## A brief mathematical revision of the Ramsey Model

We mentioned in the last post the Solow-Swan model in order to explain the importance of the specification related to theories and the regression analysis. In this post, I’m going to explain a little bit more the neoclassical optimization related to consumption, in this case, it’s going to be fundamental to the theory of Ramsey (1928) related to the behavior of savings & consumption.

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!

## References

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 in econometrics.

In the last month, while I was researching through the literature of the military expenditure and economic growth, I found a little statement from an article, which appointed one of the things less discussed in econometrics, such statement is:

“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.

### Bibliography

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.

## The budget constraints in the microeconomic approach

Following the last post which gave an example to model the Cobb-Douglas utility function regarding microeconometrics, we need to provide an important aspect related to the behavior of the consumer. That is the budget constraint (referred to as a monetary linear constraint) which limits the number of goods that the consumer can buy and use to get a certain level of utility.

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 αand B. With αand 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.

## A brief example to model the Cobb-Douglas utility function using Stata.

Regarding microeconometrics, we can find applications that go from latent variables to model market decisions (like logit and probit models) and techniques to estimate the basic approaches for consumers and producers.

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.

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.

## Handling structural breaks with logarithms

As we saw in other econometric blogs of M&S Research Hub, the use of logarithms constitutes a usual practice in econometrics, not only for the problems that can be derived from overusing them, but also it was mentioned the advantage to reduce the Heteroscedasticity -HT- (Nau, 2019) present in the series of a dataset, and some improvements that the monotonic transformation performs on the data as well.

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.

### Bibliography

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

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

## Taking Logarithms of Growth Rates and Log-based Data.

A usual practice while we’re handling economic data, is the use of logarithms, the main idea behind using them is to reduce the Heteroscedasticity -HT- of the data (Nau, 2019). Thus reducing HT, implies reducing the variance of the data. Several times, different authors implement some kind of double logarithm transformation, which is defined as taking logarithms of the data which is already in logarithms and growth rates (via differencing logarithms).

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 10th 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 9th 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.

### Bibliography

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/

## The impact of functional form over the normality assumption in the residuals

A discussed solution in order to accomplish the normality assumption in regression models relates to the correct specification of a Data Generating Process (Rodríguez Revilla, 2014), the objective here is to demonstrate how functional form might influence the distribution of the residuals in a regression model using ordinary least squares technique.

Let’s start with a Monte Carlo exercise using the theory of Mincer (1974) in which we have a Data Generating Process -DGP- of the income for a cross-sectional study of a population of a city.

With

The DGP expressed in (1) is the correct specification of income for the population of our city. Where y is the income in monetary units, schooling is the years of school of the individual, exp is the number of years of experience in the current job. Finally, we got the square of the experience which reflects by the negative sign, the decreasing returns of the variable over the income.

Let’s say we want to study the income in our city, so one might use a simple approximation model for the regression equation. In this case, we know by some logic that schooling and experience are related to the income, so we propose to use the next model in (2) to study the phenomena.

Regressing this model with our Monte Carlo exercise with the specification in (2) we got the next results, considering a sample size of 1000 individuals.

We can see that coefficients of the experience and the constant term are not so close to the DGP process, and that the estimator of schooling years on the other hand it’s approximately accurate. All variables are relevant at a 5% significance level and R^2 is pretty high.

We want to make sure if we got the right variables, so we use Ramsey RESET test to check if we got a problem of omitted variables. Let’s predict first the residuals with predict u, res of the above regression and then perform the test of omitted variables (using Ramsey omitted variable test with estat ovtest):

Ramsey test indicates no omitted variables at a 5% level of significance, so we have now an idea that we’re using the right variables. Let’s check out now, the normality assumption with a graphic distribution of the predicted residuals, in Stata we use the command histogram u, norm

Graphically the result shows that the behavior of the residuals is non-normal. In order to confirm this, we perform a formal test with sktest u and we’ll see the following results.

The test of normality of the residuals is not good. Meaning that with a 5% of the significance level of the error, the predicted residuals have a non-normal distribution. This invalidates the results of the t statistics in the coefficients in the regression of equation (2).

We should get back to our functional form in the regression model in (2), and now we should consider that experience might have some decreasing or increasing returns over the income. So, we adapt our specification including the square term of the experience to capture the marginal effect of the variable:

Now in order to regress this model in Stata, we need to generate the squared term of the experience. To do this we type gen exp_sq=experience*experience where experience is our variable.

We have now our squared variable of experience which we include the regression command as the following image presents.

We can see that coefficients are pretty accurate to the DGP of (1), which is because the specification is closer to the real relationship of the variables in our simulated exercise. The negative sign in the squared term indicates a decreasing return of experience over the income, and the marginal effect is given by:

Let’s predict our residuals of our new regression model with predict u2,res and let’s check the distribution of the residuals using histogram u2, norm

Residuals by graphic inspection presents a normal distribution, we confirm this with the formal test of normality with the command sktest u2

According to the last result we cannot reject the null hypothesis of a normal distribution in the predicted residuals of our second regression model, so we accept that residuals of our last estimates have a normal distribution with a 5% significance level.

The conclusion of this exercise is that even if we have the right variables for a regression model, just like we considered in equation (2), if the specification functional form isn’t correct then the behavior of the residuals will be not be normally distributed.

A correction in the specification form of the regression model can be considered as a solution for non-normality problems, since the interactions of the variables can be modeled better. However in real estimations, finding the right functional form is frequently harder and it’s attached to problems of the data, non-linear relationships, external shocks and atypical observations, but it worth the try to inspect the data in order to find what could be the proper functional form of the variables in order to establish a good regression model which come as accurate as possible to the data generating process.

##### References

Mincer, J. (1974). Education, Experience and the Distribution of Earnings and of employment. New York: National bureau of Economic Research (for the Carnegie Comission).

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

StataCorp (2017) Stata Statistical Software: Release 15. College Station, TX: StataCorp LLC. Avaliable in: https://www.stata.com/products/

## Discussing the Importance of Stationary Residuals in Time Series

A traditional approach of analyzing the residuals in regression models can be identified over the Classical Assumptions in Linear Models (Rodríguez Revilla, 2014), which primarily involves the residuals in aspects as homoscedasticity, no serial correlation (or auto-correlation), no endogeneity, correct specification (this one includes no omitted variables, no redundant variables, and correct functional form) and finally, normal distribution among the estimated residuals of the model with expected zero mean.

In time series context, residuals must be stationary in order to avoid spurious regressions (Woolridge, 2012), if there are no properties of
stationarity among the residuals, then basically our results tend to produce
fake relationships in our model. At this point, it is convenient to say:

“A stationary time series
process is one whose probability distributions are stable over time in the
following sense: if we take any collection of random variables in the sequence
and then shift that sequence ahead h times periods, the joint probability
distribution must remain unchanged”
(Woolridge, 2012, pág. 381)

Another definition according to Lutkepohl & Kratzig (2004) says that stationarity has time-invariant first and second moments over a single variable, mathematically:

Equation (1) simply implies that the expected values of the y process must have a constant mean, so the stationary process must fluctuate around a constant mean defined in µ, no trends are available in the process. Equation
(2) is telling us that variances are time-invariant, so the term γ, doesn’t depend on t but just on the distance h.

In order to get a better notion of stationarity, we define that a stationary process follows the pattern in the next graph. Which was generated using random values over a constant mean of 0, and with a normal probability distribution. The time period sample was n=500 observations.

The generated process fluctuates around a constant mean, and no tendency is present. How do we confirm if the series is normally distributed? Well, we can perform a histogram over the series. In Stata, the command is histogram y, norm where y is our variable.

