Econometrics Lecture 7: Functional Forms

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Questions and Answers

In the linear-log population regression function, how does a 1% change in X affect Y?

Y increases by 100 times the change in X.

Y changes by β1/100.

It results in a change in Y equal to β1.

It leads to a β1 change in Y/100. (correct)

What does the term β1 represent in the log-linear population regression function?

The percentage change in Y due to a unit change in X. (correct)

The ratio of change in Y to change in X.

The constant term in the regression equation.

The total change in Y for a given change in u.

How is β1 calculated in the linear-log population regression function?

β1 = ∆Y/Y.

β1 = ∆Y/100.

β1 = ∆Y/∆X.

β1 = ∆X/X. (correct)

In the log-linear population regression function, what does a change in X by 1 unit result in?

A 100 × β1 % increase in Y. Signup and view all the answers

What approximation is used in the linear-log and log-linear regressions for small changes?

log(Y + ∆Y) is approximately equal to ∆Y. Signup and view all the answers

What does a 1-unit increase in X1i represent in the linear regression model Yi = β0 + β1 X1i + ui?

It corresponds to a β1-unit increase in Yi. Signup and view all the answers

Which expression accurately reflects how to express compounding percent changes in economic terms?

$Y = (1 + R)^t$ Signup and view all the answers

How does the interpretation of β1 change when transforming Yi by taking the log?

It indicates the percentage change in Yi for a given unit change in X1i. Signup and view all the answers

What could the approximate equivalence of 1.2 relative to 0.7 imply in terms of percent changes?

It represents approximately 54 different 1% increases compounded. Signup and view all the answers

In the context of log transformations in regression, which model properly reflects a log-transformed dependent variable?

$log(Yi) = β0 + β1 log(X1i) + ui$ Signup and view all the answers

What does the notation $f (.)$ represent in the context of conditional expectation?

The expected value of Y given the X's Signup and view all the answers

In the approximation of the population regression function using polynomials, what form does the equation take?

$Yi = β0 + β1 Xi + β2 Xi2 + ... + βr Xir + ui$ Signup and view all the answers

What is one advantage of using logarithmic transformations in regression analysis?

It provides a direct interpretation of percentages Signup and view all the answers

Which of the following is a characteristic of polynomial regression as compared to linear regression?

Allows for curvilinear relationships Signup and view all the answers

What does the symbol $ui$ typically represent in a regression equation?

The error term Signup and view all the answers

Which method is NOT mentioned as an approach to approximate the population regression function?

Time series analysis Signup and view all the answers

What should guide the choice of the functional form in regression analysis?

Judgment, tests, and plotting predicted values Signup and view all the answers

In the equation $TestScorei = β0 + β1 Incomei + β2 Income2i + ui$, what does the term $Income2i$ represent?

The non-linear effect of income on test scores Signup and view all the answers

What is the consequence of using a linear regression when the relationship between Y and X is nonlinear?

The estimator of the effect on Y of X is biased. Signup and view all the answers

What does an F-statistic test in the context of joint hypotheses?

It tests multiple restrictions on the coefficients simultaneously. Signup and view all the answers

Which statement correctly describes the process of regression specification?

The base specification is modified to account for omitted variable bias. Signup and view all the answers

Which of the following is TRUE about the marginal effect of X when the relationship is nonlinear?

It varies depending on the values of X. Signup and view all the answers

What is the correct form of the general nonlinear population regression function?

$Y_i = f(X_1i, X_2i, ..., X_ki) + u_i$ Signup and view all the answers

What happens if a specification with the highest R-squared is chosen without further analysis?

The model may not estimate the causal effect accurately. Signup and view all the answers

What is the key assumption regarding the error term in regression?

The expected value of the error term conditioned on independent variables is zero. Signup and view all the answers

The use of the term 'omitted variable bias' refers to which issue in regression modeling?

Neglecting to include relevant variables leading to inaccurate results. Signup and view all the answers

What does the regression equation for D1 = 0 imply about the relationship between Y and X1i?

Y increases linearly with X1i. Signup and view all the answers

What does β3 represent in the fully interacted model?

The interaction effect of D1 and X1i on Y. Signup and view all the answers

What is the main purpose of the Chow Test in this context?

To test if the interaction between variables is significant. Signup and view all the answers

When β1 = 0, what does that indicate concerning the two regression lines?

They have the same intercept. Signup and view all the answers

In which situation would one prefer non-linear models over linear models?

When economic relationships are complex. Signup and view all the answers

Which statement about the relationship between X and Y in linear models is correct?

The relationship between X and Y is constant and fixed. Signup and view all the answers

How does the demand for luxury goods typically behave in relation to a rise in income?

Demand increases at an accelerating rate. Signup and view all the answers

Which hypothesis is tested by computing the t-statistic for β3 = 0?

The two lines have the same slope. Signup and view all the answers

What is the expected outcome $E(Y | D1i = 0, D2i = 0)$ when both D1i and D2i are equal to 0?

$β0$ Signup and view all the answers

In a regression with one dummy variable D1i and one continuous variable X1i, how is the intercept affected when D1i equals 1 compared to when it equals 0?

The intercept increases by $β2$ Signup and view all the answers

What does $β3$ represent in the model $E(Y | D1i = 1, D2i = 1)$?

The combined effect of D1 and D2 when both are 1 Signup and view all the answers

How does the slope of the regression line change when D1i is equal to 1 compared to when it is equal to 0?

The slope is affected by both $β1$ and $β2$ Signup and view all the answers

In the fully interacted model $Yi = β0 + β1 D1i + β2 X1i + β3 (D1i × X1i) + ui$, what happens to the intercept?

It is determined solely by $β0$ Signup and view all the answers

What effect does $D1i × X1i$ have on the model compared to the simpler model without interaction?

It modifies the slope based on D1i's value Signup and view all the answers

What is the expected outcome $E(Y | D1i = 1, X1i)$ in a model with interaction $Yi = β0 + β1 X1i + β2 D1i + ui$?

$β0 + β2 + β1 X1i$ Signup and view all the answers

In the context of regression analysis involving dummy variables, which of the following statements is true regarding interactions?

Interactions always increase the number of parameters in the model. Signup and view all the answers

Study Notes

Lecture 7: Functional Forms

Lecture for 25117 - Econometrics
Held at Universitat Pompeu Fabra
On November 11th, 2024

What We Learned Last Time

Hypothesis tests and confidence intervals for a single regression coefficient are conducted similarly to simple linear regression models (e.g., $β₁ ± 1.96SE(β₁)$ for a 95% confidence interval)
Joint hypotheses, involving multiple restrictions on coefficients, are tested using F-statistics, which relate to Fq,n-k-1 distributions.
Regression specification involves initially choosing a base specification to address omitted variable bias. The base specification can be modified by adding control variables for omitted variable bias.
Simply selecting the specification with the highest R² might not ensure estimation of the target causal effect.

Introduction

The regression function (so far) has been linear in X's, but this linear approximation isn't always optimal, leading to potential issues with misspecified models and biased estimations.
Multiple linear regression can accommodate non-linear functions of one or more X-variables.
Non-linear relationships between Y and X means the marginal effect of X depends on the value of X.
Linear regression is inappropriate for non-linear Y-X relationships. This leads to biased estimator results when the relationship between variables is non-linear.

A Linear Fit on a (Possibly) Linear Relationship

Data points display scatter with roughly a negative linear association.
The equation of the fitted line, TestScore ~ 698.9 - 2.28 × STR.

A Linear Fit on a Non-Linear Relationship

The data points display some scatter with a positive association that is not linear. Data points are suggestive of a non-linear relationship.

The General Nonlinear Population Regression Function

The general nonlinear population regression function is defined as $Y_i = f(X_{1i}, X_{2i}, ..., X_{ki}) + U_i$, where $E(U_i | X_{1i}, X_{2i}, ..., X_{ki}) = 0$.
Errors are independent and identically distributed (i.i.d.).
Outliers are rare.
No perfect multicollinearity.

Non-linear Functions of a Single Independent Variable (X)

Polynomials in X
Approximate using quadratic (2nd degree polynomial), cubic (3rd degree), or higher-degree polynomial functions.
Logarithmic Transformations
Y and/or X is transformed via logarithm, which aids in interpreting the relationship.

Polynomials in X

The data is modeled as a polynomial.
The example is quadratic, with Y = $β_0 + β_1X_i + β_2X_i^2 + u_i$.

Logarithmic Functions of Y and/or X

Approximating using the Taylor's expansion: log(x) ≈ log(x)_{x=a} + (x-a) / X_{x=a}
(Assuming a = 1, log(x) ≈ x - 1)
Compounding percentage changes.
Logarithmic transformations are used for a richer economic interpretation of variables (like interest rates, inflation, and population growth).

Linear/Log Population Regression Function

For $Y_i = b_0 + b_1 log(x_i) +u_i$, a 1% change in X is associated with a $b_1/100$ change in Y.

Log-Linear/Log-Log Population Regression Function

Similar interpretations to linear-log, but with a different interpretation based on the type of transformation applied (log on right vs log on left)

Interactions and Heterogeneous Effects

In economic models, focusing on the impact of a factor (e.g., the share of subsidized meals) might not be sufficient. It's critical to understand the heterogeneous/conditional effects of this factor, often depending on other factors (e.g., on different levels of education attainment).

Interactions Between Two Dummies

The introduction of interaction between two or more dummy variable(s).
The specification allows different intercepts and/or slopes for different combinations of dummies (e.g., a different impact when both the dummy variable and interaction factor are 1)

Interactions Between a Dummy and a Continuous Variable

The effect of a continuous variable can be different based on the value of a dummy variable.
The slopes or intercepts may vary, hence the interaction effects.

Interactions Between Two Continuous Variables

The effect of one continuous variable depends on the value of another continuous variable.
This introduces interaction effects within the model.

Wrapping up: Linear vs. Non-Linear Models

Linear models assume a constant relationship between variables, but economic relationships are rarely entirely linear.
Non-linear models allow for more flexible interpretations of relationship between economic variables.
In economic contexts, ignoring non-linear relationships can lead to inaccurate calculations and conclusions.

The Importance of Interactions

Recognizing the interaction effects between variables adds nuance to economic analysis and enhances modeling of real-world economic phenomenon.
The right functional form can impact the accuracy of predictions and have implications for policies.

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Description

This quiz covers Lecture 7 of the Econometrics course (25117) at Universitat Pompeu Fabra. It focuses on hypothesis testing, confidence intervals, F-statistics, and regression specifications. Understand the importance of selecting the correct functional forms to avoid omitted variable bias and misspecification in regression analysis.