# Econometrics

Template:Economics sidebar Econometrics is the application of mathematics, statistical methods, and computer science, to economic data and is described as the branch of economics that aims to give empirical content to economic relations. 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." An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships." The first known use of the term "econometrics" (in cognate form) was by Polish economist Paweł Ciompa in 1910. Ragnar Frisch is credited with coining the term in the sense in which it is used today.

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.Template:Sfn

## 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. 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 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 $\beta _{1}$ 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

{{#invoke:see also|seealso}} 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.

## Methods

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

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. 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. 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.

### Artificial intelligence methods

Artificial Intelligence has become important for building econometric models and for use in decision making. 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.

### 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).

### Computational methods

Computational concerns are important for evaluating econometric methods and for use in decision making. 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. A third concern is also the usability of econometric software.

### 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. 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.

A good example of structural econometrics is in the estimation of first price sealed bid auctions with independent private values. 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. An overview of econometric methods used to study this problem were provided by Card (1999).

## 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, Applied Econometrics and International Development, the Journal of Business & Economic Statistics, and the Journal of Economic and Social Measurement.