Elasticities in Regression. Elasticity is the percentage change in the dependent variable that results from a percentage change in the independent variable. Elasticity is excellent for reflecting causal relationships and calculating the amount of impact due to changes in certain variables.
There are many kinds of elasticities, including demand elasticity, supply elasticity, production elasticity, and others. Usually, the elasticity reflects the change in the independent variable. For example, demand elasticity means that it reflects changes in demand (the dependent variable) relative to price (the independent variable). Supply elasticity means that it reflects changes in supply (the dependent variable) relative to price (the independent variable). Likewise, the production variable describes the change in production (a dependent variable) with respect to land area, labor, capital, and others (independent variables).
Note: if there is an equation Y = a + bx1 + c X2 + e, then Y is called the dependent variable, X1 and X2 are called independent variables.
Elasticities in Regression: Equation Gradient
The elasticity of an equation is closely related to the gradient of the equation. A gradient describes the slope of a straight-line equation. The greater the gradient value, the more skewed the line will be.
It appears in figure D with a gradient value of 1.5 and has a steeper graph than graph C with a gradient value of 1. The steeper the gradient or slope of the line, the stronger the independent relationship with the dependent variable. This is because the change in Y, or dependent variable, will be greater due to changes in the dependent variable. For example, in graphs D and C, the gradient value of C is 1. This means that if the x value increases by 1 unit, the Y value will also increase by 1 unit. Whereas in graph D’s gradient, if the x value increases by 1 unit, the Y value will increase by 1.5 units.
In addition to the magnitude of the gradient value, we must also understand that positive or negative values on the gradient
Graph A has a positive gradient value, and the resulting graph is right-sloping (rising from left to right). While graph B has a negative gradient value, resulting in a graph that slopes to the left (down from left to right). If the graph has a positive gradient, then the relationship between Y and X is unidirectional; if x goes up, then Y also goes up. If x goes down, then Y will also go down.
Conversely, if the gradient value is negative, then the relationship between Y and X is reversed. If x goes up, Y goes down, and if x goes down, Y goes up.
What is the relationship between gradient and elasticity?
I am deliberately reminding you of one of the basic lessons in junior high school so that you understand elasticity in equations or regression. Elasticity is the same as gradient, which describes the relationship between Y and X, or the independent variable and the dependent variable. However, the fundamental difference between gradient and elasticity is: If the gradient describes the addition or subtraction of the value of Y from the addition or subtraction of the value of X, then the elasticity describes the percentage change in the value of Y due to the addition or subtraction of the percentage of the value of X. So, the elasticity takes into account the percentage, or there is an average value, to get the percentage values of Y and X.
Based on its value, elasticity can be divided into: perfectly inelastic, inelastic, unitary elastic, elastic, and perfectly elastic.
Perfect inelasticity occurs when the elasticity value is 0, meaning that there is no change in the value of Y despite a change in the value of X. This means that x has no effect on the value of Y. An example is the demand for pork by Muslims. Even though the price of pork is as cheap as possible, Muslims still do not buy the product because it is not forbidden to consume it. This means that pork commodities are perfectly inelastic to demand.
The relationship between y and x is said to be inelastic if the elasticity value is 1. This means that the changes that occur in the value of Y will be smaller than the changes that occur in the value of x. There is a great effort to increase the value of Y if you have to use variable X. An example of a case is national rice production compared to the land area of a province. Such a large national production value will have little impact if only the land area of a province is increased. The relationship may be positive, but the value will be very small. The value of an inelastic relationship can be positive or negative. Like the previous discussion on gradients, positive and negative values only explain the direction of the y and x relationships.
Unitary elasticity occurs if the elasticity value is 1. This means that the percentage increase or decrease in the value of Y will be equal to the percentage increase or decrease in the value of X.
New elasticity occurs if the elasticity value > 1, perhaps more precisely, if the absolute value of elasticity = 1 or |e| > 1. Because the elasticity value can be negative or positive. In this category, the percentage change in the value of Y is greater than the percentage change in the value of X.
Perfect elasticity occurs if the value of |e| = ∞. This means that the value of Y will always exist even if there is no value of X. You know how it came? Indeed, only a few cases occur. For example, the elasticity of land supply Whatever the price of land, no matter how expensive it is, the supply will remain. Unless the developer makes an artificial island.
Elasticity in regression
We have come to the end and the crux of this article. How do we obtain elasticity values in regression?
Regression is so popular that almost all statistical analysts know the tool. Regression produces an equation that consists of a constant and a coefficient. The coefficient on each variable is also called the gradient in the previous explanation.
The definition of elasticity is the percentage change in Y due to a percentage change in the value of X. The change in question can be positive (in the same direction) or negative (in the opposite direction), according to the sign of the coefficient in the regression.
The concept of elasticity is used to obtain a quantitative measure of the response of a function to influencing factors (Gujarati, 1995).
If the equation Y = b0 + b1X1 + b2X2, then short-term and long-term elasticity can be formulated as follows:
ESR = (∆Y/∆X) * x̅/Ȳ
Consider (∆Y/∆X) = the gradient or coefficient generated in the regression process.
ELR = ESR / (1-bt)
Let’s practice right away. To clarify this formula, I happened to have used it in the article Price Elasticity and the Effect of Soybean Imports on Domestic Production.
At the initial stage, of course, we have already regressed the equation. In this exercise, I obtained coefficients of 861.28 for X1, 1.04 for X2, and 0.70 for X3 (see row 27).
Then we determine the average for the Y value and the X value (see row 25).
Divide the average x value by the average Y (line 26). Each average x value is divided by the average Y value located in cell Y25.
Determine the SR value by multiplying the coefficient in row 27 with the division value of the average x and y in row 26. It can be seen that X1, although it has a high coefficient of 861.28, is still inelastic because the SR value is <1. This is because elasticity takes into account the percentage of Y value and X value by including the division between the average X and average Y. If the average x value is much smaller than the average Y, then it is likely to be inelastic. Conversely, if the average x value is much larger or closer to the average Y value, it is likely to be elastic. An example is the value of X2 and X3. Although the coefficients are small, they have LR > 1, meaning that in the long run they are elastic.
Or, more easily understood, the value of 861.28 means that if X1 increases by 1 unit, then Y will increase by 861.28, right? But we need to know that this value of 861.28 is only a percentage of the average Y, and it turns out that the percentage is very small. So if we calculate the elasticity value, the value is still below 1.
to get LR, just input the formula as below (cell D29):
The short-run and long-run elasticity values will help you describe or discuss the regression further. So your discussion is not only about R-squared and coefficients but can be extended to the relationship between Y values and X values. The explanation of elasticity will be easy to digest because it raises the percentage instead of the value listed on the coefficient value. This can help illustrate to your readers the variables you used during your research.