Econometrics Lecture 7: Functional Forms

The regression function we've used so far, where the relationship between the dependent variable (Y) and independent variables (X) is a straight line.

Introducing non-linearity into the regression function, allowing for relationships that are not straight lines.

The effect on the dependent variable (Y) from a change in an independent variable (X), where this effect varies depending on the value of X.

When a linear regression equation is used to model a non-linear relationship, it leads to an inaccurate representation of the true relationship.

Estimates of the effect of an independent variable (X) on the dependent variable (Y) are systematically biased when the functional form is misspecified.

The process of determining the appropriate relationship between variables in a regression model, ensuring it accurately reflects the underlying data.

Adding more independent variables to the model to control for other potential factors that could influence the dependent variable.

Using the R-squared value to select the regression model with the highest explanatory power, which might not always lead to the best model for causal analysis.

The expected difference in Y associated with a difference in X1, keeping other variables constant.

The population regression function is approximated by a quadratic, cubic, or higher-degree polynomial.

The power of X is the regressor. It is a linear multiple regression model.

Y and/or X is transformed by taking its logarithm. This provides an interpretation of the coefficients in terms of percentages (elasticities and semi-elasticities).

The choice of the functional form (linear, polynomial, logarithmic, etc.) in regression analysis should be guided by judgment, tests, and visualization of predicted values.

The coefficients of the polynomial regression model are difficult to interpret directly, but the regression function itself provides an interpretable relationship between the variables.

Plotting predicted values can help visualize the relationship between X and Y and assess the fit of the chosen functional form.

Regression models should not include variables that are perfectly collinear. This can lead to issues with estimating the model.

The relationship between a 1% increase in a variable (X) and the corresponding change in another variable (Y). It involves a compounded percent change, where the effect of the change is multiplied over time.

A way to represent a compounded percent change using logarithms. It involves taking the difference of the logarithms of the observed variable at two different points in time.

A model that assumes a constant linear relationship between the independent and dependent variables. However, this model may not always be accurate in representing real-world relationships.

Transforming variables in a regression model by using logarithms. Log transformations are often used when the relationship between variables is linear.

The impact of a one-unit change in an independent variable on the dependent variable in a regression model. This effect can be constant or vary with the value of the independent variable.

The effect of a 1% change in X on Y is given by β1/100.

The effect of a 1 unit change in X on Y is given by 100 × β1%.

In the linear-log regression model, the slope coefficient (β1) represents the change in Y for a 1% change in X.

In the log-linear regression model, the slope coefficient (β1) represents the percentage change in Y for a 1-unit change in X.

When the independent variable (X) is transformed using the natural logarithm, the regression model becomes linear-log. This transformation allows for the interpretation of the coefficient (β1) in terms of percentage changes.

The expected value of Y when both D1 and D2 are equal to 0, representing the baseline outcome.

The expected value of Y when D1 is 1 and D2 is 0. This represents the effect of D1 alone on Y, assuming D2 is not present.

The expected value of Y when D1 is 0 and D2 is 1, representing the effect of D2 alone on Y, assuming D1 is not present.

The expected value of Y when both D1 and D2 are 1. This represents the combined effect of D1 and D2 on Y, including any potential interaction between them.

The additional effect on Y caused by the combined presence of both D1 and D2, after accounting for the individual effects of each dummy variable.

The regression line when D1 is 0 is a straight line with slope β1, while the regression line when D1 is 1 is also a straight line but with a different slope (β1 + β2). The intercepts of both lines are the same.

The regression line when D1 is 0 is a straight line with slope β1, while the regression line when D1 is 1 is also a straight line but with a different intercept (β0 + β1) and slope (β1 + β2).

The regression line when D1 is 0 is a straight line with slope β1, while the regression line when D1 is 1 is also a straight line but with a different intercept (β0 + β2). The slopes of both lines are the same.

When D1 = 0, the regression line's intercept is β0 and its slope is β2. When D1 = 1, the intercept becomes (β0 + β1) and the slope becomes (β2 + β3). This means the relationship between Y and X1 is different depending on the value of D1.

A method to test if there is a significant difference in the relationship between Y and X1 for two groups defined by D1.

The Chow Test examines if the lines for the two groups are the same by testing the joint hypothesis that β1 and β3 are both equal to zero.

The t-statistic is used to specifically test if the slopes of the two lines are equal by examining if β3 is equal to zero.

The t-statistic is used to test if the intercepts of the two lines are equal by examining if β1 is equal to zero.

Linear models assume a constant relationship between variables, while non-linear models allow for more flexible and nuanced relationships. Non-linear models are better suited for capturing the complexities of real-world economic relationships.

In the context of economic phenomena, the relationship between variables can be non-linear, meaning that changes in one variable might not lead to proportional changes in another variable. For example, the demand for luxury goods might not increase linearly with income.

Non-linear models allow for more realistic representations of complex economic relationships by capturing variations in the relationship between variables, which linear models cannot.