proficiency levels. They cover both theoretical and mathematical basics of the target models and their detailed application using statistical software, leaving the researcher in no further need to search for other learning resources.

A complete DiD course that takes around 158 minutes and is recorded over 7 videos are available in the library for everyone who wants to master the DID method.

]]>The objective of this article is to present the implications of this procedures, first by analyzing what does do the logarithm to a variable, then observing what possible inferences can be done when logarithms are applied to growth rates.

There are a series of properties about the logarithms that should be considered first, we’re not reviewing them here, however the reader can check them in the following the citation (Monterey Institute, s.f). Now let’s consider a bivariate equation:

The coefficient *B* represents the marginal effect of a change of one unit in X over Y. So, interpreting the estimation with ordinary least squares estimator gives the following analysis: When *x* increases in one unit, the result is an increase of *B* in *y*. It’s a lineal equation where the marginal effect is given by:

When we introduce logarithms to the equation of (1) by modifying the functional form, the estimation turns to be non-linear. However, let’s first review what logarithms might do to the *x* variable. Suppose *x* is a time variable which follows an upward tendency, highly heteroscedastic as the next graph shows.

We can graphically appreciate that variable *x* has a positive trend, and also that has deviations over his mean over time. A way to reduce the HT present in the series is to make a logarithm transformation. Using natural logarithms, the behavior is shown in the next graph.

The units have changed drastically, and we can define that logarithm of *x* is around 2 and 5. Whereas before we had it from 10 to 120 (the range has been reduced). The reason, the natural logarithm reduces HT because the logarithms are defined as a monotonic transformation (Sikstar, s.f.). When we use this kind of transformation in econometrics like the following regression equation:

The coefficient *B* is no longer the marginal effect, to interpret it we need to divide it by 100 (Rodríguez Revilla, 2014). Therefore, the result should be read as: an increase of one unit in *x* produces a change of *B/100* in *y*.

If we use a double-log model, equation can be written as:

In this case, the elasticity is simply *B* which is interpreted in percentage. Example, if *B=0.8*. By an increase of 1% in *x*, the result would be an increase of 0.8% in *y*.

On the other hand, if we use log-linear model, equation can be written as:

In this case, *B* must be multiplied by 100 and it can be interpreted as a growth rate in average per increases of a unit of *x*. If *x=t* meaning years, then *B *is the average growth per year of *y*.

The logarithms also are used to calculate growth rates. Since we can say that:

The meaning of equation (5) is that growth rates of a variable (left hand of the equation) are approximately equal to the difference of logarithms. Returning with this idea over our *x* variable in the last graphic, we can see that the growth rate between both calculations are similars.

It’s appreciably the influence of the monotonic transformation; the growth rate formula has more upper (positive) spikes than the difference of logarithms does. And inversely the lower spikes are from the difference of logarithms. Yet, both are approximately growth rates which indicate the change over time of our *x* variable.

For example, let’s place on the above graphic when is the 10^{th} year. The difference in logarithms indicates that the growth rate is -0.38% while the growth rate formula indicates a -0.41% of the growth-related between year 9^{th} and now. Approximately it’s 0.4% of negative growth between these years.

When we use logarithms in those kinds of transformations we’ll get mathematically speaking, something like this:

Some authors just do it freely to normalize the data (in other words reducing the HT), but Would be the interpretation remain the same? What are the consequences of doing this? It’s something good or bad?

As a usual answer, it depends. What would happen if, for example, we consider the years 9 and 10 again of our original *x* variable, we can appreciate that the change it’s negative thus the growth rate it’s negative. Usually, we cannot estimate a logarithm when the value is negative.

With this exercise, we can see that the first consequence of overusing logarithms (in differenced logarithms and general growth rates) is that if we got negative values, the calculus becomes undefined, so missing data will appear. If we graph the results of such thing, we’ll have something like this:

At this point, the graphic takes the undefined values (result of the logarithm of negative values) as 0 in the case of Excel, other software might not even place a point. We got negative values of a growth rate (as expected), but what we got now is a meaningless set of data. And this is bad because we’re deleting valuable information from other timepoints.

Let’s forget for now the *x* variable we’ve been working with. And now let’s assume we got a square function.

The logarithm of this variable since its exponential would be:

and if we apply another log transformation, then we’ll have:

However, consider that if *z=*0, the first log would be undefined, and thus, we cannot calculate the second. We can appreciate this in some calculations as the following table shows.

The logarithm of 0 is undefined, the double logarithm of that would be undefined too. When z=1 the natural logarithm is 0, and the second transformation is also undefined. Here we can detect another problem when some authors, in order to normalize the data, apply logarithms indiscriminately. The result would be potential missing data problem due to the monotonic transformation when values of the data are zero.

