UGC NET Economics Unit 3 -Statistics and Econometrics

1. Probability Theory

Concept of Probability:

Probability measures the likelihood of an event occurring.
It always lies between 0 and 1.

$$P(A)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$

• 0 → Impossible event

• 1 → Certain event

Types of Probability:

1. Classical – based on equally likely outcomes (e.g., coin toss).

2. Empirical – based on past data (e.g., rainfall probability).

3. Subjective – based on personal judgment.

🔹 Important Concepts:

• Independent Events: Occurrence of one doesn’t affect another.

• Mutually Exclusive Events: Cannot occur simultaneously.

• Conditional Probability:

  $$P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$$

• Bayes’ Theorem: Used for revision of probabilities based on new information.

2. Probability Distributions

Discrete Distributions:

1. Binomial Distribution:

   $$P(x)=\left(\genfrac{}{}{0px}{}{n}{x}\right)p^x{q}^{n-x}$$

   Used for success-failure experiments.

2. Poisson Distribution:
   Used when events are rare and independent (e.g., accidents).

   $$P(x)=\frac{e^{-\lambda }{\lambda }^x}{x!}$$

🔹 Continuous Distribution:

1. Normal Distribution:
   Bell-shaped curve; symmetric around mean.
   Mean = Median = Mode.
   Used in sampling, hypothesis testing, etc.

3. Moments and Central Limit Theorem

🔹 Moments:

Moments describe shape of a distribution.

• 1st moment → Mean

• 2nd moment → Variance

• 3rd moment → Skewness (asymmetry)

• 4th moment → Kurtosis (peakedness)

🔹 Central Limit Theorem (CLT):

As sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of population distribution.

💡 This theorem justifies the use of normal probability in statistics.

4. Descriptive Statistics

🔹 Measures of Central Tendency:

• Mean (Average):

  $$\bar{X}=\frac{\sum X_i}{n}$$

• Median: Middle value when data arranged in order.

• Mode: Most frequent value.

🔹 Measures of Dispersion:

Indicate how data values are spread around the mean.

• Range

• Variance

• Standard Deviation

• Coefficient of Variation (CV)

🔹 Correlation:

Shows relationship between two variables.
Karl Pearson’s coefficient:

$$r=\frac{\sum (X-\bar{X})(Y-\bar{Y})}{\sqrt{\sum (X-\bar{X}{)}^2\sum (Y-\bar{Y}{)}^2}}$$

Values of r lie between -1 and +1.

🔹 Index Numbers:

Measure changes in price, quantity, or value over time.
Types:

• Price Index (e.g., CPI, WPI)

• Quantity Index

• Value Index

Formulas:

• Laspeyres Index: Base year weights

• Paasche Index: Current year weights

• Fisher’s Index: Geometric mean of the two (Ideal index)

5. Sampling Methods & Sampling Distribution

🔹 Sampling Methods:

1. Random Sampling – every unit has equal chance.

   • Simple Random

   • Stratified Random

   • Systematic Sampling

   • Cluster Sampling

2. Non-random Sampling – convenience or judgment-based.

🔹 Sampling Distribution:

Distribution of a statistic (like mean) from repeated random samples.
Used to estimate population parameters.

Standard Error (SE) = Standard deviation of a sampling distribution.

6. Statistical Inference and Hypothesis Testing

🔹 Estimation:

• Point Estimate: single value (e.g., sample mean).

• Interval Estimate: range of values (confidence interval).

🔹 Hypothesis Testing Steps:

1. State Null (H₀) and Alternative (H₁) hypotheses

2. Choose significance level (α)

3. Select appropriate test statistic (Z, t, χ², F)

4. Define rejection region

5. Calculate test statistic

6. Accept or reject H₀

🔹 Common Tests:

• Z-test: Large samples (n > 30)

• t-test: Small samples

• χ²-test: Goodness of fit or independence

• F-test: Compare two variances

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7. Linear Regression Models

🔹 Simple Linear Regression:

$$Y=\alpha +\beta X+u$$

where

• Y = Dependent variable

• X = Independent variable

• u = Random error term

🔹 Properties of OLS (BLUE):

OLS estimators are Best Linear Unbiased Estimators when:

1. Linear in parameters

2. Expected value of error = 0

3. Homoscedasticity (constant variance)

4. No autocorrelation

5. No perfect multicollinearity

6. Errors are normally distributed

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8. Identification Problem

Occurs in simultaneous equation systems when parameters cannot be uniquely estimated.

Identification Types:

• Under-identified: Insufficient restrictions → No unique solution

• Exactly identified: Just enough restrictions → Unique solution

• Over-identified: More restrictions than needed → Multiple estimates

9. Simultaneous Equation Models

🔹 Recursive Models:

• Equations arranged in sequence

• No feedback

• Can be solved by OLS

🔹 Non-Recursive Models:

• Feedback present (mutual dependence)

• Require Two-Stage Least Squares (2SLS) or Instrumental Variables (IV) for estimation.

10. Discrete Choice Models

Used when dependent variable is categorical (0/1, yes/no).

Types:

• Logit Model – uses logistic function

• Probit Model – uses cumulative normal distribution

Example: Probability of employment, adoption of technology, etc.

11. Time Series Analysis

Components of Time Series:

1. Trend (T): Long-term direction.

2. Seasonal (S): Regular pattern within a year.

3. Cyclical (C): Long-term up and down movements (business cycles).

4. Irregular (I): Random variations.

Models:

• Additive Model: $Y=T+S+C+I$

• Multiplicative Model: $Y=T\times S\times C\times I$
  

Stationarity:

A series is stationary when mean, variance, and covariance remain constant over time.

Autocorrelation:

Measures correlation between current and past values of a series.

 AR, MA, ARMA, ARIMA Models:

Used for forecasting and economic time series modeling.

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