Linear regression models


Notes on linear regression analysis (pdf file)

Introduction to linear regression analysis
Regression examples

Mathematics of simple regression

·         Baseball batting averages

·         Beer sales vs. price, part 1: descriptive analysis

·         Beer sales vs. price, part 2: fitting a simple model

·         Beer sales vs. price, part 3: transformations of variables

·         Beer sales vs. price, part 4: additional predictors

·         NC natural gas consumption vs. temperature

What to look for in regression output
What’s a good value for R-squared?
What's the bottom line? How to compare models
Testing the assumptions of linear regression
Additional notes on regression analysis
Stepwise and all-possible-regressions
Excel file with simple regression formulas

Excel file with regression formulas in matrix form

If you are a PC Excel user, you must check this out:
RegressIt: free Excel add-in for linear regression and multivariate data analysis


Regression example, part 4:  additional predictors

The log-log regression model for predicting sales of 18-packs from  price of 18-packs gave much better results than the original model fitted to the unlogged variables, and it yielded an estimated of the elasticity of demand for 18-packs with respect to their own price.  Specifically, the slope coefficient of 6.705 in that model implied that, on the margin, a 1 percent change in price should be predicted to yield a 6.7% change in demand in the other direction.  But what about prices of the other carton sizes?  It is reasonable to expect that there will also be substitution effects, i.e. “cross-price elasticities.”  Consumers ought to buy relatively more [fewer] 18-packs when the price of another carton size is raised [lowered].  Indeed, the correlations and scatterplots for sales of 18-packs vs. prices of 12-packs and 30-packs (taken from the original scatterplot matrix) suggest that there are such effects:

To test this hypothesis, let’s add these two variables to the previous model.  Here is the model summary report produced by RegressIt:


The coefficients of the two additional variables are significantly different from zero, and most importantly there has been a significant reduction in the standard error of the regression, from 0.356 to 0.281.  Recall that errors measured in log units can be roughly interpreted as percentages, if they are on the order of 0.2 or less.  These numbers are a bit too large for that approximation to fit very well, but still, very roughly speaking, the standard deviation of the errors in percentage terms has been reduced from around 36% to around 28%, which is a meaningful improvement.  (Return to top of page.)


The estimated coefficient of PRICE_18PK_PN (-6.331) is very close to its value in the previous model, which is good:  it means the own-price effect is fairly robust to the manipulation of the other prices.  The estimated coefficient of 0.2016 for PRICE_12PK_LN (its cross-price elasticity with respect to sales of 18-packs) means roughly that a 1% increase [decrease] in the price of 12-packs should be predicted to yield a 2% increase [decrease] in sales of 18-packs, other things being equal.  For 30-packs, whose coefficient is 2.47, the effect is a little stronger.   


Now for the plots.  The actual-and-predicted-vs-obs# plots appears to show a very nice fit:



However, the residual-vs-obs# plot indicates a bit of a problem:  there is a noticeable time pattern in the errors, namely an upward trend.  (Errors in the first half of the year are nearly all negative, while those in the second half are mostly positive.)  This pattern is also reflected in the residual autocorrelations, which are positive at lags 1 and 2: 0.18 and 0.278 respectively.  These values are actually not quite large enough in magnitude to be significant at the 0.05 level, although that is not a magical standard.  (The critical value for this test is roughly 2/SQRT(n), where n is the sample size, which is 0.28 in this case.)  The appearance of the residual plot rather than the values of the test statistics is the best diagnostic of the time pattern here.  Always look at the pictures!   (Return to top of page.)



A linear trend in the errors of a regression model can be addressed by including a time index variable (for example, the week-number variable in this file) as another regressor.  If we do that here, we obtain the following 4-variable multiple regression model:



The standard error of the regression has been further reduced by a meaningful amount, from 0.281 to 0.244.  The coefficients of the original variables have wiggled around a bit, but are in the same ballpark as before.  The coefficient of Week is significantly different from zero (t=4.079), and its value of 0.011 appears to suggest that the average trend in sales of 18-packs is about 1.1% per week.  But appearances can sometimes be deceiving:  in the context of a multiple regression model, a trend variable adjusts for differences in trends among all the variables, rather than measuring the trend in the dependent variable alone.   (Return to top of page.)


The updated plots are shown below.  The residual-vs-obs# plot now does not show any systematic time pattern, and the autocorrelations that were previously seen in the errors have essentially vanished.


Overall, this appears to be a good model, and it has a lot to say about how sales of 18-packs are affected by price.  The additional take-aways from this step of the analysis are that:

·         There is significant cross-price elasticity in the data for sales of 18-packs vs. prices of 12-packs and 30-packs.

·         The trend in sales of 18-packs is not entirely explained by trends in price

·         This exercise has illustrated the construction of a linear regression model by a series of logical steps, beginning with descriptive analysis and proceeding through the fitting of a sequence of models that are motivated by economic theory and by interpretation of patterns seen in the statistical and graphical output.


This is not quite the end of the story:  some final comments and another picture are given below.




Postscript:  this is real data, and one thing that is a bit curious about it is that there is no evidence of seasonality.  In general, beer sales in the U.S. exhibit very significant seasonality, being higher in the hotter months and having large spikes in certain holiday weeks.  Here’s a picture of total weekly sales of the top-selling brands at a much larger sample of stores:



(Return to top of page.)