6. Multiple Regression Analysis: Further Issues 6.1 Effects of Data Scaling on OLS Statistics 6.2 More on Functional Form 6.3 More on Goodness-of-Fit and Selection of Regressors 6.4 Prediction and Residual Analysis

6.1 Data Scaling and OLS -Scaling data will have NO effect on the conclusions (tests and predictions) that we obtain through OLS 1)If a dependent variable is scaled by dividing by C: -estimated coefficients and standard errors will also be divided by C (thus t stats and tests are unaffected) -R 2 will be unaffected, but SSR will be divided by C 2 and SER by C as they are unbounded

6.1 Data Scaling and OLS 2) If an independent variable is scaled by dividing by C: -the coefficient and standard error of that variable are multiplied by C (thus t statistics and tests are constant) 3) If a dependent OR independent variable in log form is scaled by C: -only the intercept is affected, due to the fact that logs in regressions deal with PERCENTAGE changes

6.1 Beta Coefficients -Due to scaling, the sizes of estimated coefficients can’t reflect the relative importance of a variable -ie: measuring in cents would create a smaller coefficient while measuring in thousands would create a larger coefficient -To avoid this, all variables can be STANDARDIZED (subtract mean and divide by standard deviation) and beta coefficients found

6.1 Beta Coefficients -to obtain beta coefficients, begin with the normal OLS regression and subtract means (note that residuals have zero sample average): -adding sample standard deviations, σhat, gives:

6.1 Beta Coefficients -Since standardizing a variable converts it to a z- score, we now have: -Where:

6.1 Beta Coefficients These new coefficients are called STANDARDIZED COEFFICIENTS or BETA COEFFICIENTS (which is confusing as the typical OLS regression uses Betas). -This regression estimates the change in y’s standard deviation when x k ’s standard deviation changes -Magnitudes of coefficients can now be obtained -note that there is no intercept in this normalized equation

6.2 Functional Form - Logs -In this course (and most economics in general) log always refers to the NATURAL LOG (ln) -a typical regression including logs is of the form: -where B 1 is the elasticity of y with respect to x 1 -where B 2 is the change in log(y) when x 2 changes by 1 -therefore 100B 2 is the approximate percentage change in y when x 2 changes by 1 -this is often called the semi-elasticity of y with respect to x 2

6.2 Log Example -Consider the following regression following the price of DVD’s: -here our simple calculation claims that the price of a DVD increases by 21% for every star the movie obtains -this approximation is inaccurate for large percentages

6.2 Log and Large Percentages -when dealing with a log-lin model, the EXACT percentage change is given by: -when percentage changes are “large”, this is a more accurate calculation -in our above example: -using the exact formula gives change of 23% as opposed to change of 21%

6.2 Logs Unfortunately, this percentage change estimation is not an unbiased estimator -it is however a consistent estimator -Since logs deal with percentage changes, they are useful in that they are invariant to scaling -If y>0, using log(y) as the dependent variable can often satisfy CLM better -in particular, heteroskedasticity or skewing can sometimes be mitigated using logs -As using logs narrows the range of a variable, it makes a study less sensitive to outliers

6.2 When to Use Logs -Although no rule for using logs is written in stone, and economic theory should ALWAYS be the basis of functional form, the following guidelines are often used: 1)Generally use logs for positive dollar amounts. 2)Generally use logs for large interval values such as population, employment, deaths, etc. 3)Variables measured in years (age, education, etc) generally DON’T use logs 4)Percentages can use logs, although their interpretation becomes a percentage change of a percentage (10% change of 60%=6%)

6.2 Log Limitations 1)Logs cannot be taken of non-zero numbers -if y is sometimes zero, one can use log(1+y), although this skews interpretation of y=0 and technically is no longer normally distributed 2)It is more difficult to predict the original variable using logs -exponents and errors must now be accounted for 3)R 2 cannot be compared across log- and lin- models

6.2 Quadratic Functions -Quadratic functions are often used to capture changing marginal effects -The simplest estimated quadratic model is: -which changes the interpretation of the coefficients such that: -as it makes no sense to analyze the effect of a change in x while keeping x 2 constant

6.2 Dynamic Quadratic Functions If it is the case that B 1 hat is positive and B 2 hat is negative, -x has a diminishing effect on y -the graph is an inverted u-shape -ie: conflict resolution -talking through a problem can work to solve it up to a certain point, where more talking is extraneous and only creates more problems -ie: Pizza and utility -eating Pizza will increase utility up to a point where additional pieces makes one sick

6.2 Dynamic Quadratic Functions -The maximum point on the graph (where y is maximized) is always at the point: -after this point, the graph is decreasing, which is of little concern if it only occurs for a small portion of the sample (ie: very few people force themselves to eat too much pizza) -this downward effect could also be found due to omitting certain variables -a similar argument goes for a u-shaped curve with a minimum point

6.2 More Quadratics and Logs -If a quadratic model has both slope coefficients either positive or negative, the model increases or decreases at an increasing rate -combining quadratics and logs allows for dynamic relationships including increasing or decreasing percentage changes: -For example, if -Then

6.2 Interaction Terms -Often a dependent variable’s impact (partial effect, elasticity, semi-elasticity) depends on the value of another explanatory variable -In these cases variables are included multiplicatively -For example, if you get a better night’s sleep on a comfortable bed, -Then

