💰Intro to Mathematical Economics Unit 10 – Econometric Models & Statistical Inference

Key Concepts and Definitions

• Econometrics combines economic theory, mathematics, and statistical inference to analyze economic phenomena and test hypotheses
• Dependent variable (Y) represents the outcome or effect being studied, while independent variables (X) are the factors believed to influence the dependent variable
• Stochastic error term (ε) captures the unexplained variation in the dependent variable not accounted for by the independent variables
• Ordinary Least Squares (OLS) is a common estimation method that minimizes the sum of squared residuals to find the best-fitting line
• Coefficient estimates (β) quantify the relationship between each independent variable and the dependent variable, holding other factors constant (ceteris paribus)
  • Interpretation depends on the functional form of the model (linear, log-linear, log-log)
• Statistical significance indicates the likelihood that the observed relationship between variables is not due to chance (p-value)
• R-squared ($R^2$) measures the proportion of variation in the dependent variable explained by the independent variables (goodness of fit)

Types of Econometric Models

• Simple linear regression models the relationship between one dependent variable and one independent variable ($Y = \beta_0 + \beta_1X + \varepsilon$)
• Multiple linear regression extends simple regression to include multiple independent variables ($Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_kX_k + \varepsilon$)
  • Allows for controlling for confounding factors and isolating the effect of each independent variable
• Logarithmic transformations (log-linear, log-log models) can be used to model non-linear relationships and interpret coefficients as elasticities
• Panel data models (fixed effects, random effects) analyze data with both cross-sectional and time-series dimensions (individuals observed over time)
• Instrumental variables (IV) estimation addresses endogeneity issues by using an instrument correlated with the independent variable but not the error term
• Time series models (autoregressive, moving average, ARIMA) analyze data collected over regular time intervals and account for temporal dependence
• Limited dependent variable models (probit, logit) are used when the dependent variable is binary or categorical

Statistical Foundations

• Gauss-Markov assumptions ensure OLS estimators are unbiased and efficient (linearity, exogeneity, homoscedasticity, no multicollinearity, normality)
• Sampling distributions describe the probability distribution of a sample statistic (mean, variance) over repeated samples
• Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population distribution
• Standard errors measure the variability of coefficient estimates and are used to construct confidence intervals and test hypotheses
• Confidence intervals provide a range of plausible values for a population parameter based on the sample estimate and desired level of confidence (90%, 95%, 99%)
• Type I error (false positive) occurs when rejecting a true null hypothesis, while Type II error (false negative) occurs when failing to reject a false null hypothesis
• Power of a test is the probability of correctly rejecting a false null hypothesis (1 - Type II error rate)

Model Specification and Estimation

• Economic theory and prior research guide the selection of relevant variables and functional form
• Omitted variable bias arises when a relevant variable is excluded from the model, leading to biased and inconsistent estimates
• Misspecification tests (RESET, Hausman) can detect omitted variables, incorrect functional form, or endogeneity issues
• Multicollinearity occurs when independent variables are highly correlated, leading to imprecise estimates and difficulty interpreting individual coefficients
  • Variance Inflation Factor (VIF) measures the degree of multicollinearity for each independent variable
• Heteroscedasticity refers to non-constant variance of the error term, which can be detected using tests (Breusch-Pagan, White) and addressed through robust standard errors or weighted least squares
• Autocorrelation in time series data can be detected using tests (Durbin-Watson) and addressed through generalized least squares or autoregressive models
• Maximum Likelihood Estimation (MLE) is an alternative to OLS that estimates parameters by maximizing the likelihood function, often used in non-linear models

Hypothesis Testing and Inference

• Null hypothesis ($H_0$) represents the default position of no effect or no difference, while the alternative hypothesis ($H_a$) represents the research claim
• Test statistic (t-statistic, F-statistic) measures the deviation of the sample estimate from the null hypothesis value, standardized by the standard error
• P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
  • Smaller p-values provide stronger evidence against the null hypothesis
• Significance level (α) is the threshold for rejecting the null hypothesis, typically set at 0.01, 0.05, or 0.10
• One-tailed tests are used when the alternative hypothesis specifies a direction (greater than or less than), while two-tailed tests are used when the alternative is non-directional (not equal to)
• Joint hypothesis tests (F-tests) evaluate the significance of multiple coefficients simultaneously
• Wald tests compare the unrestricted and restricted models to test hypotheses about subsets of coefficients
• Likelihood ratio tests compare the likelihood of the data under the null and alternative models

Interpreting Results

• Coefficient estimates represent the change in the dependent variable associated with a one-unit change in the independent variable, holding other factors constant
  • For log-transformed variables, coefficients can be interpreted as elasticities or percentage changes
• Statistical significance indicates the reliability of the estimated relationship, but does not necessarily imply economic or practical significance
• Confidence intervals provide a range of plausible values for the population parameter, with narrower intervals indicating greater precision
• Marginal effects measure the change in the dependent variable for a small change in an independent variable, holding other factors at their means
• Standardized coefficients (beta coefficients) allow for comparing the relative importance of independent variables measured on different scales
• Goodness of fit measures ($R^2$, adjusted $R^2$) indicate the proportion of variation in the dependent variable explained by the model, but do not guarantee causality or model validity
• Out-of-sample predictions can assess the model's performance on new data and guard against overfitting

Common Pitfalls and Limitations

• Endogeneity arises when an independent variable is correlated with the error term, leading to biased and inconsistent estimates
  • Sources include omitted variables, measurement error, and simultaneity (reverse causality)
• Sample selection bias occurs when the sample is not representative of the population of interest, often due to non-random sampling or self-selection
• Outliers and influential observations can disproportionately affect coefficient estimates and should be carefully examined and potentially addressed (robust regression, Cook's distance)
• Ecological fallacy involves drawing conclusions about individuals based on aggregate data, which may not hold at the individual level (Simpson's paradox)
• Extrapolation beyond the range of the data can lead to unreliable predictions, as the estimated relationships may not hold outside the sample
• Causal inference requires careful research design and strong assumptions (randomization, exogeneity, exclusion restrictions) that may not be met in observational studies
• Model uncertainty arises when multiple models fit the data equally well, and can be addressed through model averaging or Bayesian methods

Real-World Applications

• Labor economics uses econometric models to study wage determination, labor supply and demand, and the effects of policies (minimum wage, unemployment insurance)
• Environmental economics employs econometric techniques to estimate the value of non-market goods (air quality, biodiversity) and evaluate environmental policies (carbon taxes, cap-and-trade)
• Health economics applies econometric methods to analyze healthcare demand, provider behavior, and the impact of interventions (insurance expansions, drug pricing)
• Development economics uses econometrics to assess the effectiveness of poverty alleviation programs (conditional cash transfers, microfinance) and drivers of economic growth
• Public economics relies on econometric analysis to study the effects of taxation, government spending, and redistribution on individual and firm behavior
• Financial economics employs econometric models to evaluate asset pricing, risk management, and the impact of monetary policy on financial markets
• Industrial organization uses econometrics to examine market structure, firm conduct, and the effects of mergers and antitrust policies on competition and consumer welfare