All right. We're going to continue with our discussion of regressions. Remember, in this last set of data that we had was a very simple regression and the data were arranged in a very very pretty way. Here we have some more complex data. It's not super complex, but it's a little bit more complex. Here I have the, it's what I call the ice cream data. Again, this will be available to you to download and to try to replicate what I'm doing here. I've got the quantity of ice cream that's sold in various carts in Central Park, for each month, for three years, starting in 2001, 2002, and 2003. I have the average price that ice cream was being sold for during this period of time, the average temperature for the month, and the number of conventions that were within five miles of Central Park during that month that this data was being taken place. So I call this the ice cream data. So we're going do the same thing that we did before. I'm going to create a scatter plot between price and quantity to get an idea about how these data are acting together. Recall that I highlighted price first, and I'm going to do that again here, and insert a scatter plot. Now, you'll notice that when I do this, it's just you get this sort of snaky trend of data. That's because there's 36 observations here, and the sale of ice cream goes down and it goes back up and it goes down and goes back up, right? But what I really want to do is, I really want to understand the relationship between the price and the quantity of ice cream. So what I'm going to do here is, I'm going to click on this chart and I'm going to select some data, and I'm going to add as my x variables here, some quantity, the corresponding elements of quantity. Now I want to make this chart a little bit larger for you here. Now you'll notice that this information is not nearly as pretty as the previous demand schedule. That's because this is data from industry. This is data from every walk of life. There is a relationship between quantity and price here, is a downward sloping relationship here. So what happens is, as price gets lower, quantity demanded gets higher, as price gets higher, quantity demanded it gets lower. There's some kind of a line that's here, but what I want to do is, I want to try to figure out what the relationship is here. The relationship in this instance is, I want to know how does the quantity of ice cream relate to the price of ice cream. So, I'm going to estimate quantity as a function of price. So I'm going to use my regression technique to do this. So again, I go into my data analysis, I click on my regression here. Now, for my different variables here, I'm going to start by saying look, my y variable is the variable that I'm interested in trying to identify. So say, "hey, what do I want to know?" I want to know about the quantity of ice cream. That's my y variable, what we call a dependent variable. So the dependent variable is, this thing depends on something else. So, the quantity of ice cream depends on price. Quantity of ice cream depends on lots of things. So our y variable, in this case our dependent variable, is going to be quantity. So I'm going to highlight my quantity information here. Then my x variable here is price. Let say quantity is a function of price. Highlight the information in price here. Again, I'm going to click on Labels. Make sure Labels is clicked and we have this. This information is going to be put into a new worksheet. I'm going to click Okay. This is what I get. Okay. How do I interpret this? I can write out a function here, and I'm going to type it out. Quantity is 552.65 minus 128.12 times price. That's the linear equation. That's the equation that describes the relationship between price and quantity. So right here, I've just constructed what's called a demand function. I've used it using a regression technique. How do I interpret these coefficients in the demand function? Here, my coefficient of intercept is an intercept. So, this right here, this 552.65. I say if price were zero, I would consume 552 ice creams or the people would consume 552 ice creams, depending on how this data look. The coefficient of price is negative 128.12. How do I interpret this? I say, for every one dollar increase in price, quantity demanded decreases by 128.12 units. So if price goes up by a dollar, quantity demanded goes down by 128 and change. So you'll notice there's a negative relationship between price and quantity. This coefficient in front of price, will be referred to by most of your professors as Beta. They'll say, what's the Beta of price? What they're asking for is, what's the coefficient in front of your price variable. They might refer to this, the intercept here as what's your Alpha. So the general form of the equation will be quantity is equal to Alpha plus Beta price. Now, I can say Beta plus Beta price, even though my beta is negative. So when I'm describing any kind of relationship, I might describe it very generically and say y is a function of Alpha plus Beta of this variable, plus Beta two of this variable, plus Beta three of this third variable, and what have you. So we're going to get into multiple regression quite quickly, but for the first part here, it was noticed that quantity is a function of price, quantity is Alpha minus Beta price. Quantity is 552.65 minus 128.12 times price. That's how I identify and interpret these coefficients. Now, are they meaningful? Do they mean anything? In this case, what we'll do is we would look at what's called the t-statistic. The t-statistic is a statistic that captures the ratio of the coefficient to the standard error, what people refer to as the signal to the noise. So right down here, I'm going to show you that the t-statistic is in fact the coefficient and the standard error ratio. So say, let's suppose we take this coefficient here and divide it by this standard error right here, we get a number, 866. Now notice this number right here, 866, and this number, 0.866, they are the same number. That's how this t-statistic was calculated. You take the coefficient divided by the standard error. Again, the same, this t-statistic over here will be the coefficient divided by the standard error. Should I be worried about this number? Should I be paying attention to it? Yeah, the truth is that most of your professors are going to insist that you're looking at your t-statistics and the corresponding p-values, associated with those t-statistics. The t-statistic is telling you in some sense that the coefficient, 552 or negative 128, is believable, is really different than zero. If you're t-statistic, that is the ratio of the coefficient to the standard error, for a large number of observations is plus or minus 1.96, then what that's saying is that you can believe that the coefficient is not zero about 95 percent of the time. So it's either in one tail or the other tail, 95 percent of the time. Five percent of the time, maybe it might not be 552 or might not be negative 128. So what we're doing is that's what that means. The t-statistic is the ratio of the coefficient of the standard error. It corresponds to a p-value. As your t-statistic gets larger, the p-value gets smaller. Your professors are going to probably focus from time to time on what's the value of that t that tells you that the variable is significant. That is, essentially there is a statistical test that says that that coefficient is statistically different than zero. So your professors will say things like, is there a variable significant, is that variable insignificant, how significant is that variable, what have you. What they're asking for is, what's going on with that t-statistic, the ratio of the signal to the noise? The larger the signal to the noise plus noise there is, the more believable that coefficient is. So the takeaway here is, how do we run a regression on some data? We did that. How do we interpret just generally the output from the regression? We've got coefficients, we have standard errors, we have t-statistics. How do I interpret that? Now, the usefulness of this is going to be really dependent on the subject area and the background, the case that you're working on and what have you. So, your professors might be having you run data and trying to identify relationships between variables, something that's driving consumer demand, let's say, or something that's driving earnings of a particular group. In other cases, you'll be given an opportunity just to look at the output and then you'll have to make some assessments regarding what is the value of this analysis? Is there significant drivers or significant correlations between these variables. Being able to interpret coefficients along with their t-statistics, will become a very very useful tool throughout your MBA career.
