# Econometrics

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Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. [1] More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference". [2] An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships". [3] The first known use of the term "econometrics" (in cognate form) was by Polish economist Paweł Ciompa in 1910. [4] Jan Tinbergen is considered by many to be one of the founding fathers of econometrics. [5] [6] [7] Ragnar Frisch is credited with coining the term in the sense in which it is used today. [8]

In linguistics, cognates are words that have a common etymological origin. Cognates are often inherited from a shared parent language, but they may also involve borrowings from some other language. For example, the English words dish and desk and the German word Tisch ("table") are cognates because they all come from Latin discus, which relates to their flat surfaces. Cognates may have evolved similar, different or even opposite meanings, but in most cases there are some similar sounds or letters in the words, in some cases appearing to be dissimilar. Some words sound similar, but do not come from the same root; these are called false cognates, while some are truly cognate but differ in meaning; these are called false friends.

Jan Tinbergen was an important Dutch economist. He was awarded the first Nobel Memorial Prize in Economic Sciences in 1969, which he shared with Ragnar Frisch for having developed and applied dynamic models for the analysis of economic processes. He is widely considered to be one of the most influential economists of the 20th century and one of the founding fathers of econometrics. It has been argued that the development of the first macroeconometric models, the solution of the identification problem, and the understanding of dynamic models are his three most important legacies to econometrics. Tinbergen was a founding trustee of Economists for Peace and Security. In 1945, he founded the Bureau for Economic Policy Analysis (CPB) and was the agency's first director.

## Contents

A basic tool for econometrics is the multiple linear regression model. [9] Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. [10] [11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting.

The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches. Within a given approach, statistical theory gives ways of comparing statistical procedures; it can find a best possible procedure within a given context for given statistical problems, or can provide guidance on the choice between alternative procedures.

Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule, the quantity of interest and its result are distinguished.

## Basic models: linear regression

A basic tool for econometrics is the multiple linear regression model. [9] In modern econometrics, other statistical tools are frequently used, but linear regression is still the most frequently used starting point for an analysis. [9] Estimating a linear regression on two variables can be visualised as fitting a line through data points representing paired values of the independent and dependent variables.

For example, consider Okun's law, which relates GDP growth to the unemployment rate. This relationship is represented in a linear regression where the change in unemployment rate (${\displaystyle \Delta \ {\text{Unemployment}}}$) is a function of an intercept (${\displaystyle \beta _{0}}$), a given value of GDP growth multiplied by a slope coefficient ${\displaystyle \beta _{1}}$ and an error term, ${\displaystyle \varepsilon }$:

In economics, Okun's law is an empirically observed relationship between unemployment and losses in a country's production. The "gap version" states that for every 1% increase in the unemployment rate, a country's GDP will be roughly an additional 2% lower than its potential GDP. The "difference version" describes the relationship between quarterly changes in unemployment and quarterly changes in real GDP. The stability and usefulness of the law has been disputed.

${\displaystyle \Delta \ {\text{Unemployment}}=\beta _{0}+\beta _{1}{\text{Growth}}+\varepsilon .}$

The unknown parameters ${\displaystyle \beta _{0}}$ and ${\displaystyle \beta _{1}}$ can be estimated. Here ${\displaystyle \beta _{1}}$ is estimated to be −1.77 and ${\displaystyle \beta _{0}}$ is estimated to be 0.83. This means that if GDP growth increased by one percentage point, the unemployment rate would be predicted to drop by 1.77 points. The model could then be tested for statistical significance as to whether an increase in growth is associated with a decrease in the unemployment, as hypothesized. If the estimate of ${\displaystyle \beta _{1}}$ were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related. The variance in a prediction of the dependent variable (unemployment) as a function of the independent variable (GDP growth) is given in polynomial least squares.

In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis. More precisely, a study's defined significance level, denoted α, is the probability of the study rejecting the null hypothesis, given that the null hypothesis were assumed to be true; and the p-value of a result, p, is the probability of obtaining a result at least as extreme, given that the null hypothesis were true. The result is statistically significant, by the standards of the study, when . The significance level for a study is chosen before data collection, and typically set to 5% or much lower, depending on the field of study.

In mathematical statistics, polynomial least squares comprises a broad range of statistical methods for estimating an underlying polynomial that describes observations. These methods include polynomial regression, curve fitting, linear regression, least squares, ordinary least squares, simple linear regression, linear least squares, approximation theory and method of moments. Polynomial least squares has applications in radar trackers, estimation theory, signal processing, statistics, and econometrics.

