BECC-110 Solved Assignment

Title of Course: Introductory Econometrics

1. What is the difference between an estimate and an estimator? Explain all the properties of an estimator with reference to BLUE.
2. Define multicollinearity. How do we identify it? Present its characteristics and also discuss the remedial measures to handle multicollinearity.

3. Differentiate between Point Estimation and Interval Estimation.
4. Explain the assumptions of a classical regression model.
5. Explain the term ‘Normal Distribution’. What is its significance? Explain its characteristics.

6. Differentiate between the concept of association and causation.
7. Discuss critical region/s. Explain one-tail and two -tail tests.
8. Which test is employed to compare means of two groups? Explain the test with the help of an example.
9. What do you understand by Sample regression function? Explain with the help of suitable examples.
10. Explain the consequences of specification errors.

Expert Answer

Title of Course: Introductory Econometrics

What is the difference between an estimate and an estimator? Explain all the properties of an estimator with reference to BLUE.

Answer:

• Estimator as a Function: An estimator is essentially a statistical tool or function that processes sample data to generate an estimate. It can be expressed as a formula or an algorithm that takes the sample data as input and produces a numerical output that serves as the estimate of a population parameter. For example, the sample mean ($\overline{X}$$\overline{X}$bar(X)\bar{X}$\overline{X}$) is an estimator of the population mean ($\mu$$\mu$mu\mu$\mu$), and it is calculated using the formula:
  $$\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$$$$\overline{X}=\frac{1}{n}\sum _{i=1}^n X_i$$bar(X)=(1)/(n)sum_(i=1)^(n)X_(i)\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$$\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$$
  where $X_i$$X_i$X_(i)X_i$X_i$ represents the individual data points, and $n$$n$nn$n$ is the sample size.
• Estimate as a Numerical Value: An estimate is the specific numerical result obtained when an estimator is applied to a particular set of data. For example, if we have a sample of exam scores, the sample mean (say, 75) calculated from this data is the estimate of the population mean.

• Unbiasedness: An estimator is unbiased if its expected value is equal to the true value of the population parameter it is estimating. Mathematically, an estimator $\hat{\theta }$$\widehat{\theta }$hat(theta)\hat{\theta}$\hat{\theta }$ is unbiased if:
  $$E(\hat{\theta })=\theta$$$$E(\widehat{\theta })=\theta$$E( hat(theta))=thetaE(\hat{\theta}) = \theta$$E(\hat{\theta })=\theta$$
  where $\theta$$\theta$theta\theta$\theta$ is the population parameter. An unbiased estimator does not systematically overestimate or underestimate the true parameter value.
• Consistency: An estimator is consistent if, as the sample size increases, the estimate converges to the true value of the population parameter. In other words, the estimator becomes more accurate as more data is used. Formally, ${\hat{\theta }}_n$${\widehat{\theta }}_n$hat(theta)_(n)\hat{\theta}_n${\hat{\theta }}_n$ is consistent if:
  $$\underset{n\to \mathrm{\infty }}{lim}P(|{\hat{\theta }}_n-\theta |<ϵ)=1$$$$\underset{n\to \mathrm{\infty }}{lim} P(|{\widehat{\theta }}_n-\theta |<ϵ)=1$$lim_(n rarr oo)P(| hat(theta)_(n)-theta| < epsilon)=1\lim_{n \to \infty} P(|\hat{\theta}_n – \theta| < \epsilon) = 1$$\underset{n\to \mathrm{\infty }}{lim}P(|{\hat{\theta }}_n-\theta |<ϵ)=1$$
  for any positive $ϵ$$ϵ$epsilon\epsilon$ϵ$, where $n$$n$nn$n$ is the sample size.
• Efficiency: An efficient estimator has the smallest variance among all unbiased estimators of a parameter. Efficiency is crucial because it ensures that the estimator makes the best use of the available data to produce the most precise estimates. The Cramér-Rao lower bound provides a theoretical minimum variance that an unbiased estimator can achieve.
• Sufficiency: An estimator is sufficient if it captures all the information in the sample data relevant to estimating the population parameter. A sufficient estimator ensures that no other statistic provides additional information about the parameter. The concept of sufficiency is formalized by the factorization theorem, which states that a statistic $T(X)$$T(X)$T(X)T(X)$T(X)$ is sufficient for a parameter $\theta$$\theta$theta\theta$\theta$ if the likelihood function can be factored into a product where one factor depends on the data only through $T(X)$$T(X)$T(X)T(X)$T(X)$, and the other does not depend on $\theta$$\theta$theta\theta$\theta$.
• Robustness: Robustness refers to the estimator’s ability to remain effective even when the assumptions underlying the statistical model are violated. A robust estimator is not overly sensitive to outliers or deviations from normality, making it more reliable in real-world situations where data may not perfectly adhere to theoretical distributions.

