Regression: An Introduction to Econometrics Overview The goal in the econometric work is to help us move from the qualitative analysis in the theoretical work favored in the textbooks to the quantitative world in which policy makers operate. The focus in this work is the quantification of relationships. For example, in microeconomics one of the central concepts was demand. It is one half of the supply-demand model that economists use to explain prices, whether it is the price of stock, the exchange rate, wages, or the price of bananas. One of the fundamental rules of economics is the downward sloping demand curve - an increase in price will result in lower demand. Knowing this, would you be in a position to decide on a pricing strategy for your product? For example, armed with the knowledge that demand is negatively related to price, do you have enough information to decide whether a price increase or decrease will raise sales revenue? You may recall from a discussion in an intro econ course that the answer depends upon the elasticity of demand, a measure of how responsive demand is to price changes. But how do we get the elasticity of demand? In your earlier work you were just given the number and asked how this would influence your choices, while here you will be asked to figure out what the elasticity figure is. It is here things become interesting, where we must move from the deterministic world of algebra and calculus to the probabilistic world of statistics. To make this move, a working knowledge of econometrics, a fancy name for applied statistics, is extremely valuable. As you will see, this is not a place for the meek at heart. There are a number of valuable techniques you will be exposed to in econometrics. You will work hard on setting up the 'right experiment' for your study, collecting the data and specifying the equation. Fortunately, this is only the beginning. There will never be the magic button that produces 'truth' at the end of some regression, the favorite econometric technique for estimating relationships. You can also be assured you will not get it quite right the first time. There is, however, something to be learned from your 'mistakes'. To the trained eye, the summary statistics produced by any regression package paint a vivid, if somewhat blurred picture, of the problems with the model as specified. These are problems that must be dealt with because they can produce biases in the results that reduces the reliability of the regression and increases the chance we will not end up an understanding of the true relationship. With existing software packages, anyone can produce regression results so one needs to be aware of the limitations of the analysis when evaluating regression results. In this overview of econometrics we will begin with a discussion of Specification. What equation will we estimate? Does demand depend upon price alone, or does income also matter? Is demand linearly or nonlinearly related to price? These are the types of questions discussed in this section. We will then shift to Interpretation, a discussion of how to interpret the results of our regression. What if we find out demand is negatively relate to price? Should we believe the result? And what about the times where demand turns out to be positively related to price. How could we explain this result and do we actually have proof demand curves should be positively sloped. This will be followed by a discussion of the assumptions of the Classical Linear Model, all of the things that must go right if we are to have complete confidence in our results. And for those instances where we have some reason to believe there is a problem, we have a discussion of the Limitations of the Classical Linear Model where the potential problems as well as solutions are discussed. When you have completed this section, you should be well aware of the fact the estimation of 'economic relationships' has both an art and a science component. Given the technology available to people today, 1 anyone can run regressions with the use of some magic buttons. Computer programs exist that allow us to estimate the regressions, perform diagnostics to evaluate the model, and correct any problems encountered. Do not, however, be misled into thinking your empirical work will be easy. As you will find with your own work, there is a long road of painful, time-consuming work ahead of anyone who embarks on an empirical project. Furthermore, there are many places where you can take a wrong turn. This section was designed to offer you some guidance as you make the journey, to help you know in advance the obstacles you are likely to encounter and the best way of dealing with them. There is a second reason for spending the time studying regression analysis and conducting your own empirical project. The scientific advances are not a guarantee we are more likely to uncover the 'truth' that we are searching for. The world is in many respects the same as it was when was prompted to write his wonderful little book entitled, How to Lie With Statistics. In the hands of an unscrupulous researcher, the modern econometric software increases the chances someone can find the results they want. The complexities of the statistical analysis simply make it harder to find the biases in the study. Your time spent here will simply increase the chances of recognizing the biases. For an on-line overview of regression analysis you might want to check out the DAU and Stockburger sites. You should also check out the worksheet Regression, the output from an excel regression. The data on sheet simple is for years, inflation rate, unemployment rate, and interest rate appear in cells A3 - D50. Once the data set is complete, you then select Data Analysis in the Tools menu. You will then select Regression, which will bring up a dialogue box. At this time you highlight the data set for the input box. The Y variable is the variable you want to explain, in this case and it is the interest rate. The X variable is the explainer, in this case the inflation rate. We are going to use regression to see the extent to which the inflation rate explains interest rates. You then specify the top left cell of the space where you want the output to appear. For an interpretation of the results, you should check out the Interpretation page. In these results you find the coefficient of inflation to be .68 - every time the inflation rate rises by one percentage point, interest rates rise by nearly .7 percent. The t-statistic is 7.22, which indicates you should believe in this relationship, and the R2 tells you the model helps explain about one half the variation in interest rates. Mechanics Once you have decided on estimating a relationship using regression analysis, you need to decide upon the appropriate software package. There are some very useful software packages designed primarily for regression type analysis you may want to explore if you were doing some high powered regression work or you were using the software in other courses. Here, however, we will stick with Excel that allows you to run some simple regression analyses. The first step is creation of the data set, an example of which can be found on the simple tab on the Regression spreadsheet example. On the simple tab example we will be looking at a bivariate regression - a regression with only one right-side variable. The estimated equation will be of the form Y = a + bX + e, where Y is the variable being explained (dependent) and X is the variable doing the explaining (independent). To estimate the regression you simply select Data Analysis from the Tool menu and within this select Regression. You will get a dialogue box into which you need to input the relevant data. In the simple example we will be trying to identify the impact inflation has on interest rates. Because the causality runs from inflation to interest rates, the interest rate will be the dependent variable and the inflation rate will be the independent variable. You will input the dependent variable in the Input Y Range: by highlighting the interest rate column (C3:C50). You then input the independent variable in the Input X Range: by highlighting the inflation rate column (B3:B50). Because I did not use the labels you do not check off the labels box. I then tell it I would like the output to have its top left corner in cell F2. After checking off all the options you get all of the information on the simple tab. Below is the data that appears with the regression output. While all of this data gives you important information about the relationship, at this time your attention should be directed to just a few of the features 2 that are highlighted in red. The first is the adjusted R Square. This tells the reader that of all of the year - to-year variation in the interest rate, about 52% of it can be explained by movements in the independent variable (inflation rate). The second things to look for are the coefficients. In this example, the regression analysis suggests the best equation for these data would be: Interest rate = 2.44 + .68*Inflation rate What we are most interested in is the coefficient of the Inflation rate, which in this example is .68. This means every time the inflation rate rises by one percentage pint (from 4 to 5 percent), then the interest rate rises by .68 percentage points.