## Theory

Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. [10] [11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. An estimator is unbiased if its expected value is the true value of the parameter; it is consistent if it converges to the true value as the sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size. Ordinary least squares (OLS) is often used for estimation since it provides the BLUE or "best linear unbiased estimator" (where "best" means most efficient, unbiased estimator) given the Gauss-Markov assumptions. When these assumptions are violated or other statistical properties are desired, other estimation techniques such as maximum likelihood estimation, generalized method of moments, or generalized least squares are used. Estimators that incorporate prior beliefs are advocated by those who favour Bayesian statistics over traditional, classical or "frequentist" approaches.

In statistics, the bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator. Unlike the ordinary English use of the term "bias", it is not pejorative even though it's not a desired property.

In the comparison of various statistical procedures, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. This article primarily deals with efficiency of estimators.

In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ0—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ0. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ0 converges to one.

## Methods

Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting. [12]

Econometrics may use standard statistical models to study economic questions, but most often they are with observational data, rather than in controlled experiments. [13] In this, the design of observational studies in econometrics is similar to the design of studies in other observational disciplines, such as astronomy, epidemiology, sociology and political science. Analysis of data from an observational study is guided by the study protocol, although exploratory data analysis may be useful for generating new hypotheses. [14] Economics often analyses systems of equations and inequalities, such as supply and demand hypothesized to be in equilibrium. Consequently, the field of econometrics has developed methods for identification and estimation of simultaneous-equation models. These methods are analogous to methods used in other areas of science, such as the field of system identification in systems analysis and control theory. Such methods may allow researchers to estimate models and investigate their empirical consequences, without directly manipulating the system.

One of the fundamental statistical methods used by econometricians is regression analysis. [15] Regression methods are important in econometrics because economists typically cannot use controlled experiments. Econometricians often seek illuminating natural experiments in the absence of evidence from controlled experiments. Observational data may be subject to omitted-variable bias and a list of other problems that must be addressed using causal analysis of simultaneous-equation models. [16]

In addition to natural experiments, quasi-experimental methods have been used increasingly commonly by econometricians since the 1980s, in order to credibly identify causal effects. [17]

## Example

A simple example of a relationship in econometrics from the field of labour economics is:

${\displaystyle \ln({\text{wage}})=\beta _{0}+\beta _{1}({\text{years of education}})+\varepsilon .}$

This example assumes that the natural logarithm of a person's wage is a linear function of the number of years of education that person has acquired. The parameter ${\displaystyle \beta _{1}}$ measures the increase in the natural log of the wage attributable to one more year of education. The term ${\displaystyle \varepsilon }$ is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, ${\displaystyle \beta _{0}{\mbox{ and }}\beta _{1}}$ under specific assumptions about the random variable ${\displaystyle \varepsilon }$. For example, if ${\displaystyle \varepsilon }$ is uncorrelated with years of education, then the equation can be estimated with ordinary least squares.

If the researcher could randomly assign people to different levels of education, the data set thus generated would allow estimation of the effect of changes in years of education on wages. In reality, those experiments cannot be conducted. Instead, the econometrician observes the years of education of and the wages paid to people who differ along many dimensions. Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education. For example, people born in certain places may have higher wages and higher levels of education. Unless the econometrician controls for place of birth in the above equation, the effect of birthplace on wages may be falsely attributed to the effect of education on wages.

The most obvious way to control for birthplace is to include a measure of the effect of birthplace in the equation above. Exclusion of birthplace, together with the assumption that ${\displaystyle \epsilon }$ is uncorrelated with education produces a misspecified model. Another technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render ${\displaystyle \beta _{1}}$ identifiable. [18] An overview of econometric methods used to study this problem were provided by Card (1999). [19]

## Journals

The main journals that publish work in econometrics are Econometrica , the Journal of Econometrics , the Review of Economics and Statistics , Econometric Theory , the Journal of Applied Econometrics , Econometric Reviews , the Econometrics Journal , [20] Applied Econometrics and International Development , and the Journal of Business & Economic Statistics .

## Limitations and criticisms

Like other forms of statistical analysis, badly specified econometric models may show a spurious relationship where two variables are correlated but causally unrelated. In a study of the use of econometrics in major economics journals, McCloskey concluded that some economists report p-values (following the Fisherian tradition of tests of significance of point null-hypotheses) and neglect concerns of type II errors; some economists fail to report estimates of the size of effects (apart from statistical significance) and to discuss their economic importance. She also argues that some economists also fail to use economic reasoning for model selection, especially for deciding which variables to include in a regression. [21] [22]

In some cases, economic variables cannot be experimentally manipulated as treatments randomly assigned to subjects. [23] In such cases, economists rely on observational studies, often using data sets with many strongly associated covariates, resulting in enormous numbers of models with similar explanatory ability but different covariates and regression estimates. Regarding the plurality of models compatible with observational data-sets, Edward Leamer urged that "professionals ... properly withhold belief until an inference can be shown to be adequately insensitive to the choice of assumptions". [23]

## Notes

1. M. Hashem Pesaran (1987). "Econometrics," The New Palgrave: A Dictionary of Economics , v. 2, p. 8 [pp. 8–22]. Reprinted in J. Eatwell et al., eds. (1990). Econometrics: The New Palgrave, p. 1 [pp. 1–34]. Abstract Archived 18 May 2012 at the Wayback Machine (2008 revision by J. Geweke, J. Horowitz, and H. P. Pesaran).
2. P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone (1954). "Report of the Evaluative Committee for Econometrica," Econometrica 22(2), p. 142. [p p. 141-146], as described and cited in Pesaran (1987) above.
3. Paul A. Samuelson and William D. Nordhaus, 2004. Economics . 18th ed., McGraw-Hill, p. 5.
4. "Archived copy". Archived from the original on 2 May 2014. Retrieved 1 May 2014.CS1 maint: archived copy as title (link)
5. "1969 - Jan Tinbergen: Nobelprijs economie - Elsevierweekblad.nl". elsevierweekblad.nl. 12 October 2015. Archived from the original on 1 May 2018. Retrieved 1 May 2018.
6. Magnus, Jan & Mary S. Morgan (1987) The ET Interview: Professor J. Tinbergen in: 'Econometric Theory 3, 1987, 117–142.
7. Willlekens, Frans (2008) International Migration in Europe: Data, Models and Estimates. New Jersey. John Wiley & Sons: 117.
8. • H. P. Pesaran (1990), "Econometrics," Econometrics: The New Palgrave, p. 2, citing Ragnar Frisch (1936), "A Note on the Term 'Econometrics'," Econometrica, 4(1), p. 95.
• Aris Spanos (2008), "statistics and economics," The New Palgrave Dictionary of Economics , 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine
9. Greene, William (2012). "Chapter 1: Econometrics". Econometric Analysis (7th ed.). Pearson Education. pp. 47–48. ISBN   9780273753568. Ultimately, all of these will require a common set of tools, including, for example, the multiple regression model, the use of moment conditions for estimation, instrumental variables (IV) and maximum likelihood estimation. With that in mind, the organization of this book is as follows: The first half of the text develops fundamental results that are common to all the applications. The concept of multiple regression and the linear regression model in particular constitutes the underlying platform of most modeling, even if the linear model itself is not ultimately used as the empirical specification.
10. Greene, William (2012). Econometric Analysis (7th ed.). Pearson Education. pp. 34, 41–42. ISBN   9780273753568.
11. Wooldridge, Jeffrey (2012). "Chapter 1: The Nature of Econometrics and Economic Data". Introductory Econometrics: A Modern Approach (5th ed.). South-Western Cengage Learning. p. 2. ISBN   9781111531041.
12. Clive Granger (2008). "forecasting," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine
13. Wooldridge, Jeffrey (2013). Introductory Econometrics, A modern approach. South-Western, Cengage learning. ISBN   978-1-111-53104-1.
14. Herman O. Wold (1969). "Econometrics as Pioneering in Nonexperimental Model Building," Econometrica, 37(3), pp. 369-381.
15. For an overview of a linear implementation of this framework, see linear regression.
16. Edward E. Leamer (2008). "specification problems in econometrics," The New Palgrave Dictionary of Economics. Abstract. Archived 23 September 2015 at the Wayback Machine
17. Angrist, Joshua D; Pischke, Jörn-Steffen (May 2010). "The Credibility Revolution in Empirical Economics: How Better Research Design is Taking the Con out of Econometrics". Journal of Economic Perspectives. 24 (2): 3–30. doi:10.1257/jep.24.2.3. ISSN   0895-3309.
18. Pearl, Judea (2000). Causality: Model, Reasoning, and Inference. Cambridge University Press. ISBN   978-0521773621.
19. Card, David (1999). "The Causal Effect of Education on Earning". In Ashenfelter, O.; Card, D. (eds.). Handbook of Labor Economics. Amsterdam: Elsevier. pp. 1801–1863. ISBN   978-0444822895.
20. "The Econometrics Journal – Wiley Online Library". Wiley.com. Retrieved 8 October 2013.
21. McCloskey (May 1985). "The Loss Function has been mislaid: the Rhetoric of Significance Tests". American Economic Review. 75 (2).
22. Stephen T. Ziliak and Deirdre N. McCloskey (2004). "Size Matters: The Standard Error of Regressions in the American Economic Review," Journal of Socio-economics, 33(5), pp. 527-46 Archived 25 June 2010 at the Wayback Machine (press +).
23. Leamer, Edward (March 1983). "Let's Take the Con out of Econometrics". American Economic Review. 73 (1): 31–43. JSTOR   1803924.