Regression Analysis. Data Calculations Output


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1 Regression Analysis In an attempt to find answers to questions such as those posed above, empirical labour economists use a useful tool called regression analysis. Regression analysis is essentially a specific set of mathematical formulas used to calculate the effects of one or many characteristics on an individual outcome. An economist takes information on many people, runs a regression, then analyses the results of the calculations. Data Calculations Output For example, suppose we wish to determine whether someone with more years of education should expect higher wages than someone with fewer years of education. We can use regression analysis to estimate the relationship between these two variables. The steps to successful regression analysis are as follows: 1. Formulate the theory as an equation. We hypothesize that more years of education higher wages. What our hypothesis implies is that wages are a function of education. We use a linear function for simplicity. Wages = a + b*yrseducation Our hypothesis also implies that we believe b>0 The term a represents the wage you would receive if you had 0 years of education. This term is known as the intercept, or the constant In this example, b represents the returns to education. The term b is known, generally as a coefficient or parameter.
2 Graphically, our hypothesis looks something like this: Wage W=a+b*E a Slope = b YrsEducation 2. Test the hypothsesis using the appropriate data We use a statistical package such as STATA, SAS, SPSS, or Shazam to calculate b for us. These packages use the data we give it, calculate the b that best fits this data, and give us test statistics which enable us to reject or not reject our hypothesis. In regression analysis we are estimating Wages = a + b*yrseducation + e Where e represents an error term. We include this error term because we know that we are approximating a relationship. Approximations are never perfect because there are various unknown factors involved in wages such as the worker s performance, luck, and so forth. Ordinary Least Squares regression, which is the most common form of regression, finds the best fit by finding the value of b that minimizies the squared sum of all the error terms in the data sample. {slide from p.27} Note that the slide uses natural logged wages rather than just wages. This is standard practice beccause changes in variables ofen have multiplicative effects on changes in other variables. As such, the relationship may not be linear. Logging wages reduces the nonlinearity of the relationship and enables the statistical package to get a better fit. (logging also helps express results in elasticities) {slide from p.25 shows transformation} {demonstrate on slide from p. 27 better fit} 3. Interpret the regression output and make predictions Using regression output, we can take the YrsEducation of any given person and predict the wage that person will obtain in the labour market.
3 Below is an example of regression output you might find from the statistical package STATA. Note that in addition to coefficient estimates, STATA also includes standard errors, which are an estimate of the precision of the estimate b, and test statistics.. regress lnwage YrsEducation Source SS df MS Number of obs = F( 1, 998) = Model Prob > F = Residual Rsquared = Adj Rsquared = Total Root MSE = lnwage Coef. Std. Err. t P> t [95% Conf. Interval] YrsEducation _cons STATA and other packages generally test the hypothesis that b=0, so we must be careful in our interpretation of the results. In this example, we see that the estimated coefficient on the years of education variable is positive. This means that the estimation indicates a positive relationship between years of education and logged wages The return to education is 1.915, this can also be viewed as the slope of the line mapping the relation between YrsEducation and ln(wages). The standard error for YrsEducation is This number is quite small (much less than half) compared to the coefficient estimate. The tratio is the coefficient divided by the standard error. Here the tratio is large: A large tratio indicates that we are unlikely to have obtained this estimate due to chance or sampling error. The Pvalue, denoted P> t above, yeilds the probability that the tratio would take on a value as extreme as it does by chance when the true value of b is zero. P> t =0.00 implies that this probability is extremely unlikely. That is, it is extremely improbable that if b=0 we would have obtained the estimate we did by chance. The hypothesis that b=0 should be rejected. Pvalue of 0.05 implies that a sample with the coefficient estimate b^ and a tratio t c will occur only 5% of the time when the true value of b is zero.
4 The term level of significance is often associated with the Pvalue. Conventional levels of significance used by empirical labour economists are 0.01, 0.05, and 0.1. If the Pvalue is less than or equal to 0.05 we say that we reject the null hypothesis at the 5% level of significance. If we had a Pvalue of 0.07 we could reject the null hypothesis at the 7% level of significance, but we generally don t. We generally use conventional levels, so in this case, we d reject the null at the 10% level of significance because Pvalue=0.07<0.1 So what conclusions can we draw from our example above? That an estimated value of b^=1.92, with a tratio of will occur 0% of the time by chance when the true slope is b=0, therefore, it is unlikely that the true slope is b=0. Thus we reject the null hypothesis at the 1% level (the smallest conventional level). This means that our estimate b^=1.92 is likely to be closer to the truth. It is considered to be significant at the 1% level. Tests on the Model itself Other items you might notice in the regression output above are common tests for goodness of fit. The Fstatistic tells us whether a significant linear relationship exists between ln(wage) and YrsEducation In our case, the calculated F statistic is F = A relatively high value of F is good. For now, you can ignore the terms in the brackets F( 1, 998) Similar to the Pvalue, STATA gives us: Prob > F Prob > F = Here, a lower value, such as zero, indicates that a significant linear relation likely exists between ln(wage) and YrsEducation. R 2 is a measure of goodness of fit, of how well our model fits the data. In this case, the R 2 =0.1820, which is relatively low. R 2 = explained variation total variation So the highest R 2 can be is 1. In labour economics, it is not uncommon to have a low R 2, meaning that we commonly are not able to predict labour market behaviour very well. However, there are cases where the R 2 is fairly high. An R 2 of 0.18 tells us that our model explains 18% of the variation in ln(wage) from the mean of ln(wage). In other words, we cannot explain very much of the variation in ln(wage).
5 Multiple regression analysis This term simply means that you consider the effect of more than one variable on one individual outcome. For example: Wage= a + b1*yrseducation + b2*age + b3*gender + e Wage is known as the dependent variable (it is on the left hand side of the equation and is what we are trying to predict) YrsEducation, Age and Gender are known as the independent variables. Types of Data 1. Cross Sectional contains information (education, age, gender, etc.) on many individuals at one given point in time. Examples of cross sectional data sets are: SCF, CPS, LFS. 2. Time Series Data contains aggregate, economy wide measures (ex/ GDP, Unemployment Rate, etc.) for a specific area/region/country. Such data may be found on CANSIM or CITIBASE 3. Panel/Longitudinal Data contains detailed information on individuals for periods of more than one year. Generally panel data includes annual information on a person for 3 to 6 years of their life. Panel data is like cross sectional data for 2+ years. Recap: regression function just helps us calculate a and b.using real life examples (data), the regression calculations find the best fit
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