The option of ,norm is given in Stata in order to present the actual normal distribution, so we can see that real distribution it’s not far from it. We can graphically affirm that series might present a normal distribution, but in order to confirm it, we need to do a formal test, so we perform Jarque-Bera test with the command sktest y

The null hypothesis of the test is that normal distribution exists among the y variable And since p-value is bigger than a 5%significance level, we fail to reject null hypothesis and we can say that y variable is normally distributed.

Checking for unit roots also is useful when we’re trying to discover stationarity over a variable, so we perform first, the estimated ideal lag for the test, with varsoc y which will tell us what appropriated lag-length should be used in the ADF test.

Such results, indicate that ADF test over y variable must be done with one lag according to FPE, AIC, while HQIC and SBIC indicate 0 lags. It is the decision of the investigator to select the right information criteria (mostly it is selected when all error criteria are in a specific lag). However, we have a draw of FPE and AIC vs HQIC and SBIC. We will discard FPE since according to Liew (2004) this one is more suitable for samples lower than 120 observations, and thus we will select 0 lag for the test considering our sample size of 500 observations.

Null hypothesis is the existence of unit roots in the variable, so we can strongly reject this and accept that no-unit roots are present. Sometimes this test is used to define stationarity of a respective process, but we need to take in consideration that stationarity involves constant means and normal distributions. We can say for now, that y variable is stationary.

At this point, one could argue Why we need the notion of stationarity over the residuals? This is because stationarity ensures that no spurious regressions are estimated. Now let’s assume we have a model which
follows an I (0) stationary model.

And that I (0) variables are y and x, common intuition will tell us that u will be also stationary, but we need to ensure this. Proceeding with our Monte Carlo approaches, we generated the x series with a constant mean which has a normal distribution and that with u ~ (0,1) as the Data Generating Process of y expressed in equation (3). Basically u has a mean of 0, and variance of 1. Regressing y on x we got the next result.

We can see that coefficients B_0 and B_1 are approximated 1 and 2 respectively, so it’s almost close to the data generating process and both estimators are statistically significant at 1%. Let’s look at the residuals of the estimated model a little bit closer, we start by predicting the residuals using the command predict u, residuals in order to get the predicted values. Then we perform some of the tests we did before.

Graphic of the residuals with tsline u presentsthe next result, which looks like a stationary process.

A histogram over the residuals, will show the pattern
of normal distribution.

And as well, the normality test will confirm this result.

Now we need to test that the residuals don’t follow a unit root pattern, a consideration here must be done first before we use ADF test, and is that critical values of the test are not applicable to the residuals. Thus, we cannot fully rely on this test.

In Stata we can recur to the Engle-Granger distribution test of the residuals, to whether accept or reject the idea that residuals are stationary. So, we type egranger y x which provides an accurate estimate of the critical values to evaluate the residuals.

As tests evidence, Test statistic is pretty close between ADF test and Engle & Granger test but the critical values are way different. Furthermore, we should rely on the results of the Engle & Granger test. Since Test statistic is bigger than 5% critical value, we can reject the null hypothesis that x and y are not cointegrated, and we can affirm that both variables present over this estimation a long run path of equilibrium. From another view, implies that the residuals are stationary and our regression is not spurious.

This basic idea can be extended with I (1) variables, in order to test whether it exists a long run path and if the regression model in (3) turns to be super consistent. Then long-run approximations with error correction forms can be done for this model where all variables are I (1).

This idea of testing residuals in stationary models is not a formal test used in the literature, however, it can reconfirm that with I (0) models that the regression will not be spurious. And it can also help to contrast long-run relationships.

Note: The package egranger must be installed first ssc install egranger, replace should do the trick. This package parts from the regression model to be estimated, however, it has the failure it cannot be computed with time operators. So, generating first differences or lagged values must be done in separate variables.

Bibliography

Liew, V. (2004). “Which Lag Length Selection Criteria Should We Employ?”. Journal of Economics Bulletin, 1-9. Recuperated from: