# Econometrics

(Redirected from Econometric)

Econometrics is the application of mathematics, statistical methods, and, more recently, computer science, to economic data and is described as the branch of economics that aims to give empirical content to economic relations.[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] Ragnar Frisch is credited with coining the term in the sense in which it is used today.[5]

Econometrics is the intersection of economics, mathematics, and statistics. Econometrics adds empirical content to economic theory allowing theories to be tested and used for forecasting and policy evaluation.[6]

## Basic econometric models: linear regression

The basic tool for econometrics is the linear regression model. In modern econometrics, other statistical tools are frequently used, but linear regression is still the most frequently used starting point for an analysis.[7] Estimating a linear regression on two variables can be visualized as fitting a line through data points representing paired values of the independent and dependent variables.

Okun's law representing the relationship between GDP growth and the unemployment rate. The fitted line is found using regression analysis.

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 ($\Delta\ Unemployment$) is a function of an intercept ($\beta_0$), a given value of GDP growth multiplied by a slope coefficient $\beta_1$ and an error term, $\epsilon$:

$\Delta\ Unemployment= \beta_0 + \beta_1\text{Growth} + \varepsilon.$

The unknown parameters $\beta_0$ and $\beta_1$ can be estimated. Here $\beta_1$ is estimated to be −1.77 and $\beta_0$ 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 .94 points (−1.77*1+0.83). 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 $\beta_1$ were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related.

## Theory

Econometric theory uses statistical theory to evaluate and develop econometric methods. 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 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 favor Bayesian statistics over traditional, classical or "frequentist" approaches.

### Gauss–Markov theorem

The Gauss–Markov theorem shows that the OLS estimator is the best (minimum variance), unbiased estimator assuming the model is linear, the expected value of the error term is zero, errors are homoskedastic and not autocorrelated, and there is no perfect multicollinearity.

#### Linearity

The dependent variable is assumed to be a linear function of the variables specified in the model. The specification must be linear in its parameters. This does not mean that there must be a linear relationship between the independent and dependent variables. The independent variables can take non-linear forms as long as the parameters are linear. The equation $y = \alpha + \beta x^2, \,$ qualifies as linear while $y = \alpha + \beta^2 x$ can be transformed to be linear by replacing (beta)^2 by another parameter, say gamma. An equation with a parameter dependent on an independent variable does not qualify as linear, for example y = alpha + beta(x) * x, where beta(x) is a function of x.

Data transformations are often used to convert an equation into a linear form (see, however, Santos Silva and Tenreyro, 2006). For example, the Cobb–Douglas function—often used in economics—is nonlinear:

$Y=AL^{\alpha}K^{\beta}\varepsilon \,$

But it can be expressed in linear form by taking the natural logarithm of both sides:[8] $ln Y=ln A + \alpha ln L + \beta lnK + ln\varepsilon$

This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no omitted variables.

#### Expected error is zero

$\operatorname{E}[\,\varepsilon\,] = 0.$

The expected value of the error term is assumed to be zero. This assumption can be violated if the measurement of the dependent variable is consistently positive or negative. The mis-measurement will bias the estimation of the intercept parameter, but the slope parameters will remain unbiased.[9]

The intercept may also be biased if there is a logarithmic transformation. See the Cobb-Douglas equation above. The multiplicative error term will not have a mean of 0, so this assumption will be violated.[10]

This assumption can also be violated in limited dependent variable models. In such cases, both the intercept and slope parameters may be biased.[11]

#### Spherical errors

$\operatorname{Var}[\,\varepsilon|X\,] = \sigma^2 I_n,$

Error terms are assumed to be spherical otherwise the OLS estimator is inefficient. The OLS estimator remains unbiased, however. Spherical errors occur when errors have both uniform variance (homoscedasticity) and are uncorrelated with each other.[12] The term "spherical errors" will describe the multivariate normal distribution: if $\operatorname{Var}[\,\varepsilon|X\,] = \sigma^2 I_n$ in the multivariate normal density, then the equation f(x)=c is the formula for a “ball” centered at μ with radius σ in n-dimensional space.[13]

Heteroskedacity occurs when the amount of error is correlated with an independent variable. For example, in a regression on food expenditure and income, the error is correlated with income. Low income people generally spend a similar amount on food, while high income people may spend a very large amount or as little as low income people spend. Heteroskedacity can also be caused by changes in measurement practices. For example, as statistical offices improve their data, measurement error decreases, so the error term declines over time.

This assumption is violated when there is autocorrelation. Autocorrelation can be visualized on a data plot when a given observation is more likely to lie above a fitted line if adjacent observations also lie above the fitted regression line. Autocorrelation is common in time series data where a data series may experience "inertia."[14] If a dependent variable takes a while to fully absorb a shock. Spatial autocorrelation can also occur geographic areas are likely to have similar errors. Autocorrelation may be the result of misspecification such as choosing the wrong functional form. In these cases, correcting the specification is one possible way to deal with autocorrelation.

In the presence of non-spherical errors, the generalized least squares estimator can be shown to be BLUE.[15]

#### Exogeneity of independent variables

$\operatorname{E}[\,\varepsilon|X\,] = 0.$

This assumption is violated if the variables are endogenous. Endogeneity can be the result of simultaneity, where causality flows back and forth between both the dependent and independent variable. Instrumental variable techniques are commonly used to address this problem.

#### Full rank

The sample data matrix must have full rank or OLS cannot be estimated. There must be at least one observation for every parameter being estimated and the data cannot have perfect multicollinearity.[16] Perfect multicollinearity will occur in a "dummy variable trap" when a base dummy variable is not omitted resulting in perfect correlation between the dummy variables and the constant term.

Multicollinearity (as long as it is not "perfect") can be present resulting in a less efficient, but still unbiased estimate.

## Methods

Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analyzing economic history, and forecasting.[17]

Econometrics may use standard statistical models to study economic questions, but most often they are with observational data, rather than in controlled experiments. 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 by useful for generating new hypotheses.[18] Economics often analyzes 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.[19] 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.[20]

### Artificial intelligence methods

Artificial Intelligence has become important for building econometric models and for use in decision making.[21] Artificial intelligence is a nature-inspired computational paradigm which has found usage in many areas. It allows economic models to be of arbitrary complexity and also to be able to evolve as the economic environment also changes. For example, artificial intelligence has been applied to simulate the stock market, to model options and derivatives as well as model and control interest rates.

### Experimental economics

In recent decades, econometricians have increasingly turned to use of experiments to evaluate the often-contradictory conclusions of observational studies. Here, controlled and randomized experiments provide statistical inferences that may yield better empirical performance than do purely observational studies.[22]

### Data

Data sets to which econometric analyses are applied can be classified as time-series data, cross-sectional data, panel data, and multidimensional panel data. Time-series data sets contain observations over time; for example, inflation over the course of several years. Cross-sectional data sets contain observations at a single point in time; for example, many individuals' incomes in a given year. Panel data sets contain both time-series and cross-sectional observations. Multi-dimensional panel data sets contain observations across time, cross-sectionally, and across some third dimension. For example, the Survey of Professional Forecasters contains forecasts for many forecasters (cross-sectional observations), at many points in time (time series observations), and at multiple forecast horizons (a third dimension).

### Instrumental variables

In many econometric contexts, the commonly-used ordinary least squares method may not recover the theoretical relation desired or may produce estimates with poor statistical properties, because the assumptions for valid use of the method are violated. One widely used remedy is the method of instrumental variables (IV). For an economic model described by more than one equation, simultaneous-equation methods may be used to remedy similar problems, including two IV variants, Two-Stage Least Squares (2SLS), and Three-Stage Least Squares (3SLS).[23]

### Computational methods

Computational concerns are important for evaluating econometric methods and for use in decision making.[24] Such concerns include mathematical well-posedness: the existence, uniqueness, and stability of any solutions to econometric equations. Another concern is the numerical efficiency and accuracy of software.[25] A third concern is also the usability of econometric software.[26]

### Structural econometrics

Structural econometrics extends the ability of researchers to analyze data by using economic models as the lens through which to view the data. The benefit of this approach is that any policy recommendations are not subject to the Lucas critique since counter-factual analyses take an agent's re-optimization into account. Structural econometric analyses begin with an economic model that captures the salient features of the agents under investigation. The researcher then searches for parameters of the model that match the outputs of the model to the data. There are two ways of doing this. The first requires the researcher to completely solve the model and then use maximum likelihood.[27] However, there have been many advances that can bypass the full solution of the model and that estimate models in two stages. Importantly, these methods allow the researcher to consider more complicated models with strategic interactions and multiple equilibria.[28]

A good example of structural econometrics is in the estimation of first price sealed bid auctions with independent private values.[29] The key difficulty with bidding data from these auctions is that bids only partially reveal information on the underlying valuations, bids shade the underlying valuations. One would like to estimate these valuations in order to understand the magnitude of profits each bidder makes. More importantly, it is necessary to have the valuation distribution in hand to engage in mechanism design. In a first price sealed bid auction the expected payoff of a bidder is given by:

$(v-b)\Pr(b\ \textrm{wins})$

where v is the bidder valuation, b is the bid. The optimal bid $b^*$ solves a first order condition:

$(v-b^*)\frac{\partial \Pr(b^*\ \textrm{wins})}{\partial b}-\Pr(b^*\ \textrm{wins})=0$

which can be re-arranged to yield the following equation for $v$

$v=b^*+\frac{\Pr(b^*\ \textrm{wins})}{\partial \Pr(b^*\ \textrm{wins})/\partial b}$

Notice that the probability that a bid wins an auction can be estimated from a data set of completed auctions, where all bids are observed. This can be done using simple non-parametric estimators. If all bids are observed, it is then possible to use the above relation and the estimated probability function and its derivative to point wise estimate the underlying valuation. This will then allow the investigator to estimate the valuation distribution.

## Example

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

$\ln(\text{wage}) = \beta_0 + \beta_1 (\text{years of education}) + \varepsilon.$

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 $\beta_1$ measures the increase in the natural log of the wage attributable to one more year of education. The term $\varepsilon$ is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, $\beta_0 \mbox{ and } \beta_1$ under specific assumptions about the random variable $\varepsilon$. For example, if $\varepsilon$ 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 $\epsilon$ 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 $\beta_1$ identifiable.[30] An overview of econometric methods used to study this problem were provided by Card (1999).[31]

## Journals

The main journals which 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,[32] Applied Econometrics and International Development, the Journal of Business & Economic Statistics, and the Journal of Economic and Social Measurement.

## Limitations and criticisms

Like other forms of statistical analysis, badly specified econometric models may show a spurious correlation where two variables are correlated but causally unrelated. In a study of the use of econometrics in major economics journals, McCloskey concluded that economists report p values (following the Fisherian tradition of tests of significance of point null-hypotheses), neglecting concerns of type II errors; economists fail to report estimates of the size of effects (apart from statistical significance) and to discuss their economic importance. Economists also fail to use economic reasoning for model selection, especially for deciding which variables to include in a regression.[33][34]

In some cases, economic variables cannot be experimentally manipulated as treatments randomly assigned to subjects.[35] 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".[36]

Economists from the Austrian School argue that aggregate economic models are not well suited to describe economic reality because they waste a large part of specific knowledge. Friedrich Hayek in his The Use of Knowledge in Society argued that "knowledge of the particular circumstances of time and place" is not easily aggregated and is often ignored by professional economists.[37][38]

## 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 (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. ^ http://www.dziejekrakowa.pl/biogramy/index.php?id=516
5. ^ • 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.
6. ^
7. ^ Greene (2012), 12.
8. ^ Kennedy 2003, p. 110.
9. ^ Kennedy 2003, p. 129.
10. ^ Kennedy 2003, p. 131.
11. ^ Kennedy 2003, p. 130.
12. ^ Kennedy 2003, p. 133.
13. ^ Greene 2012, p. 23-note.
14. ^ Greene 2010, p. 22.
15. ^ Kennedy 2003, p. 135.
16. ^ Kennedy 2003, p. 205.
17. ^ Clive Granger (2008). "forecasting," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
18. ^ Herman O. Wold (1969). "Econometrics as Pioneering in Nonexperimental Model Building," Econometrica, 37(3), pp. 369-381.
19. ^ For an overview of a linear implementation of this framework, see linear regression.
20. ^ Edward E. Leamer (2008). "specification problems in econometrics," The New Palgrave Dictionary of Economics. Abstract.
21. ^ • Tshilidzi Marwala (2013). "Economic Modeling Using Artificial Intelligence Methods." Springer-Verlag,
22. ^ • H. Wold 1954. "Causality and Econometrics," Econometrica, 22(2), p p. 162-177.
• Kevin D. Hoover (2008). "causality in economics and econometrics," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract and galley proof.
23. ^ Peter Kennedy (economist) (2003). A Guide to Econometrics, 5th ed. Description, preview, and TOC, ch. 9, 10, 13, and 18.
24. ^ • Keisuke Hirano (2008). "decision theory in econometrics," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• James O. Berger (2008). "statistical decision theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
25. ^ B. D. McCullough and H. D. Vinod (1999). "The Numerical Reliability of Econometric Software," Journal of Economic Literature, 37(2), pp. 633-665.
26. ^ • Vassilis A. Hajivassiliou (2008). "computational methods in econometrics," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Richard E. Quandt (1983). "Computational Problems and Methods," ch. 12, in Handbook of Econometrics, v. 1, pp. 699-764.
• Ray C. Fair (1996). "Computational Methods for Macroeconometric Models," Handbook of Computational Economics, v. 1, pp. [1]-169.
27. ^ Rust, John (1987). "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher". Econometrica 55 (5): 999–1033. JSTOR 1911259.
28. ^ Hotz, V. Joseph; Miller, Robert A. (1993). "Conditional Choice Probabilities and the Estimation of Dynamic Models". Review of Economic Studies 60 (3): 497–529. JSTOR 2298122.
29. ^ Guerre, E.; Perrigne, I.; Vuong, Q. (2000). "Optimal Nonparametric Estimation of First Price Auctions". Econometrica 68 (3): 525–574. doi:10.1111/1468-0262.00123.
30. ^ Pearl, Judea (2000). Causality: Model, Reasoning, and Inference. Cambridge University Press. ISBN 0521773628.
31. ^ Card, David (1999). "The Causal Effect of Education on Earning". In Ashenfelter, O.; Card, D. Handbook of Labor Economics. Amsterdam: Elsevier. pp. 1801–1863. ISBN 0444822895.
32. ^ "The Econometrics Journal - Wiley Online Library". Wiley.com. Retrieved 2013-10-08.
33. ^ McCloskey (May 1985). "The Loss Function has been mislaid: the Rhetoric of Significance Tests". American Economic Review 75 (2).
34. ^ 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 (press +).
35. ^ Leamer, Edward (March 1983). "Let's Take the Con out of Econometrics". American Economic Review 73 (1): 34.
36. ^ Leamer, Edward (March 1983). "Let's Take the Con out of Econometrics". American Economic Review 73 (1): 43.
37. ^ Robert F. Garnett. What Do Economists Know? New Economics of Knowledge. Routledge, 1999. ISBN 978-0-415-15260-0. p. 170
38. ^ G. M. P. Swann. Putting Econometrics in Its Place: A New Direction in Applied Economics. Edward Elgar Publishing, 2008. ISBN 978-1-84720-776-0. p. 62-64

## References

(2007) v. 1: Econometric Theoryv. 1. Links to description and contents.
(2009) v. 2, Applied Econometrics. Palgrave Macmillan. ISBN 978-1-4039-1799-7 Links to description and contents.
• Pearl, Judea (2009, 2nd ed.). Causality: Models, Reasoning and Inference, Cambridge University Press, Description, TOC, and preview, ch. 1-10 and ch. 11. 5 economics-journal reviews, including Kevin D. Hoover, Economics Journal.
• Pindyck, Robert S., and Daniel L. Rubinfeld (1998, 4th ed.). Econometric Methods and Economic Forecasts, McGraw-Hill.
• Santos Silva, J.M.C. and Tenreyro, Silvana (2006), “The Log of Gravity,” The Review of Economics and Statistics, 88(4), pp. 641–658. <http://www.mitpressjournals.org/doi/pdfplus/10.1162/rest.88.4.641>
• Studenmund, A.H. (2011, 6th ed.). Using Econometrics: A Practical Guide. Contents (chapter-preview) links.
• Wooldridge, Jeffrey (2003). Introductory Econometrics: A Modern Approach. Mason: Thomson South-Western. ISBN 0-324-11364-1 Chapter-preview links in brief and detail.