Finally, if we got a range of data between 0 and 1, the logarithm transformation will induce the calculus to a negative value. Therefore, the second logarithm transformation it’s pointless since all the data in this range is now undefined.

The conclusions of this article are that when we use logarithms in growth rates, one thing surely can happen: 1) If we got potential negative values in the original growth rate, and then apply logarithms on those, the value becomes undefined, thus missing data that will occur. And the interpretation becomes harder. Now if we apply some double transformation of log values, the zero and the negative values in the data will become undefined, thus missing data problem will appear again. Econometricians should take this in considerations since it’s often a question that arises during researches, and in order to do right inferences, analyzing the original data before applying logarithms should be a step before doing any econometric procedure.

Monterey Institute. (s.f). *Properties
of Logarithmic Functions.* Obtained from:
http://www.montereyinstitute.org/courses/DevelopmentalMath/TEXTGROUP-1-19_RESOURCE/U18_L2_T2_text_final.html

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

Rodríguez Revilla, R. (2014). *Econometria I y II.*
Bogotá. : Universidad Los Libertadores.

Sikstar, J. (s.f.). *Monotonically
Increasing and Decreasing Functions: an Algebraic Approach.* Obtained from:
https://opencurriculum.org/5512/monotonically-increasing-and-decreasing-functions-an-algebraic-approach/

In response to common pitfalls of the aid system, a new setting for aid disbursements based on sector-wide approaches instead of individual projects was introduced by Kamel, et al. (1998) and was later modified by (Kanbur, et al. 1999) and evolved into a system entitled the ‘common pool approach’. Our following policy suggestions will build on these, however, we bring in some modifications for the system implementation and evaluation.

The basic rationale behind the common pool approach is that a pool of donors—instead of one— allocates unconditional funds to a recipient country’s nationally representative reform plan and its implementation strategy, instead of individual projects. Such a system would increase the recipient country’s sense of ownership and commitment while enhancing the achievability of developmental and reform goals relative to individual uncoordinated projects approach. The major drawback is the minimisation of aid received because many donors – besides political lobbies and private sector firms – might not agree to fund national plans instead of individual selective projects. In addition, donors’ ability to pursue their own interests, conditions, and opinions will dwindle. Anyhow, detailed discussion for the common pool system and sector-wide approach is found in formerly cited articles. Subsequently, we introduce a new system which combines both approaches, sector-wide and common pool, whilst including our personal reflections that will hopefully alleviate the expected drawbacks of these approaches.

Figure A1 provides a basic graphical representation of the new blended system. The graph is elaborated in the following points,

- The recipient country starts to move in the direction of prohibiting all forms of aid transferred to individual uncoordinated projects, but rather allows only aid channeled towards sectoral reform plans.

- The governmental authority with the cooperation of civil society, the private sector, policymakers, and citizens, formulate a reform plan for the target sector.

- A series of roundtable meetings are held in the recipient country capital that involves potential donors (single and multilateral), international experts, and other national parties in order to receive feedback on the preliminary proposal (sponsoring the meetings in the recipient country would ease national parties’ involvement and cooperation, which reflects in a higher sense of belonging and ownership).

- A final neat version of the proposal is then reformulated along with its implementation strategy that involves foreign and domestic shares. For instance, technical and human resources in the implementation strategy are distributed as 70% domestic and 30% by the donor’s side. One major drawback of the common pool approach is the lack of donor involvement in the implementation, which in essence is a good thing to increase the sense of ownership by the recipient. However, this is reflected in lower lobbying by the private sector and political parties in the donor country to step forward for similar approaches. We, therefore, propose a cooperative share of interests, however still managed and authorised by the domestic country and in the framework of the domestic strategy.

- The donor authority in this phase lies in accepting or rejecting the plan and the amount of fund provided, based on credibility and achievability of the plan.

- Donors together with the recipient responsible authority would agree on a set of quantifiable and measurable assessment measures that are monitored and reported by the authority itself, though donors are also allowed to intervene in the monitoring and evaluation of these measures. This is an incentive for the authority and other parties involved in the strategy to abide by the rules and the plan. Also, in the case of system corruption, which is the likely case in the majority of developing and poor countries, it is well known that foreign assessment might intervene anytime to inspect and evaluate. In addition, donors will be more relieved and secure when they have a hand in the evaluation process, unlike the common pool approach which prohibits any form of foreign intervention in the process unless requested by the recipient.

- Finally, a renewable annual funding plan is offered based on the realisation of these measures; failure to abide by the authority results in a violation of the contract. By doing this we eliminate any chances of aid misallocation, corruptive activities, and other illegal traits because the recipient knows for sure that failure will hinder any future possibilities of funding for other reform plans. Moreover, the ex-ante participation of civil society and citizens makes the government accountable to the public, which also affects their political popularity.

Eventually, let me conclude with this phrase from Kanbur, et al. (1999) “The possibility of the decline in aid will require a substantial amount of confidence on the part of recipients who adopt the approach. It requires a government with the willpower to say to donors: ‘Here is my program in this sector: if you wish to help me implement it, you are most welcome. If you wish to do something different, I regret that you are not welcome in this sector in this country.” The foremost outcome of the proposed blended system, common pool, and sector-wide approaches, is filtration of aid received by the recipient, by adopting these approaches, will be able to locate donors that endeavor no hidden, political, or ideological agendas but only support the recipient country’s development efforts.

References:

Hassan, Sherif (2016): *Seventy Years of Official Development Assistance: Reflections on the Working Age Population.* MPRA, Paper Nr. 74835.

Spurious correlation/spurious correlation occur
when a pair of variable having no (weak) causal connection appears to have
significant (strong) correlation/regression. In these meanings the term
spurious correlation/spurious has the same history as the term regression
itself. The correlation and regression analysis were invented by Sir Francis
Galton in around 1888 and in 1897, Karl Pearson wrote a paper with the
following title, ‘*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’ *(Pearson, 1897).

This title indicates number of important things about the term spurious correlation: (a) the term spurious correlation was known as early as 1897, that is, in less than 10 years after the invention of correlation analysis (ii) there were more than one types of spurious correlation known to the scientists of that time, therefore, the author used the phrase ‘On a Farm of Spurious Regression’, (c) the spurious correlation was observed in measurement of organs, a cross-sectional data (d) the reason of spurious correlation was use of indices, not the non-stationarity.

One can find in classical econometric literature that that many kinds of spurious correlations were known to experts in first two decades of twentieth century. These kinds of spurious correlations include spurious correlation due to use of indices (Pearson, 1897), spurious correlation due to variations in magnitude of population (Yule, 1910), spurious correlation due to mixing of heterogeneous records (Brown et al, 1914), etc. The most important reason, as the econometricians of that time understand, was the missing third variable (Yule, 1926).

Granger and Newbold (1974) performed a simulation study in which they generated two independent random walk time series x(t)=x(t-1)+e(t) and y(t)=y(t-1)+u(t) . The two series are non-stationary and the correlation of error terms in the two series is zero so that the two series are totally independent of each other. The two variables don’t have any common missing factor to which the movement of the two series can be attributed. Now the regression of the type y(t)=a+bx(t)+e(t) should give insignificant regression coefficient, but the simulation showed very high probability of getting significant coefficient. Therefore, Granger and Newbold concluded that spurious regression occurs due to non-stationarity.

Three points are worth considering regarding the study of Granger and Newbold. First, the above cited literature clearly indicates that the spurious correlation does exist in cross-sectional data and the Granger-Newbold experiment is not capable to explain cross-sectional spurious correlation. Second, the existing understanding of the spurious correlation was that it happens due to missing variables and the experiment adds another reason for the phenomenon which cannot deny the existing understanding. Third, the experiment shows that non-stationarity is one of the reasons of spurious regression. It does not prove that non-stationarity is ‘the only’ reason of spurious regression.

However, unfortunately, the econometric literature that emerged after Granger and Newbold, adapted the misconception. Now, many textbooks discuss the spurious regression only in the context of non-stationarity, which leads to the misconception that the spurious regression is a non-stationarity related phenomenon. Similarly, the discussion of missing variable as a reason of spurious regression is usually not present in the recent textbooks and other academic literature.

To show that spurious regression is not necessarily a time series phenomenon, consider the following example:

A researcher is interested in knowing the relationship between shoe size and mathematical ability level of school students. He goes to a high school and takes a random sample of the students present in the school. He takes readings on shoe size and ability to solve the mathematical problems of the selected students. He finds that there is very high correlation between two variables. Would this be sufficient to argue that the admission policy of the school should be based on the measurement of shoe size? Otherwise, what accounts for this high correlation?

If sample is selected from a high school having kids in different classes, the same observation is almost sure to occur. The pupil in higher classes have larger shoe size and have higher mathematical skills, whereas student in lower classes have lower mathematical skills. Therefore, high correlation is expected. However, if we take data of only one class, say grade III, we will not see such high correlation. Since theoretically, there should be no correlation between shoe size and mathematical skills, this apparently high correlation may be regarded as spurious correlation/regression. The reason for this spurious correlation is mixing of missing age factor which drives both shoe size and mathematical skills.

Since this is not a time series data, there is no question of the existence of non-stationarity, but the spurious correlation exists. This shows that spurious correlation is no necessarily a time series phenomenon. The unit root and cointegration would be just incapable to solve this problem.

Similarly, it can be shown that the unit root and cointegration analysis can fail to work even with time series data, and this will be discussed in our next blog

]]>