6.1 Dataset
How prices of paintings at auction are determined?
What information do we need to estimate the prices of a painting in an auction?
We have data on 430 auction prices for Monet paintings, with data on the dimensions of the paintings and several other variables:
- Price = Sale Price in $ (million),
- Height = Height (inches),
- Width = Width (inches),
- Signed = Binary or Dummy variable = 1 if signed, 0 if not,
- Picture = ID number (identifies repeat sales),
- House = Code for auction house where sale took place.
| PRICE | HEIGHT | WIDTH | SIGNED | PICTURE | HOUSE |
|---|---|---|---|---|---|
| 3.993780 | 21.3 | 25.6 | 1 | 1 | 1 |
| 8.800000 | 31.9 | 25.6 | 1 | 2 | 2 |
| 0.131694 | 6.9 | 15.9 | 0 | 3 | 3 |
| 2.037500 | 25.7 | 32.0 | 1 | 4 | 2 |
| 1.487500 | 25.7 | 32.0 | 1 | 4 | 2 |
| 1.870000 | 25.6 | 31.9 | 1 | 4 | 1 |
| 5.282500 | 25.5 | 35.6 | 1 | 5 | 1 |
| 5.065750 | 26.0 | 34.3 | 1 | 5 | 2 |
| 1.375000 | 25.6 | 36.2 | 1 | 5 | 2 |
| 2.530000 | 25.6 | 36.4 | 1 | 6 | 2 |
| Note: Greene, W. (2018). Econometric Analysis. 8th Edition (First 10 observations) |
- 6 variables: 3 Quantitative + 3 Qualitative.
- 1 dummy variable.
- A unit (painting) can provide more than one observations (price).
6.1.1 What Are Variables?
In statistics, a variable has two defining characteristics:
- A variable is an attribute that describes a person, place, thing, or idea.
- The value of the variable can vary from one entity to another.
For example,
- a person’s hair color is a potential variable, which could have the value of ‘’blond’’ for one person and ‘’brunette’’ for another.
- blood pressure is a variable, because it can vary across people and vary across time for a single person.
- Sex (or gender) is a variable, because it differs across people.
- Political attitude is a variable, because some people are liberals and some people are conservatives.
Qualitative vs. Quantitative Variables
Variables can be classified as qualitative (aka, categorical) or quantitative (aka, numeric).
- Qualitative variables take on values that are names or labels. The color of a ball (e.g., red, green, blue) or the breed of a dog (e.g., collie, shepherd, terrier) would be examples of qualitative or categorical variables.
- Quantitative variables are numeric. They represent a measurable quantity. For example, when we speak of the population of a city, we are talking about the number of people in the city - a measurable attribute of the city. Therefore, population would be a quantitative variable.
Discrete vs. Continuous Variables
Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.
Some examples will clarify the difference between discrete and continuous variables.
- Suppose the fire department mandates that all fire fighters must weight between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter’s weight could take on any value between 150 and 250 pounds.
- Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.3 heads. Therefore, the number of heads must be a discrete variable.
Univariate vs. Bivariate Data
Statistical data are often classified according to the number of variables being studied.
- Univariate data. When we conduct a study that looks at only one variable, we say that we are working with univariate data. Suppose, for example, that we conducted a survey to estimate the average weight of high school students. Since we are only working with one variable (weight), we would be working with univariate data.
- Bivariate data. When we conduct a study that examines the relationship between two variables, we are working with bivariate data. Suppose we conducted a study to see if there were a relationship between the height and weight of high school students. Since we are working with two variables (height and weight), we would be working with bivariate data.