Glossary

4.5 Asymptotic properties

7 min read•august 20, 2024

Asymptotic properties are crucial in econometrics, helping us understand how estimators behave as sample sizes grow. These properties, including , , and efficiency, provide a foundation for reliable statistical inference and hypothesis testing in large samples.

By studying asymptotic properties, we gain insights into the behavior of estimators like OLS, MLE, and GMM. This knowledge allows us to construct confidence intervals, perform hypothesis tests, and make informed decisions about which estimators to use in different scenarios.

Consistency of estimators

  • Consistency is a key property of estimators in econometrics that ensures the estimates converge to the true population parameter as the sample size increases
  • Consistent estimators are essential for drawing reliable inferences and making accurate predictions based on sample data

Convergence in probability

Top images from around the web for Convergence in probability

  • 6.2 The Sampling Distribution of the Sample Mean (σ Known) – Significant Statistics View original

    Is this image relevant?

  • Estimating a Population Mean (1 of 3) | Concepts in Statistics View original

    Is this image relevant?

  • 6.2 The Sampling Distribution of the Sample Mean (σ Known) – Significant Statistics View original

    Is this image relevant?

1 of 3

Top images from around the web for Convergence in probability

  • 6.2 The Sampling Distribution of the Sample Mean (σ Known) – Significant Statistics View original

    Is this image relevant?

  • Estimating a Population Mean (1 of 3) | Concepts in Statistics View original

    Is this image relevant?

  • 6.2 The Sampling Distribution of the Sample Mean (σ Known) – Significant Statistics View original

    Is this image relevant?

1 of 3
  • means that as the sample size grows, the probability of the estimator being close to the true parameter value approaches 1
  • Mathematically, an estimator θ^n\hat{\theta}_n is consistent for the parameter θ\theta if for any ϵ>0\epsilon > 0, lim⁡n→∞P(∣θ^n−θ∣<ϵ)=1\lim_{n \to \infty} P(|\hat{\theta}_n - \theta| < \epsilon) = 1
  • Intuitively, this implies that the estimator becomes more precise and less variable as more data is collected

Bias vs consistency

  • Bias refers to the difference between the expected value of an estimator and the true parameter value, i.e., Bias(θ^)=E(θ^)−θBias(\hat{\theta}) = E(\hat{\theta}) - \theta
  • An estimator can be biased but still consistent if the bias diminishes as the sample size increases
  • Consistency is a more important property than unbiasedness because it guarantees that the estimator will eventually converge to the true value, even if it is initially biased

Conditions for consistency

  • For an estimator to be consistent, it typically needs to satisfy certain regularity conditions
  • These conditions include the correct specification of the model, the independence and identical distribution of the error terms, and the existence of finite moments
  • Violating these conditions can lead to inconsistent estimators and misleading inferences (omitted variable bias, endogeneity)

Asymptotic normality

  • Asymptotic normality is a fundamental property of many estimators in econometrics that allows for the construction of confidence intervals and hypothesis tests
  • It states that as the sample size increases, the distribution of the estimator converges to a normal distribution

Central Limit Theorem

  • The (CLT) is the basis for asymptotic normality
  • It states that the sum or average of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution
  • The CLT is crucial for deriving the asymptotic properties of estimators and test statistics

Asymptotic distribution of estimators

  • Under certain regularity conditions, many estimators have an asymptotic normal distribution
  • The asymptotic distribution is characterized by the true parameter value as the mean and the , which depends on the sample size and the properties of the estimator
  • Knowing the asymptotic distribution allows for the construction of confidence intervals and hypothesis tests

Wald statistics

  • are used to test hypotheses about the parameters based on their asymptotic normal distribution
  • They are calculated as the squared difference between the estimated parameter and the hypothesized value, divided by the asymptotic variance
  • Wald statistics follow a chi-squared distribution under the null hypothesis, which enables the computation of p-values and critical values

Asymptotic efficiency

  • is a desirable property of estimators that refers to their ability to achieve the lowest possible variance among all consistent estimators
  • provide the most precise estimates and the most powerful tests

Cramér-Rao lower bound

  • The is the that an unbiased estimator can achieve
  • It serves as a benchmark for evaluating the efficiency of estimators
  • The lower bound is derived from the Fisher information matrix, which measures the amount of information the data contains about the parameters

Efficient estimators

  • An estimator is asymptotically efficient if its asymptotic variance equals the Cramér-Rao lower bound
  • Efficient estimators have the smallest possible variance among all consistent estimators
  • Examples of asymptotically efficient estimators include the maximum likelihood estimator (under certain conditions) and the generalized method of moments estimator (with optimal weighting matrix)

Relative efficiency

  • compares the efficiency of two estimators by taking the ratio of their variances
  • An estimator with a smaller variance is considered more efficient
  • Relative efficiency can help choose between competing estimators (OLS vs. GLS) and determine the required sample size for a desired level of precision

Hypothesis testing with asymptotic results

  • Asymptotic theory provides a framework for conducting hypothesis tests when the sample size is large
  • These tests rely on the asymptotic distribution of the test statistics under the null and alternative hypotheses

Wald tests

  • are based on the asymptotic normal distribution of the estimators
  • They compare the estimated parameter values to the hypothesized values, taking into account the asymptotic variance
  • Wald tests are easy to compute but may have poor finite sample properties and can be sensitive to the parameterization of the model

Likelihood ratio tests

  • Likelihood ratio (LR) tests compare the likelihood of the data under the null and alternative hypotheses
  • They are based on the difference in the log-likelihood values between the restricted and unrestricted models
  • LR tests have good asymptotic properties and are invariant to the parameterization, but they require estimating both the restricted and unrestricted models

Lagrange multiplier tests

  • Lagrange multiplier (LM) tests, also known as score tests, evaluate the gradient of the log-likelihood function at the restricted parameter values
  • They only require estimating the model under the null hypothesis, making them computationally attractive
  • LM tests are particularly useful for testing hypotheses about a subset of parameters or for detecting misspecification (heteroskedasticity, serial correlation)

Confidence intervals with asymptotic results

  • Confidence intervals provide a range of plausible values for the population parameters based on the sample estimates
  • Asymptotic theory allows for the construction of confidence intervals using the asymptotic distribution of the estimators

Standard errors of estimators

  • Standard errors measure the uncertainty associated with the parameter estimates
  • They are calculated as the square root of the asymptotic variance of the estimators
  • Smaller standard errors indicate more precise estimates and narrower confidence intervals

Asymptotic confidence intervals

  • are constructed using the point estimate and the standard error
  • For a 95% confidence interval, the lower and upper bounds are typically the point estimate plus or minus 1.96 times the standard error
  • These intervals are valid for large samples and rely on the asymptotic normality of the estimators

Bootstrapping confidence intervals

  • Bootstrapping is a resampling technique that can be used to construct confidence intervals without relying on asymptotic theory
  • It involves repeatedly drawing samples with replacement from the original data and calculating the estimates for each bootstrap sample
  • The distribution of the bootstrap estimates approximates the sampling distribution of the estimator, allowing for the construction of percentile or pivotal confidence intervals

Asymptotic properties of maximum likelihood estimators

  • Maximum likelihood estimation (MLE) is a popular method for estimating the parameters of a model by maximizing the likelihood function
  • MLE has desirable asymptotic properties, making it a powerful tool in econometrics

Consistency of ML estimators

  • Under certain regularity conditions, are consistent
  • These conditions include the correct specification of the model, the identifiability of the parameters, and the existence of a unique global maximum of the likelihood function
  • Consistency ensures that the ML estimates converge to the true parameter values as the sample size increases

Asymptotic normality of ML estimators

  • ML estimators are asymptotically normally distributed under appropriate regularity conditions
  • The asymptotic variance of the ML estimator is given by the inverse of the Fisher information matrix
  • This property allows for the construction of confidence intervals and hypothesis tests based on the normal distribution

Asymptotic efficiency of ML estimators

  • ML estimators are asymptotically efficient, meaning they achieve the Cramér-Rao lower bound for the variance
  • This efficiency property holds under the correct specification of the model and certain regularity conditions
  • Asymptotically efficient ML estimators provide the most precise estimates among all consistent estimators

Asymptotic properties of generalized method of moments estimators

  • The generalized method of moments (GMM) is a flexible estimation framework that includes many other estimators as special cases (OLS, IV, ML)
  • GMM estimators have attractive asymptotic properties under certain conditions

Consistency of GMM estimators

  • GMM estimators are consistent under the assumption that the moment conditions are correctly specified and the parameters are identified
  • Consistency requires that the number of moment conditions is at least as large as the number of parameters and that the moment conditions hold in the population
  • The consistency of GMM estimators is robust to certain types of misspecification (heteroskedasticity, serial correlation)

Asymptotic normality of GMM estimators

  • GMM estimators are asymptotically normally distributed under suitable regularity conditions
  • The asymptotic variance of the GMM estimator depends on the choice of the weighting matrix, which determines the relative importance of the moment conditions
  • The optimal weighting matrix is the inverse of the variance-covariance matrix of the moment conditions, which leads to the most efficient GMM estimator

Efficiency of GMM estimators

  • The efficiency of GMM estimators depends on the choice of the moment conditions and the weighting matrix
  • With the optimal weighting matrix, GMM estimators are asymptotically efficient and reach the Cramér-Rao lower bound
  • However, the optimal weighting matrix is typically unknown and must be estimated, which can affect the finite sample properties of the estimator (two-step GMM, iterated GMM)

Key Terms to Review (30)

Asymptotic Bias: Asymptotic bias refers to the difference between the expected value of an estimator and the true parameter value as the sample size approaches infinity. This concept highlights how estimators can behave differently with larger samples, revealing their reliability and consistency. Understanding asymptotic bias is crucial, especially when dealing with weak instruments, as it can lead to misleading conclusions in statistical inference if not properly accounted for.
Asymptotic Confidence Intervals: Asymptotic confidence intervals are statistical intervals that estimate the range within which a population parameter is likely to lie as the sample size approaches infinity. These intervals rely on the asymptotic properties of estimators, meaning that as the sample size increases, the distribution of the estimator converges to a normal distribution, which allows for easier calculation of the confidence interval.
Asymptotic Efficiency: Asymptotic efficiency refers to the property of an estimator in statistics where, as the sample size increases to infinity, it achieves the lowest possible variance among all unbiased estimators. This concept connects closely with the notions of consistency and efficiency, indicating that an estimator not only provides accurate results as more data is collected but also does so in a way that minimizes uncertainty, especially in large samples.
Asymptotic F-distribution: The asymptotic F-distribution is a probability distribution that arises in the context of hypothesis testing, particularly when comparing variances from different samples. It is the limiting distribution of the ratio of two scaled chi-squared distributions as the sample sizes grow large, often utilized in analysis of variance (ANOVA) and regression models to assess the significance of predictors.
Asymptotic Normality: Asymptotic normality refers to the property of an estimator where, as the sample size increases, its distribution approaches a normal distribution. This concept is crucial in statistics and econometrics as it allows for making inferences about population parameters using sample data, even when the underlying data does not follow a normal distribution. It connects with important statistical theories and helps ensure that estimators are reliable and valid in large samples.
Asymptotic properties of GMM estimators: Asymptotic properties of GMM (Generalized Method of Moments) estimators refer to the behavior of these estimators as the sample size approaches infinity. These properties include consistency, asymptotic normality, and efficiency, which are crucial for understanding how well GMM estimators perform in large samples. The insights gained from these properties help to ensure reliable inference and hypothesis testing in econometric models.
Asymptotic properties of maximum likelihood estimators: Asymptotic properties of maximum likelihood estimators refer to the behaviors and characteristics of these estimators as the sample size approaches infinity. These properties are important because they provide insights into the reliability and efficiency of the estimators, highlighting features such as consistency, asymptotic normality, and efficiency. Understanding these properties helps in evaluating how well the maximum likelihood estimators perform in large samples, which is crucial for statistical inference.
Asymptotic t-distribution: The asymptotic t-distribution is a statistical distribution that approximates the behavior of the t-distribution as the sample size becomes large. It reflects how the t-statistic behaves when the sample size increases, and under certain conditions, it approaches a standard normal distribution. This property is crucial for making inferences about population parameters when working with sample data, particularly in relation to confidence intervals and hypothesis testing.
Asymptotic Variance: Asymptotic variance is a measure of the variance of an estimator as the sample size approaches infinity. It helps in understanding the distributional properties of estimators, particularly in large samples, and is crucial for making inferences about population parameters. This concept is essential for evaluating the efficiency and consistency of estimators within econometric models.
Bootstrapping Confidence Intervals: Bootstrapping confidence intervals is a statistical technique that involves resampling a dataset with replacement to estimate the sampling distribution of a statistic, allowing for the construction of confidence intervals without relying on traditional parametric assumptions. This method is particularly useful in econometrics, where it helps to understand the variability and uncertainty around estimates derived from data. It leverages the power of repeated sampling to provide insights into the reliability of estimators, connecting closely to asymptotic properties by offering robust interval estimates even with small sample sizes.
Central Limit Theorem: The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the distribution of the sample mean will approach a normal distribution regardless of the original population's distribution. This fundamental theorem is crucial for understanding how random variables behave, enabling statisticians to make inferences about population parameters based on sample data.
Consistency: Consistency refers to a property of an estimator, where as the sample size increases, the estimates converge in probability to the true parameter value being estimated. This concept is crucial in various areas of econometrics, as it underpins the reliability of estimators across different methods, ensuring that with enough data, the estimates reflect the true relationship between variables.
Convergence in distribution: Convergence in distribution refers to the behavior of a sequence of random variables where the probability distribution of these variables approaches a limiting distribution as the sample size increases. This concept is crucial for understanding how estimators behave in large samples, often linked to the idea that estimators can approximate the true parameter values. In statistical analysis, it helps establish the asymptotic properties of estimators, allowing for inference based on the distributional characteristics as sample sizes grow.
Convergence in Probability: Convergence in probability refers to a statistical concept where a sequence of random variables converges to a particular value in such a way that the probability of the variables being far from that value approaches zero as the sample size increases. This concept is essential when discussing the reliability of estimators and their consistency, as well as the asymptotic properties that arise when considering large sample behavior in statistics.
Cramér-Rao Lower Bound: The Cramér-Rao Lower Bound (CRLB) provides a theoretical lower limit on the variance of unbiased estimators. This concept is crucial in statistics as it helps determine how efficient an estimator can be, specifically in large sample contexts. The CRLB states that the variance of any unbiased estimator is at least as large as the inverse of the Fisher Information, which quantifies the amount of information that a random variable carries about an unknown parameter.
Efficient Estimators: Efficient estimators are statistical estimators that have the smallest variance among all unbiased estimators for a given parameter. This means they provide the most precise estimates with the least amount of error, which is crucial in econometrics for making reliable inferences. When discussing asymptotic properties, efficient estimators become particularly important as their performance improves with larger sample sizes, highlighting their consistency and reliability in estimating population parameters as the number of observations increases.
Generalized method of moments estimators: Generalized method of moments (GMM) estimators are statistical methods used to estimate parameters in econometric models by utilizing moment conditions derived from the data. This approach is particularly useful when traditional methods, like ordinary least squares, may not be applicable or efficient due to potential violations of assumptions, such as endogeneity or heteroscedasticity. GMM estimators rely on the idea that the sample moments should match the population moments, leading to consistent and asymptotically normal estimators as the sample size increases.
Identification Conditions: Identification conditions refer to the set of criteria that must be satisfied for a statistical model to produce meaningful and unique estimates of its parameters. These conditions ensure that the model is capable of distinguishing between causal relationships and correlations, allowing for reliable inferences about the data. In the context of econometrics, identification is crucial for making valid conclusions based on the estimated models, particularly when examining causal effects.
Iid assumptions: The iid assumptions, or independent and identically distributed assumptions, refer to the condition where a set of random variables are both independent from each other and follow the same probability distribution. This concept is crucial in econometrics as it ensures that the sample data used for estimation and inference behaves consistently, allowing for valid statistical properties and reliable results.
Lagrange Multiplier Tests: Lagrange Multiplier Tests are statistical tests used to determine the presence of restrictions in a model, particularly in the context of econometric modeling. These tests help in assessing whether additional parameters are significant or if a simpler model is adequate, thus providing insights into the model's specification and its validity.
Large Sample Properties: Large sample properties refer to the behavior of statistical estimators as the sample size approaches infinity. These properties help to understand how estimators perform with a growing amount of data, ensuring that they become more accurate and consistent. Key aspects of large sample properties include consistency, asymptotic normality, and asymptotic efficiency, which are essential for making reliable inferences in econometric analysis.
Law of Large Numbers: The law of large numbers is a fundamental principle in probability and statistics that states that as the size of a sample increases, the sample mean will converge to the expected value or population mean. This concept ensures that with enough observations, the average of the results will get closer to the expected outcome, providing a foundation for making reliable inferences about larger populations from smaller samples.
Likelihood Ratio Tests: Likelihood ratio tests are statistical tests used to compare the goodness of fit of two models, where one model is a special case of the other. They are particularly useful in determining whether adding additional parameters significantly improves the model's fit to the data. In the context of asymptotic properties, likelihood ratio tests leverage large sample theory to yield asymptotic distributions that can be used for hypothesis testing.
Maximum Likelihood Estimators: Maximum likelihood estimators (MLE) are statistical methods used to estimate the parameters of a statistical model by maximizing the likelihood function. This approach is grounded in finding parameter values that make the observed data most probable under the assumed model. MLE has desirable properties, including consistency and asymptotic normality, making it a popular choice in statistical inference.
Minimum Variance: Minimum variance refers to the property of an estimator that aims to produce the lowest possible variance among all estimators. This characteristic is crucial for ensuring that the estimators are not only unbiased but also efficient, providing reliable estimates that have the least spread or uncertainty. By achieving minimum variance, estimators can be considered optimal in terms of their precision and reliability, linking them closely to concepts such as best linear unbiased estimators, overall efficiency in statistical inference, and asymptotic properties as sample sizes grow.
Relative efficiency: Relative efficiency refers to the comparison of the efficiency of different estimators in terms of their variances. It helps in determining which estimator provides more precise estimates when dealing with the same parameter, allowing us to assess their performance in statistical inference. This concept is crucial in understanding how well an estimator performs compared to others, especially when considering large sample sizes and asymptotic properties.
Slutsky's Theorem: Slutsky's Theorem is a fundamental concept in consumer theory that describes how a change in the price of a good affects the quantity demanded by separating the substitution effect from the income effect. This theorem illustrates that when the price of a good changes, consumers adjust their consumption not only because of the change in relative prices but also due to the change in their purchasing power. It serves as a foundation for understanding demand elasticity and consumer behavior in response to price changes.
Standard errors of estimators: Standard errors of estimators measure the variability or precision of an estimator in statistical analysis. They provide insight into how much an estimated parameter, like a regression coefficient, is expected to fluctuate due to sampling variability. A smaller standard error suggests a more reliable estimate, which is crucial for making inferences about a population based on sample data.
Wald Statistics: Wald statistics are a type of test statistic used in statistical inference to determine the significance of estimated parameters in a model. They help assess whether a particular parameter is significantly different from zero or another specified value. In the context of asymptotic properties, Wald statistics leverage large sample theory, meaning their distribution approaches a specific limit as sample sizes increase, leading to more reliable inferences.
Wald Tests: Wald tests are statistical tests used to assess the significance of individual coefficients or a set of coefficients in a regression model. These tests evaluate whether the estimated parameters differ significantly from zero or some other value, helping to determine if certain predictors contribute meaningfully to the model. In the context of asymptotic properties, Wald tests rely on large sample theory to make inferences about parameter estimates and their variances.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide