Basic Econometrics (807)
Q.1 Explain the concept of Best Linear Unbiased Estimator (BLUE). Prove that Ordinary Least Square (OLS) estimates are BLUE both in mathematical and matrix form.
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The concept of Best Linear Unbiased Estimator (BLUE) is an important concept in statistics, particularly in the field of econometrics. BLUE refers to a property that an estimator can possess, where it is both linear and unbiased while having the smallest variance among all other linear and unbiased estimators.
To understand BLUE, let's first define the
terms used in the concept:
- Estimator: An
estimator is a rule or formula used to estimate an unknown parameter based on
observed data.
- Linear Estimator: A
linear estimator is one that can be expressed as a linear combination of the
observed data.
- Unbiased Estimator: An
unbiased estimator is one that, on average, gives an estimate that is equal to
the true value of the parameter being estimated.
- Variance: Variance
measures the dispersion or spread of a random variable. In the context of
estimators, it represents the precision or reliability of the estimates.
Now,
let's prove that the Ordinary Least Squares (OLS) estimates are BLUE both in mathematical
and matrix form.
Mathematical Formulation:
------------------------
Consider a linear regression model with
the following equation:
Y = Xβ
+ ε
where
Y is the dependent variable, X is a matrix of independent variables, β is the
vector of unknown coefficients, and ε is the error term.
The OLS estimator is obtained by
minimizing the sum of squared errors:
minimize
Σ(Yi - Xiβ)^2
The OLS estimator can be expressed as:
β_hat
= (X'X)^(-1)X'Y
To
prove that OLS estimates are BLUE, we need to show that they are linear,
unbiased, and have the smallest variance among all other linear and unbiased
estimators.
1. Linearity:
OLS
estimates are linear because they can be expressed as a linear combination of
the observed data. The estimator β_hat is a linear function of Y and X.
2. Unbiasedness:
To
prove that OLS estimates are unbiased, we need to show that E(β_hat) = β, where
E() denotes the expectation operator.
E(β_hat)
= E[(X'X)^(-1)X'Y]
= E[(X'X)^(-1)X'(Xβ + ε)] [substituting Y = Xβ + ε]
= E[(X'X)^(-1)X'Xβ + (X'X)^(-1)X'ε]
= E[β + (X'X)^(-1)X'ε]
= β + (X'X)^(-1)X'E(ε)
= β [since E(ε) = 0]
Therefore,
OLS estimates are unbiased.
3. Minimum Variance:
To
prove that OLS estimates have the minimum variance among all linear and
unbiased estimators, we need to compare their variances with other estimators.
Consider any other linear and unbiased
estimator γ_hat of β. We can express γ_hat as:
γ_hat
= aY + b
where
a and b are constants.
The variance of γ_hat can be expressed as:
Var(γ_hat)
= Var(aY + b)
= a^2Var(Y)
Since we assume that γ_hat is unbiased,
E(γ_hat) = β. Thus, we have:
aE(Y)
+ b = β
Solving for b, we get:
b = β
- aE(Y)
The
variance of γ_hat can now be written as:
Var(γ_hat)
= a^2Var(Y)
= a^2[E(Y^2) - E(Y)^2]
Now,
we can compare the variance of γ_hat with the variance of the OLS estimator
β_hat.
Var(β_hat)
= Var[(X'X)^(-1)X'Y]
= (X'X)^(-1)X'Var(Y)X(X'X)^(-1)
= σ^2(X'X)^(-1)
where
σ^2 is the variance of the error term ε.
To
prove that OLS estimates have the minimum variance, we need to show that
Var(β_hat) ≤ Var(γ_hat) for any other linear and unbiased estimator γ_hat.
Substituting the expression for γ_hat's
variance, we have:
Var(β_hat)
≤ Var(γ_hat)
σ^2(X'X)^(-1)
≤ a^2[E(Y^2) - E(Y)^2]
To
establish the minimum variance property, we need to show that σ^2(X'X)^(-1) ≤
a^2[E(Y^2) - E(Y)^2] holds for all possible values of a.
This
condition is satisfied when a = 1/σ^2, as it minimizes the right-hand side of
the inequality. Therefore, we can conclude that:
Var(β_hat)
≤ Var(γ_hat)
This
proves that OLS estimates have the smallest variance among all other linear and
unbiased estimators, making them BLUE.
Matrix Formulation:
-------------------
The
matrix formulation of OLS allows for a concise representation of the estimators
and the proof of their BLUE properties.
Consider the linear regression model in
matrix form:
Y = Xβ
+ ε
where
Y is an n x 1 vector of observations, X is an n x k matrix of independent
variables, β is a k x 1 vector of unknown coefficients, and ε is an n x 1
vector of errors.
The OLS estimator can be expressed as:
β_hat
= (X'X)^(-1)X'Y
To
prove that OLS estimates are BLUE, we need to show that they are linear,
unbiased, and have the smallest variance among all other linear and unbiased
estimators.
1. Linearity:
OLS
estimates are linear because they can be expressed as a linear combination of
the observed data. The estimator β_hat is a linear function of Y and X.
2. Unbiasedness:
To
prove that OLS estimates are unbiased, we need to show that E(β_hat) = β, where
E() denotes the expectation operator.
E(β_hat)
= E[(X'X)^(-1)X'Y]
= E[(X'X)^(-1)X'(Xβ + ε)] [substituting Y = Xβ + ε]
= E[β + (X'X)^(-1)X'ε]
= β [since E(ε) = 0]
Therefore,
OLS estimates are unbiased.
3. Minimum Variance:
To
prove that OLS estimates have the minimum variance among all linear and
unbiased estimators, we can use the Gauss-Markov theorem.
The
Gauss-Markov theorem states that under the assumptions of the classical linear
regression model (CLRM), OLS estimates have the minimum variance among all
linear and unbiased estimators.
The
assumptions of the CLRM include linearity, strict exogeneity, no perfect multicollinearity,
and homoscedasticity.
Under
these assumptions, the OLS estimator β_hat is the Best Linear Unbiased
Estimator (BLUE) of the coefficients β.
In
conclusion, the Ordinary Least Squares (OLS) estimates are Best Linear Unbiased
Estimators (BLUE) both in mathematical and matrix form. The OLS estimators are
linear, unbiased, and have the smallest variance among all other linear and
unbiased estimators. This property makes OLS a widely used and reliable
estimation method in statistical analysis and econometrics.
Q.2 What
are properties of error term in a simple regression model? What assumption is
made about probability distribution of error term? (20)
In a
simple regression model, which is a basic form of linear regression, the error
term plays a crucial role. The error term represents the unobserved factors
that affect the dependent variable but are not captured by the independent
variable(s). Here, we will discuss the properties of the error term and the
assumption made about its probability distribution.
Properties of the Error Term:
---------------------------
1. Zero Mean: The
error term has a zero mean, E(ε) = 0. This means that, on average, the error
term does not introduce any systematic bias in the model. The positive and
negative errors cancel each other out, and the model is unbiased.
2. Constant Variance (Homoscedasticity): The
error term has a constant variance, Var(ε) = σ^2. This assumption implies that
the spread or dispersion of the errors is consistent across all values of the
independent variable(s). Homoscedasticity is important for obtaining efficient
and unbiased estimators.
3. Independence: The
error term is independent of the independent variable(s) and any other error
terms. This assumption implies that the error term for one observation does not
affect the error term for another observation. Independence ensures that the
errors are not correlated or systematically related to each other.
4. Normality: The
error term follows a normal distribution, ε ~ N(0, σ^2). This assumption is crucial
for statistical inference and hypothesis testing. It allows us to apply various
statistical techniques that rely on the assumption of normality, such as
constructing confidence intervals, conducting t-tests, and performing
hypothesis tests.
Assumption about the Probability
Distribution of the Error Term:
----------------------------------------------------------------
The
assumption made about the probability distribution of the error term in a
simple regression model is that it follows a normal distribution. This
assumption is also known as the Normality assumption or the assumption of
Normally Distributed Errors.
The assumption of normality is essential
for several reasons:
1. Least Squares Estimation: The
ordinary least squares (OLS) method, which is commonly used to estimate the
coefficients in a simple regression model, relies on the assumption of normally
distributed errors. OLS estimators are known to be efficient and have desirable
properties when the errors are normally distributed.
2. Hypothesis Testing: Many
statistical tests, such as t-tests and F-tests, are based on the assumption of
normality. These tests allow us to make inferences about the significance of
the estimated coefficients and the overall fit of the model. Violations of the
normality assumption may affect the validity of these tests.
3. Confidence Intervals:
Constructing confidence intervals around the estimated coefficients also relies
on the assumption of normality. Normality allows us to make probabilistic
statements about the range within which the true population coefficients are
likely to lie.
4. Model Interpretation: When
the errors are normally distributed, the coefficient estimates have an
intuitive interpretation. They represent the expected change in the dependent variable
for a one-unit change in the independent variable(s), assuming all other
factors are held constant.
It is
important to note that while the assumption of normality is commonly made for
simplicity and tractability, it may not always hold in practice. In such cases,
alternative estimation techniques or robust regression methods can be employed
to handle violations of the normality assumption.
In
summary, the error term in a simple regression model is assumed to have certain
properties, including zero mean, constant variance (homoscedasticity),
independence, and normality. The assumption of normality is crucial for valid
inference, estimation, hypothesis testing, and interpreting the results of the
regression model.
Q.3 Let
Ŷ = X(XʹX)-1 XʹY. Find the OLS coefficient from a regression of Ŷ on X. (20)
To
find the OLS coefficient from a regression of Ŷ on X, we first need to define
the variables and understand the notation used.
Variables:
- Ŷ:
The predicted or estimated values of the dependent variable.
- X:
The matrix of independent variables.
- Y:
The vector of observed values of the dependent variable.
Notation:
- Xʹ:
The transpose of the matrix X.
-
(XʹX)^(-1): The inverse of the matrix XʹX.
Given the formula for Ŷ, we can express it
as:
Ŷ =
X(XʹX)^(-1)XʹY
In the
context of OLS regression, we typically have a dependent variable Y and a
matrix of independent variables X. The goal is to estimate the coefficients β
that minimize the sum of squared differences between the observed Y and the
predicted values Ŷ.
To
find the OLS coefficient from a regression of Ŷ on X, we can treat Ŷ as the new
dependent variable and X as the independent variable. We can then apply the OLS
estimation procedure to obtain the coefficients.
Let's
denote the OLS coefficient from this regression as β̂'.
The
OLS estimator β̂' is obtained by minimizing the sum of squared differences
between Ŷ and the observed values of Ŷ, which is equivalent to minimizing the
sum of squared errors. Mathematically, it can be expressed as:
β̂' =
argmin(Σ(Ŷi - Ŷ)^2)
To
find the minimum of this expression, we can take the derivative with respect to
β̂' and set it equal to zero:
d/dβ̂'
(Σ(Ŷi - Ŷ)^2) = 0
Expanding the expression and simplifying,
we have:
d/dβ̂'
(Σ(Yi - X(XʹX)^(-1)XʹY)^2) = 0
Using the
properties of the derivative and matrix algebra, we can further simplify the
expression:
-2XʹY
+ 2XʹX(XʹX)^(-1)XʹY = 0
Simplifying and rearranging terms, we get:
XʹX(XʹX)^(-1)XʹY
= XʹY
Multiplying both sides by (XʹX)^(-1), we
obtain:
(XʹX)^(-1)XʹX(XʹX)^(-1)XʹY
= (XʹX)^(-1)XʹY
Simplifying the left-hand side, we have:
(XʹX)^(-1)XʹY
= β̂'
Thus,
we find that the OLS coefficient from a regression of Ŷ on X is equal to β̂',
which is the same as the OLS coefficient obtained from the original regression
of Y on X.
In
summary, when we regress Ŷ on X using the formula Ŷ = X(XʹX)^(-1)XʹY, the OLS
coefficient obtained from this regression is the same as the OLS coefficient
obtained from the original regression of Y on X. This property holds because
both regressions are based on the same underlying linear model and use the same
OLS estimation procedure to obtain the coefficient estimates.
Q.4 Explain
hypothesis. What is meaning of “accepting” or “rejecting” hypothesis? (20)
In
statistics, a hypothesis is a statement or claim about a population or a
phenomenon that we seek to investigate or test using data. It is a tentative
proposition or assumption that can be either true or false. Hypotheses are
essential for the scientific method and the process of making inferences and
drawing conclusions based on data analysis.
There are two types of hypotheses commonly
used in statistical inference:
1. Null Hypothesis (H0): The
null hypothesis represents the default position or the status quo. It states
that there is no significant difference or relationship between variables, or
no effect of a treatment or intervention. It is typically denoted as H0.
2. Alternative Hypothesis (Ha or H1): The
alternative hypothesis is the opposite of the null hypothesis. It represents
the claim or assertion that contradicts the null hypothesis. It states that
there is a significant difference or relationship between variables, or an
effect of a treatment or intervention. The alternative hypothesis can take
different forms depending on the research question and the nature of the
investigation.
When
conducting a statistical test, the goal is to gather evidence from the data to
support or reject the null hypothesis in favor of the alternative hypothesis.
This process involves making decisions based on the analysis of the data and
applying statistical techniques.
To
evaluate a hypothesis, statistical tests are performed that provide measures of
evidence against the null hypothesis. The outcome of the test leads to one of
two possible conclusions: accepting or rejecting the null hypothesis.
1. Rejecting the Null Hypothesis:
If the
evidence from the data is strong enough to contradict the null hypothesis, we
reject it in favor of the alternative hypothesis. This means that the data
provide support for the claim made in the alternative hypothesis.
When
the null hypothesis is rejected, it suggests that the observed data are
unlikely to have occurred by chance alone assuming the null hypothesis is true.
In other words, there is evidence to suggest that there is a significant
difference, relationship, or effect being investigated.
2. Accepting the Null Hypothesis:
If the
evidence from the data is not sufficient to reject the null hypothesis, we fail
to reject it. This does not mean that we accept the null hypothesis as true or
correct, but rather that there is insufficient evidence to support the
alternative hypothesis.
Accepting
the null hypothesis does not indicate that the null hypothesis is proven true
or that the variables or treatments are equal. It simply means that we do not
have enough evidence to support the alternative hypothesis and, for practical
purposes, we assume the null hypothesis to be true.
It's
important to note that the concept of "accepting" or
"rejecting" a hypothesis is based on the evidence provided by the
data, and it is always subject to uncertainty. Statistical tests assign
probabilities to the outcomes, such as p-values or confidence intervals, which
quantify the strength of evidence against the null hypothesis.
In
summary, a hypothesis is a statement or claim that is investigated or tested
using data. Accepting or rejecting a hypothesis refers to the decision made
based on the analysis of the data. Rejecting the null hypothesis indicates that
there is sufficient evidence to support the alternative hypothesis, while
accepting the null hypothesis implies that there is insufficient evidence to
reject it. These conclusions are based on statistical tests and the evaluation
of the evidence provided by the data.
Q.5 Write
notes on the following: - (20)
a) Two - stage least squares
Two-stage least squares (2SLS) is a
statistical technique used to estimate causal relationships in econometrics
when there is endogeneity or omitted variable bias. It is a method that
addresses the problem of endogeneity by using instrumental variables (IV) to
obtain consistent and efficient estimates of the coefficients.
The
basic idea behind 2SLS is to break down the estimation into two stages. In the
first stage, instrumental variables are used to estimate the endogenous
explanatory variables. Then, in the second stage, the estimated values of the
endogenous variables are used as proxies in the regression analysis to obtain
the final parameter estimates.
Here is an overview of the two stages of
the 2SLS method:
First Stage:
1. Identify endogenous variables: Identify
the explanatory variables that are endogenous, meaning they are correlated with
the error term and potentially biased in the regression analysis.
2. Find instrumental variables:
Instrumental variables are variables that are correlated with the endogenous
variables but not correlated with the error term. They are used to capture the
variation in the endogenous variables that is unrelated to the error term.
3. Estimate the first-stage regression:
Regress the endogenous variables on the instrumental variables to obtain the
predicted values of the endogenous variables. These predicted values are called
the first-stage fitted values.
Second Stage:
1. Include the first-stage fitted values: Use
the first-stage fitted values of the endogenous variables as proxies for the
actual endogenous variables in the main regression analysis.
2. Run the second-stage regression: Include
the first-stage fitted values, along with the exogenous variables, in the
regression model. Estimate the coefficients using ordinary least squares (OLS)
on the modified regression equation.
3. Obtain the final estimates: The
coefficient estimates obtained from the second-stage regression represent the
2SLS estimates, which provide consistent and efficient estimates of the causal
relationships between the independent and dependent variables.
Advantages and Limitations of Two-Stage
Least Squares:
Advantages:
1. Addresses endogeneity: 2SLS
is specifically designed to deal with endogeneity issues, where the independent
variables are correlated with the error term. It provides consistent estimates
even in the presence of endogeneity.
2. Improves efficiency: By
using instrumental variables, 2SLS takes advantage of the additional variation
in the endogenous variables that is unrelated to the error term. This leads to
more efficient estimates compared to ordinary least squares (OLS).
3. Allows for causal inference: 2SLS
helps establish causality by providing estimates that are consistent with a
causal interpretation, under the assumptions of the instrumental variables.
Limitations:
1. Requires valid instruments: The
instrumental variables used in 2SLS must satisfy certain conditions to be considered
valid. They should be correlated with the endogenous variables and have no
direct effect on the dependent variable. Finding appropriate instruments can be
challenging and may require careful consideration and knowledge of the specific
context.
2. Relies on assumptions: 2SLS
relies on certain assumptions, such as the relevance and exogeneity of the
instrumental variables. Violations of these assumptions can lead to biased
estimates. It is important to assess the validity of these assumptions before
applying the 2SLS method.
3. Loss of efficiency: While
2SLS improves efficiency compared to OLS in the presence of endogeneity, it may
still result in loss of efficiency when valid instruments are not available or
when the instrument-weak identification problem occurs.
In
summary, two-stage least squares (2SLS) is a valuable method in econometrics
for estimating causal relationships when endogeneity is a concern. It provides
consistent and efficient estimates by using instrumental variables in a
two-stage estimation process. However, it requires careful consideration of
instrument validity and assumptions and may suffer from limitations related to
instrument availability and potential loss of efficiency.
b) Three
– stage least squares
Three-stage
least squares (3SLS) is an advanced econometric technique used to estimate
simultaneous equation models with endogeneity and interdependence among the
variables. It extends the two-stage least squares (2SLS) method by
incorporating an additional stage to address the issue of simultaneous equation
bias.
Simultaneous
equation models occur when multiple equations are interrelated and jointly
determined. In such cases, the endogenous variables appear on both the
left-hand and right-hand sides of the equations, leading to endogeneity and
biased parameter estimates if not properly addressed.
The
3SLS method involves breaking down the estimation process into three stages,
each building upon the previous stage. Here is an overview of the three stages:
First Stage:
1.
Identify endogenous variables: Identify the endogenous variables in the system
of equations that are simultaneously determined and affected by other variables
in the system.
2. Find instrumental variables:
Select instrumental variables that are correlated with the endogenous variables
but are not correlated with the error terms in any of the equations.
Instrumental variables should satisfy the relevance and exogeneity assumptions.
3. Estimate the first-stage equations: For
each equation, regress the endogenous variables on their respective
instrumental variables and exogenous variables. Obtain the predicted values
(first-stage fitted values) for each endogenous variable.
Second Stage:
1. Include first-stage fitted values: Replace
the endogenous variables in each equation with their respective first-stage
fitted values obtained in the previous stage.
2. Run the second-stage regressions:
Estimate the coefficients for each equation using the modified equations with
the first-stage fitted values and exogenous variables. This stage provides
preliminary estimates of the coefficients.
Third Stage:
1. Include the second-stage residuals: Use
the residuals from the second-stage regressions as additional instrumental
variables in a system of reduced-form equations.
2. Run the third-stage regressions:
Regress the endogenous variables on the second-stage residuals, along with the
exogenous variables, to obtain the final estimates of the coefficients. These
estimates are known as the three-stage least squares estimates.
Advantages and Limitations of Three-Stage
Least Squares:
Advantages:
1. Addresses simultaneity and endogeneity: 3SLS
is specifically designed to handle endogeneity and simultaneity problems in
simultaneous equation models. It provides consistent estimates by addressing
the interdependence among the endogenous variables.
2. Efficient estimates: By
incorporating the second-stage residuals as additional instrumental variables,
3SLS utilizes the information contained in the residuals, leading to more
efficient estimates compared to 2SLS.
3. Allows for causal inference:
Similar to 2SLS, 3SLS facilitates causal inference by providing consistent
estimates that support causal interpretation under certain assumptions.
Limitations:
1. Valid instruments: As
with 2SLS, 3SLS requires valid instrumental variables that satisfy the
relevance and exogeneity assumptions. Ensuring the availability and appropriateness
of instrumental variables can be challenging.
2. Assumptions and identification: 3SLS
relies on assumptions such as instrument validity, no misspecification of
equations, and correct model specification. Violations of these assumptions can
lead to biased estimates. Additionally, identification issues may arise if the
number of instruments is limited relative to the number of endogenous variables
and equations.
3. Computational complexity: The
estimation process in 3SLS involves multiple stages and calculations, making it
computationally intensive and potentially time-consuming, particularly for
large-scale models.
In
summary, three-stage least squares (3SLS) is a powerful econometric technique
for estimating simultaneous equation models with endogeneity and
interdependence among the variables. It extends the 2SLS method by
incorporating an additional stage and utilizing second-stage residuals as
instrumental variables. While 3SLS addresses simultaneity and endogeneity, it
requires careful consideration of instrument validity, assumptions, and
identification issues. The computational complexity of the method should also
be taken into account when applying it to complex models. Dear Student,
Ye sample assignment h. Ye bilkul
copy paste h jo dusre student k pass b available h. Agr ap ne university
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0334-6483019
0343-6244948
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