6.2 Interaction Terms -If there is an INTERACTION EFFECT between two variables, the are often included multiplicatively -in order to summarize one variable’s effect on y, one must examine interesting values of the other variable (mean, lower and upper quartiles) -this can be tedious -often the examination of only one coefficient is meaningless if the interaction variable cannot be zero (ie: if comfort can’t be zero)

6.2 Reparameterization -Since the coefficients are going to be examined from at their means, it is often useful to reparameterize the model to take means into account initially: -Becomes: -In this new model, delta 2 becomes the partial effect of x 2 on y at the mean value of x 1

6.2 Reparameterization -In other words: -This is also useful in that the estimated standard errors are all estimated for the partial effects at mean values -That said, once a model considers a variety of explanatory variables with interaction terms, the typical estimation is often done with extra calculations at means later

6.3 R 2 and You -previously, R 2 was not discussed in evaluating regressions due to the initial temptation to put too much importance on R 2, which is a fallacious judgement -for example, time-series R 2 ’s can be artificially high -there are no aspects of the CLM that require a certain R 2 -R 2 simply estimates of much of y’s variation is estimated by x in the model

6.3 R 2 and You -Assumption MLR.4 (Zero Conditional Mean) determines unbiasedness and independent of the value of R 2 -however, a small R 2 implies that the error variance is small relative to y’s variance -this can make precisely estimating B J difficult -a large standard error can however be offset by a large sample size

6.3 R 2 and You -if R 2 is small, ask: Are there any variable that should be included? Are any relevant variables that haven’t been included (data may be hard to obtain) highly correlated with included variables? -If no on both counts, B j is likely reasonably precise -note that R 2’s INCREASE when a variable/ variables is/are added is important (and related to the F test for variable significance)

6.3 Adjusted R-squared -Note that the typical equation for R 2 can be written as: -If we define σ 2 y as the population variance of y and σ 2 u as the population variance in u, then R 2 is supposed to estimate the POPULATION R 2 of:

6.2 Adjusted R-squared -However SSR/n is a biased estimate of σ 2 u, and can be replaced by the unbiased estimator SSR/(n-k-1) -Likewise SST/n is a biased estimate of σ 2 y, and can be replaced by the unbiased estimator SST/(n-1) -These substitutions give us our adjusted R 2 :

6.3 Adjusted R-squared -Unfortunately, adjusted R 2 is not proven to be a better estimator -the ratio of two unbiased estimators is not necessarily itself unbiased -adjusted R 2 does add a penalty for including additional independent variables: -SSR will fall, but so will n-k-1 -therefore adjusted R 2 cannot be artificially inflated by added variables

6.3 Adjusted R-squared -When adding a variable, adjusted R 2 will increase only if that variable’s t-stat is greater than one (in absolute value) -Likewise, adding many variables only increase R 2 if the F stat for adding those variables is greater than unity -adjusted R 2 therefore gives a different answer to including/excluding variables than typical testing

6.3 Adjusted R-squared -Adjusted R 2 can also be written in terms of R 2 : -From this equation we see that adjusted R 2 can be negative -a negative adjusted R 2 indicates a very poor model fit relative to the number of degrees of freedom -note that the NORMAL R 2 must be used in the F formula of (4.41)

6.3 Nonnested Models -Sometimes it is the case that we cannot decide between two (generally highly correlated) independent variables -Perhaps they both test insignificant separately yet significant together -In deciding between the two variables (A and B), we can examine two nested models:

6.3 Nonnested Models -These are NONNESTED MODELS as neither is a special case of the other (as compared to nested restricted models in F tests) -ADJUSTED R 2 ’s can be compared, with a large difference in ADJUSTED R 2 ’s making a case for one variable other the other -a similar comparison can be done with functional forms:

6.3 Nonnested Models -In this case, adjusted R 2 ’s are a better comparison than typical R 2 ’s as the number of parameters has changed -Note that adjusted R 2 ’s CANNOT be used to choose between different functional forms of the dependent (y) variable -R 2 deals with variation in y, and by changing the functional form of y the amount of variation is also changed -6.4 will deal with ways to compare y and log(y)

6.3 Over Controlling -in the attempt to avoid omitting important variables from a model, or by overemphasizing goodness-of-fit, it is often possible to control for too many variables -in general, if changing the variable A will naturally change both the variables B and C, including all three variables would amount to OVER CONTROLLING for factors in the model

6.3 Over Controlling Examples -If one wanted to investigate the impact on reduced TV on school grades, study time should NOT be included, as -And it may be nonsensical to expect less TV not to result in more studying -If one wanted to examine the impact of increased income on recreational expenses, travel expenses should NOT be included, as they are part of recreational expenses

6.3 Reducing Error Variance In Ch. 3, we saw that adding a new x variable: 1)Increases multicollinearity (due to increased correlation between more independent variables) 2)Decreases error variance (due to removing variation from the error term) From this, we should ALWAYS include variables that affect y yet are uncorrelated with all of the explanatory variables OF INTEREST -This will not affect the biasness (of the variables of interest) but will reduce sample variance

6.3 Example Assume we are examining the effects of random Customs baggage searches on import of coral from Hawaii -since the baggage searches are random (assumed), they are uncorrelated with any descriptive variables (age, gender, income, etc.) -However, these descriptive variables may have an impact on y (coral import), they can be included and reduce error variance without making the estimation of baggage searches biased