• Best: The term "best" implies that the OLS estimator has the smallest variance among all linear unbiased estimators. This means that the OLS estimator is the most efficient in terms of minimizing the uncertainty in the estimated parameters.
• Linear: The estimator is a linear function of the observed data. In the context of a linear regression model, the OLS estimator for the coefficient $\beta$$\beta$beta\beta$\beta$ is a linear combination of the observed values of the independent variable(s).
• Unbiased: The OLS estimator is unbiased, meaning that its expected value equals the true population parameter. This ensures that, on average, the OLS estimates will not systematically deviate from the true parameter values.

• Linearity: The relationship between the dependent and independent variables must be linear.
• Independence: The errors in the regression model must be independent of each other.
• Homoscedasticity: The errors should have a constant variance (no heteroscedasticity).
• No Perfect Multicollinearity: The independent variables should not be perfectly correlated with each other.
• Normality: The errors are normally distributed (though this assumption is more related to hypothesis testing than the BLUE property itself).

Define multicollinearity. How do we identify it? Present its characteristics and also discuss the remedial measures to handle multicollinearity.

Answer:

• Correlation Matrix: A simple way to detect multicollinearity is by examining the correlation matrix of the independent variables. If the pairwise correlation coefficients between two or more independent variables are close to +1 or -1, multicollinearity is likely present. However, this method only detects pairwise multicollinearity and may miss more complex relationships.
• Variance Inflation Factor (VIF): The VIF quantifies how much the variance of a regression coefficient is inflated due to multicollinearity. It is calculated as:
  $$VIF({\beta }_i)=\frac{1}{1-R_i^2}$$$$VIF({\beta }_i)=\frac{1}{1-R_i^2}$$VIF(beta _(i))=(1)/(1-R_(i)^(2))VIF(\beta_i) = \frac{1}{1 – R_i^2}$$VIF({\beta }_i)=\frac{1}{1-R_i^2}$$
  where $R_i^2$$R_i^2$R_(i)^(2)R_i^2$R_i^2$ is the coefficient of determination obtained by regressing the $i^{th}$$i^{th}$i^(th)i^{th}$i^{th}$ independent variable against all other independent variables. A VIF value exceeding 10 is often considered indicative of significant multicollinearity, though some researchers use a threshold of 5.
• Tolerance: Tolerance is the reciprocal of VIF and is another measure used to detect multicollinearity. It is defined as:
  $$Tolerance=1-R_i^2$$$$Tolerance=1-R_i^2$$Tolerance=1-R_(i)^(2)Tolerance = 1 – R_i^2$$Tolerance=1-R_i^2$$
  A tolerance value close to 0 indicates a high level of multicollinearity.
• Eigenvalues and Condition Index: The condition index is derived from the eigenvalues of the correlation matrix of the independent variables. A condition index greater than 30 suggests multicollinearity. This method helps detect multicollinearity that may not be apparent in the correlation matrix or VIF.

• Inflated Standard Errors: Multicollinearity increases the standard errors of the regression coefficients, making the estimates less precise and reducing the statistical significance of the predictors.
• Unstable Coefficient Estimates: The coefficients of highly correlated variables can become very sensitive to changes in the model. Even small changes in the data can lead to large swings in the estimated coefficients.
• High R-squared but Few Significant Predictors: A model with multicollinearity may exhibit a high R-squared value, indicating that the model explains a large portion of the variance in the dependent variable. However, individual predictors may appear statistically insignificant due to inflated standard errors.
• Difficulty in Assessing the Contribution of Predictors: Because the independent variables are highly correlated, it becomes challenging to disentangle their individual effects on the dependent variable. The interpretation of the regression coefficients becomes less clear.

• Remove Highly Correlated Predictors: One straightforward approach is to remove one or more of the highly correlated variables from the model. This can be done based on theoretical considerations or by examining the VIF and tolerance values. By reducing the number of correlated predictors, the multicollinearity problem can be alleviated.
• Combine Variables: If two or more variables are highly correlated, they can be combined into a single variable. For example, if income and wealth are highly correlated, they can be combined into an index or a composite variable. This reduces multicollinearity while retaining the information in the original variables.
• Principal Component Analysis (PCA): PCA is a dimensionality reduction technique that transforms the correlated variables into a set of uncorrelated components. These components can then be used as predictors in the regression model. PCA helps to mitigate multicollinearity by eliminating the intercorrelation among the predictors.
• Ridge Regression: Ridge regression is a technique that adds a penalty to the regression model for large coefficients. This penalty term helps to shrink the coefficients of correlated variables, thereby reducing multicollinearity. Ridge regression is particularly useful when multicollinearity is present but it is not desirable to drop any variables from the model.
• Increase Sample Size: In some cases, increasing the sample size can reduce the impact of multicollinearity. With more data, the model has more information to accurately estimate the coefficients, thereby reducing the variance inflation caused by multicollinearity.
• Use Stepwise Regression: Stepwise regression is a method that systematically adds or removes predictors based on their statistical significance. This approach can help identify and exclude variables that contribute to multicollinearity, although it should be used with caution due to the risk of overfitting.

Differentiate between Point Estimation and Interval Estimation.

Answer:

Explain the assumptions of a classical regression model.

Answer:

Explain the term ‘Normal Distribution’. What is its significance? Explain its characteristics.

Answer:

• $\mu$$\mu$mu\mu$\mu$ is the mean,
• $\sigma$$\sigma$sigma\sigma$\sigma$ is the standard deviation,
• $x$$x$xx$x$ represents the variable.

1. Central Limit Theorem (CLT): The CLT states that the sum or average of a large number of independent and identically distributed random variables tends to follow a normal distribution, regardless of the original distribution of the variables. This makes the normal distribution a fundamental concept in inferential statistics.
2. Basis for Statistical Methods: Many statistical tests, including hypothesis testing, confidence intervals, and regression analysis, are based on the assumption of normality. This is because the properties of the normal distribution simplify the mathematical calculations underlying these methods.
3. Real-world Phenomena: Numerous natural and social phenomena, such as human heights, test scores, and measurement errors, tend to follow a normal distribution, making it a useful model for understanding and predicting real-world data.

1. Symmetry: The normal distribution is perfectly symmetrical about its mean. This implies that the left and right sides of the distribution are mirror images.
2. Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
3. Bell-shaped Curve: The graph of the normal distribution is bell-shaped, with the highest point at the mean, tapering off symmetrically on both sides.
4. Empirical Rule (68-95-99.7 Rule): Approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in understanding the spread of data around the mean.
5. Asymptotic Nature: The tails of the normal distribution curve approach, but never touch, the horizontal axis, indicating that there is a non-zero probability of observing values far from the mean.

Differentiate between the concept of association and causation.

Answer:

Discuss critical region/s. Explain one-tail and two-tail tests.

Answer:

Which test is employed to compare means of two groups? Explain the test with the help of an example.

Answer:

What do you understand by Sample regression function? Explain with the help of suitable examples.

Answer:

Explain the consequences of specification errors.

Answer:

1. Biased Estimators: If important variables are omitted, the coefficients of the included variables may be biased, as they might capture the effect of the omitted variables.
2. Inconsistent Estimators: Incorrect model specification can lead to inconsistent estimators, meaning that as the sample size increases, the estimators do not converge to the true parameter values.
3. Invalid Inferences: Specification errors can distort standard errors and test statistics, leading to incorrect conclusions in hypothesis testing and confidence intervals.
4. Reduced Predictive Power: The model’s ability to make accurate predictions is compromised, affecting its usefulness in practical applications.